Moving up and Getting Steeper: Negotiating Shared Descriptions of Linear Graphs
Author(s): Judit N. Moschkovich
Source: The Journal of the Learning Sciences, Vol. 5, No. 3, Collaborative Learning: Making
Scientific and Mathematical Meaning with Gesture and Talk (1996), pp. 239-277
Published by: Taylor & Francis, Ltd.
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OFTHELEARNING
THEJOURNAL
SCIENCES,
5(3),239-277
Inc.
Lawrence
Associates,
Erlbaum
1996,
Copyright
?
Moving Up and GettingSteeper:
Negotiating SharedDescriptions
of LinearGraphs
JuditN. Moschkovich
Institutefor Researchon Learning
Menlo Park, California
This studyexaminesmathematics
learningin the contextof peerdiscussionsby
of linesgraphedon a computer
screen.Thearticle
focusingon students'descriptions
describeshow these discussionsprovideda rich contextfor negotiatingshared
descriptions,supportedconceptualchange,and resultedin convergentmeanings
andclarifications.
Theparticipants
in thestudyused
elaborations
throughreciprocal
graphingsoftwareto explorethe connectionsbetweenlinearequationsandtheir
graphswitha peer.Thearticlepresentstheanalysisof threecasestudiesexamining
shareddescriptions
of lines.Theseconversations
howstudentsnegotiated
supported
students'constructionof shareddescriptions,but not necessarilyby presenting
conflictingideasor throughone studentguidinganother.Rather,negotiationfunctionedthroughlocalconversational
resourcessuchas the use of referenceobjects,
andcoordinated
spatialmetaphors,
gesturesandtalk.Thesecasestudiesalsopointto
animportant
roleforinstruction
in orchestrating
andsupporting
peerdiscussionsby
andmaintaining
students'focuson mathematimodelinghowto resolvenegotiations
callyproductive
learningtrajectories.
Curriculumguidelines and researchersin mathematicseducationhave endorsed
peer discussions as a context for improvingconceptual learningin mathematics
framework
forCalifornia,1992;National
1989;Mathematics
(Brown&Pallincsar,
Councilof Teachersof Mathematics[NCTM],1989;Resnick, 1989).Workingwith
peers is supposed to provide an environmentin which students can "explore,
formulateand test conjectures,prove generalizations,and discuss and apply the
results of their investigations"(NCTM, 1989, p. 128). Although there are many
Requestsfor reprintsshouldbe sent to JuditN. Moschkovich,Institutefor Researchon Learning,66
Willow Place, Menlo Park,CA 94025.
240
MOSCHKOVICH
possible ways in which a conversationwith a peer mightsupportlearning,thereare
few detailed descriptionsof such conversationsfocusing on conceptuallearning.
This article presents evidence that conversationsbetween peers can supportthe
constructionof shareddescriptionsof mathematicalobjects,describeshow students
refinedtheirdescriptionsof lineargraphsin such conversations,and examinesthe
resourcesstudentsused to constructsharedmeaningsfor theirdescriptions.
The article presents the analysis of three case studies showing how students
negotiatedthe meaningof descriptionsof lines graphedon a computerscreen and
explores the following questions:How did studentsdescribe and comparelines?
How did studentsnegotiate and constructshareddescriptionsof lines? How did
studentsrefinetheirdescriptions?Whatresourcesdid studentsuse to elaborateand
clarify theirdescriptions?
During the discussions, students grappled with problematic or ambiguous
descriptions, contesting each others' understandingsand engaging in repeated
dialogues about these descriptions.As the discussions progressed,the students
elaboratedtheir descriptions,clarified meanings,and constructedshareddescriptions. In clarifying their descriptions,they used conversationalresourcessuch as
everyday spatialmetaphors,coordinatedgestures and talk, and referenceobjects.
In the process of contesting,elaborating,andclarifyingtheirdescriptions,students
refined the meaning for many terms and developed more precise descriptionsof
lines.
These three case studies show that peer discussions can create the need for
clarification and provide a rich context for negotiating shared meanings. The
negotiationand constructionof shareddescriptionswere importantaspectsof how
these studentsmade sense of lines and their equations.The negotiationof shared
meanings that studentsengaged in, the fact that most of the students arrivedat
shareddescriptions,the ways that studentsrefinedtheirdescriptions,and the fact
thatmanystudents'descriptionscame to reflectmoreconceptualknowledge,show
thatthe negotiationof descriptionsis an importantaspect of learningthroughpeer
discussions.
These three cases also show that conversationsbetween studentscan support
conceptual change and that this progress does not necessarily happen through
conflict with a peer's perspectiveor guidanceby a more advancedother.Instead,
studentsconstructedsharedmeaningsusing constraintsand resourceslocal to the
conversationsuch as spatialmetaphors,referenceobjects,andcoordinatedgestures
and talk. Although students can and do reach agreementand make conceptual
progress during a conversation with a peer, neither resolution nor conceptual
convergenceare guaranteed.Although some conversationswith a peer are shown
to supportprogress,the last case study shows that there is also an importantrole
for instructionor guidance. Teachers can supportpeer discussions by modeling
ways to resolve negotiations,coaching studentson how to reach agreement,and
helping studentsto focus on mathematicallyproductivepaths.
NEGOTIATING
SHAREDDESCRIPTIONS
OFLINEARGRAPHS 241
THEORETICAL
FRAMEWORK
The study begins with the assumptionthatknowledge is socially constructedand
that this constructionis mediated by language (Vygotsky, 1978, 1987). I also
assume that competence in a complex domain, such as linearfunctions, involves
more thantextbookformulas,procedures,or the use of technicalterms.Following
Solomon (1989), I view competenceas knowing how to act in specific situations
involving lines and theirequations,includingknowing how to use language.The
framework for the study draws on three areas in currenttheory and research:
learningthroughcollaboration,conceptualunderstandingof linearfunctions,and
the relationbetween languageand learningmathematics.
Working collaboratively with peers is one possible context for supporting
learning;at the very leastit does not hinderlearning,andit improvesattitudesabout
subjectmatterandpeers(Brown& Pallincsar,1989;Davidson, 1985;Doise, 1985;
Sharan,1980; Webb, 1985). Researchershave begun to considerspecifically how
conversationsbetween peers might supportconceptual learning in mathematics
(Forman,1992; Forman& McPhail, 1993). Therearetwo mainparadigmsfor peer
collaboration.A neo-Piagetianperspective emphasizing sociocognitive conflict
and a neo-Vygotskianperspectiveemphasizingguidanceby a moreadvancedpeer.
This study explores a third alternativethat, ratherthan emphasizingconflict or
difference,focuses on negotiationand sharedconstructionthroughconversational,
and thus inherentlysocial, resources(Roschelle, 1992).
Although it is beyond the scope of this articleto discuss differentperspectives
on the relationbetweenlanguageandconcepts(Lucy & Wertsch,1987; Vygotsky,
1987), clearly the two arerelatedin intricateand complex ways. I assume thatthe
relationbetween language use and conceptions is a complex and dialectical one,
ratherthan unidirectionalor deterministic,without addressingthe details of this
relation.I also assumethatlearningto participatein mathematicaldiscourseis part
of learning mathematics (Durkin & Shire, 1991; Pimm, 1987). Mathematics
discourseincludesthe mathematicsregister,argumentationrulesandstyles, values,
and beliefs (Richards,1991). Learningto participatein mathematicaldiscourseis
not merely or primarilya matterof learning vocabularydefinitions. Instead, it
involves learninghow to use language while solving and discussing problemsin
differentcontexts.
Severalstudiesexploringthe relationbetweenlanguageandlearningmathematics (Cocking & Mestre, 1988; Durkin& Shire, 1991; O'Connor,in press;Pimm,
1987; Richards,1991) have focused on one aspect of mathematicaldiscourse,the
mathematicsregister(Halliday, 1978). Halliday defined register in the following
way:
A registeris a setof meaningsthatis appropriate
to a particular
functionof language,
thatexpressthesemeanings.Wecanreferto
togetherwiththewordsandstructures
242
MOSCHKOVICH
in thesenseof themeaningsthatbelongto thelanguage
the"mathematics
register,"
use of naturallanguage,thatis: notmathematics
of mathematics
(themathematical
itself),andthata languagemustexpressif it is beingusedformathematical
purposes.
(p. 195)
In light of this work, the analysis also considers possible differencesbetween
the everydayand mathematicalregistersfor this domainand shows how students'
languageuse moved closer to the mathematicsregisterby becoming more precise
and reflectingmore conceptualknowledge.
The Domainof LinearEquationsandGraphs
Linearfunctionsis a complex domainwherethe developmentof connectedpieces
of conceptualknowledge is essential for competence.In such a complex domain,
social interactionand language can play a crucial role in the development of
conceptualunderstanding.Conceptualunderstandingin this domaininvolves more
than using proceduresto manipulateequationsor graphlines; it involves understandingthe connectionsbetweenthetwo representations(algebraicandgraphical),
knowing which objects are relevant in each representation,and knowing which
objects are dependentand independent.
The complexity of descriptionsof linearequationsand graphsis a reflectionof
the conceptual complexity of this domain. Competence in the domain of linear
functions involves not only using precise descriptionsbut also understandingthe
conceptual entailmentsassociated with these descriptions.There are two initial
student conceptions documented in this domain (Moschkovich, 1992) that are
relevantto studentdescriptionsof lines: (a) Thex-interceptis relevantfor equations
of the formy = mx + b (i.e., it should appearin the equation,eitherin the place of
m or in the place of b'), and (b) m and b, or rotation and translation,are not
independent(i.e., if you changem in the equation,the y-interceptmight change in
the graph;if you change b in the equation,the slope might change in the graph;if
SStudents have also been reportedas seeing the effect of changingthe b in an equationas making
lines move fromleft to right(or rightto left), ratherthanupanddown (Goldenberg,1988;Moschkovich,
1992). Changingthe b in an equationdoes, in effect, move lines along the x-axis (as well as along the
y-axis), so thatthese descriptionsare not necessarilywrong.Whatis importantaboutthese descriptions
is that experts usually choose to focus on movementalong the y-axis as a result of changing b. This
choice reflects the fact that lines move exactly b units up or down the y-axis when, for example, an
equation is changedfrom y = mx to y = mx + b. Although it is possible to relate the parameterb in an
equationto the movementof a line along the x-axis, this is a morecomplicatedcorrelation.Lines move
either -b units along the x-axis, in the case of m = 1, or (-b)lm units, in the case of m * 1. Focusing on
movement along the y-axis is the simplest possible correlationbetween the two representationsfor
equationsof this form and is thus not an arbitrarychoice.
NEGOTIATING
SHAREDDESCRIPTIONS
OFLINEARGRAPHS 243
a line is translatedup or down on the y-axis, this is a resultof changingm; if a line
is rotatedabout the origin or the y-intercept,this is a result of changing b). As
studentsexplore this domain,they build on and refine these initialconceptions.2
Students' initial descriptionsreflected these two conceptions. The use of the
x-interceptwas reflectedin initialdescriptionsof lines as moving "right"and"left,"
either insteadof or in additionto "up"and "down,"as a resultof changingb in the
equationy = mx + b. The conflating of the role of the parametersm and b was
reflected in initial descriptionsin several ways. Initially,studentsdid not associate
a change in m with a change in the steepness of a line (or a change in b with a
translationalong the y-axis). Also, they did not separaterotationand translationas
independentproperties,describingthe effect of a change in one parameter,say m,
as possibly generatingbotha rotationanda translation.Therewere manydialogues
involving initial misunderstandingsand negotiationof the meaningof the phrases
"theline is steeper/lesssteep,""theline moved up/downthe y-axis," and "theline
moved left/right."Studentsrefinedthe meaningof relationaltermssuch as steeper
and negotiatedthese two conceptualaspectsof theirdescriptions,the separationof
rotationand translationas independentmovementsand the focus on horizontalor
vertical translation.
These students'descriptionsdid not involve technicaltermssuch as slope and
intercept.Despite the absence of technicalterms, the studentsstill discussed and
negotiated the meaning of the less technical descriptionsused in the discussion
problems.Thus, the studentsdid not simply learnto use the technicaltermsslope
and y-intercept. Instead, they constructed shared descriptions and refined the
everyday meanings of terms using conversationalresources such as gestures,
referenceobjects, and spatialmetaphorsfor clarification.
RESEARCHDESIGN
The students who participatedin the discussions were from an exemplary pilot
Ist-year algebracourse (see Table 1). The studentsworked with a peer of their
choice, using graphingsoftwareto explore linearequationsandtheirgraphswhile
being videotaped. This article reports on the analysis for three of these pairs.
Protocol analysis of the videotapeddiscussion sessions was used to explore how
studentsnegotiatedthe meaning of their descriptionsand how these descriptions
changed.
These studentsattendedan urbanschool thathas abouta 90% minoritypopulation of workingclass and lower middle class families. The studentsin this course
(Moschkovich,
butshouldbe seenas
1992),I arguethatthesearenotmisconceptions
2Elsewhere
"transitional"
thatarereasonable,
conceptions
useful,andpartof learningtrajectories.
244
MOSCHKOVICH
I
TABLE
Data Sources
observations
Classroom
two6-weekchapters,
thefirstcoveringlinearfunctions(fall)andthesecond
Observed
functions(spring).
coveringquadratic
sessions
Peerdiscussion
Sessionslastedfrom2 to 4 hrovera
Duration:Therewereno timeconstraints.
in a
periodof at least2 daysandat most4 days.Sessionswereconducted
afterschool.
classroom
of all sessions.
Data:
Videotapes
Writtenassessments.
were mostly ninthgraders,althougha few were tenthgraders.They were neither
honors nor remedial students, and the classes were heterogeneous in terms of
previousmath achievementscores. The six studentsdiscussed here speakEnglish
and one otherlanguagein theirhome (FredandHaroldspeakChinese,Marcelaand
Giselda speak Spanish, and Monica and Denise speak Tagalog). These students
were "mainstreamed"
becausethey were officially consideredproficientandfluent
in English, and they have experienced either all or most of their mathematics
educationin English.Thusmost, if not all, of theirmathematicalconversationshave
been in English.3These three pairs from three different language backgrounds,
different classroom achievement levels,4 and different scores on written assessments for this domain encounteredsimilar difficulties with descriptionsof lines
and had similardiscussions.Therefore,these particulardiscussionsof descriptions
do not seem to be linked specifically to speaking any one other language, to
achievementin mathematics,or to the students'scores on the writtenassessments.
The students were from two classrooms observed earlier in the school year
(Moschkovich, 1990) duringtwo chapterson functions,the first on equationsand
graphs of linear functions and the second covering quadraticfunctions. The two
chaptersincluded modeling of real world situations,use of graphingcalculators
andcomputersoftware,andstudentgroupworkwithsome whole-classdiscussions.
The curriculumwas designed to include exploration and discovery, focus on
mathematicsas a processratherthanon resultsor answers,supportworkin groups,
and encouragestudentsto discuss theirideas.
Their classroom work had focused on applications of linear and quadratic
functionsto a problemfrom science anddevelopinga qualitativeunderstandingof
the connections between equationsand graphs.Their classroom experiences did
not focus on the use of technical terms such as slope and intercept,or on the
andGiseldaweretheonlystudentswhouseda languageotherthanEnglishduringtheir
"Marcela
discussionsession.InthetwocaseswhentheyspokeSpanishto discusstheiranswersfortheproblems,
theSpanishandanEnglishtranslation
areprovidedin thetranscript
excerpts.
Determined
evaluation
by theircoursegradesanda qualitative
by theirteacher.
NEGOTIATINGSHAREDDESCRIPTIONSOF LINEARGRAPHS
245
definition of slope as the ratio of rise over run. In designing the problemsfor the
peer discussions, I purposefullyused the termssteeper and less steep, ratherthan
slope, to describe the difference between two lines of different slopes, and the
phrasesmove up on they-axis andmovedownon they-axis, ratherthany-intercept,
to describethe differencebetween two lines with differenty-intercepts.
Discussion Problems
In the discussionsessions, the studentsexploredslope andinterceptusing SuperPlot
(Steketee, 1985), a graphingutility that allows studentsto graphequations, and
problemsdesignedby the researcher.The problemsaddressedthe specific conceptions noted in the classroomobservations:using the x-interceptand conflatingthe
effect of changing m and b. Because the discussions were meant to resemble
classroom discussions between peers as closely as possible, interventionby the
researcherwas kept to a minimum.
The following example illustratesthe basic formatused for all the problems.In
the firstpartof each problem(such as Problem3a shown in Figure 1) studentswere
given the equationy = x and its graphand were askedto predictwhatchangingthe
equationfromy = x to a targetequation(in this case y = x+ 5) would do to the line.
In the second partof each problemthey were given the graphof y = x and a second
line and were asked to predictwhat change in the equationy = x would generate
the targetgraphedline. In some problems,Choice C read:"Theline would flip to
the otherside of they-axis"to addresstheeffect of a negativeslope. All theproblems
discussed in the transcriptshave the same formatas the problemin Figure 1.5
The introductionto the discussionsessions includeda review of basiccoordinate
graphingskills, anexplanationof how the computergraphsequations(usinga table
of values generatedby the students),a descriptionof how they were being asked
to discuss the problems,and an explanationof key words and phrasesused in the
problems(steep, steeper, less steep, origin, move up or down on the y-axis, etc.)
using examples.
To structuredialogue and discussion of differentconjecturesand predictions,
the studentsfollowed an instructionalsequence similarto the Itakuramethod for
classroomdiscussionsin science (Hatano,1988;Inagaki,1981;Inagaki& Hatano,
1977):
1. Studentswere presentedwith a questionandseveralalternativepredictions
(QuestionsA, B, and C).
2. Each studentwas directedto choose and recorda predictionon the paper.
5Figure1 is included with the first transcript.For subsequenttranscripts,I only include the target
equation.
3a. If you start with the equationy=x
then changeit to the equationy=x+5,
what wouldthat do to the graph?
Y
+8
+10
-10"
X
-8
AFTERGRAPHIN6
A. Makethe line steeper
Whyor why not?
.YES
,NO
.YES
,NO
Whyor why not?
B. Movethe line up on the y axis
Whyor why not?
.YES
,NO
.YES
,NO
Whyor why not?
C.Makethe line bothsteeper
andmoveupon the y axis.
Whyor why not?
.YES
,NO
.YES
,NO
Whyor why not?
FIGURE1 Problem3a.
246
NEGOTIATING
SHAREDDESCRIPTIONS
OFLINEARGRAPHS
247
3. The pair was asked to explain and discuss theirchoices before graphing.
4. The pair was directedto test theirpredictionsusing the computerto graph
an equation.
5. The studentswereaskedto thenchoose anagreedon answerandexplanation
once again aftergraphing.
Students were told that they did not have to agree on their choices before
graphingand thattheirindividualchoices would be recordedon the videotape,but
that they had to agree on their choices after graphing.Studentsdid follow these
instructionsand thus the conversations that ensued are labeled discussions, in
keeping with Pirie's (1991) definition of mathematicaldiscussion as "purposeful
talk on a mathematicalsubjectin which thereare genuine pupil contributionsand
interactions"(p. 143).
Analysis
The protocolanalysisof the videotapedatawas conductedby case studiesfor pairs
of students. During the classroom observations,I had noted that students had
difficultiesdescribingwhatthey saw on the computerscreen(Moschkovich,1990).
One analysisof the videotapesaddressedthis issue by coding students'descriptions
of lines. For the threecase studies I notedeach instancewhere a studentdescribed
a line, its movement,or comparedtwo lines. This analysis led to a second coding
where I analyzedthe instanceswherestudentsnegotiatedtheirdescriptions.These
instancesinvolved severalrelationalanddirectionalterms-either the ones used in
the problemsor othersgeneratedby students.The last coding of the datatracedthe
changes in students' descriptions by comparing these descriptions during two
problemsat the beginningof the sessions, duringseveral problemsin the middle,
and duringthe last two problemsin the discussion sessions.
The transcriptmaterialsaredividedintoexcerptsforeachproblem.Eachstudent
turn is numberedconsecutively within a problemexcerpt. Students'gestures,the
referentof a pronoun,or clarificationsareincludedwithin brackets.For the pairof
studentswho at times spoke Spanish(MarcelaandGiselda),the Englishtranslation
is providedin bracketsfollowing the Spanish.Any participationby the researcher
is labeledInt. for interviewer.
Thereare severalfactorsaffecting the refinementof students'descriptions,not
all of which can be attributedsolely to the peer discussionsessions. The first is that
for two semestersthese studentshad participatedin classroomwork where verbal
and written communicationabout mathematicsproblems was encouraged and
supported.Thus,they hadlearnedto participatein discussionswheretheydescribed
their solutions in detail, attemptedto understandotherstudents'explanations,and
triedto reachagreement.Also, duringthe discussionsin the classroom,the teachers
248
MOSCHKOVICH
andgraphical
intheirdescripthealgebraic
focusedoncoordinating
representations
andtreatedrotationandtranslation
as
tions,referredonly to verticaltranslation,
on a line.
transformations
independent
didchangeduringthediscusbecausethesestudents'descriptions
Nevertheless,
sion sessions,thereare also aspectsof these discussionsthat impactedtheir
thepresenceof linesgraphedon
by theresearcher,
languageuse.Theintroduction
usedin theproblems,andthefactthatseveralproblems
thescreen,thedescriptions
on
focusedon thetwoconceptionsdescribedearlierall providedsomeconstraints
seem
studentdescriptions.
However,as is seenin thecasestudies,theconstraints
thaninthepresenceof anauthoritative
to lie moreinthenatureof theconversations
text.
CASE STUDY1: USINGCOORDINATED
GESTURES
A MEANINGFOR STEEPER
ANDTALKTO NEGOTIATE
Thetermsteeperwasthefocusof manyof thediscussionsandanimportant
aspect
of the constructionof shareddescriptions.Studentsin all threecase studies
discussedthetermsteeperandnegotiated
themeaningof thisterm.Somestudents
as well as therotationof a line.Inthiscase
usedthetermto referto thetranslation
study,I showhowonepairof studentsstruggledwiththemeaningof steeper.These
two studentsinitiallyshowedsomeconfusionaboutthe meaningof steeperand
eventuallyfocusedonthetaskof explicitlyclarifyingthesemeaningstoeachother.
By the endof theirdiscussionsessiontheyuseda sharedmeaningfor steeperas
well as referredto rotationandtranslation
as independent
of a line.
properties
FredandHaroldwereworkingon theproblemshownin Figure1 providedfor
themon paper.They hadbeen instructedto firstpredictwhetherchangingthe
equationy = x to the equationy = x + 5 would make the new line steeperor not, to
graphthe equationson thecomputer,andto decidewhethertheirpredictionwas
rightaftergraphing.Whilemakingtheirpredictions,theyhadleft the answerto
QuestionA blankand answeredQuestionB as yes. Afterthey discussedtheir
interpretationsof QuestionC, Fred chose no for the answerto C, whereasHarold
thoughtthe right choice was yes (lines 14-25, transcriptnot includedhere). Next
they graphedthe equationy = x + 5 and returnedto answerQuestionsA throughC
aftergraphing.At this point, they were looking at the line for the equationy = x +
5 on the screen (see Figure 2). Excerpt 1 shows how the term steeper was a
problematicaspect of theirdescriptionsand highlightshow Haroldand Fred used
coordinatedgestures and talk as resourcesfor constructingand negotiatingtheir
descriptions.
IO
Co
FIGURE 2 Fred-line 31-"I thinkit's steeperr
Excerpt1: FredandHarold(Problem3a)
26 Harold:
Dialogue
[Reading]After graphing,
is it steeper?
27 Fred:
Isn't it steeper?No.
28 Harold:
It's not steeper,is it?
[moves handto the screen]
Are we talkingaboutthe
same thing?
Yeah ...
I think it's steeperright
here [pointsto the y-intercept of the line y = x + 5].
Cause look at it ... 1, 2,
and 1, 2 [countingup to 5
on the y-axis and then to 5
on the x-axis, the axes are
labeled with a slash every
two units]. This is the
same.
29 Fred:
30 Harold:
31 Fred:
Commentary
Haroldasks whetherthe line
on the screen is steeperthan
the line y = x.
Fredquestionswhetherthe
line is steeperand proposes
thatit is not.
Haroldquestionswhether
the line is or is not steeper.
Fredasks for a clarification.
Fredelaboratesthe meaning
of steeper,proposingthat
the the line is steeperand
justifying this claim by
pointingto two distances
along the axes.
As Fred elaboratedon his description,he proposed that the line y = x + 5 is
steeper at the y-intercept.His justificationwas based on the fact that the distance
fromthe originto the line along they-axis is the same as the distancefromthe origin
to the line along the x-axis (see Figure2).
During this elaboration,Fred was not only proposingthat the line y = x + 5 is
steeper than the line y = x, he was also using an object usually associated with
translation,the y-intercept, in his description of a steeper line. He referredto
steepness as a propertyassociatedwith a point, saying "it's steeperright here"as
he pointed to the y-intercept.In this elaboration,Fred's use of coordinatedgesture
and talk helped unpackwhathe meantby steeper:A line is steeperthananotherat
a specific location, the point where the line crosses the y-axis.
As they continued discussing their descriptionsof the line y = x + 5, Harold
proposedthat the line y = x + 5 was going up, whereasFredcontinuedto wonder
whetherthe line was steeperor not. Up to this point it is not always clear whether
FredandHaroldwere referringto QuestionA ("makethe line steeper")or Question
OFLINEARGRAPHS 251
NEGOTIATING
SHAREDDESCRIPTIONS
C ("makethe line both steeperand move up on the y-axis").Next, Fredproposed
they explore how to make a line steeper on the graph.As they took on this new
problem,they began using a pen to representsteeperlines in frontof the screen.
35 Fred:
36 Harold:
Dialogue
Can you make it deeper ...
steeper?
[Demonstratingwith the
pen.] Steeperit mightgo
like that [rotatingthe pen
counterclockwisefrom the
line y = x] or like that
[moving the pen up to
(0,5) and then rotatingthe
pen counterclockwise].
Commentary
Haroldshows two ways to
make a line steeper,using a
pen placed on the computer
screen.
In line 36, Haroldproposedtwo ways to make the line steeper:rotatingthe pen
counterclockwisefrom the line y = x or translatingthe pen up the y-axis first and
then rotatingit aboutthe y-intercept.He was referringto lines thatare steeperthan
the line y = x, which he used as a referenceobject, even thoughit was not graphed
on the screen.Thereare severalways to interpretthis proposal.He may have been
saying thatrotationis whatmakesa line steeper,regardlessof whatthe y-intercept
is. On the other hand,he may have been saying that "makinga line steeper"also
refers to translatingit up on the y-axis. In this demonstration,Haroldused coordinatedgesturesand talk concurrentlyto clarify the meaningof steeper.
37 Fred:
Dialogue
Steeperlike that [grabsthe
pen in Harold'shandand
moves it below the x-axis
so thatit is in the position
of a line thatis steeper
thany = x and also has a
negativey-intercept].This
way right?
Commentary
Fredproposesthata line
thatis below the origin and
steeperthanthe line y = x
would be steeper.
In Fred's elaborationin line 37 it is difficult to tell whetherhe was including
translationdown the y-axis as partof the description"steeperthan the line y = x"
252
MOSCHKOVICH
or showing thata line thathas a negativey-interceptcan also be steeper,regardless
of what the y-interceptmight be. However,it is strikingthatFredwas moving the
pen in Harold's hand down the y-axis as he was saying "steeperlike that."This
coordinatedgestureand utterancesuggeststhathe was associatingsteeperwith the
translationdown the y-axis.
38 Harold:
Dialogue
is
Steeper like this [puts
the pen in the position of a
line steeperthanthe line y
= x] but more like this
[keeps the pen at an inclinationsteeperthany = x as
he moves the pen up and
down the y-axis] ...
Commentary
Haroldshows a steeperline
first as a line steeperthany
= x and then also as a line
thatmoves up and down the
y-axis, presumablymeaning
a line thatis translatedup or
down as well as rotated(see
Figure3).
In this thirdelaboration,Haroldseemedto be proposingthata steeperline is one
that is rotatedcounterclockwiseabout the origin, saying "like this" as he rotated
the pen, as well as a line that is translatedup and down the y-axis, saying "more
like this"as he moved the pen up and down the y-axis. In these threeelaborations,
Fred and Haroldused coordinatedgesturesand talk to expose the details of their
understandingof the meaningof steeper. However, they did not yet seem to have
reacheda clear agreementon the meaningof this term.They returnedto answering
the questions:
40 Harold:
41 Fred:
Dialogue
So we're going up on the yaxis ... and "makethe line
both steeperand move on
the y axis" [referringto
QuestionC] ... You don't
want it steeperyou just
want it to move up on the
y-axis ... so ... yes or no?
[Looks at Fred.]
Mm ... no.
Commentary
They agree thatthe line for y
= x + 5 would not be
steeper,thatit would move
up on the y-axis, and thatthe
answerto QuestionC was
no becausethey didn't
"wantthe line to be steeper."
In sum, duringExcerpt 1, FredandHaroldwere workingon severalconcurrent
problems:answeringthe questions,decidingwhethera line can become steeperas
well as move up (or down) on the y-axis, andexemplifying what situationscan be
describedby thetermsteeper.Intheconversationjust given, they beganto negotiate
how to use the termsteeperthroughelaborationandclarification.They elaborated
(1
FIGURE 3 Harold---line38-"...
but more li
254
MOSCHKOVICH
on examples of what situationsthe descriptionsteeper refersto and clarifiedwhen
they each thoughtthe descriptionwould apply. Even while looking at the line y =
x + 5 on the screen(and apparentlyknowing wherethe line y = x would have been
located) these two studentsspent a considerableamountof time discussing and
representingthe term steeper. One way to interprettheirdialogue and gesturesis
that at differentpoints in the conversationeach of them used steeper to sometimes
referto translation,sometimesto referto rotation,and othertimes to referto both
movements.
The dialoguejust given exemplifies the use of gesturesandtalk interactivelyto
disambiguatethe meaningof a description.Throughrepeatedgesturesrepresenting
a line (or lines) on the screen and coordinatedtalk describingthese lines, each
studentelaboratedtheir understandingof the situationsin which the description
steeperwould apply.This excerpthighlightsthe importanceof gesturesin general
(McDermott,Gospodinoff,& Aron, 1978) andspecificallywhendescribinggraphical objects.Gestureswere an integralpartof these students'descriptionsof graphs,
and their language use might have been interpreteddifferentlywithout the videotape as a source of data.Anotherresourcefor elaboratingdescriptionsis the line
y ==x, which even thoughit is not graphedon the screenis implicitlypresentin each
example of a steeperline and serves as a referenceobject in severaldescriptions.
Both studentsimplicitlyreferredto this line as a referenceobject, which, because
it is shared,supportsthe constructionof a shareddescription.
By the end of Excerpt 1, Fred and Harolddid not seem to have producedan
examplethatunequivocallycommunicatedwhattheyeachmeantby steeperto their
partneror settled on a sharedunderstandingof what constitutessteeper lines. The
next excerptshows how they continuedtheirnegotiationof the meaningof the term
steeper.
Excerpt2: FredandHarold(Problem9a)
TargetEquationy = 10x
2 Fred:
Dialogue
Ten x [mumbles]y = 10x
will be up here ... so it
will be steeper,right?
It will be steeper,right ...
Yeah ... yes [writesthe answer for QuestionA].
5 Together: Steeper.
Commentary
Before graphingthe equation both Fredand Harold
predictthatthe line will be
steeper.
3 Harold:
4 Fred:
They agree on theirprediction.
NEGOTIATINGSHAREDDESCRIPTIONSOF LINEARGRAPHS
255
Duringthese first exchangesFredand Haroldcheck theirpredictionswith each
other and agree thatthe line y = 10x will be steeperthanthe line y = x. Next they
discuss whetherthe line y = 10x would move on the y-axis:
7 Fred:
Dialogue
The line will move on the
y-axis.
No ...
8 Harold:
No?
9 Fred:
Whatdo they mean on the
y-axis?
y-axis.
No, it would be still on
that ... like thatalways
cross the y ... I mean the xaxis.
6 Harold:
10 Harold:
11 Fred:
Commentary
HaroldreadsQuestionB.
Fredproposesthatthe line
will not move on the y-axis.
HaroldquestionsFred's proposal.
Fred asks for a clarification.
Fredtries to describehow
the line y = 10x would still
cross the origin.
In this dialogue, althoughFred asked for a clarificationof the meaning of the
phrase "on the y-axis," Harolddid not respond.They moved on to graphingthe
equation.
12 Fred:
13 Harold:
14 Fred:
Dialogue
Yequals ten x [he graphs
the equationy = 10x] ...
This is ten x [pointsto the
line y = 10x on the screen].
Yeah, so it's steeper.
Yeah.
Commentary
The line from the previous
problemis also on the
screen,so Fredclarifies
which is the line y = lOx.
They agreethatthe second
line is steeperaftergraphing
the equation.
After graphingthe equation,FredandHaroldeasily agreedthatthe line y = 10x
was steeper.This ease standsin contrastto the extendednegotiationin Excerpt 1.
They moved on to deciding whetherthe line moved on the y-axis or not.
19 Harold:
Dialogue
Did it go on the y-axis?
Commentary
Haroldcontinuesto question
whetherthe line went "on
the y-axis."
256
MOSCHKOVICH
20 Fred:
21 Harold:
22 Fred:
Dialogue
No ...
Did it or did it not [looks
at Fred]?... It didn'tright?
So it's on it ... passing,
right ...
The point's here [pointing
to the origin] ... See the
point ... [pointingto anotherpoint on the same
line but below the origin]
... There's a point here.
Commentary
Fredproposesthatit did not.
Haroldlooks to Fredfor an
answerand then proposes
thatit did not.
Fredoffers an elaborationinvolving the origin and anotherpoint on the line.
Even after graphing the equation, Harold remained unsure of whether the
description"moved up on the y-axis" was corrector not, referringto movement
"on the y-axis" and droppingthe word up. Fredpresenteda clarificationusing the
points on the line. It is not clearexactly whatFred'selaborationinvolved, whether
Haroldunderstoodthis explanation,or whetherHaroldwas convincedthatthe line
had not moved up. This lack of agreementstandsin contrastto theirrecentease in
agreeingthatthe line y = 10x was steeperin lines 12 to 14, showing thatalthough
they had moved closer to a sharedmeaningfor the termsteeper, they had not yet
accomplishedthis for the phrase"moveup on the y-axis."
During Excerpt 2, Fred and Harold used several resources to negotiate the
meaningof theirdescriptions.Onceagain,Fredused gesturescoordinatedwith talk
to clarify the meaningof the phrase"move on the y-axis." He also used reference
objects, the origin and points on a line, as resourcesfor his explanation.However,
in contrastto Excerpt 1, where Haroldand Fredused only one termfor steepness,
in Excerpt2 they used severaldifferentphrasesto describethe line: "itwill always
cross,""go on the y-axis," "on it", and "passing."
Fifteenminuteslater,theyworkedon thelastproblemof theirdiscussionsession,
where a line is translateddown the y-axis. This last excerpt shows how Fred and
Harold now easily reachedagreementon a descriptionwithoutresortingto challenges, elaborations,or clarifications.
Excerpt3: Fredand Harold(Problem14a)
TargetEquationy = x - 100
1 Harold:
Dialogue
[ReadingQuestionA]
"Steepnesswould change."
Commentary
NEGOTIATING
SHAREDDESCRIPTIONS
OFLINEARGRAPHS 257
Dialogue
2 Fred:
It ...
3 Together: It won't change.
4 Harold:
5 Fred:
"Move on the y-axis"
[readingQuestionB].
Yeah, cause it's coming
down.
6 Fred:
X minus 100 ...
7 Harold:
Steepnesswill not change
... it will move on the yaxis.
8 Fred:
It will come down.
Commentary
They agree in theirprediction thatthe steepnesswill
not change.
Fredproposesthatthe line
will move down on the yaxis.
They graphthe equationy =
x- 100.
Harolddescribesthe steepness as not changingand the
line as having moved on the
y-axis.
Fredspecifies thatthe line
moved down.
The dialogue in this last excerpt was very straightforward,
especially in comparisonto the initial negotiationof the meaningof steeper in Excerpt 1, and to the
later negotiationof the phrase"move on the y-axis" in Excerpt2. Duringthis last
problemeach of these studentsseemed confidentof theirdescriptionsand seemed
to assume thattheirpartnerreadilyunderstoodthese descriptions.Theirconversation proceededwithoutelaborationor clarification,both when they were makinga
predictionand when they were describinga line on the screen.The studentsthus
seemed to have constructedshared understandingsof the meaning of the term
steeper and the phrasemoveon they-axis. These sharedmeaningsallowed themto
move with ease through the cycle of predicting,graphing, and checking their
descriptionswith each other.Fredand Haroldarrivedat these sharedmeaningsby
explicitly taking on the task of clarifyingtheirdescriptionsto each other.During
this clarification,they used gestures, talk, and referenceobjects as coordinated
resourcesfor unpackingthe meaningof a description.
Towardthe end of theirdiscussion,not only did HaroldandFred'sdescriptions
flow more easily, theirdescriptionsalso reflectedsome conceptualrefinement.In
the beginningof theirconversationit was not alwaysclearwhetheror when Harold
and Fredwere consideringthe effects of changingm andchangingb as generating
independentmovementson a line. Forexample,Fredusedthey-interceptto explain
why a line was steeper.Moreover,both Fred and Haroldinitially thoughtthe line
y = x + 5 might be steeper. They also had to clarify whetherthis change in the
equationwould make a line both become steeperas well as move up on the y-axis.
By the end of theirdiscussion session, they easily and confidentlyreferredto the
258
MOSCHKOVICH
effects of changingm andchangingb, rotationandtranslation,as separateproperties
of lines.
CASE STUDY2: USINGREFERENCEOBJECTSTO
CLARIFY
ANDJUSTIFYDESCRIPTIONS
Students' initial descriptionsreflected the use of everyday meanings-meanings
thatsometimesprovedproblematicfor the conversations.Althoughthese everyday
meanings for descriptions of lines can be ambiguous, they can also serve as
resources for constructing shared descriptions. Case Study 2 shows how two
studentsused a sharedmetaphorfrom everydayexperienceand referenceobjects
to elaborate the meaning of their descriptions.The students in this case study
developed shareddescriptionsof lines by relying on a metaphorcomparinglines
to hills and by repeatedlyusing referenceobjectsto justify theirdescriptions.
Marcelaand Giseldabegan theirdiscussion with repeateddisagreementsabout
the meaningof the termsteeper.I providedthem with an explanationof steepness
comparingtwo lines on the board to two hills, explaining that steeper lines are
harder to climb. This metaphorproved to be useful for the students as they
constructeda sharedunderstandingof the meaningof steeper.Marcelapersistedin
using this metaphorto explainto Giseldawhy a line was steeperor less steep than
another.She first referredto the x-axis as "theground"and laterused both axes as
reference objects for justifying her descriptions.Reference objects were also an
importantresource for this conversation.Marcela introducedthe use of several
referenceobjects,such as thex-axis, they-axis, andthe originto clarifythe meaning
of her descriptionsto Giselda. Giselda followed Marcela's lead in using these
resourcesand laterindependentlyused these referenceobjectsin her own descriptions.
When workingon the problemwherethe targetequationwas y = x + 5 (Problem
3a, See Figure 1), Marcelaand Giseldastartedout disagreeingon whetherthe line
y = x + 5 would be steeperthanthe line y = x or not, and whetherit would move on
the y-axis. Marcelafirst proposedthatthe line would "go up five more."Giselda
proposedan alternativedescription,saying thatthe line would "touchthe middle."
They then drew the line y = x + 5 on theirpaper.Marcelawrote down "Thatwill
makethe line go up five moreandbe steeper"on the paperand checkedyes for the
answers to Questions A, B, and C. This written answer was not the same as
Marcela'sfirst proposalthatthe line would "go up five more,"because it included
a statementthatthe line will be steeper.At this point,Marcelaexplicitly disagreed
with the writtenanswer:
NEGOTIATING
SHARED
DESCRIPTIONS
OFLINEAR
GRAPHS 259
Excerpt4: MarcelaandGiselda(Problem3a)
TargetEquationy = x + 5
7 Marcela:
8 Giselda:
Dialogue
No, it's not steeper!
Look ... how come you
put yes [referringto the answers to QuestionA and
Commentary
Marcelachallengesthe written answer.
Giseldaquestionswhy Marcela had initially answered
the questionsyes,
C]?
9 Marcela:
Because it's the same line.
10 Giselda: Whatdo you mean the
same line?
Marcelajustifies her new answer, describingthe line y =
x + 5 as "thesame line."
Giselda asks for a clarification of the phrase"thesame
line."
Presumably,Marcelameantthatthe line y = x + 5 has the same slope as y = x.
They moved on to the problemof clarifyingwhatthe phrase"thesame line"meant
by pointing to the examples on the board.Therewere two examplesdrawn,y = x
andy = 8x as an example of steeper, andy = x andy = x + 6, as examplesof move
up on the y-axis. Althoughthey triedto addressthe meaningof "sameline," they
did not seem to reachany overt agreementon this aspectof Marcela's description.
They then moved on to consideringthe relationbetween a steeper line and the
origin. Giselda asked whethera line that is steeperthan anotherline has to cross
the origin and Marcelainsistedthatit does not.
At this point, I intervened,askingthem, "If two lines are parallel,do you think
one is steeperthanthe otherone?"Marcelaansweredno and Giselda said she did
not know. They both agreedthat the two lines on the board,y = x and y = x + 6,
were parallel. I attemptedto clarify the meaning of the term steeper making a
comparisonbetween lines and hills and saying thatsteeperlines or hills areharder
to climb:
25 Int:
Dialogue
Do you thinkthis one
[pointingto the line y = x +
6 on the board]is steeper
thanthis one [pointingto
the line y = x]? If you had
Commentary
260
MOSCHKOVICH
Dialogue
to climb up this hill [pointing to the line y = x + 6],
would it be harder?
26 Giselda: Yeah, the top one would
be harder.
27 Marcela: Why?
28 Giselda: Because it's sleeper,I
mean steeper[laughs].
29 Marcela: Why is it steeper?
30 Giselda: Because it is, look!
Commentary
Giselda identifiesthe line y
= x + 6 as harder to climb.
Marcelaasks for a justification.
Giselda identifiesthe line y
= x + 6 as steeper.
Marcelaagain asks for a justification.
Giselda againproposesthat
the line y = x + 6 is steeper.
In this conversation,Giseldaexplicitly identifiedthe line y = x + 6 as steeperor
harder to climb several times. Although Marcela asked Giselda to justify her
descriptionseveraltimes, Giseldaprovidedonly the line itself as evidence. After I
asked Giseldaif she thoughtthatsteepermeanthigherand she agreed,I went on to
clarify the differencebetweenthese two descriptions:
Dialogue
So if you had to climb this
hill you thinkthatwould
be harder?[pointingto y =
x + 6]
34 Giselda: [Nods her head in agreement.]
Commentary
33 Int:
35 Int:
Because ...
36 Giselda: Well, I thoughtthatsteeper
means to like high ...
higher.
37 Int:
It doesn't. This is the same
steepnessas this one
[pointingto y = x andy = x
Giselda again identifiesthe
line y = x + 6 as harderto
climb thanthe line y = x.
Giseldaclarifiesher understandingof the termsteeper.
+ 6].
Giselda seemed to accept the proposalthat there is a difference between the
meaningof the termsteeperandthe meaningof the termhigher. She also changed
her descriptionof the line y = x + 5:
NEGOTIATINGSHAREDDESCRIPTIONSOF LINEARGRAPHS
Dialogue
38 Giselda: So it's the same, it's going
in the same position, so it's
not steeper ... OK.
261
Commentary
Giselda comparesthe lines y
= x andy = x + 5.
Giseldadescribedthe line y = x + 5 as "thesame,"clarifyingthatthis means"in
the same position"and "notsteeper,"presumablymeaningthat the line y = x + 5
has the same slope as y = x. They went on to graphthe equationy = x + 5 on the
computerand answeredall the questionscorrectly.
In the next excerpt, Marcelacontinuedusing the metaphorthat lines are like
hills, comparingthe x-axis to the ground.She also beganusing referenceobjectsto
clarify and explain her descriptions,a move that Giselda followed in her later
descriptions.During this problem,Marcela and Giselda initially disagreed as to
whetherthe line for y = -0.6x was less steep than the line y = x. Giselda initially
thoughtthatthe steepnesswould not change,andMarcelatwice askedher to make
surethatwas her answer.They thengraphedthe equationon theirpaper,answered
the questions, and Marcela proceededto check Giselda's answers. Giselda had
answeredthatthe line would be steeper.
Excerpt5: MarcelaandGiselda(Problem8a)6
TargetEquationy = -0.6x
Dialogue
22 Marcela: No, it's less steeper...
23 Giselda: Why?
24 Marcela: See, it's closer to the xaxis ... [looks at Giselda]
... Isn't it?
25 Giselda: Oh, so if it's righthere ...
it's steeperright?
26 Marcela: Porqueffjate,digamosque
este es el suelo. Entonces,
si se acercamis, pues es
Commentary
MarcelacorrectsGiselda's
answer.
Marcelaclarifies the meaning of "less steeper,"using
the distancefrom the line to
the x-axis.
Marcelaintroducesthe metaphorthatthe x-axis is like
the ground.
6ChoiceC in this problemreads:"Theline would flip to the otherside of the y-axis" to addressthe
effect of a negative slope on the line.
262
MOSCHKOVICH
Dialogue
menos steep. [Because
look, let's say thatthis is
the ground,then, if it gets
closer, then it's less steep.]
Commenatry
AfterMarcelaproposedthatthe second line was "less steeper,"she clarifiedthe
meaningof this phrasefirst by using the x-axis as a referenceobject (line 24) and
then by using the metaphorthatlines are like hills (line 26), referringto the x-axis
as the ground as well as using the distance to the x-axis. Marcela continued to
describe lines using the axes (line 30). She also explained the meaning of "less
steep"using a comparisonof the distancesfrom the line to the x- andy-axes (line
32):
Dialogue
30 Marcela: ... 'cause see this one [referringto the line y = x] ...
is ... esti entreel medio de
la x y de la y [is between
the x and the y]. Right?
31 Giselda: [Nods in agreement.]
32 Marcela: This one is closer to the x
thanto they, so this one is
less steep.
33 Giselda: All right.
Commentary
Marceladescribesthe steepness of the line y = x using
the axes as referenceobjects.
Giselda agrees.
Marcelarepeatsher clarification thatthe otherline is less
steep because it is closer to
the x thanto the y-axis.
Giselda agrees.
As the discussionprogressed,Marcelarepeatedlyclarifiedherunderstandingof
the terms and phrasessteep, steeper, less steep, move on the y-axis, move up, and
move down to Giselda. Marcelacontinuedto use the x- and y-axes as reference
objects for describingthe steepness of lines, as illustratedin Excerpt 5. In subsequent discussions about steepness,Marcelacontinuedto use the metaphorthat
lines are like hills to clarify the meaningof her descriptionsto Giselda. Marcela
also alternatedbetweenusing the "ground"andthe "x-axis"or "thex" as reference
objects to clarify or justify her claims about the steepness of a line. During four
other problems,Marcelaexplainedto Giselda why a line had not "moved on the
y-axis"and why a line hadnot changedsteepness.Eachtime Marcelaused the axes
as referenceobjects, describinga line that was less steep as "closerto the x-axis"
and a steeperline as having moved "closerto the y-axis."
By Problem9a, althoughMarcelapredictedthatthe line for the equationy = 10x
would be "almoststraightup,"Giseldawas initiallynot surewhetherthe line would
be steeperor not andthenproposedthatthe line would be "less steeper."Next they
NEGOTIATINGSHAREDDESCRIPTIONSOF LINEARGRAPHS
263
workedon the questionof whetherthe line would move on the y-axis. Giseldafirst
proposedthatthis line would move on the y-axis, and Marceladisagreedwith her.
Marcelareferredto the examples on the blackboardto clarify what this question
meant. After discussing how a negative coefficient for x affects the line, they
returnedto consideringwhetherthe line would move on the y-axis:
Excerpt6: Marcelaand Giselda (Problem9a)
TargetEquationy = 10x
Dialogue
27 Marcela: The line will not move on
the y-axis. Why?
28 Giselda: Because it's not a negative
number!
29 Marcela: No! It's because it's still
crossing the x point, the
origin point.
30 Giselda: Aquf es [hereit is] because
what?
31 Marcela: Here it's because it got
closer to the x-axis ... I
mean to the ...
32 Together: y-axis!
Commentary
Marcelauses the origin as
referenceobject to justify
her claim thatthe line will
not move on the y-axis.
Giselda asks for an explanation for QuestionA: "Will
the line be steeperor less
steep?
Marcelaexplains why the
line is steeperusing the xaxis and correctsherselfto
say y-axis at the same time
thatGiseldacorrectsher.
They graphthe equationy =
10x.
Duringthisconversation,Marcelausedtwo referenceobjectsin herdescriptions.
First she used the "x point"or "the origin point"to explain why the line had not
moved on the y-axis. Second, she used the y-axis to describe a steeper line. This
last descriptionwas co-constructedby Marcelaand Giselda together.As Marcela
incorrectlydescribedtheline as closerto "thex-axis,"they changedthisto "y-axis"
in unison.
Giselda later independentlygenerateddescriptionsusing the same reference
objectsoriginallyintroducedby Marcelainto the conversation.Forexample,in the
next excerptGiselda used the origin as a referenceobject for describingsteepness
as well as verticaltranslationanddescribedsteepnessin termsof the distancefrom
264
MOSCHKOVICH
the two axes. These descriptionswere built on the use of the referenceobjects that
Marcelahad earlierintroduced.
Excerpt7: MarcelaandGiselda(Problem1la)
TargetEquationy = 0.lx
1 Giselda:
2 Marcela:
3 Giselda:
4 Marcela:
Dialogue
If you change the equation
to y = 0.1x ...
Tiene que ser bien
pegadita.[It has to be very
close.]
Right here? No, it doesn't
have to cross ...
Commentary
Marceladescribesthe line
using the x-axis as a reference object.
Giseldadescribesthe line using the origin as a reference
object.
It is! It will cross.
In this dialogue, Marcelaimplicitlyused the x-axis to describethe steepness of
the line, saying it had to be "veryclose" (to the x-axis). Giselda used the origin in
her descriptionof the line, saying thatit "doesn'thave to cross"(the origin).
As Marceladrew the line on the paper,they continueddescribingthe line and
answeringQuestionA: "Thesteepnesswould change"(the choices are steeperor
less steep). Giseldapredictedthatthe line would be less steep and Marcelaagreed.
Marcelathen providedan explanation:
Dialogue
16 Giselda: It's less because ...
17 Marcela: The line is closer to the xaxis ...
18 Giselda: Let me do it ... is closer to
the x-axis and the ... and is
furtherfrom the y-axis.
Commentary
Marceladescribesthe line's
steepnessusing the x-axis as
a referenceobject.
Giselda completesthe descriptionusing the y-axis as
a referenceobject.
They moved on to deciding whetherthe line would move on the y-axis:
19 Marcela: The line will move on the
y-axis?
20 Giselda: No.
MarcelareadsQuestionB.
NEGOTIATINGSHAREDDESCRIPTIONSOF LINEARGRAPHS
Dialogue
21 Marcela: No, no, because it will still
22 Giselda: Because it would still cross
the origin.
265
Commenatry
Marcelabegins a justification.
Giseldafinishes Marcela's
justificationusing the origin
as a referenceobject.
23 Marcela: Yeah.
Giselda was, at thatpoint,also using the axes andthe originas referenceobjects
in her descriptions(lines 18 and 22). These two descriptionswere again co-constructed.In lines 14 and 16, Giseldaprovidedthe description"less steep,"Marcela
beganthejustificationin line 17, andGiseldacompletedthejustificationin line 18.
For the answerto QuestionB, Marcelabegana justification(line 21), and Giselda
completed it (line 22). Since the dialogue in Excerpt5, Marcelaand Giselda had
moved towardco-constructingshareddescriptions.Initially,Marcelaprovidedboth
the descriptionsand the justifications, as seen in Excerpt 5. During Excerpt 6,
Giselda startedto generatea descriptionand correctedMarcela'sjustification.In
Excerpt 7, both Giselda and Marcela generateddescriptionsand justifications;
Giselda also completedjustificationsstartedby Marcela.The repeateduse of the
same reference objects in their descriptionsand justifications was an important
resourcefor moving towardthese shareddescriptions.
By problem13a,althoughneitherstudentcould predictwhatthe line y = x + 100
would look like before graphing(they wrote "we have no idea"for all answers),
once they had graphedthe equation,they easily describedthe line and agreedon
their descriptions.After they graphedthe equationy = x + 100, Giselda correctly
describedthe line as havingthe samesteepnessandcontinuedto describetranslation
using the origin as a referenceobject:
Excerpt8: MarcelaandGiselda(Problem13a)
TargetEquationy = x + 100
1 Marcela:
2 Giselda:
3 Marcela:
Dialogue
"Thesteepnesswould
change."
No ... it's still the same ...
it didn't pass throughthe
cross, cross throughthe origin point ... it went up because ...
[Writesthe explanationfor
QuestionB: Because it's
Commentary
MarcelareadsQuestionA.
Giselda uses the origin as a
referenceobject to propose
thatthe steepnessis the
same. She also proposesthat
the line went up.
266
MOSCHKOVICH
Dialogue
the same line. For Question C she writes "yes, because it's x + 100, 100
more up."]
Commentary
Case Study 2 shows that metaphorsfrom everyday experience can be useful
resourcesfor constructingshareddescriptionsof mathematicalobjects. The comparisonof lines and hills provideda way for Marcelato justify her understanding
of the meaningof steepness.Thiscase studyalso shows how studentsusedreference
objects as resources for clarifying the meaning of their descriptions and for
justifying theirdescriptions.Marcelaintroduceda metaphorcomparingthe x-axis
to the land and repeatedlyused reference objects to explain her descriptionsto
Giselda.Marcelaused the x-axis as a referenceobjectto arguewhy a line was less
steep, describedthe line y = x as being "betweenthe x and the y," and defined the
term less steep as meaning"closerto the x thanto the y." Marcelaalso introduced
the use of the origin as a reference for vertical translation.As the discussion
progressed,Giselda also came to use these referenceobjects in her descriptions,
initially completing, correcting, or justifying Marcela's descriptions, and later
generatingsimilardescriptionson her own.
Case Studies 1 and 2 show that peer discussions can successfully supportthe
constructionof shareddescriptionsof mathematicalobjects. In each of the case
studies previouslypresented,students'conversationsat the end of the discussions
reflected the constructionof shareddescriptionsof lines. That is, an individual's
descriptionwas not contestedby the partner,and the pair seemed to agree on one
description.Conversationswere no longer interruptedto negotiate,elaborate,or
clarify meanings.Althoughthe style of the conversationsin Case Study 1 and 2 are
different, both discussions resulted in the constructionof shared descriptions.
Marcelaseemed to take on the role of explainer,whereasGiseldararelyexplained
a descriptionto Marcela.In contrast,Haroldand Fredseemed to participatemore
equally in providingboth explanationsand elaborations.Although the discussion
styles were different, both pairs moved toward less elaborations,less contested
descriptions,and more sharedmeanings.
Not only did these studentsconstructshareddescriptions,theirdescriptionsalso
became more precise, and thus more mathematical,and came to reflect important
conceptualpieces. Threeof the four studentsin Case Studies 1 and 2 refinedtheir
descriptionsso that they reflected importantconceptualknowledge about linear
functions. By the end of the discussions, Harold, Fred, and Marcela explicitly
referredto translationandrotationas independentpropertiesof a line. Forexample,
in the last problem of the discussion, Marcelaexplicitly describedrotationand
translationas independentpropertiesin her writtenanswer"it only moved down."
Although Giselda's descriptions were more tentative, she had also started to
NEGOTIATINGSHARED DESCRIPTIONSOF LINEARGRAPHS
267
separatethese two properties.Thus, these conversationswith a peer supported
students'conceptualchange as well as the constructionof shareddescriptions.
CASE STUDY 3: UNRESOLVED ALTERNATIVE
DESCRIPTIONS
This last case studyis presentedas a contrastto Case Studies 1 and2. Althoughthe
studentsin Case Study 3, Monicaand Denise, attemptedto negotiatethe meaning
of theirdescriptions,they did not resolve initialambiguitiesor move towardshared
meanings for their descriptions.One characteristicof the conversationsin Case
Study 3 is thatMonica and Denise generatedmanyalternativedescriptions,rather
thanfocusing on clarifyingone or two descriptions.Moreover,they did not resolve
conflicts in the meaningof these alternativedescriptions.Forexample,Monicaand
Denise alternativelydescribedlines as having moved "to the left," "to the right,"
and "to the side(s)." The descriptionsusing left and "side(s)" were especially
ambiguous and problematicfor the conversations.Monica and Denise alternated
between using these two termsto sometimesreferto rotationaboutthe origin and
othertimes to referto horizontaltranslation.
Although choosing the axes as reference objects to describe and justify the
steepness of a line, as Marcela did in Case Study 2, might seem naturalfor
describingthe steepness of lines thatcross the origin, Monica and Denise did not
settle on any one choice of referenceobjectsto describesteepness.Monica initially
introducedthe terms left and right to referto rotation.This choice of termsmade
the discussion problematic,becausethese two studentslateralso used these terms
to referto horizontaltranslation.
The first use of the terms left and right occurredduring Problem 4a. In the
dialogueimmediatelyprecedingthisexcerpt,MonicaandDenise hadpredictedthat
the answerfor QuestionA ("Wouldthatmake the line steeper?")before graphing
the equationy = 3x, wasyes, andDenise hadexplainedthatthis was "Becausewhen
you times it (the line) goes steeper."They then proceededto graphthe equationy
= 3x and attemptedto describewhat happensto the line when b changes ("when
you add")and when m changes ("whenyou times"):
Excerpt9: MonicaandDenise(Problem4a)
TargetEquationy = 3x
16 Denise:
Dialogue
'Cause when you add it
goes on ... on ... some-
Commentary
268
MOSCHKOVICH
Dialogue
where on the line [points
to the negativeside of the
x-axis].
17 Monica: But we weren't adding!
We were timesing.
18 Denise: When you're timesing it
stays rightin the middle
[pointsto the origin].
19 Monica: It moves this way [rotates
handclockwise and counterclockwise]to the left or
to the right.
Commentary
Denise describesadditionas
affectinga line by moving it
along the x-axis.
Denise proposesthatmultiplicationdoes not change
where a line crosses the origin.
Monica uses a gestureand
the phrase"tothe left or to
the right"to describethe effect of changingm on the
line.
When Denise describedthe effect of addition,she first used the description"it
goes somewhereon the line"as she gesturedtowardthe x-axis. Monica accurately
gesturedwith her hand to representthe effect of multiplicationas rotatinga line
aboutthe originanddescribedthis movementas "tothe left"or "tothe right."Next,
they triedto clarify what they each meantby the phrase"tothe left":
Dialogue
It moved thatway [moves
handcounterclockwisebetween the two lines on
screen].
21 Monica: It moved to the left, right?
[they both point to the
lines on the screen].
22 Denise: Yeah, it moved to the left
... It moved clockwise [the
line actuallymoved counterclockwise].
23 Monica: Putyes ...
20 Denise:
24 Denise:
'Cause when ...
Commentary
Denise uses a counterclockwise gestureto clarify her
description,labeling this
movementthat way.
Monica proposesthatthe
line moved left.
Denise agreesand clarifies
the meaningof left as clockwise.
Monica proposesthatthe answer to QuestionA is yes.
NEGOTIATINGSHAREDDESCRIPTIONSOF LINEARGRAPHS
Dialogue
25 Monica: 'Cause when ... you ...
multiplythe line gets
steeper ... it moves more
to the left and makes it
steeper ... moves to the
left.
269
Commentary
Monica proposesthe explanationthatmultiplication
moves the line to the left
and makes it steeper.
This dialoguerevolved aroundthe negotiationof the meaningof the phrase"to
the left." In line 19, Monica had moved her handclockwise and counterclockwise
as she describedrotationas "to the left or to the right."Denise initially accepted
Monica's definitionof rotationas "tothe left"(line 22) andthenaddedthatthe line
moved "clockwise"when the line hadin effect movedcounterclockwise.Monica's
concludingdescriptiondescribeda changein m as havingthe effect that"it moves
more to the left and gets steeper."Althoughin line 19 Monica used "to the left or
to the right"to refer to rotation,as evidenced by her gesture, it is not clear how
Monica was using eitherleft or steeperin line 25. She may have been using left to
describe horizontaltranslation.If she was using left to describerotation,then her
last descriptionreferredto rotationtwice.7
The use of "to the right or to the left" became problematicwhen Denise
subsequentlyused rightor left to referto translationas they answeredQuestionB:
"Does it move the line up on the y-axis?" They initially went back and forth,
disagreeingas to whetherthe line had or had not moved up on the y-axis. Monica
insisted thatthe line had moved up on the y-axis. Denise insistedthatthe line had
not moved up on the y-axis becauseit still crossedthe origin.Monicaproposedthat
the line had "moved on x," and Denise answeredthat the line "didn't move on
nothing." Monica seemed to reluctantlyaccept Denise's description, and they
arguedback and forthabout who would write down the answer.As they returned
to deciding whetherthe line had moved up on the y-axis, they again disagreedon
the meaningof the phrase"moveleft":
7Thereare two aspects of this dialoguethat are relatedto students'conceptions.One is that Denise
was beginningto use the connectionbetween a change in the equationand a change in the line in her
explanations:"Whenyou add it goes on ... somewhereon the line [axis] (line 16), and "when you're
timesing it stays right in the middle [origin]"(line 18). Monica, however, neitherinitiatedthis sort of
explanationnorwas she convincedby Denise's use of this connectionbetweenthe two representations
(lines 11-15). The second aspect is that Denise focused on horizontal,ratherthan vertical,translation
(line 16).
270
MOSCHKOVICH
Excerpt10:MonicaandDenise(Problem4a, Continued)
Dialogue
It didn't move up or down
or right or left, it just got
steeper.
57 Monica: Yes, it did move left [pointing to the line]! It didn't
... Oh, gosh! [writingon
the paper].It didn't move
up or down ... because it
didn't move up or down, it
just got steeper.
56 Denise:
Commentary
Denise uses right or left to
describehorizontaltranslation.
Monica first proposesthat
the line did in effect move
left, then changes her description.
In the preceding dialogue, although Denise used left to refer to horizontal
translationalong the x-axis (line 56), Monica insistedthatthe line had moved left
(line 57), presumablymeaningthe line had gotten steeperand using the meaning
for left establishedin theirdialogue in Excerpt9. AlthoughMonica's last answer
(line 57) might seem to indicatethatthey had reachedagreementon the meaning
of theirdescription,laterconversationsshow thatthis was not the case. Monicaand
Denise continued to use several alternativedescriptions where the same term
referredto two movements.
During the next problem,Denise introducedanotherphraseto describe both
rotationandtranslation,"tothe side"(or sides), which also provedproblematicfor
the constructionof a shareddescription.Denise initiallyused the termside to refer
to horizontaltranslation,describingthe effect of additionas moving a line "to the
side" as she pointed to the segment of the x-axis left of the origin. During this
problem,Monica used left and right in associationwith addition,even thoughshe
had recently used the phrase"to the left" to referto rotationin Excerpts9 and 10.
They attemptedto clarifytheirdescriptionsof the effect of additionandagreedthat
"whenyou add it moves to the sides andparallel,"thususing "tothe sides"to refer
to translation.However,this agreementon the meaningof the phrase"movesto the
sides" was transitory,because Denise later also used the phrase"to the side" to
describerotation.
Fromthis pointon, MonicarepeatedandwrotewhatDenise said, withouteither
contributingany descriptionsof her own, asking for clarification,or contesting a
description.After working on 10 problemstogether,they had not yet arrivedat
shareddescriptionsof lines. This lackof resolutionwas due in partto the ambiguous
use of several alternativephrasesto referto both translationand rotation.In sum,
these two studentsalternatedbetweenusing"tothe left"to referto rotation(Monica
and Denise, Excerpt9) and "move right or left" to refer to horizontaltranslation
(Denise, Excerpt 10; Monica, transcriptnot shown). They also used alternative
NEGOTIATINGSHARED DESCRIPTIONSOF LINEARGRAPHS
271
descriptionsusing the term side(s) to refer to rotation,such as "tilts to the side"
(Denise) as well as horizontaltranslation,"to the side" (Denise), "to the sides"
(Monica and Denise). The discussion betweenMonica and Denise continuedto be
characterizedby this repeateduse of alternativedescriptionswithoutagreementon
sharedmeanings.
Duringsubsequentproblems,Monica and Denise generatedseveralmore alternative ways to describe a line, but failed to agree on the meaning of these
descriptionsor settle on shareddescriptions.Forexample,while describingthe line
y = x + 6, Denise used the phrase "it will just go down," focusing on vertical
translation,whereas Monica said the line would "just move over," focusing on
horizontal translation.Denise described this line as having "the same angle,"
whereasMonicadescribedthe line as staying"onthe axes."Monicaintroducedyet
a thirddescriptionusing the term side. She describedthe effect of changing the
coefficient of x as making the line "go to the other side," which could refer to
translation,rotation,or reflectionaboutthey-axis. Monicaalso introducedthe term
lower, which can be interpretedas referringto either translationor rotation,to
describe the effect of multiplicationby a negative number.By the end of their
discussion sessions, Monica and Denise had not agreedon shareddescriptionsfor
eitherrotationor translationandhadnot resolvedthe conflictinguses of theirmany
alternativedescriptionsinvolving the termsleft or side(s).
AlthoughMonica and Denise did not arriveat sharedmeanings,their descriptions did come to reflectsome conceptualchanges.Theirlaterdescriptionsreflected
an increased coordinationbetween the algebraic and graphicalrepresentations.
They describeda change in the equationas generatinga change in the line, saying
"addand it will go down," or "multiplyit will go down and to the other side." In
herlaterdescriptions,Denise focusedon verticaltranslationas theresultof a change
in b. She also began to separaterotationand translationas independentproperties,
for example, stating that "when you add it went up, maybe when you subtractit
will go down, it [the steepness] will be the same" and saying that a line with a
differenty-interceptwould "justgo down."
Monica's descriptionsalso reflected a greatercoordinationof the two representations.For example, she describedthe effect of a coefficient thatis less than 1
as "it [the coefficient] will make it steeperbut not thatmuch because multiplying
changes the steepness, right?"On the other hand, Monica continuedto focus on
horizontaltranslationas the resultof a changein b, to referto horizontalandvertical
translationconcurrently,and at times combinedreferencesto rotationand translation in her descriptions.
The conversationsin this last case study illustratethe importanceof not only
negotiating meanings but also resolving these negotiations.Although these two
studentsengaged in repeatednegotiations,they did not resolve the conflicts among
their alternativedescriptions.Unlike the students in Case Studies 1 and 2, who
addressed conflicts directly and resolved their negotiations by the end of the
272
MOSCHKOVICH
discussion session, Monica and Denise persistedin using eitherdifferentdescriptions or the same descriptionwith differentor ambiguousmeanings.
When comparedwith the studentsin Case Studies 1 and 2, Monica and Denise
generatedmany more alternativedescriptionsfor the same situation,ratherthan
focusing on elaborating and clarifying one or two descriptions. Although the
students in Case Studies 1 and 2 generated some alternativedescriptions,they
usually returnedto the descriptionsprovidedin the problems(steeper,less steep,
and "moves up/downon the y-axis")to describethe lines on the screen.Moreover,
Monica insisted on focusing on horizontaltranslationeven though the problems
only referredto verticaltranslation.The studentsin the first two case studiesthus
stayed on mathematicallyproductivepathsby focusing on only a few descriptions
and using the descriptionsprovidedin the problems.These comparisonspoint to
three importantcharacteristicsof a conversationwith a peer, whether and how
students(a) addressconflicts, (b) resolve negotiations,and (c) maintaina focus on
mathematicallyproductivepaths.
CONCLUSIONS
The analysis presentedin this articleshows thatpeer discussions can successfully
support the constructionof shared descriptionsof mathematicalobjects. These
conversationscan create the need for clarificationand provide a rich context for
negotiatingsharedmeanings.Studentsuse manyresourcesto elaborateandclarify
theirdescriptions:everydaymeaningsand metaphors,referenceobjects,and coordinatedgesturesand talk. Althoughstudentscan and do reach agreementduringa
conversationwith a peer, neitherresolutionnorconceptualconvergenceis guaranteed. The role of instructionin orchestratingand supportingpeer discussionlies in
modeling how to resolve negotiations and in maintaining students' focus on
mathematicallyproductivepaths.
Reachingconversationalclarityand moving towardagreementwere important
goals during these discussions. Students initially used descriptions that were
sometimes ambiguousand other times problematicfor their conversations.They
often did not use terms referringto the translationand rotationof lines with the
same meaningas thatintendedby the researcheror, perhapsmore importantly,by
their partners.Students'descriptionswere sometimes problematic,as in the case
of the term steeper to mean higher (Case Study 2) or to refer to translation
concurrentlywith rotation(Case Study 1). Other instances were the uses of the
terms left or side(s) to referto both translationand rotation(Case Study 3). In the
first two case studies,as the discussionproceeded,the studentsincreasinglysettled
on shareddescriptionsof the lines on the screen.
The negotiationof meaning that studentsengaged in, the fact that two of the
pairsarrivedat shareddescriptions,and the ways thatsome of the studentsrefined
their descriptionsall show that peer discussions can be a productivecontext for
NEGOTIATINGSHAREDDESCRIPTIONSOF LINEARGRAPHS
273
transformingstudents'languageuse. Peerdiscussionsmay motivatetherefinement
of students'descriptionsby creatingsituationsin which theirdescriptionsare not
clear or preciseenoughto communicatesuccessfullywith anotherstudent.In much
the same way thattheremust be some motivationfor changingor giving up one's
initial conceptions about a domain (or the conceptions that work in everyday
situations), so also, there must be some motivation for changing the everyday
languageone uses to describeobjectsin thatdomain.If the initial languageused is
not precise enough or is too ambiguousto communicatesuccessfully with another
student,reachingconversationalclaritycan be a motivationfor the negotiationof
meaning,the elaborationof descriptions,and the refinementof languageuse.
Although the constructs of sociocognitive conflict or guidance by a more
advanced other have contributedto the understandingof peer discussion, the
analysis presented in this article focused on conversationsand conversational
resourcesas a way to understandthe processof learningthroughpeer discussions.
One importantreasonforfocusingon conversationalprocessesis thatconversations
are inherentlysocial phenomena.This move shifts learning from an individual
location, as in the sociocognitive conflict model, to a social site. Studentsused
several local conversationalresourcesto elaborateand disambiguatedescriptions
such as the coordinationof talk and gestures,the use of referenceobjects, and the
use of spatialmetaphorsfrom everydayexperience.
The analysis of peer conversationspresentedhere draws on neo-Vygotskian
theories in some importantways. The assumptionsthat learningis mediatedby
language, that social interactionis integralto the learningprocess, that learning
involves the constructionof socially sharedmeaning,and that learningin school
involves a shift from everydayto "scientific"concepts are all centralto neo-Vygotskianperspectives.However,this analysisalso divergesfromthese frameworks
in takinga perspectivethatmakestheco-constructionof sharedmeaningsas central
as guidance.Thus,ratherthanidentifyingwho is the more advancedparticipantor
privileging the contributionsof one participantas the source of expertise, the
contributionsof each participantare considered equally in the negotiation and
constructionof meanings.
This accountof learningmathematicsshows thatunderstandingthe connection
between the algebraicand graphicalrepresentationsof linear functions includes
refiningdescriptions.The negotiationandrefinementof students'descriptionswere
an importantaspect of making sense of lines and their equations.This learning
process involved, in part,a shift fromeverydayto moremathematicaland precise
descriptions. One importantdifference between the everyday and the school
mathematicsregistersmay be the meaningof relationaltermssuch as steeper and
less steep, and phrases such as moves up the y-axis and moves down the y-axis.
Meaningsfor these termsandphrasesthatmay be sufficientlyprecisefor everyday
purposesproved to be ambiguousfor describinglines in the context of a mathematicaldiscussion.
274
MOSCHKOVICH
Although the differencebetween the everydayand mathematicalregistersmay
sometimes be an obstacle for describing lines in mathematicallyprecise ways,
everydaymeaningsand metaphorscan also be resourcesfor understandingmathematicalconcepts. Ratherthanemphasizingthe limitationsof the everydayregister
in comparisonto the mathematicsregister,it is more importantto understandhow
the two registersserve differentpurposesand how everydaymeaningscan provide
resourcesfor conceptualchange.
Each of the studentsdiscussed hererefinedtheirdescriptionsof lines in at least
some conceptualways. This refinementin students'descriptionscan be understood
as a movement toward the mathematicsregister, where descriptionsof lines are
precise and reflect conceptual knowledge central to this domain. However, the
mathematicsregistertranscendsthe use of technicaltermsanddoes notconsistonly
of technicaltermssuch as slope and intercept.These studentsdid not simply learn
to use the technicaltermsslope andy-intercept.Instead,they refinedthe meaning
of theirdescriptionsby connectingeven nontechnicalphrasessuch as "theline will
be steeper"or "theline will move up on the y-axis"to conceptualknowledgeabout
lines and equations.
Mathematicaldescriptionsof lines involve conceptualknowledge such as the
interdependencyof the two representations,what is necessary and sufficient for
describinglines and theirmovement,and which propertiesof lines are dependent
or independent.Some of the core assumptionsone makeswhen using the termsand
phrasessteeper, less steep, moves up, or moves down to describethe movementof
lines are:
1. Rotationand translationare necessaryand sufficientto describethe movement of all lines.
2. Rotationand translationare independentof each other.
3. The movementof lines is describedin termsof a preferredreferenceobject.
In the case of y = mx + b, the preferredreferenceobjects for describingthe
effect of changingb arepointson they-axis. The preferredreferenceobjects
for describingthe effect of changingm are one or both of the axes.
These assumptionsareembeddedin mathematicaldescriptionsof lines andtheir
movement in a plane. Studentsin this studyrefinedtheirdescriptionsso that they
reflectedsome aspectsof these conceptualpieces. The refineddescriptionsof five
of the six students reflect the following conceptual knowledge: an increasing
coordinationbetween the algebraicand graphicalrepresentations,a separationof
the parametersm and b (and the correspondingmovement of lines), omitting
horizontaltranslation,andfocusingon verticaltranslationas a resultof changingb.
NEGOTIATINGSHARED DESCRIPTIONSOF LINEARGRAPHS
275
One model for supportingthe refinementof mathematicaldescriptionsin the
classroom might be to present vocabularyitems and explain these explicitly to
students. However, this study suggests that there may be importantdifferences
between a discussion with a peer and a presentationby an adult. One of the
differencesbetweenthe peerdiscussionsand the presentationby an adultwas that,
when workingwith a peer, studentshad many opportunitiesto generatetheirown
descriptions,elaboratethe meanings of these descriptions,and negotiate shared
descriptions.Thus, one of the beneficialprocesses in peer discussions may be the
occurrenceof such conversationalcycles of elaborationand clarification.
However, some discussions were more successful thanothers.Although negotiation was an importantprocess, the resources students bring to bear on these
negotiations and the characterof their discussions seems to be related to the
constructionof shareddescriptions.The two studentsin Case Study3 did not reach
agreement,resolve discrepanciesin theirdescriptions,or move towardconceptual
convergence. These two studentsalso seemed to generatemany alternativeways
to describesituationsratherthanpersevereat understandinga few descriptions,like
the studentsin Case Studies 1 and 2. This difference in the natureof these peer
conversations points to the importantrole of instruction in orchestratingand
supportingpeer discussions. This role lies in modeling how to resolve negotiation
and in maintainingstudents'focus on productivequestions.
The analysis presentedhere raises questionsregardinghow studentslearnwith
peers, specifically in termsof the role of authorityand languageuse. If one of the
peers is identified as an authority,then peer discussions are much like adult
guidanceandcan be describedin termsof peertutoring,scaffolding,andso on. But
these discussions were also differentfrom adultguidance.Studentsused theirown
termsand meanings,and they engaged in extensive discussions of the meaningof
terms. These two activities may be in contrastto the way that studentsengage in
discussionswith adults.Exploringthe differencesbetweenadultguidanceandpeer
discussions, especially in termsof how languageis used, elaborated,andclarified,
is an importantfocus for furtherresearch.
Thereareseveralotherissues raisedby thisstudythatmeritfurtherinvestigation.
There are importantquestions in terms of language use and the mathematics
register.The differences between the vernacularand mathematicaluses of terms
need to be exploredin more detail. Whatthe advantagesand disadvantagesof the
use of everyday spatial metaphorsin mathematicalcontexts might be remainsan
open question. In particular,furtherresearchshould address how students who
speak a language other than English develop competence in the mathematics
register in English. Although the metaphorthat "learningmathematicsis like
learninga second language"may be useful, it is not clear whatthe similaritiesand
differences might be between learninga second language, learningmathematics,
and learningmathematicsin a second language.
276
MOSCHKOVICH
ACKNOWLEDGMENTS
The researchreportedhere was supportedin partby the NationalScience Foundation and a SpencerDissertationFellowship. I thankAlan Schoenfeld,MaryBrenner, M. C. O'Connor,Andee Rubin, and the membersof the FunctionsResearch
Groupat the Universityof California,Berkeley,for theircommentson a previous
version of this article.
REFERENCES
Brown, A., & Pallincsar,A. (1989). Guidedcooperativelearningandindividualknowledgeacquisition.
In L. Resnick (Ed.), Knowing,learning, and instruction:Essays in honor of Robert Glaser (pp.
393-451). Hillsdale, NJ: LawrenceErlbaumAssociates, Inc.
Cocking, R., & Mestre, J. (Eds.). (1988). Linguisticand culturalinfluenceson learningmathematics.
Hillsdale, NJ: LawrenceErlbaumAssociates, Inc.
Davidson, N. (1985). Small group learning and teaching in mathematics:A selective review of the
research.In R. Scmuck (Ed.), Learning to cooperate, cooperating to learn (pp. 221-230). New
York:Plenum.
Doise, W. (1985). On the social developmentof the intellect. In V. Shulman,L. Restaino-Baumann,&
L. Butler (Eds.), Thefuture of Piagetian theory: The neo-Piagetians (pp. 99-121). New York:
Plenum.
Durkin,K., & Shire, B. (Eds.). (1991). Language in mathematicaleducation:Researchand practice.
Philadelphia:Open UniversityPress.
Forman,E. (1992). Discourse, intersubjectivityand the developmentof peer collaboration:A Vygotskianapproach.In L. T. Winegar& J. Valsiner(Eds.), Children'sdevelopmentwithinsocial context:
Vol. 1. Metatheoryand theory(pp. 143-159). Hillsdale, NJ: LawrenceErlbaumAssociates, Inc.
Forman,E., & McPhail,J.(1993). A Vygotskianperspectiveon children'scollaborativeproblemsolving
activities. In E. Forman, N. Minick, & C. Stone (Eds.), Contextsfor learning: Sociocultural
dynamicsin children's development(pp. 213-229). New York:OxfordUniversityPress.
Goldenberg, P. (1988). Mathematics,metaphors,and human factors: Mathematical,technical, and
pedagogicalchallenges in the educationaluse of graphicalrepresentationsof functions.Journalof
MathematicalBehavior, 7, 135-173.
Halliday, M. A. K. (1978). Sociolinguistics aspects of mathematicaleducation.In M. Halliday(Ed.),
Language as social semiotic: The social interpretationof language and meaning (pp. 195-204).
London:UniversityParkPress.
Hatano, G. (1988). Social and motivationalbases for mathematicalunderstanding.In G. Saxe & M.
Gearhart(Eds.),New directionsforchild development:Vol.41. Children'smathematics(pp.55-70).
San Francisco:Jossey-Bass.
Inagaki,K. (1981). Facilitationof knowledgeintegrationthroughclassroomdiscussion. The Quarterly
Newsletterof the Laboratoryof ComparativeHumanCognition,3(2), 26-28.
Inagaki, K., & Hatano,G. (1977). Amplificationof cognitive motivationand its effect on epistemic
observation.AmericanEducationalResearchJournal,14, 485-491.
Lucy, J., & Wertsch,J. (1987). Vygotsky and Whorf:A comparativeanalysis. In M. Hickman(Ed.),
Social andfunctional approachesto language and thought(pp. 67-86). Orlando,FL: Academic.
Mathematicsframeworkfor Californiapublic schools. (1992). Sacramento:CaliforniaDepartmentof
Education.
NEGOTIATINGSHARED DESCRIPTIONSOF LINEARGRAPHS
277
McDermott,R., Gospodinoff,K., & Aron,J. (1978). Criteriafor an ethnographicallyadequatedescription of concertedactivities and theircontexts.Semiotica,24, 247-275.
Moschkovich, J. N. (1990, July). Students'interpretationsof linear equationsand their graphs. Paper
presentedat the 14th annualmeetingof the InternationalGroupfor the Psychologyof Mathematics
Education,Oaxtepec,Mexico.
Moschkovich, J. N. (1992). Making sense of linear equations and graphs: An analysis of students'
conceptionsand language use. Unpublisheddoctoraldissertation.
NationalCouncilof Teachersof Mathematics.(1989). Curriculumand evaluationstandardsfor school
mathematics.Reston, VA: Author.
O'Connor,M. C. (in press). Negotiateddefining:Speech activitiesand mathematicsliteracies.In J. G.
Greeno& S. Goldman(Eds.), Thinkingpractices. Mahwah,NJ:LawrenceErlbaumAssociates, Inc.
Pimm, D. (1987). Speaking mathematically:Communicationin mathematicsclassrooms. London:
Routledge.
Pirie, S. (1991). Peer discussion in the context of mathematicalproblemsolving. In K. Durkin& B.
Shire (Eds.),Languagein mathematicaleducation:Researchandpractice (pp. 143-161). Philadelphia:Open UniversityPress.
Resnick,L. (1989). Treatingmathematicsas an ill-structureddiscipline.InR. Charles& A. Silver(Eds.),
Research agendafor mathematicseducation: Vol. 3. The teachingand assessing of mathematical
problem solving (pp. 32-60). Hillsdale,NJ: LawrenceErlbaumAssociates, Inc.
Richards,J. (1991). Mathematicaldiscussions. In E. von Glasersfeld(Ed.), Radical constructivismin
mathematicseducation(pp. 13-51). Dordrecht,The Netherlands:KluwerAcademic.
Roschelle, J. (1992). Learningby collaborating:Convergentconceptualchange. The Journal of the
LearningSciences, 2, 235-276.
Sharan,S. (1980). Cooperativelearningin small groups.Reviewof EducationalResearch,50, 241-271.
Solomon, Y. (1989). Thepractice of mathematics.New York:Routledge.
Steketee, S. (1985). SuperPlot[Computersoftware].Berkeley,CA: EduSoft.
Vygotsky, L. S. (1978). Mind in society: The developmentof higherpsychological processes. Cambridge, MA: HarvardUniversityPress.
Vygotsky, L. S. (1987). Thinkingand speech. In R. Rieber& A. Carton(Eds.), Thecollected worksof
L. S. Vygotsky:Vol. 1. Problemsof generalpsychology (pp. 39-288). New York: Plenum.
Webb, N. (1985). Verbal interactionsand learningin peer-directedgroups. Theory in Practice, 24,
32-39.