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Co-constructed Narratives in Online,
Collaborative Mathematics Problem-Solving
Johann SARMIENTO=
, Stefan
TRAUSAN-MATU, Gerry STAHL
Virtual Math Teams Project, the Math Forum @
3=
210
Cherry Street, Philadelphia, PA 19104, USA
<=
span
style=3D'mso-spacerun:yes'> 1-(215) 895 2188, Fax: 1- (215) 895=
2964
jsarmi@drexel.edu, stefan.trausan-matu@cis.drexel.=
edu,
gerry.stahl@cis.drexel.edu
Abstract=
b>. Our approach to the study o=
f learning
of mathematical problem-solving extends the notion of narrative learning
environments to include the dynamics of collaborative dialogs and related e=
mergent
narratives. This perspective favours the conception of the dialogical aspec=
ts
of interaction as shared achievements of co-participants and as central
meaning-making procedures, based on our qualitative analysis of transcripts
from online collaborative math problem-solving interactions. From these
observations we attempt to establish a link between narrative learning
environments and dialogical perspectives and explore relevant implications =
for
the design of the Virtual Math Teams collaborative learning environment.
Introduction
&=
nbsp; Research
in the field of Narrative Learning Environments (NLEs) is concerned with qu=
estions
such as how to characterize the contribution of narratives and narration to=
learning,
and how to use knowledge of narratives to design learning environments. As =
part
of the Virtual Math Teams research project (mathforum.org/wiki/VMT/), we ha=
ve
investigated talk-in interaction within the context of online collaborative
mathematical problem solving and have found similarities between the narrat=
ive
approach and a dialogical perspective on sense-making and interaction.
Therefore, we propose to ex=
tend
the idea of NLEs to encompass collaborative learning environments which, in
addition to using narrative structures, also offer the possibility of joint
participation and interaction with a diverse set of linguistic and
extra-linguistic objects (e.g. mathematical objects and their derivative
properties).
1. Narrative
Learning Environments (NLE)
Research
and development on NLEs explores intelligent learning environments where
“narrative is approached and applied” to support learning and t=
he
construction of meaning [1]. As such, NLEs build and extend the long held
interest in AI for the structuring power that narratives and narration exer=
t on
cognition (e.g. [2], [3]). A narrative learning environment is expected to promote three main kinds of
activities for learners: co-construction (the ability to
participate in the construction of a narrative), exploration (engage=
ment
in active exploration of the learning tasks, following a narrative approach=
and
trying to understand and reason about an environment and its elements), and=
reflection
(consequent analysis of what happened within the learning session). To
date, research on NLEs has concentrated on the analysis and use of narrative
elements such as virtual storytelling, interactive drama, and participatory
narratives, mostly within the context of literacy development and language
learning (e.g. [4]) and the exploration of points of intersection between A=
I,
educational technologies and narratology. Generally, this approach treats
narrative as an object and a fixed structure of interaction.
2. The Dialogical Perspectiv=
e on
Learning
&=
nbsp; The
dialogical perspective pursues
meaning-making as an interactional achievement of co-participants, rather t=
han
as a fixed property of linguistic objects. Theorists of the dialogical aspe=
ct
of language and meaning (e.g. Bakhtin [5,6,7], Harré [8], Sacks [9],
Schegloff [10]) point to the features of talk as action, and of shared acti=
on
in itself, as core processes of human meaning-making. These socially shared
procedures might point to general sense-making strategies with applicabilit=
y to
particular domains (e.g. fictional storytelling, or math problem solving). =
As Wegerif stress=
es [11],
the dialogical perspective on learning attempts to access the creative spac=
e of
“the interanimation of more than one perspective” that emerges =
in
the dynamics of interactive narratives and collaborative meaning-making. Wh=
at
is common to both narration and collaborative dialogues is the discourse; the emergent coherence of the sequencing, projec=
tion
and referencing of utterances generated within meaning making shared with
others and with meaningful artefacts [14]. As such, narration and dialogues as
interactive events open up opportunities for participants to engage in co-construction of possible worlds, to explore them in dialogue, and to reflect together o=
n the
experience. Participation and engagement are then central to the learning
processes conceived as a socio-cultural practice [12], speech and interacti=
on
being extremely important mediators in this process. Furthermore, as Vygots=
ky
states in his concept of the Zone of Proximal Development [13],
children’s potential learning abilities are especially accessible wit=
hin
their interactions with others, a fact that adds practical and theoretical
support to the use of collaborative learning.
Participatory or
interactive narratives offer opportunities for co-construction of meaning
precisely based on the dialogic principle of interactivity resulting on an
intermix of classical narrative structures and other frameworks of shared
participation, a point we seek to illustrate within the domain of collabora=
tive
mathematical problem solving. In summary, we
propose to connect narrative learning environments and collaborative learni=
ng
environments by virtue of their common concern for the role of discourse and
interaction in learning and its potential support via designed artefacts.
3. Collaborative Math Problem-solving: Co-construc=
tion,
exploration and reflection
The
Virtual Math Teams (VMT) research program investigates the innovative use of
online collaborative environments to support effective K-12 mathematics
learning as part of the research and development activities of the Math For=
um (mathforum.org)
at
3.1. Data sources and Methodology
&=
nbsp; As part of the initial exploratory phase of
research, the VMT offered more than 20, one to one and a half hour online
sessions in which small groups of students used AOL Instant Messenger©
technologies to interact and collaboratively attempt to solve a mathematical
problem provided. Through these events we have collected a corpus of chat
transcripts that constitute our main source of data. The VMT implements a
multidisciplinary approach to the analysis of these transcripts, which
integrates quantitative modelling of students’ interactions as well as
ethnographic and conversation analytical studies of collaborative problem
solving. A coding scheme has been developed for the quantitative analysis of
the sequential organization of postings recorded in a chat log. This coding
scheme includes nine content and threading dimensions (e.g. conversation,
problem-solving content and threads) of each chat line (see [16] and [17] for further discussion). The
analysis presented here represents an example of the complementary ethnogra=
phic
analysis of these same data.
Several researche=
rs
have explored the interdependencies between discourse, narratives, and
mathematics in general (Cocking & Ch=
ipman
[18]) as well as the role of narr=
atives
in mathematics learning (
3.2. Emergent Narrative Elements from Shared
Participation.
 =
; The
following analysis illustrates the ideas proposed by using data from one of=
the
online transcripts of a VMT collaborative problem-solving session. The sess=
ion
presented here has three main participants, SKI, YAG and GOH. “Press for Time” is the probl=
em
assigned for the session:
The Rational Rea=
der,
a popular daily newspaper, has to be printed by 5 a.m. so that it can be
distributed. Late one night, a major story broke and the front page had to =
be
rewritten, which delayed the start of the printing process until 3 a.m. To =
try
to get the printing done on time, the Reader used both their new printing p=
ress
and their old one. The new press is three times as fast as the old one, and
with both of them running, the printing was finished exactly on time. How l=
ong
does it take to print a normal edition of the paper using only the new pres=
s?
From the transcript we can infer that, at least two of the participan= ts (SKI and YAG) had worked on the problem prior to their joint participation = in the online collaborative session and, as a result, the group members orient themselves to an “expository” mode of interaction in which repo= rts of “ways” to solve the problem are offered in the form of story= -like narrations. The process of narrating, the constitutin= g of narrator and narratee voices as well as the resulting narrative, however, a= re to be considered as an interactional achievement of all the participants. On the other hand, an interactive narrative within the speech genre of mathema= tics problem solving (in the Bakhtinian sense [7]), has specific characteristics that govern the space of possible transformations of the different “events” of the narra= tive being produced. The following excerpts allow us to illustrate these ideas:<= o:p>
1.
7:26:10 SKI i started and solved with a syste=
m 2.
7:26:12 SKI of equations 3.
7:26:14 YAG let SKI explain... 4.
7:26:24 SKI lets just say x is the time for t=
he old
machine and y is for the new 5.
7:26:29 GOH ok 6. 7:26:35 SKI our first equation is like this <= o:p> 7.
7:26:41 SKI if we atke the recip of x 8.
7:26:45 YAG *choughSHOWOFFc=
hough*
9.
7:26:55 YAG :P 10.
7:26:57 YAG :-D 11.
7:26:59 SKI thats how much of the job the old one d=
oes
in one hour 12.
7:27:02 YAG yep 13.
7:27:12 SKI and the reciprocal of y is how much of =
the
job the new one does in one hour 14.
7:27:16 YAG recip [of] y is the new one 15.
7:27:24 SKI ok 16.
7:27:29 SKI recip=3Dreciprocal 17.
7:27:33 SKI anyways 18.
7:27:38 YAG and, recip y+ recip x =3D 1/2 19.
7:27:43 SKI we add 1/x and 1/y 20.
7:27:48 SKI ya 21.
7:27:50 SKI what YAG said 22.
7:27:53 SKI 1/2 23.
7:27:56 YAG in hours and fraction of work 24.
7:28:04 YAG needed to be done 25.
7:28:05 SKI cuz they together get half the job done=
in
one hour 26.
7:28:09 YAG :P 27.
7:28:13 SKI are u getting our first equation?=
... |
59.
7:29:47 YAG ummm 60.
7:29:50 YAG pure luck! 61.
7:29:51 SKI 1/x is how much the o=
ld one
does in one hour 62.
7:29:57 GOH right. 63.
7:29:58 SKI how much of the job i=
t does
in an hour 64.
7:30:01 YAG (frac of job done) 65.
7:30:03 SKI 1/y is for the new ma=
chine 66.
7:30:08 GOH right 67.
7:30:11 SKI add those up 68.
7:30:18 YAG and since they do it
together at 3-5 69.
7:30:20 SKI thats how much of the=
job
they do together in one hour 70.
7:30:22 YAG it took 2 hrs 71.
7:30:25 SKI ya 72.
7:30:29 SKI listen to [YAG] ... 84.
7:31:06 SKI the whole job took 2 =
hours 85.
7:31:14 YAG with both machines 86.
7:31:19 SKI so in one hour they d=
id 1/2
of the job 87.
7:31:34 YAG and in the 2nd hour t=
hey
did the other half 88.
7:31:54 GOH Okay, I got it. 1/2 i=
s how
much of the job they do together in one hour 89.
7:31:58 SKI rite 90.
7:32:00 YAG yepyepyep 91.
7:32:06 SKI u know what x and y
represent rite? ... |
As can be seen in these excerpts, even in this “expository=
221;
orientation, co-participants take active roles in co-constructing the
explanation. Even though SKI initiates his story-like report with the form =
of a
first person narrative (“i started and solved with a system of
equations“), the shared narrative space of this interaction is
constituted with YAG and GOH’s uptake of SKI’s narrator voice
(lines 3 and 5) and their subsequent participation. SKI’s narration s=
eems
to shift to the first person plural (“our first equation is like
this”) and subsequently we can observe how SKI and YAG share the
narrator role by completing each other postings or interjecting new ones (e=
.g.
lines 23 and 25). SKI and YAG have, at this point, constituted themselves a=
s a
recognizable collectivity (Lerner [22]) oriented towards the task of produc=
ing
an intelligible narrative explanation for GOH (e.g. line 27).
<= o:p>
On the other hand, by virtue of the interactional nature of the
conversation being produced, GOH is by no means restricted to a passive
audience role. One of the interesting peculiarities of our attempt to inter=
sect
the framework of narratology and the domain of collaborative mathematical
problem-solving, results in a unique instantiation of the idea of
“possible worlds.” The
complex world of linguistic and mathematical objects which SKI, YAG and GOH
both access and co-construct (e.g. the proposition “The new press is
three times as fast as the old one” included in the problem statement, and
SKI’s posting “the reciprocal of y is how muc=
h of
the job the new one does in one hour ), their individual perspectives, =
and
the transformations that they exert on such objects (e.g. SKI use of
“cuz” - because - on line 25) are governed not by strict=
logical
laws (as is sometimes assumed in narrative semantics) but by the local
sense-making procedures of the co-participants and their orientation to
joint-activity. For, instance, SKI in line 27 asks GOH for an assessment of=
her
state of participation, and GOH eventually (line 57) requests that the co-c=
onstructed
narrative be reoriented towards a further sense-making on the mathematical =
and
narrative objects so far established (e.g. 1/x, “the old one,R=
21;
“how much of the job they do
together in one hour,” etc.).
In addition to the co-construction of the narrative explanation in
itself, the dialogical orientation opens the space for the exploration of p=
ossibilities
of the local world of mathematical objects and, what is perhaps even more
interesting as far as learning is concerned, to anticipate the intelligibil=
ity
of the co-constructed narrative. In line 91, SKI’s question to GOH se=
ems
to represent, both an orientation towards a prerequisite for the
intelligibility of the mathematical narrative being produced, as well as an
anticipation of a potential problem of understanding. It is in these instan=
ces
of dialogical interaction where we are able to observe the power of what
Feurenstein [23], elaborating on
Vygotsky, has characterized as “mediated learning experiences:”=
interactions through which co-parti=
cipants
place themselves between each other and the world, and co-construct the mea=
ning
of their joint activity (i.e. verbal or otherwise). In mediation, stimuli a=
nd
responses are selected, changed, amplified and interpreted in complex ways =
that
represent a "type of organization (which) is basic to all higher
psychological processes” ([13], p. 40). N=
eedless
to say this role is also shared among co-participants.
Although we have referr=
ed
to this context as collaborative problem solving, it might appear that the =
work
being done is closer to an “explanation” than to co-constructio=
n of
knowledge. Yet, the participants, perhaps influenced by the very nature of
dialogic interactions, make such explanations interactive and participatory=
for
all members of the group. The outcome of this approach is that there is a
constant interchange between first person singular and third person plural
narration, and a consequent change in agency and authorship embedded within=
certain
mathematical objects: “my way”=
(e.g “I started and solved with a system of equations”) <=
span
lang=3DEN-GB style=3D'font-family:"Times New Roman";color:black;letter-spac=
ing:
-.15pt;mso-ansi-language:EN-GB;mso-bidi-font-style:italic'>contrasted to
“your way” (e.g. “YAG its kinda hard to understand ur =
way”),
and sometimes becoming “our way” (e.g. “so 8 hours is =
480
minute[s], divide by 3, to get 160 minutes our answer!!!!”). Of central interest to our analysis are the methods used by
co-participants to orient themselves to certain forms of participation that
guide them in their collaborative sense making. The use of the
“expository” mode of interaction here differs slightly from
Mercer’s [24] conception of =
the
three kinds of inter-subjective talk: disputational, cumulative=
i>,
and exploratory. In Mercer’s framework, disputational t=
alk
is characterized by the speakers being concerned with defending their own
selves, at the possible expense of any attempt at a solution . In cumulative talk, each
speaker seeks to support the other's self but fails to explore facts and
solutions. Exploratory talk, according to Mercer occurs when speakers
"engage critically but constructively with each other's ideas" (p=
.98).
For a more complete analysis of the two main “participation
frameworks” identified in VMT research see [16].
Although one could argue that
the structure of the task itself (a word or “story” problem) mi=
ght
contribute to the emergence of narrative elements in the dialogical interac=
tions
among participants, similar phenomena has been observed on geometry and oth=
er
non-word problems.
4. Implications for design, future research.
 =
; The analysis
presented in the previous section illustrates how certain narrative structu=
res
may emerge from the dialogical interactions and the ways participants orient
themselves to their shared sense-making during mathematical problem-solving=
. Although
we have presented a single in-depth case, we seek to identify a diverse arr=
ay
of patterns of participation, through discourse and conversation analysis in
parallel with statistical natural language processing techniques (e.g. [25]=
, [17]),
with the goal of informing the design of the appropriate learning supports =
for
online, collaborative math problem-solving. Engagement, participation, and
ultimately, learning might be emergent aspects of distributed activity syst=
ems
that offer rich opportunities for the learners to construct meaning through
language and interaction in true dialogical contexts. Further research and
development is necessary to integrate, in the design of future learning
environment, theories of sense-making that account for the narrative and di=
alogical
aspects of individual, small-group and community interactions. Additional text processing is envisioned, such as automated narrati=
ve
summarization and intelligent indexing with the specific intent of facilita=
ting
the re-usability of collaborative problem-solving dialogs for specific lear=
ning
purposes, including the potential support for an online community of math
problem-solvers represented as a “narrative of dialogues”.
Acknowledgements
=
T=
he
authors wish to express their gratitude to all the member of the Virtual Ma=
th
Teams research project who actively participated in the discussion of the i=
deas
presented here including Math Forum staff and other members of the
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