Dialogical Perspectives on Narratives
in Collaborative Mathe=
matics
Problem-Solving
Johann SARMIENTO 1,
Stefan TRAUSAN-MATU 1,2, Gerry STAHL1
3210 Cherry Street, Philadelphia, PA 19104, USA
1-(215) 895 2188, Fax: 1- (215) 895 2964
13, Calea 13 Septembrie
and
Politehnica University of Bucharest,
313, Splaiul Independentei,
Bucharest, ROMANIA
jsarmi@drexel.edu, trausan@racai.ro, gerry.stahl@cis=
.drexel.edu
Abstract. Our
approach to the study of learning of mathematical problem-solving extends t=
he
notion of narrative learning environments to include the dynamics of
collaborative dialogs and related emergent narratives. This perspective fav=
ours
the conception of the dialogical aspects of interaction as shared achieveme=
nts
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 establi=
sh a
link between narrative learning environments and dialogical perspectives and
explore relevant implications for the design of the Virtual Math Teams
collaborative learning environment
<=
span
style=3D'font-size:9.0pt;font-family:"Times New Roman";mso-ansi-language:EN=
-US;
mso-fareast-language:EN-US'>Truth is not to be found inside the head of an
individual person, it is born between people collectively searching for tru=
th,
in the process of their dialogic interaction. (Bakhtin, [1], p.110)
Introduction
&=
nbsp; Research
in the field of Narrative Learning Environments (NLEs) is concerned with
questions 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 (VMT, see
http://mathforum.org/wiki/VMT) research project, we have investigated talk-=
in
interaction within the context of collaborative mathematical problem-solving
online and have found similarities and differences between the narrative
approach and a dialogical perspective on sense-making and interaction.
Therefore, we propose to extend the concept of NLEs to encompass collaborat=
ive learning
environments for mathematics which, in addition to using narrative structur=
es,
offer also 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).
In the following
sections we present these perspectives and offer some ideas for future rese=
arch
and development. The next section briefly presents the problematic of narra=
tive
learning environments. Section 2 introduces the main ideas of the dialogical
theory of Mikhail Bakhtin and its relevance for narrative learning environm=
ents.
Section 3 presents in detail a qualitative analysis of a chat transcript as
part of the VMT project. Section 4 concludes with some implications for des=
ign
and future research.
1. Narrative Learning Environments (NLE)
=
; Theori=
sts
of the narrative aspect of cognition (e.g. Jerome Bruner [2, 3], Walter Fis=
her[4,
5], Roger Schank[6], etc.) argue that the narrative form is the primary mea=
ns
through which human beings create and convey meanings about the world. The
interest in narrative that AI and Cognitive Science have shown revolves aro=
und
the ability of narratives to structure and mediate knowledge [7]. As such,
major areas of AI work include story understanding and generation as well as
the development of interactive environments structured as narrative spaces.=
Research
and development on Narrative Learning Environments (NLEs), a field of work =
at
the intersection of AI, educational technologies and narratology, is concer=
ned
with intelligent learning environments where “narrative is approached=
and
applied” to support learning and the construction of meaning [8]. NLEs
are expected to promote three main kinds of activities for learners:
(1) co-construction: [the ability to]
participate in the construction of a narrative;
(2) exploration: engage in active
exploration of the learning tasks, following a narrative approach and tryin=
g to
understand and reason about an environment and its elements;
(3) reflection: engage in consequent
analysis of what happened within the learning session [8].
2. The dialogical perspective on Learning
=
; The
dialogical perspective sees meaning-making as an interactional
achievement of co-participants, rather than a property of narratives or oth=
er
linguistic objects. Theorists of the dialogical aspect of language and mean=
ing
(e.g. Bakhtin [10, 11], Harré [12], Sacks [13, 14], and Schegloff [1=
5]) point
to the features of talk as action, and of shared action in itself, as the c=
ore
processes of human meaning-making. These socially shared procedures might p=
oint
to general sense-making strategies with applicability within particular dom=
ains
(e.g. fictional storytelling, or mathematical problem-solving).
As Wegerif stresses [16], the dialogical
perspective on learning attempts to access the creative space of “the
interanimation of more than one perspective” that emerges in the dyna=
mics
of interactive narratives and collaborative meaning-making. Bakhtin in
particular, considers any human language related activity, be it in the for=
m of
oral speech or writing, as dialogic— i.e. containing more than one vo=
ice
([10, 11]). This is of no surprise if we realize that narratives, as
interaction, contain not only the voice of the narrator but also, at least,=
the
voice of the listener. When telling a story, the narrator anticipates the
listener, for instance possible aspects that might require elaboration (esp=
ecially
in learning contexts). This ideas are very important because they move the
emphasys of learning and other sense making activities from an individual
knowledge acquisition perspective (as in cognitive science) to a dialogic,
collaborative, social activity of knowledge building.
From this perspective, narratives resemble,=
as
well, processes of collaborative scientific discourse. The procedures used =
in
structuring a narrative and, for example, writing a proof of a theorem, or
presenting a solution for a problem exhibit significant similarities in the=
ir
communicative structures. What is common to both narratives and theorem pro=
ving,
or collaborative problem-solving is the discourse; the emergent sens=
e-making
of the sequencing of utterances generated within joint interactions with ot=
hers
and with meaningful artefacts. Furthermore, when we refer to these activiti=
es
in the context of learning, it is interesting to note that “r=
ather
than speaking only about acquisition of knowledge”, we =
also
view learning as “becoming a participant in a certain discourse”
[17], or of mastering a certain (e.g. mathematics) speech genre [11].
3. Collaborative Math Problem-solving: Co-construction, 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 Drexel University. VMT extends the Math Forum’s
“Problem of the Week (PoW)” service by bringing together groups=
of
3 to 5 students in grades 6th to 11th to collaborate online in discussing a=
nd
solving non-routine mathematical problems. Currently, participants interact
using a computer-supported collaborative learning environment which combines
quasi-synchronous text-based communication (e.g. chat) and a shared whitebo=
ard
among other interaction tools. At the core of VMT research is the premise t=
hat
primarily, group knowledge arises in discourse and is preserved in linguist=
ic
artifacts whose meaning is co-constructed within group processes ([19]). Key
issues addressed by the VMT include the design challenge of structuring the
online collaborative experience in a meaningful and engaging way, and the
methodological challenge of finding appropriate methodological approaches t=
o study
the forms of collaboration and reasoning that take place.
3.1. Data sources and Methodology
=
; As
part of the initial exploratory phase of research, the VMT offered more tha=
n 20,
1-hour online sessions in which small groups of students used AOL Instant M=
essenger©
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 [20] and [21] f=
or
further discussion). The analysis presented here represents an example of t=
he
complementary ethnographic analysis of these same data.
3.2. Emergent Narrative Elements from Shared Participation.
&=
nbsp; The
following analysis illustrates the above ideas 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. “Pre=
ss
for Time” is the problem assigned for the session, which by virtu=
e of
its presentation as a word problem, could contribute to the display of
narrative elements in the dialogical interactions among participants:
The Rational Reader, 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 pro=
cess
until 3 a.m. To try to get the printing done on time, the Reader used both
their new printing press 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 finish=
ed
exactly on time. How long does it take to print a normal edition of the pap=
er
using only the new press?
1. SKI i
started and solved with a system 57. =
GOH =
how
come 1/x and 1/y added equal 1/2? 59. =
YAG =
ummm
72. =
SKI =
listen
to [YAG] ... 84. =
SKI =
the
whole job took 2 hours ... As can be seen in these excerpts, even in t=
his
“expository” 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 SKI̵=
7;s
narrator voice (lines 3 and 5) and their subsequent participation. SKI̵=
7;s
narration seems to shift to the first person plural (“our first equat=
ion
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 [27]) oriented towards the task of produc=
ing
an intelligible narrative explanation for GOH (e.g. line 27). 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 intersect the framework of narratology and the domain of
collaborative mathematical problem-solving, results in a unique instantiati=
on
of the idea of “possible worlds.” The complex world of linguist=
ic
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 much of the job the new one does in one
hour), their individual perspectives, and the transformations that they exe=
rt
on such objects (e.g. SKI use of “cuz” - because - on li=
ne
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-constructed narrative be reorient=
ed
towards a further sense-making on the mathematical and narrative objects so=
far
established (e.g. 1/x, “the old one,” “how much=
of
the job they do together in one hour,” etc.). In addition to t=
he
co-construction of the narrative explanation in itself, the dialogical
orientation opens the space for the exploration of possibilities of the loc=
al
world of mathematical objects and, what is perhaps even more interesting as=
far
as learning is concerned, to anticipate the intelligibility of the
co-constructed narrative (in Bakhtin’s ideas, the narrator’s vo=
ice
is combining with the listener’s voice, with, for example, hers possi=
ble
questions, in what he utters). In line 91, SKI’s question to GOH seem=
s to
represent, both an orientation towards a prerequisite for the intelligibili=
ty
of the mathematical narrative being produced, as well as an anticipation of=
a
potential problem of understanding. It is in these instances of dialogical
interaction where we are able to observe the power of what Feurenstein [28],
elaborating on Vygotsky, has characterized as “mediated learning
experiences:” interactions through which co-participants place themse=
lves
between each other and the world, and co-construct the meaning of their joi=
nt
activity (i.e. verbal or otherwise). In mediation, stimuli and 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). Needless to say this role is also shared am=
ong
co-participants. Although we have
referred to this context as collaborative problem solving, it might appear =
that
the work being done is closer to an “explanation” than to
co-construction 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 =
220;I
started and solved with a system of equations”) 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 inter=
est
to our analysis are the methods used by co-participants to orient themselve=
s 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 [26] conception of the three kinds of
inter-subjective talk: disputational, cumulative, and exploratory. In
Mercer’s framework, disputational talk is characterized by the speake=
rs
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 [20]. Although one could argue that the structure of the t=
ask
itself (a word or “story” problem) might contribute to the
emergence of narrative elements in the dialogical interactions among partic=
ipants,
similar phenomena has been observed on geometry and other non-word problems=
. We have seen that
two of the central elements proposed for narrative learning environments:
co-construction and exploration are clearly visible in the dialogical inter=
actions
illustrated through the transcript presented. The third characteristic elem=
ent
of a narrative learning environment, that of reflection or engagemen=
t in
“consequent analysis of what happened within the learning
session” [8] seems to present itself differently in the
un-moderated experiences captured in our data, a fact that would suggest a
potential area where explicit support from a pedagogical environment might =
be
specially fruitful. Having access to, at least, a partial record of the
interaction in the same way that we as researchers have had through the
analysis presented here might be a unique advantage of an electronic
environment. In addition, we are interested in fostering reflection,
particularly, at the community level, i.e. at the level where the activity =
of
small-groups gets reified into one diverse and collective narrative, a
narrative of dialogues. 4. Implications for design, future research. =
The analysis presented in the
previous section illustrates how certain narrative structures may emerge fr=
om
the dialogical interactions and the ways participants orient themselves to
their shared sense-making during mathematical problem-solving. Moreover, fr=
om a
Bakhtin’s, dialogistic perspective, narratives are always multi-voiced
(when we build a narrative, the voice of the potential listener will be
virtually present, at least, for example, by our concern for plausibility
and/or usefulness of the narration). &=
nbsp; The
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 Drexel
University community. In addition two anonymous reviewers provided insightf=
ul
comments regarding the relevance and utility of the ideas presented. The
research described here has been supported by NSF grants REC 0325447 and DUE
0333493 and the Fulbright Program. Any opinions, findings, or recommendatio=
ns
expressed are those of the authors and do not necessarily reflect the views=
of
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