Dialogical Perspectives on Narratives
in Collaborative Mathematics
Problem-Solving
Johann SARMIENTO 1,
Stefan TRAUSAN-MATU 1,2, Gerry STAHL1
1 Virtual Math Teams Project, the
Math Forum @
3210 Cherry Street, Philadelphia, PA 19104, USA
1-(215) 895 2188, Fax: 1- (215) 895 2964
2Research Institute for Artificial Intelligence,
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
the notion of narrative learning environments to include the dynamics of
collaborative dialogs and related emergent narratives. This perspective favours
the conception of the dialogical aspects 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
Truth is not to be found inside the head of an
individual person, it is born between people collectively searching for truth,
in the process of their dialogic interaction. (Bakhtin, [1], p.110)
Introduction
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
collaborative learning environments for mathematics which, in addition to using
narrative structures, 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 research
and development. The next section briefly presents the problematic of narrative
learning environments. Section 2 introduces the main ideas of the dialogical
theory of Mikhail Bakhtin and its relevance for narrative learning
environments. 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 design and future research.
1. Narrative Learning Environments (NLE)
Theorists
of the narrative aspect of cognition (e.g. Jerome Bruner [2, 3], Walter
Fisher[4, 5], Roger Schank[6], etc.) argue that the narrative form is the
primary means through which human beings create and convey meanings about the
world. The interest in narrative that AI and Cognitive Science have shown
revolves around 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 concerned 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 trying to
understand and reason about an environment and its elements;
(3) reflection: engage in consequent
analysis of what happened within the learning session [8].
To
date, research and development in the field of narrative learning environments
has concentrated on the analysis and use of narrative elements such as virtual
storytelling, interactive drama, and participatory narratives, primarily within
the context of literacy development and language learning (e.g. [9]). We
propose to extend the idea of the usage of narratives in two dimensions: the
use of co-constructed narratives in Computer-Supported Collaborative Learning
(CSCL) and its use in mathematics learning.
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 other
linguistic objects. Theorists of the dialogical aspect of language and meaning
(e.g. Bakhtin [10, 11], Harré [12], Sacks [13, 14], and Schegloff [15]) point
to the features of talk as action, and of shared action in itself, as the core
processes of human meaning-making. These socially shared procedures might point
to general sense-making strategies with applicability within particular domains
(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 dynamics of
interactive narratives and collaborative meaning-making. Bakhtin in particular,
considers any human language related activity, be it in the form of oral speech
or writing, as dialogic— i.e. containing more than one voice ([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 (especially in learning
contexts). These ideas are very important because they move the emphasis 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 their
communicative structures. What is common to both narratives and theorem
proving, or collaborative problem-solving is the discourse; the emergent
sense-making of the sequencing of utterances generated within joint interactions
with others and with meaningful artefacts. Furthermore, when we refer to these
activities in the context of learning, it is interesting to note that “rather 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].
Participation
in the learning processes is usually a social activity, language being an
extremely important cultural artefact. As Vygotsky states in his concept of the
Zone of Proximal Development [18], children’s potential learning abilities are
especially accessible within their interactions with others. Participatory or
interactive narratives offer opportunities for co-construction of meaning
precisely based on the dialogic principle (through which Bakhtin extends
Vygotsky’s theory) of interactivity resulting in an intermix of classical
narrative structures and other frameworks of shared participation, a point we
seek to illustrate within the domain of collaborative mathematical problem
solving. In summary, we propose to connect narrative learning environments and
collaborative learning 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-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 Forum
(mathforum.org) at
3.1. Data sources and Methodology
As
part of the initial exploratory phase of research, the VMT offered more than
20, 1-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 [20] and [21] for
further discussion). The analysis presented here represents an example of the
complementary ethnographic analysis of these same data.
Several
researchers have explored the interdependencies between narratives and
mathematics (Cocking & Chipman [22]) as well as the role of narrative in
mathematics learning (
3.2. Emergent Narrative Elements from Shared Participation.
The following analysis illustrates the
above ideas by using data from one of the online transcripts of a VMT
collaborative problem-solving session. The session presented here has three
main participants, SKI, YAG and GOH. “Press for Time” is the problem
assigned for the session, which by virtue 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 process
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 finished
exactly on time. How long does it take to print a normal edition of the paper
using only the new press?
From
the transcript we can infer that at least two of the participants (SKI and YAG)
had worked on the problem prior to their joint participation in the online
collaborative session, and as a result, orient themselves to an “expository”
mode of interaction in which reports of “ways” to solve the problem are offered
in the form of story-like narrations. The form in which a way of solving a
problem is then made accessible during this collaborative problem solving
interaction is, to a certain extent, similar to that of the narration of a
story. The process of narrating and the resulting narrative, however, are to be
considered as an interactional achievement of all the participants despite the
apparent fact of an established narrator voice or the references made by
participants to the authorship of particular ways of proceeding with their
joint work. On the other hand, an interactive narrative within the speech genre
of mathematics problem solving (in the Bakhtinian sense [11]), has specific
characteristics that govern the space of possible transformations of the
different “events” of the narrative being produced. In fact, exercising
narration is obviously a way of supporting learning as mastering a given speech
genre. In this process, dialogues are essential (at least because mastering a
speech genre implies being able to dialogize in it). The following excerpts
allow us to illustrate these ideas:
|
1. SKI i started
and solved with a system 2. SKI of
equations 3. YAG let SKI
explain... 4. SKI lets just
say x is the time for the old machine and y is for the new 5. GOH ok 6. SKI our first
equation is like this 7. SKI if we atke the recip of x 8. YAG *choughSHOWOFFchough*
9. YAG :P 10. YAG :-D 11. SKI thats how much of the job the old one does in one
hour 12. YAG yep 13. SKI and the reciprocal of y is how much of the job the
new one does in one hour 14. YAG recip [of]
y is the new one 15. SKI ok 16. SKI recip=reciprocal
17. SKI anyways 18. YAG and, recip
y+ recip x = 1/2 19. SKI we add 1/x
and 1/y 20. SKI ya 21. SKI what YAG
said 22. SKI 1/2 23. YAG in hours
and fraction of work 24. YAG needed to
be done 25. SKI cuz they together get half the job done in one hour 26. YAG :P 27. SKI are u
getting our first equation? ... |
57. GOH how come
1/x and 1/y added equal 1/2? 58. SKI ok 59. YAG ummm 60. YAG pure
luck! 61. SKI 1/x is
how much the old one does in one hour 62. GOH right. 63. SKI how much
of the job it does in an hour 64. YAG (frac of
job done) 65. SKI 1/y is
for the new machine 66. GOH right 67. SKI add those
up 68. YAG and since
they do it together at 3-5 69. SKI thats how
much of the job they do together in one hour 70. YAG it took 2
hrs 71. SKI ya 72. SKI listen to
[YAG] ... 84. SKI the whole
job took 2 hours 85. YAG with both
machines 86. SKI so in one
hour they did 1/2 of the job 87. YAG and in
the 2nd hour they did the other half 88. GOH Okay, I
got it. 1/2 is how much of the job they do together in one hour 89. SKI rite 90. YAG yepyepyep
91. SKI u know
what x and y represent rite? ... |
As can be seen in these excerpts, even in this
“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 SKI’s narrator voice (lines 3 and 5) and their
subsequent participation. SKI’s narration seems 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 as a recognizable collectivity (Lerner [27]) oriented
towards the task of producing 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 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 much 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-constructed narrative be reoriented 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 the
co-construction of the narrative explanation in itself, the dialogical
orientation opens the space for the exploration of possibilities of the local
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 voice is combining
with the listener’s voice, with, for example, her possible questions, in what
he utters). In line 91, SKI’s question to GOH seems 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 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 themselves between each other and the world, and
co-construct the meaning of their joint 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 among 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 “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 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 [26] conception of the three kinds of inter-subjective talk:
disputational, cumulative, and exploratory. In Mercer’s framework,
disputational talk 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 [20]. Although one
could argue that the structure of the task itself (a word or “story” problem)
might contribute to the emergence of narrative elements in the dialogical
interactions among participants, similar phenomena have been observed with
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
interactions illustrated through the transcript presented. The third
characteristic element of a narrative learning environment, that of reflection
or engagement 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 from
the dialogical interactions and the ways participants orient themselves to
their shared sense-making during mathematical problem-solving. Moreover, from
Bakhtin’s dialogical 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).
Although
we have presented a single in-depth case, we seek to identify a diverse array
of patterns of participation in narratives, through discourse and conversation
analysis in parallel with statistical natural language processing techniques
(e.g. [21, 29]), with the goal of informing the design of the appropriate
learning supports for online, collaborative math problem-solving, that
integrates the ideas of NLEs and CSCL. Engagement, participation, and
ultimately, learning might be emergent aspects of distributed activity systems
that offer rich opportunities for the learners to construct meaning through
language and interaction in true dialogical contexts [30]. Further research and
development is necessary to integrate, in the design of future learning
environments, theories of sense-making that account for the narrative and
dialogical aspects of individual, small-group and community interactions.
Additional text processing is envisioned, such as automated narrative
summarization and intelligent indexing with the specific intent of facilitating
the re-usability of collaborative problem-solving dialogs for specific learning
purposes, including the potential support for an online community of math
problem-solvers characterized as a “narrative of dialogues”.
Acknowledgements
The authors wish to express their
gratitude to all the member of the Virtual Math Teams research project who
actively participated in the discussion of the ideas presented here including
Math Forum staff and other members of the
References
[1] Bakhtin, M.M. (1984) Problems of Dostoevsky’s Poetics.
Edited and trans. by Caryl Emerson.
[2] Bruner, Jerome. 1986. Actual Minds, Possible Worlds.
[3] Bruner, J. S. (1991). The narrative construction of reality. Critical
Inquiry, 18, 1-21.
[4] Fisher, W. R. "The Narrative Paradigm: An
Elaboration." Communication Monographs, 52, December (1985):
347-367.
[5] Fisher, W. R. "Narrative Rationality and the Logic of
Scientific Discourse." Argumentation 8 (1994): 21-32
[6] Schank, R. C. (1995). Tell me a story: Narrative and
intelligence.
[7] Herman, D., Editor. (2003) Narrative Theory and the
Cognitive Sciences.
[8] Narrative Learning Environments. Workshop at The 12th
International Conference on Artificial Intelligence in Education (AIED 2005). Retrieved
online on April 26, 2005 from: http://gaips.inesc-id.pt/aied05-nle/
[9] Machado,
I., Brna, P., Paiva, A. (2004). 1, 2, 3 .... Action! Directing Real Actors and Virtual Characters.
Lecture Notes in Computer Science - Proceedings of Technologies for
Interactive Digital Storytelling and Entertainment: Second International
Conference, TIDSE 2004,
[10] Bakhtin, M.M.(1981).The
dialogic imagination: Four essays by M. M. Bakhtin (M .Holquist, Ed.; C.
Emerson & M. Holquist, Trans.).
[11] Bakhtin, M.M.(1986).Speech genres & other late essays
(Caryl Emerson and Michael Holquist Eds.; Vern W. McGee, Trans.) (pp.60–102).
[12] Harre, R.,
& Gillett, G. (1994). The discursive mind. Thousand Oaks, CA: Sage
[13] Sacks, H. (1992a, b). Lectures on
conversation, vols. I and II. Edited by Gail Jefferson with an introduction
by Emanuel A. Schegloff.
[14] Sacks, H., Schegloff, E. & Jefferson,
G. (1974). A simplest systematics for the organization of turn-taking in
conversation. Language, 50, 697-735.
[15] Schegloff, E. A. (1997). "Narrative Analysis" Thirty
Years Later. Journal of Narrative and Life History, 7(1-4), 97-106.
[16] Wegerif, R. (2005). A dialogical understanding of the
relationship between CSCL and teaching thinking skills. In T. Koschman, D.
Suthers, & T.W. Chan (Eds.). Computer Supported Collaborative Learning
2005: The Next 10 Years!
[17] Sfard, A. (1998). On two metaphors for learning and the dangers
of choosing just one. Educational Researcher, 27, 4-13.
[18] Vygotsky, L.S. (1978). Mind in society: The development of
higher psychological processes.
[19] Stahl, G. (2004). Building collaborative knowing: Elements of a
social theory of CSCL. In J.-W. Strijbos, P. Kirschner & R. Martens (Eds.),
What we know about CSCL ... and implementing it in higher education.
(pp. 53-86).
[20] Zemel, A., Xhafa, F., Stahl, G. (2005). Analyzing the
Organization of Collaborative Math Problem-solving in Online Chats using
Statistics and Conversation Analysis. Paper submitted to the 11th
International Workshop on Groupware. September 25-29, 2005. Porto de Galinhas (
[21] Cakir,
M., Xhafa, F. Zhou, N., Stahl, G. (2005) Thread-based analysis of patterns of
collaborative interaction in chat. Paper presented at the 12th International
Conference on Artificial Intelligence in Education (AIED 2005),
[22] Cocking,
R. R., and Chipman, S. (1998). Conceptual Issues Related to Mathematics
Achievement of Language Minority Children. In Cocking, R. R., and Mestre, J.P. Linguistic
and Cultural Influences on Learning Mathematics. Hillsdale: Erlbaum. 1988,
pp. 17-46.
[23]
[24]
[25] Ten
Have, P. (1999). Doing Conversation Analysis: A Practical Guide.
[26] Mercer,
Neil (2000) Words and Minds: How We Use Language to Think Together.
Routledge.
[27] Lerner,
G. H. (1993). Collectivities in action: Establishing the relevance of conjoined
participation in conversation. Text, 13(2), 213-245.
[28] Feuerstein
, R., Hoffman, M., & Miller, R. (1980). Instrumental Enrichment.
[29] Juravsky,
D., Martin, J., Speech and Language Processing: An Introduction to Natural Language
Processing, Computational Linguistics and Speech Recognition, Prentice
Hall: San Francisco 2000
[30] Stahl,
G. (2002). The complexity of a collaborative interaction. Paper presented at
the International Conference of the Learning Sciences (ICLS '02),