Co-constructed narratives in Online,
Collaborative Mathematics Problem-Solving
Johann SARMIENTO
Virtual Math Teams Project, the Math Forum @
3210
Cherry Street, Philadelphia, PA 19104, USA
Stefan TRAUSAN-MATU
Address
City - Country,
Gerry STAHL
Address
City - Country,
Abstract. Our approach to the study of the narrative
role in the learning of mathematical problem-solving departs slightly from the
conception of narrative as the central artefact of interest while favouring the
conception of the dialogical aspects of interaction as shared achievements of
co-participants and central meaning-making procedures. On the other hand, our qualitative analysis
of transcripts of collaborative problem-solving interactions online revealed
striking resemblances with the narrative form.
Based on these observations we attempt to establish a link between the
narrative 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 such learning environments. As part of the Virtual Math Teams (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. In the following sections we present these perspectives and offer some reflectiosn for future research and development.
1. Narrative Learning Environments (NLE)
Theorist
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 humans 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 of
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].
According to the organizers of this workshop, a narrative learning
environment is 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.
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, mostly within the context of
literacy development and language learning (e.g. [9]).
.
2. The dialogical perspective
Related
to the narrative perspective but particularly concerned with the interactive
and participatory aspects of the joint talk and activity, the discursive or dialogical perspective pursues
meaning-making as an interactional achievement of co-participants, more than a
property of narratives or other linguistic objects. Theorists of the dialogical aspect of
language and meaning (e.g. Mikhail Bakhtin [10],[11], Rom Harré [12], and conversation
analyst such as Harvey Sacks [13],[14] and Emanuel Schegloff[15]) point to the
features of talk as action and of shared action as the core processes of human meaning-making. These socially shared procedures might
suggest general sense-making strategies that are instantiated within particular
domains (e.g. fictional storytelling, or. mathematical problem-solving).
Participatory
or interactive narratives offer opportunities for co-construction of meaning
precisely based on the dialogic principle of interactivity, a point we seek to
illustrate within the domain of collaborative mathematical
problem-solving. Some of the narrative
elements visible in this specific domain are attributable to the sequential
unfolding of the problem-solving task, a condition that gets confounded with
dialogical and interactional participation schemes adopted by the
participants. We would like to suggest
that elements intersecting the narrative and dialogical perspectives,
discovered through our analysis in the particular context of collaborative
mathematics problem-solving, could represent an extended standpoint for the analysis
and design of learning environments in general.
3. Virtual Math Teams (VMT)
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 at
As part of the initial
exploratory phase of research, the VMT offered almost 20 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 data
source. 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.
3.1. Collaborative Problem-solving: Co-construction,
exploration and reflection.
Several researchers
have explored the interdependencies between narratives and mathematics (Cocking & Chipman [18]) as well as the role of narrative in
mathematics learning (
The following analysis illustrates these ideas
by using data from one of the online transcripts of a collaborative
problem-solving session. The VMT 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?
“Press
for Time” - Problem of The Week
As it turns out, 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, in
addition to the orientation to the narrative structure mediated by the
narrative presentation of the problem, the participants orient themselves to an
“expository[1]”
mode of interaction in which reports of “ways” to solve the problem are
presented in 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. An interactive narrative
within the speech genre of mathematics
problem solving, however, has specific characteristics that govern the space of
possible transformations of the different “episodes” of a story. Let us illustrate an example in which
participants allude to this resource:
1. 7:26:10 SKI i started
and solved with a system
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 the 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
7. 7:26:41 SKI if we atke
the recip of x
8. 7:26:45 YAG
*choughSHOWOFFchough*
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 does 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=reciprocal
17. 7:27:33 SKI
anyways
18. 7:27:38 YAG
and, recip y+ recip x = 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?
As can be seen in this excerpt even in this “expository” orientation,
co-participants can take active roles in co-constructing the explanation. Even though SKI initiates his story-like
report with the form of a firs person narrative (“i started and solved with
a system of equations“), the shared narrative space of this dialogical form
gets transformed after YAG and GOH’s interactional acceptance of SKI’s narrator
voice (lines 3 and 5). As a result, we
see a transformation in the structure of SKI’s narrative 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. SKI
and YAG have, at this point, constituted themselves as a recognizable collectivity
(Lerner [22]) 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 that 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 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, when SKI in line 27 asks GOH,
indirectly but unequivocally, for an assessment of her state of
participation, GOH eventually requests a
clarification of the current state of the co-constructed narrative which, as
can be seen in the following excerpt, is also co-produced and results in
further re-organization of the meaning of 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.):
1. 7:29:38 GOH how come 1/x and
1/y added equal 1/2?
2. 7:29:42 SKI ok
3. 7:29:47 YAG ummm
4. 7:29:50 YAG pure luck!
5. 7:29:51 SKI 1/x is how
much the old one does in one hour
6. 7:29:57 GOH right.
7. 7:29:58 SKI how much of
the job it does in an hour
8. 7:30:01 YAG (frac of job
done)
9. 7:30:03 SKI 1/y is for
the new machine
10. 7:30:08 GOH right
11. 7:30:11 SKI add those up
12. 7:30:18 YAG and since they
do it together at 3-5
13. 7:30:20 SKI thats how
much of the job they do together in one hour
14. 7:30:22 YAG it took 2 hrs
15. 7:30:25 SKI ya
16. 7:30:29 SKI listen to
[YAG]
17. 7:30:38 YAG so 1/2 =0.5
18. 7:30:42 YAG :P
19. 7:30:44 SKI ya
20. 7:30:47 SKI u getting
that?
21. 7:30:52 YAG slow
22. 7:30:53 GOH I think so....
23. 7:30:54 YAG down
24. 7:30:55 SKI hmm
25. 7:30:57 YAG [S-K-
26. 7:30:58 SKI i will
27. 7:30:59 YAG I]
28. 7:31:06 SKI the whole job
took 2 hours
29. 7:31:14 YAG with both
machines
30. 7:31:19 SKI so in one
hour they did 1/2 of the job
31. 7:31:34 YAG and in the 2nd
hour they did the other half
32. 7:31:54 GOH Okay, I got it.
1/2 is how much of the job they do together in one hour
33. 7:31:58 SKI rite
34. 7:32:00 YAG yepyepyep
35. 7:32:06 SKI u know what x
and y represent rite?
In addition to the co-construction of the narrative explanation, the
dialogical participatory orientation opens the space for exploration of the
possibilities of the local world of mathematical objects and, what is perhaps
even more interesting as far as learning goes, to anticipate the
intelligibility of the co-constructed narrative. In line 35, SKI’s question to GOH seems to
represent, both, an orientation towards a prerequisite for the intelligibility
of the mathematical narrative being constructed, 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 [23] has labelled the “mediated learning
experience” where the mediating agent “selects, changes, amplifies and
interprets both the stimuli that come to the learner and the learner’s
responses” so as to produce an type of experience that leads to conceptual
change. Needless to say this role is
also shared among co-participants.
Although we have refered to this context as collaborative problem
solving, it is clear from the interactions and the references made by the
participants that the work being done is closer to an “explanation” than to
co-construction of knowledge, and 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 being that there is a constant interchange
between first person singular and third person plural narration and a
consequent change in agency and authorship embedded with objects of unique
mathematical characteristics: “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!!!!”).
Interactionally, a “way” might be equivalent to a trajectory of
constituting collective understanding.
There might be methods to start a way, to stop it, to abandon it, to “start
over”, etc., there might be interactional features that make a way "work",
and there might possibly be “degrees” of achieving the constituting of such
collective understanding which open up the space for different things to
happen, such as following a different narrative and problem-solving trajectory.
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 element characteristic of a
narrative learning environment, that of reflection or engagement in “consequent analysis
of what happened within the learning session” 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 or dialogues. Although we have not explicitly used AI
techniques to shape the learning environment where the interactions presented
here take place, we expect that such methods might play a role at this level,
in areas 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.
4. Implications for design, future research.
Participation is what we seek. How do we organize to more symmetric
participatory framework?
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[1] The use of the “expository” participation framework
here differs slightly from Mercer’s [21] conception of the three kinds of
inter-subjective of 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 or an approach to truth. In cumulative
talk, each speaker seeks to support the other's self but fail to explore
facts and solutions. Exploratory talk,
acoording to Mercer occurs when speakers "engage critically but
constructively with each other's ideas" (p.98).