Co-constructed narratives in online collaborative mathematics problem-solving

Co-constructed Narratives in Online, Collaborative Mathematics Problem-Solving

Johann SARMIENTO= , Stefan TRAUSAN-MATU, Gerry STAHL

Virtual Math Teams Project, the Math Forum @ Drexel University,=

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. 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 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 6^th to 11^{th to
collaborate online in discussing and solving non-routine mathematical probl=
ems.
Currently, participants interact using a computer-supported collaborative
learning environment which combines quasi-synchronous text-based communicat=
ion
(e.g. chat) and a shared whiteboard among other interaction tools. At the c=
ore
of VMT research is the premise that primarily, group knowledge arises in
discourse and is preserved in linguistic artifacts whose meaning is co-cons=
tructed
within group processes ([15]). 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 to study the forms of collaboration a=
nd
reasoning that take place.}

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 (Burto= n, [19],[20]). Our qualitative analysis of collaborative mathematical problem-= solving, based on the conversation analysis (e.g. [9],[10]), seeks to understand the methods that co-participants use to organize their shared interactions. The object of inquiry in conversation analysis (CA) is not exclusively conversa= tion as a linguistic entity, but rather talk and social interaction. The interes= t of CA is “with the local production of [social] order and with ‘members’ methods’ for doing so” ([21], p.19). Using the methods of CA, our analysis of transcripts of online collaborative problem-solving revealed, in particular instances, narrative elements—e.g. the emergence of a narrator and a = narratee as well as structured sequences of events, that participants oriented to in their collaborative production of problem solutions.

=

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 <= /p>
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?=

...

57.       7:29:38     GOH    how come 1/x and 1/y = added equal 1/2?

58.       7:29:42     SKI    ok =

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, 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.<= /p>
  = ;        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 Drexel University community. In addition two anonymous reviewers provided insightful 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 recommendations expressed= are those of the authors and do not necessarily reflect the views of the sponso= rs.

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