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

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 coparticipants 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 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 research project (mathforum.org/wiki/VMT/), we have investigated talk-in interaction within the context of online collaborative mathematical problem solving and have found similarities between the narrative approach and a dialogical perspective on sense-making and interaction. Therefore, we propose to extend 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 the construction of meaning [1]. As such, NLEs build and extend the long held interest in AI for the structuring power that narratives and narration exert 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 (engagement 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 AI, educational technologies and narratology. Generally, this approach treats narrative as an object and a fixed structure of interaction. 2. The Dialogical Perspective on Learning The dialogical perspective pursues meaning-making as an interactional achievement of co-participants, rather than as a fixed property of linguistic objects. Theorists of the dialogical aspect 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 action in itself, as core processes of human meaning-making. These socially shared procedures might point to general sense-making strategies with applicability to particular domains (e.g. fictional storytelling, or math problem solving). As Wegerif stresses [11], 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. What is common to both narration and collaborative dialogues is the discourse; the emergent coherence of the sequencing, projection 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 on the experience. Participation and engagement are then central to the learning processes conceived as a socio-cultural practice [12], speech and interaction being extremely important mediators in this process. Furthermore, as Vygotsky states in his concept of the Zone of Proximal Development [13], children’s potential learning abilities are especially accessible within 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 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 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 to 11 to collaborate online in discussing and solving non-routine mathematical problems. Currently, participants interact using a computer-supported collaborative learning environment which combines quasisynchronous text-based communication (e.g. chat) and a shared whiteboard among other interaction tools. At the core of VMT research is the premise that primarily, group knowledge arises in discourse and is preserved in linguistic artifacts whose meaning is co-constructed 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 and reasoning that take place. 3.1. Data sources and Methodology 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 ethnographic analysis of these same data. Several researchers have explored the interdependencies between discourse, narratives, and mathematics in general (Cocking & Chipman [18]) as well as the role of narratives in mathematics learning (Burton, [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 conversation as a linguistic entity, but rather talk and social interaction. The interest 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 session presented here has three main participants, SKI, YAG and GOH. “Press for Time” is the problem assigned for the session: 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, the group members 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 process of narrating, the constituting of narrator and narratee voices as well as the resulting narrative, however, are to be considered as an interactional achievement of all the participants. On the other hand, an interactive narrative within the speech genre of mathematics problem solving (in the Bakhtinian sense [7]), has specific characteristics that govern the space of possible transformations of the different “events” of the narrative being produced. The following excerpts allow us to illustrate these ideas: 1. 7:26:10 SKI i started and solved 57. 7:29:38 GOH how come 1/x and 1/y