Shared Deictic Referencing in Online Mathematics Discourse

Recent research on the impact of new communication and information technologies stress the transformation that new forms of mediated interactions are having on the ways that teenagers, especially, participate and make sense of different aspects of their lives, including education. The Virtual Math Teams (VMT) project is an NSF-funded research program that investigates the innovative use of online collaborative environments to support effective K-12 mathematics learning. The ethnographic case study presented here explores the sustained interactions of five virtual teams of teenagers distributed across the U.S. as they engaged in mathematical problem solving throughout a series of four successive sessions online. In particular, our ethnomethodological analysis highlights the “member methods” displayed and developed by these teams in their collective sense-making. More specifically, we concentrate our analysis on the deictic referencing methods used by the participants as they collaboratively construct and evolve a space of mathematical objects, reason about them, and constitute their own sense of collectivity. We explore implications for understanding learning and interaction of virtual teams and online communities, as well as for designing effective activities and supports for them. Epistemology of Referencing Referencing is a primary means for humans to establish joint attention and to make shared meaning. Vygotsky, in a particularly rich passage, described the interactional origin of pointing as an example of how gestures become meaningful artifacts for individual minds through social interaction: A good example of this process may be found in the development of pointing. Initially [e.g., for an infant], this gesture is nothing more than an unsuccessful attempt to grasp something, a movement aimed at a certain object which designates forthcoming activity.... When the mother comes to the child’s aid and realizes this movement indicates something, the situation changes fundamentally. Pointing becomes a gesture for others. The child’s unsuccessful attempt engenders a reaction not from the object he seeks but from another person. Consequently, the primary meaning of that unsuccessful grasping movement is established by others.... The grasping movement changes to the act of pointing. As a result of this change, the movement itself is then physically simplified, and what results is the form of pointing that we may call a true gesture. (Vygotsky, 1930/1978, p. 56, italics added) The pointing gesture is perhaps the most basic form of deictic referencing. In its origin where the infant begins to be socialized into a shared world, the meaning of the gesture emerges interactionally as the participants orient to the same object and recognize that they are doing so jointly. In grasping, the infant’s being-in-the-world is intentionally directed at the object; it is a being-at-the-object (Husserl, 1929/1960). When the mother joins the infant by transforming this grasp into a joint engagement with the object, the intentionality of the infant’s grasp becomes intersubjective intentionality, constituting the infant and child as being-there-together-at-the-object (Heidegger, 1927/1996, §26). This fundamental act of collaborative existence simultaneously comes to be symbolized for them by the pointing gesture, which is practiced, repeated and abstracted by them together over time. The mother and infant become a small group, caring for shared objects by being-in-the-world-together and understanding as collaborative practice the symbolic meaning of the physical gesture as a referencing artifact. As researchers, we can see new referencing gestures being created within interactions among collaborating people, particularly when their interaction is taking place via a new medium that they must learn how to use. In the analysis below, a chat posting—“What is the area of this shape?”—constitutes the participants in the chat as a group by designating them as the intended collective recipient and as the expected respondent to the question (Lerner, 1993). The group is the intended agent who will work out the mathematics of the proposal to compute the area. Simultaneously, by referencing a mathematical object (“this shape”), the posting constitutes the group as a beingthere-together-at-the-object. We shall see that both these aspects of being a group may necessitate considerable interactional work by the participants. Before the elicited answer about area can be given in response to the question, the group has to negotiate what it as a group takes the object to be. Also, it may require a number of actions for group participants to co-construct the shared object and their being-there-together-at-the-object. In attempting to do this, they constitute themselves as a group and they may also establish referential gestures or terms that take on a shared meaning of intending the new math object. The interactional work of the group involves making use of the resources of the environment that mediates their interaction. This is particularly noticeable in online interaction. Vygotsky’s infant and mother could use fingers, gaze, touch, voice. Online participants are restricted to exchanging textual postings and to using features of the mediating software (Garcia & Jacobs, 1999; Stahl, 2005b). As designers of online education, we are interested in understanding how students collaboratively create new communicative gestures or interactional methods, including ways of referencing objects for joint consideration. More generally, an interactional understanding of referencing and meaning making leads to a theory of group cognition—rather than individual cognition based on mental representations—as a basis for studying collaborative learning (Stahl, 2006). Technology for Referencing in a Chat Environment In our design-based research at the Virtual Math Teams project (Stahl, 2005a), we started by conducting chats in a variety of commercially available environments: AOL Instant Messenger, Babylon, Blackboard, WebCT. Based on these early investigations, we concluded that we needed to add a shared whiteboard for drawing geometric figures and for persistently displaying notes. We also found a need to minimize “chat confusion” by supporting explicit referencing of response threads (Cakir et al., 2005; Fuks, Pimentel, & de Lucena, 2006). We decided to try ConcertChat, a research chat environment with special referencing tools (Mühlpfordt & Wessner, 2005). By collaborating with the software developers, our educational researchers have been able to successively try out versions of the environment with groups of students and to gradually modify the environment in response to what we find by analyzing the chat logs. ConcertChat provides a variety of referencing supports for math chats: • A shared whiteboard allows chat participants to create drawings. As new objects appear in the drawing, an implicit form of referencing occurs. Participants typically refer with a deictic term in their textual chat to a new addition to the drawing, whose recent appearance for the group makes it salient. • When someone types a new chat message, they can select and point to a rectangular area in the whiteboard. When that message appears in the chat as the last posting or as a selected posting, a bold line appears connecting the text to the area of the drawing (see figure 1). • Similarly, a chat message can point to one or more earlier textual postings. ConcertChat includes a threaded view of the chat postings that, based on the explicit references between postings, displays them like a typical threaded discussion with responses indented under the posting that they reference. • Of course, one can also make all the usual verbal references: using deictic terms (that, it, his, then); quoting part of an earlier posting; or citing the author of a previous posting. In May 2005, we conducted a series of chats using ConcertChat. We formed five virtual math teams, each containing about four middle-school students selected by volunteer teachers at different schools across the USA. The teams engaged in online math discussions for four hour-long sessions over a two-week period. They were given a brief description of a non-traditional geometry environment: a grid-world where one could only move along the lines of a grid (Krause, 1986). The students were encouraged to come up with their own questions about the gridworld, such as questions about shortest paths between points A and B in this world. The chats were facilitated by a member of our research project team. The facilitator welcomed students to the chat, pointed them toward the task, briefly demonstrated the graphical referencing tool and then kept generally quiet until it was time to end the session. We are now analyzing the resultant chat logs in order to draw design implications for a full-scale online math discussion service. Figure 1. Screen view of ConcertChat with referencing. The image has been modified to show graphical references from chat lines 1, 5, 10 and 12 to the whiteboard. Only the reference from a single selected chat line would actually appear at any given time. An Analysis of a Case of Referencing The chat log excerpt visible in figure 1 is reproduced in figure 2 (with line numbers added for referencing in this paper). In this interactional sequence, two students discuss parts of a drawing that has already been constructed in the shared whiteboard. The students had created the drawing as part of discussions about shortest paths between points A and B in a grid-world. In particular, a red triangle, ABD, was drawn with sides of length 4, 6 and 2√13. A thick black staircase line was drawn as a path on the grid from A to B. In this excerpt, the students propose a math problem involving this drawing. The message in line 1 of the chat (see figure 2) proposes a mathematical question for the group to consider: “What is the area of this