How a Virtual Math Team Structured its Problem Solving

To develop a theory of small-group interaction in CSCL settings, we need an approach to analyzing the structure of computer-mediated discourse. Conversation Analysis examines informal face-to-face talk in terms of a fine structure of adjacency pairs, but needs to be adapted to online textual interaction and extended to analyze longer sequences built on adjacency pairs. This paper presents a case study of students solving a math problem in an online chat environment. It shows that their problem-solving discourse consists of a sequence of exchanges, each built on a base adjacency pair and each contributing a move in their collaborative problem-solving process. Structuring Group Cognition at Multiple Levels A year ago in my opening keynote talk (Stahl, 2009a) at the International Conference of Computers in Education (ICCE 2009) in Hong Kong, I claimed that the discourse of group cognition (Stahl, 2006) has a hierarchical structure, typically including the following levels, as illustrated with a particular case study from the Virtual Math Teams (VMT) Project (Stahl, 2009c): a. Group event: E.g., Team B’s participation in the VMT Spring Fest 2006. b. Temporal session: Session 4 of Team B on the afternoon of May 18, 2006. c. Conversational topic: Determining the number of sticks in a diamond pattern (lines 1734 to 1833 of the chat log of Session 4). d. Discourse move: A stage in the sequence of moves to accomplish discussing the conversational topic (e.g., lines 1767-1770—see Logs 1-10 below). e. Adjacency pair: The base interaction involving two or three utterances, which drives a discourse move (lines 1767 and 1769). f. Textual utterance: A text chat posting by an individual participant, which may contribute to an adjacency pair (line 1767). g. Indexical reference: An element of a textual utterance that points to a relevant resource. In VMT, actions and objects in the shared whiteboard are often referenced in the chat. Mathematical content and other resources from the joint problem space and from shared past experience are also brought into the discourse by explicit or implicit reference in a chat posting. The multi-layered structure corresponds to the multiplicity of constraints imposed on small-group discourse—from the character of the life-world and of culture (which mediate macro-structure) to the semantic, syntactic and pragmatic rules of language (which govern the fine structure of utterances). A theory of group cognition must concern itself primarily with the analysis of mid-level phenomena—such as how small groups accomplish collaborative problem solving and other conversational topics. The study of mid-level group-cognition phenomena is a realm of analysis that is currently underdeveloped in the research literature. For instance, many CSCL studies focus on coding individual (microlevel) utterances or assessing learning outcomes (macro-level), without analyzing the group processes (midlevel). Similarly, Conversation Analysis (CA) centers on micro-level adjacency pairs while socio-cultural Discourse Analysis is concerned with macro-level identity and power, without characterizing the interaction patterns that build such macro phenomena out of micro-elements. Understanding these mid-level phenomena is crucial to analyzing collaborative learning, for it is this level that largely mediates between the interpretations of individuals and the socio-cultural factors of communities. The analysis in this paper illustrates the applicability of the notion of a ‘long sequence’ as vaguely suggested by both Sacks (1962/1995, II p. 354) and Schegloff (2007, pp. 12, 213). A longer sequence consists of a coherent series of shorter sequences built on adjacency pairs. This multi-layered sequential structure will be adapted in this paper from the informal face-to-face talk-in-interaction of CA to the essentially different, but analogous, context of groupware-supported communication and group cognition, such as the text chat of VMT. I will show how a small group of students collaborating online constructed a coherent long sequence, through which they solved the problem that they had posed for themselves. Methodologically, it is important to note that the definition of the long sequence—like that of the other levels of structure listed above—is oriented to by the discourse of the students and is not simply a construct of the researcher. An Analytic Method Recently, I have been trying to apply the CA perspective and techniques in a systematic way to the analysis of VMT chat logs. Schegloff’s (2007) book on Sequence Organization in Interaction represents the culmination of decades of CA analysis. As indicated by its subtitle, it provides a useful primer in CA. My goal is to transform CA to apply to online chat and to extend it to analyze the larger scale interactions of group cognition. Schegloff’s presentation makes clear the central role of the adjacency pair as the primary unit of sequence construction according to CA. An adjacency pair is composed of two conversational speaking turns by two different people, with an interactional order, such as a question followed by an answer to the question. The simple two-turn pair can be extended with secondary adjacency pairs that precede, are inserted between or follow up on the base pair, recursively. This yields “extensive stretches of talk which nonetheless must be understood as built on the armature of a single adjacency pair, and therefore needing to be understood as extensions of it” (p. 12). These “extensive stretches of talk” are still focused on a single interaction of meaning making, and not a larger cognitive achievement like problem solving. However, both Sacks and Schegloff provide vague suggestions about the analysis of longer sequences. These suggestions have not been extensively developed within CA. This paper is an attempt to explore them in an online text-chat context. As I have frequently argued (e.g., Stahl, 2006; 2009c; Stahl, Koschmann & Suthers, 2006), I believe that adapting CA to computer-mediated communication offers the best prospects for analysis of interaction in groupware—i.e., for a theory of small groups appropriate to CSCL. I designed and directed the Virtual Math Teams (VMT) Project from 2003 to the present in order to produce a corpus of data that could be analyzed in as much detail as needed to determine the structure of group cognition, that is, of collaborative knowledge building through interaction at the group unit of analysis. In looking at the VMT data corpus, the VMT research team has clearly seen the differences between online text chat and verbal conversation. The system of turn taking so important in CA (Sacks, Schegloff & Jefferson, 1974) does not apply in chat. Instead, chat participants engage in ‘reading’s work’ (Zemel & Çakir, 2009), in which “readers connect objects through reading’s work to create a ‘thread of meaning’ from the various postings available for inspection” (p. 274f). The first and second parts of an adjacency pair may no longer be literally temporally adjacent to each other, but they still occur as mutually relevant, anticipatory and responsive. The task of reading’s work—for both participants and analysts—is to reconstruct the threading of the adjacency pair response structure (Stahl, 2009b). We have tried to explore the larger sequential structure of problem-solving chat by using the CA notion of openings and closings (Schegloff & Sacks, 1973). VMT researchers looked at several math chats from 2004, which used a simple chat tool from AOL. We coded and statistically analyzed the fine-structure threading of adjacency pairs (Çakir, Xhafa & Zhou, 2009). In addition, we defined long sequences based on when opening and closing adjacency pairs achieved changes in topic (Zemel, Xhafa & Çakir, 2009). These long sequences were graphed to show their roles in constituting the chat sessions, but their internal sequential structures were not investigated. My colleagues and I have subsequently conducted numerous case studies from the VMT corpus. We have been particularly drawn to the records of Team B and Team C in the VMT Spring Fest 2006. These were particularly rich sessions of online mathematical knowledge building because these teams of students met for over four hours together and engaged in detailed explorations of interesting mathematical phenomena. However, partially because of the richness of the interactions, it was often hard for analysts to determine a clear structure to the student interactions. Despite access to everything that the students knew about each other and about the group interaction, it proved hard to unambiguously specify the group-cognition processes at work (Medina, Suthers & Vatrapu, 2009; Stahl, 2009b; Stahl, Zemel & Koschmann, 2009). Therefore, in the following case study, I have selected a segment of Team B’s final session, in which the structure of the interaction seems to be clearer. The interaction is simpler than in earlier segments partially because two of the four people in the chat room leave. Thus, the response structure is more direct and less interrupted. In addition, the students have already been together for over four hours, so they know how to interact in the software environment and with each other. Furthermore, they set themselves a straightforward and well-understood mathematical task. The analysis of this relatively simple segment of VMT interaction can then provide a model for subsequently looking at other data and seeing if it may follow similar patterns. The Case Study Three anonymous students (Aznx, Bwang, Quicksilver) from US high schools met online as Team B of the VMT Spring Fest 2006 contest to compete to be “the most collaborative virtual math team.” They met for four Figure 1. VMT interface with stair-step pattern of horizontal and vertical sticks. hour-long sessions during a two-week period in May 2006. A facilitator (Gerry) was present in the chat room to help with technical issues, but not to instruct in mathematics. In their first session, they solved a given problem, finding a mathematical formula for the growth pattern of the number of squares and the number of sticks making up a stair-step arrangement of squares. They determined the number of sticks by drawing just the horizontal sticks together and then just the vertical ones (see Figure 1). They noticed that both the horizontals and the verticals formed the same pattern of 1 + 2 + 3 + ... + n + n sticks at the n stage of the growth pattern. They then applied the well-known Gaussian formula for the sum of consecutive integers, added the extra n, and multiplied by 2 to account for both the horizontal and vertical sets of sticks. In the second session, they explored problems that they came up with themselves, related to the stairstep problem, including 3-D pyramids. Here they ran into problems drawing and analyzing 3-D structures. However, they managed to approach the problem from a number of perspectives, including decomposing the structure into horizontal and vertical sticks. In the third session, Team B was attracted to a diamond-shaped variation of the stair-step figure, as explored by Team C in the Spring Fest. They tried to understand how the other team had derived its solution. They counted the number of squares by simplifying the problem through filling in the four corners surrounding the diamond to make a large square; the corners turned out to follow the stair-step pattern from their original problem. In the fourth session, they discovered that the other team’s formula for the number of sticks was wrong. In the following, we join them an hour and 17 minutes into the fourth session, when one of the students as well as the facilitator had to leave. Problem-Solving Moves In this section of the paper, the interaction is analyzed as a sequence of moves in the problem-solving interaction between Bwang and Aznx, the two remaining students. Each move is seen to include a base adjacency pair (in bold face), which provides the central interaction of the move and accomplishes the focal problem-solving activity. The captions of log segments indicate the aim of the move, according to the analysis. In line 1734 (see Log 1), Bwang states that the team is close to being able to solve the problem of the number of sticks in the n stage of the diamond pattern, suggesting that they might stay and finish it up. Note that this is the end of the last of the scheduled four sessions for the contest, despite some arrangements underway to allow the team to continue to meet. Aznx responds in line 1736, indicating—and implicitly endorsing the suggestion—that the team could indeed continue to work on the current topic. This opens the topic for the group. Quicksilver apologetically stresses that he must leave immediately. He just wants to know the location of the new chat room that the facilitator is setting up for the team to continue its math explorations on a future date. The facilitator supplies this information and everyone says goodbye to Quicksilver. Aznx expresses uncertainty about how to proceed now that Quicksilver has gone and the facilitator has arranged things for the future. Line 1749 (see Log 2) questions whether he and Bwang need to go as well. Bwang then reiterates his suggestion that they could stay and finish solving the problem. He argues that it should not take much longer. Bwang directly asks Aznx if he wants to solve the problem now. Aznx agrees by responding to Bwang’s question in the affirmative. This effects a decision by the pair of students to start working on the problem right away. Bwang continues to argue for starting on the problem now—posting line 1754 just 3 seconds after Aznx’ agreement, probably just sending what he had already typed before reading Aznx’ response. Bwang then notes the response. Log 1. Open a Topic LINE TIME AUTHOR TEXT OF CHAT POSTING 1734 08.17.20 bwang8 i think we are very close to solving the problem here 1735 08.17.35 Quicksilver Oh great...I have to leave 1736 08.17.39 Aznx We can solve on that topic. 1737 08.17.42 Quicksilver Sorry guys 1738 08.17.45 bwang8 oh 1739 08.17.46 Aznx It shouldn’t take much time. 1740 08.17.47 bwang8 ok 1741 08.17.50 Aznx k, bye Quicksilver 1742 08.17.52 Quicksilver Just tell me the name of the