Submission to the Human Computer Interaction Consortium (HCIC 06)</t= itle> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"place"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"PlaceName"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"PlaceType"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"City"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"country-region"/>  <style>  </style>  </head> <body lang=3DEN-US link=3Dblue vlink=3Dpurple style=3D'tab-interval:.5in'> <div class=3DSection1> Presentation to the CSCL SIG workshop: Nordic Analysis of Interaction and Learning (NAIL 2005), <st1:place w:st=3D"on"><st1:City w:st=3D"on">Goth= enburg</st1:City>, <st1:country-region w:st=3D"on">Sweden</st1:country-region></st1:place>,De= cember2005<o= :p></o:p> <o:p> </o:p> <o:p> </o:p> Group cognition in chat: Methods of interaction / Methodologies of analys= is <o:p> </o:p> Gerry Stahl The Virtual Math Teams Project, Math Forum and College of Information Science & Technology, <st1:pl= ace w:st=3D"on"><st1:PlaceName w:st=3D"on">Drexel</st1:PlaceName> <st1:PlaceTyp= e w:st=3D"on">University</st1:PlaceType></st1:place> <o:p> </o:p> <o:p> </o:p> How do groups construct their shared experience of collaborating online? While answers to many questions in human-computer interaction have been formulated largely in terms of individual psychology, questions of collaborative experience require consideration of the group as= the unit of analysis. Naturally, groups include individuals as contributors and interpreters of content, but the group interactions have structures and ele= ments of their own that call for different analytic approaches. In the Virtual Math Teams project, we are studying how middle school students do mathematics collaboratively in online chat environments. We are particularly interested in the methods that they develop to conduct their interactions in such= an environment. Taken together, these methods define a culture, a shared set of ways to make sense together. The methods are subtly responsive to the chat medium, the pedagogical setting, the social atmosphere and the intellectual resources that are available to the participants. These methods help define= the nature of the collaborative experie= nce for the small groups that develop and adopt them. We have adapted the scientific methodology of conversa= tion analysis to the micro-analysis of online, text-based, mathematical discours= e. In this paper, we share some of our preliminary findings about how small gr= oups make sense collaboratively in the settings we study (see Acknowledgments). = For instance, we distinguish between expository and exploratory modes of narrat= ive, show how individual and group knowledge is intertwined, analyze a proposal-response pair that is typical in math chats and look at referencing patterns that determine chat threading. Through the use of the kinds of methods analyzed in th= is paper, small groups construct their collaborative experience. The chat take= s on a flow of interrelated ideas for the group, analogous to an individual̵= 7;s stream of consciousness. The referential structure of this flow provides a basis for the group’s experience of intersubjectivity, common ground = and a shared world. We call this experience group cognition—a form of distributed cognition that involves advanced levels of cognition like mathematical problem solving. As designers of educational chat environments, we are particularly interested in how small groups of students construct their interactions in chat media that have different technical features. How do t= he students learn about the meanings that designers embedded in the environment and how do they negotiate the methods that they adopt to turn technological possibilities into practical means for mediating their interactions? Ultimately, how can we design with students the technologies, pedagogies and communities that will result in desirable collaborative experiences for the= m? <h1>Participant methods for “doing” collaboration</h1> In ord= er to understand the experience of people and groups collaborating online in our Virtual Math Teams service at the Math Forum, we look in detail at the interactions as captured in the computer log. In particular, we are studying groups of three to six middle- or high-school students discussing mathemati= cs in chat rooms. The logs that we collect allow us to see what the participan= ts see to a good approximation. <o:p></o:p> We conceptu= alize the patterns of interaction that we observe as methods. This is a concept that we take from ethnomethodology <= /span>(Garfinkel, 1967; Heritage, 1984; Livingston, 1987). Ethnomethodology is a phenomenologic= al approach to sociology that tries to describe the methods that members of a culture use to accomplish what they do, such as how they carry on conversat= ions (Sacks, Schegloff, & Jefferson, 19= 74) or how they do mathematics <!-= -[if supportFields]> ADDIN EN.CITE <EndNote><Cite><Author>Livingston</Author><Year&= gt;1986</Year><RecNum>562</RecNum><MDL><REFERENC= E_TYPE>1</REFERENCE_TYPE><REFNUM>562</REFNUM><AUTHO= RS><AUTHOR>Livingston, Eric</AUTHOR></AUTHORS><YEAR>1986</YEAR><TITLE&g= t;The Ethnomethodological Foundations of Mathematics</TITLE><PLACE_PUBLISHED>London, UK</PLACE_PUBLISHED><PUBLISHER>Routledge & Kegan Paul</PUBLISHER></MDL></Cite></EndNote><![endif]-->(Livingston, 1986). In particular, the branch of ethnomethodology known as conversation analysis (Psathas, 1995; Sacks, 1992; ten Have,= 1999) has developed an extensive and detail= ed scientific literature about the methods that people deploy in everyday info= rmal conversation and how to analyze what is going on in examples of verbal interaction. <o:p></o:p> Methods are= seen as the ways that people produce social order and make sense of their shared wo= rld. For instance, conversation analysis has shown that there are well-defined procedures that people use to take turns at talk. There are ways that people use to determine when they can speak and how they can signal that others may take a turn at conversation (Sacks et al., 1974).<o:p></o:p> We adopt the general approach of conversation analysis, but we must make many adaptation= s to it given the significant differences between our chat logs and informal conversation. Our data consists of chat logs of student messages about mathematics. The messages are typed, not spoken, so they lack intonation, verbal stress, accent, rhythm, personality. The participants are not face-to-face, so their bodily posture, gaze, facial expression and physical engagement are missing. Only completed messages are posted; the halting pro= cess of producing the messages is not observable by message recipients <!= --[if supportFields]> ADDIN EN.CITE <EndNote><Cite><Author>Garcia</Author><Year>1= 999</Year><RecNum>533</RecNum><MDL><REFERENCE_TY= PE>0</REFERENCE_TYPE><REFNUM>533</REFNUM><AUTHORS&g= t;<AUTHOR>Angela Garcia</AUTHOR><AUTHOR>Jennifer Baker Jacobs</AUTHOR></AUTHORS><YEAR>1999</YEAR><TITLE= >The Eyes of the Beholder: Understanding the Turn-Taking System in Quasi-Synchro= nous Computer-Mediated Communication</TITLE><SECONDARY_TITLE>Researc= h on Language and Social Interaction</SECONDARY_TITLE><VOLUME>34</VOLUME><NUMBE= R>4</NUMBER><PAGES>337-367</PAGES></MDL></Cit= e><Cite><Author>Garcia</Author><Year>1998</Ye= ar><RecNum>532</RecNum><MDL><REFERENCE_TYPE>0<= ;/REFERENCE_TYPE><REFNUM>532</REFNUM><AUTHORS><AUTH= OR>Angela Garcia</AUTHOR><AUTHOR>Jennifer Baker Jacobs</AUTHOR></AUTHORS><YEAR>1998</YEAR><TITLE= >The Interactional Organization of Computer Mediated Communication in the College Classroom</TITLE><SECONDARY_TITLE>Qualitative Sociology</SECONDARY_TITLE><VOLUME>21</VOLUME><NUMBER&= gt;3</NUMBER><PAGES>299-317</PAGES></MDL></Cite&= gt;</EndNote><![endif]-->(Garcia & Jacobs, 1998, 1999)</spa= n>. The messages are displayed in a part= icular software environment and the messages are designed by their posters to be r= ead and responded to in that environment (Livingston, 1995; Zemel, 2005)= . The textual messages are persistent = and may be read or ignored at will, and may be re-read later—although they scroll off-screen after several other postings appear. Several participants= may be typing messages at the same time, and the order of posting these messages may be unpredictable by the participants (Cakir et al., 2005). Consequently, messages do not necess= arily appear immediately following the messages to which they may be responding. = In addition to these features of chat, our logs are concerned with mathematics= and are created within educational institutional contexts—such as the Math Forum website and sometimes school-related activities or motivations. Thus,= the chats may involve building mathematical knowledge, not just socializing and conversing about opinions or everyday affairs. <o:p></o:p> These diffe= rences between our chats and normal conversation mean that the rules of turn-takin= g, etc. have all been transformed. What remains, however, is that people still develop methods for creating and sustaining social order and shared meaning making. Chat participants are skilled at creating and adapting sophisticated methods that accomplish their tasks in these unique environments. It is the analyst’s job to recognize and describe these methods, which are gene= rally taken for granted by the participants.<o:p></o:p> Among the s= tudent chat methods of interest to us are the interactional means that the students use:<o:p></o:p> <![if !supportLists]>· <![endif]>To adapt to institutional settings<o:p></o:p> <![if !supportLists]>· <![endif]>To socialize; to have fun; to flirt<o:p></o:p> <![if !supportLists]>· <![endif]>T= o get to know each other better<o:p></o:p> <![if !supportLists]>· <![endif]>To establish interpersonal relations or roles<o:p></o:p> <![if !supportLists]>· <![endif]>T= o form themselves into groups<o:p></o:p> <![if !supportLists]>· <![endif]>To define a problem to work on<o:p></o:p> <![if !supportLists]>· <![endif]>To start working on a problem<o:p></o:p> <![if !supportLists]>· <![endif]>To agree on how to proceed<o:p></o:p> <![if !supportLists]>· <![endif]>To bring in math resources<o:p></o:p> <![if !supportLists]>· <![endif]>To agree on solutions<o:p></o:p> <![if !supportLists]>· <![endif]>T= o stop problem solving<o:p></o:p> <h1>Research methodology for studying collaboration</h1> In the= chat context, participants exchange textual postings. This is the sole visible b= asis for interaction, communication, mutual understanding and collaborative knowledge building within a generic chat environment. We are developing a c= hat environment that supplements this with some social awareness features and w= ith a shared whiteboard for drawing geometric figures, but for the moment let us consider a generic chat room. In addition to the content of the typed posti= ngs, their order, sequentiality and timing typically play a significant role in = how the postings are understood. The participants log in with a chat “handle” that is associated with their postings; the wording of this handle may imply something about the person so named. The postings by a given participant are linked together as his (or hers?) via the handle. Furthermore, we assume that the participants come to the chat room with specific expectations and motivations—in our case, because it is part= of the Math Forum site and may be recommended by a teacher, parent or friend. Thus, there is an open-ended set of factors that may enter the chat from its socio-cultural context. There is also more-or-less shared language (e.g., English and basic math terminology) and culture (e.g., contemporary teen subculture and classroom math practices) that can play a role in the chats.= <o:p></o:p> To study wh= at takes place among students in chat rooms, we hold data sessions (Jordan & Henderson, 1995)<= !--[if supportFields]>= <![endif]-->. These are meetings in which a number= of researchers take a careful look at chat logs and discuss what appears to be taking place. Focus is directed toward brief extracts that seem to present interactions that are of analytic interest to the research group. The chat = log reveals to the researchers most of what was visible to the student participants. The researchers can take into account the institutional conte= xt in which the chat took place when it is made relevant within the chat. As members of the broader society to which the students also belong, the researchers share to a large extent a competent understanding of the culture and language of the chat. Thus, they are capable of making sense of the chat because they see the same things that the participants saw and can understa= nd them in similar ways. Moreover, by repeatedly studying the persistent log of the chat and by bringing their analytic skills to it, researchers who have = made themselves familiar with this genre can make explicit many aspects of the interaction that were taken for granted by participants in the flow of the moment. By working collaboratively, the researchers can minimize the likeli= hood of idiosyncratic analyses. <o:p></o:p> Ethnomethod= ology provides a further theoretical justification for the ability of researchers= to produce rigorous analyses of recorded interactions. This has to do with the notion of accountability </span= >(Garfinkel, 1967; Livingston, 1987)</s= pan>. When people interact, they typically construct social order (such as conducting a fun chat or developing a math solution) and may produce social objects (like textual postings). These obj= ects are accountable in the sense that they are designed to reveal their own significance. A brief text, for instance, is written to be read in a certain way; its choice of wording, syntax, references and placement in the larger = chat are selected to show the reader how to read it (Livingston, 1995). The account that a chat posting give= s of itself for the other students in the chat can also be taken advantage of by= the researchers. The researchers in a data session discuss the log in order to agree on the accounts of the postings, individually and in their interactive unity.<o:p></o:p> The Virtual= Math Teams service at the Math Forum is being developed by the VMT research proj= ect (Stahl et al., 2005). We are building on the established Problem-of-the-Week service at the Math Forum digital library (<a href=3D"http://mathforum.org/">http://mathforum.org</a>) by systematically opening this service up to small groups of students, rather than primarily = to individual students. We have taken a design-based research approach = (Design-Based Research Collective, 200= 3) to co-evolve the software, pedagogy, mathematics and service through an iterative process of trial, analysis and design modification. The software started with generic, commercial and educational chat systems and now involves development of a research prototy= pe. The pedagogy started with principles of mathematics education and computer-supported collaborative learning and is now incorporating efforts = to build a user community engaged in discussing math and facilitating collaborative practices. The math problems started out using the same Problems-of-the-Week offered to individuals and are now providing opportuni= ties for groups to explore open-ended mathematical worlds as well as to work on issues that the participants generate themselves. The service started as occasional offerings and is now gearing up for continuous availability supported by as-needed monitoring and feedback. <o:p></o:p> As the tria= ls progress, we analyze the resultant logs in ways described in this paper and= use our results to inform our redesign of the software, pedagogy, mathematics a= nd service. Thereby, ethnomethodologically-informed video analysis or interact= ion analysis provides the analytic component of design research, a component th= at is not often specified in discussions of design-based research (Koschmann, Stahl, & Zemel, 2006)<= /span>. The usage of our insights into how students interact in chat is at odds with the usual practices of ethnomethodology and conversation analysis, which claim not to impose researcher or designer interests on their data. While we try to understand = what the student participants are up to in their own terms and how they are maki= ng sense of the activity structure that we provide for them, we are doing this= in order to motivate design decisions. Our goal is not just to understand the student meaning-making processes, but to use that understanding to modify t= he VMT service to allow groups to engage in more effective math discourse.<o:p= ></o:p> <h1>Expository and exploratory discourse</h1> Althou= gh our ethnomethodological chat analysis methodology modeled on conversation analy= sis has so far yielded the most insight into our data, we are pursuing a variet= y of approaches including coding (Strijbos & Stahl, 2005)<!-= -[if supportFields]>= <![endif]-->, statistical (Zemel, Xhafa, & Cakir, 2005)</spa= n> and ethnographic (Sarmiento, Cakir, & Stahl, 2006)<= /span> investigations. These independent approaches can shed important light on the data and inform each other. Ethnographic analyses of the socio-cultural context, such as the classroom experiences of individual chat participants or their other activities in the Math Forum community help to clarify the personal motivations and the math resources that students bring into the chat (Renninger & Shumar, 1998)<= !--[if supportFields]>= <![endif]-->. <o:p></o:p> In our proj= ect, a statistical analysis led to an interesting conversation analytic result. A statistical comparison of codes between chats in which students had time to work on math problems individually prior to the chats (condition A) and tho= se where they first saw the problem in the collaborative chat context (conditi= on B) led to a puzzling anomaly (Zemel, Xhafa, & Stahl, 2005)</spa= n>. While most of the chats in both cond= itions were clustered together, one chat from each condition clustered more with t= he chats from the other condition. A conversation analysis of the two anomalous chats led to a distinction between = expository narrative and exploratory inqui= ry. In conversation analytic terms, this is largely a difference in turn-taking methods. In exposition, one person makes a bid to “tell a story” about how they solved a problem. The other group members offer the exposito= r an extended turn at talking (or posting). The expositor dominates the discours= e, providing a sequential account across several unusually long turns. The oth= er group members listen (read) attentively, provide brief encouraging exclamations, pose questions and provide an audience. In a math problem-sol= ving session, there may be multiple expositions concerning subsequent parts of t= he problem solution, possibly by different people. In exploratory inquiry, the turns are more equally shared as the group collectively investigates the problem and constructs a solution path. The steps in exploration may each involve several participants, with one person proposing a move and others agreeing, making the move or challenging it. The distinction of exposition versus exploration parallels that between cooperation (people dividing up t= asks to reach a common goal) and collaboration (people working together on each task) (Dillenbourg, 1999).<o:p></o:p> The statist= ical quandary was resolved by noticing that the anomalous chat from condition A consisted largely of collaborative exploration despite the fact that the students may have had a chance to produce their own solutions in advance. In the anomalous chat from condition B, the students took time in the chat to first work out at least partial solutions on their own before contributing = to the chat; they then provided expositions on what they found. These examples demonstrate that external conditions do not mechanically determine the meth= ods that people use to interact. In fact, it is common for students in a chat to alternate between cooperative expository and collaborative exploratory sequences of interaction.<o:p></o:p> <h1>The group of individuals</h1> The di= fference between cooperative exposition and collaborative exploration in math problem solving chats is related to the difference between individual solution and group solution. A given math chat log can be ambiguous as to whether it sho= uld be analyzed as a set of contributions from individual thinkers or whether it should be analyzed as a group accomplishment. Often, it is helpful to view = it both ways and to see an intertwining of these two perspectives at work </sp= an>(Stahl, 2005d).<o:p></o:p> We tried an experiment where we had students solve standard math problems individually = and then solve the same problems in chat groups. In the group that we tracked, = the group not only correctly solved all the problems that were solved by any one member of their group individually, but also solved some that no one did by themselves. Here is one that was solved by the group:<o:p></o:p> Three years ago, men made up two out of every three intern= et users in <st1:place w:st=3D"on"><st1:country-region w:st=3D"on">America</st= 1:country-region></st1:place>. Today the ratio of male to female users is about 1 to 1. In that time the number of American females using the internet has grown by 30,000,000, while the number of males who use the internet has grown by 100%. By how much has= the total internet-user population increased in <st1:place w:st=3D"on"><st1:cou= ntry-region w:st=3D"on">America</st1:country-region></st1:place> in the past three yea= rs? (A) 50,000,000 (B) 60,000,000 (C) 80,000,000 (D) 100,000,0= 00 (E) 200,000,000 When we fir= st looked at the log, it appeared that one student (Mic) who seemed particular= ly weak in math was clowning around a lot and that another (Cosi) managed to s= olve the problem herself despite this distraction in the chat room. We synthesiz= ed her contributions to the chat, putting them into the format of a coherent paragraph:<o:p></o:p> I think it̵= 7;s more than 60,000,000. It can’t be exactly 60,000,000 because the men and w= omen cannot increase equally and even out from an unequal starting point to a 1-= to-1 ratio. . . . Oh, no wait, I mean it’s less than 60,000,000. It must be 50,000,000. Yeah, I’m pretty sure that is what it is, because the wom= en population had to grow more than the men in order to equal out—so the= men must have grown less than 30,000,000. So the total must be less than 60,000= ,000 and the only answer like that is 50,000,000. <Cosi><o:p></o:p>= In thinking= about why Cosi could solve this problem in the group context but not alone, we noticed that she was not simply solving the problem as one would in isolati= on (e.g., setting up algebraic equations), but was interacting with the group effort. In particular, Dan, Mic and Hal had set up a certain way of thinking about the problem and of exploring possible solutions. Cosi was reflecting = on the group approach and repairing problems in its logic. The numbers, words = and considerations that she used were supplied by the context of on-going interactive activities and shared meanings.<o:p></o:p> If we combine the proposals from Mic, Dan, Hal and Cos= i, they read like the cognitive process of an individual problem solver: How can I figure out the increase in users without knowing= the total number of internet users? <Mic> It seems to all come from the 30,000,000 figure. <Dan> 30,000,000 is the number of increase in Amer= ican females. Since the ratio of male to female is 1 to 1, <Mic> the total= of male and female combined would be 60,000,000. <Hal> No, I think it mu= st be more than 60,000,000 because the male and female user populations can’t get higher at equal rates and still even out to a 1 to 1 ratio after starting uneven. No, I made a mistake, the total must be less than 60,000,000. It could be 50,000,000, which is the only multiple choice option less than 60,000,000. <Cosi> Very smart. <Dan> Clearly, Cosi made some contributions to the group tha= t were key to the group solution. They were acknowledged as such. Cosi was termed “very smart” — although this could equally well be said of the group as a whole. While no individual in the group could see how to sol= ve the problem, everyone contributed to exploring it in a way that rather effi= ciently led to a solution. In fact, Mic’s clowning around can be seen to be an extremely effective facilitation of the group process. By joking and laughi= ng a lot, the group relieved some of the pressure to solve a problem that was be= yond any individual’s reach and to open a social space in which ideas coul= d be put forward without fear of being harshly judged. Mathematical problem solving is a paradigm case of hum= an cognition. It is common to say of someone who can solve math problems that = he or she is smart. In fact, we see that taking place in the chat. Here, the g= roup has solved the problem by constructing an argument much like what an indivi= dual might construct. So we can attribute group cognition or intelligence to the group. Attributing the solution to the group rather than to the sum of the individuals in the group can be motivated by seeing that the construction of mathematical meaning in the solution process was done across individuals. T= hat is, meaning was created by means of interactions among individual contribut= ions (postings) to the chat — such as through what are called adjacency pairs in conversation an= alysis — more than by individual postings construed as expressing a series of personal mental representations. <h1>Math proposal adjacency pairs</h1> In an = early chat of the VMT project using AOL’s Instant Messanger, a popular chat environment, we observed a repeated pattern of interaction that we have sin= ce found to be common in math chats (Stahl, 2005c). Here is an excerpt from that chat (l= ine numbers added; handles anonymized):<o:p></o:p> 17. </span= >Avr (8:23:27 PM):= i think we have to figure out the h= eight by ourselves 18. = Avr (8:23:29 PM):= if possible 19. = pin (8:24:05 PM): <= /span>i know how 20. = pin (8:24:09 PM): <= /span>draw the altitude' 21. = Avr (8:24:09 PM):= how? 22. = Avr (8:24:15 PM):= right 23. = Sup (8:24:19 PM): proportions? 24. = Avr (8:24:19 PM):= this is frustrating 25. = Avr (8:24:22 PM):= I don't have enough paper In this log we see several examples of a three-step pa= ttern: <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l13 lev= el1 lfo19; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>a. = <![endif]>A proposal bid is made by Avr in lines 17 and 18 for the group to work on: “I think we have to ….” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l13 lev= el1 lfo19; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>b. </spa= n><![endif]>The bid is taken up by someone else (Pin in line 19) on behalf of the group: “I know how” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l13 lev= el1 lfo19; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>c. = <![endif]>There is an elaboration of the proposal by members of the group. The proposed wor= k is begun, often with a secondary proposal for the first sub-step, such as Pin’s new proposal bid in line 20. The third step initiate= s a repeat of the three-step process: <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l19 lev= el1 lfo21; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>a. = <![endif]>A proposal bid is made by Pin in line 20 for the group to work on: “Draw the altitude” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l19 lev= el1 lfo21; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>b. </spa= n><![endif]>An acceptance is made by someone else (Avr in line 22) on behalf of the group: “Right!” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l19 lev= el1 lfo21; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>c. = <![endif]>There is an elaboration of the proposal by members of the group. The proposed wor= k is begun, often with a secondary proposal for the first sub-step, such as Sup’s new proposal bid in line 23. But here the pattern breaks = down. It is unclear to us as analysts what Sup’s proposal bid, “Proportions?” is proposing. Nor is it responded to by the other group members as a proposal. Avr’s lines 24 and 25 ignore it and seem= to be reporting on Avr’s efforts to work on the previous proposal to draw the altitude. Breakdown situations are often worth analyzing carefully, for they can expose in the breach practices that otherwise go unnoticed, taken = for granted in their smooth execution. Our analysis of Sup’s “failed proposalR= 21; helps to specify—by way of counter-example—the conditions that promote successful proposals in math chats: (a) a clear semantic and syntac= tic structure, (b) careful timing within the sequence of postings, (c) a firm interruption of any other flow of discussion, (d) the elicitation of a resp= onse, (e) the specification of work to be done and (f) a history of helpful contributions. In addition, there are other interaction characteristics and mathematical requirements. For instance, the level of mathematical backgrou= nd knowledge assumed in a proposal must be compatible with the expertise of the participants and the computational methods must correspond with their train= ing. We call the three-step pattern described above a math proposal adjacency pair (Stahl, 2005a). It seems to be a common interaction pattern in collaborative problem solving of mathematics in our chats. As we see in other chats, however, not all student groups adopt this method. We call this a form of “adjacency pair” in keeping with conversation analysis terminology (Duranti, 1998; Schegloff, 1991), even though in chat logs the two parts of the pair may not appear adjacent = due to the complexities of chat postings: e.g., line 22 responds to line 20, wi= th line 21 intervening as a delayed response to line 19. <h1>References and threading</h1> The mo= re we study chat logs, the more we see how interwoven the postings are with each other and with the holistic Gestalt of the interactional context that they form. There are many ways in which a posting can reference elements of its context. The importance of indexicality to creating shared meaning was stre= ssed by Garfinkel (1967). Vygotsky also noted the central role= of pointing for mediating intersubjectivity in his analysis of the genesis of = the infant-and-mother’s pointing gesture (1930/1978, p. 56). Our analysis of face-to-face collabo= ration emphasized that spoken utterances in collaborative settings tend to be elliptical, indexical and projective ways of referencing previous utterance= s, the conversational context and anticipated responses (Stahl, 2006, chapter 12).<o:p></o:p> We have rec= ently developed VMT-Chat, a chat environment that not only includes a shared whiteboard, but has functionality for referencing areas of the whiteboard from chat postings and for referencing previous postings (see figure 1). The shared whiteboard= is necessary for supporting most geometry problems. (This will save Avr the frustration of running out of paper, and also let Pin and Sup see what she = is drawing and add to it or reference it.) Sharing drawings is not enough; students must be able to reference specific objects or areas in the drawing. (Sup could have pointed to elements of the triangles that he felt to be significantly proportional.) The whiteboard also provides opportunities to = post text where it will not scroll away. (Sup could have put his failed proposal= in a text box in the whiteboard, where he or the others could come back to it later.) The graphical references (see the blue line from a selected posting= to an area of the drawing) can also be used to reference one or more previous postings from a new posting, in order to make the threads of responses clea= rer in the midst of “chat confusion” (Pimentel, Fuks, & Lucena, 2005)</= span>.<o:p></o:p> In one of o= ur first chats using VMT-Chat, the students engaged in a particularly complex interaction of referencing a figure in the whiteboard whose mathematics they wanted to explore (Stahl, Wessner et al., 2006). Here is the chat log from figure 1 (graphical references to the whiteboard are indicated by “[REF= TO WB]” in the log.):<o:p></o= :p> 1 ImH:<= span style=3D'mso-tab-count:1'> what is the area = of this shape? [REF TO WB] 2 &= nbsp; Jas: which shape? 3 &= nbsp; ImH: woops 4 &= nbsp; Imh: ahh! 5 &= nbsp; Jas: kinda= like this one? [REF TO WB] 6 &= nbsp; Jas: the o= ne highlighted in black and dark red? 7 &= nbsp; ImH: between th stairs and the hypotenuse 8 &= nbsp; Jas: oh 9 &= nbsp; Jas: that = would be a tricky problem, each little “sector” is different 10 = Jas: this section [REF TO WB] 11 = ImH: perimeter is 12ro= ot3 12 = Jas: is sm= aller than this section [REF TO WB] 13 = ImH: assume those line= s are on the blocks 14 = Jas: the staircase lines? 15 = ImH: yea 16 Jas: they already are on the blocks Line 1 of t= he chat textually references an abstract characteristic of a complex form in the whiteboard: “the area of this shape.” The software function to support this refere= nce failed, presumably because the student, ImH, was not experienced in using i= t and did not cause the graphical reference line to point to anything in the draw= ing. Line 5 provides a demo of how to use the referencing tool. Using the tool’s line, a definite textual reference (“the one”) and the use of line color and thickness in the = drawing, lines 5 and 6 propose an area to act as the topic <![if !vml]><img width=3D623 height=3D421 src=3D"nail_files/image002.gif" align=3Dleft hspace=3D12 alt=3D"Text Box: Figure 1. Screen view of VMT-Chat with referenc= ing. Line 12 of the chat is selected. " v:shapes=3D"_x0000_s1038"><![endif]>of the chat. Line 7 makes explicit in text the definition of a sub-area of the proposed area. Line 8 accepts the new definition and line 9 starts to work = on the problem concerning this area. Line 9 references the problem as “that” and notes that it is tricky because the area defined does not consist of standard forms whose area would be easy to compute and add u= p. It refers to the non-uniform sub-areas as little “sectors”. Lin= e 10 then uses the referencing tool to highlight (roughly) one of these little sectors or “sections”. Line 12 continues line 10, but is interrupted in the chat log by line 11, a failed proposal bid by ImH. The c= hat excerpt continues to reference particular line segments using deictic prono= uns and articles as well as a growing vocabulary of mathematical objects of concern: sectors, sections, lines, blocks. <o:p></o:p> Progress is= made slowly in the collaborative exploration of mathematical relationships, but having a shared drawing helps considerably. The students use multiple textual and graphical means to reach a shared understanding of mathematical objects that they find interesting but hard to define. In this excerpt, we start to get a sense of the complex ways in which brief textual postings weave dense webs = of relationships among each other and with other elements of the collaborative context. <o:p></o:p> <h1>Constructing the collaborative experience</h1> Our go= al in the VMT Project is to provide a service to students that will allow them to have a rewarding experience collaborating with their peers in online discussions of mathematics. We can never know exactly what kind of subjecti= ve experience they had, let alone predict how they will experience life under conditions that we design for them. Our primary access to information relat= ed to their group experiences comes from our chat logs. The logs capture most = of what student members see of their group on their computer screens. We can e= ven replay the logs so that we see how they unfolded sequentially in time. Of course, we are not engaged in the interaction the way the participants were= and recorded experiences never quite live up to the live version because the engagement is missing. We do test out the environments ourselves and enjoy = the experience, but we experience math and collaboration differently than do middle-school students. We also interview students and their teachers, but teenagers rarely reveal much of their life to adults.<o:p></o:p> <![if !vml]><img width=3D219 height=3D503 src=3D"nail_files/image004.gif" align=3Dright hspace=3D12 alt=3D"Text Box: Figure 2. Threading of adjacency pairs= and other uptake. " v:shapes=3D"_x0000_s1037"><![endif]>So we try to understand how collaborative experiences are structured as interp= ersonal interactions. The focus is not on the individuals as subjective minds, but = on the human, social group as constituted by the interactions that take place = in the group.<o:p></o:p> <h2>Replies, up-take, pairs and triplets</h2> Figure= 2 provides a diagram of the responses of postings in the chat discussed above involving Avr, Pin and Sup (Stahl, 2005b). The numbered posts from each partici= pant are placed in chronological order in a column for that participant. Math proposal adjacency pairs are indicated with red arrows and other kinds of responses are indicated with green arrows. Note that Sup’s failed proposal bid (line 23) is isolated. Most of the chat has coherence, flow or motion due to the fact that most postings are responses to previous message= s. This high level of responses is due to the fact that many postings elicit responses, the way that a greeting invariably calls forth another greeting = in response, or a question typically produces an answer. In a healthy conversation, most contributions by one participant are taken up by others. Conversationalists work hard to fit their offerings into the timing and evolving focus of the on-going interaction. In chat, the timing, rules and practices are different, but the importance of up-take remains. <o:p></o:p>= The fact th= at the group process and the cross-ties between people are central to collaborative experiences does not contradict the continuing importance of the individual= s. The representation of figure 2 uses columns to indicate the connections and implicit continuity within the sequence of contributions made by an individ= ual. We may project psychological characteristics onto the unity of an individual’s postings, attributing this unity to personal interests, personality, style, role, etc. Such attributions may change as the chat unfolds. The point is that the individual coherence of each participantR= 17;s contributions adds an important dimension of implicit connections among the postings.<o:p></o:p> Adjacency p= airs like math proposals, greetings and questionings provide important ties that= cut across the connections of individual continuities. They form the smallest u= nits of meaning precisely by binding together postings by different people. A proposal bid that is not taken up is not a meaningful proposal, but at best= a failed attempt at a proposal. A one-sided greeting that is not recognized by the other is not an effective greeting. An interrogative expression that do= es not call for a responce is no real questioning of another. These are all interactional moves whose meaning consists in a give-and-take between two o= r more people. When we hear something that we recognize as a proposal, a greeting = or a question, we feel required to attempt an appropriate response. We may ignore the proposal, snub the greeter or refuse to answer the question, but then o= ur silence is taken as a response of ignoring, snubbing or refusing—and = not simply a lack of response or up-take. <o:p></o:p> In fact, th= e way that a response is taken is also part of the interaction itself. In discuss= ing the building of “common ground,” Clark argues that shared under= standing by A and B of A’s utterance involves not only B believing that he understands A, but also A believing that B understands (Clark & Brennan, 1991). This requires an interaction spannin= g at least multiple utterances. Thus, for instance, the most prevalent interacti= on in classroom discourse is when a teacher poses a question, a student provid= es an answer demonstrating understanding and then the teacher acknowledges the student response as such an understanding (Lemke, 1990). Here, the elemental cell of interact= ional meaning making is a sequence of contributions by at least two different peo= ple. It is clear that the meaning is constructed through the interaction of mult= iple people, and is not a simple expression of pre-existing mental representatio= ns in any one individual’s head.<o:p></o:p> <h2>Longer sequences</h2> Althou= gh much attention has been given to adjacency pairs in conversation analysis and although such pairs can be thought of as the cells of meaning making in collaborative interaction, they form only one of many levels of analysis. F= or instance, there are longer sequences, episodes and topics in dialogs and ch= ats that provide layers of structure and sense (Linell, 2001; Zemel, Xhafa, & Cak= ir, 2005). An hour-long chat is not a homogeneous interchange. A typical math chat might start with a period of introductions, greetings, socializing. Th= en there could be some problem-solving work. This might be periodically interrupted by joking, playing around, or silliness. People may come and go, requiring catching up and reorganizing. Each of these episodes has boundari= es during which the group members must negotiate whether to stop what they were doing and start something else. These transitions may themselves be longer sequences of interaction, especially in large groups. We have barely begun = to explore these different layers. <o:p></o:p> The chat ex= cerpt from VMT-Chat above was from the second hour-long session in a series of fo= ur chats with the same groups. The sessions referred back to previous sessions= and prepared for future ones. We hope to foster a community of Math Forum users= who come back repeatedly to math chats, potentially with their friends. Their c= hats will reference other chats and other online experiences, building connectio= ns at the community level. This adds more layers of interconnections.<o:p></o:= p> <h2>Constructing proofs</h2> In our= chats, students work on math problems and themes. In solving problems and exploring math worlds or phenomena, the groups construct sequences of mathematical reasoning that come close to proofs. Proofs in mathematics have an interest= ing and subtle structure. One must distinguish: the problem situation; the exploratory search for the solution; the effort to reduce a haphazard solut= ion path to an elegant, formalized proof; the statement of the proof; and the l= ived experience of following the proof (Livingston, 1986, 1987). Each of these has its own structures= and practices. Each necessarily references the others. To engage in mathematics= is to become ensnarled in the intricate connections among them. To the extent = that these aspects of doing math have been distinguished and theorized, it has b= een done as though there is simply an individual mathematician at work. There h= as been virtually no research into how these could be accomplished and experie= nced collaboratively—despite the fact that talking about math has for some= time been seen as a priority in math education (NCTM, 1989).<o:p></o:p> <h2>The stream of group consciousness</h2> Psycho= logists like Williams James and novelists like Jack Kerouac have described narrativ= es that we tell ourselves silently about what we are doing or observing as our stream of consciousness. This “inner voice” rattles on even as = we sleep, making connections that Sigmund Freud found significant (if somewhat shocking in his day). In what sense might online chats—with their meanderings, flaming, associative referencing, unpredictable meaning making= and unexpected images—deserve equal status as streams of (group) consciou= sness? Group cognition can be self-conscious.<o:p></o:p> Our sense o= f time and the rhythms of life are largely reliant upon the narratives we tell ourselves (Trausan-Matu, Stahl, & Sarmiento,= in preparation). We know that we have already lived t= hrough a certain part of the day or of our life because we place the present in the nexus of its ties to our memories of the past or our hopes for the future. = In similar ways, the web of references in a chat that connects postings to pri= or postings to which they respond and to future postings that they elicit defi= nes a temporality of the chat. This is a lived sense of time that is shared by = the group in the chat. Like our individual internal clocks, the group temporali= ty must be attuned to the larger world outside—the world of family life = that calls the students away from the chat for dinner or the world of school that interrupts a chat with class changes or homework pressures. The temporality that is constructed as a dimension of the collaborative experience is constrained by nature of the social situation and technological environment= .<o:p></o:p> <h2>Group cognition in chat</h2> The fa= ct that meaning is created at the group unit of analysis rather than by particular individuals suggests the notion of group cognition (Stahl, 2006). The traditional view of cognition, particularly in Western philosophy since Descartes, is that meaning, ideas = and thoughts are created in individual minds. Theories in cognitive science formulated this in terms of mental representations in the heads of individuals—an approach that has been critiqued by more recent theori= es of situated and distributed cognition. The mental contents of individual cognition—in the traditional view—can subsequently be expressed= in language and communicated through the external world, to be then interprete= d in the minds of other individuals. Meaning, in this view, exists only in individual minds, and cognition is always personal.<o:p></o:p> Whether or = not one accepts some version of the cognitivist view in general, it seems that in situations of collaboration notions of shared meaning and group cognition a= re useful and important. Here, “shared meaning” has a deeper significance than what seems to underlie Clark’s analysis of common ground, where sharing is reduced to coordination among the individual mental contents of several minds (Clark & Brennan, 1991). Shared meaning is constructed across= pairs or triplets of postings by more than one participant. It is not that an ans= wer to a question implies that the answerer has in mind the same thing as the questioner, but that the answer and the question by themselves are fragment= ary; they have meaning only as part of the question-answer interaction. The unit= of meaning is the interaction itself, and this is a group phenomenon not an individual one. Moreover, with adequate capture of collaborative interactio= ns, it is possible to see the construction of meaning in the traces of interact= ion; it is not necessary to hypothesize about hidden mental operations or conten= ts.<o:p></o:p> Of course, = in some sense it is easier to visualize individuals than groups as cognitive agents= . As Vygotsky’s analysis of the infant’s gesture shows, we are used = to identifying other individuals as meaning-expressing agents. Given our perceptual orientation to a primarily visual world (Merleau-Ponty, 1945/2002), it is more natural for us to assign = agency to physical objects like human bodies than to more abstract entities like online groups. What, we may well wonder, is a collaborative group?<o:p></o:p> Groups cons= titute themselves. We can see how they do this in the chat logs. At one level the = Math Forum service brings several students together and locates them in a chat r= oom together. It may supply a math problem for them to work on and it may provi= de a facilitator who introduces them to the environment. At this point, they are= a potential group with a provisionally defined membership. The facilitator mi= ght say something like, “Welcome to our first session of Virtual M= ath Teams! I am the facilitator for your session. . . . As a group, decide which question you would like to work on.” (This is part of the facilitator script from the session involving ImH and = Jas excerpted above.) Here we can see that the facilitator has defined the group (“as a group … you”) and distinguished her = own role as outside the group (“I= am the facilitator … your session”). The potential group projected by the facilitator need not necessarily materialize. Individual students many come to the setting, look around, decide it is lame, and leave as individuals. However, this rarely happens. Sometimes an individual will leave without ever interacting, but as long as there are enough students there, a group will emerge.<o:p></o:p></s= pan> Students co= me to the chat environment with certain motivations, expectations and experiences. These are generally sufficient to get the group started. One can see the gr= oup form itself. This is often reflected in the shift from singular to plural pronouns: “Let’s get started. Let us do some math.” We saw this in Avr’s proposal:</s= pan> “I= think we have to figure out the height b= y ourselves.” The proposal bid= comes from the individual, but the projected work is for the group. Through her u= se of “we,” Avr constitutes the group. Through her proposal bid, s= he constitutes the group as a recipient of the bid and elicits a response from= them. Someone other than Avr must respond to the bid on behalf of the group. When= Pin says, “I know how: draw the altitude,” he is accepting Avr̵= 7;s proposal as a task for the group to work on and in so doing he makes a prop= osal about how the group should go about approaching this task (by making a geometric construction). In this interchange, the group (a) is projected as= an agent in the math work (Lerner, 1993), and (b) is actually the agent of mea= ning making because the meaning of Avr’s proposal is defined by the interaction within the group. <o:p></o:p> If the group experience is a positive one for the participants, they may want to return. Some chats end with people making plans to get together again. In some experiments, the same groups attended multiple sessions. We would like to s= ee a community of users form, with teams re-forming repeatedly and with old-time= rs helping new groups to form and learn how to collaborate effectively.<o:p></= o:p> The recogni= tion that collaborative groups constitute themselves interactionally and that th= eir sense making takes place at the group unit of analysis has fundamental methodological implications for the study of collaboration. The field of computer-supported collaborative learning (CSCL) was founded a decade ago to pursue the analysis of group meaning making (Stahl, Koschmann, & Suthers, 2006= ). We view the research described here = as a contribution to the CSCL tradition.<o:p></o:p> <h1>How groups construct their experience</h1> We are designers. Our goal is to design an exciting mathematical group experience = for students. We want to design an online collaborative service, with strong pedagogical direction and effective computer support. We approach this by trying to understand how groups of students construct their experience in s= uch settings. Because we are designing a computer-supported experience that has never bef= ore existed and because we want our design to be based on detailed study of how students actually created their collaborative experience in the environment= we are designing, we follow a highly iterative try-analyze-redesign cycle.<o:p= ></o:p> When studen= ts enter our website, they are confronted by a densely designed environment. The lob= by to our chat rooms is configured to help students find their way to a room t= hat will meet their needs. In the room, there is a bewildering array of software functionality for posting and displaying chat notes, drawing geometric forms and annotating them, keeping track of w= ho is doing what and configuring the space to suit oneself. There may be a sta= tement of a math problem to solve or an imaginary world to explore mathematically.= The service, problems and software are all designed to enhance the user’s experience. But how can a student who is new to all this understand the meanings of the many features and affordances that have been built into the environment?<o:p></o:p> Groups of s= tudents spontaneously develop methods for exploring and responding to their environments. They try things out and discuss what happens. A new group may doodle on the whiteboard and then joke about the results. They bring with t= hem knowledge of paint and draw programs and skills from video games, SMS and I= M. The individuals may have considerable experience with single-user apps, but react when someone else erases their drawing; they must learn to integrate coordination and communication into their actions. The math problems they f= ind in the chat rooms may be quite different than the drill-and-practice proble= ms they are used to in traditional math textbooks. It may take the group a whi= le to get started in productive problem solving, so the group has to find ways= to keep the group together and interacting in the meantime. There may be vario= us forms of socializing, interspersed with attempts to approach the math. As unaccustomed as the math may be, the students always have some knowledge and experience that they can bring to bear. They may apply numerical computatio= ns to given values; try to define unknowns and set up equations; graph relationships; put successive cases in a table; use trigonometric relations= hips or geometric figures; draw graphical representations or add lines to an existing drawing. Mainly, they put proposals out in the chat stream and res= pond to them. Sometimes the flow of ideas wanders without strong mathematical reflection. Other times, one individual can contribute substantial progress= and engage in expository narrative to share her contribution with the group.<o:= p></o:p> Groupware i= s never used the way its designers anticipated. The designers of VMT-Chat thought t= hat its referencing tools would immediately clarify references to elements of drawings and transform chat confusion into logical threaded chat. But our studies of the actual use of these designed functions tell a quite different and more interesting story. The shared whiteboard with graphical references from the chat may allow more complex issues to be discussed, but they do not make pointing problem free. We saw above how much work ImH and Jas engaged = in to clarify for each other what they wanted to focus on. In the excerpt and = in the longer chat, they used a variety of textual, drawing and referencing methods. In the process, they learned how to use these methods and they tau= ght each other their use. In a matter of a fraction of a minute, they were able= to reach a shared understanding of a topic to work on mathematically. In that brief time, they used dozens of indexical methods, some that would prove mo= re useful than others for the future.<o:p></o:p> Chat is a h= ighly constrained medium. Participants feel various pressures to get their indivi= dual points of view out there. In a system like VMT-Chat, there is a lot to keep track of: new postings, changes to the whiteboard, signs that people are joining, leaving, typing, drawing. Small details in how something is writte= n, drawn or referenced may have manifold implications through references to present, past or future circumstances. Students learn to track these detail= s; apply them creatively; acknowledge to the group that they have been recogni= zed; check, critique and repair them. Each group responds to the environment in = its own way, giving group meaning to the features of the collaborative world and thereby putting their unique stamp on their group experience.<o:p></o:p></s= pan> In the proc= ess, they create a group experience that they share. This experience is held together with myriad sorts of references and ties among the chat postings a= nd drawings. Often, what is not said is as significant as what is. Individual postings are fragmentary, wildly ambiguous, and frequently confusing. In li= vely chats, much of what happens remains confusing for most participants. Clarity comes only through explicit reflections, up-takes, appreciations, or probin= g. The interactions among postings, at many levels, coheres into a stream of g= roup consciousness, a flow of collaboration, a shared lived temporality and, with luck, an experience of mathematical group cognition.<o:p></o:p> As we have = seen in this paper, when students enter into one of our chats they enter into a com= plex social world. They typically quickly constitute a working group and begin to engage in activities that configure a group experience. This experience is conditioned by a social, cultural, technological and pedagogical environment that has been designed for them. Within this environment, they adopt, adapt= and create methods of social practice for interacting together with the other students who they find in the chat environment. Over time, they explore the= ir situation together, create shared meaning, decide what they will do and how they will behave, engage in some form of mathematical discourse, socialize = and eventually decide to end their session. Then our job begins: to analyze what has happened and how the environment we are designing conditions the collaborative experiences that groups construct there.<o:p></o:p> <h1>Workshop Issues</h1> <h2>How does one transform recordings of interaction into analytic accounts?<o:p></= o:p></h2> This p= aper provides a case study of how a group of researchers, technology designers a= nd educational service providers collaboratively analyze logs of students interacting in an evolving chat environment. We try to describe the methods that chat groups spontaneously develop to make sense of their designed situation and to establish social order for their group interaction as it unfolds locally within that activity structure.<o:p></o:p> <h2>What is the relation between studies of interaction and theories of learning?<o:= p></o:p></h2> We vie= w the term learning with suspicion. I= t is an accountable member’s matter, a child’s concern in responding= to the parental inquiry, “What did you learn in school today?” The psychological theory of changes in mental representations is particularly suspect when applied to collaborative learning. With luck, groups interact = and make meaning. Learning for groups consists in having more methods available= as interactional resources and being equipped with more developed meanings. Through rigorous studies of interaction, we can observe elemental acts of meaning making and the development of member methods and other social pract= ices.<o:p></o:p> <h2>How does one make the analyses relevant for curricular design?<o:p></o:p></span= ></h2> Our cu= rricular goals are to foster enjoyable experiences of discussion of mathematical the= mes in group discourse – making sense of mathematical objects and constructing shared mathematical meanings. We are not interested in the transfer or memorization of facts, but in the development by students of th= e ability to use the resources of a fledgling mathematician, to be able to “do” school math. An ultimate goal might be that a group of students experiences the wonder of understanding an elegant mathematical re= lationship or proof. But we are far from such goals. For now, we need to understand how groups interact in various chat settings and we need to clear away some of = the overwhelming barriers that prevent groups from working together better.<o:p= ></o:p> The worksho= p question is itself problematic. The term “curricular design” presupposes= a theory of learning and instruction. It presupposes categories of student, instructor, pedagogical designer, interaction analyst. In our work, we bring together a group of people who want to work on the VMT service/research project. As individuals, we have a variety of skills, motivations, trainings and interests. But as a group, we set group goals and perspectives, such as trying out a design concept or analyzing an excerpt from a trial chat. What= we learn in one activity informs what we do in another. Although we use the te= rm “student” for our target user, our service is designed for open access on the Web and will be available for school drop-outs, plumbers, professional mathematicians, soccer moms and ethnomethodologists – however they define their own status. <o:p></o:p> <h1>Acknowledgments</h1> This paper is an attempt to reflect the thrust of research being conducted in a multi-disciplinary project, informed by studies conducted us= ing several methodologies and theories derived from various traditions. The argument and presentation of the paper are those of the author. Many details and formulations in the paper would be put differently by other members of = the research group; the paper does not necessarily represent their individual or professional views accurately. For more detailed discussions, see the cited papers.<o:p></o:p> The Virtual= Math Teams Project is a collaborative effort at <st1:place w:st=3D"on"><st1:Plac= eName w:st=3D"on">Drexel</st1:PlaceName> <st1:PlaceType w:st=3D"on">University</= st1:PlaceType></st1:place>. The Principal Investigators are Gerry Stahl, Stephen Weimar and Wesley Shum= ar. A number of Math Forum staff work on the project, especially Stephen Weimar, Annie Fetter and Ian Underwood. The graduate research assistants are Murat Cakir, Johann Sarmiento, Ramon Toledo and Nan Zhou. Alan Zemel is the post-= doc; he facilitates weekly conversation analysis data sessions. The following visiting researchers have spent 3 to 6 months on the project: Jan-Willem Strijbos (<st1:country-region w:st=3D"on">Netherlands</st1:country-region>), Fatos Xhafa (<st1:country-region w:st=3D"on">Spain</st1:country-region>), S= tefan Trausan-Matu (<st1:country-region w:st=3D"on">Romania</st1:country-region>), Martin Wessner (<st1:country-region w:st=3D"on">Germany</st1:country-region= >), Elizabeth Charles (<st1:place w:st=3D"on"><st1:country-region w:st=3D"on">C= anada</st1:country-region></st1:place>). The VMT-Chat software was developed at the Fraunhofer Institute IPSI in <st= 1:place w:st=3D"on"><st1:City w:st=3D"on">Darmstadt</st1:City>, <st1:country-region= w:st=3D"on">Germany</st1:country-region></st1:place>, by Martin Wessner, Martin Mühlpfordt and colleagues based on their ConcertChat. The VMT project = is supported by grants from the NSDL, IERI and SoL programs of the US National Science Foundation. <o:p></o:p> <h1>References</h1> Cakir, M., Xhafa, F., Zhou, N= ., & Stahl, G. (2005). Thread-based analysis of patterns of collaborative interaction in chat. Paper presen= ted at the international conference on AI in Education (AI-Ed 2005), Amsterdam, Netherlands.<o:p></o:p> Clark, H., & Brennan, S. (1991). Grounding = in communication. In L. Resnick, J. Levine & S. Teasley (Eds.), Perspectives on socially-shared cognit= ion (pp. 127-149). Washington, DC: APA.<o:p></o:p> Design-Based Research Collective. (2003). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32 (1), 5-= 8.<o:p></o:p> Dillenbourg, P. (1999). What do you mean by "collaborative learning"? In P. Dillenbourg (Ed.), Collaborative learning: Cognitive and computational approaches (pp. 1-16). Amsterdam, NL: Pergamon, Elsevier Science.<o:p></o:p> Duranti, A. (1998). 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MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C5FA3B.A985CD60" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C5FA3B.A985CD60 Content-Location: file:///C:/2688918C/nail.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Submission to the Human Computer Interaction Consortium (HCIC 06)</t= itle> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"place"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"PlaceName"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"PlaceType"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"City"/> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"country-region"/>  <style>  </style>  </head> <body lang=3DEN-US link=3Dblue vlink=3Dpurple style=3D'tab-interval:.5in'> <div class=3DSection1> Presentation to the CSCL SIG workshop: Nordic Analysis of Interaction and Learning (NAIL 2005), <st1:place w:st=3D"on"><st1:City w:st=3D"on">Goth= enburg</st1:City>, <st1:country-region w:st=3D"on">Sweden</st1:country-region></st1:place>,De= cember2005<o= :p></o:p> <o:p> </o:p> <o:p> </o:p> Group cognition in chat: Methods of interaction / Methodologies of analys= is <o:p> </o:p> Gerry Stahl The Virtual Math Teams Project, Math Forum and College of Information Science & Technology, <st1:pl= ace w:st=3D"on"><st1:PlaceName w:st=3D"on">Drexel</st1:PlaceName> <st1:PlaceTyp= e w:st=3D"on">University</st1:PlaceType></st1:place> <o:p> </o:p> <o:p> </o:p> How do groups construct their shared experience of collaborating online? While answers to many questions in human-computer interaction have been formulated largely in terms of individual psychology, questions of collaborative experience require consideration of the group as= the unit of analysis. Naturally, groups include individuals as contributors and interpreters of content, but the group interactions have structures and ele= ments of their own that call for different analytic approaches. In the Virtual Math Teams project, we are studying how middle school students do mathematics collaboratively in online chat environments. We are particularly interested in the methods that they develop to conduct their interactions in such= an environment. Taken together, these methods define a culture, a shared set of ways to make sense together. The methods are subtly responsive to the chat medium, the pedagogical setting, the social atmosphere and the intellectual resources that are available to the participants. These methods help define= the nature of the collaborative experie= nce for the small groups that develop and adopt them. We have adapted the scientific methodology of conversa= tion analysis to the micro-analysis of online, text-based, mathematical discours= e. In this paper, we share some of our preliminary findings about how small gr= oups make sense collaboratively in the settings we study (see Acknowledgments). = For instance, we distinguish between expository and exploratory modes of narrat= ive, show how individual and group knowledge is intertwined, analyze a proposal-response pair that is typical in math chats and look at referencing patterns that determine chat threading. Through the use of the kinds of methods analyzed in th= is paper, small groups construct their collaborative experience. The chat take= s on a flow of interrelated ideas for the group, analogous to an individual̵= 7;s stream of consciousness. The referential structure of this flow provides a basis for the group’s experience of intersubjectivity, common ground = and a shared world. We call this experience group cognition—a form of distributed cognition that involves advanced levels of cognition like mathematical problem solving. As designers of educational chat environments, we are particularly interested in how small groups of students construct their interactions in chat media that have different technical features. How do t= he students learn about the meanings that designers embedded in the environment and how do they negotiate the methods that they adopt to turn technological possibilities into practical means for mediating their interactions? Ultimately, how can we design with students the technologies, pedagogies and communities that will result in desirable collaborative experiences for the= m? <h1>Participant methods for “doing” collaboration</h1> In ord= er to understand the experience of people and groups collaborating online in our Virtual Math Teams service at the Math Forum, we look in detail at the interactions as captured in the computer log. In particular, we are studying groups of three to six middle- or high-school students discussing mathemati= cs in chat rooms. The logs that we collect allow us to see what the participan= ts see to a good approximation. <o:p></o:p> We conceptu= alize the patterns of interaction that we observe as methods. This is a concept that we take from ethnomethodology <= /span>(Garfinkel, 1967; Heritage, 1984; Livingston, 1987). Ethnomethodology is a phenomenologic= al approach to sociology that tries to describe the methods that members of a culture use to accomplish what they do, such as how they carry on conversat= ions (Sacks, Schegloff, & Jefferson, 19= 74) or how they do mathematics <!-= -[if supportFields]> ADDIN EN.CITE <EndNote><Cite><Author>Livingston</Author><Year&= gt;1986</Year><RecNum>562</RecNum><MDL><REFERENC= E_TYPE>1</REFERENCE_TYPE><REFNUM>562</REFNUM><AUTHO= RS><AUTHOR>Livingston, Eric</AUTHOR></AUTHORS><YEAR>1986</YEAR><TITLE&g= t;The Ethnomethodological Foundations of Mathematics</TITLE><PLACE_PUBLISHED>London, UK</PLACE_PUBLISHED><PUBLISHER>Routledge & Kegan Paul</PUBLISHER></MDL></Cite></EndNote><![endif]-->(Livingston, 1986). In particular, the branch of ethnomethodology known as conversation analysis (Psathas, 1995; Sacks, 1992; ten Have,= 1999) has developed an extensive and detail= ed scientific literature about the methods that people deploy in everyday info= rmal conversation and how to analyze what is going on in examples of verbal interaction. <o:p></o:p> Methods are= seen as the ways that people produce social order and make sense of their shared wo= rld. For instance, conversation analysis has shown that there are well-defined procedures that people use to take turns at talk. There are ways that people use to determine when they can speak and how they can signal that others may take a turn at conversation (Sacks et al., 1974).<o:p></o:p> We adopt the general approach of conversation analysis, but we must make many adaptation= s to it given the significant differences between our chat logs and informal conversation. Our data consists of chat logs of student messages about mathematics. The messages are typed, not spoken, so they lack intonation, verbal stress, accent, rhythm, personality. The participants are not face-to-face, so their bodily posture, gaze, facial expression and physical engagement are missing. Only completed messages are posted; the halting pro= cess of producing the messages is not observable by message recipients <!= --[if supportFields]> ADDIN EN.CITE <EndNote><Cite><Author>Garcia</Author><Year>1= 999</Year><RecNum>533</RecNum><MDL><REFERENCE_TY= PE>0</REFERENCE_TYPE><REFNUM>533</REFNUM><AUTHORS&g= t;<AUTHOR>Angela Garcia</AUTHOR><AUTHOR>Jennifer Baker Jacobs</AUTHOR></AUTHORS><YEAR>1999</YEAR><TITLE= >The Eyes of the Beholder: Understanding the Turn-Taking System in Quasi-Synchro= nous Computer-Mediated Communication</TITLE><SECONDARY_TITLE>Researc= h on Language and Social Interaction</SECONDARY_TITLE><VOLUME>34</VOLUME><NUMBE= R>4</NUMBER><PAGES>337-367</PAGES></MDL></Cit= e><Cite><Author>Garcia</Author><Year>1998</Ye= ar><RecNum>532</RecNum><MDL><REFERENCE_TYPE>0<= ;/REFERENCE_TYPE><REFNUM>532</REFNUM><AUTHORS><AUTH= OR>Angela Garcia</AUTHOR><AUTHOR>Jennifer Baker Jacobs</AUTHOR></AUTHORS><YEAR>1998</YEAR><TITLE= >The Interactional Organization of Computer Mediated Communication in the College Classroom</TITLE><SECONDARY_TITLE>Qualitative Sociology</SECONDARY_TITLE><VOLUME>21</VOLUME><NUMBER&= gt;3</NUMBER><PAGES>299-317</PAGES></MDL></Cite&= gt;</EndNote><![endif]-->(Garcia & Jacobs, 1998, 1999)</spa= n>. The messages are displayed in a part= icular software environment and the messages are designed by their posters to be r= ead and responded to in that environment (Livingston, 1995; Zemel, 2005)= . The textual messages are persistent = and may be read or ignored at will, and may be re-read later—although they scroll off-screen after several other postings appear. Several participants= may be typing messages at the same time, and the order of posting these messages may be unpredictable by the participants (Cakir et al., 2005). Consequently, messages do not necess= arily appear immediately following the messages to which they may be responding. = In addition to these features of chat, our logs are concerned with mathematics= and are created within educational institutional contexts—such as the Math Forum website and sometimes school-related activities or motivations. Thus,= the chats may involve building mathematical knowledge, not just socializing and conversing about opinions or everyday affairs. <o:p></o:p> These diffe= rences between our chats and normal conversation mean that the rules of turn-takin= g, etc. have all been transformed. What remains, however, is that people still develop methods for creating and sustaining social order and shared meaning making. Chat participants are skilled at creating and adapting sophisticated methods that accomplish their tasks in these unique environments. It is the analyst’s job to recognize and describe these methods, which are gene= rally taken for granted by the participants.<o:p></o:p> Among the s= tudent chat methods of interest to us are the interactional means that the students use:<o:p></o:p> <![if !supportLists]>· <![endif]>To adapt to institutional settings<o:p></o:p> <![if !supportLists]>· <![endif]>To socialize; to have fun; to flirt<o:p></o:p> <![if !supportLists]>· <![endif]>T= o get to know each other better<o:p></o:p> <![if !supportLists]>· <![endif]>To establish interpersonal relations or roles<o:p></o:p> <![if !supportLists]>· <![endif]>T= o form themselves into groups<o:p></o:p> <![if !supportLists]>· <![endif]>To define a problem to work on<o:p></o:p> <![if !supportLists]>· <![endif]>To start working on a problem<o:p></o:p> <![if !supportLists]>· <![endif]>To agree on how to proceed<o:p></o:p> <![if !supportLists]>· <![endif]>To bring in math resources<o:p></o:p> <![if !supportLists]>· <![endif]>To agree on solutions<o:p></o:p> <![if !supportLists]>· <![endif]>T= o stop problem solving<o:p></o:p> <h1>Research methodology for studying collaboration</h1> In the= chat context, participants exchange textual postings. This is the sole visible b= asis for interaction, communication, mutual understanding and collaborative knowledge building within a generic chat environment. We are developing a c= hat environment that supplements this with some social awareness features and w= ith a shared whiteboard for drawing geometric figures, but for the moment let us consider a generic chat room. In addition to the content of the typed posti= ngs, their order, sequentiality and timing typically play a significant role in = how the postings are understood. The participants log in with a chat “handle” that is associated with their postings; the wording of this handle may imply something about the person so named. The postings by a given participant are linked together as his (or hers?) via the handle. Furthermore, we assume that the participants come to the chat room with specific expectations and motivations—in our case, because it is part= of the Math Forum site and may be recommended by a teacher, parent or friend. Thus, there is an open-ended set of factors that may enter the chat from its socio-cultural context. There is also more-or-less shared language (e.g., English and basic math terminology) and culture (e.g., contemporary teen subculture and classroom math practices) that can play a role in the chats.= <o:p></o:p> To study wh= at takes place among students in chat rooms, we hold data sessions (Jordan & Henderson, 1995)<= !--[if supportFields]>= <![endif]-->. These are meetings in which a number= of researchers take a careful look at chat logs and discuss what appears to be taking place. Focus is directed toward brief extracts that seem to present interactions that are of analytic interest to the research group. The chat = log reveals to the researchers most of what was visible to the student participants. The researchers can take into account the institutional conte= xt in which the chat took place when it is made relevant within the chat. As members of the broader society to which the students also belong, the researchers share to a large extent a competent understanding of the culture and language of the chat. Thus, they are capable of making sense of the chat because they see the same things that the participants saw and can understa= nd them in similar ways. Moreover, by repeatedly studying the persistent log of the chat and by bringing their analytic skills to it, researchers who have = made themselves familiar with this genre can make explicit many aspects of the interaction that were taken for granted by participants in the flow of the moment. By working collaboratively, the researchers can minimize the likeli= hood of idiosyncratic analyses. <o:p></o:p> Ethnomethod= ology provides a further theoretical justification for the ability of researchers= to produce rigorous analyses of recorded interactions. This has to do with the notion of accountability </span= >(Garfinkel, 1967; Livingston, 1987)</s= pan>. When people interact, they typically construct social order (such as conducting a fun chat or developing a math solution) and may produce social objects (like textual postings). These obj= ects are accountable in the sense that they are designed to reveal their own significance. A brief text, for instance, is written to be read in a certain way; its choice of wording, syntax, references and placement in the larger = chat are selected to show the reader how to read it (Livingston, 1995). The account that a chat posting give= s of itself for the other students in the chat can also be taken advantage of by= the researchers. The researchers in a data session discuss the log in order to agree on the accounts of the postings, individually and in their interactive unity.<o:p></o:p> The Virtual= Math Teams service at the Math Forum is being developed by the VMT research proj= ect (Stahl et al., 2005). We are building on the established Problem-of-the-Week service at the Math Forum digital library (<a href=3D"http://mathforum.org/">http://mathforum.org</a>) by systematically opening this service up to small groups of students, rather than primarily = to individual students. We have taken a design-based research approach = (Design-Based Research Collective, 200= 3) to co-evolve the software, pedagogy, mathematics and service through an iterative process of trial, analysis and design modification. The software started with generic, commercial and educational chat systems and now involves development of a research prototy= pe. The pedagogy started with principles of mathematics education and computer-supported collaborative learning and is now incorporating efforts = to build a user community engaged in discussing math and facilitating collaborative practices. The math problems started out using the same Problems-of-the-Week offered to individuals and are now providing opportuni= ties for groups to explore open-ended mathematical worlds as well as to work on issues that the participants generate themselves. The service started as occasional offerings and is now gearing up for continuous availability supported by as-needed monitoring and feedback. <o:p></o:p> As the tria= ls progress, we analyze the resultant logs in ways described in this paper and= use our results to inform our redesign of the software, pedagogy, mathematics a= nd service. Thereby, ethnomethodologically-informed video analysis or interact= ion analysis provides the analytic component of design research, a component th= at is not often specified in discussions of design-based research (Koschmann, Stahl, & Zemel, 2006)<= /span>. The usage of our insights into how students interact in chat is at odds with the usual practices of ethnomethodology and conversation analysis, which claim not to impose researcher or designer interests on their data. While we try to understand = what the student participants are up to in their own terms and how they are maki= ng sense of the activity structure that we provide for them, we are doing this= in order to motivate design decisions. Our goal is not just to understand the student meaning-making processes, but to use that understanding to modify t= he VMT service to allow groups to engage in more effective math discourse.<o:p= ></o:p> <h1>Expository and exploratory discourse</h1> Althou= gh our ethnomethodological chat analysis methodology modeled on conversation analy= sis has so far yielded the most insight into our data, we are pursuing a variet= y of approaches including coding (Strijbos & Stahl, 2005)<!-= -[if supportFields]>= <![endif]-->, statistical (Zemel, Xhafa, & Cakir, 2005)</spa= n> and ethnographic (Sarmiento, Cakir, & Stahl, 2006)<= /span> investigations. These independent approaches can shed important light on the data and inform each other. Ethnographic analyses of the socio-cultural context, such as the classroom experiences of individual chat participants or their other activities in the Math Forum community help to clarify the personal motivations and the math resources that students bring into the chat (Renninger & Shumar, 1998)<= !--[if supportFields]>= <![endif]-->. <o:p></o:p> In our proj= ect, a statistical analysis led to an interesting conversation analytic result. A statistical comparison of codes between chats in which students had time to work on math problems individually prior to the chats (condition A) and tho= se where they first saw the problem in the collaborative chat context (conditi= on B) led to a puzzling anomaly (Zemel, Xhafa, & Stahl, 2005)</spa= n>. While most of the chats in both cond= itions were clustered together, one chat from each condition clustered more with t= he chats from the other condition. A conversation analysis of the two anomalous chats led to a distinction between = expository narrative and exploratory inqui= ry. In conversation analytic terms, this is largely a difference in turn-taking methods. In exposition, one person makes a bid to “tell a story” about how they solved a problem. The other group members offer the exposito= r an extended turn at talking (or posting). The expositor dominates the discours= e, providing a sequential account across several unusually long turns. The oth= er group members listen (read) attentively, provide brief encouraging exclamations, pose questions and provide an audience. In a math problem-sol= ving session, there may be multiple expositions concerning subsequent parts of t= he problem solution, possibly by different people. In exploratory inquiry, the turns are more equally shared as the group collectively investigates the problem and constructs a solution path. The steps in exploration may each involve several participants, with one person proposing a move and others agreeing, making the move or challenging it. The distinction of exposition versus exploration parallels that between cooperation (people dividing up t= asks to reach a common goal) and collaboration (people working together on each task) (Dillenbourg, 1999).<o:p></o:p> The statist= ical quandary was resolved by noticing that the anomalous chat from condition A consisted largely of collaborative exploration despite the fact that the students may have had a chance to produce their own solutions in advance. In the anomalous chat from condition B, the students took time in the chat to first work out at least partial solutions on their own before contributing = to the chat; they then provided expositions on what they found. These examples demonstrate that external conditions do not mechanically determine the meth= ods that people use to interact. In fact, it is common for students in a chat to alternate between cooperative expository and collaborative exploratory sequences of interaction.<o:p></o:p> <h1>The group of individuals</h1> The di= fference between cooperative exposition and collaborative exploration in math problem solving chats is related to the difference between individual solution and group solution. A given math chat log can be ambiguous as to whether it sho= uld be analyzed as a set of contributions from individual thinkers or whether it should be analyzed as a group accomplishment. Often, it is helpful to view = it both ways and to see an intertwining of these two perspectives at work </sp= an>(Stahl, 2005d).<o:p></o:p> We tried an experiment where we had students solve standard math problems individually = and then solve the same problems in chat groups. In the group that we tracked, = the group not only correctly solved all the problems that were solved by any one member of their group individually, but also solved some that no one did by themselves. Here is one that was solved by the group:<o:p></o:p> Three years ago, men made up two out of every three intern= et users in <st1:place w:st=3D"on"><st1:country-region w:st=3D"on">America</st= 1:country-region></st1:place>. Today the ratio of male to female users is about 1 to 1. In that time the number of American females using the internet has grown by 30,000,000, while the number of males who use the internet has grown by 100%. By how much has= the total internet-user population increased in <st1:place w:st=3D"on"><st1:cou= ntry-region w:st=3D"on">America</st1:country-region></st1:place> in the past three yea= rs? (A) 50,000,000 (B) 60,000,000 (C) 80,000,000 (D) 100,000,0= 00 (E) 200,000,000 When we fir= st looked at the log, it appeared that one student (Mic) who seemed particular= ly weak in math was clowning around a lot and that another (Cosi) managed to s= olve the problem herself despite this distraction in the chat room. We synthesiz= ed her contributions to the chat, putting them into the format of a coherent paragraph:<o:p></o:p> I think it̵= 7;s more than 60,000,000. It can’t be exactly 60,000,000 because the men and w= omen cannot increase equally and even out from an unequal starting point to a 1-= to-1 ratio. . . . Oh, no wait, I mean it’s less than 60,000,000. It must be 50,000,000. Yeah, I’m pretty sure that is what it is, because the wom= en population had to grow more than the men in order to equal out—so the= men must have grown less than 30,000,000. So the total must be less than 60,000= ,000 and the only answer like that is 50,000,000. <Cosi><o:p></o:p>= In thinking= about why Cosi could solve this problem in the group context but not alone, we noticed that she was not simply solving the problem as one would in isolati= on (e.g., setting up algebraic equations), but was interacting with the group effort. In particular, Dan, Mic and Hal had set up a certain way of thinking about the problem and of exploring possible solutions. Cosi was reflecting = on the group approach and repairing problems in its logic. The numbers, words = and considerations that she used were supplied by the context of on-going interactive activities and shared meanings.<o:p></o:p> If we combine the proposals from Mic, Dan, Hal and Cos= i, they read like the cognitive process of an individual problem solver: How can I figure out the increase in users without knowing= the total number of internet users? <Mic> It seems to all come from the 30,000,000 figure. <Dan> 30,000,000 is the number of increase in Amer= ican females. Since the ratio of male to female is 1 to 1, <Mic> the total= of male and female combined would be 60,000,000. <Hal> No, I think it mu= st be more than 60,000,000 because the male and female user populations can’t get higher at equal rates and still even out to a 1 to 1 ratio after starting uneven. No, I made a mistake, the total must be less than 60,000,000. It could be 50,000,000, which is the only multiple choice option less than 60,000,000. <Cosi> Very smart. <Dan> Clearly, Cosi made some contributions to the group tha= t were key to the group solution. They were acknowledged as such. Cosi was termed “very smart” — although this could equally well be said of the group as a whole. While no individual in the group could see how to sol= ve the problem, everyone contributed to exploring it in a way that rather effi= ciently led to a solution. In fact, Mic’s clowning around can be seen to be an extremely effective facilitation of the group process. By joking and laughi= ng a lot, the group relieved some of the pressure to solve a problem that was be= yond any individual’s reach and to open a social space in which ideas coul= d be put forward without fear of being harshly judged. Mathematical problem solving is a paradigm case of hum= an cognition. It is common to say of someone who can solve math problems that = he or she is smart. In fact, we see that taking place in the chat. Here, the g= roup has solved the problem by constructing an argument much like what an indivi= dual might construct. So we can attribute group cognition or intelligence to the group. Attributing the solution to the group rather than to the sum of the individuals in the group can be motivated by seeing that the construction of mathematical meaning in the solution process was done across individuals. T= hat is, meaning was created by means of interactions among individual contribut= ions (postings) to the chat — such as through what are called adjacency pairs in conversation an= alysis — more than by individual postings construed as expressing a series of personal mental representations. <h1>Math proposal adjacency pairs</h1> In an = early chat of the VMT project using AOL’s Instant Messanger, a popular chat environment, we observed a repeated pattern of interaction that we have sin= ce found to be common in math chats (Stahl, 2005c). Here is an excerpt from that chat (l= ine numbers added; handles anonymized):<o:p></o:p> 17. </span= >Avr (8:23:27 PM):= i think we have to figure out the h= eight by ourselves 18. = Avr (8:23:29 PM):= if possible 19. = pin (8:24:05 PM): <= /span>i know how 20. = pin (8:24:09 PM): <= /span>draw the altitude' 21. = Avr (8:24:09 PM):= how? 22. = Avr (8:24:15 PM):= right 23. = Sup (8:24:19 PM): proportions? 24. = Avr (8:24:19 PM):= this is frustrating 25. = Avr (8:24:22 PM):= I don't have enough paper In this log we see several examples of a three-step pa= ttern: <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l13 lev= el1 lfo19; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>a. = <![endif]>A proposal bid is made by Avr in lines 17 and 18 for the group to work on: “I think we have to ….” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l13 lev= el1 lfo19; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>b. </spa= n><![endif]>The bid is taken up by someone else (Pin in line 19) on behalf of the group: “I know how” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l13 lev= el1 lfo19; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>c. = <![endif]>There is an elaboration of the proposal by members of the group. The proposed wor= k is begun, often with a secondary proposal for the first sub-step, such as Pin’s new proposal bid in line 20. The third step initiate= s a repeat of the three-step process: <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l19 lev= el1 lfo21; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>a. = <![endif]>A proposal bid is made by Pin in line 20 for the group to work on: “Draw the altitude” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l19 lev= el1 lfo21; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>b. </spa= n><![endif]>An acceptance is made by someone else (Avr in line 22) on behalf of the group: “Right!” <p class=3DMsoNormal style=3D'margin-top:0in;margin-right:0in;margin-bottom= :0in; margin-left:.25in;margin-bottom:.0001pt;text-indent:-.25in;mso-list:l19 lev= el1 lfo21; tab-stops:list .25in left 1.0in 2.0in 3.0in 4.0in dotted 4.9in'><![if !supp= ortLists]>c. = <![endif]>There is an elaboration of the proposal by members of the group. The proposed wor= k is begun, often with a secondary proposal for the first sub-step, such as Sup’s new proposal bid in line 23. But here the pattern breaks = down. It is unclear to us as analysts what Sup’s proposal bid, “Proportions?” is proposing. Nor is it responded to by the other group members as a proposal. Avr’s lines 24 and 25 ignore it and seem= to be reporting on Avr’s efforts to work on the previous proposal to draw the altitude. Breakdown situations are often worth analyzing carefully, for they can expose in the breach practices that otherwise go unnoticed, taken = for granted in their smooth execution. Our analysis of Sup’s “failed proposalR= 21; helps to specify—by way of counter-example—the conditions that promote successful proposals in math chats: (a) a clear semantic and syntac= tic structure, (b) careful timing within the sequence of postings, (c) a firm interruption of any other flow of discussion, (d) the elicitation of a resp= onse, (e) the specification of work to be done and (f) a history of helpful contributions. In addition, there are other interaction characteristics and mathematical requirements. For instance, the level of mathematical backgrou= nd knowledge assumed in a proposal must be compatible with the expertise of the participants and the computational methods must correspond with their train= ing. We call the three-step pattern described above a math proposal adjacency pair (Stahl, 2005a). It seems to be a common interaction pattern in collaborative problem solving of mathematics in our chats. As we see in other chats, however, not all student groups adopt this method. We call this a form of “adjacency pair” in keeping with conversation analysis terminology (Duranti, 1998; Schegloff, 1991), even though in chat logs the two parts of the pair may not appear adjacent = due to the complexities of chat postings: e.g., line 22 responds to line 20, wi= th line 21 intervening as a delayed response to line 19. <h1>References and threading</h1> The mo= re we study chat logs, the more we see how interwoven the postings are with each other and with the holistic Gestalt of the interactional context that they form. There are many ways in which a posting can reference elements of its context. The importance of indexicality to creating shared meaning was stre= ssed by Garfinkel (1967). Vygotsky also noted the central role= of pointing for mediating intersubjectivity in his analysis of the genesis of = the infant-and-mother’s pointing gesture (1930/1978, p. 56). Our analysis of face-to-face collabo= ration emphasized that spoken utterances in collaborative settings tend to be elliptical, indexical and projective ways of referencing previous utterance= s, the conversational context and anticipated responses (Stahl, 2006, chapter 12).<o:p></o:p> We have rec= ently developed VMT-Chat, a chat environment that not only includes a shared whiteboard, but has functionality for referencing areas of the whiteboard from chat postings and for referencing previous postings (see figure 1). The shared whiteboard= is necessary for supporting most geometry problems. (This will save Avr the frustration of running out of paper, and also let Pin and Sup see what she = is drawing and add to it or reference it.) Sharing drawings is not enough; students must be able to reference specific objects or areas in the drawing. (Sup could have pointed to elements of the triangles that he felt to be significantly proportional.) The whiteboard also provides opportunities to = post text where it will not scroll away. (Sup could have put his failed proposal= in a text box in the whiteboard, where he or the others could come back to it later.) The graphical references (see the blue line from a selected posting= to an area of the drawing) can also be used to reference one or more previous postings from a new posting, in order to make the threads of responses clea= rer in the midst of “chat confusion” (Pimentel, Fuks, & Lucena, 2005)</= span>.<o:p></o:p> In one of o= ur first chats using VMT-Chat, the students engaged in a particularly complex interaction of referencing a figure in the whiteboard whose mathematics they wanted to explore (Stahl, Wessner et al., 2006). Here is the chat log from figure 1 (graphical references to the whiteboard are indicated by “[REF= TO WB]” in the log.):<o:p></o= :p> 1 ImH:<= span style=3D'mso-tab-count:1'> what is the area = of this shape? [REF TO WB] 2 &= nbsp; Jas: which shape? 3 &= nbsp; ImH: woops 4 &= nbsp; Imh: ahh! 5 &= nbsp; Jas: kinda= like this one? [REF TO WB] 6 &= nbsp; Jas: the o= ne highlighted in black and dark red? 7 &= nbsp; ImH: between th stairs and the hypotenuse 8 &= nbsp; Jas: oh 9 &= nbsp; Jas: that = would be a tricky problem, each little “sector” is different 10 = Jas: this section [REF TO WB] 11 = ImH: perimeter is 12ro= ot3 12 = Jas: is sm= aller than this section [REF TO WB] 13 = ImH: assume those line= s are on the blocks 14 = Jas: the staircase lines? 15 = ImH: yea 16 Jas: they already are on the blocks Line 1 of t= he chat textually references an abstract characteristic of a complex form in the whiteboard: “the area of this shape.” The software function to support this refere= nce failed, presumably because the student, ImH, was not experienced in using i= t and did not cause the graphical reference line to point to anything in the draw= ing. Line 5 provides a demo of how to use the referencing tool. Using the tool’s line, a definite textual reference (“the one”) and the use of line color and thickness in the = drawing, lines 5 and 6 propose an area to act as the topic <![if !vml]><img width=3D623 height=3D421 src=3D"nail_files/image002.gif" align=3Dleft hspace=3D12 alt=3D"Text Box: Figure 1. Screen view of VMT-Chat with referenc= ing. Line 12 of the chat is selected. " v:shapes=3D"_x0000_s1038"><![endif]>of the chat. Line 7 makes explicit in text the definition of a sub-area of the proposed area. Line 8 accepts the new definition and line 9 starts to work = on the problem concerning this area. Line 9 references the problem as “that” and notes that it is tricky because the area defined does not consist of standard forms whose area would be easy to compute and add u= p. It refers to the non-uniform sub-areas as little “sectors”. Lin= e 10 then uses the referencing tool to highlight (roughly) one of these little sectors or “sections”. Line 12 continues line 10, but is interrupted in the chat log by line 11, a failed proposal bid by ImH. The c= hat excerpt continues to reference particular line segments using deictic prono= uns and articles as well as a growing vocabulary of mathematical objects of concern: sectors, sections, lines, blocks. <o:p></o:p> Progress is= made slowly in the collaborative exploration of mathematical relationships, but having a shared drawing helps considerably. The students use multiple textual and graphical means to reach a shared understanding of mathematical objects that they find interesting but hard to define. In this excerpt, we start to get a sense of the complex ways in which brief textual postings weave dense webs = of relationships among each other and with other elements of the collaborative context. <o:p></o:p> <h1>Constructing the collaborative experience</h1> Our go= al in the VMT Project is to provide a service to students that will allow them to have a rewarding experience collaborating with their peers in online discussions of mathematics. We can never know exactly what kind of subjecti= ve experience they had, let alone predict how they will experience life under conditions that we design for them. Our primary access to information relat= ed to their group experiences comes from our chat logs. The logs capture most = of what student members see of their group on their computer screens. We can e= ven replay the logs so that we see how they unfolded sequentially in time. Of course, we are not engaged in the interaction the way the participants were= and recorded experiences never quite live up to the live version because the engagement is missing. We do test out the environments ourselves and enjoy = the experience, but we experience math and collaboration differently than do middle-school students. We also interview students and their teachers, but teenagers rarely reveal much of their life to adults.<o:p></o:p> <![if !vml]><img width=3D219 height=3D503 src=3D"nail_files/image004.gif" align=3Dright hspace=3D12 alt=3D"Text Box: Figure 2. Threading of adjacency pairs= and other uptake. " v:shapes=3D"_x0000_s1037"><![endif]>So we try to understand how collaborative experiences are structured as interp= ersonal interactions. The focus is not on the individuals as subjective minds, but = on the human, social group as constituted by the interactions that take place = in the group.<o:p></o:p> <h2>Replies, up-take, pairs and triplets</h2> Figure= 2 provides a diagram of the responses of postings in the chat discussed above involving Avr, Pin and Sup (Stahl, 2005b). The numbered posts from each partici= pant are placed in chronological order in a column for that participant. Math proposal adjacency pairs are indicated with red arrows and other kinds of responses are indicated with green arrows. Note that Sup’s failed proposal bid (line 23) is isolated. Most of the chat has coherence, flow or motion due to the fact that most postings are responses to previous message= s. This high level of responses is due to the fact that many postings elicit responses, the way that a greeting invariably calls forth another greeting = in response, or a question typically produces an answer. In a healthy conversation, most contributions by one participant are taken up by others. Conversationalists work hard to fit their offerings into the timing and evolving focus of the on-going interaction. In chat, the timing, rules and practices are different, but the importance of up-take remains. <o:p></o:p>= The fact th= at the group process and the cross-ties between people are central to collaborative experiences does not contradict the continuing importance of the individual= s. The representation of figure 2 uses columns to indicate the connections and implicit continuity within the sequence of contributions made by an individ= ual. We may project psychological characteristics onto the unity of an individual’s postings, attributing this unity to personal interests, personality, style, role, etc. Such attributions may change as the chat unfolds. The point is that the individual coherence of each participantR= 17;s contributions adds an important dimension of implicit connections among the postings.<o:p></o:p> Adjacency p= airs like math proposals, greetings and questionings provide important ties that= cut across the connections of individual continuities. They form the smallest u= nits of meaning precisely by binding together postings by different people. A proposal bid that is not taken up is not a meaningful proposal, but at best= a failed attempt at a proposal. A one-sided greeting that is not recognized by the other is not an effective greeting. An interrogative expression that do= es not call for a responce is no real questioning of another. These are all interactional moves whose meaning consists in a give-and-take between two o= r more people. When we hear something that we recognize as a proposal, a greeting = or a question, we feel required to attempt an appropriate response. We may ignore the proposal, snub the greeter or refuse to answer the question, but then o= ur silence is taken as a response of ignoring, snubbing or refusing—and = not simply a lack of response or up-take. <o:p></o:p> In fact, th= e way that a response is taken is also part of the interaction itself. In discuss= ing the building of “common ground,” Clark argues that shared under= standing by A and B of A’s utterance involves not only B believing that he understands A, but also A believing that B understands (Clark & Brennan, 1991). This requires an interaction spannin= g at least multiple utterances. Thus, for instance, the most prevalent interacti= on in classroom discourse is when a teacher poses a question, a student provid= es an answer demonstrating understanding and then the teacher acknowledges the student response as such an understanding (Lemke, 1990). Here, the elemental cell of interact= ional meaning making is a sequence of contributions by at least two different peo= ple. It is clear that the meaning is constructed through the interaction of mult= iple people, and is not a simple expression of pre-existing mental representatio= ns in any one individual’s head.<o:p></o:p> <h2>Longer sequences</h2> Althou= gh much attention has been given to adjacency pairs in conversation analysis and although such pairs can be thought of as the cells of meaning making in collaborative interaction, they form only one of many levels of analysis. F= or instance, there are longer sequences, episodes and topics in dialogs and ch= ats that provide layers of structure and sense (Linell, 2001; Zemel, Xhafa, & Cak= ir, 2005). An hour-long chat is not a homogeneous interchange. A typical math chat might start with a period of introductions, greetings, socializing. Th= en there could be some problem-solving work. This might be periodically interrupted by joking, playing around, or silliness. People may come and go, requiring catching up and reorganizing. Each of these episodes has boundari= es during which the group members must negotiate whether to stop what they were doing and start something else. These transitions may themselves be longer sequences of interaction, especially in large groups. We have barely begun = to explore these different layers. <o:p></o:p> The chat ex= cerpt from VMT-Chat above was from the second hour-long session in a series of fo= ur chats with the same groups. The sessions referred back to previous sessions= and prepared for future ones. We hope to foster a community of Math Forum users= who come back repeatedly to math chats, potentially with their friends. Their c= hats will reference other chats and other online experiences, building connectio= ns at the community level. This adds more layers of interconnections.<o:p></o:= p> <h2>Constructing proofs</h2> In our= chats, students work on math problems and themes. In solving problems and exploring math worlds or phenomena, the groups construct sequences of mathematical reasoning that come close to proofs. Proofs in mathematics have an interest= ing and subtle structure. One must distinguish: the problem situation; the exploratory search for the solution; the effort to reduce a haphazard solut= ion path to an elegant, formalized proof; the statement of the proof; and the l= ived experience of following the proof (Livingston, 1986, 1987). Each of these has its own structures= and practices. Each necessarily references the others. To engage in mathematics= is to become ensnarled in the intricate connections among them. To the extent = that these aspects of doing math have been distinguished and theorized, it has b= een done as though there is simply an individual mathematician at work. There h= as been virtually no research into how these could be accomplished and experie= nced collaboratively—despite the fact that talking about math has for some= time been seen as a priority in math education (NCTM, 1989).<o:p></o:p> <h2>The stream of group consciousness</h2> Psycho= logists like Williams James and novelists like Jack Kerouac have described narrativ= es that we tell ourselves silently about what we are doing or observing as our stream of consciousness. This “inner voice” rattles on even as = we sleep, making connections that Sigmund Freud found significant (if somewhat shocking in his day). In what sense might online chats—with their meanderings, flaming, associative referencing, unpredictable meaning making= and unexpected images—deserve equal status as streams of (group) consciou= sness? Group cognition can be self-conscious.<o:p></o:p> Our sense o= f time and the rhythms of life are largely reliant upon the narratives we tell ourselves (Trausan-Matu, Stahl, & Sarmiento,= in preparation). We know that we have already lived t= hrough a certain part of the day or of our life because we place the present in the nexus of its ties to our memories of the past or our hopes for the future. = In similar ways, the web of references in a chat that connects postings to pri= or postings to which they respond and to future postings that they elicit defi= nes a temporality of the chat. This is a lived sense of time that is shared by = the group in the chat. Like our individual internal clocks, the group temporali= ty must be attuned to the larger world outside—the world of family life = that calls the students away from the chat for dinner or the world of school that interrupts a chat with class changes or homework pressures. The temporality that is constructed as a dimension of the collaborative experience is constrained by nature of the social situation and technological environment= .<o:p></o:p> <h2>Group cognition in chat</h2> The fa= ct that meaning is created at the group unit of analysis rather than by particular individuals suggests the notion of group cognition (Stahl, 2006). The traditional view of cognition, particularly in Western philosophy since Descartes, is that meaning, ideas = and thoughts are created in individual minds. Theories in cognitive science formulated this in terms of mental representations in the heads of individuals—an approach that has been critiqued by more recent theori= es of situated and distributed cognition. The mental contents of individual cognition—in the traditional view—can subsequently be expressed= in language and communicated through the external world, to be then interprete= d in the minds of other individuals. Meaning, in this view, exists only in individual minds, and cognition is always personal.<o:p></o:p> Whether or = not one accepts some version of the cognitivist view in general, it seems that in situations of collaboration notions of shared meaning and group cognition a= re useful and important. Here, “shared meaning” has a deeper significance than what seems to underlie Clark’s analysis of common ground, where sharing is reduced to coordination among the individual mental contents of several minds (Clark & Brennan, 1991). Shared meaning is constructed across= pairs or triplets of postings by more than one participant. It is not that an ans= wer to a question implies that the answerer has in mind the same thing as the questioner, but that the answer and the question by themselves are fragment= ary; they have meaning only as part of the question-answer interaction. The unit= of meaning is the interaction itself, and this is a group phenomenon not an individual one. Moreover, with adequate capture of collaborative interactio= ns, it is possible to see the construction of meaning in the traces of interact= ion; it is not necessary to hypothesize about hidden mental operations or conten= ts.<o:p></o:p> Of course, = in some sense it is easier to visualize individuals than groups as cognitive agents= . As Vygotsky’s analysis of the infant’s gesture shows, we are used = to identifying other individuals as meaning-expressing agents. Given our perceptual orientation to a primarily visual world (Merleau-Ponty, 1945/2002), it is more natural for us to assign = agency to physical objects like human bodies than to more abstract entities like online groups. What, we may well wonder, is a collaborative group?<o:p></o:p> Groups cons= titute themselves. We can see how they do this in the chat logs. At one level the = Math Forum service brings several students together and locates them in a chat r= oom together. It may supply a math problem for them to work on and it may provi= de a facilitator who introduces them to the environment. At this point, they are= a potential group with a provisionally defined membership. The facilitator mi= ght say something like, “Welcome to our first session of Virtual M= ath Teams! I am the facilitator for your session. . . . As a group, decide which question you would like to work on.” (This is part of the facilitator script from the session involving ImH and = Jas excerpted above.) Here we can see that the facilitator has defined the group (“as a group … you”) and distinguished her = own role as outside the group (“I= am the facilitator … your session”). The potential group projected by the facilitator need not necessarily materialize. Individual students many come to the setting, look around, decide it is lame, and leave as individuals. However, this rarely happens. Sometimes an individual will leave without ever interacting, but as long as there are enough students there, a group will emerge.<o:p></o:p></s= pan> Students co= me to the chat environment with certain motivations, expectations and experiences. These are generally sufficient to get the group started. One can see the gr= oup form itself. This is often reflected in the shift from singular to plural pronouns: “Let’s get started. Let us do some math.” We saw this in Avr’s proposal:</s= pan> “I= think we have to figure out the height b= y ourselves.” The proposal bid= comes from the individual, but the projected work is for the group. Through her u= se of “we,” Avr constitutes the group. Through her proposal bid, s= he constitutes the group as a recipient of the bid and elicits a response from= them. Someone other than Avr must respond to the bid on behalf of the group. When= Pin says, “I know how: draw the altitude,” he is accepting Avr̵= 7;s proposal as a task for the group to work on and in so doing he makes a prop= osal about how the group should go about approaching this task (by making a geometric construction). In this interchange, the group (a) is projected as= an agent in the math work (Lerner, 1993), and (b) is actually the agent of mea= ning making because the meaning of Avr’s proposal is defined by the interaction within the group. <o:p></o:p> If the group experience is a positive one for the participants, they may want to return. Some chats end with people making plans to get together again. In some experiments, the same groups attended multiple sessions. We would like to s= ee a community of users form, with teams re-forming repeatedly and with old-time= rs helping new groups to form and learn how to collaborate effectively.<o:p></= o:p> The recogni= tion that collaborative groups constitute themselves interactionally and that th= eir sense making takes place at the group unit of analysis has fundamental methodological implications for the study of collaboration. The field of computer-supported collaborative learning (CSCL) was founded a decade ago to pursue the analysis of group meaning making (Stahl, Koschmann, & Suthers, 2006= ). We view the research described here = as a contribution to the CSCL tradition.<o:p></o:p> <h1>How groups construct their experience</h1> We are designers. Our goal is to design an exciting mathematical group experience = for students. We want to design an online collaborative service, with strong pedagogical direction and effective computer support. We approach this by trying to understand how groups of students construct their experience in s= uch settings. Because we are designing a computer-supported experience that has never bef= ore existed and because we want our design to be based on detailed study of how students actually created their collaborative experience in the environment= we are designing, we follow a highly iterative try-analyze-redesign cycle.<o:p= ></o:p> When studen= ts enter our website, they are confronted by a densely designed environment. The lob= by to our chat rooms is configured to help students find their way to a room t= hat will meet their needs. In the room, there is a bewildering array of software functionality for posting and displaying chat notes, drawing geometric forms and annotating them, keeping track of w= ho is doing what and configuring the space to suit oneself. There may be a sta= tement of a math problem to solve or an imaginary world to explore mathematically.= The service, problems and software are all designed to enhance the user’s experience. But how can a student who is new to all this understand the meanings of the many features and affordances that have been built into the environment?<o:p></o:p> Groups of s= tudents spontaneously develop methods for exploring and responding to their environments. They try things out and discuss what happens. A new group may doodle on the whiteboard and then joke about the results. They bring with t= hem knowledge of paint and draw programs and skills from video games, SMS and I= M. The individuals may have considerable experience with single-user apps, but react when someone else erases their drawing; they must learn to integrate coordination and communication into their actions. The math problems they f= ind in the chat rooms may be quite different than the drill-and-practice proble= ms they are used to in traditional math textbooks. It may take the group a whi= le to get started in productive problem solving, so the group has to find ways= to keep the group together and interacting in the meantime. There may be vario= us forms of socializing, interspersed with attempts to approach the math. As unaccustomed as the math may be, the students always have some knowledge and experience that they can bring to bear. They may apply numerical computatio= ns to given values; try to define unknowns and set up equations; graph relationships; put successive cases in a table; use trigonometric relations= hips or geometric figures; draw graphical representations or add lines to an existing drawing. Mainly, they put proposals out in the chat stream and res= pond to them. Sometimes the flow of ideas wanders without strong mathematical reflection. Other times, one individual can contribute substantial progress= and engage in expository narrative to share her contribution with the group.<o:= p></o:p> Groupware i= s never used the way its designers anticipated. The designers of VMT-Chat thought t= hat its referencing tools would immediately clarify references to elements of drawings and transform chat confusion into logical threaded chat. But our studies of the actual use of these designed functions tell a quite different and more interesting story. The shared whiteboard with graphical references from the chat may allow more complex issues to be discussed, but they do not make pointing problem free. We saw above how much work ImH and Jas engaged = in to clarify for each other what they wanted to focus on. In the excerpt and = in the longer chat, they used a variety of textual, drawing and referencing methods. In the process, they learned how to use these methods and they tau= ght each other their use. In a matter of a fraction of a minute, they were able= to reach a shared understanding of a topic to work on mathematically. In that brief time, they used dozens of indexical methods, some that would prove mo= re useful than others for the future.<o:p></o:p> Chat is a h= ighly constrained medium. Participants feel various pressures to get their indivi= dual points of view out there. In a system like VMT-Chat, there is a lot to keep track of: new postings, changes to the whiteboard, signs that people are joining, leaving, typing, drawing. Small details in how something is writte= n, drawn or referenced may have manifold implications through references to present, past or future circumstances. Students learn to track these detail= s; apply them creatively; acknowledge to the group that they have been recogni= zed; check, critique and repair them. Each group responds to the environment in = its own way, giving group meaning to the features of the collaborative world and thereby putting their unique stamp on their group experience.<o:p></o:p></s= pan> In the proc= ess, they create a group experience that they share. This experience is held together with myriad sorts of references and ties among the chat postings a= nd drawings. Often, what is not said is as significant as what is. Individual postings are fragmentary, wildly ambiguous, and frequently confusing. In li= vely chats, much of what happens remains confusing for most participants. Clarity comes only through explicit reflections, up-takes, appreciations, or probin= g. The interactions among postings, at many levels, coheres into a stream of g= roup consciousness, a flow of collaboration, a shared lived temporality and, with luck, an experience of mathematical group cognition.<o:p></o:p> As we have = seen in this paper, when students enter into one of our chats they enter into a com= plex social world. They typically quickly constitute a working group and begin to engage in activities that configure a group experience. This experience is conditioned by a social, cultural, technological and pedagogical environment that has been designed for them. Within this environment, they adopt, adapt= and create methods of social practice for interacting together with the other students who they find in the chat environment. Over time, they explore the= ir situation together, create shared meaning, decide what they will do and how they will behave, engage in some form of mathematical discourse, socialize = and eventually decide to end their session. Then our job begins: to analyze what has happened and how the environment we are designing conditions the collaborative experiences that groups construct there.<o:p></o:p> <h1>Workshop Issues</h1> <h2>How does one transform recordings of interaction into analytic accounts?<o:p></= o:p></h2> This p= aper provides a case study of how a group of researchers, technology designers a= nd educational service providers collaboratively analyze logs of students interacting in an evolving chat environment. We try to describe the methods that chat groups spontaneously develop to make sense of their designed situation and to establish social order for their group interaction as it unfolds locally within that activity structure.<o:p></o:p> <h2>What is the relation between studies of interaction and theories of learning?<o:= p></o:p></h2> We vie= w the term learning with suspicion. I= t is an accountable member’s matter, a child’s concern in responding= to the parental inquiry, “What did you learn in school today?” The psychological theory of changes in mental representations is particularly suspect when applied to collaborative learning. With luck, groups interact = and make meaning. Learning for groups consists in having more methods available= as interactional resources and being equipped with more developed meanings. Through rigorous studies of interaction, we can observe elemental acts of meaning making and the development of member methods and other social pract= ices.<o:p></o:p> <h2>How does one make the analyses relevant for curricular design?<o:p></o:p></span= ></h2> Our cu= rricular goals are to foster enjoyable experiences of discussion of mathematical the= mes in group discourse – making sense of mathematical objects and constructing shared mathematical meanings. We are not interested in the transfer or memorization of facts, but in the development by students of th= e ability to use the resources of a fledgling mathematician, to be able to “do” school math. An ultimate goal might be that a group of students experiences the wonder of understanding an elegant mathematical re= lationship or proof. But we are far from such goals. For now, we need to understand how groups interact in various chat settings and we need to clear away some of = the overwhelming barriers that prevent groups from working together better.<o:p= ></o:p> The worksho= p question is itself problematic. The term “curricular design” presupposes= a theory of learning and instruction. It presupposes categories of student, instructor, pedagogical designer, interaction analyst. In our work, we bring together a group of people who want to work on the VMT service/research project. As individuals, we have a variety of skills, motivations, trainings and interests. But as a group, we set group goals and perspectives, such as trying out a design concept or analyzing an excerpt from a trial chat. What= we learn in one activity informs what we do in another. Although we use the te= rm “student” for our target user, our service is designed for open access on the Web and will be available for school drop-outs, plumbers, professional mathematicians, soccer moms and ethnomethodologists – however they define their own status. <o:p></o:p> <h1>Acknowledgments</h1> This paper is an attempt to reflect the thrust of research being conducted in a multi-disciplinary project, informed by studies conducted us= ing several methodologies and theories derived from various traditions. The argument and presentation of the paper are those of the author. Many details and formulations in the paper would be put differently by other members of = the research group; the paper does not necessarily represent their individual or professional views accurately. For more detailed discussions, see the cited papers.<o:p></o:p> The Virtual= Math Teams Project is a collaborative effort at <st1:place w:st=3D"on"><st1:Plac= eName w:st=3D"on">Drexel</st1:PlaceName> <st1:PlaceType w:st=3D"on">University</= st1:PlaceType></st1:place>. The Principal Investigators are Gerry Stahl, Stephen Weimar and Wesley Shum= ar. A number of Math Forum staff work on the project, especially Stephen Weimar, Annie Fetter and Ian Underwood. The graduate research assistants are Murat Cakir, Johann Sarmiento, Ramon Toledo and Nan Zhou. Alan Zemel is the post-= doc; he facilitates weekly conversation analysis data sessions. The following visiting researchers have spent 3 to 6 months on the project: Jan-Willem Strijbos (<st1:country-region w:st=3D"on">Netherlands</st1:country-region>), Fatos Xhafa (<st1:country-region w:st=3D"on">Spain</st1:country-region>), S= tefan Trausan-Matu (<st1:country-region w:st=3D"on">Romania</st1:country-region>), Martin Wessner (<st1:country-region w:st=3D"on">Germany</st1:country-region= >), Elizabeth Charles (<st1:place w:st=3D"on"><st1:country-region w:st=3D"on">C= anada</st1:country-region></st1:place>). The VMT-Chat software was developed at the Fraunhofer Institute IPSI in <st= 1:place w:st=3D"on"><st1:City w:st=3D"on">Darmstadt</st1:City>, <st1:country-region= w:st=3D"on">Germany</st1:country-region></st1:place>, by Martin Wessner, Martin Mühlpfordt and colleagues based on their ConcertChat. 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