Sustaining online collaborative problem solving with math proposals<= /title> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"country-region"/> <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"PlaceName"/> <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"time"/>  <style>  </style>  </head> <body lang=3DEN-US link=3Dblue vlink=3Dpurple style=3D'tab-interval:.5in'> <div class=3DSection1> BEST PAPER AWARD at the International Conference on Computers in Education (ICCE 2005), <st1:country-region w:st=3D"on"><st1:place w:st=3D"on">Singapore</st= 1:place></st1:country-region>, December 2005 <o:p> </o:p> <h1 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>Sustaining Online = Collaborative Problem Solving with Math Proposals<o:p></o:p></h1> <o:p> </o:p> <o:p> </o:p> Gerry STAHL<o:p></o:p> <st1:place w:st=3D"on"><st1:PlaceName w:st=3D"on">Drexel</st1:PlaceName> <st1:PlaceType w:st=3D"on">Universit= y</st1:PlaceType></st1:place><o:p></o:p> <o:p> </o:p> <o:p> </o:p> Abst= ract. Learning takes place over long periods of time= that are hard to study directly. Even the learning experience involved in solvin= g a challenging math problem in a collaborative online setting can be spread ac= ross hundreds of utterances during an hour or more. Such long-term interactions = are constructed out of utterance-level interactions, such as the strategic proposing of a next step. This paper identifies a pattern of exchange of utterances that it terms math propo= sal adjacency pair, and describes its characteristics. Drawing on the metho= dology of conversation analysis, the paper adapts this approach to mathematical problem-solving communication and to the computer-mediated circumstances of online chat. In particular, a failed proposal is contrasted with successful proposals in the log of an actual chat. Math proposal adjacency pairs constitute the collaborative group as a working group, give direction to th= eir problem solving and help to sustain their interaction. <o:p></o:p></= p> <o:p> </o:p> <o:p> </o:p> <h2 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>1. Doing mathematics together onlin= e<o:p></o:p></h2> <o:p> </o:p> Computers offer many opportunities for innovation= in education. One of the major avenues is by supporting the building of collab= orative knowledge [1]. For instance, it is now possible for students around the world to work together on challenging math problems. Through online discussion, they can share problem-solving experie= nces and gain fluency in communicating mathematically. In a research project at the Math Forum @ Drexel (<a href=3D"http://mathforum.org/">http://mathforum.org</a>), we have begun to = invite middle school students to participate in online chats about interesting problems in beginning algebra and geometry. The following problem, discusse= d in the example in this paper, is typical: If two equilateral triangles have edge-lengths of 9 cubits and 12 cubits, what is = the edge-length of the equilateral triangle whose area is equal to the sum of t= he areas of the other two? We rely on a variety of methods from the learning scie= nces to guide our research and to analyze the results of our trials. In particul= ar, we use conversation analysis <!= --[if supportFields]> ADDIN EN.CITE <EndNote><Cite><Author>Sacks</Author><Year>19= 74</Year><RecNum>526</RecNum><MDL><REFERENCE_TYP= E>0</REFERENCE_TYPE><REFNUM>526</REFNUM><AUTHORS>= ;<AUTHOR>Harvey Sacks</AUTHOR><AUTHOR>Emanuel A. Schegloff</AUTHOR><AU= THOR>Gail Jefferson</AUTHOR></AUTHORS><YEAR>1974</YEAR><TI= TLE>A Simplest Systematics for the Organization of Turn-Taking for Conversation</TITLE><SECONDARY_TITLE>Language</SECONDARY_TIT= LE><VOLUME>50</VOLUME><NUMBER>4</NUMBER><PAGE= S>696-735</PAGES><DATE>December 1974</DATE><URL>www.jstor.org</URL></MDL></Cite&= gt;<Cite><Author>Pomerantz</Author><Year>1991</Y= ear><RecNum>521</RecNum><MDL><REFERENCE_TYPE>7&l= t;/REFERENCE_TYPE><REFNUM>521</REFNUM><AUTHORS><AUT= HOR>Pomerantz, Anita</AUTHOR><AUTHOR>Fehr, B.J.</AUTHOR></AUTHORS>= <YEAR>1991</YEAR><TITLE>Conversation Analysis: An Approach to the Study of Social Action as Sense Making Practices</TITLE><SECONDARY_AUTHORS><SECONDARY_AUTHOR>van Dijk, Teun A.</SECONDARY_AUTHOR></SECONDARY_AUTHORS><SECONDARY_TITLE>= ;Discourse as Social Interaction: Discourse Studies, A Multidisciplinary Introduction, Volume 2</SECONDARY_TITLE><PLACE_PUBLISHED>London, UK</PLACE_PUBLISHED><PUBLISHER>Sage</PUBLISHER><PAGES&= gt;64-91</PAGES></MDL></Cite><Cite><Author>Ps= athas</Author><Year>1995</Year><RecNum>469</RecN= um><MDL><REFERENCE_TYPE>1</REFERENCE_TYPE><REFNUM&g= t;469</REFNUM><AUTHORS><AUTHOR>Psathas, George</AUTHOR></AUTHORS><YEAR>1995</YEAR><TITLE= >Conversation Analysis: The Study of Talk-in-Interaction</TITLE><PLACE_PUBLISHED>Thousand Oaks, CA</PLACE_PUBLISHED><PUBLISHER>Sage</PUBLISHER></MDL&g= t;</Cite><Cite><Author>ten Have</Author><Year>1999</Year><RecNum>471</RecNu= m><MDL><REFERENCE_TYPE>1</REFERENCE_TYPE><REFNUM>= ;471</REFNUM><AUTHORS><AUTHOR>ten Have, Paul</AUTHOR></AUTHORS><YEAR>1999</YEAR><T= ITLE>Doing Conversation Analysis: A Practical Guide</TITLE><PLACE_PUBLISHED>Thousand Oaks, CA</PLACE_PUBLI= SHED><PUBLISHER>Sage</PUBLISHER></MDL></Cite><= ;Cite><Author>Sacks</Author><Year>1992</Year><= ;RecNum>246</RecNum><MDL><REFERENCE_TYPE>1</REFEREN= CE_TYPE><REFNUM>246</REFNUM><AUTHORS><AUTHOR>Har= vey Sacks</AUTHOR></AUTHORS><YEAR>1992</YEAR><TITLE&= gt;Lectures on Conversation</TITLE><SECONDARY_AUTHORS><SECONDARY_AUTHOR>= G. Jefferson</SECONDARY_AUTHOR></SECONDARY_AUTHORS><PLACE_PUBLI= SHED>Oxford, UK</PLACE_PUBLISHED><PUBLISHER>Blackwell</PUBLISHER><N= UMBER_OF_VOLUMES>2</NUMBER_OF_VOLUMES></MDL></Cite><= ;/EndNote><![endif]-->[2; 3; 4; 5; 6]<!-= -[if supportFields]><![endif]--> to interpret the intera= ctions that take place in the student chats. In this paper, we adapt the findings = of conversation analysis to math chats and develop a specific form of adjacency pairs that seem to be important for math chats. Before presenting this, it = may be useful to describe briefly how the notion of adjacency pairs differs from naïve conceptions of conversation. There is a widespread common-sense or folk-theory [7; 8] view of conversation as= the exchange of propositions [9]. This view was refined = and formalized by logicians and cognitive scientists as involving verbal expres= sion in meaningful statements by individuals, based on their internal mental representations. Speech served to transfer meanings from the mind of a spea= ker to the mind of a listener, who then interpreted the expressed message. Following Wittgenstein [10] in critiquing this view, speech act theory [11; 12] argued that the utteran= ces spoken by individuals were ways of acting in the world, and were meaningful= in terms of what they accomplished through their use and effects. Of course, t= he expression, transmission and interpretation of meaning by individuals can be problematic, and people frequently have to do some interactional work in or= der to re-establish a shared understanding. The construction of common ground h= as been seen as the attempt to coordinate agreement between individual understandings [13]. Conversation analysis takes a different view of conversation. It looks at how interactional mechanisms, like the use of adjacency pairs [14; 15], co-construct inter-subjectivity. Adjacency pairs are common sequences of utterances by different people—such as mutual greetings or question/answer interchanges—that form a meaningful speech act spanning multiple utterances that cannot be attributed to an individual or to the expression = of mental states. We are interested in what kinds of adjacency pairs are typic= al for math chats. Online math chats differ from ordinary informal conver= sation in a number of ways. They are focused on the task of solving a specific pro= blem and they take place within a somewhat formal institutional setting. They involve the doing of mathematic= s [16]. And, of course, they a= re computer-mediated rather than being face-to-face. The approach of conversat= ion analysis is based on ethnomethodology [17], which involves the stu= dy of the methods that people use to accomplish what they are doing. So, we are interested in working out the methods that are used by students in online m= ath chats. In this paper we discuss a particular method of collaboration in math chats that we have elsewhere called exploratory participation: participants engage each other in the conjoint discovery= and production of both the problem and possible solutions [18]. The medium of online chat has its own peculiarities. M= ost importantly, it is a text-based medium, where interaction takes place by the sequential response of brief texts to each other [19; 20]. As a quasi-synchronous medium [21], chat causes confusion because several people can be typing at once and their texts can appear in = an order that obscures what they are responding to. Furthermore, under time pr= essure to submit their texts so that they will appear near what they are responding to, some chat participants break their messages into several short texts. Because of these peculiarities of chat, it is necessary for researchers to carefully reconstruct the intended threading of texts that respond to each other before attempting to interpret the flow of interaction [22; 23]. <o:p> </o:p> <o:p> </o:p> 2. Math proposal adj= acency pairs <o:p> </o:p> In order to begin to analyze the methods that stu= dents use in math chats, we take a close look at an excerpt from an actual chat. Figure 1 shows an excerpt from near the beginning of the log of one of our first online collaborative math problem-solving sessions. Three students—named Avr, Sup and Pin—hav= e just entered the chat room, said hello to each other and read the problem involv= ing three triangles. The first thing to notice here is a pattern of proposa= ls, discussions and acceptances similar to what takes place in face-to-face discourse. Proposals about steps in solving the math problem are made by <s= pan class=3DSpellE>Avr in lines 1, 3, 8, 17 and by Pin in lines 20, 27. = These proposals are each affirmed by someone else in lines 2, 6, 10, 19, 22, 28, respectively. To avoid chat confusion, note that line 21 responds to= line 19, while line 22 responds to line 20. The timestamps show that lines 20 an= d 21 effectively overlapped each other chronologically: Avr= was typing line 21 before she saw line 20. Similarly, lines 24 and the following were responses to line 20, not line 23. We will correct for these confusions in Figure 2, which reproduces a key passage in this excerpt. <![if !vml]><img width=3D589 height=3D661 src=3D"icce_files/image001.gif" align=3Dleft hspace=3D12 alt=3D"Text Box: 1. Avr (8:21:46 PM): Okay, I think we should start with th= e formula for the area of a triangle 2. Sup (8:22:17 PM): ok &= #10;3. Avr (8:22:28 PM): A =3D 1/2bh 4. Avr (8:22:31 PM): I believ= e 5. pin (8:22:35 PM): yes 6. pin (8:22:37 PM): i concue&= #13; 7. pin (8:22:39 PM): concur* 8. Avr (8:22:42 PM): then fi= nd the area of each triangle 9. Avr (8:22:54 PM): oh, wait = 0;10. Sup (8:23:03 PM): the base and heigth are 9 and 12 right? 11= . Avr (8:23:11 PM): no 12. Sup (8:23:16 PM): o 13. Avr (8= :23:16 PM): that's two separate triangles 14. Sup (8:23:19 PM): oo= o 15. Sup (8:23:20 PM): ok 16. Avr (8:23:21 PM): right= 3; 17. Avr (8:23:27 PM): i think we have to figure out the height by ou= rselves 18. Avr (8:23:29 PM): if possible 19. pin (8:24:0= 5 PM): i know how 20. pin (8:24:09 PM): draw the altitude' = 0;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 f= rustrating 25. Avr (8:24:22 PM): I don't have enough paper = 0;26. pin (8:24:43 PM): i think i got it 27. pin (8:24:54 PM): its= a 30/60/90 triangle 28. Avr (8:25:06 PM): I see 29. pin = (8:25:12 PM): so whats the formula Figure 1. Excerpt of = 3½ minutes from a one-hour chat log. Three students chat about a geo= metry problem. Line numbers have been added and screen-names anonymized; ot= herwise the transcript is identical to what the participants saw on their s= creens. " v:shapes=3D"_x0000_s1026"><![endif]>In Figure 1, we see several examples of= a three step pattern: <![if !supp= ortLists]>1.&n= bsp; <![endif]>A proposal is made by an individual for the group to work on: “I think we should ….” <![if !supp= ortLists]>2.&n= bsp; <![endif]>An acceptance is made on behalf of the group: “Ok,” “right” <![if !supp= ortLists]>3.&n= bsp; <![endif]>There is an elaboration of the proposal by members = of the group. The proposed work is begun, often with a secondary proposal for = the first sub-step. This suggests that collabora= tive problem-solving of mathematics may often involve a particular form of adjac= ency pair. We will call this a math prop= osal adjacency pair. Many adjacency pairs allow for insertion of other pairs between the two parts of the original pair, delaying completion of the pair= . For instance, a question/answer pair may be interrupted by utterances seeking clarification of the question; the clarification interaction may itself con= sist of question/answer pairs, possibly with their own clarifications—this= may continue recursively. With math proposal adjacency pairs, the subsidiary pa= irs seem to come after the completion of the original pair, in the form of secondary proposals, questions or explanations that start to do the work th= at was proposed in the original pair. Proposals seem to lead to some kind of further mathema= tical work as a response to carrying out what was proposed. Often—as seen in the current example—that work consists of making further proposals. I= t is striking that the proposed work is not begun until there is agreement with = the proposal. This may represent consent by the group as a whole to pursue the proposed line of work. Of course, it is not so clear in the current example, where there are only three participants and the interaction often seems to = take place primarily between pairs of participants. As confirmed by other chat examples, however, the proposal generally seems to be addressed to the whole group and opens the floor for participants other than the proposer to respo= nd. The use of “we” in “we should” or “we have to” (stated or implied) constitutes the multiple participants as a pl= ural subject, an effective unified group [24]. Any one other than the proposer may respond on behalf of the group. Moreover, there seems to be what in conversation analy= sis is called an interactional preference [25] for acceptance of the proposal. That is, if one accepts a proposal, it suffices to briefly indica= te agreement: “ok.” If one wants to reject a proposal, then one ha= s to account for this response by giving reasons. We would like to characterize in more detail the metho= d of making math proposal adjacency pairs. Often, the nature of an interactional method= is seen most clearly when it is breached [17]. Methods are generally = taken for granted by people; they are not made visible or conducted consciously. = It is only when there is a breakdown</= i> [26] in the smooth, tacit performance of a method that people focus on its characteristics in order to overcome the breakdown. The normally transparent method becomes visible in = its breach. We can interpret Sup’s posting in= line 23 as a failed proposal. Given = the mathematics of the triangle problem, a proposal like S= up’s related to proportionality might have been fruitful. However, in this chat, = line 23 was effectively ignored by the group. While its character as a fail= ed proposal did not become visible to the participants, it can become clear to= us by comparing it to successful proposals in the same chat and by reflecting = on its situation in the chat in order to ask why it was not a successful propo= sal. <o:p> </o:p> <o:p> </o:p> <h2 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>3. A failed proposal<o:p></o:p></h2> <o:p> = </o:p> Let us look= at line 23 in its immediate interactional context in Figure 2. We can distinguish a number of ways in which it differed from successful math proposals that sol= icited responses and formed math proposal adjacency pairs. <![if !vml]><img width=3D589 height=3D197 src=3D"icce_files/image002.gif" align=3Dleft hspace=3D12 alt=3D"Text Box: 17, 18. Avr (8:23: 29 PM): i think we have to figure out t= he height by ourselves … if possible 19. pin (8:24:05 PM): i= know how 21. Avr (8:24:09 PM): how? 20. pin (8:24:09 PM)= : draw the altitude' 22. Avr (8:24:15 PM): right 24. Avr = (8:24:19 PM): this is frustrating 23. Sup (8:24:19 PM): proportion= s? Figure 2. Part of the chat log excerpt in figure 1, w= ith order revised for threading. " v:shapes=3D"_x0000_s1028"><![endif]>(a) All the other proposals (1, 3, 8, 1= 7, 20, 27) were stated in relatively complete sentences. Additionally, some of the proposals were introduced with a phrase to indica= te that they were the speaker’s proposal (1. “I think we should …,” 17. “I think we have to …,” 20. “<s= pan class=3DSpellE>i know how …” = and 27. “i think i got it …”). The exceptions to these were simply continuations of previous proposals: line 3 provided the formula proposed in line 1 and line 8 proposed to “then” use that formu= la. Line 23, by contrast, provided a single word with a question mark. There wa= s no syntactic context (other than the question mark) within the line for interpreting that word and there was no reference to semantic context outsi= de of the line. Line 23 did not respond in any clear way to a previous line and did not provide any alternative reference to a context in the original prob= lem statement or elsewhere. For instance, Sup could have said, “I think we should compute the proportion of the height to the base of those equilateral triangles.” (b) The timing of line 23 w= as particularly unfortunate. It exactly overlapped a line from Avr. Since Avr had been setting = the pace for group problem solving during this part of the chat, the fact that she w= as involved in following a different line of inquiry spelled death for an alternative proposal at the time of line 23. Pin either seemed to be continuing on his own thread without acknowledgi= ng anyone else at this point, or else he was responding too late to previous postings. So a part of the problem for Sup was that there was little sense = of a coherent group process—and what sense there was did not include him. = If he was acting as part of the group process, for instance posing a question = in reaction to Pin and in parallel to Avr, he was not doing= a good job of it and so his contribution was ignored in the group process. It= is true that a possible advantage of text-based interaction like chat over face-to-face interaction is that there may be a broader time window for responding to previous contributions. In face-to-face conversation, turn-ta= king rules may define appropriate turns for response that expire in a fraction o= f a second as the conversation moves on. In computer-based chat, the turn-taking sequence is more open. However, even here if one is responding to a posting that is several lines away, it is important to make explicit somehow to what one is responding. He could have said, “I = know another way to find the height – using proportions.” Sup’s posting does not do anything like that; it relies purely upon sequential timing to establish its context, and that fai= ls in this case. (c) Su= p’s posting 23 came right after Pin’s proposal 20: “draw the altitude.” Avr had responded to this with= 22 (“right”) but Pin seems to have ignored that. Pin’s proposal had opened up work= to be done and both Avr and Pin responded after li= ne 23 with contributions to this work. So Sup’s proposal came in the middle of an ongoing line of work without relating to = it. In conversational terms, he made a proposal when it was not time to make a proposal. It is like trying to take a conversational turn when there is not= a pause that creates a turn-taking opportunity. Now, it is possible—especially in chat—to introduce a new proposal at any time. However, to do so effectively, one must make a special effort to bring the on-going work to a temporary halt and to present one’s new propos= al as an alternative. Simply saying “proportions?” will not do it.= Sup could have said, “Instead of drawing the altitude, let’s use <s= pan class=3DSpellE>propostions to find it.” (d) To get a response to a proposal= , one must elicit at least an affirmation or recognition. Line 23 does not really solicit a response. For instance, Avr’s question, 21: “how?” called for an answer—that was given = by Pin in line 20, which actually appeared in the chat window just prior to the question and with the same time stamp. But Sup’s= posting does not call for a specific kind of answer. Even Sup’s own previous proposal in line 10 ended with “right?”—requiring agreement or disagreement. Line 10 el= icited a clear response from Avr, line 11(“no= 221;) followed by an exchange explaining why Sup’s</sp= an> proposal was not right. (e) Other proposals in the = excerpt are successful in contributing to the collaborative knowledge building or g= roup problem solving in that they open up a realm of work to be done. One can lo= ok at Avr’s successive proposals on lines 1,= 3, 8 and 17 as laying out a work strategy. This elicits a response from Sup tryi= ng to find values to substitute into the formula and from Pin trying to draw a graphical construction that will provide the values for the formula. Sup’s proposal in line 23, however, neither cal= ls for a response nor opens up a line of work. There is no request for a reaction = from the rest of the group and the proposal is simply ignored. Since no one responded to Sup, he could have continued by doing some work on the proposal himself. He could have come back and made the proposal more explicit, reformulated it more strongly, taken a first step in working on it, or pose= d a specific question related to it. But he did not—at least not until much later—and the matter was lost. (f) Another serious hurdle for Sup was his status in t= he group at this time. In lines 10 through 16, Sup had made a contribution that was taken as an indication that he did not have a strong grasp of the math problem. He offered the lengths of the two given triangles as the base and height of a single triangle (line 10). Avr immediately and flatly stated that he was wrong (line 11) and then proceede= d to explain why he was wrong (line 13). When he agreed (line 15), Avr summarily dismissed him (line 16) and went on to = make a new proposal that implied his approach was all wrong (lines 17 and 18). Then Pin, who had stayed out of the interchange, re-entered, claiming to know ho= w to implement Avr’s alternative proposal (lin= es19 and 20) and Avr confirmed that (line 22). Sup’s legitimacy as a source of useful proposal= s had been totally destroyed at precisely the point just before he made his ineffective proposal. Less than two minutes later, Sup tries again to make a contribution, but realizes himself that what he = says is wrong. His faulty contributions confirm repeatedly that he is a drag on = the group effort. He makes several more unhelpful comments later and then drops= out of the discourse for most of the remaining chat. The weaknesses of line 23 as a proposal suggest some characteristics for successful proposals: (a) a clear semantic and syntactic structure, (b) careful timing within the sequence of postings, (c) a firm interruption of any other flow of discussion, (d) the elicitation of a response, (e) the specification of work to be done and (f) a history of hel= pful 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 trai= ning. Other characteristics will become visible in other examples of chats. At this time, the notion of math proposal adjacency pa= irs is just a preliminary proposal based on a single chat log excerpt. It calls for extensive conversation analysis of a corpus of logs of collaborative online math problem solving to establish whether this is a fruitful way of interpreting the data. If it turns out to be a useful approach, then it wil= l be important to determine what interactional methods of producing such proposa= ls are effective (or not) in fostering successful knowledge building and group cognition. An understanding of these methods can guide the design of activi= ty structures for collaborative math. As we are collecting a corpus of chat lo= gs, we are evolving computer support through iterative trials and analyses. <o:p> </o:p> <o:p> </o:p> <h2 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>4. Designing computer support<o:p></o:p></h2> <o:p> </o:p> If the failure of Sup’s proposal about proportions is considered deleterious to the collaborative knowledge building around the triangles problem, then what are the implications of this for the design of education= al computer-based environments? One response would be to help students like Sup formulate stronger proposals. Presumably, giving him positive experiences of interacting with students like Avr and Pin, who= are more skilled in chat proposal making, would provide Sup with models and examples from which he can learn—assuming that he perseveres. Another approach to the problem would be to build functionality into the software and structures into the activity that scaff= old the ability of weak proposals to survive. As students like Sup experience success with their proposals, they may become more aware of what it takes to make a strong proposal. [16] Professional mathematicians rely heavily upon inscript= ion: the use of specialized notation, the inclusion of explicit statements of all deductive steps and the format of the formal proof to support the discussio= n of math proposals—whether on an informal whiteboard, a university blackb= oard or in an academic journal. Everything that is to be indexed in the discussi= on is labeled unambiguously. To avoid ellipsis, theorems are stated explicitly, with all conditions and dependencies named. The projection of what is to be proven is encapsulated in the form of the proof, which starts with the give= ns and concludes with what is proven. Perhaps most importantly, proposals for = how to proceed are listed in the proof itself as theorems, lemmas, etc.—organized sequentially. One could imagine a chat system supplemented with a wi= ndow containing an informal list of proposals analogous to the steps of a proof. After Sup’s proposal, the list might look= like Figure 3. When Sup made a proposal in the chat, he would enter a statement of it in the proof window in logical sequence. He could cross out his own proposal w= hen he felt it had been convincingly argued against by the group. <o:p> </o:p> <![if !vml]><img width=3D540 height=3D228 src=3D"icce_files/image003.gif" alt=3D"Text Box: Given: 2 equilateral triangles of edge-length 9 cubits and= 12 cubits formula for a triangle: A =3D 1/2bh = Area of each triangle =3D ? b, h =3D 9, 12 draw the alti= tude use proportions for ratio of altitude to base &= #10;Find: The edge-length of the equilateral triangle whose area is equal t= o the sum of the areas of the other two triangles Figure= 3. A list of proposals " v:shapes=3D"_x0000_s1029"><![endif]> <o:p> </o:p> The idea is that important proposals that were made wo= uld be retained in a visible way and be shared by the group. Of course, there are = many design questions and options for doing something like this. Above all, would students understand this functionality and would they use it? The design indicated in Figure 3 is only meant to be suggestive. Another useful tool for group mathematics would be a s= hared drawing area. In the chat environment used by Sup, Pin and Avr, there was no shared drawing, but a student could create a drawing and send = it to the others. Pin did this twelve minutes after= the part of the interaction shown in the excerpt. Before the drawing was shared, much time was lost due to confusion about references to triangles and verti= ces. For math problems involving geometric figures, it is clearly important to be able to share drawings easily and quickly. Again, there are many design iss= ues, such as how to keep track of who drew what, who is allowed to erase, how to point to items in the drawing and how to capture a record of the graphical interactions in coordination with the text chatting. <o:p> </o:p> <o:p> </o:p> References <o:p> </o:p> <![if !supportLists]>[= 1] </sp= an><![endif]>Stahl, G. (2006) Group cognition: Computer support for building collaborative knowle= dge, MIT Press, Cambridge, MA. Available at: <a href=3D"http://www.cis.drexel.edu/faculty/gerry/mit/">http://www.cis.drexel= .edu/faculty/gerry/mit/</a>. <![if !supportLists]>[= 2] </sp= an><![endif]>Sacks, H., Schegloff, E. A., & Jefferson, G. (1974= ) A simplest systematics for the organization of turn-taking for conversation, Langu= age, 50 (4), pp. 696-735. Available at: <a href=3D"http://www.jstor.org/">www.jstor.org</a>. <![if !supportLists]>[= 3] </sp= an><![endif]>Pomerantz, A. & Fehr,= B. J. 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MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C5FA3C.698D5920" 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_01C5FA3C.698D5920 Content-Location: file:///C:/268618C5/icce.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sustaining online collaborative problem solving with math proposals<= /title> <o:SmartTagType namespaceuri=3D"urn:schemas-microsoft-com:office:smarttags" name=3D"country-region"/> <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"PlaceName"/> <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"time"/>  <style>  </style>  </head> <body lang=3DEN-US link=3Dblue vlink=3Dpurple style=3D'tab-interval:.5in'> <div class=3DSection1> BEST PAPER AWARD at the International Conference on Computers in Education (ICCE 2005), <st1:country-region w:st=3D"on"><st1:place w:st=3D"on">Singapore</st= 1:place></st1:country-region>, December 2005 <o:p> </o:p> <h1 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>Sustaining Online = Collaborative Problem Solving with Math Proposals<o:p></o:p></h1> <o:p> </o:p> <o:p> </o:p> Gerry STAHL<o:p></o:p> <st1:place w:st=3D"on"><st1:PlaceName w:st=3D"on">Drexel</st1:PlaceName> <st1:PlaceType w:st=3D"on">Universit= y</st1:PlaceType></st1:place><o:p></o:p> <o:p> </o:p> <o:p> </o:p> Abst= ract. Learning takes place over long periods of time= that are hard to study directly. Even the learning experience involved in solvin= g a challenging math problem in a collaborative online setting can be spread ac= ross hundreds of utterances during an hour or more. Such long-term interactions = are constructed out of utterance-level interactions, such as the strategic proposing of a next step. This paper identifies a pattern of exchange of utterances that it terms math propo= sal adjacency pair, and describes its characteristics. Drawing on the metho= dology of conversation analysis, the paper adapts this approach to mathematical problem-solving communication and to the computer-mediated circumstances of online chat. In particular, a failed proposal is contrasted with successful proposals in the log of an actual chat. Math proposal adjacency pairs constitute the collaborative group as a working group, give direction to th= eir problem solving and help to sustain their interaction. <o:p></o:p></= p> <o:p> </o:p> <o:p> </o:p> <h2 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>1. Doing mathematics together onlin= e<o:p></o:p></h2> <o:p> </o:p> Computers offer many opportunities for innovation= in education. One of the major avenues is by supporting the building of collab= orative knowledge [1]. For instance, it is now possible for students around the world to work together on challenging math problems. Through online discussion, they can share problem-solving experie= nces and gain fluency in communicating mathematically. In a research project at the Math Forum @ Drexel (<a href=3D"http://mathforum.org/">http://mathforum.org</a>), we have begun to = invite middle school students to participate in online chats about interesting problems in beginning algebra and geometry. The following problem, discusse= d in the example in this paper, is typical: If two equilateral triangles have edge-lengths of 9 cubits and 12 cubits, what is = the edge-length of the equilateral triangle whose area is equal to the sum of t= he areas of the other two? We rely on a variety of methods from the learning scie= nces to guide our research and to analyze the results of our trials. In particul= ar, we use conversation analysis <!= --[if supportFields]> ADDIN EN.CITE <EndNote><Cite><Author>Sacks</Author><Year>19= 74</Year><RecNum>526</RecNum><MDL><REFERENCE_TYP= E>0</REFERENCE_TYPE><REFNUM>526</REFNUM><AUTHORS>= ;<AUTHOR>Harvey Sacks</AUTHOR><AUTHOR>Emanuel A. Schegloff</AUTHOR><AU= THOR>Gail Jefferson</AUTHOR></AUTHORS><YEAR>1974</YEAR><TI= TLE>A Simplest Systematics for the Organization of Turn-Taking for Conversation</TITLE><SECONDARY_TITLE>Language</SECONDARY_TIT= LE><VOLUME>50</VOLUME><NUMBER>4</NUMBER><PAGE= S>696-735</PAGES><DATE>December 1974</DATE><URL>www.jstor.org</URL></MDL></Cite&= gt;<Cite><Author>Pomerantz</Author><Year>1991</Y= ear><RecNum>521</RecNum><MDL><REFERENCE_TYPE>7&l= t;/REFERENCE_TYPE><REFNUM>521</REFNUM><AUTHORS><AUT= HOR>Pomerantz, Anita</AUTHOR><AUTHOR>Fehr, B.J.</AUTHOR></AUTHORS>= <YEAR>1991</YEAR><TITLE>Conversation Analysis: An Approach to the Study of Social Action as Sense Making Practices</TITLE><SECONDARY_AUTHORS><SECONDARY_AUTHOR>van Dijk, Teun A.</SECONDARY_AUTHOR></SECONDARY_AUTHORS><SECONDARY_TITLE>= ;Discourse as Social Interaction: Discourse Studies, A Multidisciplinary Introduction, Volume 2</SECONDARY_TITLE><PLACE_PUBLISHED>London, UK</PLACE_PUBLISHED><PUBLISHER>Sage</PUBLISHER><PAGES&= gt;64-91</PAGES></MDL></Cite><Cite><Author>Ps= athas</Author><Year>1995</Year><RecNum>469</RecN= um><MDL><REFERENCE_TYPE>1</REFERENCE_TYPE><REFNUM&g= t;469</REFNUM><AUTHORS><AUTHOR>Psathas, George</AUTHOR></AUTHORS><YEAR>1995</YEAR><TITLE= >Conversation Analysis: The Study of Talk-in-Interaction</TITLE><PLACE_PUBLISHED>Thousand Oaks, CA</PLACE_PUBLISHED><PUBLISHER>Sage</PUBLISHER></MDL&g= t;</Cite><Cite><Author>ten Have</Author><Year>1999</Year><RecNum>471</RecNu= m><MDL><REFERENCE_TYPE>1</REFERENCE_TYPE><REFNUM>= ;471</REFNUM><AUTHORS><AUTHOR>ten Have, Paul</AUTHOR></AUTHORS><YEAR>1999</YEAR><T= ITLE>Doing Conversation Analysis: A Practical Guide</TITLE><PLACE_PUBLISHED>Thousand Oaks, CA</PLACE_PUBLI= SHED><PUBLISHER>Sage</PUBLISHER></MDL></Cite><= ;Cite><Author>Sacks</Author><Year>1992</Year><= ;RecNum>246</RecNum><MDL><REFERENCE_TYPE>1</REFEREN= CE_TYPE><REFNUM>246</REFNUM><AUTHORS><AUTHOR>Har= vey Sacks</AUTHOR></AUTHORS><YEAR>1992</YEAR><TITLE&= gt;Lectures on Conversation</TITLE><SECONDARY_AUTHORS><SECONDARY_AUTHOR>= G. Jefferson</SECONDARY_AUTHOR></SECONDARY_AUTHORS><PLACE_PUBLI= SHED>Oxford, UK</PLACE_PUBLISHED><PUBLISHER>Blackwell</PUBLISHER><N= UMBER_OF_VOLUMES>2</NUMBER_OF_VOLUMES></MDL></Cite><= ;/EndNote><![endif]-->[2; 3; 4; 5; 6]<!-= -[if supportFields]><![endif]--> to interpret the intera= ctions that take place in the student chats. In this paper, we adapt the findings = of conversation analysis to math chats and develop a specific form of adjacency pairs that seem to be important for math chats. Before presenting this, it = may be useful to describe briefly how the notion of adjacency pairs differs from naïve conceptions of conversation. There is a widespread common-sense or folk-theory [7; 8] view of conversation as= the exchange of propositions [9]. This view was refined = and formalized by logicians and cognitive scientists as involving verbal expres= sion in meaningful statements by individuals, based on their internal mental representations. Speech served to transfer meanings from the mind of a spea= ker to the mind of a listener, who then interpreted the expressed message. Following Wittgenstein [10] in critiquing this view, speech act theory [11; 12] argued that the utteran= ces spoken by individuals were ways of acting in the world, and were meaningful= in terms of what they accomplished through their use and effects. Of course, t= he expression, transmission and interpretation of meaning by individuals can be problematic, and people frequently have to do some interactional work in or= der to re-establish a shared understanding. The construction of common ground h= as been seen as the attempt to coordinate agreement between individual understandings [13]. Conversation analysis takes a different view of conversation. It looks at how interactional mechanisms, like the use of adjacency pairs [14; 15], co-construct inter-subjectivity. Adjacency pairs are common sequences of utterances by different people—such as mutual greetings or question/answer interchanges—that form a meaningful speech act spanning multiple utterances that cannot be attributed to an individual or to the expression = of mental states. We are interested in what kinds of adjacency pairs are typic= al for math chats. Online math chats differ from ordinary informal conver= sation in a number of ways. They are focused on the task of solving a specific pro= blem and they take place within a somewhat formal institutional setting. They involve the doing of mathematic= s [16]. And, of course, they a= re computer-mediated rather than being face-to-face. The approach of conversat= ion analysis is based on ethnomethodology [17], which involves the stu= dy of the methods that people use to accomplish what they are doing. So, we are interested in working out the methods that are used by students in online m= ath chats. In this paper we discuss a particular method of collaboration in math chats that we have elsewhere called exploratory participation: participants engage each other in the conjoint discovery= and production of both the problem and possible solutions [18]. The medium of online chat has its own peculiarities. M= ost importantly, it is a text-based medium, where interaction takes place by the sequential response of brief texts to each other [19; 20]. As a quasi-synchronous medium [21], chat causes confusion because several people can be typing at once and their texts can appear in = an order that obscures what they are responding to. Furthermore, under time pr= essure to submit their texts so that they will appear near what they are responding to, some chat participants break their messages into several short texts. Because of these peculiarities of chat, it is necessary for researchers to carefully reconstruct the intended threading of texts that respond to each other before attempting to interpret the flow of interaction [22; 23]. <o:p> </o:p> <o:p> </o:p> 2. Math proposal adj= acency pairs <o:p> </o:p> In order to begin to analyze the methods that stu= dents use in math chats, we take a close look at an excerpt from an actual chat. Figure 1 shows an excerpt from near the beginning of the log of one of our first online collaborative math problem-solving sessions. Three students—named Avr, Sup and Pin—hav= e just entered the chat room, said hello to each other and read the problem involv= ing three triangles. The first thing to notice here is a pattern of proposa= ls, discussions and acceptances similar to what takes place in face-to-face discourse. Proposals about steps in solving the math problem are made by <s= pan class=3DSpellE>Avr in lines 1, 3, 8, 17 and by Pin in lines 20, 27. = These proposals are each affirmed by someone else in lines 2, 6, 10, 19, 22, 28, respectively. To avoid chat confusion, note that line 21 responds to= line 19, while line 22 responds to line 20. The timestamps show that lines 20 an= d 21 effectively overlapped each other chronologically: Avr= was typing line 21 before she saw line 20. Similarly, lines 24 and the following were responses to line 20, not line 23. We will correct for these confusions in Figure 2, which reproduces a key passage in this excerpt. <![if !vml]><img width=3D589 height=3D661 src=3D"icce_files/image001.gif" align=3Dleft hspace=3D12 alt=3D"Text Box: 1. Avr (8:21:46 PM): Okay, I think we should start with th= e formula for the area of a triangle 2. Sup (8:22:17 PM): ok &= #10;3. Avr (8:22:28 PM): A =3D 1/2bh 4. Avr (8:22:31 PM): I believ= e 5. pin (8:22:35 PM): yes 6. pin (8:22:37 PM): i concue&= #13; 7. pin (8:22:39 PM): concur* 8. Avr (8:22:42 PM): then fi= nd the area of each triangle 9. Avr (8:22:54 PM): oh, wait = 0;10. Sup (8:23:03 PM): the base and heigth are 9 and 12 right? 11= . Avr (8:23:11 PM): no 12. Sup (8:23:16 PM): o 13. Avr (8= :23:16 PM): that's two separate triangles 14. Sup (8:23:19 PM): oo= o 15. Sup (8:23:20 PM): ok 16. Avr (8:23:21 PM): right= 3; 17. Avr (8:23:27 PM): i think we have to figure out the height by ou= rselves 18. Avr (8:23:29 PM): if possible 19. pin (8:24:0= 5 PM): i know how 20. pin (8:24:09 PM): draw the altitude' = 0;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 f= rustrating 25. Avr (8:24:22 PM): I don't have enough paper = 0;26. pin (8:24:43 PM): i think i got it 27. pin (8:24:54 PM): its= a 30/60/90 triangle 28. Avr (8:25:06 PM): I see 29. pin = (8:25:12 PM): so whats the formula Figure 1. Excerpt of = 3½ minutes from a one-hour chat log. Three students chat about a geo= metry problem. Line numbers have been added and screen-names anonymized; ot= herwise the transcript is identical to what the participants saw on their s= creens. " v:shapes=3D"_x0000_s1026"><![endif]>In Figure 1, we see several examples of= a three step pattern: <![if !supp= ortLists]>1.&n= bsp; <![endif]>A proposal is made by an individual for the group to work on: “I think we should ….” <![if !supp= ortLists]>2.&n= bsp; <![endif]>An acceptance is made on behalf of the group: “Ok,” “right” <![if !supp= ortLists]>3.&n= bsp; <![endif]>There is an elaboration of the proposal by members = of the group. The proposed work is begun, often with a secondary proposal for = the first sub-step. This suggests that collabora= tive problem-solving of mathematics may often involve a particular form of adjac= ency pair. We will call this a math prop= osal adjacency pair. Many adjacency pairs allow for insertion of other pairs between the two parts of the original pair, delaying completion of the pair= . For instance, a question/answer pair may be interrupted by utterances seeking clarification of the question; the clarification interaction may itself con= sist of question/answer pairs, possibly with their own clarifications—this= may continue recursively. With math proposal adjacency pairs, the subsidiary pa= irs seem to come after the completion of the original pair, in the form of secondary proposals, questions or explanations that start to do the work th= at was proposed in the original pair. Proposals seem to lead to some kind of further mathema= tical work as a response to carrying out what was proposed. Often—as seen in the current example—that work consists of making further proposals. I= t is striking that the proposed work is not begun until there is agreement with = the proposal. This may represent consent by the group as a whole to pursue the proposed line of work. Of course, it is not so clear in the current example, where there are only three participants and the interaction often seems to = take place primarily between pairs of participants. As confirmed by other chat examples, however, the proposal generally seems to be addressed to the whole group and opens the floor for participants other than the proposer to respo= nd. The use of “we” in “we should” or “we have to” (stated or implied) constitutes the multiple participants as a pl= ural subject, an effective unified group [24]. Any one other than the proposer may respond on behalf of the group. Moreover, there seems to be what in conversation analy= sis is called an interactional preference [25] for acceptance of the proposal. That is, if one accepts a proposal, it suffices to briefly indica= te agreement: “ok.” If one wants to reject a proposal, then one ha= s to account for this response by giving reasons. We would like to characterize in more detail the metho= d of making math proposal adjacency pairs. Often, the nature of an interactional method= is seen most clearly when it is breached [17]. Methods are generally = taken for granted by people; they are not made visible or conducted consciously. = It is only when there is a breakdown</= i> [26] in the smooth, tacit performance of a method that people focus on its characteristics in order to overcome the breakdown. The normally transparent method becomes visible in = its breach. We can interpret Sup’s posting in= line 23 as a failed proposal. Given = the mathematics of the triangle problem, a proposal like S= up’s related to proportionality might have been fruitful. However, in this chat, = line 23 was effectively ignored by the group. While its character as a fail= ed proposal did not become visible to the participants, it can become clear to= us by comparing it to successful proposals in the same chat and by reflecting = on its situation in the chat in order to ask why it was not a successful propo= sal. <o:p> </o:p> <o:p> </o:p> <h2 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>3. A failed proposal<o:p></o:p></h2> <o:p> = </o:p> Let us look= at line 23 in its immediate interactional context in Figure 2. We can distinguish a number of ways in which it differed from successful math proposals that sol= icited responses and formed math proposal adjacency pairs. <![if !vml]><img width=3D589 height=3D197 src=3D"icce_files/image002.gif" align=3Dleft hspace=3D12 alt=3D"Text Box: 17, 18. Avr (8:23: 29 PM): i think we have to figure out t= he height by ourselves … if possible 19. pin (8:24:05 PM): i= know how 21. Avr (8:24:09 PM): how? 20. pin (8:24:09 PM)= : draw the altitude' 22. Avr (8:24:15 PM): right 24. Avr = (8:24:19 PM): this is frustrating 23. Sup (8:24:19 PM): proportion= s? Figure 2. Part of the chat log excerpt in figure 1, w= ith order revised for threading. " v:shapes=3D"_x0000_s1028"><![endif]>(a) All the other proposals (1, 3, 8, 1= 7, 20, 27) were stated in relatively complete sentences. Additionally, some of the proposals were introduced with a phrase to indica= te that they were the speaker’s proposal (1. “I think we should …,” 17. “I think we have to …,” 20. “<s= pan class=3DSpellE>i know how …” = and 27. “i think i got it …”). The exceptions to these were simply continuations of previous proposals: line 3 provided the formula proposed in line 1 and line 8 proposed to “then” use that formu= la. Line 23, by contrast, provided a single word with a question mark. There wa= s no syntactic context (other than the question mark) within the line for interpreting that word and there was no reference to semantic context outsi= de of the line. Line 23 did not respond in any clear way to a previous line and did not provide any alternative reference to a context in the original prob= lem statement or elsewhere. For instance, Sup could have said, “I think we should compute the proportion of the height to the base of those equilateral triangles.” (b) The timing of line 23 w= as particularly unfortunate. It exactly overlapped a line from Avr. Since Avr had been setting = the pace for group problem solving during this part of the chat, the fact that she w= as involved in following a different line of inquiry spelled death for an alternative proposal at the time of line 23. Pin either seemed to be continuing on his own thread without acknowledgi= ng anyone else at this point, or else he was responding too late to previous postings. So a part of the problem for Sup was that there was little sense = of a coherent group process—and what sense there was did not include him. = If he was acting as part of the group process, for instance posing a question = in reaction to Pin and in parallel to Avr, he was not doing= a good job of it and so his contribution was ignored in the group process. It= is true that a possible advantage of text-based interaction like chat over face-to-face interaction is that there may be a broader time window for responding to previous contributions. In face-to-face conversation, turn-ta= king rules may define appropriate turns for response that expire in a fraction o= f a second as the conversation moves on. In computer-based chat, the turn-taking sequence is more open. However, even here if one is responding to a posting that is several lines away, it is important to make explicit somehow to what one is responding. He could have said, “I = know another way to find the height – using proportions.” Sup’s posting does not do anything like that; it relies purely upon sequential timing to establish its context, and that fai= ls in this case. (c) Su= p’s posting 23 came right after Pin’s proposal 20: “draw the altitude.” Avr had responded to this with= 22 (“right”) but Pin seems to have ignored that. Pin’s proposal had opened up work= to be done and both Avr and Pin responded after li= ne 23 with contributions to this work. So Sup’s proposal came in the middle of an ongoing line of work without relating to = it. In conversational terms, he made a proposal when it was not time to make a proposal. It is like trying to take a conversational turn when there is not= a pause that creates a turn-taking opportunity. Now, it is possible—especially in chat—to introduce a new proposal at any time. However, to do so effectively, one must make a special effort to bring the on-going work to a temporary halt and to present one’s new propos= al as an alternative. Simply saying “proportions?” will not do it.= Sup could have said, “Instead of drawing the altitude, let’s use <s= pan class=3DSpellE>propostions to find it.” (d) To get a response to a proposal= , one must elicit at least an affirmation or recognition. Line 23 does not really solicit a response. For instance, Avr’s question, 21: “how?” called for an answer—that was given = by Pin in line 20, which actually appeared in the chat window just prior to the question and with the same time stamp. But Sup’s= posting does not call for a specific kind of answer. Even Sup’s own previous proposal in line 10 ended with “right?”—requiring agreement or disagreement. Line 10 el= icited a clear response from Avr, line 11(“no= 221;) followed by an exchange explaining why Sup’s</sp= an> proposal was not right. (e) Other proposals in the = excerpt are successful in contributing to the collaborative knowledge building or g= roup problem solving in that they open up a realm of work to be done. One can lo= ok at Avr’s successive proposals on lines 1,= 3, 8 and 17 as laying out a work strategy. This elicits a response from Sup tryi= ng to find values to substitute into the formula and from Pin trying to draw a graphical construction that will provide the values for the formula. Sup’s proposal in line 23, however, neither cal= ls for a response nor opens up a line of work. There is no request for a reaction = from the rest of the group and the proposal is simply ignored. Since no one responded to Sup, he could have continued by doing some work on the proposal himself. He could have come back and made the proposal more explicit, reformulated it more strongly, taken a first step in working on it, or pose= d a specific question related to it. But he did not—at least not until much later—and the matter was lost. (f) Another serious hurdle for Sup was his status in t= he group at this time. In lines 10 through 16, Sup had made a contribution that was taken as an indication that he did not have a strong grasp of the math problem. He offered the lengths of the two given triangles as the base and height of a single triangle (line 10). Avr immediately and flatly stated that he was wrong (line 11) and then proceede= d to explain why he was wrong (line 13). When he agreed (line 15), Avr summarily dismissed him (line 16) and went on to = make a new proposal that implied his approach was all wrong (lines 17 and 18). Then Pin, who had stayed out of the interchange, re-entered, claiming to know ho= w to implement Avr’s alternative proposal (lin= es19 and 20) and Avr confirmed that (line 22). Sup’s legitimacy as a source of useful proposal= s had been totally destroyed at precisely the point just before he made his ineffective proposal. Less than two minutes later, Sup tries again to make a contribution, but realizes himself that what he = says is wrong. His faulty contributions confirm repeatedly that he is a drag on = the group effort. He makes several more unhelpful comments later and then drops= out of the discourse for most of the remaining chat. The weaknesses of line 23 as a proposal suggest some characteristics for successful proposals: (a) a clear semantic and syntactic structure, (b) careful timing within the sequence of postings, (c) a firm interruption of any other flow of discussion, (d) the elicitation of a response, (e) the specification of work to be done and (f) a history of hel= pful 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 trai= ning. Other characteristics will become visible in other examples of chats. At this time, the notion of math proposal adjacency pa= irs is just a preliminary proposal based on a single chat log excerpt. It calls for extensive conversation analysis of a corpus of logs of collaborative online math problem solving to establish whether this is a fruitful way of interpreting the data. If it turns out to be a useful approach, then it wil= l be important to determine what interactional methods of producing such proposa= ls are effective (or not) in fostering successful knowledge building and group cognition. An understanding of these methods can guide the design of activi= ty structures for collaborative math. As we are collecting a corpus of chat lo= gs, we are evolving computer support through iterative trials and analyses. <o:p> </o:p> <o:p> </o:p> <h2 style=3D'margin:0in;margin-bottom:.0001pt;text-indent:0in'>4. Designing computer support<o:p></o:p></h2> <o:p> </o:p> If the failure of Sup’s proposal about proportions is considered deleterious to the collaborative knowledge building around the triangles problem, then what are the implications of this for the design of education= al computer-based environments? One response would be to help students like Sup formulate stronger proposals. Presumably, giving him positive experiences of interacting with students like Avr and Pin, who= are more skilled in chat proposal making, would provide Sup with models and examples from which he can learn—assuming that he perseveres. Another approach to the problem would be to build functionality into the software and structures into the activity that scaff= old the ability of weak proposals to survive. As students like Sup experience success with their proposals, they may become more aware of what it takes to make a strong proposal. [16] Professional mathematicians rely heavily upon inscript= ion: the use of specialized notation, the inclusion of explicit statements of all deductive steps and the format of the formal proof to support the discussio= n of math proposals—whether on an informal whiteboard, a university blackb= oard or in an academic journal. Everything that is to be indexed in the discussi= on is labeled unambiguously. To avoid ellipsis, theorems are stated explicitly, with all conditions and dependencies named. The projection of what is to be proven is encapsulated in the form of the proof, which starts with the give= ns and concludes with what is proven. Perhaps most importantly, proposals for = how to proceed are listed in the proof itself as theorems, lemmas, etc.—organized sequentially. One could imagine a chat system supplemented with a wi= ndow containing an informal list of proposals analogous to the steps of a proof. After Sup’s proposal, the list might look= like Figure 3. When Sup made a proposal in the chat, he would enter a statement of it in the proof window in logical sequence. He could cross out his own proposal w= hen he felt it had been convincingly argued against by the group. <o:p> </o:p> <![if !vml]><img width=3D540 height=3D228 src=3D"icce_files/image003.gif" alt=3D"Text Box: Given: 2 equilateral triangles of edge-length 9 cubits and= 12 cubits formula for a triangle: A =3D 1/2bh = Area of each triangle =3D ? b, h =3D 9, 12 draw the alti= tude use proportions for ratio of altitude to base &= #10;Find: The edge-length of the equilateral triangle whose area is equal t= o the sum of the areas of the other two triangles Figure= 3. A list of proposals " v:shapes=3D"_x0000_s1029"><![endif]> <o:p> </o:p> The idea is that important proposals that were made wo= uld be retained in a visible way and be shared by the group. Of course, there are = many design questions and options for doing something like this. Above all, would students understand this functionality and would they use it? The design indicated in Figure 3 is only meant to be suggestive. Another useful tool for group mathematics would be a s= hared drawing area. In the chat environment used by Sup, Pin and Avr, there was no shared drawing, but a student could create a drawing and send = it to the others. Pin did this twelve minutes after= the part of the interaction shown in the excerpt. Before the drawing was shared, much time was lost due to confusion about references to triangles and verti= ces. For math problems involving geometric figures, it is clearly important to be able to share drawings easily and quickly. 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