From Jsarmi

Session 4 - Team 5: Collective Remembering

This excerpt illustrates a case in which a team is collectively engaged in trying to reconstruct parts of their previous session in order to initiate their current problem solving activity. Remembering of past activity unfolds as a collective engagement in which different team members participate dynamically. Some of the current team members were not present in the previous session and yet, they are instrumental in the reconstruction of that past and in shaping its current relevance.

Last Tuesday you worked on finding a formula

This was the fourth session of team five. Towards the start of the session, with only two participants present (Gdo and Meets), the moderator stated that he was a "new" moderator and asked them if they were part of the chat "last time." Gdo and Meets assent and add that they remember other people being present. Ten minutes had passed since the start of the session when the facilitator stated that maybe Gdo and Meets were going to the "the team" for the night. A few minutes later, Gdo joined the room while the facilitator is explaining general aspects of how and why the chat sessions are offered. Later on, Extrick joined the room as well.

After the introduction, the facilitator suggested that during the summer the team members could work with their friends on a new problem he introduced: the "circle problem." He also posted that they could "pursue the circle questions in this chat" if they wanted or "any other questions and worlds" that they thought of. The team seems disoriented about what to do and after more than a minute of silence the following sequence takes place:

 111  8:25:38 PM  meets:  ert
 112  8:25:42 PM  meets:  whoops
 113  8:25:52 PM  meets: we're doing the problem he juts gave us right?
 114  8:26:05 PM  Mod:    Last Tuesday you worked on finding a formula for the number of shortest paths 
                          between any two points A and B on the grid.  You explored multiple possibilities 
                          and figured out that x+y and x^2+y^2 work (where x and y correspond to the # of units 
                          you need to travel along x and y axis to get from A to B) but only for some points, not all.  
                          You may want to continue exploring more cases and see if you can find a general formula.
 115  8:26:31 PM  Mod: or you can work on the problem i posted earlier
 116  8:26:50 PM  drago: ok
 117  8:27:04 PM Mod: I can also post all the original questions if you would like to see them
 118  8:27:17 PM  gdo: post the original
 119  8:27:42 PM  drago: ok

 (Whiteboard activity by the facilitator and by Gdo, from 8:27:35 PM to 8:29:18 PM to post and arrange a textbox 
  with “all the original questions”)

The problem he just gave *US* Last Tuesday YOU workED FINDING a formula explorED figurED out YOU MAY WANT TO CONTINUE EXPLORING and SEE IF YOU CAN FIND

 120  8:30:11 PM    gdog:      where did u guys last leave off (Points to Message 119)
 121  8:31:20 PM   MFmod:      I think that the above section I wrote is where the group last was  (Points to Message 114)
 122  8:31:36 PM   MFmod:      yes?
 123  8:31:42 PM   drago:      well
 124  8:31:48 PM   gdo:        i dont remember that
 125  8:31:51 PM   drago:      actually, my internet connection broke on Tuesday
 126  8:31:56 PM   drago:      so I wasn't here
 127  8:32:12 PM   MFmod:      so maybe that is not the best place to pick up
 128  8:32:14 PM   estrick:    i wasnt able to be here on tuesday either
 129  8:32:50 PM   gdo:        how bout u meets
 130  8:33:01 PM   meets:      uh...
 131  8:33:11 PM   meets:      where'd we meet off....
 132  8:33:16 PM   meets:      i remember
 133  8:33:22 PM   gdo:        i was in ur group
 134  8:33:24 PM   meets:      that we were trying to look for a pattern
 135  8:33:27 PM   gdo:        but i didn't quite understand it
 136  8:33:34 PM   gdo:        can u explain it to us again meets

 137 meets:      with the square, the 2by 2 square, and the 3by2 rectangle
 138 meets:      sure...
 139 meets:      so basically...
 140 gdo:        o yea
 141 gdo:        i sort of remember
 142 meets:      we want a formula for the distance between poitns A and B
 143 drago:      yes...
 144 meets:      ill amke the points
 145 MFmod:      since some folks don't remember and weren't here why don't you
                  pick up with this idea and work on it a bit
 146 meets:      okay
 147 meets:      so there are those poitns A and B
 148 meets:      (that's a 3by2 rectangle
 149 meets:      we first had a unit square
 150 meets:      and we know that there are only 2 possible paths......

One of the things that are remarkable about the way this interaction unfolds is the fact that although it might appear as if it is Meets who remembered what they were doing last time, the actual activity of remembering unfolds as a collective engagement in which different team members participate dynamically. In fact, later in this sequence there is a point where Meets remembers the fact that they had discovered that there are 6 different shortest paths between the corners of a 2-by-2 grid but he reports that he can only “see” four at the moment. Even though Drago did not participate in the original work leading to that finding, he was able to see the six paths when Meets presented the 2- by-2 grid on the whiteboard and proceeded to invent a method of labeling each point of the grid with a letter so that one can name each path and help others see it (e.g., “from B to D there is BAD, BCD …”). After this, Meets was able to see again why it is that there are six paths in that small grid and together with Drago, they proceeded to investigate, in parallel, the cases of a 3-by-3 and a 4-by-4 grid using the method just created. The result can be seen on the whiteboard:

[1]

Despite the fact that this picture is a restrictively static representation of the team’s use of the whiteboard, it allows us to illustrate some unique aspects of this remarkable creative organization of their collective activity. First, we see again the crucial role of indexicals and referencing activity in the collective construction of the mathematical ideas of the team (e.g., through the use of labels, the witnessing of actions on the whiteboard, and the coordination of parallel activity).

The use of the whiteboard represents an interesting way of making visible the procedural reasoning behind a concept (e.g., shortest path). The fact that a newcomer can use the persistent history of the whiteboard to re-trace the team’s reasoning seems to suggest a possible strategy towards preserving complex results of problem-solving activities. However, the actual meaning of these artifacts is highly situated in the doings of the co-participants, a fact that challenges the ease of their reuse despite the availability of detailed records such as those provided by the whiteboard history.

Despite these technical limitations, we could view the artifacts created by this team as “bridging” objects which, in addition to being a representation of the teams’ moment-to-moment joint reasoning, could also serve for their own future work and for other members of the VMT online community. These particular objects are constructed in situ as a complex mix of resources that “bridge” different points in their own problem-solving and, potentially, those of others. As can be seen in Figure 4, the two team members combined the depiction of the cases being considered, the labeling and procedural reasoning involved in identifying each path, a summary of results for each case (i.e., the list of paths expressed with letter sequences) and a general summary table of the combined results of both cases. The structure of these artifacts represents the creative work of the team but also documents the procedural aspects of such interactions in a way that can be read retrospectively to document the past, or “projectively” to open up new possible next activities.