Dataset1/D1T2SS
From Jsarmi
Contents |
Team 2: Trajectory
mi and bb come to all sessions, most stable team (dyad + audience?) In session 3, mi and bb talk about "personal" things so it is obvious they know each other outside the chat, qwer participates less? qwer does not return for the last session
Session I
The team produces the following set of 5 questions recorded in a textbox on the whiteboard:
- (C/W) What is the shortest way to get from A to B? (lv)
- (C/W) What is the longest way? (qw)
- What is the straight distance between A and B? (lv)
- What is the measure of angle B? (qw)
- How many ways are there to get from A to B in rectangle ABCD? (mi)
- Clarification questions?:
- (C/ ) Are there any remean loopholes, or is the grid just flat? (qw)
- (C/ ) Can you fold the plane? (qw)
Activity develops quickly after the moderator presents the task. Questions are proposed and some solutions for them are put forward quickly. At the beginning participation is very equal, except that Bb does not offer a potential question himself. Mi's participation grows progressively, at the beginning just asking a clarification question to qwer (about overlap) and correcting a value in a proposed solution for the straight distance (*52; *2 not 4). Very participatory sequence that goes from a long silence to quick postings:
(1'34 of silence) SithLordVader91 5/10/05 8:31:38 PM EDT: what is the straight distance between A and B? (suppose each line is 1 unit bob123 5/10/05 8:31:53 PM EDT: 2rt(13) bob123 5/10/05 8:32:13 PM EDT: or 10 bob123 5/10/05 8:32:18 PM EDT: depending on what you mean (diagonal distance between A and B drawn by lordVader) qwer 5/10/05 8:32:49 PM EDT: Following the lines, I think it's 10. If you don't follow them, it's the root of 48. Pythag. mathisfun 5/10/05 8:33:04 PM EDT: *52 SithLordVader91 5/10/05 8:33:13 PM EDT: yeah SithLordVader91 5/10/05 8:33:22 PM EDT: square root of 52 qwer 5/10/05 8:33:24 PM EDT: 52 what? qwer 5/10/05 8:33:29 PM EDT: oh SithLordVader91 5/10/05 8:34:46 PM EDT: 4 square root 13 SithLordVader91 5/10/05 8:34:54 PM EDT: in radical form mathisfun 5/10/05 8:35:00 PM EDT: *2 not 4 SithLordVader91 5/10/05 8:35:36 PM EDT: yeah... 2 sry (36 of silence)
Notice how mi does two other-initiated repairs: *52 and *2 not 4
When they get to the fourth question (posted by qw in #201) bb and mi get much more involved (we can use tan, sin or cos). Lv and qw state that they have not taken trigonometry classes (only algebra) but mi and bb seem very comfortable with it, and bb takes on the development of that approach. Bb posts what looks like some basic definitions of trigonometric concepts/formulas (e.g. tangent=opp/adj, sine=opp/hyp, etc.) but when lv attempts to participate (so,... that would be 6/4=3/2) not much direct attention is given to his attempt (he closes with oh... sry, didnt see that). At this point, it seems as if, participation footing has shifting and now bb and, to a lesser extent, mi are driving the collaboration, since they know about trigonometry. Here you see this pattern:
| OPEN QUESTION TO ALL by A: | 201 qwer: What about, what's the angle of B? I think it involves a sin. |
| ANSWER WITH PROCEDURAL EXPLANATION by B-> | 203 bob123: tan of angle b=3/2, so you do tan^-1(3/2)=56.30993247 |
| COLLECTIVE WITHDRAWAL FROM ASSESSMENT by A and C (A COMPLAINT? A REQUEST FOR EXPLANATION?) | (20 secs.) 209 SithLordVader91: im only in algebra 1.... i havent covered sine, cosine, and tangent yet |
| CONCEPTUAL EXPLANATION by B | (4 secs.) 211 bob123: tangent=opp/adj |
| ACCEPTANCE OF EXPLANATION: (Withdrawal of complaint?) | 209 SithLordVader91: ohh... i c ... 219 SithLordVader91: ok... thx |
| ATTEMPT AT APPLYING PRESENTED EXPLANATION | (35 secs) 221 SithLordVader91:so,... that would be 6/4=3/2 |
| CONNECTING CONCEPTUAL AND PROCEDURAL EXPLANATIONS | (41 secs.) 222 bob123: and then you set tan(angle B)=3/2 |
| ASSESSMENT | (30 secs.) 225 SithLordVader91: oh... sry, didnt see that |
(What is the role of D?)
So in the session, these two patterns of interaction seem to position Bob as an "expert" but one who only furnishes explanations if required, and even so, one whose explanations are divided in conceptual and procedural bits (?). One wonders whether this position develops cumulative through the two interactional episodes. In any event, after a long silence, Mi transitions to a new activity by posting the fifth question. After another long silence (57 secs.) Bob offers a sort of vague answer ("infinite if you count overlap / otherwise a lot) which are supported by lv and qw. Mi then makes a couple of postings that seem to open up/invite participation of the whole group towards considering this question further:
226 mathisfun: How many ways are there to get from A to B? ... 236 mathisfun: does everyone see the ACBD rectangle? ... 239 mathisfun: letz find the amount of way in that small rectangle
Bb responds with a candidate answer (56) and a series of quick postings that document his calculations but do not seem to amount to a full explanation. Here we see a similar pattern than the one explored earlier:
| OPEN QUESTION TO ALL by A: | 226 mathisfun: How many ways are there to get from A to B? ... |
| ANSWER WITH PROCEDURAL EXPLANATION by B: | 238 bob123: using only the shortest way, then it would be 56 ... |
| REQUEST FOR AN EXPLANATION by C: (a Challenge?) | 248 qwer: what does that mean? |
| RESPONSE WITH CONCEPTUAL EXPLANATION by C: | 249 bob123: the number of ways to choose 5 things from 8 things 250 bob123: it's a formula |
| ACCEPTANCE OF EXPLANATION: (Withdrawal of challenge?) | 251 : ok |
Lv asks for a clarification on what X choose Y means and bb explains (the formula to find the number of ways in an n by m rectangle like this is (n+m-2) choose (m-1)). The group does not object to that being the answer (although bob explains that it works "assuming you can only go right and down"). Activity in this episode follows much more quickly than before but at the end it seems to stall (more than 30 secs. of silence), so they see if there are more questions but time is up and the facilitator engages them in the final conversation about the session.
Session II
Session II starts with a lot of socializing, people joining in and leaving. Then the facilitator initiates this sequence:
94 Moderator_G: Remember what you were doing Tuesday?
95 mathisfun: sorta
96 qwer: Yes.
97 Moderator_G: What?
98 mathisfun: i'm pretty sure
99 qwer: There were two points on a grid in which one could only travel
on the lines and we made and answerd questions about it.
100 Moderator_G: Right. There were several groups like yours and they each came up
with several ideas about this grid world
101 Moderator_G: I listed some of their questions ...
102 Moderator_G: ... and also some questions that these group questions raised
for the moderators
Then a textbox gets posted with 9 questions, 6 from "the groups" and 3 more from the moderators. They start with question 7. Qw proposes a formula, Mi tests it with two points and finds that it works. (Interestingly, this is a sort of new pattern of interaction for the team since bob was dominating most of the answers in session I. Mi, however, seems to be consistent in his orientation towards collaboration and specifically, testing answers)
Bob interjects that the points Mi chose are not the same as last time (bob123: wasn't it 4 and 6 yesterday?) However before the group responds to Bobs posting, Marisol, a new participant, proposes a formula (for 7, the shortest way is the square root of [(x2-x1)**2 + (y2-y1)**2]) to which Bob disagrees (but that wouldn't be along the grid) as well as others. Then, since question 7 seems done, they move on to question 8 (How many shortest paths are there from A to B and how does this vary with changes in the positioning of A relative to B):
144 mathisfun: letz start working on number 8 145 bob123: we already did that yesterday 146 qwer: we did? 147 mathisfun: but we did it so that there was only right and down 148 bob123: i mean tuesday 149 mathisfun: i guess we will do it with left and up? 150 qwer: It would be almost the same. 151 bob123: it's (|x2-x1|+|y2-y1|-2) choose (|x2-x1|-1) 152 bob123: try it if you like 153 mathisfun: nah 154 mathisfun: if you are so sure... 155 bob123: i'm not 156 bob123: actually 157 bob123: take out the -2 and the -1 158 mathisfun: then letz check it
Then they engage in this activity of "cheking" bob's formula. There are two interesting things here. One is that Mi seems to drive this process, changing his participation style in this session and possibly so far in both sessions. Also, there is an interesting and subtle way in which Mi engages in a sequence of tests for different cases and how he indexes different "shapes". THIS NEEDS TO BE UNPACKED! He only uses the word "shape" when Qw make some posts that seem to be oriented towards a different case/shape and then Mi seems to hint to the fact that they might not be looking at the same thing (are u at 100%?). They had had that breakdown in session I when someone was not looking at the whiteboard at 100% magnification and because of that, was not seeing the points A and B on the grid lines. Also, Mi had already done this in this session with Qw's formula for the distance: he created a case, tested the formula and concluded that it worked.
At the end of this session bob makes repeated erratic lines on the whiteboard, states that he is bored, and proceed to make more drawings turning into a very disruptive participant. Is he not challenged enough? At the end, while he is still in the room (but not paying attention?) the moderator asks the group about the formula they had and they say that they did not understand it fully:
309 Moderator_G: I also found your solution of 8 very creative. I do not understand how you got it or why it works 310 qwer: Neither do I. 311 mathisfun: I was something that bob found out 312 mathisfun: I guess we should ask him next time... 313 mathisfun: or unless he responds 314 Moderator_G: Maybe we can all think about it some more ... and make sure it really works 315 Moderator_G: See you on Tuesday!
Session III
Bob comes early and the facilitator asks him to explain the formula he proposed in the previous session for the number of shortest paths. He states that it is based on Pascal's triangle. The facilitator does not understand Bob's long explanation and suggest that he should wait to explain it to all. When they arrive he tries to remember his formula with some difficulty but at the end posts an elaborated textbox which based on some of the team members suggestions (to use x1, y1, x2, y2) gets finalized as this:
In an n by m rectangle (n>m), the number of ways to get from the top left corner to the bottom right corner without retracing steps or going backwards (where going backwards is defined as making a move such that it would result in making the distance between the point and the top left corner shorter) is: (m+n) choose (m) or (m+n) choose (n) or (m+n)!/(m!n!) Using coordinates (x1,y1) and (x2,y2) with those acting as the top left corner and the lower right corner respectively, the number of ways to get from (x1,y1) to (x2,y2) is: (|x1-x2|+|y1-y2|)! choose (|x1-x2|)! or (|x1-x2|+|y1-y2|)! choose (|x1-x2|)! Definition of choose: n choose k is n!/((n-k)!k!)
No further work is done on this formula. Did they understand it now? We don't know, no further work is done to achieve understanding! That kind of work is very different than agreement work or acceptance work. Then they move to work on the cylinder problem. Here there is a combination of agreement work (since Bob position himself as knowing the right answer) and exploration/shared-understanding work. Mathisfun engages in the shared work of developing understanding: creates a cylinder, a case, a bird's eye view, etc. Bob states a procedure to get to an answer: "think right angles / and circumference of base circle / finding the distance would involve the law of cosines, but that's ok"). Later, he treats the exploration as checkup: "give the coordinates of the points and I can give you the distance after you roll it up". That approach appears very different than "letz say that AB was that on the cylinder / and that on the paper / everyone see them?" posted by mi while making some drawings on the whiteboard). At the end, bob engages with mi's exploratory talk, in part, because mi is always asking for assessment, but at the end closes with "the rest is self-explanatory (unless you don't know what a 30-60-90 triangle is)". Then they moved to another "case."
The idea of case is very important, and how a sense of procedure is being developed (e.g. Mi says "so do we go through the same steps again?). Bob and mathisfun dominate this dialog with bob still as the expert. Qwer only comes much later when bob uses the word "chord" and she asks "What's a chord?" At then end, bob and mi seem satisfied with the procedure they developed and prompted by the facilitator, write a textbox that describes their method:
the way to find the shortest length between two points on the lateral side of a cylinder can be done by first finding the vertical lenght of the two points by using the distance between the two points and then finding then finding the horizontal will use the law of cosines in with the equation is c^2 = a^2+b^2-2ab*cos(c). With this we are able to determine the horizontal side. By using the pythagorean theorem to find out the length between the two.
When asked about what they would like to do next time Qw post "maybe use points on another shape" Interesting because of his little participation but it seems as if there is interest in maybe trying to understand with other cases?
Session IV
Starts with the facilitator being very active (long message on a textbox). Bob remarks that "the other 2 aren't here yet though" The facilitator has prepared a problem definition based on their ideas (is it?). Bob starts by saying "the way i see it you do the same thing you did with the circle". Mi draws the prism and then asks "so should I make the bird's eye view?"
36 mathisfun, 20:16 (19.05): k letz get started 37 bob123, 20:16 (19.05): the way i see it, you do the same thing you did with the circle 38 mathisfun, 20:16 (19.05): alright 39 mathisfun, 20:16 (19.05): so letz draw the triangular prism 40 mathisfun, 20:18 (19.05): there 41 mathisfun, 20:19 (19.05): so should i make the bird's eye view? 42 bob123, 20:19 (19.05): yeah 43 mathisfun, 20:19 (19.05): k 44 mathisfun, 20:20 (19.05): there
I said it earlier
bob123, 20:29 (19.05): i said use the law of cosines earlier [Refering to: bob123, 20:24 (19.05): using law of cosines]
Implicit references to prior work? (e.g. "again")
98 mathisfun, 20:33 (19.05): lets go on 99 bob123, 20:33 (19.05): ok 100 bob123, 20:34 (19.05): so you use the length that you find 101 bob123, 20:34 (19.05): as the leg of the right triangle 102 bob123, 20:34 (19.05): once again 103 bob123, 20:34 (19.05): and then use the height as the other leg 104 mathisfun, 20:34 (19.05): phythageoran right? 105 bob123, 20:34 (19.05): yep
They work on the prism problem and come to a solution. The facilitator is not asked for assessment but for another problem. He then asks a question that they respond and after that the session is brought to a close.
Noticings
- There is continuity of some interactional reasoning procedures such as using a bird's eye view, or testing cases
- Strong role played by bob, and bob and mi. Qwer did not return to session IV... a coincidence?
- When asked about their understanding of a solution they did not feel ready to take ownership but ascribed authorship to the original person who posted it
Group composition: Stable
Session 1: mi bb qw lv Session 2: mi bb qw mr(L) Session 3: mi bb qw [ho] Session 4: mi bb (L) Late, little participation after being corrected after joining, some postings towards the end [ ] zero participation
Grid-World vs. Diagonals
Session 1: define shortest way (grid) and straight distance (diagonal), work on grid mostly, high engagement! Session 2: continue work on number of paths, move to 3D grid (some of it does not follow rules but is new problem) Session 3: continue work on number of paths, continue work on 3D-cylinder grid Session 4: 3D grid and other variations (e.g. prism)
