participation frameworks and decision-making in problem-solving chats progress report january 12,...

Participation Frameworks Participation Frameworks and Decision-making in and Decision-making in Problem-Solving ChatsProblem-Solving Chats

Progress ReportProgress Report

January 12, 2005January 12, 2005

Problem SolvingProblem Solving

We have observed in the chat transcripts We have observed in the chat transcripts that that problem-solvingproblem-solving is a set of is a set of mathematical actions whereby actors mathematical actions whereby actors come to constitute the problem as the come to constitute the problem as the activities performed to produce a solutionactivities performed to produce a solution. .


This notion of the problem is different from the This notion of the problem is different from the “problem” posted on the website. “problem” posted on the website.

With an interactional approach, “With an interactional approach, “the problemthe problem” ” participants are concerned with is participants are concerned with is their own their own emergent sense of what they are doingemergent sense of what they are doing. .

In this view, the “problem” emerges by working In this view, the “problem” emerges by working to discover what the problem is.to discover what the problem is.


The discovery of the problem The discovery of the problem may or may notmay or may not occur during the chat as a collective occur during the chat as a collective achievement. achievement. Participants in chats who have engaged in this Participants in chats who have engaged in this

discovery “discovery “off-lineoff-line” typically report their “work” in ways ” typically report their “work” in ways that do no implicate the participation of recipients in that do no implicate the participation of recipients in the performance of the math activities described in the performance of the math activities described in the report.the report.

Alternatively, participants may organize themselves to Alternatively, participants may organize themselves to collectively discover the problem by inviting and collectively discover the problem by inviting and allocating the participation of actors allocating the participation of actors in and as the in and as the discovery of the problemdiscovery of the problem. .


Whether they do the discovery offline or with the Whether they do the discovery offline or with the participation of chat participants, participation of chat participants, the “problem” is the “problem” is something that actors must produce for themselvessomething that actors must produce for themselves using the resources afforded them by: using the resources afforded them by:

their individual (and, when appropriate distributed) their individual (and, when appropriate distributed) competencecompetence to recognize what mathematical actions may be appropriate to to recognize what mathematical actions may be appropriate to perform and to actually specify and perform those actions, perform and to actually specify and perform those actions,

the the problem informationproblem information posted on the website, and posted on the website, and

the the technical affordancestechnical affordances of the chat environment itself. of the chat environment itself.


In the next slide, we see an example of how an In the next slide, we see an example of how an actor works to produce the extra problem as a actor works to produce the extra problem as a collectively endorsed “interpretation” of the collectively endorsed “interpretation” of the posted text on the website. posted text on the website.

What is apparent in this excerpt is that whatever What is apparent in this excerpt is that whatever is posted as the “problem text” is not what is posted as the “problem text” is not what participants take to be the problem, but serves participants take to be the problem, but serves as a resource by which they come to constitute as a resource by which they come to constitute the problem in terms of projected or achieved the problem in terms of projected or achieved mathematical activities. mathematical activities.

119 MCP What's this extra saying? Like, if both of the smaller triasngles aresitting on their bases, the base of one is 5 and the base of theother is 7? Is that the interpretation?

120 REA I guess it is 10121 MCP If they're oriented with corresp angles in corresp locations?122 REA or should I say 9.8123 REA what do you think124 REA i used the proportions125 MCP Oh, I guess this is where that 7 from AH3's answer came from,

way back there. I didn't know where that came from.126 MCP I still need to make sure I know what the wording is saying. Am I

interpreting the q right?127 MCP MUR, can you say whether I'm interpreting the extra question's

set-up correctly?128 MUR Hmm, I am really not allowed to do that. Why don't you discuss

this with your partner?129 REA Probably130 MCP REA, is what I said earlier the way you're reading it?131 REA my dad got 10.6132 MCP Hmm. Ok, I'll go with what I think it's saying, and think out loud.133 MCP Here are two similartriangles, oriented the same way, both sitting

on their bases.134 REA what it means is that one side is 5 and 7135 MCP One's base is 5 and the other's is 7.136 MCP OPne's height is, who knows, so call it h. The other's is (7/5)h137 REA wait138 MCP THe area of the first is .5*5*h, and the other is .5*7*(7/5)h.139 MCP What?140 MCP THey're similar, so their heights are in the proportion 7 to 5.141 MCP So the larger has a height that's 7/5 as big as the smaller's height.142 REA I might have gotten it143 REA the answer is 10.57

(Powwow 2-2)


In the next example, AVR’s encounter with the problem In the next example, AVR’s encounter with the problem text leads to the formulation of a “first” problem for text leads to the formulation of a “first” problem for participants, i.e., determining the area of each triangle participants, i.e., determining the area of each triangle referenced in the problem text. referenced in the problem text.

This excerpt also shows that in certain circumstances, This excerpt also shows that in certain circumstances, problems emerge in the interaction.problems emerge in the interaction.

One method actors use in problem solving situations is to use One method actors use in problem solving situations is to use available resources to produce an “initial” problem in terms of available resources to produce an “initial” problem in terms of projected mathematical activities.projected mathematical activities.

In other words, a “problem” may actually consist of a sequence In other words, a “problem” may actually consist of a sequence or series of problems constituted by an actor or by actors each or series of problems constituted by an actor or by actors each of which is made provisionally relevant by prior problems, math of which is made provisionally relevant by prior problems, math activities and produced results.activities and produced results.

31 GER Here's the problem that you'll be working on tonight:32 GER If two equilateral triangles have edgelengths of 9 cubits and 12

cubits, what's the edgelength of the equilateral triangle whosearea is equal to the sum of the areas of the other two?

33 GER You can also read the problem athttp://mathforum.org/pow/vmt/feb2604/problem.html

34 AVR hmmm35 AVR interesting36 GER If you create a picture that you would like to share with your

group, there are instructions on the problem page about how todo that.

37 PIN very38 AVR I think we can crack it, though39 AVR **begins to scribble on paper**40 AVR or should I not do that?41 PIN doesnt matter42 AVR got it43 AVR **proceeds with scribbling..**44 AVR Okay, I think we should start with the formula for the area of a

triangle45 SUP ok46 AVR A = 1/2bh47 AVR I believe48 PIN yes49 PIN i concue50 PIN concur*51 AVR then find the area of each triangle

(PowWow 2-1)

Participation FrameworksParticipation Frameworks

Close inspection of the chat transcripts Close inspection of the chat transcripts has made evident that has made evident that chat participants chat participants organize their participation to achieve what organize their participation to achieve what they consider to be the task at handthey consider to be the task at hand. .

The “task at hand” is not the same in all The “task at hand” is not the same in all chats, or even in the same chat at different chats, or even in the same chat at different times or locations in sequence of postings times or locations in sequence of postings (What was it Heraclitus said?)(What was it Heraclitus said?)..

Participation FrameworksParticipation Frameworks While actors may engage in a wide variety of activities, While actors may engage in a wide variety of activities,

there are two kinds of participation frameworks, which there are two kinds of participation frameworks, which appear to be characteristic features of PowWows, to which appear to be characteristic features of PowWows, to which participants in chats appear to be oriented: participants in chats appear to be oriented:

““ExpositoryExpository” participation framework is one in which an actor reports on ” participation framework is one in which an actor reports on mathematical actions that have yielded relevant and meaningful results in mathematical actions that have yielded relevant and meaningful results in ways that ways that do not implicate the participation of other actorsdo not implicate the participation of other actors in the in the specification of or performance of these actions or the production of these specification of or performance of these actions or the production of these results.results.

““ExploratoryExploratory” participation framework is one in which actors iteratively: ” participation framework is one in which actors iteratively: • participate in participate in decision-makingdecision-making regarding what mathematical activities regarding what mathematical activities

to perform, to perform, • specify and perform decided-upon specify and perform decided-upon mathematical activitiesmathematical activities, and , and • assess these activities and the resultsassess these activities and the results obtained through them. obtained through them.

Expository ParticipationExpository Participation

An actor reports on the sense he or she has An actor reports on the sense he or she has made of a problem. made of a problem.

Reporting is done through a sequence of Reporting is done through a sequence of multiple postings in which a sequence of multiple postings in which a sequence of mathematical activities that produced a relevant mathematical activities that produced a relevant result is described.result is described.

Such reporting has many of the characteristics Such reporting has many of the characteristics of storytelling or news reporting (Sacks, 1992; of storytelling or news reporting (Sacks, 1992; Jefferson, 1978).Jefferson, 1978).

Expository ParticipationExpository Participation

In the next excerpt, AH3 produces a report using an In the next excerpt, AH3 produces a report using an organization remarkably similar to the organization of a organization remarkably similar to the organization of a story as described by Jefferson (1978):story as described by Jefferson (1978):

““[S]torytelling can involve a story preface with which a teller [S]torytelling can involve a story preface with which a teller projects a forthcoming story, a next turn in which teller projects a forthcoming story, a next turn in which teller produces the story, a next in which a coparticipant aligns produces the story, a next in which a coparticipant aligns himself as a story recipient, a next in which teller produces the himself as a story recipient, a next in which teller produces the story, and a next in which story recipient talks by reference to story, and a next in which story recipient talks by reference to the story. Further, the story preface can have consequences the story. Further, the story preface can have consequences for the story’s reception, and thus a rather extended series of for the story’s reception, and thus a rather extended series of turns at talk can be seen as a coherent conversational unit.”turns at talk can be seen as a coherent conversational unit.”

(Jefferson 1978, p. 219)(Jefferson 1978, p. 219)

22 MCP Probably a straight area compute. B4 I do it, I want to guess, ok?23 REA Have any of you guys learend of the 30-60-90 concept24 AH3 I think I have the solution!25 REA what26 MCP I guess 1527 REA k28 MCP I think it's like the Pythagorean idea, applying to triangles.29 AH3 sqrt(5^2 + 7^2) = sqrt(74)30 MCP Yes, 30-60-90 is needed fact31 AH3 The solution is sqrt(74)32 REA how33 MCP 7?34 AH3 Go to...35 AH3 http://mathforum.org/dr.math/faq/form ulas/faq.triangle.html36 AH3 Under scalene triangle, the formula for the area of any triangle is...37 AH3 K = a^2 * sin(B) * sin(C)/[2 sin (A)]38 AH3 Why is that smiley their39 AH3 K = a^2 * sin( B )* sin(C)/[2 sin (A)]40 AH3 Where a = an edgelength of an isosceles triangle41 REA MUR, do we solve the extra42 AH3 What do you think

(PowWow 2-2)

102 AME They could learn from me 103 KOH ahem 104 KOH so Ameem, please explain your way... 105 KIL yea 106 KIL please? 107 AME well 108 AME I just used some equations 109 AME equation 1- x + r = 8 110 KIL which are? 111 KOH ok... simple equations or complicated ones? 112 KIL yes 113 AME Simple 114 KOH ok... 115 AME As I was Saying 116 KOH where did that 1 came from? 117 AME The first equation is x + r = 8 118 KOH yes! 119 KIL i get it... 120 AME The second is 16 + x^2 = r^2 121 AME Now we substitute 122 KIL i c 123 AME And we are DONE!!! 124 KOH thats my way!!! 125 KIL i c now! 126 AME My way just makes more sense 127 KIL i c

(powwow 18)

(powwow 18)

258 KOH I want to say that the center of the circleis not the center of the square

259 AME so what is it260 ROB yes261 KOH you guys ever heard of pythagorean

theorem?262 KIL yes263 AME ya264 ROB of course265 KOH good266 AME what do you think we are dumb267 KIL a^2 + b^2 = c^2268 AME go on269 KOH so you draw a line from C to A270 AME yes271 KIL yes272 KOH which is the radius that you wanna find273 AME yes274 KIL yes275 ROB yes276 KOH you draw a perpendicular line from

point C to AE277 KOH the intersection will be marked D278 ROB yes279 AME ok280 KIL ok281 AME AD = 4 right?282 KOH AD = 4283 KOH yes284 KIL yes285 KOH so you wanna find DC, too286 ROB yes287 KIL so 16 = a ^2 + b ^2288 KOH please note that D, C, B, and F are all

on the same line289 AME ok290 KIL yes291 KIL noted292 ROB ok293 KOH and that DF equals 8294 ROB ya295 KIL yes296 AME yes297 KIL then...

(powwow 18)

298 KOH since DC + CF = 8, and we know CF(which is r), so DC = 8-r

299 AME ok300 ROB ok301 AME so its basically my way302 AME just shorter?303 KOH substitude : r^2 = 4^2 + (8-r)^2304 KIL ok305 ROB uh306 KOH and simplifying : never mind...307 AME r^2 = 16 + 64 +r^2 -16r308 KOH I am too lazy to type out my whole thing309 AME ok310 KIL right311 KOH wow...312 ROB jeez313 AME Well does everyone get it314 KOH cancel r^2 from both sides315 KIL 16r = 80316 KIL ?317 KOH yes318 KIL r = 80/16319 KOH cool!320 AME yup321 KIL =?322 AME r = 5323 KIL ya324 KIL i knew that325 KOH hey!326 AME ok327 ROB o?328 KOH ok, nevermind...329 KIL yay!330 AME So does everyone get how to do this?331 KIL R = 5332 KIL !333 KOH I am still developing my second way for

Extra

Exploratory ParticipationExploratory Participation

Actors may engage in what is recognizable and Actors may engage in what is recognizable and observable as observable as problem-solvingproblem-solving and and decision-makingdecision-making, in , in which:which:

Candidate projected mathematical activitiesCandidate projected mathematical activities are identified (to some are identified (to some degree of relevant specificity) and put forward for consideration by degree of relevant specificity) and put forward for consideration by others in the chat.others in the chat.

Decisions are made with respect to uptakeDecisions are made with respect to uptake of (i.e., allocating of (i.e., allocating participation in) these projected mathematical activities.participation in) these projected mathematical activities.


In the next example, AME invites recipients to consider a In the next example, AME invites recipients to consider a projected set of math activities. projected set of math activities.

The matter is considered but is identified as possibly The matter is considered but is identified as possibly problematic by AZN. problematic by AZN.

Rather than abandon the projected course, AME Rather than abandon the projected course, AME modifies his proposal and suggests a different allocation modifies his proposal and suggests a different allocation of participation in which he takes up the projected set of of participation in which he takes up the projected set of activities while other participants “continue” with the activities while other participants “continue” with the activities in which they were engaged. activities in which they were engaged.

LIF, on the other hand, offers an alternative set of math LIF, on the other hand, offers an alternative set of math activities for recipients to consider.activities for recipients to consider.

(PowWow 9)

310 AME i have an idea 311 FIR k 312 AZN yes? 313 AME Why don't try putting num. in place of n 314 FIR huh/ 315 AME and see if it works out 316 AZN ok...but that might take long 317 FIR what does that do? 318 AME we could find a range 319 FIR oh 320 FIR how 321 LIF yah 322 EEF ya 323 AME Or how about i work on that 324 AME and you continue 325 AZN n is a variable, meaning that we can replace it with

numbers and see wut happens 326 AZN r u sure? 327 AME sure 328 LIF yes but we are trying to solve this ALGEBRAICALLY

not guess and check 329 AME I'll be here 330 AZN true 331 AME I say guess and check goes faster 332 FIR that's what I did 333 LIF not necesarilly 334 AME any way to solve the problem will work 335 LIF especially when you are dealing with 2 336 AME why not try multiple 337 EEF guess and check isnt a good strategy 338 FIR well if you keep going upwards it is 339 AME ok 340 LIF ohk so we know taht n2 + 4n + 4 + 9 > n2 + 5n 341 AME Lets forget about what i jest said 342 AME *just 343 AZN ok then


If and when a projected set of mathematical activities is If and when a projected set of mathematical activities is endorsed (i.e., decided upon, usually through a endorsed (i.e., decided upon, usually through a consensus-like set of decision procedures), actors: consensus-like set of decision procedures), actors:

Specify and engage in the activitiesSpecify and engage in the activities that had been projected, and that had been projected, and

Assess their ongoing actions and any results obtainedAssess their ongoing actions and any results obtained in in performing these actions for how they constitute members’ performing these actions for how they constitute members’ sense of the problem. sense of the problem.

Participation FrameworksParticipation Frameworks

Both exploratory and expository Both exploratory and expository participation frameworks may occur in the participation frameworks may occur in the same chatsame chat..

Which framework members elect to deploy Which framework members elect to deploy

is a members’ matter that is is a members’ matter that is worked out worked out in in situsitu and is and is independent of the prior independent of the prior accessibility or availability of the “problem accessibility or availability of the “problem information” as posted on the websiteinformation” as posted on the website..

Next StepsNext Steps

Examine the way decision making is done in Examine the way decision making is done in these chats, especially with respect to the these chats, especially with respect to the organization of participation in the chat.organization of participation in the chat.

Continue to detail how purportedly “solvable Continue to detail how purportedly “solvable math problems” emerge in terms of: math problems” emerge in terms of: inviting participation in projected math activities, inviting participation in projected math activities, specifying of what both the math activities and specifying of what both the math activities and

participation consist, and participation consist, and assessment of participation in math activities in terms assessment of participation in math activities in terms

of results and their relevance for subsequent action.of results and their relevance for subsequent action.

Next StepsNext Steps

Continue to detail how reports are produced.Continue to detail how reports are produced.

Continue to examine the chats for the Continue to examine the chats for the interactional basis of observed statistical results.interactional basis of observed statistical results.

participation frameworks and decision-making in problem-solving chats progress report january 12,...

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