Resolving Differences: Twists and Turns in a Quasi ‑ Synchronous Online Collaborative Mathematics...

Resolving Differences: Twists and Resolving Differences: Twists and Turns in a Quasi‑Synchronous Turns in a Quasi‑Synchronous

Online Collaborative Mathematics Online Collaborative Mathematics Problem-Solving SessionProblem-Solving Session

Ramon Prudencio S. ToledoRamon Prudencio S. ToledoAlan R. ZemelAlan R. Zemel

Gerry StahlGerry StahlThe Virtual Math Teams &The Virtual Math Teams &

College of Information Science and TechnologyCollege of Information Science and Technologyat Drexel University, Philadelphia, USAat Drexel University, Philadelphia, USA

16 July 200716 July 2007 The VMT at Drexel UniversityThe VMT at Drexel University

ProblemProblem


DataData

AIM® log of PoWwow 10AIM® log of PoWwow 10 Participants: self-selected, self-described Participants: self-selected, self-described

middle-school studentsmiddle-school students Problem available before the sessionProblem available before the session Moderator available to assist in the Moderator available to assist in the

technology but not in problem solvingtechnology but not in problem solving Excerpt: 010 – 077 of roughly 580 linesExcerpt: 010 – 077 of roughly 580 lines


An An excerptexcerpt from VMT chat from VMT chat010 Alice (7:01:05 PM): Is this everyone?011 MFPowwow (7:01:15 PM): If you create a picture that you would like to share with your group, you can mail it to powwow or you can make a direct connection with me.012 MFPowwow (7:01:24 PM): This is everyone tonight.013 Alice (7:01:36 PM): ok014 Alice (7:02:04 PM): so…..015 MFPowwow (7:02:12 PM): So you’ve all seen the problem. If you’ve got any ideas, now’s the time to start. :-) 014 Alice (7:02:29 PM): Ok017 Alice (7:02:38 PM): Anyone have a pic?018 Mario (7:02:58 PM): Just the one with the problem statement019 Alice (7:03:17 PM): lol020 Mario (7:03:25 PM): Should we label some points?021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?024 Alice (7:04:38 PM): The center?025 Mario (7:04:47 PM): No, down at the vertex026 Alice (7:04:52 PM): oh027 Alice (7:05:00 PM): That might help028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.029 Mario (7:05:20 PM): You name where the green line meets the base030 Alice (7:05:30 PM): B031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see what BV is033 Alice (7:06:40 PM): Lets not 034 Mario (7:06:46 PM): What's your idea?035 Alice (7:07:01 PM): It states that something is wrong with the pic. 036 Alice (7:07:08 PM): so we can't find what BV is037 Mario (7:07:31 PM): Yeah, and I think if we 'found' BV, it would be something not possible


How do participants in an online How do participants in an online quasi-synchronous collaborative quasi-synchronous collaborative

mathematics problem solving mathematics problem solving resolve their differences?resolve their differences?

How are resources achieved?How are resources achieved? How are choices and order of approach How are choices and order of approach

determined in problem solving?determined in problem solving? How is the validity of an approach collaboratively How is the validity of an approach collaboratively

assessed?assessed? How is the appropriateness of a proposed How is the appropriateness of a proposed

solution collaboratively assessed?solution collaboratively assessed?


How are resources achieved?How are resources achieved?

Resources are situational: they have to be brought into the problem Resources are situational: they have to be brought into the problem solvingsolving020 Mario (7:03:25 PM): Should we label some points?020 Mario (7:03:25 PM): Should we label some points?021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?024 Alice (7:04:38 PM): The center?024 Alice (7:04:38 PM): The center?025 Mario (7:04:47 PM): No, down at the vertex025 Mario (7:04:47 PM): No, down at the vertex026 Alice (7:04:52 PM): oh026 Alice (7:04:52 PM): oh027 Alice (7:05:00 PM): That might help027 Alice (7:05:00 PM): That might help028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.029 Mario (7:05:20 PM): You name where the green line meets the base029 Mario (7:05:20 PM): You name where the green line meets the base030 Alice (7:05:30 PM): B030 Alice (7:05:30 PM): B031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong 031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong

with the pic.with the pic.


How are resources achieved?How are resources achieved?

Resources are situational: they have to be Resources are situational: they have to be brought into the problem solvingbrought into the problem solving shapesshapes

069 Mario (7:12:14 PM): Otherwise, the red lines 069 Mario (7:12:14 PM): Otherwise, the red lines and the base are almost an and the base are almost an equilateral triangleequilateral triangle

070 Alice (7:12:32 PM): I think this requires trig 070 Alice (7:12:32 PM): I think this requires trig

071 Mario (7:12:50 PM): So, one possible wrong 071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really a thing is, this is really a hexagonhexagon


How are choices and order of How are choices and order of approach determined in problem approach determined in problem

solving?solving?

Cooperation vs. collaborationCooperation vs. collaboration Participant invitationsParticipant invitations Participants choose:Participants choose:

Collaborating participantsCollaborating participants Which approaches are to be usedWhich approaches are to be used The order in which the approaches are usedThe order in which the approaches are used


Collaboration vs. cooperationCollaboration vs. cooperation

Management of collective as a collaborative group which Management of collective as a collaborative group which will work together instead of subdividing the problem will work together instead of subdividing the problem among its membersamong its members017 Alice (7:02:38 PM): Anyone have a pic?017 Alice (7:02:38 PM): Anyone have a pic?018 Mario (7:02:58 PM): Just the one with the problem statement018 Mario (7:02:58 PM): Just the one with the problem statement019 Alice (7:03:17 PM): lol019 Alice (7:03:17 PM): lol020 Mario (7:03:25 PM): Should we label some points?020 Mario (7:03:25 PM): Should we label some points?021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?024 Alice (7:04:38 PM): The center?024 Alice (7:04:38 PM): The center?025 Mario (7:04:47 PM): No, down at the vertex025 Mario (7:04:47 PM): No, down at the vertex026 Alice (7:04:52 PM): oh026 Alice (7:04:52 PM): oh027 Alice (7:05:00 PM): That might help027 Alice (7:05:00 PM): That might help


Participant invitationsParticipant invitations

For co-participantsFor co-participants Non-directiveNon-directive DirectiveDirective

However, there can be self-selectionHowever, there can be self-selection



Participant invitation: non-directiveParticipant invitation: non-directive Non-directive = invitation does not prescribe a task for Non-directive = invitation does not prescribe a task for

invited participantinvited participant010 Alice (7:01:05 PM): Is this everyone?010 Alice (7:01:05 PM): Is this everyone?011 MFPowwow (7:01:15 PM): If you create a picture that you 011 MFPowwow (7:01:15 PM): If you create a picture that you

would like to share with your group, you can mail it to would like to share with your group, you can mail it to powwow or you can make a direct connection with me.powwow or you can make a direct connection with me.

012 MFPowwow (7:01:24 PM): This is everyone tonight.012 MFPowwow (7:01:24 PM): This is everyone tonight.013 Alice (7:01:36 PM): ok013 Alice (7:01:36 PM): ok014 Alice (7:02:04 PM): so…..014 Alice (7:02:04 PM): so…..015 MFPowwow (7:02:12 PM): So you’ve all seen the problem. 015 MFPowwow (7:02:12 PM): So you’ve all seen the problem.

If you’ve got any ideas, now’s the time to start. :-) If you’ve got any ideas, now’s the time to start. :-) 014 Alice (7:02:29 PM): Ok014 Alice (7:02:29 PM): Ok



Participant invitation: non-directiveParticipant invitation: non-directive Non-directive = invitation does not prescribe a Non-directive = invitation does not prescribe a

task for invited participanttask for invited participant034 Mario (7:06:46 PM): What's your idea?034 Mario (7:06:46 PM): What's your idea?

035 Alice (7:07:01 PM): It states that something is 035 Alice (7:07:01 PM): It states that something is wrong with the pic. wrong with the pic.

036 Alice (7:07:08 PM): so we can't find what BV is036 Alice (7:07:08 PM): so we can't find what BV is

037 Mario (7:07:31 PM): Yeah, and I think if we 037 Mario (7:07:31 PM): Yeah, and I think if we 'found' BV, it would be something not possible'found' BV, it would be something not possible



Expected response to non-directive invitation: Expected response to non-directive invitation: availabilityavailability

010 Alice (7:01:05 PM): Is this everyone?010 Alice (7:01:05 PM): Is this everyone?

011 MFPowwow (7:01:15 PM): If you create a picture that you 011 MFPowwow (7:01:15 PM): If you create a picture that you would like to share with your group, you can mail it to would like to share with your group, you can mail it to powwow or you can make a direct connection with me.powwow or you can make a direct connection with me.

012 MFPowwow (7:01:24 PM): This is everyone tonight.012 MFPowwow (7:01:24 PM): This is everyone tonight.

013 Alice (7:01:36 PM): ok013 Alice (7:01:36 PM): ok

014 Alice (7:02:04 PM): so…..014 Alice (7:02:04 PM): so…..

015 MFPowwow (7:02:12 PM): So you’ve all seen the problem. 015 MFPowwow (7:02:12 PM): So you’ve all seen the problem. If you’ve got any ideas, now’s the time to start. :-) If you’ve got any ideas, now’s the time to start. :-)

014 Alice (7:02:29 PM): Ok014 Alice (7:02:29 PM): Ok



Participant invitation: DirectiveParticipant invitation: Directive Directive = expects a particular form of Directive = expects a particular form of

participation from other participantsparticipation from other participants017 Alice (7:02:38 PM): Anyone have a pic?017 Alice (7:02:38 PM): Anyone have a pic?018 Mario (7:02:58 PM): Just the one with the 018 Mario (7:02:58 PM): Just the one with the

problem statementproblem statement019 Alice (7:03:17 PM): lol019 Alice (7:03:17 PM): lol020 Mario (7:03:25 PM): Should we label some 020 Mario (7:03:25 PM): Should we label some

points?points?021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that



Participant invitation: DirectiveParticipant invitation: Directive Expected response: follow through of directiveExpected response: follow through of directive

017 Alice (7:02:38 PM): Anyone have a pic?017 Alice (7:02:38 PM): Anyone have a pic?

018 Mario (7:02:58 PM):018 Mario (7:02:58 PM): Just the one with the Just the one with the problem statementproblem statement

019 Alice (7:03:17 PM): lol019 Alice (7:03:17 PM): lol

020 Mario (7:03:25 PM): Should we label some 020 Mario (7:03:25 PM): Should we label some points?points?

021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O

022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that


Self-selectionSelf-selection

Self-selection = Self-selection = participant volunteersparticipant volunteers through repetitions through repetitions of claims or proposalsof claims or proposals017 Alice (7:02:38 PM): Anyone have a pic?017 Alice (7:02:38 PM): Anyone have a pic?

* * ** * *028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.

* * ** * *031 Alice (7:06:19 PM): I have an idea that might help us find 031 Alice (7:06:19 PM): I have an idea that might help us find

whats wrong with the pic.whats wrong with the pic.* * ** * *

035 Alice (7:07:01 PM): It states that something is wrong with the 035 Alice (7:07:01 PM): It states that something is wrong with the pic. pic.

* * ** * *062 Alice (7:11:28 PM): I know whats wrong with the pic062 Alice (7:11:28 PM): I know whats wrong with the pic



solving?solving? Constitution of the problem as a shared problemConstitution of the problem as a shared problem

Identification of resources, approaches and solutionsIdentification of resources, approaches and solutions017 Alice (7:02:38 PM): Anyone have a pic?017 Alice (7:02:38 PM): Anyone have a pic?018 Mario (7:02:58 PM): Just the one with the problem statement018 Mario (7:02:58 PM): Just the one with the problem statement019 Alice (7:03:17 PM): lol019 Alice (7:03:17 PM): lol020 Mario (7:03:25 PM): Should we label some points?020 Mario (7:03:25 PM): Should we label some points?021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?024 Alice (7:04:38 PM): The center?024 Alice (7:04:38 PM): The center?025 Mario (7:04:47 PM): No, down at the vertex025 Mario (7:04:47 PM): No, down at the vertex026 Alice (7:04:52 PM): oh026 Alice (7:04:52 PM): oh027 Alice (7:05:00 PM): That might help027 Alice (7:05:00 PM): That might help028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.029 Mario (7:05:20 PM): You name where the green line meets the base029 Mario (7:05:20 PM): You name where the green line meets the base030 Alice (7:05:30 PM): B030 Alice (7:05:30 PM): B031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see what BV is032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see what BV is033 Alice (7:06:40 PM): Lets not 033 Alice (7:06:40 PM): Lets not



solving?solving? Revelation of some strategyRevelation of some strategy

Produce steps which describe the strategy.Produce steps which describe the strategy. Build on the formulation of the problem.Build on the formulation of the problem.

017 Alice (7:02:38 PM): Anyone have a pic?017 Alice (7:02:38 PM): Anyone have a pic?018 Mario (7:02:58 PM): Just the one with the problem statement018 Mario (7:02:58 PM): Just the one with the problem statement019 Alice (7:03:17 PM): lol019 Alice (7:03:17 PM): lol020 Mario (7:03:25 PM): Should we label some points?020 Mario (7:03:25 PM): Should we label some points?021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?024 Alice (7:04:38 PM): The center?024 Alice (7:04:38 PM): The center?025 Mario (7:04:47 PM): No, down at the vertex025 Mario (7:04:47 PM): No, down at the vertex026 Alice (7:04:52 PM): oh026 Alice (7:04:52 PM): oh027 Alice (7:05:00 PM): That might help027 Alice (7:05:00 PM): That might help028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.029 Mario (7:05:20 PM): You name where the green line meets the base029 Mario (7:05:20 PM): You name where the green line meets the base030 Alice (7:05:30 PM): B030 Alice (7:05:30 PM): B031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see what BV is032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see what BV is033 Alice (7:06:40 PM): Lets not 033 Alice (7:06:40 PM): Lets not



solving?solving? The pronominal ‘we’ and its variants are used to solicit participation.The pronominal ‘we’ and its variants are used to solicit participation. Co-participants are solicited to participate in one’s approach.Co-participants are solicited to participate in one’s approach.

Produce steps which describe the strategy.Produce steps which describe the strategy. Build on the formulation of the problem.Build on the formulation of the problem.

017 Alice (7:02:38 PM): Anyone have a pic?017 Alice (7:02:38 PM): Anyone have a pic?018 Mario (7:02:58 PM): Just the one with the problem statement018 Mario (7:02:58 PM): Just the one with the problem statement019 Alice (7:03:17 PM): lol019 Alice (7:03:17 PM): lol020 Mario (7:03:25 PM): Should 020 Mario (7:03:25 PM): Should wewe label some points? label some points?021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?024 Alice (7:04:38 PM): The center?024 Alice (7:04:38 PM): The center?025 Mario (7:04:47 PM): No, down at the vertex025 Mario (7:04:47 PM): No, down at the vertex026 Alice (7:04:52 PM): oh026 Alice (7:04:52 PM): oh027 Alice (7:05:00 PM): That might help027 Alice (7:05:00 PM): That might help028 Alice (7:05:16 PM): 028 Alice (7:05:16 PM): LetsLets find out whats wrong with the pic first. find out whats wrong with the pic first.029 Mario (7:05:20 PM): 029 Mario (7:05:20 PM): You name where the green line meets the baseYou name where the green line meets the base030 Alice (7:05:30 PM): B030 Alice (7:05:30 PM): B031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.032 Mario (7:06:30 PM): 032 Mario (7:06:30 PM): WeWe could use good ol' Pythag thm to see what BV is could use good ol' Pythag thm to see what BV is033 Alice (7:06:40 PM): 033 Alice (7:06:40 PM): LetsLets not not



solving?solving? Resolution may be through tokens of agreementResolution may be through tokens of agreement

066 Alice (7:11:45 PM): The diagnol is not 4.6066 Alice (7:11:45 PM): The diagnol is not 4.6067 Mario (7:11:51 PM): Right067 Mario (7:11:51 PM): Right068 Fatima (7:12:02 PM): exactly068 Fatima (7:12:02 PM): exactly069 Mario (7:12:14 PM): Otherwise, the red lines and the base are 069 Mario (7:12:14 PM): Otherwise, the red lines and the base are

almost an equilateral trianglealmost an equilateral triangle070 Alice (7:12:32 PM): I think this requires trig 070 Alice (7:12:32 PM): I think this requires trig 071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really 071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really

a hexagona hexagon072 Alice (7:12:56 PM): No072 Alice (7:12:56 PM): No073 Mario (7:13:01 PM): Right073 Mario (7:13:01 PM): Right074 Alice (7:13:09 PM): Im talking about the triangle diagnol074 Alice (7:13:09 PM): Im talking about the triangle diagnol075 Mario (7:13:11 PM): Let'sd stick with octagon075 Mario (7:13:11 PM): Let'sd stick with octagon076 Mario (7:13:24 PM): So we assume 4 is right076 Mario (7:13:24 PM): So we assume 4 is right077 Alice (7:13:32 PM): yes077 Alice (7:13:32 PM): yes



solving?solving? Resolution may be through accommodation to an Resolution may be through accommodation to an

alternative approachalternative approach028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.029 Mario (7:05:20 PM): You name where the green line meets the 029 Mario (7:05:20 PM): You name where the green line meets the

basebase030 Alice (7:05:30 PM): B030 Alice (7:05:30 PM): B031 Alice (7:06:19 PM): I have an idea that might help us find whats 031 Alice (7:06:19 PM): I have an idea that might help us find whats

wrong with the pic.wrong with the pic.032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see 032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see

what BV iswhat BV is033 Alice (7:06:40 PM): Lets not033 Alice (7:06:40 PM): Lets not034 Mario (7:06:46 PM): What's your idea?034 Mario (7:06:46 PM): What's your idea?035 Alice (7:07:01 PM): It states that something is wrong with the pic. 035 Alice (7:07:01 PM): It states that something is wrong with the pic. 036 Alice (7:07:08 PM): so we can't find what BV is036 Alice (7:07:08 PM): so we can't find what BV is037 Mario (7:07:31 PM): Yeah, and I think if we 'found' BV, it would be 037 Mario (7:07:31 PM): Yeah, and I think if we 'found' BV, it would be

something not possible something not possible


How is the validity of an approach How is the validity of an approach collaboratively assessed?collaboratively assessed?

Appeal to the problem formulation (Appeal to the problem formulation (appeal to the writer of the problemappeal to the writer of the problem))035 Alice (7:07:01 PM): It states that something is wrong with the pic. 035 Alice (7:07:01 PM): It states that something is wrong with the pic. 036 Alice (7:07:08 PM): so we can't find what BV is036 Alice (7:07:08 PM): so we can't find what BV is037 Mario (7:07:31 PM): Yeah, and I think if we 'found' BV, it would be something not 037 Mario (7:07:31 PM): Yeah, and I think if we 'found' BV, it would be something not

possiblepossible Appeal to a familiar method (Appeal to a familiar method (Pythagorean TheoremPythagorean Theorem))

020 Mario (7:03:25 PM): Should we label some points?020 Mario (7:03:25 PM): Should we label some points?021 Mario (7:03:56 PM): Like, center is O021 Mario (7:03:56 PM): Like, center is O022 Alice (7:04:04 PM): We could do that022 Alice (7:04:04 PM): We could do that023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?023 Mario (7:04:21 PM): Vertex where red line meets is, what, V?024 Alice (7:04:38 PM): The center?024 Alice (7:04:38 PM): The center?025 Mario (7:04:47 PM): No, down at the vertex025 Mario (7:04:47 PM): No, down at the vertex026 Alice (7:04:52 PM): oh026 Alice (7:04:52 PM): oh027 Alice (7:05:00 PM): That might help027 Alice (7:05:00 PM): That might help028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.028 Alice (7:05:16 PM): Lets find out whats wrong with the pic first.029 Mario (7:05:20 PM): You name where the green line meets the base029 Mario (7:05:20 PM): You name where the green line meets the base030 Alice (7:05:30 PM): B030 Alice (7:05:30 PM): B031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.031 Alice (7:06:19 PM): I have an idea that might help us find whats wrong with the pic.032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see what BV is032 Mario (7:06:30 PM): We could use good ol' Pythag thm to see what BV is


How is the appropriateness of a How is the appropriateness of a proposed solution collaboratively proposed solution collaboratively

assessed?assessed?058 Mario (7:11:10 PM): With the numbers given, BV would be 058 Mario (7:11:10 PM): With the numbers given, BV would be 059 Mario (7:11:11 PM): yeah059 Mario (7:11:11 PM): yeah060 Alice (7:11:14 PM): I think thats wrong060 Alice (7:11:14 PM): I think thats wrong061 Fatima (7:11:19 PM): how so?061 Fatima (7:11:19 PM): how so?062 Alice (7:11:28 PM): I know whats wrong with the pic062 Alice (7:11:28 PM): I know whats wrong with the pic063 Mario (7:11:31 PM): base would be twice that063 Mario (7:11:31 PM): base would be twice that064 Fatima (7:11:33 PM): what064 Fatima (7:11:33 PM): what065 Mario (7:11:41 PM): 4.54 ish065 Mario (7:11:41 PM): 4.54 ish066 Alice (7:11:45 PM): The diagnol is not 4.6066 Alice (7:11:45 PM): The diagnol is not 4.6067 Mario (7:11:51 PM): Right067 Mario (7:11:51 PM): Right068 Fatima (7:12:02 PM): exactly068 Fatima (7:12:02 PM): exactly069 Mario (7:12:14 PM): Otherwise, the red lines and the base are almost an equilateral triangle069 Mario (7:12:14 PM): Otherwise, the red lines and the base are almost an equilateral triangle070 Alice (7:12:32 PM): I think this requires trig 070 Alice (7:12:32 PM): I think this requires trig 071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really a hexagon071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really a hexagon072 Alice (7:12:56 PM): No072 Alice (7:12:56 PM): No073 Mario (7:13:01 PM): Right073 Mario (7:13:01 PM): Right074 Alice (7:13:09 PM): Im talking about the triangle diagnol074 Alice (7:13:09 PM): Im talking about the triangle diagnol075 Mario (7:13:11 PM): Let'sd stick with octagon075 Mario (7:13:11 PM): Let'sd stick with octagon076 Mario (7:13:24 PM): So we assume 4 is right076 Mario (7:13:24 PM): So we assume 4 is right077 Alice (7:13:32 PM): yes077 Alice (7:13:32 PM): yes


How is the appropriateness of a proposed How is the appropriateness of a proposed solution collaboratively assessed?solution collaboratively assessed?

In this case, appropriateness is based on mutual In this case, appropriateness is based on mutual agreement(!) agreement(!)

058 Mario (7:11:10 PM): With the numbers given, BV would be 058 Mario (7:11:10 PM): With the numbers given, BV would be 059 Mario (7:11:11 PM): yeah059 Mario (7:11:11 PM): yeah060 Alice (7:11:14 PM): I think thats wrong060 Alice (7:11:14 PM): I think thats wrong061 Fatima (7:11:19 PM): how so?061 Fatima (7:11:19 PM): how so?062 Alice (7:11:28 PM): I know whats wrong with the pic062 Alice (7:11:28 PM): I know whats wrong with the pic063 Mario (7:11:31 PM): base would be twice that063 Mario (7:11:31 PM): base would be twice that064 Fatima (7:11:33 PM): what064 Fatima (7:11:33 PM): what065 Mario (7:11:41 PM): 4.54 ish065 Mario (7:11:41 PM): 4.54 ish066 Alice (7:11:45 PM): The diagnol is not 4.6066 Alice (7:11:45 PM): The diagnol is not 4.6067 Mario (7:11:51 PM): Right067 Mario (7:11:51 PM): Right068 Fatima (7:12:02 PM): exactly068 Fatima (7:12:02 PM): exactly069 Mario (7:12:14 PM): Otherwise, the red lines and the base are almost an equilateral triangle069 Mario (7:12:14 PM): Otherwise, the red lines and the base are almost an equilateral triangle070 Alice (7:12:32 PM): I think this requires trig 070 Alice (7:12:32 PM): I think this requires trig 071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really a hexagon071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really a hexagon072 Alice (7:12:56 PM): No072 Alice (7:12:56 PM): No073 Mario (7:13:01 PM): Right073 Mario (7:13:01 PM): Right074 Alice (7:13:09 PM): Im talking about the triangle diagnol074 Alice (7:13:09 PM): Im talking about the triangle diagnol075 Mario (7:13:11 PM): Let'sd stick with octagon075 Mario (7:13:11 PM): Let'sd stick with octagon076 Mario (7:13:24 PM): So we assume 4 is right076 Mario (7:13:24 PM): So we assume 4 is right077 Alice (7:13:32 PM): yes077 Alice (7:13:32 PM): yes


How is the appropriateness of a proposed How is the appropriateness of a proposed solution collaboratively assessed?solution collaboratively assessed?

The agreement is based on a result which The agreement is based on a result which neither proposal obtained, but the group moves neither proposal obtained, but the group moves on. on.

058 Mario (7:11:10 PM): With the numbers given, BV would be 058 Mario (7:11:10 PM): With the numbers given, BV would be 059 Mario (7:11:11 PM): yeah059 Mario (7:11:11 PM): yeah060 Alice (7:11:14 PM): I think thats wrong060 Alice (7:11:14 PM): I think thats wrong061 Fatima (7:11:19 PM): how so?061 Fatima (7:11:19 PM): how so?062 Alice (7:11:28 PM): I know whats wrong with the pic062 Alice (7:11:28 PM): I know whats wrong with the pic063 Mario (7:11:31 PM): base would be twice that063 Mario (7:11:31 PM): base would be twice that064 Fatima (7:11:33 PM): what064 Fatima (7:11:33 PM): what065 Mario (7:11:41 PM): 4.54 ish065 Mario (7:11:41 PM): 4.54 ish066 Alice (7:11:45 PM): The diagnol is not 4.6066 Alice (7:11:45 PM): The diagnol is not 4.6067 Mario (7:11:51 PM): Right067 Mario (7:11:51 PM): Right068 Fatima (7:12:02 PM): exactly068 Fatima (7:12:02 PM): exactly069 Mario (7:12:14 PM): Otherwise, the red lines and the base are almost an equilateral triangle069 Mario (7:12:14 PM): Otherwise, the red lines and the base are almost an equilateral triangle070 Alice (7:12:32 PM): I think this requires trig 070 Alice (7:12:32 PM): I think this requires trig 071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really a hexagon071 Mario (7:12:50 PM): So, one possible wrong thing is, this is really a hexagon072 Alice (7:12:56 PM): No072 Alice (7:12:56 PM): No073 Mario (7:13:01 PM): Right073 Mario (7:13:01 PM): Right074 Alice (7:13:09 PM): Im talking about the triangle diagnol074 Alice (7:13:09 PM): Im talking about the triangle diagnol075 Mario (7:13:11 PM): Let'sd stick with octagon075 Mario (7:13:11 PM): Let'sd stick with octagon076 Mario (7:13:24 PM): So we assume 4 is right076 Mario (7:13:24 PM): So we assume 4 is right077 Alice (7:13:32 PM): yes077 Alice (7:13:32 PM): yes


Summative findingsSummative findings Each proponent of a proposal wants the Each proponent of a proposal wants the

collective to use their proposal.collective to use their proposal. SpurnedSpurned proponents find alternate ways to proponents find alternate ways to

reallocate participation by recycling a proposal reallocate participation by recycling a proposal or presenting a new step which reorients or presenting a new step which reorients participants to the participants to the spurnedspurned proposal. proposal.

Participants use Participants use achieved shared resourcesachieved shared resources to to build competing proposals.build competing proposals.

Due to the quasi-synchronous nature of chat, Due to the quasi-synchronous nature of chat, participants may select parts of competing participants may select parts of competing proposals and append them to their own proposals and append them to their own proposals.proposals.


Next stepsNext steps

Investigation of resolution of differences in Investigation of resolution of differences in quasi-synchronous online collaborative quasi-synchronous online collaborative environments with dual interaction spaces environments with dual interaction spaces ((shared whiteboardshared whiteboard and and chat windowchat window))

Resolving Differences: Twists and Turns in a Quasi ‑ Synchronous Online Collaborative Mathematics...

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