Resolving Differences: Twists and Turns in a QuasiSynchronous Online Collaborative Mathematics Probl

1
Resolving Differences Twists and Turns in a
Quasi-Synchronous Online Collaborative
Mathematics Problem-Solving Session

• Ramon Prudencio S. Toledo
• Alan R. Zemel
• Gerry Stahl
• The Virtual Math Teams
• College of Information Science and Technology
• at Drexel University, Philadelphia, USA

2
Problem
3
Data

• AIM log of PoWwow 10
• Participants self-selected, self-described
  middle-school students
• Problem available before the session
• Moderator available to assist in the technology
  but not in problem solving
• Excerpt 010 077 of roughly 580 lines

4
An excerpt from VMT chat
010 Alice (70105 PM) Is this everyone? 011
MFPowwow (70115 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 (70124 PM)
This is everyone tonight. 013 Alice (70136 PM)
ok 014 Alice (70204 PM) so.. 015 MFPowwow
(70212 PM) So youve all seen the problem. If
youve got any ideas, nows the time to start.
-) 014 Alice (70229 PM) Ok 017 Alice
(70238 PM) Anyone have a pic? 018 Mario
(70258 PM) Just the one with the problem
statement 019 Alice (70317 PM) lol 020 Mario
(70325 PM) Should we label some points? 021
Mario (70356 PM) Like, center is O 022 Alice
(70404 PM) We could do that 023 Mario (70421
PM) Vertex where red line meets is, what, V? 024
Alice (70438 PM) The center? 025 Mario
(70447 PM) No, down at the vertex 026 Alice
(70452 PM) oh 027 Alice (70500 PM) That
might help 028 Alice (70516 PM) Lets find out
whats wrong with the pic first. 029 Mario
(70520 PM) You name where the green line meets
the base 030 Alice (70530 PM) B 031 Alice
(70619 PM) I have an idea that might help us
find whats wrong with the pic. 032 Mario (70630
PM) We could use good ol' Pythag thm to see what
BV is 033 Alice (70640 PM) Lets not 034 Mario
(70646 PM) What's your idea? 035 Alice
(70701 PM) It states that something is wrong
with the pic. 036 Alice (70708 PM) so we
can't find what BV is 037 Mario (70731 PM)
Yeah, and I think if we 'found' BV, it would be
something not possible
5
How do participants in an online
quasi-synchronous collaborative mathematics
problem solving resolve their differences?

• How are resources achieved?
• How are choices and order of approach determined
  in problem solving?
• How is the validity of an approach
  collaboratively assessed?
• How is the appropriateness of a proposed solution
  collaboratively assessed?

6
How are resources achieved?

• Resources are situational they have to be
  brought into the problem solving
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that
• 023 Mario (70421 PM) Vertex where red line
  meets is, what, V?
• 024 Alice (70438 PM) The center?
• 025 Mario (70447 PM) No, down at the vertex
• 026 Alice (70452 PM) oh
• 027 Alice (70500 PM) That might help
• 028 Alice (70516 PM) Lets find out whats wrong
  with the pic first.
• 029 Mario (70520 PM) You name where the green
  line meets the base
• 030 Alice (70530 PM) B
• 031 Alice (70619 PM) I have an idea that might
  help us find whats wrong with the pic.

7
How are resources achieved?

• Resources are situational they have to be
  brought into the problem solving
• shapes
• 069 Mario (71214 PM) Otherwise, the red lines
  and the base are almost an equilateral triangle
• 070 Alice (71232 PM) I think this requires
  trig
• 071 Mario (71250 PM) So, one possible wrong
  thing is, this is really a hexagon

8
How are choices and order of approach determined
in problem solving?

• Cooperation vs. collaboration
• Participant invitations
• Participants choose
• Collaborating participants
• Which approaches are to be used
• The order in which the approaches are used

9
Collaboration vs. cooperation

• Management of collective as a collaborative group
  which will work together instead of subdividing
  the problem among its members
• 017 Alice (70238 PM) Anyone have a pic?
• 018 Mario (70258 PM) Just the one with the
  problem statement
• 019 Alice (70317 PM) lol
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that
• 023 Mario (70421 PM) Vertex where red line
  meets is, what, V?
• 024 Alice (70438 PM) The center?
• 025 Mario (70447 PM) No, down at the vertex
• 026 Alice (70452 PM) oh
• 027 Alice (70500 PM) That might help

10
Participant invitations

• For co-participants
• Non-directive
• Directive
• However, there can be self-selection

11
Participant invitations

• Participant invitation non-directive
• Non-directive invitation does not prescribe a
  task for invited participant
• 010 Alice (70105 PM) Is this everyone?
• 011 MFPowwow (70115 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 (70124 PM) This is everyone
  tonight.
• 013 Alice (70136 PM) ok
• 014 Alice (70204 PM) so..
• 015 MFPowwow (70212 PM) So youve all seen the
  problem. If youve got any ideas, nows the time
  to start. -)
• 014 Alice (70229 PM) Ok

12
Participant invitations

• Participant invitation non-directive
• Non-directive invitation does not prescribe a
  task for invited participant
• 034 Mario (70646 PM) What's your idea?
• 035 Alice (70701 PM) It states that something
  is wrong with the pic.
• 036 Alice (70708 PM) so we can't find what BV
  is
• 037 Mario (70731 PM) Yeah, and I think if we
  'found' BV, it would be something not possible

13
Participant invitations

• Expected response to non-directive invitation
  availability
• 010 Alice (70105 PM) Is this everyone?
• 011 MFPowwow (70115 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 (70124 PM) This is everyone
  tonight.
• 013 Alice (70136 PM) ok
• 014 Alice (70204 PM) so..
• 015 MFPowwow (70212 PM) So youve all seen the
  problem. If youve got any ideas, nows the time
  to start. -)
• 014 Alice (70229 PM) Ok

14
Participant invitations

• Participant invitation Directive
• Directive expects a particular form of
  participation from other participants
• 017 Alice (70238 PM) Anyone have a pic?
• 018 Mario (70258 PM) Just the one with the
  problem statement
• 019 Alice (70317 PM) lol
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that

15
Participant invitations

• Participant invitation Directive
• Expected response follow through of directive
• 017 Alice (70238 PM) Anyone have a pic?
• 018 Mario (70258 PM) Just the one with the
  problem statement
• 019 Alice (70317 PM) lol
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that

16
Self-selection

• Self-selection participant volunteers through
  repetitions of claims or proposals
• 017 Alice (70238 PM) Anyone have a pic?
• 028 Alice (70516 PM) Lets find out whats wrong
  with the pic first.
• 031 Alice (70619 PM) I have an idea that might
  help us find whats wrong with the pic.
• 035 Alice (70701 PM) It states that something
  is wrong with the pic.
• 062 Alice (71128 PM) I know whats wrong with
  the pic

17
How are choices and order of approach determined
in problem solving?

• Constitution of the problem as a shared problem
• Identification of resources, approaches and
  solutions
• 017 Alice (70238 PM) Anyone have a pic?
• 018 Mario (70258 PM) Just the one with the
  problem statement
• 019 Alice (70317 PM) lol
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that
• 023 Mario (70421 PM) Vertex where red line
  meets is, what, V?
• 024 Alice (70438 PM) The center?
• 025 Mario (70447 PM) No, down at the vertex
• 026 Alice (70452 PM) oh
• 027 Alice (70500 PM) That might help
• 028 Alice (70516 PM) Lets find out whats wrong
  with the pic first.
• 029 Mario (70520 PM) You name where the green
  line meets the base
• 030 Alice (70530 PM) B
• 031 Alice (70619 PM) I have an idea that might
  help us find whats wrong with the pic.
• 032 Mario (70630 PM) We could use good ol'
  Pythag thm to see what BV is
• 033 Alice (70640 PM) Lets not

18
How are choices and order of approach determined
in problem solving?

• Revelation of some strategy
• Produce steps which describe the strategy.
• Build on the formulation of the problem.
• 017 Alice (70238 PM) Anyone have a pic?
• 018 Mario (70258 PM) Just the one with the
  problem statement
• 019 Alice (70317 PM) lol
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that
• 023 Mario (70421 PM) Vertex where red line
  meets is, what, V?
• 024 Alice (70438 PM) The center?
• 025 Mario (70447 PM) No, down at the vertex
• 026 Alice (70452 PM) oh
• 027 Alice (70500 PM) That might help
• 028 Alice (70516 PM) Lets find out whats wrong
  with the pic first.
• 029 Mario (70520 PM) You name where the green
  line meets the base
• 030 Alice (70530 PM) B
• 031 Alice (70619 PM) I have an idea that might
  help us find whats wrong with the pic.
• 032 Mario (70630 PM) We could use good ol'
  Pythag thm to see what BV is

19
How are choices and order of approach determined
in problem solving?

• The pronominal we and its variants are used to
  solicit participation.
• Co-participants are solicited to participate in
  ones approach.
• Produce steps which describe the strategy.
• Build on the formulation of the problem.
• 017 Alice (70238 PM) Anyone have a pic?
• 018 Mario (70258 PM) Just the one with the
  problem statement
• 019 Alice (70317 PM) lol
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that
• 023 Mario (70421 PM) Vertex where red line
  meets is, what, V?
• 024 Alice (70438 PM) The center?
• 025 Mario (70447 PM) No, down at the vertex
• 026 Alice (70452 PM) oh
• 027 Alice (70500 PM) That might help
• 028 Alice (70516 PM) Lets find out whats wrong
  with the pic first.
• 029 Mario (70520 PM) You name where the green
  line meets the base
• 030 Alice (70530 PM) B
• 031 Alice (70619 PM) I have an idea that might
  help us find whats wrong with the pic.

20
How are choices and order of approach determined
in problem solving?

• Resolution may be through tokens of agreement
• 066 Alice (71145 PM) The diagnol is not 4.6
• 067 Mario (71151 PM) Right
• 068 Fatima (71202 PM) exactly
• 069 Mario (71214 PM) Otherwise, the red lines
  and the base are almost an equilateral triangle
• 070 Alice (71232 PM) I think this requires
  trig
• 071 Mario (71250 PM) So, one possible wrong
  thing is, this is really a hexagon
• 072 Alice (71256 PM) No
• 073 Mario (71301 PM) Right
• 074 Alice (71309 PM) Im talking about the
  triangle diagnol
• 075 Mario (71311 PM) Let'sd stick with octagon
• 076 Mario (71324 PM) So we assume 4 is right
• 077 Alice (71332 PM) yes

21
How are choices and order of approach determined
in problem solving?

• Resolution may be through accommodation to an
  alternative approach
• 028 Alice (70516 PM) Lets find out whats wrong
  with the pic first.
• 029 Mario (70520 PM) You name where the green
  line meets the base
• 030 Alice (70530 PM) B
• 031 Alice (70619 PM) I have an idea that might
  help us find whats wrong with the pic.
• 032 Mario (70630 PM) We could use good ol'
  Pythag thm to see what BV is
• 033 Alice (70640 PM) Lets not
• 034 Mario (70646 PM) What's your idea?
• 035 Alice (70701 PM) It states that something
  is wrong with the pic.
• 036 Alice (70708 PM) so we can't find what BV
  is
• 037 Mario (70731 PM) Yeah, and I think if we
  'found' BV, it would be something not possible

22
How is the validity of an approach
collaboratively assessed?

• Appeal to the problem formulation (appeal to the
  writer of the problem)
• 035 Alice (70701 PM) It states that something
  is wrong with the pic.
• 036 Alice (70708 PM) so we can't find what BV
  is
• 037 Mario (70731 PM) Yeah, and I think if we
  'found' BV, it would be something not possible
• Appeal to a familiar method (Pythagorean Theorem)
• 020 Mario (70325 PM) Should we label some
  points?
• 021 Mario (70356 PM) Like, center is O
• 022 Alice (70404 PM) We could do that
• 023 Mario (70421 PM) Vertex where red line
  meets is, what, V?
• 024 Alice (70438 PM) The center?
• 025 Mario (70447 PM) No, down at the vertex
• 026 Alice (70452 PM) oh
• 027 Alice (70500 PM) That might help
• 028 Alice (70516 PM) Lets find out whats wrong
  with the pic first.
• 029 Mario (70520 PM) You name where the green
  line meets the base
• 030 Alice (70530 PM) B
• 031 Alice (70619 PM) I have an idea that might
  help us find whats wrong with the pic.
• 032 Mario (70630 PM) We could use good ol'
  Pythag thm to see what BV is

23
How is the appropriateness of a proposed solution
collaboratively assessed?

• 058 Mario (71110 PM) With the numbers given,
  BV would be
• 059 Mario (71111 PM) yeah
• 060 Alice (71114 PM) I think thats wrong
• 061 Fatima (71119 PM) how so?
• 062 Alice (71128 PM) I know whats wrong with
  the pic
• 063 Mario (71131 PM) base would be twice that
• 064 Fatima (71133 PM) what
• 065 Mario (71141 PM) 4.54 ish
• 066 Alice (71145 PM) The diagnol is not 4.6
• 067 Mario (71151 PM) Right
• 068 Fatima (71202 PM) exactly
• 069 Mario (71214 PM) Otherwise, the red lines
  and the base are almost an equilateral triangle
• 070 Alice (71232 PM) I think this requires
  trig
• 071 Mario (71250 PM) So, one possible wrong
  thing is, this is really a hexagon
• 072 Alice (71256 PM) No
• 073 Mario (71301 PM) Right
• 074 Alice (71309 PM) Im talking about the
  triangle diagnol
• 075 Mario (71311 PM) Let'sd stick with octagon
• 076 Mario (71324 PM) So we assume 4 is right

24
How is the appropriateness of a proposed solution
collaboratively assessed?

• In this case, appropriateness is based on mutual
  agreement(!)
• 058 Mario (71110 PM) With the numbers given,
  BV would be
• 059 Mario (71111 PM) yeah
• 060 Alice (71114 PM) I think thats wrong
• 061 Fatima (71119 PM) how so?
• 062 Alice (71128 PM) I know whats wrong with
  the pic
• 063 Mario (71131 PM) base would be twice that
• 064 Fatima (71133 PM) what
• 065 Mario (71141 PM) 4.54 ish
• 066 Alice (71145 PM) The diagnol is not 4.6
• 067 Mario (71151 PM) Right
• 068 Fatima (71202 PM) exactly
• 069 Mario (71214 PM) Otherwise, the red lines
  and the base are almost an equilateral triangle
• 070 Alice (71232 PM) I think this requires
  trig
• 071 Mario (71250 PM) So, one possible wrong
  thing is, this is really a hexagon
• 072 Alice (71256 PM) No
• 073 Mario (71301 PM) Right
• 074 Alice (71309 PM) Im talking about the
  triangle diagnol
• 075 Mario (71311 PM) Let'sd stick with octagon

25
How is the appropriateness of a proposed solution
collaboratively assessed?

• The agreement is based on a result which neither
  proposal obtained, but the group moves on.
• 058 Mario (71110 PM) With the numbers given,
  BV would be
• 059 Mario (71111 PM) yeah
• 060 Alice (71114 PM) I think thats wrong
• 061 Fatima (71119 PM) how so?
• 062 Alice (71128 PM) I know whats wrong with
  the pic
• 063 Mario (71131 PM) base would be twice that
• 064 Fatima (71133 PM) what
• 065 Mario (71141 PM) 4.54 ish
• 066 Alice (71145 PM) The diagnol is not 4.6
• 067 Mario (71151 PM) Right
• 068 Fatima (71202 PM) exactly
• 069 Mario (71214 PM) Otherwise, the red lines
  and the base are almost an equilateral triangle
• 070 Alice (71232 PM) I think this requires
  trig
• 071 Mario (71250 PM) So, one possible wrong
  thing is, this is really a hexagon
• 072 Alice (71256 PM) No
• 073 Mario (71301 PM) Right
• 074 Alice (71309 PM) Im talking about the
  triangle diagnol
• 075 Mario (71311 PM) Let'sd stick with octagon

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Summative findings

• Each proponent of a proposal wants the collective
  to use their proposal.
• Spurned proponents find alternate ways to
  reallocate participation by recycling a proposal
  or presenting a new step which reorients
  participants to the spurned proposal.
• Participants use achieved shared resources to
  build competing proposals.
• Due to the quasi-synchronous nature of chat,
  participants may select parts of competing
  proposals and append them to their own proposals.

27
Next steps

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