Group Cognition in Online Collaborative Math Problem
Solving

Gerry Stahl

Drexel
University

This is a case study of online collaboration on an algebra
problem. It adapts the methodology of conversation analysis to
quasi-synchronous, text-based chat room technology. The analysis is conducted
within the context of a design-based research effort, so a primary goal is to
identify technological barriers caused by standard chat technology with an eye
to designing a more appropriate and supportive online collaborative learning
environment.

“Group cognition” is a theoretical framework in
which cognitive processes are identified as resulting from the dynamic
interaction of multiple personal interpretive perspectives within contexts of
group discourse and collaboration. The analysis is conducted within a
theoretical framework that focuses attention on the small group unit of
analysis as the site of problem-solving agency, rather than on cognitive
processes of the individual participants. The analysis results in the
identification of interactive methods of “doing mathematics” as a
group. This, in turn, reflects back on the theoretical framework and refines
the notion of group cognition.

The analysis aims to motivate the following theoretical,
methodological and design-based claims:

·
The discourse displays elements of mathematical
understanding, problem-solving strategies and logical rationality by the group that
parallel those of individual students.

·
Interaction among the student participants can
be conceptualized as an instance of “group cognition.”

·
Excerpts of online collaborative math problem
solving can productively be analyzed at the small group unit of analysis.

·
The methodology of conversation analysis can
effectively be adapted to interpret text-based online interaction.

·
Group cognition displays the potential to
achieve more than the individual participants seem capable of accomplishing on
their own, but also displays interactional problems that prevent the group from
achieving its full problem-solving potential.

·
Conclusions can be drawn from such an analysis
that are relevant to the design of improved computer-supported collaborative
learning environments.

Introduction

A micro-analysis is conducted of a three-and-a-half
minute long excerpt from an online interaction. The interaction took place
among three students participating in the Virtual Math Teams service of the
Math Forum. Conversation analysis of the interaction highlights various methods
that the group of students engages in—both mathematical and social interaction
methods. It identifies interactions through which the group members constitute
the group as a problem-solving agent, define individual roles within the group,
establish the style of communication, define their individual and group
identities, articulate the problem, suggest math strategies, make mathematical
proposals and negotiate math knowledge. Close analysis of the dialogical work
done by sequences of chat postings reveals complex social and mathematical
moves that can be characterized as instances of group cognition.

In the chat log, members’ postings constitute the
group discourse as such and orient to it as salient. For instance, individuals
make proposals to be shared by the group. These proposals are often explicitly
presented as the individual’s personal opinion, but acceptance by another
group member makes them part of the shared flow of considerations and sets them
up for being built upon by anyone in the group. Subsequent postings reference
them, identifying them as integral to the group discourse.

Doing mathematics together online

Computers offer many opportunities for innovation in
education. One of the major avenues is by supporting the building of
collaborative knowledge (Stahl, 2006). For instance, it is now
possible for students around the world to work together on challenging math
problems. Through online discussion, they can share problem-solving experiences
and gain fluency in communicating mathematically.

In a research project at the Math Forum @ Drexel (http://mathforum.org), we have begun to invite
middle school students to participate in online chats about interesting
problems in beginning algebra and geometry. The following problem, discussed in
the example in this paper, is typical:

If two equilateral triangles have edge-lengths of 9 cubits and
12 cubits, what is the edge-length of the equilateral triangle whose area is
equal to the sum of the areas of the other two?

We rely on a variety of methods from the learning sciences
to guide our research and to analyze the results of our trials. In particular,
we use conversation analysis (Pomerantz & Fehr,
1991; Psathas, 1995; Sacks, Schegloff, & Jefferson, 1974; Sacks, 1992; ten
Have, 1999)
to interpret the interactions that take place in the student chats. In this
paper, we adapt the findings of conversation analysis to math chats and develop
a specific form of adjacency pairs that seem to be important for math chats.
Before presenting this, it may be useful to describe briefly how the notion of adjacency
pairs differs from naïve conceptions of conversation.

There is a widespread common-sense or folk-theory (Bereiter, 2002;
Dennett, 1991)
view of conversation as the exchange of propositions (Shannon & Weaver,
1949).
This view was refined and formalized by logicians and cognitive scientists as
involving verbal expression in meaningful statements by individuals, based on
their internal mental representations. Speech served to transfer meanings from
the mind of a speaker to the mind of a listener, who then interpreted the
expressed message. Following Wittgenstein (1953) in critiquing this view,
speech act theory (Austin, 1952; Searle,
1969)
argued that the utterances spoken by individuals were ways of acting in the
world, and were meaningful in terms of what they accomplished through their use
and effects. Of course, the expression, transmission and interpretation of
meaning by individuals can be problematic, and people frequently have to do
some interactional work in order to re-establish a shared understanding. The
construction of common ground has been seen as the attempt to coordinate
agreement between individual understandings (Clark & Brennan,
1991).

Conversation analysis takes a different view of
conversation. It looks at how interactional mechanisms, like the use of adjacency pairs (Duranti, 1998;
Schegloff, 1991),
co-construct inter-subjectivity. Adjacency pairs are common sequences of
utterances by different people—such as mutual greetings or
question/answer interchanges—that form a meaningful speech act spanning
multiple utterances that cannot be attributed to an individual or to the expression
of mental states. We are interested in what kinds of adjacency pairs are
typical for math chats.

Online math chats differ from ordinary informal conversation
in a number of ways. They are focused on the task of solving a specific problem
and they take place within a somewhat formal institutional setting. They
involve the doing of mathematics (Livingston, 1986). And, of course, they are
computer-mediated rather than being face-to-face. The approach of conversation
analysis is based on ethnomethodology (Garfinkel, 1967), which involves the study of
the methods that people use to accomplish what they are doing. So, we are
interested in working out the methods that are used by students in online math
chats. In this paper we discuss a particular method of collaboration in math
chats that we have elsewhere called exploratory
participation: participants engage each other in the conjoint discovery and
production of both the problem and possible solutions (Zemel, Xhafa, &
Stahl, 2005).

The medium of online chat has its own peculiarities. Most
importantly, it is a text-based medium, where interaction takes place by the
sequential response of brief texts to each other (Livingston, 1995;
Zemel, 2005).
As a quasi-synchronous medium (Garcia & Jacobs,
1999),
chat causes confusion because several people can be typing at once and their
texts can appear in an order that obscures what they are responding to.
Furthermore, under time pressure to submit their texts so that they will appear
near what they are responding to, some chat participants break their messages
into several short texts. Because of these peculiarities of chat, it is
necessary for researchers to carefully reconstruct the intended threading of
texts that respond to each other before attempting to interpret the flow of
interaction (Cakir et al., 2005; Strijbos & Stahl,
2005).

Math proposal adjacency pairs

In order to begin to analyze the methods that students
use in math chats, we take a close look at an excerpt from an actual chat.
Figure 1 shows an excerpt from near the beginning of the log of one of our
first online collaborative math problem-solving sessions. Three
students—named Avr, Sup and Pin—have just entered the chat room,
said hello to each other and read the problem involving three triangles.

The first thing to notice here is a pattern of proposals,
discussions and acceptances similar to what takes place in face-to-face
discourse. Proposals about steps in solving the math problem are made by Avr in
lines 1, 3, 8, 17 and by Pin in lines 20, 27. These proposals are each affirmed
by someone else in lines 2, 6, 10, 19, 22, 28, respectively.

To avoid chat confusion, note that line 21 responds to line
19, while line 22 responds to line 20. The timestamps show that lines 20 and 21
effectively overlapped each other chronologically: Avr was typing line 21
before she saw line 20. Similarly, lines 24 and the following were responses to
line 20, not line 23. We will correct for these confusions in Figure 2, which
reproduces a key passage in this excerpt.

In Figure 1, we see several examples of a
three step pattern:

1.
A proposal is made by an individual for the
group to work on: “I think we should ….”

2.
An acceptance is made on behalf of the group:
“Ok,” “right”

3.
There is an elaboration of the proposal by
members of the group. The proposed work is begun, often with a secondary
proposal for the first sub-step.

This suggests that collaborative problem-solving of
mathematics may often involve a particular form of adjacency pair. We will call
this a math proposal adjacency pair.

Many adjacency pairs allow for insertion of other pairs
between the two parts of the original pair, delaying completion of the pair.
For instance, a question/answer pair may be interrupted by utterances seeking
clarification of the question; the clarification interaction may itself consist
of question/answer pairs, possibly with their own clarifications—this may
continue recursively. With math proposal adjacency pairs, the subsidiary pairs
seem to come after the completion of the original pair, in the form of
secondary proposals, questions or explanations that start to do the work that
was proposed in the original pair.

Proposals seem to lead to some kind of further mathematical
work as a response to carrying out what was proposed. Often—as seen in
the current example—that work consists of making further proposals. It is
striking that the proposed work is not begun until there is agreement with the
proposal. This may represent consent by the group as a whole to pursue the
proposed line of work. Of course, it is not so clear in the current example,
where there are only three participants and the interaction often seems to take
place primarily between pairs of participants. As confirmed by other chat
examples, however, the proposal generally seems to be addressed to the whole
group and opens the floor for participants other than the proposer to respond.
The use of “we” in “we should” or “we have
to” (stated or implied) constitutes the multiple participants as a plural
subject, an effective unified group (Lerner, 1993). Any one other than the
proposer may respond on behalf of the group.

Moreover, there seems to be what in conversation analysis is
called an interactional preference
(Schegloff, Jefferson,
& Sacks, 1977)
for acceptance of the proposal. That is, if one accepts a proposal, it suffices
to briefly indicate agreement: “ok.” If one wants to reject a
proposal, then one has to account for this response by giving reasons.

We would like to characterize in more detail the method of
making math proposal adjacency pairs. Often, the nature of an interactional
method is seen most clearly when it is breached (Garfinkel, 1967). Methods are generally taken
for granted by people; they are not made visible or conducted consciously. It
is only when there is a breakdown (Heidegger, 1927/1996) in the smooth, tacit
performance of a method that people focus on its characteristics in order to
overcome the breakdown. The normally transparent method becomes visible in its
breach. We can interpret Sup’s posting in line 23 as a failed proposal. Given the mathematics
of the triangle problem, a proposal like Sup’s related to proportionality
might have been fruitful. However, in this chat, line 23 was effectively
ignored by the group. While its character as a failed proposal did not become
visible to the participants, it can become clear to us by comparing it to
successful proposals in the same chat and by reflecting on its situation in the
chat in order to ask why it was not a successful proposal.

A failed proposal

Let us look at line 23 in its immediate interactional
context in Figure 2. We can distinguish a number of ways in which it differed
from successful math proposals that solicited responses and formed math
proposal adjacency pairs.

(a) All the other proposals (1, 3, 8, 17, 20,
27) were stated in relatively complete sentences. Additionally, some of the
proposals were introduced with a phrase to indicate that they were the
speaker’s proposal (1. “I think we should …,” 17.
“I think we have to …,” 20. “i know how …”
and 27. “i think i got it …”). The exceptions to these were
simply continuations of previous proposals: line 3 provided the formula
proposed in line 1 and line 8 proposed to “then” use that formula.
Line 23, by contrast, provided a single word with a question mark. There was no
syntactic context (other than the question mark) within the line for
interpreting that word and there was no reference to semantic context outside
of the line. Line 23 did not respond in any clear way to a previous line and
did not provide any alternative reference to a context in the original problem
statement or elsewhere. For instance, Sup could have said, “I think we
should compute the proportion of the height to the base of those equilateral
triangles.”

(b) The timing of line 23 was particularly unfortunate. It
exactly overlapped a line from Avr. Since Avr had been setting the pace for
group problem solving during this part of the chat, the fact that she was
involved in following a different line of inquiry spelled death for an
alternative proposal at the time of line 23. Pin either seemed to be continuing
on his own thread without acknowledging anyone else at this point, or else he
was responding too late to previous postings. So a part of the problem for Sup
was that there was little sense of a coherent group process—and what
sense there was did not include him. If he was acting as part of the group
process, for instance posing a question in reaction to Pin and in parallel to
Avr, he was not doing a good job of it and so his contribution was ignored in
the group process. It is true that a possible advantage of text-based
interaction like chat over face-to-face interaction is that there may be a
broader time window for responding to previous contributions. In face-to-face
conversation, turn-taking rules may define appropriate turns for response that
expire in a fraction of a second as the conversation moves on. In
computer-based chat, the turn-taking sequence is more open. However, even here
if one is responding to a posting that is several lines away, it is important
to make explicit somehow to what one is responding. He could have said,
“I know another way to find the height – using proportions.”
Sup’s posting does not do anything like that; it relies purely upon
sequential timing to establish its context, and that fails in this case.

(c) Sup’s posting 23 came right after Pin’s
proposal 20: “draw the altitude.” Avr had responded to this with 22
(“right”) but Pin seems to have ignored that. Pin’s proposal
had opened up work to be done and both Avr and Pin responded after line 23 with
contributions to this work. So Sup’s proposal came in the middle of an
ongoing line of work without relating to it. In conversational terms, he made a
proposal when it was not time to make a proposal. It is like trying to take a
conversational turn when there is not a pause that creates a turn-taking
opportunity. Now, it is possible—especially in chat—to introduce a
new proposal at any time. However, to do so effectively, one must make a
special effort to bring the on-going work to a temporary halt and to present
one’s new proposal as an alternative. Simply saying
“proportions?” will not do it. Sup could have said, “Instead
of drawing the altitude, let’s use propostions to find it.”

(d) To get a
response to a proposal, one must elicit at least an affirmation or recognition.
Line 23 does not really solicit a response. For instance, Avr’s question,
21: “how?” called for an answer—that was given by Pin in line
20, which actually appeared in the chat window just prior to the question and
with the same time stamp. But Sup’s posting does not call for a specific
kind of answer. Even Sup’s own previous proposal in line 10 ended with “right?”—requiring
agreement or disagreement. Line 10 elicited a clear response from Avr, line
11(“no”) followed by an exchange explaining why Sup’s
proposal was not right.

(e) Other proposals in the excerpt are successful in
contributing to the collaborative knowledge building or group problem solving
in that they open up a realm of work to be done. One can look at Avr’s
successive proposals on lines 1, 3, 8 and 17 as laying out a work strategy.
This elicits a response from Sup trying to find values to substitute into the
formula and from Pin trying to draw a graphical construction that will provide
the values for the formula. Sup’s proposal in line 23, however, neither
calls for a response nor opens up a line of work. There is no request for a
reaction from the rest of the group and the proposal is simply ignored. Since
no one responded to Sup, he could have continued by doing some work on the
proposal himself. He could have come back and made the proposal more explicit,
reformulated it more strongly, taken a first step in working on it, or posed a specific
question related to it. But he did not—at least not until much
later—and the matter was lost.

(f) Another serious hurdle for Sup was his status in the
group at this time. In lines 10 through 16, Sup had made a contribution that
was taken as an indication that he did not have a strong grasp of the math
problem. He offered the lengths of the two given triangles as the base and
height of a single triangle (line 10). Avr immediately and flatly stated that
he was wrong (line 11) and then proceeded to explain why he was wrong (line
13). When he agreed (line 15), Avr summarily dismissed him (line 16) and went
on to make a new proposal that implied his approach was all wrong (lines 17 and
18). Then Pin, who had stayed out of the interchange, re-entered, claiming to
know how to implement Avr’s alternative proposal (lines19 and 20) and Avr
confirmed that (line 22). Sup’s legitimacy as a source of useful
proposals had been totally destroyed at precisely the point just before he made
his ineffective proposal. Less than two minutes later, Sup tries again to make
a contribution, but realizes himself that what he says is wrong. His faulty
contributions confirm repeatedly that he is a drag on the group effort. He
makes several more unhelpful comments later and then drops out of the discourse
for most of the remaining chat.

The weaknesses of line 23 as a proposal suggest some
characteristics for successful proposals: (a) a clear semantic and syntactic
structure, (b) careful timing within the sequence of postings, (c) a firm
interruption of any other flow of discussion, (d) the elicitation of a
response, (e) the specification of work to be done and (f) a history of helpful
contributions. In addition, there are other interaction characteristics and
mathematical requirements. For instance, the level of mathematical background
knowledge assumed in a proposal must be compatible with the expertise of the
participants, and the computational methods must correspond with their
training. Other characteristics will become visible in other examples of chats.

At this time, the notion of math proposal adjacency pairs is
just a preliminary proposal based on a single chat log excerpt. It calls for
extensive conversation analysis of a corpus of logs of collaborative online
math problem solving to establish whether this is a fruitful way of
interpreting the data. If it turns out to be a useful approach, then it will be
important to determine what interactional methods of producing such proposals
are effective (or not) in fostering successful knowledge building and group
cognition. An understanding of these methods can guide the design of activity
structures for collaborative math. As we are collecting a corpus of chat logs,
we are evolving computer support through iterative trials and analyses.

Designing computer support

If the failure of Sup’s proposal about
proportions is considered deleterious to the collaborative knowledge building
around the triangles problem, then what are the implications of this for the
design of educational computer-based environments? One response would be to
help students like Sup formulate stronger proposals. Presumably, giving him
positive experiences of interacting with students like Avr and Pin, who are
more skilled in chat proposal making, would provide Sup with models and
examples from which he can learn—assuming that he perseveres.

Another approach to the problem would be to build
functionality into the software and structures into the activity that scaffold
the ability of weak proposals to survive. As students like Sup experience
success with their proposals, they may become more aware of what it takes to
make a strong proposal. (Livingston, 1986)

Professional mathematicians rely heavily upon inscription:
the use of specialized notation, the inclusion of explicit statements of all
deductive steps and the format of the formal proof to support the discussion of
math proposals—whether on an informal whiteboard, a university blackboard
or in an academic journal. Everything that is to be indexed in the discussion
is labeled unambiguously. To avoid ellipsis, theorems are stated explicitly,
with all conditions and dependencies named. The projection of what is to be
proven is encapsulated in the form of the proof, which starts with the givens
and concludes with what is proven. Perhaps most importantly, proposals for how
to proceed are listed in the proof itself as theorems, lemmas,
etc.—organized sequentially.

One could imagine a chat system supplemented with a window
containing an informal list of proposals analogous to the steps of a proof.
After Sup’s proposal, the list might look like Figure 3. When Sup made a
proposal in the chat, he would enter a statement of it in the proof window in
logical sequence. He could cross out his own proposal when he felt it had been
convincingly argued against by the group.

The idea is that important proposals that were made would be
retained in a visible way and be shared by the group. Of course, there are many
design questions and options for doing something like this. Above all, would
students understand this functionality and would they use it? The design
indicated in Figure 3 is only meant to be suggestive.

Another useful tool for group mathematics would be a shared
drawing area. In the chat environment used by Sup, Pin and Avr, there was no
shared drawing, but a student could create a drawing and send it to the others.
Pin did this twelve minutes after the part of the interaction shown in the
excerpt. Before the drawing was shared, much time was lost due to confusion
about references to triangles and vertices. For math problems involving
geometric figures, it is clearly important to be able to share drawings easily
and quickly. Again, there are many design issues, such as how to keep track of
who drew what, who is allowed to erase, how to point to items in the drawing
and how to capture a record of the graphical interactions in coordination with
the text chatting.

Conclusions

Some methods of contributing proposals are effective,
others are not. We can identify several problems with a particular failed
proposal in the excerpt: (a) it lacks semantic clarity and has a weak syntactic
structure; (b) its timing in being followed immediately by a stronger proposal is
unfortunate; (c) the proposer has a history of distracting from the flow of the
group problem-solving rather than contributing to it; (d) the proposer lacks
alignment with the group focus of discussion. Due to the quasi-synchronous
nature of the chat medium, there is a competition to time and structure
postings and social relationships so as to increase the likelihood of a posting
being taken up into the group discourse; this particular posting faired poorly
in this competition.

Methods for effectively doing math collaboratively integrate
skills of text chatting (typing, abbreviating, posting quickly, referencing
other postings), socializing (establishing roles, legitimacy, social
relations), formulating proposals mathematically (proper level of abstraction,
use of symbols, strategies, explanations) and interacting (making effective
proposals, leading the group discussion, eliciting desired responses). Some of
the math methods are simultaneously methods of interacting socially and
constituting group identity (e.g., who can play what role, whose proposals will
be taken seriously). Many common methods of doing math appear in the chat:
selecting levels of abstraction, using formulas, substituting numeric values.

Online methods tend to parallel face-to-face methods, but
technical mediation makes a difference, making some things easier (like working
in parallel) and other things harder (reconciling computational differences,
repairing losses from overlap). Whether face-to-face or online, the flow of the
discourse from one proposal to the next, from method to method and from problem
to strategy to sub-problem to solution defines the logic of mathematical group
discourse as a specific form of group cognition.

The mathematical discourse of the chat log follows quite
closely the sequential format of a proof. Student proposals are strategic steps
in a proof that would, if successfully completed, derive an answer to the
problem from its givens. The accountability of each step means that the group cannot
continue after a proposal is made until that proposal is accepted. Permissible
responses to a proposal are to accept it because one recognizes it as
legitimate, to reject it, or to question it. If one rejects it, then the
rejecter is accountable for providing a convincing reason for rejection.

Considered at the small group unit of analysis, the group of
students accomplished cognitive achievements. The group underwent cognitive
change and pursued a math problem. It reflected upon and selected
problem-solving strategies. It built knowledge and fashioned symbolic artifacts,
including formulae. It thought as a group. Viewing its behavior as a thinking
group constituted by interacting interpretive individuals, the analysis makes
visible how shared meaning was constructed. It analyzes the group discourse and
looks at issues of sequentiality, accountability, sociality and shared
meaning-making through the negotiation and acceptance of proposals. Many of the
questions concerning mathematics education and thinking that arise for
individual students present themselves at the group unit of analysis as well.

If sequentiality and accountability are the hallmarks of
high-order rational thought, then group discourse meets the criteria for being
considered an important form of cognition. Moreover, there is reason to hope
that computer-supported collaboration can produce high-order cognitive
achievements that rival and ultimately surpass those of individuals. This is
not to deny the problems that can arise in groups, that may even hold back
individual accomplishments. Nor is it to claim that current technologies
provide the required forms of support. But it does point to a potential that
transcends the limitations of the individual human mind and allows people to
think together. The analysis summarized here suggests several software features
that would help to avoid the kinds of problems that arose in the studied chat
and could foster the greater potential of online collaborative math problem
solving: a separate list of proposals— perhaps structured in a proof
format—could allow the group to periodically review what proposals were
made, responded to, accepted, built upon; a shared whiteboard would allow
diagrams to be shared and annotated; a threading mechanism might reduce chat
confusion caused by overlapping postings hiding sequential relationships and
references.

The notion of group cognition developed here may be considered
a strong form of distributed cognition. It goes considerably beyond
Norman’s (1993) argument that the individual
mind extends outside the head to artifacts in the world as forms of external
memory, like a reminder string on the finger. It is similar to Hutchins’ (1996) example of the cognition that
steers a large ship being distributed across people, artifacts and procedures. The
case analyzed here shows creative high-order problem-solving and mathematical
knowledge building taking place as group cognition.

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