BEST
PAPER AWARD at the International Conference on Computers in Education (ICCE
2005), Singapore,
December 2005
Sustaining Online =
Collaborative
Problem Solving with Math Proposals
Gerry STAHL
Drexel Universit=
y
Abst=
ract. Learning takes place over long periods of time=
that
are hard to study directly. Even the learning experience involved in solvin=
g a
challenging math problem in a collaborative online setting can be spread ac=
ross
hundreds of utterances during an hour or more. Such long-term interactions =
are
constructed out of utterance-level interactions, such as the strategic
proposing of a next step. This paper identifies a pattern of exchange of
utterances that it terms math propo=
sal
adjacency pair, and describes its characteristics. Drawing on the metho=
dology
of conversation analysis, the paper adapts this approach to mathematical
problem-solving communication and to the computer-mediated circumstances of
online chat. In particular, a failed proposal is contrasted with successful
proposals in the log of an actual chat. Math proposal adjacency pairs
constitute the collaborative group as a working group, give direction to th=
eir
problem solving and help to sustain their interaction. =
p>
1. Doing mathematics together onlin=
e
Computers offer many opportunities for innovation=
in
education. One of the major avenues is by supporting the building of collab=
orative
knowledge [1]. 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 experie=
nces
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, discusse=
d 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 t=
he
areas of the other two?
We rely on a variety of methods from the learning scie=
nces
to guide our research and to analyze the results of our trials. In particul=
ar,
we use conversation analysis ADDIN EN.CITE
<EndNote><Cite><Author>Sacks</Author><Year>19=
74</Year><RecNum>526</RecNum><MDL><REFERENCE_TYP=
E>0</REFERENCE_TYPE><REFNUM>526</REFNUM><AUTHORS>=
;<AUTHOR>Harvey
Sacks</AUTHOR><AUTHOR>Emanuel A. Schegloff</AUTHOR><AU=
THOR>Gail
Jefferson</AUTHOR></AUTHORS><YEAR>1974</YEAR><TI=
TLE>A
Simplest Systematics for the Organization of Turn-Taking for
Conversation</TITLE><SECONDARY_TITLE>Language</SECONDARY_TIT=
LE><VOLUME>50</VOLUME><NUMBER>4</NUMBER><PAGE=
S>696-735</PAGES><DATE>December
1974</DATE><URL>www.jstor.org</URL></MDL></Cite&=
gt;<Cite><Author>Pomerantz</Author><Year>1991</Y=
ear><RecNum>521</RecNum><MDL><REFERENCE_TYPE>7&l=
t;/REFERENCE_TYPE><REFNUM>521</REFNUM><AUTHORS><AUT=
HOR>Pomerantz,
Anita</AUTHOR><AUTHOR>Fehr, B.J.</AUTHOR></AUTHORS>=
<YEAR>1991</YEAR><TITLE>Conversation
Analysis: An Approach to the Study of Social Action as Sense Making
Practices</TITLE><SECONDARY_AUTHORS><SECONDARY_AUTHOR>van
Dijk, Teun
A.</SECONDARY_AUTHOR></SECONDARY_AUTHORS><SECONDARY_TITLE>=
;Discourse
as Social Interaction: Discourse Studies, A Multidisciplinary Introduction,
Volume 2</SECONDARY_TITLE><PLACE_PUBLISHED>London,
UK</PLACE_PUBLISHED><PUBLISHER>Sage</PUBLISHER><PAGES&=
gt;64-91</PAGES></MDL></Cite><Cite><Author>Ps=
athas</Author><Year>1995</Year><RecNum>469</RecN=
um><MDL><REFERENCE_TYPE>1</REFERENCE_TYPE><REFNUM&g=
t;469</REFNUM><AUTHORS><AUTHOR>Psathas,
George</AUTHOR></AUTHORS><YEAR>1995</YEAR><TITLE=
>Conversation
Analysis: The Study of
Talk-in-Interaction</TITLE><PLACE_PUBLISHED>Thousand Oaks,
CA</PLACE_PUBLISHED><PUBLISHER>Sage</PUBLISHER></MDL&g=
t;</Cite><Cite><Author>ten
Have</Author><Year>1999</Year><RecNum>471</RecNu=
m><MDL><REFERENCE_TYPE>1</REFERENCE_TYPE><REFNUM>=
;471</REFNUM><AUTHORS><AUTHOR>ten
Have, Paul</AUTHOR></AUTHORS><YEAR>1999</YEAR><T=
ITLE>Doing
Conversation Analysis: A Practical
Guide</TITLE><PLACE_PUBLISHED>Thousand Oaks, CA</PLACE_PUBLI=
SHED><PUBLISHER>Sage</PUBLISHER></MDL></Cite><=
;Cite><Author>Sacks</Author><Year>1992</Year><=
;RecNum>246</RecNum><MDL><REFERENCE_TYPE>1</REFEREN=
CE_TYPE><REFNUM>246</REFNUM><AUTHORS><AUTHOR>Har=
vey
Sacks</AUTHOR></AUTHORS><YEAR>1992</YEAR><TITLE&=
gt;Lectures
on
Conversation</TITLE><SECONDARY_AUTHORS><SECONDARY_AUTHOR>=
G.
Jefferson</SECONDARY_AUTHOR></SECONDARY_AUTHORS><PLACE_PUBLI=
SHED>Oxford,
UK</PLACE_PUBLISHED><PUBLISHER>Blackwell</PUBLISHER><N=
UMBER_OF_VOLUMES>2</NUMBER_OF_VOLUMES></MDL></Cite><=
;/EndNote>[2; 3; 4; 5; 6] to interpret the intera=
ctions
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 [7; 8] view of conversation as=
the
exchange of propositions [9]. This view was refined =
and
formalized by logicians and cognitive scientists as involving verbal expres=
sion
in meaningful statements by individuals, based on their internal mental
representations. Speech served to transfer meanings from the mind of a spea=
ker
to the mind of a listener, who then interpreted the expressed message.
Following Wittgenstein [10] in critiquing this view,
speech act theory [11; 12] argued that the utteran=
ces
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, t=
he
expression, transmission and interpretation of meaning by individuals can be
problematic, and people frequently have to do some interactional work in or=
der
to re-establish a shared understanding. The construction of common ground h=
as
been seen as the attempt to coordinate agreement between individual
understandings [13].
Conversation analysis takes a different view of
conversation. It looks at how interactional mechanisms, like the use of adjacency pairs [14; 15], 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 typic=
al
for math chats.
Online math chats differ from ordinary informal conver=
sation
in a number of ways. They are focused on the task of solving a specific pro=
blem
and they take place within a somewhat formal institutional setting. They
involve the doing of mathematic=
s [16]. And, of course, they a=
re
computer-mediated rather than being face-to-face. The approach of conversat=
ion
analysis is based on ethnomethodology [17], which involves the stu=
dy 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 m=
ath
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 [18].
The medium of online chat has its own peculiarities. M=
ost importantly,
it is a text-based medium, where interaction takes place by the sequential
response of brief texts to each other [19; 20]. As a quasi-synchronous
medium [21], 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 pr=
essure
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 [22; 23].
2. Math proposal adj=
acency
pairs
In order to begin to analyze the methods that stu=
dents
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—hav=
e just
entered the chat room, said hello to each other and read the problem involv=
ing
three triangles.
The first thing to notice here is a pattern of proposa=
ls,
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 an=
d 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.&n=
bsp;
A proposal is made by an individual for the group to
work on: “I think we should ….”
2.&n=
bsp;
An acceptance is made on behalf of the group:
“Ok,” “right”
3.&n=
bsp;
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 collabora=
tive
problem-solving of mathematics may often involve a particular form of adjac=
ency
pair. We will call this a math prop=
osal
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 con=
sist
of question/answer pairs, possibly with their own clarifications—this=
may
continue recursively. With math proposal adjacency pairs, the subsidiary pa=
irs
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 th=
at
was proposed in the original pair.
Proposals seem to lead to some kind of further mathema=
tical
work as a response to carrying out what was proposed. Often—as seen in
the current example—that work consists of making further proposals. I=
t 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 respo=
nd.
The use of “we” in “we should” or “we have
to” (stated or implied) constitutes the multiple participants as a pl=
ural
subject, an effective unified group [24]. Any one other than the
proposer may respond on behalf of the group.
Moreover, there seems to be what in conversation analy=
sis is
called an interactional preference [25] for acceptance of the
proposal. That is, if one accepts a proposal, it suffices to briefly indica=
te
agreement: “ok.” If one wants to reject a proposal, then one ha=
s to
account for this response by giving reasons.
We would like to characterize in more detail the metho=
d of making
math proposal adjacency pairs. Often, the nature of an interactional method=
is
seen most clearly when it is breached [17]. Methods are generally =
taken
for granted by people; they are not made visible or conducted consciously. =
It
is only when there is a breakdown=
i> [26] 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 S=
up’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 fail=
ed
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 propo=
sal.
3. 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 sol=
icited
responses and formed math proposal adjacency pairs.
(a) All the other proposals (1, 3, 8, 1=
7, 20,
27) were stated in relatively complete sentences.
Additionally, some of the proposals were introduced with a phrase to indica=
te
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 formu=
la.
Line 23, by contrast, provided a single word with a question mark. There wa=
s no
syntactic context (other than the question mark) within the line for
interpreting that word and there was no reference to semantic context outsi=
de
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 prob=
lem
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 w=
as
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 w=
as
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 acknowledgi=
ng
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-ta=
king
rules may define appropriate turns for response that expire in a fraction o=
f 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 fai=
ls
in this case.
(c) Su=
p’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 li=
ne 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 propos=
al
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 el=
icited
a clear response from Avr, line 11(“no=
221;)
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 g=
roup
problem solving in that they open up a realm of work to be done. One can lo=
ok
at Avr’s successive proposals on lines 1,=
3, 8
and 17 as laying out a work strategy. This elicits a response from Sup tryi=
ng
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 cal=
ls 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 pose=
d 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 t=
he
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 proceede=
d 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 ho=
w to
implement Avr’s alternative proposal (lin=
es19
and 20) and Avr confirmed that (line 22). Sup’s legitimacy as a source of useful proposal=
s 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 hel=
pful
contributions. In addition, there are other interaction characteristics and
mathematical requirements. For instance, the level of mathematical backgrou=
nd
knowledge assumed in a proposal must be compatible with the expertise of the
participants, and the computational methods must correspond with their trai=
ning.
Other characteristics will become visible in other examples of chats.
At this time, the notion of math proposal adjacency pa=
irs 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 wil=
l be
important to determine what interactional methods of producing such proposa=
ls
are effective (or not) in fostering successful knowledge building and group
cognition. An understanding of these methods can guide the design of activi=
ty
structures for collaborative math. As we are collecting a corpus of chat lo=
gs,
we are evolving computer support through iterative trials and analyses.
4. 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 education=
al
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 scaff=
old
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. [16]
Professional mathematicians rely heavily upon inscript=
ion:
the use of specialized notation, the inclusion of explicit statements of all
deductive steps and the format of the formal proof to support the discussio=
n of
math proposals—whether on an informal whiteboard, a university blackb=
oard
or in an academic journal. Everything that is to be indexed in the discussi=
on
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 give=
ns
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 wi=
ndow
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 w=
hen
he felt it had been convincingly argued against by the group.
The idea is that important proposals that were made wo=
uld 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 s=
hared
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 verti=
ces.
For math problems involving geometric figures, it is clearly important to be
able to share drawings easily and quickly. Again, there are many design iss=
ues,
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.
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