SUSTAINING INTERACTION IN A MATHEMATICAL COMMUNITY
OF PRACTICE
João Filipe de Lacerda Matos†, Yishay Mor*, Richard Noss*, Madalena Santos†
†: Faculdade de Ciências da Universidade de Lisboa, *: London Knowledge Lab

Fourth Congress of the
European Society for Research in Mathematics Education
Work Group 9
17 - 21 February 2005 in Sant Feliu de Guíxols, Spain

ABSTRACT. This paper focuses on an activity in which students explore sequences
through a game, using ToonTalk programming and a web-based collaboration
system. Our analytical framework combines theory of communities of practice with
domain epistemology. We note three factors which influence the length and quality of
interactions: facilitation, reciprocation and audience-awareness.
Introduction
This paper tells the story of an experiment to design a mathematical community of
practice, in the course of the WebLabs Project, a 3 year EU-funded educational
research project oriented towards finding new ways of representing and expressing
mathematical and scientific knowledge in communities of young learners. Our work
focuses on the iterative design of exploratory activities in domains such as numeric
sequences, cardinality, probabilistic thinking, fundamental kinematics, and ecological
systems. In this paper, we will focus on an activity called Guess my Robot, which is
aimed at advancing students’ understanding of number sequences. We use that
activity to explore the following question:
What are the factors that sustain interaction in a mathematical activity over a
web-based collaboration medium?
Our analysis is informed by the notion of ‘community of practice’ as it is used within
the situated approach to learning (Lave and Wenger, 1991; Wenger 1998). The
insights we gain from this analysis are fed into the next iteration of the activity
design. Thus, we have built on our initial observations of communities to actively
cultivate their existence.
Wenger proposes three dimensions of practice as the property of a community:
• Mutual engagement: a sense of “working together”. Sharing ideas and
artefacts, with a common commitment to the interactions between members of
the community.
• Joint enterprise: having some object as an agreed common goal, defined by the
participants in the very process of pursuing it, not just a stated agenda but
something that creates among participants relations of mutual accountability; that
become an integral part of the practice.
• Shared repertoire: agreed resources for negotiating meanings. This includes
routines, words, tools, procedures, stories, gestures, symbols, and so on.
Artefacts that the community has produced or adapted in the course of its

existence and have become part of its practice. The repertoire combines both
reificative and participative aspects. It includes the discourse members use to
create meaningful statements about the world as well as the styles in which they
express their forms of membership and their identities as members.
To these we add an epistemological dimension, in that we intend to encourage the
formation of mathematical communities. That is, we are trying to generate
communities of practice – both physically and virtually – in which there are agreed
socio-mathematical norms, where it is natural to make conjectures, test hypotheses,
offer counter-examples and so on. By restricting our attention to a specific domain of
mathematical activity, we commit ourselves to make specific and concrete claims.
Our focus on design provides us with a unique opportunity to go beyond explanatory
observations. We can verify our claims by changing the activity system and
monitoring predicted change.
WebLabs, ToonTalk, WebReports and the Guess my Robot game
WebLabs utilizes two main media for its activities: ToonTalk (a programming
environment) and WebReports (a web-based collaboration system). We see
programming as playing a key role in individual and group learning. Students explore
and test their conceptions of the phenomena through programming. Furthermore, by
sharing programmed models, they can communicate ideas in a concrete yet rigorous
form. We are programming with ToonTalk1 (Kahn, 1996; 1999; Mor et al., 2004) a
language used in the past with younger children to construct video games (Hoyles,
Noss & Adamson, 2002). ToonTalk is a computer game, programming environment
and programming language in one. In ToonTalk programs take the form of animated
cartoon robots. Programming is done by training these robots: leading them through
the task they are meant to perform. After training, programs are generalised by
“erasing” superfluous detail from robots' “minds”.
The individual and collaborative facets of learning are intertwined at all stages of our
activities. The WebReports2 system was set up to support both. The primary aim of
this system is to allow learners to reflect on each others work by sharing working
models of their ideas. The “atomic unit” of content in the system is a web report: a
document containing formatted text, multi-media objects and most importantly –
ToonTalk models. Reports are edited using a visual editor. Students can grab any
model constructed in their ToonTalk environment, and copy it instantaneously into
their report. These models are embedded in the report as images, which link to the
actual code object. When clicked, they automatically open in the reader’s ToonTalk
environment – which could be in another classroom or another country. The reader
can then manipulate the object, modify it, and even respond with a comment that may
include her own model. This last point is crucial: rather than simply discussing what
each other thinks, students can share what they have built and rebuild each others’
attempts to model any given task or object.

1 http://www.ToonTalk.com
2 http://www.weblabs.org.uk/wlplone/.

Our activity design methodology exploits the affordances of the system. The initial
discussion of a phenomenon can lead to the group’s publishing a report on their
observations, conjectures, and suggested path of inquiry. Finally, when a task or
activity is completed, a concluding report will be published by either individuals or
the group, to share conclusions with remote peers.
One of the experiments we have conducted in the course of the WebLabs project was
a game called Guess my Robot. The activity we designed was based on the “Guess
my rule” game, an activity well-known to many teachers and researchers as a way of
encouraging students to discuss and compare the formulation of rules, and in
particular the equivalence (or not) of their algebraic symbolism. It has also been
employed in the context of Logo and spreadsheets (c.f. Healy & Sutherland, 1990). In
its classical form, it has been used as an introduction to functions and to formal
algebraic notation. As Carraher and Earnest (2003) have recently reported, even
children in younger grades enjoy participating in this game, and can be drawn into a
discussion of algebraic nature through using it.
We first experimented with the Guess my Robot activity in 2002/3 (Mor & Sendova,
2003). Our experience from this pilot informed both the design of the activity and of
the collaboration system. In 2003/4 we expanded the experiment, with significantly
greater response. This iteration included 33 students from 6 sites (in different
European countries). There are several differences between our version of the game
and other variations. Most notable is the media by which it is conducted, and the
specific rules of game inspired by those. In our game, proposers (students) invent a
rule for a number sequence and model it as a ToonTalk robot (procedure) that
generates that sequence. They then collect the first few terms of its output in a
ToonTalk box and embed it in a web report. Responders can click on the image of the
box, and explore its contents in their own ToonTalk environment. They use a variety
of tools to uncover the rule of the sequence: ToonTalk programming, Excel and
(even!) paper and pencil. Once they succeed, they respond to the challenge by posting
a comment on the report, which includes a robot they created for generating the same
sequence.

Figure 1: Rita's Guess my Robot page

Figure 1 shows an example of such a challenge. It was posted by Rita3, a 14 year old
girl from Portugal. This example will accompany us throughout this paper. Rita’s
challenge provoked several different solutions, which led to long threads of
interaction, some of which included fairly sophisticated mathematical arguments. Not
all of our data is so impressive: overall, 45 challenges and 33 responses were posted.
However, only 17 of the challenges received any response at all. A lot can be said
about those challenges and responses – their mathematical structure and its relation to
the tools used; the forms of expression which evolved through the game; how
students construct their challenges, and how they select a challenge to respond to; the
evidence all these present on questions of meta-cognitive skills and practices and so
on.
Data and methods
The present dataset encompasses 33 students from 6 sites, 15 girls and 18 boys, ages
10 (2), 11 (10), 12 (16), 13 (2) and 14 (3). Challenges were posted between 26th
December 2003 and 5th May 2004. The last response was submitted on 28th May
2004. Overall, 45 challenges and 33 responses were posted. Only 17 of the challenges
received a response (obviously, some received more than one – a maximum of three
per challenge). However, there are 114 comments altogether, up to 30 per a single
report (3rd quartile at 3.25). The subject group is highly diverse. Each site had its own
characteristics in terms of student selection, class setting, age, ethnic background,
gender, and teacher-student ratio.
From a methodological point of view, one of the advantages of using a web-based
collaborative system is that it is a self-documenting medium. All the challenges and
responses posted by students, as well as any verbal comments, are archived and dated
on the system. This data is abundant and easily accessible. Yet at the same time it is
shallow: it does not record the classroom interactions or the problem-solving
strategies used by the students. Analyzing this data cannot provide answers about
personal and group learning trajectories, but it can point to interesting questions, such
as:
• Students developed an ability to flow between different representations of the
same sequence. In what ways does this ability affect their understanding of the
mathematical objects they manipulate and the methods they use?
• The structure of the game requires participants to make conjectures, model
them by programming, and test them. Does this facet of the activity influence
students’ mathematical argumentation?
• We identified several canonical structures of sequences which appeared in
many challenges and in different sites. These structures are notably different then
those taught in standard curricula. What are the epistemological sources of this
difference, and what are their implications?

3 We use the aliases, or “handles” children chose for themselves in the web reports system. With the system’s access
restrictions in mind, we can use these as anonymized identifiers.

These questions are then explored by looking at field notes, session recordings and
interviews across sites. In this paper we wish to focus on one theme, the issue of
sustaining interaction in a mathematical game, within a web-based collaborative
system. The next section elaborates this question.
Sustaining mathematical interaction
It is clear that sustaining the kind of interaction we seek is strongly contingent on the
domain, the activity structures, and, of course, the tools that we offer to students.
Nevertheless, as in any learning environment, the epistemological, cultural and social
factors are intertwined. Thus, our answers cannot be detached from social and
cultural considerations.
Asking how to sustain interaction implicitly suggests that it is a positive force. Yet
this is itself a claim that needs to be scrutinized. In the case of Rita’s challenge, the
first responses were bare robots. As the interaction developed (in fact, in several
concurrent threads) students went deeper and deeper into the questions that emerged
from the situation: equivalence of models, solution strategies and even notions of
proof. Participants shifted from the competitive and somewhat technical base level of
the game to a collaborative effort of understanding the mathematical structure of their
models, and sharing of analytical tools.
Assuming we accept sustained interaction as a desirable phenomenon, we need to
look closely at the cases were it occurs and try to identify their unique characteristics.
We should obviously pay closest attention to cases were the interaction is
distinguished not only by quantity but also by quality. That is, quality of the
mathematical and meta-mathematical discussion exhibited in the interaction. There
are 3 main themes that have emerged from our preliminary observations: facilitation,
reciprocation and audience-awareness.
Facilitation
Our first conjecture regards the role of the facilitator. As Wenger et al. (2002) note,
“Alive communities, whether planned or spontaneous, have a ‘coordinator’ who
organizes events and connects community”. We assert that this role of coordination,
or facilitation in our terms, is critical in maintaining the dynamics of the game.
Facilitation takes on three forms:
• Technical: providing technical apprenticeship on how to use the system, e.g.
how to post a response; pointing teachers and students to interesting postings.
• Pedagogical: setting new challenges to participants; noting the mathematical or
computational aspect of postings to teachers and students.
• Sociomathematical: shifting the conversation towards mathematical content. In
the terminology of Yackel & Cobb (1995), establishing the sociomathematical
norms of the game.
At first, the Bulgarian students posted their response in a separate report. Yishay
copied the text and the robots from their reports and posted them as comments on
Rita’s challenge. He then e-mailed the teachers at both sites about this. Obviously,

this is not a very interesting event to report. Nevertheless, none of the following
discussions about sophisticated mathematical ideas would have occurred without it.
As an example of promoting sociomathematical norms, consider the following
comment posted by the London researchers:
This is a question from the London team (Richard, Celia, Ken, Yishay and Gordon) to all three
of you:
We think your robots will generate the same sequence for ever, but how can we be sure?
This question provoked students in both sides to think about the question of
equivalence. The Bulgarians approached this question by working it out algebraically
in a group. Rita considered this option, but thought that the rules of the game
restricted her to using ToonTalk. Her solution was to construct a robot that compares
two sequences by subtracting respective terms. She explains:
Clearly that this is not a prove of that robot produces the same sequence, that is only one
conjecture, or either, I have 99% of sure that they are equal, but still did not can to get a
demonstration.
One of the responses to Rita’s difference robot is an example of a pedagogical
intervention. Gordon comments:
Wow - this is really great work! Did you know that you could actually create other sequences
using the difference robot that you built? I.e. if the two robots you send off in the trucks don't
generate the same sequence, then your difference robot will generate a sequence of non-zero
numbers. Try it!
Gordon suggests a new challenge, based on the work that Rita had published.
Unfortunately, at this point we have to report a lack of success. Rita responded
politely, but did not pick up the challenge. Her teacher’s field notes reveal an
explanation: she answered the comment, and was disappointed not to receive a
response from Gordon. It was not a lack of interest in the mathematical problem, but
rather a suspicion that Gordon would not maintain the interaction on his side. We will
return to this important observation later, when we mention the issue of presence.
Using a web-based medium eliminates constraints of organizational structure. An
expert in London or Portugal can facilitate activity in a classroom in Cyprus. The
WebReports system includes several features which aid facilitation. For instance,
challenges are listed automatically, with the number of comments they received. The
facilitator can identify challenges which have not been responded to, and use the
system’s messaging facility to invoke other participant’s awareness to them.
Whenever the facilitator identifies a common technical or conceptual problem, she
can publish a tutorial which addresses it.
Reciprocation
A second theme we identify is reciprocation. Under some circumstances, students
feel a stronger obligation to reply than others. These circumstances may have a social
element, for instance the sense of obligation is stronger when a comment is posted by
a group of students or by a teacher. On the other hand, a very strong element in
reciprocation is a socio-mathematical factor: participants sense they should “give
something in return” for a positive experience, and solving a tough challenge is seen
as such. Thus, participants’ tendency to respond rises with the difficulty of the

challenge. This conjecture addresses not only the frequency of responses, but also
their quality: when the challenge was gratifying, students respond with more then
their solution, adding unexpected levels of mathematical discourse to the interaction.
When Nasko posts his response to Rita’s challenge, he adds:
Here is also a sequence generated by the same robot. Two questions:
1. What was the input of my robot?
2. Can your robot generate it?

Nasko’s response dissects the process of generating the sequence from its initial
conditions, giving rise to the idea that the same process can produce different
mathematical objects.
Rita responds in two stages. First, she reciprocates on the social level –
congratulating Nasko on his response, and sharing her original model with him. She
explains to her teacher that she should respond immediately so as not to discourage
him. Only then does she set on solving his challenge. After she does that, she
reciprocates on a domain knowledge level, by posting her solutions.
The flip side of this phenomenon is that students do not respond to challenges they
see as uninteresting. Sometimes, a student might pick up a simple challenge as a
“drilling challenge”, but will not invest in posting her solution. At the end of the
activity, we asked students to publish a concluding reflective report. When asked
about the responses to her challenge, one girl responded:
I don’t receive any comments to my sequence, because is to easy...
Reciprocation is so natural in classroom practice that it goes unnoticed: a teacher
acknowledges a student’s remark; students support each other’s claims. In a web-
based environment it raises tensions which we need to accommodate. Teachers need
to actively seek students’ contributions and react to them, less the students feel
unnoticed. Other issues arise from the need to adjust to asynchronous
communication: at the beginning of one session, Rita posted a comment and then sat
back, waiting for a reply, growing frustrated by the minute. Her teacher had to
explain that although she could see several members of the community on-line, they
might be occupied with other activities and unaware of her comment.
On the positive side, streamlining the ToonTalk objects into the text of the reports
had the effect of enriching students’ interactions. When Nasko posted his robot as a
response to Rita’s challenge, she reciprocated by posting hers. This gave rise to the
question of comparing the robots and asserting their equivalence. Since robots, as
coded objects, are by nature formal structures, the discussion took a much more
formal tone than may have been the case with bare text.
Audience-awareness
Our last conjecture is perhaps the most socially-oriented. We find that two
characteristics of a participant provoke response to her contributions: cordiality and

presence. The first is almost trivial – participants respond more eagerly to friendly,
inviting comments. The second is accentuated by the medium we chose, and in a way
related to the issue of reciprocity. We find that participants prefer to interact with
peers who project a strong presence. (e.g. appear on the “active users” list, post
frequent comments, have a rich home page). Our conjecture is that this stems from
the fact that participants are in fact interested in sustained interactions, and thus
prefer to communicate with peers (or researchers) from whom they expect a higher
probability of response. This entails immediate implications for us: participants are
set back by one-off comments, and researchers should refrain from commenting if
they do not intend to participate in subsequent discussion.
An example of this idea has been mentioned above: Rita did not attempt to solve
Gordon’s challenge because she suspected he might not be available to appreciate her
response.
On the positive side, a team of Cypriot students replied to Rita’s challenge nearly a
month after the previous interactions. Because they identified themselves as a team,
Rita felt a stronger commitment to her audience. She felt obliged to reply to the
Cypriots, and to do so thoughtfully. The Cypriots volunteer an explanation of their
solution strategy:
1. We copied Rita's numbers in Excel, to be easier to find relations between the numbers and
especially the differences.
2. We found the differences between the numbers on that sequence.
3. We noticed that differences between numbers could be calculated if we multiply every one
difference by 4.
4. So, we decided that we could work with formula 4* number.
5. To get Rita’s sequence, we had to add 8 to the previous formula. The final formula is
4*number+8
Best
Cyprus Mathematics WebLabs Team
And Rita responds by taking the role of the facilitator, and elevating the discussion:
I can prove that my sequence and your sequence are equal with the process of algebraic
representation used by Sofia group.
Rita's sequence:
A1 = 2
An+1 = (An + 2) x 4, but if I using the distributive property of the multiplication relatively to the
addition I can write that:
A1 = 2
An+1 = An x 4 + 8
That is the algebraic representation of the Cyprus’s sequence. Then I can prove that two
sequences are equal.
Conclusions
In this paper we have explored the question of sustaining interaction in a
mathematical activity over a web-based collaboration medium. Our approach

attempts to interleave the theoretical framework of communities of practice with
epistemological observations arising from the specific knowledge domain of number
sequences. As a case study, we have chosen one of our experiments involving a game
called Guess my Robot. Our analysis suggests several factors which contribute to the
extent and to the quality of interactions: facilitation, reciprocation and audience-
awareness. Supporting these elements has guided our design of the webreports
system. Nevertheless, along with its potentials the technology raises challenges –
which need to be addressed by adjusting patterns of behaviour as well as social
norms. The fundamental elements of a community of practice are reflected both in
our analysis and in the design of the tools, the rules and the roles in our activities.
Mutual engagement, in the sense of sharing and discussing artefacts, is afforded by
the features built into the WebReports system; its support of joint and individual
authoring of documents, the ease of commenting on others’ document, and most
importantly – the ability to include models of ideas as manipulable objects in these
documents. The notions of facilitation and reciprocation elaborate on the idea of
mutual engagement. Implicit rules of engagement emerge by which, for example,
harder challenges are more esteemed and provoke richer responses.
A sense of joint enterprise is valuable in motivating students to engage in the activity.
This motivation is related to participants’ audience-awareness; a factor that is easy to
neglect in traditional environments, but takes prominence in a web-based
environment, where the communication channels are thin. As the accepted value of
the enterprise rises, in terms of its mathematical richness, so does the level of
collaboration.
The concept of shared repertoire is related to that of sociomathematical norms, but
also the domain-specific questions, such as the implicit agreement on what
constitutes a hard challenge and the positive value of one. Using programming
(specifically ToonTalk) as a taken-as-shared resource enriches the repertoire with a
language that is both rigorous and expressive. As students master the multiple facets
of their repertoire, the boundaries between the verbal and computational languages
they use are blurred. Their argumentation is shaped by the tools, while at the same
time they shape the tools to express their arguments.
Synergising distinct paradigms is always a challenging task. In our case, we still see
more questions than answers before us, but these questions are enough to make the
effort worthwhile.
We acknowledge the support of the European Union, Grant # IST-2001-32200, directed by Prof. Richard Noss and
Prof. Celia Hoyles. (http://www.weblabs.eu.com.)
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