Context

Mathematical content	number and algebra; sequences; target audience: 11-14;  Skill domain:  Argumentation, Problem solving
Learning and instruction	Mathematical content: explicit, embedded in rules; Charecteristics of toy: Transparent; Pragmatic support: Saving data, Toy helps perform comutations; Charecteristics of rules: Ambigue; Educational objectives: Meta cognitive objectives; Metaphor: participatory; Mathematical education theories: Constructivism
Educational context	production: academic prototype; local: player to PC or competative - group as unit; role of educator: facilitator; teacher: subject teacher; location where game is played: one classroom or inter-classroom, border oc curriulum; type of computers: PCs; ration of devieces per students: 1 or pairs or groups; connectivity: internet
games	Game instance:ToonTalk, Webreports, Game rules and shifting group of players. set-up phase: create 'game ground' on collaborative web space. set-down phase: analysis. Game session:one for proposer, one for responder. set-up phase: create on-line identity, learn game rules. set-down phase: reflect on game and learning experience.
Interface and interaction	controls: mouse, text based; mode of manipulation: direct, symboloc; pace: self-defined; user-space: first-person; navigation: flexible; feedback: form - windows, 2D/3D,  content: result; skill level: intermidiate; user setting - number: multiplayer, role: opponent
Software design	Platform: web browser, Desktop; Development Methodology: Exerimental Design; Development environment - Programming language: ToonTalk, Plone, Python. Customization: practicaly none built-in, but easy to replicate and reconfigure Testing: Field testing

Aims

The Guess my Robot game (GmR) was designed and tested by the WebLabs project . It was a pivotal activity in the project's explorations of number sequences.  Most students entered it with very little formal knowledge of sequences, and minimal ToonTalk experience. After GmR they moved on to more advanced topics, such as the Fibonacci sequence, convergence and divergence, and cryptography.

Details

The activity we designed was based on the well-known “Guess my rule” game, an activity well-known to many teachers and researchers (see Mor & Sendova, 2003). This game has been used in the context of spreadsheets and logo activities (Hoyles, Healy & Sutherland, 1991). It has also been used in many classrooms in UK over many years to provoke children to discuss and compare the formulation of rules, and in particular the equivalence (or not) of their algebraic symbolism. 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. It is an excellent context for students to come to understand that different articulations of their constructions can indeed yield the same results and it does this as children feel some ownership of their construction and are willing therefore to engage with others to compare and contrast solutions.

The "Guess my robot" game is designed to encourage students: to build sequences with robots and to challenge others to program the robot that made the sequence; to compare the robots used and to discuss the different methods; and to take a robot that "encapsulates the process" and use it to generate new sequences by using new inputs.

Our version of the game differs from classical variants in three aspects, which we will try to show are fundamental to the knowledge students construct:

• In terms of the mathematical content, our game involves sequences rather than functions. Of course, a sequence is defined by a function, often defined as a function, but still as cognitive constructs these are separate entities.
• In our game, the rule of the sequence is formulated as a small program. On one hand, this imposes rigour. On the other, it preserves a sense of process.
• Using WebReports as the game arena provides a self-documenting medium. Participants’ contributions are available for inspection and reflection at any time.

In this game, proposers (students) train a robot to generate a numerical sequence, and publish its first few terms as a ToonTalk “box” in a WebReport, using the template shown in Figure 3. Responders build a robot that will produce this sequence, and thus show that they have worked out the underlying rule. The game activity is orchestrated through a main game page, shown in Figure 4. This page lists the existing challenges and points to the rules of the game (Figure 5) and the template.

 

Figure 4: Guess my Robot main page.

 

Figure 5: Guess my Robot rules.

 

Figure 6 shows a typical challenge page, and Figure 7 shows a response posted to that challenge.

Figure 6: Example Guess my Robot challenge.

 

Figure 7: Example Guess my Robot response.

We first experimented with the Guess my Robot activity in 2002/3 with 8 students in London and Sofia (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). 

Observations 

This activity has been tested over 3 consecutive years, in 6 sites and 4 countries.

We first experimented with the Guess my Robot activity in 2002/3 with 8 students in London and Sofia (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), 15 girls and 18 boys, ages 10 (2), 11 (10), 12 (16), 13 (2) and 14 (3). Challenges were posted between 26^th December 2003 and 5^th May 2004. The last response was submitted on 28^th 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 (3^rd 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.

The Portuguese sites were Vale de Milhaços School, St Peter’s School and Corroios School all middle schools on the south margin of river Tagus, of Portugal. Each school as a computer room equipped with computers connected to the Internet and 3.26 ToonTalk version was installed in all computers. The WebLabs groups work in these rooms once a week in sessions of 45 to 90 minutes.

The groups were selected by their maths teachers among all the students of their classes. Sometimes, some of the students go to the Resources School Centre, to work in WlPlone and be on-line at the same time of students from other Portuguese schools and other countries.

In autumn 2004 we tested the final version of the design. In London, we worked with a group of 10 boys, age 13-14, in an after school club for “gifted and talented”. This group was instructed by Gordon and Yishay. We administered a pre-activity individual assessment, which consisted of a questionnaire and a follow-up interview. We conducted in-activity probes: short interviews (up to a couple of minutes) asking students about the activity they are engaged with. We also conducted a post-activity individual assessment, which was similar in structure the pre-activity one. We collected field notes and products (webreports and ToonTalk models).

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?

These questions are then explored by in-depth analysis of selected field notes, session recordings and interviews across sites.

In 2003 / 2004 we had a relatively large group of students from several sites participating in the game concurrently. This allowed us to obtain some broad numeric estimates of the activity’s success.

Overall, 45 challenges and 33 responses were posted. 17 of the challenges received a response (obviously, some received more – a maximum of 3 per challenge).

20 of the challenges were completely compliant with the rules of the game (detailed in Figure 5, whereas the other 28 had deviated – typically posting the challenge sequence as text instead of a ToonTalk object. These deviations were due to technical difficulties or lack of teacher guidance.

We found is useful to organize learning outcomes under four headings: a shift in complexity, emergence of concepts of proof and equivalence, programming as mathematical narrative and sustaining mathematical interaction.

The first two are concrete mathematical topics dealing with the evolution of mathematical knowledge. The later two are overarching themes which we found conducive to the first two, as well as to other activities and their outcomes.

A Shift in and about complexity

In an analysis of the challenges and responses posted in 2003, we found a trend towards students devising more difficult or complex sequences, with a concurrent trend away from sequences that they found too hard to solve. We also see convergence to a common game culture, with close compliance to the rules of the game and emergent programming conventions, which we discuss separately below.

We identified seven dominant classes of sequences proposed by students (see detailed analysis below). Three of these classes are familiar from the standard school setting: the trivial sequence, i.e. the natural numbers; arithmetic and geometric progression. The remaining four were considerably more complex than the structures most students encounter in school. Notably, when modeling sequences of their choice, the more complex classes were the more popular among students. Furthermore, this tendency towards complexity seems to grow over time.

We attribute this result to three intertwined factors. The first is the general affordance of the programming environment as an exploratory and expressive medium. We expand on that issue below. The second is shaped by structure of the activity, which encouraged students to formalize their intuitive view, rather than try to ignore their intuitions and replace them by unrelated knowledge. We claim that the intuitive view is primarily recursive in form. Students define sequences as a function from one term to the next (a_n = f(a_n-1)) rather than the “school maths” view of a sequence as a function of the natural numbers (a_n = f(n)). By acknowledging this preference and building on it we could free the students to develop their own formalizations and understandings, allowing them to engage with mathematical structures far more complex then they would in their regular curriculum.

The third factor regards the community aspect of the activity design. Students developed a culture which rewarded complexity and rigor. This culture emerged out of two elements: the embedding of the activity in a social space, in which students interacted with an audience of peers, and the time which allowed concepts and conventions to ripen. Both elements are often removed from the regular school setting, in which the students’ primary (legitimate) interaction is with the teacher, and the class is often too pressed by the curriculum to allow ideas to mature in their natural pace. Although the activity failed to form a joint enterprise in the sense of an explicit common goal, student’s mutual commitment to the game meant that it the quality of mathematical interaction had become an implicit joint enterprise in itself. Perhaps the key motivating characteristic of this social space was that it made students feel they were equal members in a diverse community which brought together teachers, researchers, and children form different ages, cultures, knowledge, people that have very different interests and ways of approaching challenges and problems.

This dynamic manifests itself in the trend towards complexity. Out of the 12 cases in which the same student posted a second or third challenge, in 8 the subsequent challenge was more complex than the first, and in 1 case the student shifted from an unknown type (i.e. challenge no one could solve) to one of the advanced types. This result points at a tendency to converge towards the convention of sequence complexity mentioned above. However, it should be regarded cautiously in light of the small sample size. In a few cases, students from site A which posted ill-formed challenges or ill-formed responses received suggestions from students or researchers from site B, and later posted well formed game entries.

The group learning trajectory is probably more relevant than the personal one in the case of this activity. Although each student was engaged with the game for a short period, the project community as a whole was involved in it for several months. We have sporadic evidence of knowledge in transit between sites, and therefore are interested in the extent to which the group knowledge evolved over time.

At first, we looked at the sequence types over time (and across sites). The ratio of data points to categories resulted in inconclusive evidence. In an attempt to obtain more trustworthy results, we decided to track difficulty over time, rather than type. We derived a crude measure of computational complexity of sequences, which takes into account the mathematical operations involved in computing its terms. We grouped the challenges and responses into 3 “difficulty bins”, and tracked the percentage of challenges in each bin over time. Our results show a clear shift from less to more difficult sequences, with a minor convergence to mid-range difficulty. To phrase this in less technical terms, we witness the emergence of a convention of sharing challenges which are “hard, but not too hard”. This phenomenon supports our claim that the community element was a strong driving force behind the emergent complexity. Students were motivated by their desire to be appreciated by the community. In the framework of this game, appreciation is manifested in the responses a challenge receives.

Students’ growing ability to cope with complexity is acknowledged in their reflections on the activity: “it is very interesting that using simple robots we could create complex number sequences”. This should not be seen as a sign that things go easily as they refer that “actually we do not like the fact that most of our predictions did not come true”. It is visible here an emerging acceptance of the partiality of  knowledge (e.g. the notion that one’s prediction in science may be not really true) interpreted as a positive contribute to the knowledge of the community as a whole group and not as sign of ‘not knowing’ things.

A Detailed review of the sequence categories

We identified 7 dominant classes of sequences proposed by students. We labeled them as trivial, arithmetic, geometric, combined, interleaved, compound and complex. Out of 45 challenges in the 2003 experiment, we found 12 combined, 6 geometric, 6 interleaved, and between 1 and 4 instances of the other categories (12 challenges were not categorized).

	challenges	responded		responses		cross site responded		cross site responses
			%		%		%		%
trivial	2	0	70	0	0	0	0	0	0
arithmetic	3	2	70	4	130	1	30	2	70
geometric	6	4	50	9	150	4	70	6	100
combined	12	6	50	12	100	1	10	2	20
interleave	6	3	50	4	70	2	30	2	30
compound	4	2	0	4	100	2	50	3	80
other	12	0	0	0	0	0	0	0	0
total	45	17		33		10		15

Table 1: Challenge types and responses. Most common shaded.

 

The trivial type refers to the sequence {1, 2, 3, 4…}. Two students posted this as a challenge. Since this sequence was included as an example, we suspect that at least one of these pages was published by mistake. The other had a note by the author saying “The objective is to make the sequence 1, 2, 3, 4 but so that the numbers are kept”. This leads us to deduce that the student saw the challenge is in the programming technique rather than the sequence itself. Both students were from the same site. Both later posted more advanced challenges.

Arithmetic sequences are commonly presented in their closed form:

            a_n = C + B*n

However, we prefer the equivalent recursive form:

            a₀ = C, a_n = a_n-1 + B

This form is closer to the way students describe these sequences verbally before instruction. We claim that the recursive form also has the advantage of a narrative and embodied structure, ideas we discuss below. Mathematically, the recursive form extends to an interesting generalization observed in the combined sequence type below. ToonTalk facilitates a direct and straightforward translation of this form into program code.  Indeed, most (if not all) students choose this path.

Arithmetic sequences are probably the easiest (non trivial) to pose and solve. Interestingly, only 3 students chose to post such sequences, one of them with a mocking comment indicating that he thought this sequence was too easy to take seriously (“is a difficult sequence that they can manage to discover It!!!!!.  I find that you go to have to think very before discovering...”).  One student later posted a more advanced challenge. The other two did not post any other challenge.

The simplicity of the challenge did not induce many responses either. Out of the four responses to arithmetic sequences, one was posted by a researcher and two did not include a robot.

As with arithmetic sequences, geometric sequences can be presented in their closed form:

            a_n = C * n^B

Or in the equivalent recursive form:

            a₀ = C, a_n = a_n-1 * B

Note that in the recursive form a structural similarity between arithmetic and geometric sequences emerges, a similarity which can prove pedagogically advantageous. Indeed, in the core sequences activities in London 2004, students identified this affinity of structure and explained how the same robot constructed to generate arithmetic sequences could be used to generate geometric sequences. Geometric sequences are considered mathematically more complex than arithmetic ones. Nevertheless, they were twice as popular in challenges as well as in responses. In relative terms, this category scored the highest in responses. This can be attributed to the emerging culture of posing sequences which are solvable yet challenging, mentioned above, which a direct consequence of students’ interaction with their audience. This culture fosters a very productive dynamics of learning.

The analogy can be carried further, as we demonstrate in the “small change challenge” and “add up surprises”. There we provoke students to use or modify robots designed for arithmetic sequences and their sums to generate power sequences, the Fibonacci sequence and factorials. Students successfully completed these challenges and reported that they were highly rewarding.

Combined sequences are best described in their recursive form:

a_n = p*(a_n-1 +q) =  p*a_n-1 +p*q

For example,   1, 5, 13, 29… (a_n = 2a_n-1 +3, a₀ = 1).

A bit of algebra yields the following closed form:

            a_n = p^n-1* a₁ + q*(1- p^n-1)/(1-p)

In 2003, this type of sequences was by far the most popular among challenges, appearing twice as many times as the next popular categories. It is also the most popular in responses, although not so in relative terms – suggesting that in part, the popularity in responses is reflects the high probability of selecting one by chance.

Students approach sequences of this type intuitively, trying various values for X and Y. Interestingly, solving it formally is not as hard as you might expect. If we take three consecutive terms, we can write 2 formulas:

a_n =  p*a_n-1 +q
a_n+1 =  p*a_n +q

We do a bit of algebra:

q =  an - p*an-1
q = an+1 -  p*an
an  - p*an-1 = an+1 - p*an
p*(a_n - a_n-1) = a_n+1 - a_n

Finally, we get:

p = (a_n+1 - a_n) / (a_n - a_n-1)
q= a_n+1 - a_n*p

Structurally, this type of sequence is a generalization of arithmetic and geometric sequences. In the former, you add a constant to the current term to obtain the next; in the later you multiply it. In a combined sequence you add one constant to the current term and multiply it by another.

This type of sequence is not taught in schools, probably due to the complexity of its closed form. However, using programming media such as ToonTalk or Excel, it is a natural step forward from the classic arithmetic and geometric sequences. We would speculate that the prominence of this type is related to the medium used. Unfortunately, we do not have control data on the frequency of the combined type in paper-and-pencil experiments, and therefore cannot verify our conjecture.

The interleaved category includes sequences which are generated by applying different rules in tandem. It is best understood by examples (all taken from student challenges):

 

{-7, 0, 20, 27…}	ao = -7, an =  an-1 +7, an+1 =  an +20
{1, 2, 4, 12, 15, 60, 64…}	a1 =1, an = n*an-1, an+1 = an-1 + n
{22, 44, 59, 118, 133…}	ao = 22, an =  an-1 *2, an+1 =  an +15

 

4 out of the 6 instances of this type incorporated a multiplicative and an additive factor. In that sense, they are similar to the combined type. In fact, if you take every other term of such an interleaved sequence you obtain a combined sequence.

However, these are harder to describe in closed form and at the same time easier to solve. Note that programming a ToonTalk robot to produce such a sequence is almost the same as programming one for the combined sequence: the only difference is that the robot sends out the intermediate result (between the first and second operations) as a sequence term. As with the combined type, this type is afforded by the medium, and it would be interesting to see if its prominence is affected by the choice of tool.

A compound sequence is one that is obtained by modulating another sequence. Formally, we can characterize these sequences as:

a_n = f( a_n-1 ,b_n)

These sequences are pedagogically interesting because they require the student to manipulate a sequence as an object, and not just as a process. There were 4 instances of this type (Table 2) – more than the trivial or arithmetic type but less than the other advanced types (geometric, combined and interleaved). Notably, the compound sequences score high on the relative response rate. Nevertheless, this result has to be considered with caution, due to the small sample size.

0, -2, -5, -9	a0 = 0, an = an-1 –2 – n
9, 45, 180, 540, 1080	ao = 9, an =  an-1 *(5 – n%5)
1, 3, 26, 260, 2606	an = åbn, b1 = 1, bn = 10*bn-1 +n
0.066667, 0.0625, 0.05882353, 0.055555556, 0.052631579	an =  1 /(15+n)

Table 2: compound type sequence

Equivalence and the emergence of proving and justification (steps towards 'proof').

When responding to a sequence, it is common that the response robot will be different from the robot used to create the challenge. This phenomenon was predicted by us, and designed to stimulate a discussion about equivalence of sequences. Indeed, students were intrigued by this fact, and were easily drawn into discussions about “how do you know that the two robots generate the same sequence?” Quite surprisingly, in several cases across different sites, students proposed a ToonTalk-based solution to this problem. Often this proposal begins with an intuitive view base on associative link: ToonTalk has a tool called scales, which can compare atomic objects (such as numbers or characters). Students proposed using this tool to compare sequences.

Such a proposal seems to indicate that students are looking for external authority for mathematical truth: The sequences are equivalent if the computer (or the teacher) says so. This attitude is discouraged by challenging students to implement their solution. Once they observe that the tool only compares single elements, they often design a next-level solution: programming a robot iteratively to compare terms from two sequences. In some cases, this is done using the scales (and the robot stops on the first non-equal pair of terms); in others cases, by subtracting respective terms and sending out the sequence of difference, thus implicitly defining a new operation on the domain of sequences.

The same pattern of events emerged in several distinct cases, and in other activities. When exploring the Fibonacci sequence, for example, two students noticed that their robots are different. A comment from researchers in another country provoked them to use the same method for assessing equivalence.

Alex: We can place each robot in a different house and find out their outputs. If the two outputs are the same, that means they are equivalent.

Chris: We could also use a third robot … Which will compare the output from the two robots and return a “Yes” if the two outputs are the same or a “No” if they are different.

Researcher: Well done! Your ideas are really productive. Can you think of something else?

Alex: I think Chris’ solution is quite difficult. It might be easier to find the differences. The new robot will subtracts the output numbers from the two Fibonacci robots and generates a sequence with the differences.

Chris: The new sequence will contain only 0s.

The next level of learning is driven by discussion. We ask students if the robot they have constructed is a proof that the other two robots are equivalent[vale3] . One girl responded:

“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.”

This discussion was initiated by teachers or researchers, who highlighted key mathematical issues. However, over time the ownership shifted to the students. Again, this relates to the social structure of the community, which puts diverse participants on equal grounds. This structure is embedded in the technology: the WebReports system gives the students ownership over the discussion in a technical sense: it is conducted on a web page which is created by a student, managed by her, and listed under her name.

This realization was the first step in acknowledging the need for formal representation and formal methods of proof. Our observation that such acknowledgment emerged from these activities is supported by evidence from other activities, such as the convergence. As one boy explained:

Aaron: um, like the debates were great, about um, is there a limit and can we prove it and um, also I’ve learnt a lot more on how to prove things in Algebra, with the a_k for the terms and everything about which I didn’t know before. I’ve learnt a lot more about that.

And concluded:

You gotta have a proper method instead of just, like, um, try and fail

While his friend remarked:

Lewis: if you program a robot to do it rather than having to write down each thing you can get a lot more results done and also you can see exactly sort of what’s going on its not just like “just DO it” and like “that’s just what you have to do”. You have to build the robot yourself and so you know exactly what it’s doing.

And later in the same interview:

its about logically thinking things through, rather than just um, like you being told that this is the equation you have to do and say “oh, yeah”. So, what we’re being told is right, instead we sort of have to discover for ourselves, and you then have to think through things more logically. Think more in depth into things.

Appropriating programming as mathematical narrative

The game, in its design, promotes alternative representations of numeric sequences as ToonTalk objects and programs. As students begin to master the game, they also develop their ability to manipulate sequences in these new representations. Students also invoked, from their own initiative, alternative representations as appropriate to solving the problems at hand. These include Excel and emergent graphic notations for ToonTalk programs.

Students used ToonTalk to generate sequences and to test their hypotheses. In rare cases, students also initiated unexpected uses of ToonTalk to demonstrate mathematical arguments. We did not find any evidence for the use of ToonTalk as a tool to automate sequence analysis. However, students did report using Excel for this purpose.

Students’ competence in using the tools (both ToonTalk and the webreports system) varied dramatically between sites. In some sites, they show perfect conviviality. In others, they fail to utilize the power of the tools, resort to alternatives and do not comply with the rules of the game. Needless to say, the activity was far more successful in the former sites than in the later. The deep gaps between levels of control leads us to believe that the problem does not stem from the activity or tool design, but from a failure to communicate these to all sites. On the other hand, to some extent it might be a result of technical difficulties (lack of bandwidth and computing resources). Finally, we should consider the possibility of conflicting agendas: our measure of success is students’ engagement with deep mathematical ideas. For them, and sometimes for their teachers, success may be interpreted in terms of participation in an international community, regardless of the content.

In (Mor & Noss, 2004) we have argued that programming, and ToonTalk programming in particular, provides a form of expression which can be considered as “mathematical narrative”: it retains the rigor of a formal representational system, while attaining the approachability of natural discourse. This approachability is supported by three key elements:

• Context: A program, like a narrative, exists in a particular context. At the marco level, ToonTalk robots operate in the context of the ToonTalk city. At the micro level, a robot is contextualized by its input box.
• Voice: a narrative is told by a narrator, which is always present – explicitly or implicitly. Likewise, a ToonTalk program is dominated by a narrator in to levels. First, as with any programming language, one can talk of programming style, attributed to individuals and groups. Second, in the case of ToonTalk, when running the program one can observe the actions of the programmer as she created the robot. It is as if the robot enacts the programmer’s process of solving a problem.
• Plot: by plot we mean a sequence of events, bound to each other temporally and causally. This is the fundamental structure of a narrative. It is also the fundamental structure of computer programs. Advances in neuro-psychology suggest that our mind is particularly adapted to constructing and acquiring knowledge in sequential form.

These three elements are removed from traditional mathematical representational systems, as they are used in school. An algebraic expression, for example, is completely self sufficient in its detachment from context and person. It is static and timeless in nature. These features, which give Algebra its power, also make it inapproachable by many students.

The Guess my Robot activity addressed the issues of context, voice and plot in several levels. At the level of the game structure and the supporting infrastructure, individual pages and comments are contextualized and personalised – both automatically by the system and by participants in their text. The game has a plot (challenge – respond – discuss) which is accentuated by the chronological listing of reports and comments.

At a structural level, the stream method of generating sequences follows students’ intuitive recursive view, which has a narrative structure. The sequence is generated by starting from a given context and moving from one term to the next.

The narrative structure of the game in general and the programming paradigm in detail, allowed students to express their ideas in an intuitive manner without compromising rigor. Thus they were able to engage with structures that would have appeared far too complex in traditional classroom settings.