The problem / intent

Initiating and sustaining a mathematical discussion in a learning community is vital to the establishment of socio-mathematical norms and to the collaborative construction of knowledge in the community. This goal is always difficult to achieve, especially in geographically distributed communities. We address this by A Challenge exchange game of Build this puzzles, using a League chart to orchestrate social interaction.

The context

Guess my X assumes a degree of social and technical sophistication which suggests it would be suitable for young teens and above. It can, however, be adapted for younger children.

The game requires flexibility in time to allow learning dynamics to emerge. It can be interleaved with other activities.

It is suitable for concrete, well-bounded content domains, such as computation, modelling or analysis. It uses these as a stratum for developing meta-cognitive skills.

mathematical content	content domain: any; target audience: 11-14 (could fit younger & older children with careful planning); Skill Domain: Reasoning, Argumentation, Problem solving, Computational;
Learning and Instruction	Mathematical content: explicit, embedded in rules / embedded in toy; Didactical functionalities: Characteristics of rules - Defined / Ambigue; Metaphor: participatory; Rationale: Games as meaningful contexts for pupils to develop mathematical contents, Games as attractors to motivate and involve pupils; Grand theory: Socio-Constructivism;
Educational Context	role of educator: facilitator, subject teacher; ratio of learning to playing: most of the learning occurs directly through playing; collaborative learning; orchestration: gaming as competition;
Games	players: single or teams; game facilitator: human facilitated, although theoretically could be assisted by artificial agent; game time: turn-based, a-synchronous, ideally spread over several weeks; Intended use: pure educational; games as genres: Puzzle, programming games; game as media: Computer game or Paper based game; games as social activities: both face-2-face and computer-mediated, multi-player;

The pattern

Guess my X is a pattern of game structure, which can be adapted to a wide range of mathematical topics. It extends the Challenge exchange pattern to encourage discussion and collaborative learning, and to break down classroom hierarchies. It uses the Build this pattern to engender reflection and discussion about the relationships between mathematical objects and the processes that produce them. The core of the pattern is described in figure 1.

Game instance

The game is facilitated by the teacher. Students play two different roles: proposers and responders. A proposer sets a challenge, in the form of a mathematical object which she constructed. The explicit rules of the game define the nature of the process by which this object can be created, but not its details. The aim of a responders is to reproduce this object, by uncovering the hidden process that generates it. If successful, the responder publishes the details of this process, typically in the form of a computer program which implements it. For example, the objects can be graphs, and the processes the functions that generate them.

The rules of the game are intentionally left vague, in the sense that the evaluation function used to determine the responders' success is not fully specified. This requires students to negotiate what constitutes a correct answer, and in doing so collaboratively refine the underlying mathematical concepts. These negotiations can lead to discussions of issues such as proof, equivalence and formal descriptions. The quality and extent of these discussions depends on the scaffolding and provocations provided by the teacher.

It is important to keep the Mathematical content explicit from the start. The game is not a sugar-coating to disguise the mathematics: it is a game with mathematical game-pieces.

set-up phase

• Before the game begins, the teacher needs to verify that the players have a minimal competence in analysing and constructing the mathematical objects to be used.
• Teacher introduces the rules of the game and the game environment.
• Teacher simulates one or two game rounds as a whole class discussion.
• Students may need to initialize their game space on the chosen collaborative medium.
• If the game uses separate media for construction and communication, consider using a Task in a box to streamline the transition between them.
  

set-down phase

The Post ludus discussion should highlight the issue of the evaluation function and its resolution.

Game session

The game sessions for the proposer and the responder are different, although the same player can play both parts in parallel.

• Proposer initiates the game, by constructing and object according the game rules and publishing it. She then waits for responses.
• Responder chooses an attractive challenge, and attempts to resolve it. If she believed she has succeeded, she responds to the challenge by posting the object she constructed and the method she used.
• Proposer reviews the response, and confirms or rejects it. If the response is rejected, an argument needs to be provided.

Play session

Each play session involves one iteration of the game. Students tend to prolong their interaction in the game, by providing secondary challenges, etc. Since the iterations are a-synchronous, there may be a time gap of several days between turns.
The communication medium chosen for the game should afford Narrative spaces for the proposer and the responder. Although the rules of the game only call for an exchange of mathematical objects, the ability to augment these with personal narratives is crucial for personal reflection as well as for collaborative knowledge building.

Related patterns

(see more details in Figure 2)

Leads to: Implementing a Behavior, Post ludus
Elaborates: Challenge Exchange; Build this
Follows: sugar-coating (anti-pattern)

Notes

Both proposers and responders tend to converge to challenges which are hard but not too hard. When the environment encourages social cohesion, players seem to reciprocate 'good' challenges. This tendency has several advantages:

• It ensures that the difficulty level students encounter is optimal for learning.
  
• It encourages gradual escalation of mathematical difficulty.
  
• It provides the teacher with a non-invasive monitoring mechanism to assess students' performance.
  

Examples

• Guess my Robot
• Guess my garden