Context

Stacey is a year 4 (age 9) teacher in a primary school in inner London. She teaches her class all subjects apart from music and ICT. She is especially fond of Mathematics, and tries to encourage what she calls "mathematical talk". She is eager to integrate new tools in her teaching and is willing to experiment together with her students. She does not have any advanced training in Maths education and educational technologies.

 

mathematical content	content domain: number and algebra; target audience: 9-11; Skill Domain: Argumentation, Computational;
Learning and Instruction	Mathematical content: explicit, embedded in toy, embedded in rules;
Educational Context	production: non-academic, freeware/opensource; role of educator: facilitator;
Games	players: teams; games as genres: quiz;

Aims

To engage students in debate about mathematical claims.

Details

Stacey is a teacher in a primary school in inner London. This vignette starts with an observation of a way she had devised for using on-line drill & practice games to facilitate mathematical discussion. Drill & Practice games are usually regarded as having very little educational merit. Nevertheless, they dominate the ‘educational game’ market in terms of availability. This is probably due to several factors:

 

• A ‘folk perception’ of learning, which leans towards behaviourism.
• Simplicity of design – the flow of a D&P game is strictly linear. The player has little control over the sequencing of the game, and thus cannot initiate unexpected branching.
• Lack of mathematical depth – building mathematical ideas into the core structure of a game is a hard problem. It is much easier to overlay a synthetic mathematical layer over an existing game structure. Such a layer needs to be shallow, and thus can hardly go beyond drilling of factual or technical knowledge.

Yet, with careful attention to the design of the situation in which such games are used, they can be leveraged to provoke reflection and argumentation, and support important meta-cognitive skills.

In the autumn of 2005 Stacey had an electronic whiteboard installed in her class. At first she experimented with the class in various uses of the board, and with time integrated it deeply into her daily practice. One of the uses she had developed was to play on-line games in whole class settings. Usually, this was done at the end of the session, when she had a few minutes to fill and did not find it useful to introduce a new topic.

Stacey uses a simple on-line game with a quiz-style structure, such as Mathionaire¹ and displays it on the whiteboard. Students raise their hands to answer the questions that the game presents. After Stacey chooses a student and he or she responds, the rest of the class is given a chance to object to the answer. If no hands are raised, she enters the response into the game interface, and proceeds to the next question. However, if some students raise their hands to object, Stacey selects one and facilitates a discussion between her and the original responder. After the students argue for their solutions, Stacey lets the class vote between them and enters the majority solution into the software. Thus, Stacey had embedded a rigid, shallow and individualistic on-line game in a flexible, dynamic and social game of her own. To distinguish between the two, we will call the former 'the quiz' and the later 'the game'.

In an unstructured interview, Stacey explained that the initial motivation for this game came from the frequent need to fill in dead time at the end of a session, when a topic had been exhausted or the students saturated. However, her interest in what she calls ‘mathematical talk’ led her to this structure, which emphasizes discussion over plain drill. Her intuitive concept of ‘mathematical talk’ is akin to the notions of socio-mathematical norms (Yakel & Cobb, 1995) and argumentation (Schwarz et al, 2002).

The value of such a game as a learning tool relies heavily on the teacher's sensitivity of the teacher to manage the whole class situation, and not simply to allow the most vociferous children to control the debate. It also relies on her ability to scaffold the students' argumentation and promote positive norms. Given these conditions, we see several virtues in this design.

• Using the quiz provides the teacher with a ready-made set of questions to provoke discussion, preparing such a set to have at hand would be infeasible for many teacher, simply due to time pressures.
  
• Using the quiz mechanism to display questions frees the teacher from technical duties and allows her to focus on the students – both as individuals and as a group.
  
• Driving the flow and speed of the game by students' objections allows her to monitor the class's ability and difficulties. Rather than assuming which questions are simple, she lets the class decide and move on quickly to the next.
  
• Having the software judge the answers dissolves the class power structure and eliminates tensions; it is not the teacher who found you wrong.
  
• The different roles in the game (responder, objector, voter) allows students to engage at the level of participation they are comfortable with, so that even peripheral participants are intent and focused on the discussion.

The list could go on, and each one of these comments deserves elaboration. This small example is emblematic of good practice in many ways. Yet the short interview that we conducted indicated that Stacey did not deliberately weave all these features into the design the game. In fact, she is probably not aware of most of the educational theories we would bring to bear on our analysis. Stacey did what most good teachers do – as indeed do most experts in their field of expertise: she applied her intuitions to make the most of the resources at hand. These intuitions are a product of years of experience in tackling similar problems and carefully monitoring her successes and failures. Needless to say, if a teacher would share such a game with another, she would convey the details, quite likely as a narrative, without making explicit the generalizable principles. A learning pattern could serve two purposes. It could illuminate a valuable consideration in the game design in a way that can be transferable to other, similar, situations. In doing so it can improve Stacey's practice by making her aware to the theoretical considerations emerging from her design, and it would allow her to share this design with her colleagues in the most efficient way.

References

Erna Yackel and Paul Cobb (1995). Classroom Sociomathematical Norms and Intellectual Autonomy . Nineteenth International Conference for the Psychology of Mathematics Education, 264--271, Program Committee of the 19th PME Conference, Recife, Brazil.

Baruch B. Schwarz, Yair Neuman, Julia Gil and Merav Ilya (2002). Construction of Collective and Individual Knowledge in Argumentative Activity . The Journal of the Learning Sciences, (12)2.

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¹: Mathionaire was developed by Duncan Keith, an English maths teacher. It is avialable for free at: http://www.subtangent.com/maths/mathionaire.php