Numbers and Maps: The Dynamic Interaction of Internal Meanings and External Resources in Use Dave Pratt University of Warwick <dave.pratt@warwick.ac.uk> Amanda Simpson University of Warwick <a.r.simpson@warwick.ac.uk> We describe the work of two groups of six 8 year-old children as they plan Father Christmas’s epic journey on December 24 th . We trace how the children draw upon formal and personal knowledge to make connections with a range of external representations. We conclude that the context, despite its distracting potential, is critical in supporting engagement, while the representations act in mutual support of the construction of a utility for directed number (Ainley & Pratt, 2002). Introduction Mathematics in text books and indeed in conventional classrooms is often presented as exercises or worksheets in which the mathematics itself has been processed into a form that is easily digested. This McDonald’s version of mathematics ensures that the mathematical skill or technique is laid bare and typically the sole focus of attention. In this paper the mathematical focus is directed number though the reader will soon become aware that the children’s activity spans a rich panoply of disciplines. By directed number, we refer to positive and negative numbers. In this context, McDonald’s mathematics might take the form of an exercise in which the children are presented with a series of additions of one negative number to another. Ainley and Pratt (2002) have argued that, as a result, mathematics learning often becomes sterile. Children gain no sense of the bigger picture and activity is not driven by the task itself. In contrast Ainley and Pratt wish to promote tasks in which children do construct a purpose for the activity. Ainley and Pratt propose the construct of utility of a mathematical concept. They argue that a crucial aspect of a concept that is often given insufficient attention is how that concept might be useful. The McDonald’s approach is unlikely to generate utility since the skills are intentionally isolated from any context. Ainley and Pratt see the teacher’s planning problem as one of constructing tasks that are likely to be both purposeful and yet lead to the construction of intended utilities. However, when children work on tasks that encompass meaningful contexts, the mathematical ideas have necessarily to sit alongside a whole panoply of knowledge and ideas drawn from everyday experience and knowledge from other disciplines. We recognise that such knowledge is double-edged. It may support the learning or it may act as a distraction. Students may overgeneralise either on the basis of everyday knowledge brought into the mathematical problem, or because they apply rules blindly, having failed to make those rules truly meaningful (Ben-Zeev, 1996; VanLehn, 1986). The following advice, which seems to go some way towards advocating the McDonald’s approach to mathematics teachers, is taken from the Department for Education and Skills TeacherNet website (2004): The structured nature of mathematical knowledge suits a structured teaching style. Break down content into relatively small chunks and ensure that students have fully mastered each one before going on to the next step. This will build students confidence about their ability… Children easily develop misconceptions about the meaning of mathematical concepts. Primary school pupils will often acquire a rule and then overgeneralise it to situations in which it is not applicable. 470