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The functional mathematics specifications require students to make sense of situations, represent them, analyse them, use appropriate mathematics, interpret results and communicate (Ofqual, 2012). These descriptors suggest a different way of thinking about mathematics and experiencing the subject in classrooms from a traditional didactic approach (Wake, 2005) since problems need to be discussed, analysed and interpreted (QCA, 2007). Research in schools suggests that teaching methods based on knowledge

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transfer from teacher to student are unlikely to produce the desired outcomes of being able to use and apply mathematics in a range of situations (Boaler, 1998). Developing the skills required for functional mathematics seems to demand a more connected exploration of mathematics rather than traditional pproaches to teaching the subject (Swan, 2006).

Three possible approaches to functionality have been suggested: content focus, process focus or a context focus that involves the ma of real situations (Roper et al., 2006). The content focus (J. Evans, 2000) that assumes the transfer of knowledge and application of skills will be possible if concepts are soundly grasped. This strategy, as discussed earlier, has been considered inadequate for learning that needs to be applied across a range of contexts. Alternatively, adopting a process focus adds competency with processes to content knowledge but neither of these approaches would seem to adequately capture the purpose of functional mathematics as a tool to be used confidently in a range of different contexts (QCA, 2007) and therefore these methods seem unlikely to produce the appropriate outcomes.

T ‘

Mathematics Education approach (RME) and this may promise better alignment to the demands of developing functionality. By using life situations as the starting point for learning this approach also provides space for developing the conceptual understanding that

functionality requires (Wake, 2005) rather than simply teaching applications. Based on a

(Van Den Heuvel-Panhuizen, 2003, p.13), realistic contexts are used as the starting point for the construction of mathematical models that then

(Van den Heuvel-Panhuizen & Drijvers, 2001, p.2). In this

detail. The process moves from the context to a model and then back to the original situation. Mathematical features are abstracted from the situation and a mathematical model is constructed, from which the problem is solved before applying the solution back into the context from which it originated. In this process Treffers and Vonk (1987) identified

mathematical tools are used to organise the problem whilst still located in the context, followed by the vertical process of making connections and finding solutions within the

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mathematical system (Van Den Heuvel-Panhuizen, 2003). In this way the modelling process serves as a means of organising and working mathematically with real situations.

The use of mathematical modelling, in this sense, seems suited to an approach for teaching functional mathematics since it promotes a way of thinking that aligns to the intended purposes of being able to interpret and make sense of situations (QCA, 2007).

Student modelling in functional mathematics involves students in making mathematical sense of a situation using their mathematical knowledge and in using their mathematics to explore the situation purposefully. (Roper et al., 2006, p.96)

There are, however, three particular characteristics of a functional approach to mathematics suggested by Ofqual (2012) that may present challenges for teachers and are worth some further consideration:

 the use of realistic contexts, scenarios and problems  tasks that are relevant to the context

 the development of problem-solving skills

The use of realistic contexts, scenarios and problems (Ofqual, 2012, p.2) is a potential

means of m

be particularly difficult to achieve (Aydin & Monaghan, 2011). Using in a classroom usually involves written or verbal descriptions of scenarios without actually being within the situation described and research with children has shown how their

interpretations of contextualised problems varied according to the assumptions that they made (Cooper, 2004; Cooper & Harries, 2002). Although some children did relate questions

non-mathematical knowledge or experience, others interpreted the problem as a

mathematics question in a superficial disguise (Cooper & Harries, 2002). It seems that rather than making mathematics more related to real life from a student perspective, the effect can be the opposite and written scenarios that appear contrived may only serve to reinforce perceptions that mathematics is unrelated to real life.

Developing and using tasks that are relevant to the context (Ofqual, 2012, p.2) is also more complex than simply selecting a context to fit around a mathematical problem. Using

contexts may only disguise the mathematical relations (Lave, 1988) and randomly inserting context into a question may not serve any helpful purpose (Boaler, 1993). Wiliam (1997) makes a distinction between contextualised mathematics questions that involve solving a

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real problem, a realistic one or performing a task that relates very little to the mathematics being taught. Alternatively, the use of context in a question may be described as authentic or artificial (Drake et al., 2012) depending on the match, or mismatch, between the mathematical problem and the context in which it is situated.

Essential to using mathematics to model a real world situation or problem is the genesis of the activity in the real world itself.

(Wake, 2005, p.9)

Authentic contextualised tasks arise from the situation, using a scenario in which the mathematical problem is naturally occurring, rather than using the context as simply a metaphor to illustrate an aspect of pure mathematics (Wiliam, 1997). For a functional

only do the scenarios themselves have to be authentic descriptions of life situations but the tasks also need to arise naturally, representing problems that would need to be solved in the situation.

The development of problem-solving skills is also central to the functional skills curriculum but multiple meanings have been attributed to the term. The word roblem

mathematics may be used as a label for any question involving mathematics or be reserved for a particularly challenging task (Schoenfeld, 1992). There is some agreement that

mathematical problem-solving is concerned with performing non-routine tasks (Burkhardt & Bell, 2007) and requires the development of independent thinking (Lester, 1994). This demands sound, conceptual knowledge and a connected understanding of mathematics so students can make decisions on how to approach the problem and which mathematics to use (Burkhardt & Bell, 2007). Polya (1985) outlines four phases of solving a problem:

understanding the problem, drawing up a plan, implementing the plan and reviewing. In this structured approach there is a need for mathematical knowledge in order to devise and implement the plan but the role of the teacher in supporting students whilst encouraging the development of independence is also important (Polya, 1985). In addition to a secure knowledge-base, students need to be familiar with appropriate tools or techniques to use in solving problems and develop the ability to manage their own progress (Schoenfeld, 1992).

‘ (Wood, Bruner, &

Ross, 1976) where students make personal decisions but still within the security of a

teacher- -and-

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gain confidence. These strategies may help develop the resilience needed to be successful in problem-solving (Dweck, 2000) and the self-beliefs that influence their perceptions and choices of action (Schoenfeld, 1992).

The type of mathematical thinking required in problem-solving (Chapman, 1997; Swan, 2006) is not easily developed through the use of routine, closed questions and transmission approaches to learning (Boaler, 1998). The use of open questions and non-routine problems provide opportunities for exploration rather than replication of knowledge. Thinking

mathematically is a dynamic process which enables the individual to handle more complex ideas and expands understanding (Mason, 1988) whilst a collaborative approach, in which students are engaged in tasks that encourage questioning and reflection, has the potential to develop mathematical thinking (Swan, 2006) in a way that functional mathematics seems to require.

These considerations of the curriculum, however, cannot be separated from the needs of students. In the following section, two particular student characteristics, arising from their experiences and responses to mathematics, will be highlighted and some possible teaching approaches considered that may be suitable to address these problems.