relationships and their effects on “outputs” such as mortgage repayments, and to have a language to communicate this to customers.

Turning to formal 14-19 education, especially vocational qualifications which generally have a specific work context, we believe the same broad requirements for understanding apply. There should be some focus in mathematical education – the “functional mathematics” part – on developing understanding of the process of modelling, and on the skills needed to interpret and communicate about model outputs. This should be considered as universal, and not restricted to the most basic mathematics. Conversely, modelling can no longer be left to the small numbers of expert employees who will build mathematical models. Many employees will need to interpret and understand something about the range of applicability of mathematical models . Our point of view fits with the current discussion of functional mathematics as introduced in the White Paper. The submission by the Association of Teachers of Mathematics (ATM) draws heavily on our earlier work (Hoyles et al, 2002):

Mathematical literacy is much more than the ability to carry out and understand calculations and we hope that a functional mathematics course would help to prepare young people to meet the demands of the workplace by developing skills such as complex modelling, interpreting different representations of data, extrapolating, monitoring and communicating.

# (ATM, 2005)

There is some debate as to whether functional mathematics should be considered as a basic mathematics qualification for students who cannot access higher-level mathematics qualifications. From our point of view looking at workplaces, it is evident that all school and college mathematics courses could benefit from a more “functional” approach involving the use of mathematics as a tool to solve a variety of problems. The effect of being presented with problems to solve and tools to face up to them in collaborative settings can be genuinely functional, and most crucially motivational, for all students, especially when they are required to justify their solutions in group- working.

A key finding of our TLRP project is the central importance of making invisible techno-mathematical skills visible, making the output from technologies explicit so that the connection between mathematical ideas and workplace objectives becomes apparent. As our example also shows, it is possible for people to progress successfully through the hierarchy of mathematics qualifications even to postgraduate level without knowing how to pull out the invisible mathematics and use it as a basis for action and communication across different communities in the workplace or beyond the workplace to the customer.

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