Building towards mastery

‘The man who removes a mountain begins by carrying away small stones.’

                                                                                                      (Chinese proverb)

If learners are to become secure in their mathematical skills, they will need opportunities to practise and apply the skills in a range of contexts, sometimes by consolidating the functional mathematics skills in manageable bites of learning. It may be that, before learners can tackle a big problem, they need to practise on smaller, more focused problems first. To complete the following activity, learners need to apply their process skills and use a range of mathematical skills and techniques.

Running a car

Scenario

Running a car is an expensive business. According to the RAC, the average cost of running a new car in 2006 was more than £5,500 a year, of which about £2,400 was depreciation. Nevertheless, many people, especially young people, plan to buy and run a car.

Task

Investigate the cost of running a car. Compare it with your budget, or what you think your budget may be in the future. Learners would need to consider the many costs involved, including insurance, tax, petrol, repairs, servicing and MOT costs, etc. While a complex problem such as this can motivate and enthuse some learners, some may need to tackle it step by step. More focused activities would help learners who tackle it step by step to develop and practise their process skills, and enable them to build the skills needed to complete the activity.

The following activity focuses on one aspect of running a car – the cost of insurance. This will enable learners to practise the process skills in a less technically demanding activity. Learners could be given support by being provided with data or with the sources where information can be found. Should you take the risk?

Scenario

A major cost of running a car is the insurance premium you have to pay. You are planning to buy a car (or to change your present car).

Task

Choose three cars that you would like to buy. Make sure they vary in price, age and engine size. Get quotes for insurance on each, with you as the named driver, from different sources, for example using the internet, or by telephone. Activities that provide opportunities to build skills in small steps will develop learners’ autonomy in applying functional mathematics, enabling them to tackle more demanding activities as they become more confident.

Learners will also need opportunities to transfer their skills and apply them in other contexts. Problem-centred activities could be developed that will enable learners to make the links between a familiar context and one that is less familiar, as shown in the example ‘Temperatures for tender plants’ below. This activity requires learners to use their functional mathematics skills to inform decision making.

Temperatures for tender plants

Scenario

In February, March, April and May you have some tender plants in the greenhouse at the garden centre where you work. You need to have heating on if the overnight temperature is likely to fall below 4°C.

Task

Using information about temperatures in previous years, investigate the key periods when overnight heating is likely to be required in the greenhouse. It is important that, at the end of each session, you encourage learners to identify the mathematical processes and skills they have used, and check that they understand how these can be applied in wider contexts. Learners will need to become familiar with the language used to describe the processes of problem solving in mathematics; they will need repeated opportunities to reflect on their practices and to develop their skills in describing these, using appropriate language. Eventually, learners should be so competent in recognising similarities and differences between the processes in different contexts that they are able to select suitable approaches for new situations.

This is a necessary stepping stone to transferability. Using formative assessment in this way will enable you to determine the next step for your learners, whether this is to consolidate the skills developed in a session through further activities that transfer to other contexts, or to plan for progression. Progression can be either horizontal, where the skills are practised until learners are secure in their application, or it could be vertical, where the skills are developed further towards the next level of functionality. The activities provides learners with the opportunity to apply their functional mathematics skills in a more challenging context. Learners are expected to be more autonomous about identifying the relevant information required, the mathematical techniques are likely to be more technically demanding and, although the context may be familiar, learners are less likely to be familiar with the task.