In order to challenge pupils who are gifted in mathematics, teachers should set activities that expect pupils to use a range of techniques accurately and efficiently, provide for a higher level of abstraction and lead to more advanced thinking.

When working with gifted pupils, teachers should aim to:

• lay stronger foundations;
• develop deeper understanding;
• cultivate a willingness to reflect on the connections between different aspects of mathematics;
• foster a desire to understand fully the mathematical concepts studied and the reasons why particular methods are correct;
• develop higher-level thinking skills.

Teachers at key stages 1 to 3 generally use the national curriculum and the appropriate Framework for teaching mathematics as the starting point for planning their mathematics lesson. Teachers need to be sensitive to the following points:

• activities should have clear goals, and should aim to increase pupils' ability to analyse and solve problems, stimulate originality and encourage initiative and self-direction;
• activities should challenge pupils to develop their thinking through, for example, observing, comparing, classifying, hypothesising, criticising, interpreting and summarising;
• when open-ended tasks are used, teachers need to be clear what lines pupils are likely to pursue, what processes are involved and what outcomes are achievable and expected;
• when providing additional work for gifted pupils, care should be taken to ensure that the pupils do not come to view the work as an imposition;
• gifted pupils should not be expected to work unsupported and undirected for extended periods.

All schools should have a coherent programme for their mathematically most able pupils. The challenge is to provide a mathematics programme that nurtures pupils' special talents while both extending and motivating them. The programme does not need to involve extensive additional provision, but it should provide a mixture of challenge and encouragement. At all times it is important to remember that gifted pupils need time to relax and grow, as well as opportunities to be stimulated and challenged.

In some cases, where a year group includes a number of pupils who are gifted in mathematics, it may be possible to run a withdrawal group for, say, one lesson a week. Schools have found this a useful strategy because it gives pupils more human interaction -- both with a teacher and with able peers -- and so can help to bring challenging mathematics to life. It is helpful to make it clear that the pupils involved need to complete all their appropriate classwork and catch up on any work that they miss while taking part in the extra group.

Whatever programme is used, it should have an explicit mathematical focus, identifying clearly the main topics within the mathematics curriculum which:

• gifted pupils should be routinely expected to master in greater depth than their peers, for example arithmetic, algebra, fractions, ratio and proportion, the links between algebra and geometry (including applications of Pythagoras), solving harder word problems;
• provide additional stimulus for gifted and talented pupils, for example number puzzles; work involving different numeral systems or number bases; the use of factors, powers, cancellation and approximation to simplify calculations with very large numbers; time and calendar problems; iteration and infinity; problems involving local deduction (for example cryptic cross-number puzzles, letter sums, and strategies for winning simple two-person games);
• offer opportunities for extended research into topics of interest.

It is helpful for schools to identify and make available the books and resources that teachers are expected to use to achieve these goals.

Managing provision in the general guidance

Matching teaching to pupils' needs in the general guidance

Transfer and transition in the general guidance