Conclusion

A few key points are emphasised below:

In considering the most appropriate forms of provision for mathematically able children it seems important to keep in mind the nature of the mathematical abilities we are trying to develop. The work of Krutetskii and others provides a valuable starting point for discussion and is summarised below.

Able children aged approximately 11 and over have the ability to:

grasp the formal structure of a problem in a way that leads to ideas for action - establish the direction of the problem;

generalise from the study of examples; search for and recognise pattern; explore special cases in a systematic way leading to conjectures about possible relationships; generalise approaches to problem solving;

reason in a logical way and as a consequence develop chains of reasoning: explaining, verifying, justifying, proving;

use mathematical symbols as part of the thinking process; represent mathematical situations using algebraic notation;

think flexibly; adapt their ways of approaching problems and to switch from one mode of thought to another;

reverse their direction of thought; work forwards and backwards in an attempt to solve a problem;

leave out intermediate steps in a logical argument and think in abbreviated mathematical forms; take valid shortcuts;

remember generalised mathematical relationships, problem types, generalised ways of approaching problems and patterns of reasoning; utilise relational thinking.

A discerning classroom based approach to the provision of challenge is one that simultaneously recognises and promotes mathematical ability.

Challenge can be provided through:

problem solving, enquiry and a focus on mathematical proof;

mathematical discussion, the initiation of sustained dialogue in which questioning by the teacher promotes the growth of pupils' mathematical abilities and understanding.

Organisational approaches can support the provision of challenge through:

team approaches to planning and the design of programmes of study that seek to build challenge in;

developing criteria for the selection of challenging materials including the most effective way to utilise commercially available schemes;

puzzle and problem displays;

leadership that creates a consensus through review meetings and co-teaching;

enlivening the curriculum with extra challenge via a mathematics club, maths and logic festivals, competitions and quizzes, industry links and mathematical magazines.

The way children are grouped per se will not guarantee that they are challenged.

Case studies that reveal the context and nature of children's mathematical thinking provide a valuable resource for teachers seeking to achieve agreement about the most appropriate forms of provision.