Linton Waters welcomes advice on providing a challenge for able children. There is surprisingly little literature on the teaching of mathematically able children accessible to classroom teachers. So I start by welcoming any stimulus to the debate. Roy Kennard’s booklet aims to analyse the issues and to offer teachers practical guidance on providing “a challenging mathematical environment for their most able students”.

The author acknowledges the difficulty of specifying whom we mean by the most able and rejects the notion of defining a specific percentage of higher achievers. Instead he begins by summarising the work of Kruteskii and subsequent researchers attempting to identify the characteristics of mathematically able children. One of the clear messages is that mathematical ability is predominantly identified through how children tackle problems and investigate situations; the way they reason and use generalised approaches which allow them to transcend routine responses. In other words, it’s more about how they do mathematics than how much they remember or how fast they work. This has profound implications for the way the mathematics curriculum is designed and teaching is organised in many schools.

The author proposes, reasonably, that in school recognition of mathematical ability goes hand in hand with its development. He has some suggestions for promoting both. Curriculum design must ensure that able pupils are challenged. Teaching and organisation must provide plenty of opportunities for pupils to discuss mathematical ideas. He demonstrates that attainment target 1 reflects the process and problem solving abilities identified earlier. So the responsibility of schools is to ensure that pupils regularly have opportunities to develop and demonstrate competence in these abilities. This is consistent with the non-statutory guidance for the national curriculum which says that attainment target 1 should permeate the rest of the curriculum.

All these messages have been articulated since the time of the Cockcroft Report in the early Eighties and I am confident there will be support among teachers for these ambitions, although many will see them as desirable for all pupils and not just the most able. The booklet offers guidance on teaching, organisation and curriculum planning to promote challenge. While helpful in some schools, none of this is particularly original or insightful but is easy to comprehend and apply. It reinforces the important idea that what able children need is good teaching and not something mystical or beyond the reach of competent teachers.

Three case studies of mathematically able pupils, aged six and nine, responding to three different situations, illustrate many of the points well. They would make an excellent focus for discussion in primary schools looking at challenge in mathematics.

Although the booklet is detailed in parts, some important issues are barely touched on. The first concerns the identification of pupils who choose not to demonstrate their abilities. Even when we have provided the opportunities and offered incentives some pupils underachieve. Many good schools are seeking help in identifying such pupils and developing strategies to raise their attainment. Second, there is little on encouraging pupils to take more responsibility for their learning and making judgments about the value of pieces of work in promoting their learning. For example, by asking pupils to decide how many questions in a routine exercise they need to do before they try to apply the skills or concepts in a more challenging task.

The third issue concerns the crucial role of teacher expectations. How can schools develop a culture in which teachers’ expectations are routinely challenged in a constructive manner?

Each of these issues deserves consideration when looking at how to improve a school’s work with more able pupils.

Overall, the book reinforces messages about good teaching, differentiation and the raising of standards. It should be read by all teachers of mathematics. For some it may be provocative, for many it will be constructive and helpful, and for a small but, I hope, increasing number, it will be confirmatory but may leave them wanting more.

Linton Waters is county adviser for mathematics for Shropshire