Introduction

Effective provision for the top 10% in mathematics should ensure that such students are constantly provided with:

• challenging problems to solve;
• tasks which reinforce and deepen their understanding of standard curriculum topics;
• encouragement to see mathematics as a rich and exciting subject to explore and enthuse about.

These requirements are no different from the provision that is appropriate for all pupils, except that expectations about the level of mathematical knowledge and understanding should be very much higher for the top 10%.

There are three approaches to the problem of provision:

• Enrichment for Added Depth
• Enrichment for Added Breadth
• Acceleration

Enrichment for Added Depth

The most important aspect of provision is to strengthen able pupils' conceptual understanding of standard curriculum topics and to extend their ability to apply that knowledge by presenting them with challenging problems. This should take place in the context of their everyday classwork and homework by providing differentiated tasks or alternatively by providing tasks which are sufficiently open to allow for responses at different levels.

• Start a new section of work by posing a problem to the class as a whole so that pupils can work on it for themselves before being told how to do it. This gives all pupils an opportunity to think for themselves and the ideas of those who get further can be used in subsequent class discussion.
• The more demanding tasks in miscellaneous exercise and extension exercises in standard text books can often provide fruitful starting points, particularly if the problems are presented in an unstructured format.
• When able pupils have mastered a new idea and have used it in a variety of ways, pose a problem which relates to the next stages in the topic without providing them with a formal method. For example, when they have been working on trigonometry with right angled triangles, ask them to try and find the angles in a triangle with sides of 5, 7 and 8 without at any point drawing attention to the cosine formula to avoid encroaching on future lesson material.
• Open ended tasks offer great scope for able pupils to think for themselves and to go further than others.
• Encourage able pupils to explain what they are doing carefully using correct mathematical language and constantly challenge them by asking the question 'why?'. For example, if the class have been considering a test for divisibility by 3 or 9 ask them to explain why it works and to try and find a test for 11 by themselves.
• Ask abler pupils to do alternate questions in routine parts of exercises so that they get on to more demanding questions more quickly.
• Offer practice of routine skills in contexts which prompt further questions related to generalising and explaining patterns. For example, calculating , , and so on gives plenty of practice in adding fractions, but the pattern in the results raises interesting questions and further generalisation arises by looking at fractions where the difference between numerator and denominator is other than 1.
• Challenge able pupils to do more demanding mental calculations which extend their mathematical understanding rather than just their ability to hold a lot of numbers in their heads simultaneously. Calculations that can be done using algebraic identities like the difference of two squares or which are aided by an awareness of factors are particularly relevant. Suggest that pupils find alternative ways of doing the same calculation.
• Provide a class with a variety of questions to do on a topic some easy and some more difficult and let them choose which ones to do. Able pupils will often choose for themselves to go for the harder ones.
• Always include one or two more demanding problems in homework tasks, besides offering consolidation of routine skills and standard problems to solve.
• Frequently throw out ideas for pupils to pursue on their own at home - see if you can find some more examples like this; what else can you find out about Pythagoras; here is a nice puzzle for you to try at home; try exploring this with a spreadsheet or a program on your graphical calculator.

We would particularly recommend two sets of publications - one recent and one much older - as sources of appropriate material for the top 10%:

• Maths Challenge Books 1, 2 and 3
  Edited by Tony Gardiner and published by Oxford University Press.
  These books provide material linked to standard curriculum topics specifically designed to extend and deepen the understanding of able pupils.
• School Mathematics Project Books 1 to 5
  Published by Cambridge University Press in the 1960s.
  The old 'hard back' SMP books have long been out of print, but are often to be found languishing in stock cupboards and department libraries. They are rich in ideas linked to many current curriculum topics and are ideal for able pupils.

Enrichment for Added Breadth

There are many ways of providing enrichment for individuals outside the classroom by stimulating an interest in mathematics beyond the confines of standard curriculum topics. Able pupils should be made aware of the wide range of books on mathematics that are available and the great variety of material that is available on the world wide web. Some further details will be found in the last section of this document. The NRICH website at www.nrich.maths.org.uk is a particularly valuable source for able pupils.

Mathematical competitions are another valuable activity for the top 10%. Secondary teachers can find details of competitions on the website of the United Kingdom Mathematics trust at www.mathcomp.leeds.ac.uk and primary teachers can find out about the Primary Mathematics Challenge from The Mathematical Association.

Acceleration

Schools often accelerate pupils in their top sets in that they may be working at one or two National Curriculum levels above those in other sets. Such moderate acceleration is entirely acceptable and appropriate when it forms part of a planned progression in mathematics through to the end of the sixth form. In very exceptional circumstances it may be appropriate for a pupil to work in a class with a higher age group, but only if careful thought has been given to mathematical progression through to the end of that sixth form.

It is unacceptable to accelerate pupils if they are not going to acquire a deep understanding of the material they are studying which goes beyond that necessary to achieve good examination results. There is a very real danger that such pupils do not continue with their study of mathematics at sixth form level, because the faster pace and narrower focus has put them off the subject or appears to have left them with little that is new because of superficial 'coverage'. The Mathematical Association has campaigned actively against government policies designed to encourage schools to accelerate more of their able pupils by entering them for GCSE in Year 9 or even earlier. Correspondence with ministers on this issue will be found on the President's Page.

Books about Mathematics

There are a wide range of books about mathematics other than standard text books. Look in the school library, the public library, book shops and on-line book shops such as: Amazon, Blackwell's and WH Smith.

Here are a few suggested authors and titles:

Mathematics for Gifted Pupils by Anita Straker is a useful source of ideas and resources for the teacher. It was written nearly 20 years ago and is now out of print, but it is still very relevant if you can find a copy.

Books by Brian Bolt, published by Cambridge University Press are full of ideas and activities written for pupils, but with answers and commentary included. Here are some of his titles:
Mathematical Activities
More Mathematical Activities
Even More Mathematical Activities
101 Mathematical Projects
Mathematics meets Technology

Making polyhedra can be very rewarding and you can learn a lot of mathematics from such activity. These books, the first of which was written over 40 years ago, are all excellent and have enough ideas to keep anybody busy for a long time!

Many books have been written for adults as a way of communicating mathematics to the non-specialist, but they are full of stimulating ideas which are accessible to many interested pupils and will often be found on the shelves in a public library.

Mathematical Web Sites

A rich variety of mathematical material can be found on the world wide web.

Here are a few sites that offer interesting material for able pupils.

Finally, do not forget to look at the Links section of the website.

Doug French (Chair of MA Teaching Committee)