Teaching Mathematics to Able Children

Valsa Koshy

• Publisher: David Fulton Publishers
• Year: 2012
• ISBN:
  9781136626135
• Edition: 1
• Topic: Éducation
• Subtopic: Éducation générale

This book enables teachers to effectively meet the needs of their most able mathematicians. Using a tried and tested set of principles developed and used by The Able Children's Education Unit at Brunel University, the author demonstrates how to: identify high mathematical ability in a pupil, plan suitably challenging activities and teach them most effectively within the existing National Numeracy framework, make the most of the classroom resources available, including ICT and external agencies, implement strategies for differentiation, illustrated with real-life classroom examples.Accessible in style and featuring practical case studies throughout, this book will give teachers and student teachers the confidence and knowledge to effectively challenge and develop the skills of the most able mathematician.

Chapter 1

Mathematically able Pupils: Setting the Scene

The importance given to identifying and nurturing mathematical talent is not new. Several years ago, Straker (1982) convincingly stressed the importance of making provision for ‘mathematically gifted pupils’. She pointed out that under the provision of the Education Act 1944, children are entitled to receive an education appropriate to their age, ability and aptitude and that children with a high level of general or specific ability are no exception. Straker maintains that:

Gifted pupils have a great deal to contribute to the future well-being of the society, provided their talents are developed to the full during their formal education. There is a pressing need to develop the country’s resources to the fullest extent, and one of our most precious resources is the ability and creativity of all children.

(1982: 7)

Concerns that the needs of mathematically able pupils are not being met have been raised many times in the past two decades. HMI (1978) expressed the view that ‘there was a widespread tendency to underestimate the capabilities of all groups of children, particularly the able’. The need for policies for providing for different ability levels was highlighted by HMI (1979) in the context of secondary education.

The need for making provision for children who have high mathematical ability was expressed clearly in Mathematics Counts (Cockcroft 1982) in the following statements:

It is not sufficient for such children to be left to work through a text book or a set of workcards; nor should they be given repetitive practice of processes which have already been mastered.

The report goes on to say that:

The statement that able children can take care of themselves is misleading; it may be true that mathematically such children can take care of themselves better than the less able, but this does not mean that they should be entirely responsible for their own programming; they need guidance, encouragement and the right kind of opportunities and challenges to fulfil their promise.

(Cockroft 1982: para. 332)

Ten years on, concerns regarding adequate provision for able pupils continued to be raised. Alexander et al. (1992) pointed out that there was an ‘obsessive fear’ in some schools of being deemed ‘elitist’ and as a consequence, ‘the needs of some of our most able children have quite simply not been met’. Mackintosh, Her Majesty’s Inspector of Schools has this advice to offer those who feel uneasy about highlighting the needs of the able:

There is very clear evidence that focusing sharply on what the most able children can achieve raises the expectations generally, because essentially it involves careful consideration of the organisation and management of teaching and learning.

(OFSTED 1994a: 13)

More recently, schools have contacted the centre at Brunel University to seek advice when their school OFSTED inspection reports have highlighted concerns regarding the lack of challenge in the mathematics curriculum offered to children, the nature of repetitive work they are given, and over reliance on text books.

Why is it that the concerns raised in official reports have remained the same for 20 years? In a data gathering exercise during one of my in-service sessions on teaching mathematically able pupils, I asked teachers if they thought that their schools made adequate provision for their mathematically able pupils. The answer was an astounding ‘no’, but more interesting are the following related findings:

• Of the 74 schools from 13 LEAs represented, only three had a policy for teaching mathematically able pupils, whereas 54 schools had well defined policy statements on teaching mathematics to children with some learning difficulties. Although no one will disagree that the needs of children who experience learning difficulties must be met, it is surprising that there seem to be little appreciation that the most able also have special needs which are related to their special abilities, attributes and learning styles.
• Many of the participants felt that the publication of league tables made it inevitable that efforts and resources had to be targeted at raising the level of achievement of pupils who were borderline to reach the average Level 4 in the National Curriculum tests.
• There seemed to be an overwhelming concern about the lack of in- service support provided for teachers to enable them to explore issues of high ability in mathematics and analyse the needs of able pupils so that effective provision could be made.
• Lack of resources, especially not knowing what to do with most of what is available, was cited as a problem.
• Teachers’ lack of subject knowledge in mathematics was perceived to be a barrier in making adequate provision for mathematically able pupils

This book aims to address the above concerns by offering principles and the strategies to put the principles into practice. The contents of this book should enable a coordinator to design a policy and to implement that policy with greater confidence and understanding.

Able Children and Mathematically Able Children: Who are they?

Although there is no simple way of defining an ‘able child’ or a ‘mathematically able child’, it is useful to consider some issues relating to high ability in order to acquire a better understanding of how one may identify highly able mathematicians and make provision for them.

What are the complexities associated with aspects of identifying an ‘able’ cohort of mathematicians? First, how many children are we talking about? It is useful to consider the different perspectives that exist. In the past, it was assumed that the top 25 per cent of 11-year-olds who qualified for grammar school places were ‘able’; HMI (1992) refers to 5 per cent of children as ‘very able’ and a model presented in OFSTED (1994b), as shown in Figure 1.1, refers to 2 per cent of pupils being ‘exceptionally able’. The diversity of terminology used (George 1990) and its implications also create confusion. How do we refer to ‘able’ pupils? Do we refer to them as ‘able’, ‘exceptional’, ‘gifted’ or ‘talented’? The terminology used by the government, ‘gifted and talented’, to identify the top 5-10 per cent of the ability range may perhaps reduce some of the tension regarding the choice of words. The term ‘higher ability’ pupils fits in with the ‘continuum’ model (Koshy and Casey 1997) shown in Figure 1.2 for referring to more able pupils; as this model facilitates more flexibility both in the identification of ability and in making provision.

Second, what do we mean by ability? Different perspectives exist.

Figure 1.1 Distribution of ability in pupils in schools (OFSTED 1994b)

Figure 1.2 The ability continuum (Koshy and Casey 1997)

Perspectives on Ability

Efforts to identify and nurture ability have been around for many decades. Early definitions of ability (Terman and Oden 1947) were based on a measure of intelligence, the Intelligence Quotient (IQ), described as the ‘general intelligence factor’, indicating pupils’ ability to reason and make connections. In 1921, Terman and colleagues selected pupils with IQs over 140 for a study of giftedness. IQ testing is used for many purposes; for example, some schools use it for selection purposes. IQ tests are also given to pupils for diagnostic purposes. In spite of some useful purposes served by the IQ concept, many educationists and psychologists believe that a single measure of intelligence does not acknowledge the diverse talents, aptitudes and abilities of pupils. They believe that a broader concept of giftedness over a range of abilities is likely to achieve a more enhanced quality of provision for able pupils.

Ogilvie (1973) proposed that we need to consider a range of talents and abilities in children. Six categories of giftedness were listed:

• leadership
• high intelligence
• artistic talent
• creativity
• physical talent
• mechanical ingenuity.

Howard Gardner’s (1983, 1993) theory of multiple intelligences describes ability in domain-specific terms. Gardner proposed seven intelligences:

• linguistic intelligence (relating to language)
• logical-mathematical intelligence (relating to mathematics and sciences)
• bodily kinaesthetic intelligence (physical – dance, sport)
• musical intelligence (music and rhythm)
• spatial intelligence (space)
• interpersonal intelligence (interpersonal skills, leadership skills)
• intrapersonal intelligence (ability to reflect on oneself).

In the context of what this book is trying to achieve, we are concerned with the second one in the list: logical-mathematical intelligence. Gardner considers two essential aspects of logical-mathematical intelligence. In a person who possesses this intelligence, the process of problem solving is often remarkably rapid. Such a person is able to cope with many variables at once and creates numerous hypotheses and evaluates them before accepting or rejecting them. Gardner (1993) maintains that logical-mathematical reasoning with the companion skill of language provides the basis of IQ tests. IQ tests assess problem-solving skills; this may explain why pupils with high IQ scores often show high mathematical ability. Aspects of high mathematical ability are discussed, in more detail, in Chapter 3.

A broader definition of ‘giftedness’ or ‘intelligence’ is recommended for the identification and fulfilment of mathematical talent by a task force in the USA. Sheffield explains:

One of the charges to the Task Force was to determine mathematical promise in a more inclusive fashion than the traditional definition of gifted and talented that frequently only included 3 to 5% of the students.

(1999: 15)

In the task force report (Sheffield et al. 1995), mathematical promise is described as a function of ability, motivation, belief and experience or opportunity. None of these variables are considered to be fixed, but are areas that need to be developed so that mathematical success can be maximised for an increased number of promising pupils. The aspects listed in the report offer much food for thought for teachers in their efforts to make better and more effective provision for pupils who have the potential to do well. The report recognises that ability can be enhanced and developed and acknowledges the impact of experiences on nurturing ability. The role of motivation plays a part; careful consideration should be given to those students who may disguise their ability due to the fear of being subject to peer ridicule. Belief in one’s ability as well as in the importance of mathematical success by the students, teachers and parents is also important.

Case Studies of Mathematically Promising Pupils

In the following section the reader is...

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• Acknowledgements
• Introduction
• 1. Mathematically able pupils: setting the scene
• 2. Identifying mathematically promising pupils
• 3. Effective provision for mathematically able pupils
• 4. The National Numeracy Strategy and the able mathematician
• 5. Organisation for teaching and learning mathematics
• 6. Using ICT to teach able pupils
• 7. Selecting resources for mathematically able pupils
• References
• Index