Mental Mathematics for the Numeracy Hour

Tony Harries, Mike Spooner

• Publisher: Routledge
• Year: 2013
• Print ISBN:
  9781138420625
• eBook ISBN:
  9781134116010
• Topic: Education
• Subtopic: Education General

This text is designed to help coordinators and teachers of mathematics in primary schools to plan and develop an appropriate, differentiated scheme of mental mathematics activity to support numeracy and other mathematical work throughout the primary years.

CHAPTER 1

Introduction

Introduction

What is 75 – 38? How would you approach this calculation? Would you immediately reach for pencil and paper and carry out a standard written algorithm, would you use a mental method such as counting on, or would you work mentally and make jottings of your working as you go along? Do you have any pictures or images in your head as you work? In many ways it doesn’t really matter how you approach the calculation since you will have worked out an efficient way of working for yourself. However when we are working with children we want to help them work towards efficient ways of calculating so that they are not prevented from exploring mathematics because their methods of calculation are long and time-consuming activities. In order to do this we need to start with a recognition of pupils’ own strategies and work through these towards more efficient ways of working.

It could be argued that all mathematics is a mental activity and that written work is simply an articulation on paper of this activity; this articulation we try to do in as succinct a form as possible. In this book we explore the area of mental mathematics and provide activities which will help teachers develop the mathematical competence and confidence of their pupils and at the same time help pupils to enjoy and value what they are doing.

A number of principles underpin the way in which the book has evolved:

• The environment or classroom ethos within which the pupils can develop both their mathematics and themselves as learners, is important. Generally it needs to be both non-competitive and non-judgemental so that pupils can feel confident about taking risks in their exploration of mathematics.

• Pupils need time in order to develop their mathematics. They need time to think, time to explore, time to discuss, and time to explain their reasoning.

• Pupils are constructors of knowledge and part of the role of the teacher is to ‘scaffold’ the development of that knowledge and the meaning that the pupils give to the development and application of that knowledge.

• In terms of progression this is viewed as a recognition of what the pupils bring with them and the intuitive ways they have of working, and using this to help the pupils work towards more flexible and efficient ways of working.

• Mathematics can be seen as a mental activity that needs to be communicated, and as such the pupils need to develop a language they can use in order to facilitate this. The aim will be to develop greater precision and clarity of communication.

• The National Numeracy Strategy (NNS) provides a clear framework within which pupils’ mathematics can develop. It provides both challenges and support for teachers and needs to be seen as a dynamic framework through which development can occur for both teachers and pupils.

It is probably true that in the past mental mathematics has been perceived as uni-dimensional in that it has been associated with memory tests such as table tests and speed tests – something that for many pupils has been a demoralising and humiliating experience which has created a barrier to the learning of mathematics. In the chapters that follow we try to develop a wider view of mental mathematics which explores the two aspects of mental mathematics identified by Askew (1998) – instant recall (memory activity) and figuring out (strategic activity). In undertaking this task the following themes are explored in a variety of ways.

Variety of approach

The National Numeracy Strategy (DfEE 1999a) has advocated the use of the daily mathematics lesson. In order to make this effective then variety within and across lessons is vital in order that interest and motivation are maintained. Pupils need clear structures within which to work and the daily mathematics lesson will provide this, but within that lesson teachers will need to seek ways in which the pupils remain active and purposeful in their approach to their learning. There are many skills to be developed in the lessons. In addition to technical skills, there are practical skills, investigative skills, and the skill of explanation, to name but a few. Further, the teachers will need to be aware of the increased knowledge about the working of the brain now available and the way in which this knowledge can help us to present ideas to pupils in a way that can be meaningful and appropriate to their way of learning.

Involvement and ethos

In the past, feelings about mathematics among many pupils in schools have been negative. Many students, when asked to talk about their memories and feelings about mathematics, use words such as panic, humiliation, mental multiplication table tests, punishment. It is clear that for them mental mathematics is about short tests – usually of multiplication tables – that depend very much on speed and memory. This has often been a distressing experience for them.

As the NNS is developed in schools it is important that the environment within which the pupils work is positive. Further, it needs to be somewhere where the pupils feel comfortable in addition to being challenged. In developing this kind of ethos within the classroom we need to recognise that working with the whole class is an exercise involving risks to the self-esteem of some pupils. The challenge for the teacher is to create an environment where pupils are prepared to share their thoughts and to see risk-taking as an essential aspect of the learning process (see Figure 1.1).

Figure 1.1

Language and communication

One of the most quoted passages from the Cockcroft Report (1982) is that concerning communication:

We believe that all these perceptions of the usefulness of mathematics arise from the fact that mathematics provides a means of communication which is powerful, concise and unambiguous. (p. 2)

While the question of ambiguity could be questioned, there can be no argument with the idea that mathematics can be both a powerful and precise way of communicating ideas. Thus we need to be able to help pupils both to develop a language that allows them to talk about mathematics, and also to use mathematical language precisely to explain the solution to problems. Mental mathematics becomes a vehicle through which the linguistic side of mathematics can be developed, as pupils seek ways to explain what they are doing in as precise a way as possible. In the NNS there is a strong emphasis on communication and within this, progression is seen as the development of greater precision.

Developing skills which underpin calculation

In The National Numeracy Strategy (DfEE 1999a) a number of skills of calculation are identified. These skills are underpinned by:

Figure 1.2

• Some number facts need to be remembered so that there can be instant recall without hesitation. These will include addition/subtraction facts as well as multiplication/division facts. The number of these facts that the pupils need to know will develop from year to year but there can be no doubt that having a set of remembered facts is an advantage in pursuing efficient methods of calculation.

• Some strategies make use of known facts in order to complete other calculations. These will include an understanding that knowing a single digit addition fact will enable other calculations to be undertaken speedily. For example knowing 5 + 7 = 12 facilitates the calculation of 50 + 70, 35 + 7 … 12 – 7, 120 – 70 … Similarly, knowing a simple multiplication fact enables other calculations to be undertaken. For example, knowing that 6 × 2 = 12 facilitates the calculation of 60 × 2, 6 × 20, 600 × 2 … 6 × 4, 6 × 8 … 12 ÷ 2, 120 ÷ 2 …

This second skill requires the pupils to be able to look at a number and/or a calculation and to use their knowledge of the structure of the number system to identify related calculations. It also means that it is important for pupils to be able to look at numbers and see through them so that the richness of the number is appreciated and used.

Competence and efficiency

In working with numbers our objective has to be the ability to calculate both competently and efficiently. In pursuing this, much research (Bierhoff 1996, Gravemeijer 1994) and observation of practice in other countries suggests that working mentally and not rushing into formal written methods is important. This does not suggest that mental and written methods are unconnected, but that confidence and competence with mental methods will actually help in understanding formal written methods. Often when doing mental calculations pupils will need to make informal jottings both to help their progress and to remind them of what they have done. There are two ways of looking at the mental/written methods interface. One is to see it as a progression or a link – a formal, succinct way of efficiently expressing on paper approximately what was being followed ‘in head’. The other is to see it as a switch – that a formal written method is an alternative way of performing a calculation which in some circumstances is more efficient. Recognising whether, for them, the methods are linked or alternative is important for pupils.

Developing links

In the King’s College study (Askew et al. 1997) which explored effective teaching of numeracy, the researchers concluded that numeracy teaching was most effectively undertaken by teachers who saw connections in mathematics and sought to help pupils to see the connections and so become efficient workers with numbers. It is thus important that when pupils learn a fact they are also able to give related facts which can also be ‘known’. For example, the idea of linking addition and subtraction is crucial in developing efficiency so that pupils know that once they know a fact such as 5 + 7 = 12, then they also know that 12 – 7 = 5, 12 – 5 = 7 and 7 + 5 = 12. In this way one known fact becomes four known facts. Similarly with multiplication and division, knowing one fact such as 6 × 4 = 24 immediately becomes four known facts with 4 × 6 = 24, 24 ÷ 4 = 6, 24 ÷ 6 = 4 added to the original one.

The idea about connections goes further with clear links being made between addition and multiplication and between subtraction and division. In this way the whole of mathematical calculation becomes a series of interconnecting activities rather than disconnected ones.

Creating images

One way in which pupils can be helped to progress efficiently and competently with their calculations is through the images that are used to ‘scaffold’ their development. This idea of scaffolding will be addressed in Chapter 2. These images are broadly of three types.

• Firstly we have sound images. When young children are learning anything involving words they will keep repeating the word until it sounds right. Similarly with pupils working with number they will work on counting until the sequence they are using sounds right. Thus with young children the use of rhymes will be an important activity which focuses attention on the sound of the words or sequences.

• Secondly there are concrete images which are broadly of two types which are linked to seeing and touching. In this way the learner can link sound to either/both something they can see and/or something that they can touch. The most commonly used resources for this are beads and blocks.

• Finally there are the symbolic images which are the mathematical symbols which the pupils need to know and which are the symbolic representations of the sound, sight and touch images with which they are familiar. These are the symbols which they will then need to learn to manipulate in order to develop their mathematical competence.

It should be noted that although the images are listed sequentially it is not a hierarchical sequence. The pupils need to be able to move confidently between images and not feel the need to leave one type of image behind when they move on’.

The structure of the book

We believe that good practice in the classroom needs to be underpinned by an understanding of the theories about the way in which children learn mathematics. Thus the second chapter explores the development of theories about the learning of mathematics and seeks to place the role of mental mathematics within that context. The chapter stresses the need to move from the uni-dimensional view of mental mathematics which emphasises the recall of number facts to explore present day understandings of the multi-dimensional aspects of working in one’s head. It also explores the way in which the development of mental strategies helps in the understanding of concepts and aids the development of written methods.

Chapters 3 to 6 focus on particular themes within mental mathematics and will follow the same format. A brief review of research on the theme of the chapter is followed where appropriate by a consideration of the curriculum and classroom management issues related to the theme. The last part of the chapter consists of a set of activities which illustrate the theme of the chapter and can be used with children at different stages in their schooling, or as INSET activities with teachers.

Chapter 3 focuses on the development of a classroom ethos for mental mathematics. Initially the research in this area is outlined and discussed – providing a rationale for the activities that follow. Here there is a strong emphasis on involvement activities, using such resources as flashback cards and number petals as ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Dedication
• Preface
• 1 Introduction
• 2 Perspectives on the teaching and learning of mathematics
• 3 Working with the whole class: developing a supportive classroom ethos
• 4 The use of images in the development of mathematical understanding: focusing on addition and subtraction
• 5 Developing concepts of multiplication and division
• 6 Using and applying mental mathematics
• 7 Supporting activities
• Bibliography
• Index