Mathematics in the Early Years

David Clemson, Wendy Clemson

• Publisher: Routledge
• Year: 2006
• Print ISBN:
  9780415096287
• eBook ISBN:
  9781134864539
• Topic: Education
• Subtopic: Education General

The Clemsons' clear and readable book takes the reader from debates about how children learn and what children know and can do when they start school; through to a discussion of how mathematics can be managed, assessed and evaluated in the school and classroom. Linking these two parts of the book is a section on the subject of mathematics itself, from which the non-specialist reader can gain a view of what mathematics is, what needs to be thought about in planning and offering a curriculum and the special dilemmas faced in teaching and learning mathematics as a subject. A bank of case studies offers an opportunity to see mathematics in action in a variety of classrooms.

Part I

A context for mathematics

GENERAL INTRODUCTION

Teachers carry out their teaching on the basis of a complex set of personal and professional values and beliefs. All the actions that we take as teachers are based on our theories. In some cases we adopt the theories of others – writers, colleagues or friends – but in all cases we adapt them to our own beliefs and preferences, those ways of operating which we feel good about, and those which we think efficacious.

It is one of the major planks of our educational thinking that equal opportunities for learning should be offered to all children. To offer all the children in our care these opportunities in mathematics we need to evaluate our beliefs about how children learn. We can then implement those theoretical perspectives which are appropriate in learning situations. We also need to take cognisance of the value of what children have already learned, and continue to learn at home. Account needs to be taken of the contribution parents can make to the mathematical education of their children, both before their schooldays and when they are of school age. It is important to examine prejudices we may harbour about children’s potential. We may all have made assumptions about children, or justified our treatment of them, in terms of the homes and families they come from and according to whether they are girls or boys. These ideas need exploration to see if there is any foundation for them. Finally, as teachers we cannot escape our culpability in the provision of children’s first impressions of school mathematics. If children quickly acquire the view that mathematics is ‘sums’ in books, or are denied access to the whole panoply of mathematics possibilities, it is the teacher’s fault. These are the themes for this section and form the backcloth for what is said in the rest of the book.

In order to unpick all these issues we have organised Part I into two chapters. Chapter 1 explores how theoretical ideas about how children learn can have affected our personal theories of learning in mathematics. In Chapter 2 we look at the two most influential sets of people in young children’s lives, namely their parents and their first teachers. Our contention is that when children are making a start on school mathematics, their parents have already affected their mathematics knowledge and experience. We also believe that parents can continue to make a contribution once their children start school. It is then that the chief responsibility for children’s mathematical progress rests with the teachers. They can make or mar young mathematicians by their attitudes and expectations.

Chapter 1

How children learn

INTRODUCTION

In this chapter our main focus is on children’s intellectual development in school, for it is with this that teachers have a particular concern. Even though it is difficult to measure and quantify intellectual growth and achievement, it is through assessment of these qualities that teachers and schools are commonly judged. We are focusing on that period of children’s lives which extends from before they begin school to when they are 8 years old. During this short time there are monumental changes in what children know, understand and can do. To try to give the flavour of the learning children undertake, here are two snapshots of children at either end of our chronological time scale. We must stress that they are only examples and cannot do more than give a glimpse of this period of vital mathematical education. The first is part of a conversation between a researcher and Patrick (aged 4 years and one month):

Mh	How many is two and one more?
Patrick	Four.
Mh	Well, how many is two lollipops and one more?
Patrick	Three.
Mh	How many is two elephants and one more?
Patrick	Three.
Mh	How many is two giraffes and one more?
Patrick	Three.
Mh	So how many is two and one more?
Patrick	Six.

(Hughes 1986: 47)

The second is part of a session in which one of the authors was working with 8 year olds, using Logo. Katie had drawn a robot and decided she wanted to instruct the turtle to draw in a diagonal. In order to achieve this it was suggested that she drew a square on the screen. This she did, using 100 turtle steps for the length of the side, finishing with the turtle pointing vertically up the screen and in the bottom left hand corner.

Katie	We’ll turn it 45.
Dc	OK – left or right?
Katie	Right.
Dc	What next?
Katie	Forward … erm … well the square is 100 so forward 100.

(When the diagonal proved too short, Katie tried again.)

Katie	Forward 150.
Dc	150 more or where you started from?
Katie	50 more (when this proved too long, she had another go) … 40 more … just a tiny bit more. We’ll do 141. (Katie sat for a short while and then said) … If we make the square 200 then the diagonal will be 282.
Dc	Let’s try it.

(This Katie did to good effect. At this point DC thought Katie would now double the 200 square, but no …)

Katie

If we make a 300 it will be 141 add 282 … 423.

(Clemson 1992: 6)

Here is evidence that Katie, who is thought by her teacher and DC to be a typical 8 year old, can, without ‘special’ work, find the fundamental ideas underpinning Pythagoras’ theorem.

In our debate about children’s learning we have to consider children like Patrick and Katie and what we ask of them when doing mathematics. We have to take account of children like Sharon who entered school at almost 5 years old and who was unable to count beyond three, not able to give the word for ‘yellow’ and unable to give names to some things commonly found on a breakfast table. While providing tasks which present progression for Sharon, we also probably find children like Stefan starting school at the same time. Stefan already enjoyed making collections, could build inventive and complex models from construction kits, sang counting rhymes with accuracy and gusto and, through an interest in cars, already knew about speed limits and common road signs. One of the important backdrops to the provision of learning opportunities for Patrick and Katie and Sharon and Stefan is a view of how these children learn.

There is no single comprehensive theory that explains how children develop intellectually or how they learn. There are two main schools of thought and each has something to offer the teacher in the infant classroom. There are those who attempt to construct models of the internal thought processes and associations these have with learning (cognitive), and those whose starting point is the observation of behaviour (behaviourist).

We shall examine some facets of the work of a few influential individuals from each of the schools of thought, using as ‘pegs’ on which to hang the argument the notions of children being ‘ready, willing and able’ to learn what the teacher intends and the idea that learning conforms to a pattern which has a structure. We have therefore organised the main body of this chapter in relation to ideas about:

— the readiness of children to tackle new learning;

— the motivation of children in relation to mathematics and new learning;

— the ability of children to tackle new ideas;

— the structuring of learning in mathematics.

READINESS

The concept of ‘readiness’ is prevalent in primary schools and may be seen to pervade all that teachers of infants do in offering children learning opportunities in all areas of the curriculum. We may say things like ‘I wonder if Bobby is ready for that book yet?’ or ‘I moved blue group on today because they were ready for tens and units’. The idea that children develop cognitively, rather as they do physically, is part of the belief system of many infant teachers. This comparison with physical development is important. The visibility and measurability of physical change has clear advantages for teachers for whom there is an imperative to do two things. The first is that they are required to justify their approach to teaching and learning. Secondly, they are asked to indicate the efficacy of the process through the assessment of change and progress. The desire to draw on substantive theories and observable and measurable change is absolutely understandable in our scientific culture. Indeed, it is that desire that probably led to the ready adoption of many of the cognitive ideas put forward by Jean Piaget.

Stages and ages

From the early nineteenth through to the second half of the twentieth century, there has developed in infant schools a philosophy of child-centredness based on the ideas of Rousseau, Froebel, Montessori and Dewey. It is therefore not surprising that the work of Piaget, with its focus on individual development in the child, should have profoundly affected the work of infant teachers. Indeed, between the First and Second World Wars the Consultative Committee to the Board of Education for England and Wales was much influenced by the work of Piaget as well as many notable progressive practitioners (Evans 1985).

Piaget’s work, including his idea of ‘stages’ in development, has had a powerful impact on mathematics education. He suggests that children think differently from adults and that their thinking passes through five main stages which are equated to chronological age bands: sensori-motor period (birth to 2 years), pre-operational thought (2–4 years), intuitive thought (4–7 years), concrete operations (8–11 years), and formal operations (11–14 years). Of these, therefore, the suggestion is that the infant teacher is concerned with pre-operational thought and intuitive thought.

In coming to his views on the development of knowledge, Piaget worked with small numbers of children. He offered these children carefully constructed challenges and recorded their responses to the set tasks. These tasks and challenges were designed to ‘test’ the child’s ability in particular aspects of perception and understanding. Mathematics figured prominently in the tasks that Piaget set.

Examples of children’s thinking

An example of a challenge that Piaget offered to the small sample of children he worked with is to do with spatial concepts and perception (Piaget and Inhelder 1956). The conclusion drawn from this experiment was that children are unable to see things from any viewpoint except their own. This has, however, been vigorously challenged by a number of other researchers. For example, Donaldson (1978) reports studies made by Hughes in which, by setting the task in a modified way, it was shown that young children are able to see things from another person’s point of view. Clearly the way a task is presented to a child can have a marked effect on the outcome. Another example serves to support this point.

In a series of experiments to do with the conservation of number¹ (Piaget 1952), the results obtained led Piaget to determine three stages of performance, and it was only at stage three, demonstrated largely by older children, that conservation of number was achieved. There has also been strong criticism of these particular experiments, including important comments about the language used and the way the task was presented to the child. For example:

Children should not therefore be written off as ‘non-conservers’ if they fail on a Piagetian conservation task; they may well conserve in more favourable circumstances and may have a concept of number which is adequate for many basic number situations.

(Dickson et al. 1984: 185)

Stages, ages and mathematics

Piaget’s theory offered a template for children’s acquisition of knowledge, and a ‘scientific’ explanation of children’s understanding and capacity for tackling new concepts. Piagetian influence is still entrenched, leading to a ready acceptance of the possibility of developmental stages in respect of the acquisition of mathematical knowledge. Piaget’s work also raises questions to regarding preparedness of children to tackle new ideas or novel formulations of existing information. We feel sure there are some teachers who, in talking about, for example, Patrick or Stefan (whom we talked of in the introduction to this chapter), would say they are ‘ready’ or not yet ‘ready’ to learn addition, on account of their ages.

There are so many critics of Piagetian theory today that the idea of ‘readiness’ based on his work, so warmly embraced in the 1960s, is now unacceptable. We cannot withhold opportunities from children on the basis that they are not yet ready to learn from them due to the stage of learning development that they might have reached. This is particularly so in relation to a child’s age – one cannot state that 6 year old children have to wait until they are 7 or 8 in order to encounter particular ideas.

It is still the case that many mathematics schemes continue to evidence the influence of Piaget’s stages. A recognition of that can help us to offset the limitations imposed on children’s learning by the ‘book, level or phase a year’ approach, or the idea that children must ‘do all of A before they are ready for B’. The teacher who remarked to us, ‘I think Brian is really at Level four² in some parts of his mathematics but I must get him to do all the Levels two and three work first’, seems to us to be adopting an age and stage approach which insists upon all work at a given level having, of necessity, to be undertaken before it is possible or even permissible for children to tackle work at the frontiers of their understanding. Whilst appreciating the reasons for this we would argue for support for children in doing what they are able to do rather than taking on a scheme wholesale along with principles of ‘ages and stages’ and all that this implies in relation to readiness.

The emphases in the theories of Bruner, another leading cognitive theorist, have much to do with ‘stages’ but little to do with ‘ages’. In this respect they are different from those of Piaget. However, they may have just as important an effect on how we view children’s learning.

Spirals in learning

Jerome Bruner’s name is generally less well known in UK...

Table of contents

• Cover Page
• Half Title page
• Title Page
• Copyright Page
• Contents
• List of illustrations
• Editor's preface
• Foreword
• Acknowledgements
• Part I A context for mathematics
• Part II The subject mathematics
• Part III Managing mathematics
• Part IV Mathematics in action
• Part V Assessment, record keeping and evaluation
• Part VI In-service education
• Bibliography
• Author index
• Subject index