Developing Understanding In Primary Mathematics

Key Stages 1 & 2

Deirdre Pettitt, Andrew Davis, A. Davis, A. Davis

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781135719685
• Edition: 1
• Topic: Bildung
• Subtopic: Bildung Allgemein

This text presents a philosophical yet classroom approach to mathematics teaching, and examines how mathematics is taught across the curriculum and age range in primary school. It explores the role of play, story, drama, pattern, sound, and children's drawings and games in maths teaching.

1
Introduction

Andrew Davis

‘What is the capital of France, Smith?’

‘Er…Madrid, sir.’

‘No, Smith. Stand up! Paris is the capital city of France. What is the capital of France, Davis?’

I had a copy of ‘The Hobbit’ on my knees. I was making furtive efforts to read it, raising my head every so often to masquerade as an attentive pupil.

‘Edinburgh, sir.’

‘STAND UP, DAVIS!’

The book slid to the floor with a crash, observed grimly by the teacher.

‘What is the capital of France, Fisher?’

‘Paris, sir.’

‘Thank you, Fisher.’

The teacher slowly approached my desk. I knew what would happen next….

Such scenes were a dreary and frequent part of my childhood at primary school. In the nineteen nineties, it would be difficult to discover living examples of this teaching style. Yet its model of learning still persists. Many people, including some politicians think that learning is a kind of copying. There are items to be copied; ‘facts’ about number, or historical events, or gravity, for instance. The learner takes a copy of these facts into her head. She can then easily demonstrate her learning; on request, she can say or write it. Teachers in this account are thought to ‘transmit’ that which is to be learned: to perform the necessary actions for the pupil to acquire a copy of the teacher’s version of the facts.

What might these ‘necessary actions’ be? On the face of it, standing up at the front and simply ‘telling’ should be quite sufficient.Teachers stuff the rucksack (the pupil mind) with contents (facts) or perhaps pour knowledge into children (who start off empty). Shuard remarked in a famous broadcast that pupils had proved to be very leaky vessels.

In effect learning per se is being equated with one of its particular varieties, namely rote learning (Bailey, 1984). Now arguably it is a pity that we no longer ask children to memorize poems or historical dates. Most teachers have always expected children to learn number bonds and tables; their expectations have now been backed up by National Curriculum requirements. However, the bulk of primary mathematics should not only be taken into the memory but also digested thoroughly. Information which is merely memorized cannot be used or applied in the everyday world, for it can be ‘in’ the mind without being understood.

Rote learning is a relatively simple phenomenon to grasp. Other varieties of learning, where the ‘digesting’ is more comprehensive, are in many ways deeply mysterious. It is much easier to attack incorrect accounts than to tell a positive story. This book tries to rise to such a positive challenge, for Key Stages 1 and 2 mathematics in particular. The next chapter makes some general observations about mathematics learning, subsequently developed and applied in practical ways.

The ‘transmission’ model of teaching, we will contend, fails to take account of the nature of much mathematical knowledge and understanding, and of the clear implications for the ways in which mathematics must be learned. How then should we help children to develop mathematical understanding? What are the appropriate roles for the teacher?

We argue that certain kinds of classroom contexts together with particular teaching styles, are especially likely to stimulate pupils to ‘construct’ or ‘invent’ their own mathematical understanding. A selection of these ideas for classroom application are discussed in detail. The latter constitute the heart of this book. To attack transmission theories of learning is scarcely original and we devote only a section of the second chapter to doing so. Nevertheless, we are conscious that criticism of such theory has had frustratingly little effect on educational policy or classroom practice in the last fifteen years; hence we do not apologize too abjectly for raising objections in print yet again.¹

Our experience tells us that some primary teachers agree with our approach. Those who do and who enjoy labels could say they were ‘constructivist’ in outlook and that they believed in ‘constructivism’. We make much use of this term. A well-thought out constructivism am be a valuable acquisition for teachers whose convictions were previously at the tacit intuitive level. Views of teaching and learning held by politically influential groups are in stark contrast to constructivism, and in such a climate it is especially important to be able to support practice with a coherent and readily justifiable theoretical position.

Transmission culture was thought to be very much in retreat in the fifties and sixties. It is often claimed that the scene was dominated then by child-centred ideas. Child-centredness meant different things to different people. It referred to styles of teaching, to a set of moral attitudes to children and in particular to the view that pupils should exercise substantial control over their own learning. Confusion abounded, but lurking behind some of the opposition to the then prevailing fashion for formal whole class teaching was an intuitive rejection of transmission as a model of teaching and learning. However, even that rejection itself was confused with a superficial hostility to teachers talking to large groups of children. We will see later that a constructivist teacher may well choose at times to work directly with the whole class.

Many of us now know that child-centred thinking in whatever guise actually made only a modest impact on the practice of most primary schools. Transmission remained a potent model for learning at the classroom level. Whole class teaching, with which transmission was and still is confused became politically incorrect on some initial teacher education courses and in some LEAs (Alexander, 1992). For the last fifteen years or so transmission has once more become politically correct in the context of national education policy.

Chapter 2 outlines a version of constructivism which underlies the rest of the book. Constructivism is quite fashionable at present among theorists, much of it vague or incoherent. Our objectives are not primarily academic or scholarly, and we have therefore refrained from treating critically and in detail the substantial ‘constructivist’ writings in the literature. To explore the exotic landscapes of von Glasersfeld and his ilk would be of little help to those closer to classroom practice, and for whom this book is written. We hope to offer a straightforward and robust version of constructivism, with clearly specifiable implications for classroom mathematics.

Chapter 3 proceeds to examine ways in which story making, narrative and drama can contribute to the growth of children’s mathematical understanding. The role of the teacher is a key issue, and this is explored in detail through many examples.

The discussion then touches on the state of health of teachers’ own mathematics. Adults, it is suggested, might profitably experience constructing or reconstructing aspects of their own knowledge of mathematics, thereby improving their flexibility,confidence, and capacities to help children to learn. This thought is examined with the help of some examples in chapter 4.

In chapter 5 some links between investigation, the search for pattern and constructing mathematical understanding are explored. Within investigative contexts, children can be helped to search for patterns, and enabled to discern similarities, differences, repetitions, and regularities given appropriate interactions with teachers.

Games have long been advocated as valuable aids in primary mathematics; chapter 6 illustrates ways in which games could aid children’s mathematical construction. It is argued that it is the package of games plus teachers playing certain roles which provides the key to effective learning here.

Children may be helped in their mathematical invention through engaging in practical activities using sounds, and the next chapter illustrates ways in which this might be done. There is little which is intrinsically ‘constructivist’ about sounds and so much discussion is devoted to the kinds of interactions between teachers and children in these contexts which should provide good catalysts for mathematical invention.

Chapter 8 looks at young children’s drawing behaviour and focuses on the surprisingly rich mathematical aspects which are developed through this. It is argued that a deeper understanding of children’s drawing systems will enable teachers to heighten their own understanding of art and mathematics and therefore to be able to improve the quality of their work with children.

Children’s experiences outside school play a substantial role in fostering their individual creation of mathematical structures; this affects the level of mathematical knowledge possessed by the new Reception pupil and continues to influence developments in their mathematical understanding while they are at school. Important questions arise concerning the relative contribution of school and the ‘real world’ to the growth of mathematical understanding. These issues are examined in the final chapter. Moreover we ask about the senses in which mathematics should be ‘real’ to young learners, and about the extent to which this can actually be the case. The last words in the book are left to a class teacher who might not call herself a constructivist but who has reflected long and hard on the challenges presented by teaching mathematics in the primary school. Her insights are sampled to provide a concluding commentary.

Note

1 Child-centred thinkers in the fifties and sixties effectively criticized ‘Gradgrind’ teaching (a familiar reference to the Dickensian character who was obsessed with the attempt to transmit facts to his pupils). The objection was not, or should not have been to facts as such, but to Gradgrind’s vision of the nature of facts and of how they might be suitably acquired.

2
Constructivism

Andrew Davis

Introduction

In chapter 1 we met an old enemy: the ‘transmission’ view of learning. We regard this particular model as committing three major categories of mistake (among others).

1. It is wrong about the knowledge which is to be gained by pupils.
2. It fails to understand pupil’s minds and the way they must ‘store’ what they learn.
3. The idea of transmission itself cannot properly capture how pupils learn from teachers or indeed from anyone.

In this chapter we cast light on these three interconnected mistakes as we focus on the true nature of mathematical knowledge. We will see that a proper understanding of this has clear implications for the ways in which mathematics may be effectively taught. We look at these implications in general terms in this chapter, while reserving detailed discussion of their classroom application for later.

What is Mathematics?

It is less controversial than in the sixties to insist that a necessary condition for effective teaching is the teacher’s adequate grasp of the subject. (But it is not sufficient; sadly pupils have sometimes gained little mathematics even after spending long periods with very competent mathematicians.) Furthermore, what teachers take a subject to be significantly influences how they teach it.

For example, if history for you consisted of a collection of discrete facts’ which would include dates and statutes together with the definitive account of the ‘causes of important events’ you might well deem exposition to be your single most effective teaching style. Admittedly the history-as-facts model does not make such a decision inevitable. Neither however does it encourage alternatives. For instance, why on such a model bother to provide opportunities for groups of children to examine evidence and create interpretations? At the end of the day effective teaching would presumably be ensuring that individual pupils had acquired the relevant facts. Investigating, discussing, and interpreting would seem unlikely to promote this efficiently.

Suppose, on the other hand, the subject is seen as a collective understanding of ‘stories’ about the past. ‘Doing history’ would include attempts to create and grasp such narratives in the light of the evidence. According to this perspective pupils could not be treated as ‘rucksacks’ for historical facts; instead they would require guidance in the processes of entering into these stories; in developing for themselves appropriate narrative explanations of past events while taking account of relevant evidence.

This is but one vivid example of the links between favoured methods for teaching a given subject and views about its nature. Of course, the character of history itself is controversial. Vast passions were expended on drafts of the National Curriculum for history. The subject is inherently political and ideological; differing sets of values and visions of society are linked to particular ‘stories’ about past events. If history students are allowed a measure of autonomy, then there is a risk that they may come up with ‘stories’ which conflict with those preferred by certain political groupings.

In sharp contrast, mathematics is viewed as ‘absolute’, as outside political preferences (and for most, pretty uninteresting). The minor ripples around the time of the publication of the interim report of the mathematics National Curriculum working group and the resignation of a right-wing Professor of Economics, cannot hold a candle to the heady events preceding the official history ring-binder. We can imagine much more readily that teachers carry around with them differing understandings of history than that they might possess distinctive conceptions of mathematics.

Indeed, the very idea that there might be more than one view about the essence of mathematics itself may seem eccentric. Surely, it may be argued, there is no room for dispute. Its essential features are quite obvious. There are mathematical facts to be understood and learned, for example, that 2+2=4 or that regular hexagons can be used to cover a space without gaps whereas regular pentagons cannot. There are also skills to be acquired such as those of using a compass or protractor. The facts are immutable, and, so to speak, ‘out there’; they are part of the reality with which children gradually come to terms as they gain experience and education.

Some traditionalists place greater emphasis on the ‘basics’ of number, and could wish that primary schools spent less time on ‘trendy’ topics such as shape and probability. To that extent there is disagreement, but the dispute is about curriculum content, rather than the status of mathematics itself.

However, I think that we need to ponder more deeply on the nature and status of mathematics. Once we do, possibilities of alternative and indeed rival conceptions begin to emerge. Let us begin with the thought that mathematics is somehow ‘out there’, independently of us. There is an insidious mixture of truth and falsity in this vague and rather philosophical-sounding idea. The false elements in the mixture can distort our thinking about mathematics learning.

It IS true in some sense that, for instance, 2+2=4, whether I think so or not. I recall an argument with my father when I was a boy; in a spirit of mischief I denied that God could contrive that 2 and 2 were 5. My father, concerned that I was setting limits on God’s power, asserted that He jolly well could. My father was wrong, I was right, but no doubt at the wrong time and in the wrong way as usual. I was not undermining divine omnipot...