The Practice of Mathematics

Yvette Solomon

• Publisher: Routledge
• Year: 2013
• Edition: 1
• eBook ISBN:
  9781134677252
• Topic: Psychology
• Subtopic: Social Psychology
• Index: Psychology

The psychological description and explanation of how children learn to work with numbers is dominated by the theories of Piaget. Yvette Solomon suggests an alternative approach to the child's conception of number.

1

The development of the number concept as a field of psychological investigation

The central idea of this book involves the argument that current psychological theories of number development share common assumptions about the kind of knowledge that is involved in knowing about number, and that those assumptions can be questioned. The major concern of such theories may be said to be the generation of explanations of how children come to have knowledge, but there is, generally, little discussion of what it is that children know when they know about numbers. This is, apparently, due to the dominance of Piaget's (1952) approach to the psychology of number development; critical discussion centres on Piaget's claims about when children have an understanding of number, and on his account of how this understanding develops, but it rarely challenges his conceptualization of knowing about number itself.

But what a theory says about the development of knowledge is related to what it says, or assumes, about the nature of that knowledge; thus the flaws in Piaget's account of the growth of number understanding may be a result of his assumptions regarding what is involved in knowing number. In Chapters 2 to 5, therefore, I will consider two broad areas: (i) the influences on Piaget's theory which led him to develop the kind of account of knowing number that he proposes; and (ii) his resultant theory of number development in terms of its adequacy as an account of the growth of understanding. Chapter 2 examines the background to Piaget's account and his proposed genetic epistemology, while Chapters 3 and 4 set his theory of the development of the number concept within me context of his epistemological concerns. Chapter 4 also examines Piaget's account of the growth of understanding in terms of its adequacy as a psychological account, while Chapter 5 discusses the general problems of describing and explaining the growth of knowledge in Piaget's theory. I will suggest that such problems are inherent in any theory which, like Piaget's, assumes that knowledge gain is a question of the solitary individual's construction of concepts defined in what I shall call ‘essentialist’ terms: that is, in terms of the possession of certain logical notions which, it is claimed, are necessary and sufficient conditions for understanding number.

This is an argument which I pursue in Chapters 6 to 8 where I suggest that recent developments in the field, whether critical of Piaget or not, do not show a significant departure from the Piagetian approach. In Chapter 7, I show that attempts to improve on Piaget's theory from within the Piagetian framework do not avoid the problems of his original account; this is also true of Piaget's critics, as I demonstrate in Chapters 6 and 8. Following on from the suggestions made in Chapter 5, Chapter 6 also develops the claim that knowledge is intrinsically social, and that coming to know about number is more fruitfully characterized as entering into the social practices of number use; in Chapter 8 I illustrate this by means of a reinterpretation of the data generated by some recent research into number development and arithmetic skills. Finally, I will make some suggestions in Chapter 9 for a reorientation of the field which focuses on the social practices of number use and an analysis of how children enter into those practices.

The book thus concerns five main themes: (i) the context of Piaget's theory; (ii) the problems of explaining the development of knowledge about number; (iii) the development of the field since Piaget; (iv) a reconceptualization of number understanding; and (v) an analysis of how children are inducted into the social practices of number. For the remainder of this chapter I will consider each of these in turn.¹

The context of Piaget's theory

Piaget's original intention in studying the development of thought in the child was to answer problems of epistemology concerning the justification of knowledge claims (see Piaget 1966, 1972a, 1972b, for instance). Specifically, in developing an account of the child's conception of number, Piaget intended to contribute to a debate dating back to Kant and concerning the nature and status of mathematical propositions: due to a series of developments in the foundations of arithmetic which are described in the first section of Chapter 2 the question, ‘What makes mathematical statements true?’ was at the centre of the debate.

For Piaget, the psychological genesis of ideas provides the solution to this question, by transforming it into one which asks, ‘What makes mathematics psychologically possible?’ His introduction of psychological development into epistemology to form his own genetic epistemology is the subject of the second section of Chapter 2. Piaget argues for a special relationship between logic and psychology in terms of a parallelism between thought structures and the structures of mathematics or logic, while his main aim in genetic epistemology is to synthesize the opposing ideas of mathematics as creation and mathematics as discovery. Here are seen the beginnings of Piaget's psychological accounts of knowing and coming to know as logically structured knowledge arrived at by a process which synthesizes genesis and structure.

Piaget's epistemological concerns lead him to develop a particular kind of theory of the child's conception of number, as Chapter 3 shows: his criticisms of the attempts of the logicist and intuitionist schools to answer the question, ‘What makes mathematical statements true?’ provide a basis for his account. The first section of Chapter 3 shows how Piaget (1966, 1968a, 1972a) brings empirical evidence to bear on the logicist and intuitionist examinations of the foundations of arithmetic, and judges both to be inadequate on this basis. The result is Piaget's synthesis of order and class described in the second section of the chapter.

In noting the context of Piaget's theory in terms of his intentions and the influences upon him, an important question can be raised regarding die status of the theory as a psychological account. Piaget's theory is clearly a product of his epistemological aims, and as such it lays great emphasis on logical structure and its development. Not only does this lead to problems in explaining development, as I show in Chapters 4 and 5, but it also suggests that Piaget's assumptions regarding the nature of knowing about number should be questioned. The form taken by his theory is not the only possible one, and it may be that, while Piaget is describing what must be true for a judgement to be objective, he does not thereby provide an account of the attainment of that objectivity.

Problems of explaining the development of knowledge about number

Piaget's epistemological concerns extend beyond the account of the foundations of number to a criticism of empiricism and rationalism (1966, 1972a, 1972b) and a restatement of his intention to find a middle way between ‘genesis without structure’ and ‘structure without genesis’, which I describe in the first section of Chapter 4. The next section discusses the resultant synthesis of genesis and structure; Piaget is concerned to avoid both the simple mirroring of empiricism and the innate structures of rationalism by means of the assumption of a biological drive towards higher states of equilibrium in which new structures are created from prior ones without being identical to them. Examination of the details of Piaget's (1966, 1968b, 1972a, 1978a, 1978b, 1980) account shows a fundamental problem in the equilibration model which concerns the recognition and resolution of disequilibrium. It apparently begs die question as to how disequilibrium can be recognized in die first place without presupposing die knowledge which is, ostensibly, die product of die resolution of disturbance, not its cause. This is a problem which functions at all levels of development and affects Piaget's account of the separation of subject and object, die formation of concepts, and die completion of operational thought which is necessary for possession of the number concept. As I will show in Chapter 5, Piaget's (1966) emphasis on the solitary abstraction of structure from action as the basis for logico-mathematical knowledge leads to major problems in his theory.

The problems encountered by Piaget's theory are not unique to his account of development. They apply, as I argue in the first section of Chapter 5, to any theory which relies on a model of transition from weaker to stronger logics in order to describe development. Essentially, the problem is that concepts belonging to a stronger logic, if they really do belong to that logic, cannot possibly be generated from the concepts of a weaker logic since, by definition, the stronger logic can express ideas inexpressible in the terms of the weaker. This criticism directly threatens Piaget's account, which relies on an idea of reconstruction and qualitative change in order to achieve its middle way explanation.

Piaget hopes to avoid the threatened collapse into nativism by means of the device of reflective abstraction of new content from the forms of preexisting knowledge or actions. But, as I argue in the second section of Chapter 5, his notion of reflective abstraction fails to present a real alternative to ordinary abstraction because it is not possible that someone could abstract the concept of, say, one-one correspondence from the activity of playing with pebbles without the concept of one-one correspondence being presupposed. Furthermore, while the action that the child engages in in playing with pebbles could be described as that of putting things into one-one correspondence, this does not necessarily mean that the child sees it as that action, or could ever see it as that action while he remains outside of a mathematical context.

If the meaning of action is given by the context in which it occurs, then understanding of number use cannot be abstracted from action by the individual alone; in the third section of Chapter 5 I argue that Piaget's equilibration model cannot account for the growth of knowledge that is objective. Thus I present the first part of the argument that knowledge is intrinsically social and that the development of number understanding must therefore be reconceptualized as entering into social practices: understanding numbers is a question of knowing how and when to use them appropriately in different contexts. This argument is pursued in Chapter 6.

Development of the field since Piaget

Piaget's general approach tends to dominate the field of investigation into development of the number concept; even theorists who are critical of Piaget appear to share his assumptions regarding the nature of understanding number, as Chapters 6 and 8 show. Piaget's account of concept possession is an essentialist one: it assumes that all uses of number have a common factor and that the term ‘number’ has a unitary meaning such that understanding of the number concept can be attributed to someone if they fulfil certain necessary and sufficient conditions, namely, understanding of the logical notions of class, order, and one-one correspondence. In the first section of Chapter 6 I will criticize this account of number understanding and argue instead that knowing about numbers involves a series of applications of number in different situations, and that this knowledge can be shown in a variety of ways, through a variety of behaviours.

Although theories of number development since Piaget have followed new trends, for instance the information-processing approach or the move into arithmetic, an essentialist account of concept possession remains. Information-processing models represent an attempt to express the dynamism of development which is missing in Piaget's theory by means of the statement of knowledge in procedural rather than declarative terms. But as the second section of Chapter 7 shows, Case's (1978a, 1978b, 1982, 1985) attempt to fill the gaps in Piaget's account of growth results in a theory which relies on a basic abstraction model of development in which die abstraction of new knowledge presupposes the very knowledge to be gained. Its effect is more to emphasize the social nature of knowledge man to produce an adequate account of its development. In the first section of Chapter 7 I discuss another recent development of Piagetian theory; this is Doise and Mugny's (1984) and Perret-Clermont's (1980) attempt to introduce social factors into the Piagetian account of development by means of a socio-cognitive model of conflict. They do not, however, produce a theory which answers the criticisms of Piaget's original account: their model of conflict introduces social factors into the growth process, but fails to recognize the essentially social nature of knowing itself. Furthermore, there is reason to doubt whether children really experience the conflict on which Doise, Mugny and Perret-Clermont's model relies; this is also a criticism of Case's information-processing version of Piaget's theory.

Essentialism remains in the accounts of number development given by Piaget's critics too: in the second and third sections of Chapter 6 I discuss two theories, Bryant's (1972, 1974) and Gelman and Gallistel's (1978), in these terms. Even though it opposes Piaget's theory, Bryant's account shows basic similarities: it assumes that having the number concept entails having access to essential logical principles and that the possession of these can be demonstrated by success in any task isomorphic in structure to those principles. Gelman and Gallistel's account also opposes Piaget in that their initial aim is to give a positive account of what children actually know, rather than a negative one couched, as Piaget's is, in terms of logical lacunae. They do not, however, fulfil this purpose, ending with an account which merely sets a lesser criterion for having the number concept while retaining the Piagetian concept of knowing. For Gelman and Gallistel, failure on a test is the result of a failure of cognitive organization, a lack of ‘algebraic reasoning’, or the overwhelming nature of certain experimental materials; their account assumes that children understand what is required of them in the experimental situation. Relatedly, Bryant's theory also assumes that the part played by context in knowledge is unimportant, so that a task isomorphic in structure to a particular logical principle can be solved solely by reference to that structure with no other information or understanding necessary; the possibility is not considered that understanding of the situation itself in terms of participating in its social practices is in fact intrinsic to success in the task in question.

This is also true of the work reported by Donaldson (1978) and her coworkers, and discussed in the first section of Chapter 8. These researchers maintain that the design of Piagetian tasks is such that they produce ‘false negative results’: linguistic, non-linguistic, and social factors contribute to masking the child's true competence. Thus Donaldson's main conclusion is that children are unable to show their competence because they cannot attend to the language of the test question and so ‘disembed’ it from the non-linguistic aspects of die situation. Defences of Piaget, however, claim that the modified tasks designed by Donaldson and her co-workers simply produce ‘false positive results’: they allow children to respond correctly without recognition of the logical necessity of their answers. Both of these views, however, assume that correct solution of Piagetian tasks demands a certain underlying logical competence. They fail to capture the relationship between children's behaviour in such situations and their interpretation of the experimental context and die way in which language is used in it to indicate me experimenter's intention. In particular, while Donaldson does pay some attention to die child's interpretation of die experimental situation, she assumes, contrary to the position I develop in die first section of Chapter 6, mat language understanding can be separated from the context in which it is used and that correct solution of Piagetian tasks is a qu...

Table of contents

• Cover
• Half Title
• International Library of Psychology
• Full Title
• Copyright
• Contents
• Acknowledgements
• 1. The development of the number concept as a field of psychological investigation
• 2. Why does Piaget's theory take its particular form?
• 3. The child's conception of number
• 4. Piaget's account of the growth of understanding
• 5. Does Piaget give an adequate account of growth?
• 6. Do number theorists give adequate accounts of knowing?
• 7. Can a Piagetian perspective be defended?
• 8. Knowing how and when to use numbers
• 9. ‘The development of mathematical thinking’: entering into the social practices of number use
• Notes
• References
• Index