The Sociology of Mathematics Education

Mathematical Myths / Pedagogic Texts

Paul Dowling

• Publisher: Routledge
• Year: 2002
• eBook ISBN:
  9781135710026
• Topic: Education
• Subtopic: Education General

Until the 1960s, maths was studied as an academic subject in a desire to have more mathematicians. The current trend, however, has moved away from viewing maths as a purely intellectual endeavour and towards developing a more mathematically competent workforce and citizenry. This trend has seen a large increase in the number of maths schemes being produced by the major educational publishers, which attempt to make maths easier and more approachable by using language instead of symbols. So why do so many children still fail at maths? The author contends that to understand this, teachers need to analyze and evaluate the maths textbooks they are currently using. The author shows the reader how to systematically analyze and evaluate these textbooks. This interrogation of classroom resources, should have important implications for teaching strategies and for textbook design and use.

Chapter 1
Mathematical Myths

This book is, as its title indicates, a sociology. By the use of this term I mean that the theoretical space in which I am interested is concerned with patterns of relationships between individuals and groups and the production and reproduction of these relationships in cultural practices and in action. My principal aim in this book is to introduce a theorizing of this space. This theorizing is a language of description which has been designed to enable the analysis of empirical data.¹ I shall refer to my particular language of description as social activity theory. I shall present its detailed structure in Chapter 6, following a discussion, in the previous chapter, of work which has been influential in its formulation. Social activity theory has also been designed to be consistent with my general conception of the nature of sociological analysis. I shall refer to this general methodological position as constructive description. This position will also be elaborated in Chapter 6.

As well as occupying or establishing a theoretical space, the book is concerned with an empirical space. This space is school mathematics. I shall be making reference to a range of data relating to school mathematics. Principally, however, I shall produce an analysis of the secondary school mathematics scheme, SMP 11–16². Arguably, it is through its institutionalized texts that the specificity of any activity is most clearly visible. The social activity analysis of these pedagogic texts will, therefore, enable me to make a number of statements about the nature of the school mathematics activity to which they relate. These statements have to do with the ways in which school mathematics is established as a set of practices and, specifically, the divisions and distributions within mathematics and between mathematics and other practices.

Because the principal data form that I shall be using is textual, this work is also intended to be a contribution to textual analysis. In the main data analysis chapters— Chapters 7–11—I shall introduce a number of approaches to textual analysis. The emphasis will be on a semiotic approach, but I shall also incorporate quantitative content analysis in Chapters 7 and 9.

So, the book is concerned with particular theoretical and empirical spaces— sociology (social activity theory) and school mathematics, respectively—and with general and research methodology—constructive description and textual analysis. I have titled this book: The Sociology of Mathematics Education. My contention, however, is that constructive description and social activity theory are of far more general applicability in sociological work.Their generalizability can, I want to claim, extend beyond the empirical space—school mathematics—and beyond the analysis of pedagogic texts. In other words, the book is a localized introduction to a more general language. I hope to be able to demonstrate this as well in the book.

Nevertheless, mathematics in general and school mathematics in particular have been absolutely central in the generation of the more general theory. The choice of school mathematics as the empirical domain is not arbitrary.There are at least three respects in which school mathematics may, on the face of it, at least, be referred to as a critical case. Firstly, it exhibits a highly explicit grammar in respect of what can count as a mathematical utterance and what can count as a true mathematical utterance. This is evidenced by the fact that very few of the answers in the SMP 11–16 Teacher’s Guides or in school mathematics answer books more generally include variations.

Secondly, school mathematics text is at least as distinctive as and arguably more distinctive than that of any other academic school discipline. Halliday (1978) and, following him, Pimm (1987) have made reference to a ‘mathematical register’ incorporating specialized and reinterpreted terms. The recognizability of elements of this register is substantially a function of the high degree of explicitness of the grammar of mathematics and this has certainly penetrated secondary school mathematics, so that mathematical texts can, generally, instantly be classified as such.

Both the explicitness of its grammar and the consequent recognizability of its texts contribute to the suitability of school mathematics as a critical case for my purposes. This is because an important feature of my analytic work entails distinguishing between mathematical and other practices. The particular forms of realization of the relationship between mathematical and other practices give rise to a mythology: mathematics is a mythologizing activity to a degree that is probably unparalleled on the school curriculum. This is the third critical feature of school mathematics.³

As a school discipline, mathematics has maintained a prominent place on the curriculum at least since the advent of compulsory schooling in England early on in this century. This prominence is most commonly justified on utilitarian grounds. That is, mathematics is claimed to be useful, to comprise ‘use-values’ with respect to diverse economic and domestic practices. This interpretation of mathematics is generally opposed to the elitist view of mathematics as an intellectual endeavour which is substantially isolated from other activities. This position is famously put by G.H.Hardy in his A Mathematician’s Apology:

I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. […] The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them. (Hardy, quoted by Davis and Hersh, 1981; pp. 85–6)

It is, of course, not universally accepted that society ought to deploy any of its scarce resources towards subsidizing Cambridge dons in order that they might create memorials to themselves. Curriculum developments associated with the modern mathematics movement in the 1950s and 1960s generally located the utility of mathematics within mathematics as an academic discipline: what was wanted was more mathematicians. Such a position differs from Hardy’s, but is not necessarily operationally incompatible with it. The current trend, however, is oriented more towards the widespread dissemination of mathematical use-values: not more mathematicians, but a more mathematically competent workforce and citizenry. I want to argue that this latter position, in particular, fails to recognize the social basis of its own epistemology. That is, it generalizes its own practices beyond the context in which they are elaborated, failing to recognize the fundamental implications of moving between contexts.

Alongside the aim of dissemination there has been an increasing amount of writing within the general field of mathematics education that claims mathematics as an essential feature of all human cultures. This view is often explicitly opposed to the elitist view which is often associated with the eurocentric understanding of mathematics as an exclusively European product. Again, I shall claim that this position is no more than a recycling of precisely the same eurocentrism in its failure to acknowledge the social and cultural specificities of the practices that it redescribes in terms of European school mathematics. It might be appropriately referred to as a crude form of mathematical anthropology.

The utilitarian views of mathematics—whether or not mathematics is valued in its own terms—and mathematical anthropology may be associated with a mythology comprising three myths of school mathematics. The myths are, as I hinted at above, concerned with the relationship between mathematics and other cultural practices. I shall introduce these myths in the next three sections of this chapter. In the fourth section, I shall consider a governmental text from a critical political context, the new South Africa. This text represents an attempt to recruit education—clearly including mathematics education—to restructure the socioeconomic sphere via the elimination of its divisions and hierarchies. The text can do this, however, only by establishing its own myth. The question that is raised in the concluding section of the chapter is how can we describe the conditions which make possible the construction of these myths and which constitute them as myths. This is the question that I shall begin to address in Chapter 2 and which is the most central concern of the fundamental sociological thrust of this book.

The Myth of Reference

Mike Cooley (1985) relates the following anecdote:

At one aircraft company they engaged a team of four mathematicians, all of PhD level, to attempt to define in a programme a method of drawing the afterburner of a large jet engine. This was an extremely complex shape, which they attempted to define by using Coon’s Patch Surface Definitions. They spent some two years dealing with this problem and could not find a satisfactory solution. When, however, they went to the experimental workshop of the aircraft factory, they found that a skilled sheet metal worker, together with a draughtsman had actually succeeded in drawing and making one of these. One of the mathematicians observed: They may have succeeded in making it but they didn’t understand how they did it.’ (Cooley, M., 1985; p. 171)

I want to suggest that what is going on here is the propagation of a myth. Cooley is, of course, ironizing the mathematician’s utterance in his recruitment of it as reported speech. But the mathematician’s voice prior to this refraction is constituting intellectual mathematics as definitive of understanding and so a higher order of activity than the manual tasks of drawing and metalwork. This hierarchy is not to be gainsaid by the contingency of practical success. A division is being established between the intellectual and the manual and the former, represented by mathematics, is constituted as generating commentary upon the latter. Mathematics is mythologized as being, at least potentially, about something other than itself.

But this myth has a basis in social relations. In its way—and from the opposite perspective—the mathematician’s pronouncement is as dramatic an illustration of social class hierarchy as is Molière’s exclamation by Monsieur Jourdain upon discovering that he had been speaking prose for forty years without suspecting a thing. The successful ‘manual’ workers’ alleged failure to recognize the mathematical nature of their task is constituted as the extent of their lack of understanding. Prose by any other name may perhaps evoke sweet or acrid odours depending upon the relation between utterer and interlocutor. Here, it is more than a question of style. The moral shock expressed by the mathematician is a response to the possibility of an overturning of the social order, that the hand might succeed very nicely without the head.

Some time ago, I came across another industrial illustration which relates specifically to school mathematics. The Cockcroft Committee of Enquiry into the teaching of mathematics in schools was set up in 1978 as part of James Callaghan’s Labour government’s response to widespread criticisms of state education, especially from the engineering industry and the writers of the Black Papers (see Kogan, 1978; Lawton, 1980; Salter & Tapper, 1981). The Cockcroft Committee was instructed to pay:‘…particular regard to the mathematics required in further and higher education, employment and adult life generally…’ (Cockcroft et al, 1982, p. ix).

As part of their response, the Committee set up three research studies. One of these studies was conducted by a team from Bath University. This study focused on the mathematical requirements of the working practices of sixteen-to-nineteen-year-olds. The researchers collected data which they categorized as ‘specific tasks incorporating mathematics’ (STIM) and ‘mathematics incorporated in specific tasks’ (MIST). They found that a great many young employees—a Vast army of people’—did not appear to require any formal mathematics, not even counting or recording numbers. Nevertheless, they claimed that ‘all these occupations involve actions which could be described in mathematical terms’ (Bailey et al, 1981; p. 12). The researchers presented a list of these mathematical terms (MIST) as follows:

A set, dis-joint sets.
Mappings, one-to-one, one-to-many, many-to-one correspondences.
Symmetry, bilateral and rotational.
Rotation, reflection, translation and combinations of these [.] Tessellating patterns.
Logical sequences (if…then…). (Bailey et al, 1981; p. 26)

The terminology used here is very much out of the ‘modern mathematics’ tradition of the era. The tasks (STIM) to which these MIST items correspond are:

1. Articles are sorted into separate collections for packing or on an accept/ reject basis.
2. Articles are moved into particular orientations involving moving sideways, turning over or round.
3. Articles, such as wine glasses [,] are checked for uniformity of shape.
4. Packed articles form regular patterns.
5. Assembly tasks can involve matching parts, such as connecting wires to correct terminals.
6. Tasks often have to be carried out in particular orders sometimes requiring simple decisions, but which would not often be verbalised. For example, a creeler in a carpet factory: ‘Is the spool empty? Yes! Replace with another of the same colour’. Awareness of the consequences of not following the prescribed order may be important, (ibid; pp. 25–6)

The researchers are not claiming that creelers need to study formal logic in order to be able to replace a carpet spool when it’s empty rather than when it’s full. On the contrary, they are clear that such tasks are successfully carried out in the absence of mathematical knowledge. Yet in making this claim, they are, like Cooley’s mathematician, establishing a division between the mathematicalintellectual and the manual and constituting the former as generative of commentaries upon the latter. It is as if the mathematician casts a knowing gaze upon the non-mathematical world and describes it in mathematical terms. I want to claim that the myth is that the resulting descriptions and commentaries are about that which they appear to describe, that mathematics can refer to something other than itself. I shall refer to this myth as the myth of reference.

This myth is endemic in school mathematics. These mathematical tasks, for example, are taken from the ‘Y’ series of books in the secondary school mathematics scheme, SMP 11–16:

Shopkeeper A sells dates for 85p per kilogram. B sells them at 1.2 kg for £1.

1. Which shop is cheaper?
2. What is the difference between the prices charged by the two shopkeepers for 15 kg of dates? (SMP 11–16 Book Y1, p. 55)

Britannia best British flour cost £0.71 for 12.5 kg.

Uncle Sam’s best American flour cost $1.30 for 3.5 lb.

If 1kg=2.2 lb and £1=$1.85, which brand of flour was cheaper, and by how much per kilogram? (SMP 11–16 Book Y1, p. 56)

These tasks both recruit a domestic shopping setting. However, it is quite apparent that the tasks are mathematical rather than domestic. This is made clear through a number of devices. The use of the letters A and B to stand, algebra-style, for the shopkeepers, for example, and the reference to an unlikely purchase (15 kg of dates) and an unlikely comparison (cheaper flour hardly justifies a trip across the Atlantic). The stylizing of the names of the brands of flour is also quite obvious. A division between mathematics and domestic activity is maintained so that mathematics is again constituted as generating non-mathematical referents.

The myth of reference, we might say, constructs mathematics as a system of exchange-values, a currency. Mathematical commentaries can be exchanged for the practices which they describe: Coon’s Patch Surface Definitions for making an afterburner; formal logic for carpet creeling; ratio and proportion for domestic shopping. The myth encourages us to move between two spheres of activity, one of which is always mathematics. The range of other activities is a measure of the power of mathematics as a currency. School mathematics textbooks often incorporate a considerable diversity of non-mathematical settings, with sequences of tasks moving rapidly from one setting to another. But mathematics is ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• List of Figures and Tables
• Acknowledgments
• Preface By Series Editor
• Chapter 1: Mathematical Myths
• Chapter 2: Juggling Pots and Texts
• Chapter 3: Sociology, Education and the Production of ‘Ability’
• Chapter 4: The Analysis of School Texts: Some Empirical Antecedents
• Chapter 5: Towards a Language of Description: Some Theoretical Antecedents
• Chapter 6: Constructive Description and Social Activity Theory
• Chapter 7: An Introduction to the Empirical Text
• Chapter 8: The Textualizing of Algebra
• Chapter 9: Genres of Production
• Chapter 10: Setting and the Public Domain
• Chapter 11: Interpellating the Teacher
• Chapter 12: Disturbing and Re-Establishing Equilibrium
• References and Bibliography