The Pragmatics of Mathematics Education

Vagueness and Mathematical Discourse

Tim Rowland

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781135698331
• Edition: 1
• Topic: Education
• Subtopic: Education General

Drawing on philosophy of language and recent linguistic theory, Rowland surveys several approaches to classroom communication in mathematics. Are students intimidated by the nature of mathematics teaching? Many students appear fearful of voicing their understanding - is fear of error part of the linguistics of mathematics? The approaches explored here provide a rationale and a method for exploring and understanding speakers' motives in classroom mathematics talk. Teacher-student interactions in mathematics are analysed, and this provides a toolkit that teachers can use to respond to the intellectual vulnerability of their students.

1
Preview and Methodology

It [APU practical testing] afforded an opportunity to hold a prolonged mathematical conversation with a child. My understanding of children’s thought processes when solving problems has been considerably extended. (A teacher, quoted in Foxman et al., 1980, p. 73)

A mathematics teacher, Judith, has introduced her class of 14-year-olds to a mathematical investigation about, line segments between points on a square-dot grid. Some way into the activity, she asks Allan, one of the pupils: ‘Right. Can you make any predictions before you start?’ Allan considers the question, and answers: ‘The maximum will probably be, er, the least ’ll probably be ’bout fifteen.’ A first reading may suggest that there is nothing unusual about this interchange; we could, indeed, find many more like it. But step back; view the interchange through the lenses of an anthropologist. If, as it seems, Judith’s intention is to request information, why does her question address Allan’s ability to supply it? And why is his answer, ostensibly a mathematical utterance, so notably devoid of precision? What may we infer about Allan’s attitude to his prediction from the manner in which he formulates it?

I set out on this research with a clear aim: to access and describe the mathematical frameworks and private constructions locked away in children’s minds. My concern was to uncover what they ‘knew’ and how they structured that knowledge. I saw this as the most likely kind of outcome (in the spirit of what I shall call the ‘linguistic principle’) of the mathematical conversations in which I planned to engage them. In other words, I began with my attention focused on ‘transactional’ functions of language:

That function which language serves in the expression of content we describe as transactional, and that function involved in expressing social relations we will describe as interactional.

Whereas linguists, philosophers of language and psycholinguists have, in general, paid attention to the use of language for the transmission of ‘factual propositional information’, sociologists and psycholinguists have been particularly concerned with the use of language to negotiate role-relationships, peer-solidarity, the exchange of turns in a conversation, the saving of face of both speaker and hearer. (Brown and Yule, 1983, pp. 1–4)

I suggest that the interactional function of language is paramount in the exchange between Judith and Allan. Their interactional purposes are coded in the form of the language they use in the question and in the answer. The purpose of this book is to elucidate how that linguistic coding works, and to demonstrate the didactic value of knowledge of that code. Michael Halliday leaves us in no doubt as to its educational importance:

If we consider the language of a child, there is good evidence to suggest that control over language in its interpersonal function is as crucial to educational success as is control over the expression of content, for it is through this function that the child learns to participate, as an individual, and to express and develop his own personality and his own uniqueness. (Halliday, 1976, pp. 197–198)

The two categories of language function are not exclusive, and both are of the utmost importance in talk about mathematics. My initial and continuing interest in transactional elements is evident in this book, in my implementation of variants of Piaget’s clinical interview in many conversations with children. This research orientation is most strongly represented in Chapter 5.

Thereafter, the analytical focus shifts towards interactional components of mathematics talk. I shall explore how speakers in such conversations show their concern for a number of pragmatic¹ goals, principally those to do with the saving of ‘face’ (Goffman, 1967).

Two Principles

My approach to this research is guided by two fundamental, related principles, which I name ‘linguistic’ and ‘deictic’.

Linguistic principle: language is a means of access to thought. One corollary of this principle is that talk with children has potential for insight into the structure of fragments of their mathematical thinking. The roots of the principle are in Freudian psychoanalysis and its provenance as a research method in education goes back at least to Piaget and famously bears fruit in mathematics education in Ginsburg (1977). The work of Douglas Farnham (1975) belongs to a strand of work with an explicitly linguistic foundation. Farnham drew on contemporary work of Barnes, Coulthard and others on patterns of classroom interaction to account for the child’s development of mathematical understanding in terms of social sense-making. The linguistic principle, as I use it, makes no claim to ‘transparency’—that, in some direct sense, one person’s speech is a direct channel for the undistorted communication of their thought to others. Such a view is incompatible with my position regarding the construction of meaning. Subsequently, I supplemented the linguistic principle with:

Deictic principle: speakers use language for the explicit communication of thought, and as a code to express and point to concepts, meanings and attitudes. As I began systematically to tape-record and transcribe teaching sessions and one-toone interviews with children, I experienced a growing awareness (which I describe first in Chapter 5) that the language the children used when talking about mathematics was of considerable interest in its own right—not in the sense that I had originally expected (for example, by providing descriptions of images), but in the subtle ways that the children used language to point to private concepts, meanings, beliefs, feelings or attitudes in the context of their mathematical thinking. I try to capture the essence of this language function in the term ‘deictic’, which derives from the Greek deiknumi, meaning ‘to show’ or ‘to point’. The Greek root is associated with a linguistic term, ‘deixis’, which features in Chapter 5.

The centrality of the linguistic and deictic principles to my research orientation is examined later in this chapter, in a discussion of the clinical method, and in Chapter 4, where I shall set out some linguistic interpretive tools.

The deictic principle is at the heart of a paper in which the linguist Michael Stubbs (1986) draws attention to some ways in which speakers use language to convey beliefs and attitudes, or to distance themselves from the propositions they make. (A new presentation of the paper is available in Chapter 8 of Stubbs, 1996.) This epistemic subtext is sometimes summed up in the phrase ‘prepositional attitude’ (Ginsburg et al., 1983, p. 26), glossed by Sperber and Wilson (1986a, pp. 10–11) as follows:

Utterances are used not only to convey thoughts but to reveal the speaker’s attitude to, or relation to, the thought expressed; in other words, they express ‘prepositional attitudes’ […]

Stubbs claims that no utterance is neutral with regard to the belief and commitment of the speaker, and urges the importance of the study of markers of propositional attitude:

whenever speakers (or writers) say anything, they encode their point of view towards it: whether they think it a reasonable thing to say, or might be found to be obvious, questionable, tentative, provisional, controversial, contradictory, irrelevant, impolite, or whatever. […] All sentences encode such a point of view […] and the description of the markers of such points of view and their meanings should therefore be a central topic for linguistics. (1986, p. 1)

Stubbs identifies vagueness and indirect language as a principal means of encoding propositional attitude. The opening interaction between Judith and Allan exemplifies both indirectness (on Judith’s part) and vagueness (Allan).

In fact, vagueness is the linguistic feature which unifies most of the data which I analyse in this book; rather, it is vague aspects of the language of participants in mathematical conversation that I shall single out for analytical attention. My principal reason for choosing that particular focus is that I came to see the significance, for mathematics talk, of Stubbs’s insight about the encoding of propositional attitude. More surprisingly, I came to perceive how vagueness, suitably deployed, can also assist the transactional purposes of mathematics talk. The main and subordinate aims of my book are best understood in the light of these surprising perceptions.

Aims and Themes

My overall aim is to expose and understand some of the ways that participants in mathematics talk use language—especially vague language—to achieve their communicative and affective purposes. This comprehensive aim guides my choice of subject matter, and finds empirical expression in work that I present as four studies, reported in Chapters 5 to 8. Each study was motivated by a particular subaim, related to the main one.

First, in this chapter and in Chapter 4, I review the methodological and linguistic matters which underpin the design and interpretation of each of the four studies. Given the unifying theme of this book, I discuss some mathematical and philosophical dimensions of vagueness in Chapter 3. The mathematical process of generalization features strongly in three of the four studies; since I hold the view that this process encapsulates the essence of mathematical thought, I have devoted Chapter 2 to an exploration and exposition of its special character.

Chapter 5 is principally a detailed study of one 9-year-old child, Susie. My aim in that chapter is to demonstrate the transactional effectiveness of the pronouns ‘it’ and ‘you’ in our mathematical conversations. The vagueness of these words is associated with reference indeterminacy. I show how the first of these pronouns enables Susie to introduce certain concepts and generalizations into our conversation, despite the fact that she has no name for them. I demonstrate that the second is associated with vagueness-as-generality, and that ‘you’ surfaces in children’s mathematics talk as a natural language pointer to generalization.

The study in Chapter 6 is based on several similar conversations with pairs of children aged 9 to 11. The similarity lies in the fact that each begins from the same numerical-combinatorial task, designed to provoke generalization. A paper of George Lakoff (1972) had first alerted me to a linguistic feature of the transcripts of these conversation, namely ‘hedges’ (such as ‘I think’, ‘maybe’, ‘about’ and ‘around’). My aim in this study is to identify the use and prevalence of hedges in connection with conjectures. I show, as Stubbs suggests, that such hedges are powerful indicators of propositional attitude. In particular they point to vulnerability, they protect against loss of face. I shall introduce a construct which I call the Zone of Conjectural Neutrality, a space in which conjectures can be tested whilst minimizing the affective risk to their originators.

In Chapter 7, I report a study which aims to trace the development, between the ages of 4 and 11, of modal language competence, especially in the use of modal auxiliaries and hedges. The mathematical activities entailed in this study are counting and estimation. I identify a trend towards a developing ability to indicate propositional attitude in these ways in the primary years, and increasing awareness that vagueness is essential to estimation. I identify an anomaly in this trend, and account for it by reference to institutional factors.

The final empirical study, in Chapter 8, examines a disparate collection of teaching episodes across a wide age range, and involving eight different teachers. Here the aim is to validate three claims which arise from the findings in the earlier chapters. First, it is a demonstration of the applicability of the linguistic toolkit which I have assembled to the analysis of transactional and interactional features of transcripts of talk in mathematical interaction. Second, it confirms the prevalence and interactional significance of a number of previously identified (in the earlier studies) aspects of vague and indirect language in mathematics talk, across a wide age range. Third, by involving a group of teachers to work with me for that study, I was able to validate the relevance of my methods and findings to their day-to-day work in the classroom.

A Context in Language Research

The linguist Joanna Channell concludes her book on vague language (1994, p. 209) with a call for more research in ‘variation study’, including ‘the study of occurrence of vagueness in different registers or genres’. The term ‘register’ refers to the specialized language peculiar to certain user-groups (Halliday et al., 1964). Such studies of aspects of vague spoken language exist in certain academic fields, for example medicine (Prince et al., 1982) and biology (Dubois, 1987). One aspect of variation not specifically included by Channell in her call for research is agevariation: the informants in every empirical study of vagueness of which I am aware are adults. This book makes a contribution to the study of vague language in these two dimensions of variation—register (mathematical) and age (especially 5-to 11-year-olds).

Mathematics education is necessarily and beneficially an eclectic discipline; the relevance of linguistics to this book is principally interpretive, in that certain organizing principles of language use, particularly those which have become associated with the young linguistic science of ‘pragmatics’ (Levinson, 1983; Mey, 1993), are applied to make sense of some vague features of mathematics talk identified in various corpora.

Hymes (1972) has claimed:

Studying language in the classroom is not really ‘applied’ linguistics; it is really basic research. Progress in understanding language in the classroom is progress in linguistic theory, (p. xviii)

As it happens, most of the language data for my study were gathered in the course of ‘interviews’ in classrooms, rather than in naturalistic classroom settings per se. I see this study as basic research, not in linguistic theory, but in mathematics education. A number of regularities of speech found in the data will be described and interpreted as phenomena observable in the interaction between individuals as they talk about mathematics. Some evidence of the presence and broader consideration of the pedagogic significance of these phenomena in the interaction between practising teachers and pupils is provided in Chapter 8. The book concludes with some proposals for the application of this basic research in the cause of improving the teaching of mathematics, informed by my commitment to a constructivist view of learning and a quasi-empiricist philosophy, to be described in Chapter 3.

The ability of speakers to encode ideas, commitments, attitudes and beliefs in their utterances—not least in teachers’ and pupils’ mathematical utterances—would be of little use if their interlocutors were unable to decode, interpret and understand the intended subtext in such utterances. The communication and interpretation of prepositional attitudes are central to mathematics education because the articulation of beliefs, conjectures and even ‘answers’ in mathematics is notoriously a risk-taking activity; this point is developed further in Chapter 6. Acknowledging that some relevant groundwork has been done in linguistics, David Pimm identifies interpretation, within clas...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• List of Figures
• Series Editor’s Preface
• Preface
• Acknowledgments
• 1. Preview and Methodology
• 2. Generalization
• 3. Perspectives on Vagueness
• 4. Discourse and Interpretation
• 5. Pointing with Pronouns
• 6. Hedges
• 7. Estimation and Uncertainty
• 8. Pragmatics, Teaching and Learning
• 9. Summary and Review
• Appendix 1. Transcript Conventions
• Appendix 2. Index of Transcripts
• References
• Author Index
• Subject Index