Analysis of classroom mathematics discourse: shifting attention from transaction to interaction

Tim Rowland

Homerton College, University of Cambridge

Paper presented at the British Educational Research Association Conference, Cardiff University, 7-10 September 2000.

There is a long tradition of research in mathematics education which positions researchers as mathematicians or cognitive psychologists whose task is to find out what kind of mathematics and thinking about mathematics is in the heads of their ‘subjects’. The focus is on how individuals structure their mathematical thinking, and this is often reported within a broader framework of cognitive development. Within this tradition, discourse analysis has focused on the ‘transactional’ function of language in the expression of ‘factual’ propositional content.

The purpose of this paper is to emphasise how language also fulfils an essential ‘interactional’ function in expressing social relations and propositional attitude, and to argue for the significance of such dimensions in the analysis of discourses of a mathematical nature. Much of the paper is in the nature of a survey, setting out a selection from the various analytical frameworks that I have found relevant in my own work. The paper concludes with a short fragment of text for illustrative purposes.

Discourse

The terms ‘text’ and ‘discourse’ are used with a variety of meanings within linguistics (Crystal, 1991). Within mathematics education, Morgan (1998) aligns her approach to that of Hodge and Kress (1988), for whom ‘text’ refers to a piece of written or spoken language with "a socially ascribed unity" (ibid, p. 6), such as an article, a conversation, an interview or a diary entry, whereas ‘discourse’ comprises such texts and the wider set of social and linguistic practices within which they are situated, embracing the social process in which texts are embedded. This more comprehensive sense of ‘discourse’ has consequences for the ways that analysts might ‘interpret’ or make sense of a text..

Interpreting a particular text necessitates knowing who composed it and who is intended to receive it, as well as the kinds of interpretations that are possible within the discourse. (Morgan, 1998, p. 4)

Elsewhere (Brown and Yule, 1983) ‘text’ is used in a narrower sense, to refer to a physical ‘product’ (written or transcribed) deriving from discourse, which in turn is viewed as a more dynamic process of expression and interpretation. This sense and

that of Hodge and Kress differ only in that the latter would seem to include, as ‘text’, an on-going classroom episode, for example, or a video of such an event, whereas Brown and Yule would restrict the term to a transcript of the event. The now-widespread use of video, with the range of means of interrogation made possible by media such as CD Rom, seem to make Morgan’s use (following Hodge and Kress) more in keeping with the possibilities made available by digital technology. Indeed, the existence of electronic text media, from word processor files to packages such as NUD*IST for the analysis of qualitative data, demonstrates the futility of restricting the meaning of ‘text’ to hard copy.

Media

Conversations can be preserved, for later scrutiny, in the form of videotapes, audiotapes, field notes or transcripts (electronic or hard copy). Each of these media has advantages and disadvantages. The videotape, for example, includes non-verbal data (such as gestures and actions on materials) and seems to facilitate subsequent group consideration of features such as critical moments in the discourse; the audiotape preserves speech features such as intonation, pauses, and voice tone; the transcript, a transformation of the primary record of the event, focuses attention on the spoken word, or coded speech features. The transcript as electronic text file is invaluable to the computational linguist with an interest in (say) the relative frequency of use of certain words or grammatical structures. Since I choose to focus on spoken language in my own work, I usually audiotape, and transcribe using a word processor.

Analysis

Deborah Schiffrin (1994) sets out to reconcile many of the differences between alternative traditions in discourse analysis. She identifies six such approaches, including those associated with Speech Act Theory, with the Gricean cooperative framework, with Interactional Sociolinguistics (embracing Politeness Theory) and with Ethnomethodology. Schiffrin argues that

All [these approaches] attempt to answer some of the same questions, e.g. How do we organise language into units that are larger than sentences? How do we use language to convey information about the world, ourselves and our social relationships? (p. viii)

[...] all the approaches to discourse view language as social interaction, and all are compatible with a functionalist rather than a formalist paradigm. (p. 415)

One of six unifying principles identified by Schiffrin (p. 416) asserts that analysis of discourse is empirical; analyses are predictive, they produce hypotheses. This accurately characterises my own approach to transcripts of mathematics talk.

Brown and Yule (1983, p. 22) claim that:

The discourse analyst, with his ‘ordinary language data’ [...] may wish to discuss, not ‘rules’, but regularities, simply because his data constantly exemplifies non-categorical phenomena. (author’s emphasis)

This interest in regularities is indicative of my own approach to the analysis of the transcripts of mathematics talk. But, for me, those regularities must have some inferential significance; I cannot be content merely to describe them. I do want to try to get "behind conversation", to make inferences and conjectures about "what is really going on" (Levinson, 1983, p. 287). But this includes a great deal of a social and interpersonal character, extending beyond pure semantic meaning to meanings that are embedded in the context of the discourse. The meanings to be considered include meanings of a ‘pragmatic’ kind. Thus, as Brown and Yule conclude:

An analytic approach to linguistics which involves contextual considerations, necessarily belongs to that area of language study called pragmatics. ‘Doing discourse analysis’ certainly involves ‘doing syntax and pragmatics’, but it primarily consists of ‘doing pragmatics’ … In discourse analysis, as in pragmatics, we are concerned with what people using language are doing, and accounting for the linguistic features in the discourse as the means employed in what they are doing. (Brown and Yule, 1983, p. 26).

The terms ‘semantics’ and ‘pragmatics’ have been used above. The following section clarifies the sense in which I shall use and distinguish between them.

Meaning

The term ‘semantics’ is interpreted in a number of different ways in different fields. Broadly speaking, semantics has to do with the relationship between signs – in mathematics, principally words, symbols and various kinds of graphics – and what they mean or refer to. Thus the concepts of definition, meaning, truth and empirical verification of an assertion are all semantic, since they involve more than the purely internal structure – the syntax – of the language in use. When this language is a formal one, the propositional calculus for example, semantics can be made into an exact study (Tarski, 1956; Carnap, 1948). By contrast, the semantic characteristics of natural languages such as English are comparatively ill-defined and user-dependent. In part this is due to the vagueness inherent in natural languages. Russell (1923) has argued that all language is vague. For example, the word ‘red’ is vague because

there are shades of colour concerning which we shall be in doubt whether to call them red or not, not because we are ignorant of the meaning of the word "red", but because it is a word the extent of whose application is essentially doubtful. (1923, p. 85)

Russell makes an elaborate case for the vagueness of all words, including names and even logical connectives, and this lexical vagueness, in turn, infects all propositions. One vague word is enough to entail the vagueness of a sentence.

Similarly, there exist a number of differing accounts of ‘meaning’ within natural language. In part, accounts differ because of different perspectives deriving from linguistics, philosophy, psychology, sociology literary criticism, theology, and so on. In discussing meaning in this paper, I have in mind a view of communication that focuses on human intention, and my starting point is what the philosopher Paul Grice (1957) has called ‘speaker meaning’.

Grice distinguishes between two kinds of meaning: natural and non-natural. The first of these is meaning of a conventional, literal kind, as in, for example

I like mathematics

a square has four edges

if x>2 then x2>4

This might be called the semantic or truth-conditional meaning of such utterances. Of course, such a notion is not unproblematic, pre-supposing as it does that words refer to things in an unambiguous way, and that the syntax of the sentence then takes care of the meaning. A more recent view of reference among linguists is that it is the speaker who refers, by use of some expression (Lyons, 1977, p. 177). Here, reference is viewed as an action on the part of the speaker/writer (Brown and Yule, 1983, p. 28). Successful reference is achieved by the speaker if the hearer (as interpreter of the utterance) is able to recover the intended referent(s). This brings us to Grice’s second kind of meaning, which is more closely aligned to the focus of this paper.

Non-natural meaning (which Grice denotes ‘meaning-nn’) is associated with (human) communication which is intended to be received in a particular way by a recipient. This is the ‘pragmatic’ meaning, which may be conventional, but certainly need not be in the case, for example, of ironic, metaphorical and indirect communications. It is not hard to invent examples of the kind (respectively)

I just love mathematics.

John is an absolute pig

Would you like to pass me that book?

A critical feature of meaning-nn is not so much the conventional propositional content of an utterance, but the particular way that it is intended to be recognised and interpreted by a recipient. As Grice states:

‘A meant-nn something by x’ is (roughly) equivalent to ‘A intended the utterance of x to produce some effect in an audience by recognition of this intention’. (1957, p. 58)

Pragmatics deals with language in use, and with the ways that people use and

exploit language (especially spoken language) to convey more than, or other than, the logical content of their utterances. The distinction between the truth-conditional, semantic meaning and the pragmatic meaning of an utterance is demonstrated by B’s turn in the following interchange.

A: Do you think you’ll get back home in time to cut the grass?

B: Well, I’ll try, but there are road works on the A14 this week.

The logical force of ‘but’ is indistinguishable from that of ‘and’, both being symbolised by ‘^’ or ‘&’ (Mey, 1993, p. 25). The semantic content of the reply is to the effect that B will try to get home on time, and that there are road works on a particular road. In practice, we read much more into it. Even in the absence of ‘well’, we might infer that:

B will travel by road;

The road works on the A14 might cause B to be delayed;

B is not confident that s/he will be home before Mary has to leave.

The inclusion of ‘well’ adds an additional dimension to the reply: that is, B’s anticipation that his or her reply is not the one that A would like to hear. Wierzbicka (1976) analyses ‘well’ as a ‘pragmatic particle’, a word whose function is to express a pragmatic meaning at minimal cost. She uses the term ‘pragmatic meaning’ to refer to factors of propositional attitude such as assumptions, attitudes and intentions. Another perspective on this particle is offered later in this paper.

Grice’s characterisation of meaning-nn entails a second intention and a condition which is necessary for the communication of meaning-nn to be successful: that the hearer recognise the speaker’s communicative intention. This recognition might be assisted by a range of facts and information already available to the hearer, and also by speaker stress or intonation (or coding conventions associated with such prosodic features in written text). Under these conditions, speakers routinely make utterances whose intended meanings either vary from, or are not restricted to, their semantic meanings. Grice’s notion of meaning-nn draws attention to the possibility that speaker meaning might be relatively free of conventional meaning.

Functions of language

In an account of the ‘transactional’ and ‘interactional’ functions of language, Brown and Yule touch on a distinction which seems to parallel that between semantic and pragmatic meaning.

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 sociolinguists 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)

These two particular functions of language are both of the utmost importance in talk about mathematics, although this might not be inferred from a reading of the ‘standards’ for Initial Teacher Training in England (DfEE, 1998), where reference to language is restricted to its transactional function and semantic meaning.

As part of all courses trainees must be taught the importance of ensuring that pupils progress from using informal mathematical vocabulary, to using precise and correct mathematical vocabulary, notation and symbols. (Annexe D, A2b)

Trainees must be taught that pupils’ progress in mathematics depends upon them teaching their pupils the correct use of mathematical language, including the importance of using:

i. mathematical definitions and vocabulary precisely – especially where a word, e.g. similar, has an everyday meaning and a mathematical meaning which is more subject-specific;

ii. mathematical sentences and connectives like "therefore" and "because" to indicate the logical connections between consecutive sentences;

iii. qualifiers and quantifiers correctly, e.g. ,"and", " or", "not", "if", "then", "for all", there exists". (Annexe G, A2a)

Whilst precision sits well with semantic meaning in the mathematics classroom, pragmatic meaning draws heavily on the exploitation of vagueness (Rowland, 1999a). In that setting, moreover, the interactional function merits as much attention as the transactional. Indeed, Michael Halliday, leaves us in no doubt as to its educational significance.

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 its 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. 197-8)

Pragmatic meaning, or meaning-nn, is the means frequently (though not necessarily consciously) used by speakers to convey affective meanings to do with social relations, attitudes and beliefs. 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. This subtext is captured by the phrase ‘propositional attitude’, glossed by Sperber and Wilson (1986, 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 ‘propositional attitudes’ [...]

Stubbs (1986, p. 1) 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.

Shiffrin (1994) and Jaworski and Coupland (1999) give excellent surveys of a wide range of approaches to discourse which are particularly sensitive to pragmatic meaning and interactional function. The following sections of this paper give a brief overview of two major contributions to pragmatic analysis: Austin’s speech act theory and Grice’s cooperative principle. Both have their roots in philosophy of language, and each will play a part in the pragmatic analysis of a teaching episode towards the end of this paper.

Speech Acts

In the late 1950s, the Oxford philosopher John Austin gave some lectures on how speakers "do things with words", and so invented a theory which now occupies a central place in pragmatics. Austin (1962) called non-propositional requests, wishes, warnings and the like ‘speech acts’. The essential property of speech acts (‘performative’ utterances) is that they bring about (or have the potential to bring about) a change in some state of affairs. Paradigm examples include the naming of a ship, the joining of two persons in marriage, and the sentencing of a criminal. The name, the marriage and the sentence are what they are because an authorised person has declared them to be so. Austin distinguished between the locution of a speech act (the words uttered), its illocution (the intention of the speaker in making the utterance) and its perlocution (its effects, intended or otherwise).

Whereas declarative utterances typically have truth conditions, speech acts must satisfy certain ‘felicity conditions’ in order to ‘count’ as an action. In the case of orders and requests, for example, the felicity conditions include the ‘preparatory’ requirements that the speaker believes that the hearer is capable of carrying out the indicated action, and that they would not necessarily do it without being asked; and also a ‘sincerity’ condition, that the speaker actually wants the hearer to do what s/he is being asked to do (Levinson, 1983, p. 240).

The felicity conditions for a question include the expectation that the enquirer doesn’t know the answer, that s/he would like to know it, and has reason to believe the hearer is able to supply it. Questions in classroom situations are curiously exempted from these rules, and the conditions governing appropriateness in the answers to such questions may differ in that the enquirer (A) already has the

information sought in the question, and so "the request is for B to display whether or not s/he has the information" (Labov and Fanshel, 1977, p. 79).

One aspect of speech act theory with significant pragmatic implications concerns indirectness. Three broad illocutionary categories are normally identified - statement, question and command - having typical realisations in declarative, interrogative and imperative verb forms. These agreements between intended function and realised form break down in ‘indirect speech acts’, in which the outward (locutionary) form of an utterance does not correspond with the intended illocutionary force of the speech act which it performs (Levinson, 1983, pp. 263ff). Common forms of this are to declare a preference or to use an interrogative form in order to convey an order or request. For example:

Teacher: I’d like to take in your exercise books.

Diner: Can you bring me the wine list?

These are both instances of how speakers frequently accomplish an indirect speech act by stating or questioning one of the felicity conditions (Gordon and Lakoff, 1971). The teacher explicitly states his wish to receive the books i.e. that s/he meets the felicity condition to do with speaker sincerity; the diner questions the ability of the waiter to provide the list i.e. s/he questions one of the preparatory pre-conditions. In these cases, the indirectness is strongly associated with the concern of the speaker to observe social ‘politeness’ conventions. Here, politeness has a very specific and theoretically well-developed sense (Brown and Levinson, 1987). Indirectness is one of a range of strategies with the purpose of avoiding so-called face threatening acts which impose on, or might give offence to, the addressee.

The use of indirect speech acts in the mathematics classroom is discussed at greater length in Rowland (1996). The following utterances of one teacher, Hazel, are examples of this linguistic regularity.

17 Hazel: Shall we try it out and see what happens? Do you want to each choose your own set of consecutive numbers?

66 Hazel: Right would you like to try out with ten, twelve and fourteen one of you and the other one can try another jump.

130 Hazel: Can you tell me what the difference in the answers of the two sums that, the two multiplications you're doing, would be when you have a difference of four between each number?

In each case, Hazel questions a felicity condition associated with her instruction to

the children.

Conversational Implicature

The notion of conversational implicature is due to Grice (1975), who argued that the pragmatic ‘logic’ of language in use is different from the logic of formal semantics. Grice proposed that ordinary conversation is posited on a Cooperative Principle (CP), embodied in four sub-principles or ‘maxims’ of conversation. These maxims specify what participants need to do in order to converse rationally and cooperatively. The requirements are, essentially:

maxim of Quality: let your contribution be truthful: do not say what you believe to be false;

maxim of Quantity: let your contribution be as informative as is required (for the current purposes), and not more informative than is required;

maxim of Manner: let your contribution be clearly expressed - e.g. be brief, orderly, unambiguous;

maxim of Relevance: let your contribution be relevant to the matter in hand.

Now it is evidently not the case that all participants in all conversations observe all of these four maxims in all contributions. How, then, do hearers interpret speech which is supposedly cooperative, yet which is for example, superficially untrue or irrelevant? Viewed from the other side of the coin, how do speakers successfully violate the maxims in order to communicate fine nuances of meaning, to enable the hearer to read ‘between the lines’, as it were?

The genius of Grice’s theory is the following recognition. Whilst speakers do not always observe the maxims at the surface level, nevertheless hearers interpret the contributions of other participants in conversation as if they were intended to observe the maxims at some level of meaning other than that contained in the semantic content of the utterance. Such a view of communication underpins a means of pragmatic inference identified by Grice, and which he named ‘conversational implicature’. An implicature is rather like a hint. One of the examples given by Grice concerns a reference which appears not to address the intellectual criteria which might be thought most essential to the job (in philosophy) in question, but elaborates on marginal qualities such as punctuality and good manners. The communication violates the maxims of Quality and Relevance, and thereby implicates the view of the referee that the applicant is not suited to the job.

A useful way of looking at the Gricean interpretive framework would be to say that either speakers are overtly cooperative because they observe the maxims, or else they are covertly cooperative by setting up implicatures by means of maxim violations. In this technical sense, though not in the everyday sense, one can argue

that every contribution to conversation is ‘cooperative’.

Grice’s theory offers an analytical framework for the pragmatic meaning of the particle ‘well’, which was mentioned earlier in this paper. Brockway (1981) describes ‘well’ as a ‘maxim hedge’, suggesting that the speaker is serving notice to the hearer that the contribution about to come will in some respect fall short of the requirements of one or more of the maxims of cooperation. For examples of this in mathematical contexts, see Rowland (1999a).

Hedges

Some recent approaches to the problem of vagueness within the field of linguistics originate in consideration of the meaning and function of a class of words and phrases called ‘hedges’. These include words such as ‘sort of’, ‘about’ and ‘approximately’ - words which have the effect of blurring category boundaries or otherwise-precise measures - as well as words and phrases such as ‘I think’, ‘maybe’ and ‘perhaps’, which hedge the commitment of the speaker to that which s/he asserts.

The work of Zadeh (1965) and Goguen (1969) laid the foundation for fuzzy interpretation of vague language, setting the scene for work by George Lakoff (1972), who concludes that:

one need not throw up one’s hands in despair when faced by the problems of vagueness and fuzziness. Fuzziness can be studied seriously within formal semantics [...] For me some of the most interesting questions are raised by the study of words whose meaning implicitly involves fuzziness - words whose job is to make things fuzzier or less fuzzy. I will refer to such words as ‘hedges’. (1973, p. 471, emphasis added).

Whilst much of Lakoff’s paper is taken up with technical details in mathematical logic, he begins from, and frequently returns to, the issue of the meaning of vague language in use.

Hedges can be usefully separated into four basic types. This observation was initially made in a study (Prince, Frader and Bosk, 1982) of paediatric clinicians, whose spoken language in case-conferences turned out to be unusually rich in hedging - about one hedge every 15 seconds.

The first major type of hedge - a SHIELD - is exemplified in Prince et al. (1982) by "Well. I think that ... " and "There is evidence that’s been presented ....". These indicate some uncertainty in the mind of the speaker in relation to some proposition. Prince et al. subdivide Shields into two kinds. The first of these is termed a Plausibility Shield, typified by ‘I think’, ‘probably’ and ‘maybe’. A Plausibility Shield ‘implicates’, in the Gricean sense, a position held, a belief to be considered - as well as indicating some doubt that it will be fulfilled by events, or stand up to evidential scrutiny.The second kind, an Attribution Shield, implicates

some degree, or quality, of knowledge to a third party. A favourite Attribution Shield with the clinicians, with evident attendant suspicion, was "According to the mother ... ".

The second major category of hedge – an APPROXIMATOR - includes ‘about’ and ‘a little bit’. In distinction to Shields, these Approximator-hedges are located inside the proposition itself. The effect is to modify (as opposed to comment on) the proposition, making it more vague. A sub-category of Approximators - called Rounders - consists of the standard adverbs of estimation, such as ‘about’, ‘around’ and ‘approximately’, which are commonplace in the domain of measurements and quantitative data. Another type of Approximator is called an Adaptor. Words or phrases such as ‘a little bit’, ‘somewhat’, ‘sort of’, attach vagueness to nouns, verbs or adjectives associated with class membership.

More often than not, Grice’s maxims can be applied to account for pragmatic meaning conveyed in hedged assertions. For example, the promise:

Maybe I’ll come and visit you next week.

flouts the maxim of Manner, and implicates the related proposition

I may fail to come to see you next week.

because, if I were firm in my desire and intent to come, I would have made an unambiguous promise without recourse to a plausibility shield. Grice’s theory of implicature thus accounts for the effectiveness of hedges in conveying uncertainty and lack of commitment. Such hedges are a powerful means of interpreting the propositional attitude of the speaker.

Hedges can also play a part in politeness strategies. The compounding of hedges, indirectness, particles, and so on, increases the relative politeness of expressions (Brown and Levinson, 1987).

There follows a brief fragment from one mathematics lesson.

[At BERA 2000, the unedited transcript will be distributed for discussion in the session].

The discussion of the interaction here focuses on the pragmatic meanings of some of the utterances within the text, with concern for the beliefs and attitudes of the three participants towards the subject-matter and each other.

RACHEL

Rachel teaches 16 to 18-year-old students at a sixth form college with a strong academic reputation. The text to be considered is the transcript of a teaching episode involving Rachel and two 18-year-old students who were following an Advanced Supplementary (AS) course in mathematics.

The students were presented with the well-known ‘stairs’ investigation in written form, as follows.

You need to climb a staircase with n steps. You are allowed to go up the steps taking either 1 or 2 steps at a time. In how many ways can you go up n steps? As an extension consider being able to take 1, 2 or 3 steps.

The transcript is one of a pair supplied by Rachel, who describes Juliette and Di as bright students who work well together in class. In the event, both gained A grades in the AS examination. Rachel begins by checking that the task is clear.

1 Rachel: So, do you think you understand what it means?

Rachel’s question is softened by its indirectness. Rather than question her students’ understanding directly, she opts to enquire about their propositional attitude towards their understanding of the task. Rachel’s reticence seems to derive from her wish to support the students in getting started, whilst respecting that they are ‘bright students’ who might prefer to be left alone, to sort it out themselves. Brown and Levinson remark that the more effort a speaker extends in face-saving work, the more he (sic) will be seen as trying to satisfy the face wants of the hearer (1987, p. 143).

By contrast, Di’s reply is direct, if not exactly reassuring in its clarity.

2 Di: Yes, n steps.

Juliette, however, takes up Rachel’s implicit offer to clarify the task, seeking confirmation with the tag-question "can’t you?" [3, 7]

3 Juliette: So you can have a combination of ones and twos, can’t you?

4 Di: It’s going to take n steps or n over two steps.

5 Juliette: Or a combination. You could look at ...

6 Di: You could take n as being one [writing a table] x, y

7 Juliette: If n is two, you could do either two steps or one, can’t you?

n equals three ... [writing down 2 1, 1 2] Does it count if you do two and one and one and two?

8 Rachel: Yes, they’re different.

In [3, 5] and throughout the interview, Juliette and Di freely use ‘you’ for generalisation and/or detachment (Rowland, 1999b). Note that the students’ use of ‘you’ is almost invariably of this kind, whereas Rachel uses the pronoun mainly to

address the students. Di quickly goes into ‘investigation mode’ [6] with a table of (x, y) values. Juliette, seeking further clarification of the task, enquires about the rules on ‘sameness’ ("Does it count ..?"). This time Rachel adjudicates [8], having chosen not to respond to an earlier opportunity to contribute. Perhaps she judges that a correct understanding of sameness is crucial to the success of the activity in terms of the generalisation she anticipates, and is reluctant to leave it to chance.

Having listed and counted the number of ways for one, two, three steps, Di makes a prediction for four.

10 Di: Yeah, [writing out the (x, y) table] so you have one, one; two, two; three, three; so four is four.

In fact, Di’s prediction is false (there are five ways for four steps), yet (unusually) it is not marked in any way as regards uncertainty. Because predictions are essentially enthymematic (information-extending), they tend to be marked with a hedge (‘maybe …’) or a question (‘are there …?’). Di needs to be more cautious than she appears to be. What is the teacher to do?

11 Rachel: I think maybe you need a few more before you can generalise.

The double plausibility Shield [11] is, presumably, play-acting on Rachel’s part. She wants to avoid over-directing their work but also to indicate, but not directly, that Di is in error. Rachel wrote a reflective account of her two interviews after she had transcribed them (Williams, 1995) and comments:

I feel I have to interrupt and prompt her to consider a few more cases. In retrospect, perhaps I should have waited to see if Juliette did that.

Acting on Rachel’s prompt, the students list the possibilities for the case n=4, and find five ways Soon, the two students are working on the case n=5:

18 Juliette: ... Two, one, one, one, one, two, one, one, one, one two, one [more writing]. That’s it. So, we’ve got one, two, three, five, eight and now you’re going to get twelve.

19 Rachel: Why?

20 Juliette: It’s a series. You add one. You add one, that’s that’s ... you add one. Oh, I don’t know.

21 Rachel: Yes, yes, you add one.

22 Juliette: You add one, then you add two … then you add four, the interval between. No, that’s right now. It’s something to do with the series it goes up with.

23 Rachel: Mmm.

This time it is Juliette who is misled to a faulty prediction (12 ways for 5 steps) by an alternative regularity [22] in the first few terms. She seems to be looking at first differences, observing that they increase by one each time (if one ignores the first term). Again, her prediction [18] is unhedged. She freely decentres with ‘you’ [18, 20, 22]. Note that "we’ve" got one, three, five etc. (actual data) whereas "you’re" going to get twelve (prediction). Finally "I" don’t know (a personal epistemic state). Once again, Rachel has a dilemma. In the face of Juliette’s apparent conviction, Rachel asks for an explanation [19], which is taken to be a request for an account of the perceived regularity. Juliette’s account [20] is hesitant, with several false starts. Rachel enthusiastically offers encouragement [21] and prompts Juliette to resume her explanation, despite the fact that it is unclear where it might get her. The continuation [22] is elliptic, perpetuating the faulty prediction, but this time seems to entail some internal debate. Rachel’s response [23] is minimal. She wants Juliette to sort this one out for herself. Later, she was to reflect (Williams, 1995):

Juliette has the right sort of idea [22] and says "It’s something to do with ... " but she will not commit herself. I am trying hard not to interfere and biting my tongue with "Mmm".

In the end Di is surprised to find more ways than Juliette had predicted [31] and Rachel can restrain herself no longer:

31 Di: [counting all the combinations] ten, eleven, twelve, thirteen.

32 Rachel: That’s all right.

33 Juliette: [puzzled] That’s O.K?

Rachel’s comment:

I had to confirm that she was correct [32], I couldn’t bear the uncertainty and wanted them to know they had got to the correct number of ways. Looking back it would have been better to let them sort it out. (ibid.)

Rachel brings out another affective dimension in the conjecturing atmosphere. The student is required to take risks, but the teacher may have to "bear the uncertainty" when s/he judges that the pupil must resolve it for him/herself. That is not to say that the teacher cannot participate in the struggle, but her/his role may be best restricted to light, indirect, linguistic scaffolding.

The two students are hooked onto familiar sequences (arithmetic, geometric) and are somewhat inflexible in their search for regularity in the (Fibonacci) sequence 1, 2, 3, 5, 8, 13 ... In the end, Rachel finds it hard to allow them to flounder:

35 Juliette: Times two, no, no direct ... no common difference no common ratio.

36 Rachel: But when you were doing your thing of adding on each time it worked. Well ...

Rachel later commented:

I sense that I am hesitant in letting them search too long for a solution and that I assume they want to get to the answer quickly. I am in some way anxious that it is taking them a long time, but in interrupting, I interrupt interpreting conversations they could have in getting to the answer ...

I was anxious that it should work and I think that bright students should be able to get to the answer without too much difficulty. I did have expectations of the students ...

I felt that I intervened too quickly and didn’t let them struggle enough. Perhaps I was more nervous of the tape than they were.

I felt that their knowledge got in the way of their intuition. They felt that the question should fit into an arithmetic or geometric progression type question.

This is a curious and, in many ways, an untypical classroom text. The teacher seems to be more anxious and less at ease than her students. Rachel is prepared to go to some lengths to demonstrate her respect for her students’ ability, yet this very fact limits her ability to allow them to be stuck for very long. Her dilemma is exacerbated by the students’ confidence. They rarely question their own predictions, so it falls to Rachel to do so, but in ways that avoid threats to their self-esteem as mathematicians.

conclusion

Classroom talk is a rich resource for the analysis of students’ cognitive structuring of mathematics, in which student errors are a particularly rich basis for conjectures about fundamental mathematical misconceptions. Such analyses provide essential diagnostic insights into individual knowledge construction. Reports of such analyses are typically set within a framework of knowledge about mathematical cognition. This may be of a general kind, to do with concept formation, abstraction and so on, or related to knowledge about the construction of knowledge in particular topic areas such as fractions or functions.

Concern for more interactional features of the classroom, such as students’ propositional attitudes and teachers’ sensitivity to their students’ self-esteem, necessitates rather a different set of lenses and analytical tools from which to view texts. Linguistic tools which focus on pragmatic meaning have significant potential for text analysis in mathematics education, especially in research into social and affective factors in the teaching and learning of mathematics.

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Dr Tim Rowland

University of Cambridge

Homerton College

Cambridge CB2 2PH

UK

tr202@cam.ac.uk