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INDUCTING PUPILS INTOMATHEMATICAL DISCOURSEJenni Back aa King's College , University of London ,Published online: 14 Apr 2008.

To cite this article: Jenni Back (2000) INDUCTING PUPILS INTO MATHEMATICALDISCOURSE, Research in Mathematics Education, 2:1, 33-44, DOI:10.1080/14794800008520066

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3 INDUCTING PUPILS INTO MATHEMATICAL DISCOURSE Jenni Back

King's College, University of London

This paper looks at examples of teacher/pupil talk taken from transcripts of primary school mathematics lessons and explores the relationships between the educational discourse of the classroom, mathematical discourse and the discourse of school mathematics. Some conjectures are made about the teachers' approaches to their teaching on the basis of transcript evidence.

This paper seeks to explore the relationship between teachers, their pupils and the talk in which they engage in the mathematics classroom. I see classroom talk as an ongoing dialogue between the pupils and their teacher in which their thinking is developed in the context of the interaction. For this reason the talk has a historical dimension as well as the social and psychological; in attempting to describe the nature of this talk it seems to me essential to integrate all these aspects.

My perspective has been particularly influenced by the work of Neil Mercer (1 995) who considers the talk of classrooms from the perspective of teacher and pupils undertaking a joint construction of knowledge. He states that his theory about this joint, or guided, construction of knowledge has its origins in the work of both Bruner and Vygotsky, and he believes that language especially the talk of the classroom is central to this process. His theory seeks to take account of the role of language in creating knowledge and understanding, how people help other people to learn and the special nature and purpose of formal education.

The inter-personal nature of the construction of knowledge is highlighted by his emphasis on the social and cultural aspects of classroom events, and this in turn stresses the role which language plays in the classroom. Mercer sees talk as social action and says that in classroom talk we can see pupils, and their teachers, working out what they know and achieving what they can. He suggests that in order to understand how talk is used to create knowledge and understanding, both the context of the talk and the continuity of the talk need to be considered. The context, he suggests, involves both the physical objects around the participants and anything else which they use or respond to in their talk. The continuity of the talk occurs within the ongoing relationship between the teacher and her pupils and may involve reference by the speakers to meanings established in previous conversations. This idea takes into account the historical aspect of classroom talk.

Mercer stresses Vygotsky's emphasis on language influencing the structure of thought and on cognitive development as a social and communicative process. He also draws out Vygotsky's idea that "the limits of a person's learning or problem solving ability can be expanded if another person provides the right kind of cognitive support" (Mercer, 1995, p. 72). This implies that with the support of a teacher, learners can achieve levels of understanding that they would never achieve alone. It

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is from this viewpoint that Bruner's concept of scaffolding is developed. Much of the description of scaffolding in the literature deals with motherlchild or expertlnovice dyads and practical activities but Mercer suggests that scaffolding can offer a metaphor for teachers' participation in pupils' learning. Mercer quotes Bruner's description of scaffolding as referring "to the steps taken to reduce the degrees of freedom in carrying out some task so that the child can concentrate on the difficult skill she is in the process of acquiring" (quoted in Mercer, 1995, p. 73).

Mercer suggests that the concept of scaffolding needs to be reinterpreted if it is to be useful in analysing classroom practice and suggests that it may involve helping students to apply frames of reference that they only partially grasp and are inexperienced in applying. It also needs to be a process which involves more than two people to be generally applicable to a classroom context. To illustrate this he quotes a piece of dialogue from a lesson on subtraction in which a teacher is working with two boys on the method of decomposition. This exchange shows the teacher talking through the problem with her pupils and asking them to tell her what they are doing. Through prompts, confirmations and queries she supports them through an activity which they lack the confidence or skill to tackle unaided. After the exchange they were able to do more subtractions without assistance. Mercer suggests that this model of scaffolding learning through the talk of the classroom might be helpful in understanding classroom processes and links it with the use of language to 'frame' the experience of the classroom.

Mercer then goes on to explore the connections between the talk of the classroom and the concept of discourse. He uses discourse in the sense of language used to carry out the social and intellectual life of a community (p. 79) and refers to the work of Swales (1990) who describes a 'community of discourse' as a group of individuals who have agreed some common goals, and share some established networks of communication and some distinctive terminologies and ways of using language. This community and its discourse may transcend the comings and goings of participating members. This analysis seems to fit the context of mathematics classrooms. Mercer goes on to distinguish between the different forms of discourse which are evident in classrooms and underlie educational practice. The discourse of 'teaching-and-learning' in classrooms, characterised by Initiation-Response- Feedback (IRF) exchanges, cued elicitations, repetition and re-formulation, he calls 'educational discourse'. Participation in this form of discourse is certainly not the aim or purpose of education but it does seem to be necessary to the process. Mercer suggests that the goal is to enable pupils to be independent participants in wider communities of 'educated discourse'. I accept this analysis and suggest that the talk of the classroom should seek to offer pupils opportunities to practise being users of 'educated discourse'. In the context of mathematics lessons then, the teacher should be seeking to offer pupils the opportunity to practise being users of 'mathematical discourse'.

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The question which arises from this analysis is: in what sense can teachers 'scaffold' their pupils' induction into educated discourse through the medium of the educational discourse of the classroom? In seeking to answer this, Mercer refers to the work of Northedge (1990) who suggests that discussion within classrooms will tend to gravitate towards 'everyday' discourse without the help of the teacher to translate some of what is said into the terms of the academic discourse within which the group is aiming to participate. The teacher's function in this is to 'frame' the discussion by raising the salient features of the 'educated' discourse, posing questions, asking for evidence, questioning lines of argument and so on. Mercer suggests that teachers have to start from where their pupils are, use what they already know and help them to go "back and forth across the bridge from 'everyday discourse' into 'educated discourse"' (1 995, p. 83).

But what is the nature of mathematical discourse? Mathematical discourse is characterised by features such as logical deduction, the need for reasons and reference to rules which distinguish it even at an elementary level. Pupils seem to be aware even at around the age of six that it is inappropriate to talk about feelings or recount personal experiences in a mathematical context. Morgan (1998) observes that mathematical 'texts' use specialist and semi-specialist vocabulary, symbolism, an abstract impersonal style and the construction of an argument. She draws a distinction between 'text' and 'discourse', to which I subscribe, suggesting that 'discourses' include the wider social and linguistic practices within which 'text' arises. 'Texts' in turn refer to a piece of written or spoken language which has its own unity arising from a specific social context. As such the talk within a lesson would be a 'text', and so would a shopping list. Another point raised by Morgan is that 'discourses' are not distinct and isolated but overlap and compete. In the context of classroom talk pupils and teachers may be participating in mathematical discourse and educational discourse at the same and different moments, and the discourse of school mathematics may be quite different from that of academic mathematics even when they share common features.

The focus then of this paper is an exploration of 'texts' arising from mathematics lessons in classrooms in order to consider the teachers' induction of their pupils into discourse which might be described as mathematical. In this setting, the work of Voigt (1985) is illuminating. His paper, with its focus on the patterns and routines of the classroom, is close to my own work and his methodology has similarities to mine. He is concerned with the construction of mathematical meanings and trying to make sense of what happens in mathematics classrooms. He studies the patterns of interaction in these classrooms in an attempt to see the real and potential consequences of these patterns for the pupils' learning behaviour. He works from a micro-ethnographical perspective concentrating on the detailed description of a small sample of behavioural records. From this he draws attention to a number of features of classroom discourse which I have also found in my own fieldwork. Voigt suggests (p. 80) that the structure of the pattern of interactions in a lesson , which he calls the 'elicitation pattern' often has three phases. These start with the teacher opening the

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task, move on through the teacher guiding a development of a solution and fixing the solution and end with an evaluation of the methods and of the results, reflecting on the context and interpreting the method. In this development the teacher may use a number of strategies to structure the discourse such as markers, to draw attention to a change in subject, pace or question, non-verbal cues, such as pointing to numbers or artefacts, and metaphors, to elucidate meaning. Voigt also describes teachers' use of repetition to confirm pupils' utterances and indexical expressions whose meaning is located in the situated context in which they arise.

METHODOLOGY

My work involves detailed and close analysis of the text of lesson transcripts. It has arisen from time spent in teachers' mathematics classrooms initially 'helping out' as a parent volunteer and subsequently collecting field notes and audio recordings of lessons. In these classrooms then I would see myself in the role of participant observer, a familiar figure to teachers and pupils alike and, as such, as unintrusive as possible with, hopefully, minimal distortion of the usual patterns of interaction of the classrooms. The teachers with whom I have been working are self-selected and enthusiastic about teaching mathematics. As such they may not be considered to be a representative sample but the analysis of their interactions does nonetheless show some interesting features of these teachers' attempts to develop their pupils' participation in some kind of mathematical discourse.

My analysis focuses on the words as they are spoken, although there are occasions on which the indexicality of the text requires some reference to intonation or gesture. My interest is in the ways in which the teachers use the talk of the classroom to encourage and develop their pupils' mathematical thinking and expression. In analysing the text, I went through transcripts line by line to attribute the function of each phrase or turn by pupil or teacher using descriptors such as initiation, response, feedback, question, answer, affirmation, repetition and so concentrating on methods akin to those of discourse analysis (Potter and Wetherell, 1987). After this, I looked for patterns associated with the meaning of the on-going exchange drawing on grounded theory methods (Strauss and Corbin, 1990) to refine and develop categories for the strategies which the teachers were using to encourage their pupils' mathematical talk. I was searching for evidence of a joint construction of mathematical meaning between the teachers and their pupils, and evidence in the pupils' talk of their mathematical understanding.

This paper focuses on evidence from two lessons. The first involved work with a group of six high attaining Year 1 pupils (aged 6 to 7 years) on a worksheet entitled Triangular Walls (See Figure 1). Most of the text was tied to the sheet and the process of finding answers by counting on or counting back. The task was to find the missing number in an addition sum using a 'hundred square' as an aid if necessary. The missing number could be the sum or one of the addends or any of the numbers being added i.e. in a+b=c, a, b, or c might be missing. Furthermore, the pupils had to identifi which numbers were linked in this way and to determine which number

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could be found i.e. where two numbers were given in the 'wall' from which the third could be found. The task was presented in the form of the worksheet with minimal explanation given on the sheet.

Triangular Walls The number in the top brick is found by adding the two below it.

Using this rule i t is possible to build triangular walls of any size

For example , 2&l ,

I Complete the following triangular walls

Figure 1: Extracts from the Triangular Walls worksheet. The numbers shown in italicised bold were subsequently entered by hand

The second lesson was on Factors and Multiples. It came at the end of a series of lessons on these topics and involved a class of 26 Year 4 pupils (aged 8 to 9 years) of mixed age, sex and attainment. It took the form of a whole class discussion and started with a verbally given set of ten questions. The first five questions asked for the factors of five large numbers e.g. "Write down two factors of 132" and the other five asked for multiples of numbers less than ten with three or more digits e.g. "Write down a number over 100 that has 4 as a factor". It was unusual in that the teacher made no written record of the answers or questions but concentrated on the verbal exchange.

FINDINGS

In analysing the texts from the lessons a number of common features emerged. These can be divided into three categories:

1. Clarifying the task.

2. Explicating the pupils' responses.

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3. Providing a 'template' or specific form of words.

Often the teachers are involved in all three of these activities over a short piece of text but the three areas do seem to be distinct behaviours on the part of the teacher. To illustrate these categories I shall take a number of passages from the two transcripts and look at them in detail. The first example is taken from the beginning of the Triangular Walls lesson at the beginning when the teacher was setting the pupils off on the task.

Example 1: Triangular Walls

Conventions: [ denotes overlapping speech

T:

John:

T:

Children:

T:

John:

T:

Sev. ch.:

T:

Lenny :

T:

Sev. ch.:

T:

Lenny :

T:

... denotes an inaudible word or words

Look at the very first one and we'll have a look at this one together

1,4,3, now [that must be

[Do you remember how to do it?

Yes

Which numbers do you take first of all, John?

That's 8 so you put the answer ... Ah ah no! Do you take all the numbers at once?

No.

Which numbers do you take?

First you take that and that ... You take 1 and 4 and put it in the square above there.

[Yes

[Don't you? 1 and 4 makes what, Lenny?

5

Right so you put that in the square above the 1 and 4.

In this example the teacher opens the exchange by drawing the pupils' attention to the worksheet. She gets a response from John: "1, 4, 3, now [that must be" and responds by asking if they remember how to do it: "[Do you remember how to do it?" to which the pupils give a positive response. In response to John's offer of the three numbers taken together 1,4 and 3, she asks him directly: "Which numbers do you take first of all, John?" and he persists with his incorrect interpretation. She tells him he is wrong and goes on to ask him a question: "Ah ah no! Do you take all the numbers at once?" The rest of the group answer "no" clearly and she follows this up with "Which numbers do you take?" All these contributions serve to clarzfi the task. It is clear from John's suggestion that he will take 1, 3 and 4 to make 8 that he, at least but maybe some of the others too, is unclear about what he is intended to do.

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The teacher picks up this cue and works with her pupils and what they say to make the task clear to them all.

Her next turn occurs in response to Lenny's suggestion: "First you take that and that . . . " and in this case she explicates the pronouns he has used so that the rest of the group can understand what he is saying by filling in the numbers he meant by "that and that": "You take 1 and 4 and put it in the square above there". There is no doubt that Lenny knew which numbers he was referring to but the teacher needs to make his statement clear for the rest of the group so that all the participants can follow the lesson. There may often be some tension between communicating with the group and with the individuals who are part of it - teaching involves making judgements about how to balance the needs of both individuals and the group as a whole from moment to moment.

The teacher's statement about 1 and 4 is affirmed by the group, and the teacher now goes on to ask for the answer: "1 and 4 makes what, Lenny?" In doing this she provides a template in the form of a pattern of words which the pupils pick up and use through the rest of the lesson. "(a number) and (a number) makes (a number)" becomes a word pattern which is repeated through out the lesson by pupils and teacher alike. In the phrase, any one of the three numbers may be missing and the task involves working out the missing number.

The following example comes slightly further on in the same lesson transcript

Example 2: Triangular Walls

Lenny: Two plus three

T : Right. two and what makes seven?

Lenny: Right. Can you get ... T: Right what number do you have to add to two to get seven?

Here the teacher again provides a template by saying "two and what makes seven?" to which the pupil gives a hesitant response which prompts her to go on to clarzfi the task by expressing it in the form "what number do you have to add to 2 to get 7?" The pupil goes on to answer the question successfully, even though the teacher had ignored his earlier line of thought by focusing on the template she had in mind.

About halfway through the same lesson, the teacher continues to use the same approaches in talking to her pupils.

Example 3: Triangular Walls

F: ' What next?

T : Now you can see 20 is eight plus what?

F: Eight add what is 20

T : Yes.

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F: Well if eight add two is 10 then if add two, eight add two is ten then it must be eight add ten

T: Yes, you said eight add two

F: [Eight add er er 12

T: [You said eight add two

F: That must be 8 add 12

T: Yes that's right.

F: If it was eight add two, it must be eight add 12.

T : Yes, you're right. Just right, well done. That's good working out, good.

The teacher's response to her pupil's question is to offer a template which the pupil re-arranges into the same order as the teacher had used previously: "eight add what is 20" with which the teacher agrees. The pupil goes on to work out the answer explaining his thinking out loud. In this explication much of the meaning is still implicit. The teacher offers the pupil a reminder of what he has said: "Yes, you said eight add two" when she feels, probably wrongly, that he may be losing track of his reasoning, so again explicating the talk.

Similar instances of these three actions occur in the Factors and Multiples lesson which was with a different teacher. The following extract occurs during a sequence discussing the factors of 132.

Example 4: Factors and Multiples T :

Anna:

T:

Ch?

Anna:

T :

Martin:

T:

Simon:

T:

Colin:

T:

Colin:

T:

Another factor? Anna?

How do you know that 66 is a factor of 132? [pause] You're right but how [do you know that.

[Double 66

I'm not sure.

You're not sure - why? Why do you think?

If you double er whatever it is it [makes that.

[Well done. Yes. 2 lots of 66 would make 132. You should know why you've chosen it, it shouldn't just be a lucky guess (to Anna ) should it? Well done. Simon?

Is 4 right?

Is 4 right? Is 4 a factor of 132, Colin?

Yes

How do you know?

Cos 32 comes in your 4 times table.

Well done.

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The passage is particularly interesting as it shows the teacher explicating pronouns for some of the pupils' contributions but not doing so in others. I will consider possible reasons for this later but first wish to look at the actions themselves. The teacher asks Anna for a factor and follows this up by asking for a reason why she was able to give that factor. This pattern of question and follow up question was used throughout the process of gathering answers for the factors questions and serves to clariJSl the task for the pupils. The teacher ignores the contribution by an unidentified child "Double 66", but when Martin offers "If you double er whatever it is it [makes that", she follows up by explicating the pronoun 'it' and replacing "double": "[Well done. Yes. 2 lots of 66 would make 132". The last five lines of this example show the same pattern as before with the teacher asking for a factor, verifying the answer, asking for reasons why it is right and affirming the reason. The reasons offered by the pupils have been established as legitimate in earlier lessons on the same topic. Colin's phrase "Cos 32 comes in your 4 times table" refers back to earlier lessons in which the teacher had established the template "(the last two digits) are in your 4 times table" as a legitimate reason for establishing 4 as a factor of a given number.

The following passage follows immediately after the above example and shows a pupil using the same template as in the previous example but this time inappropriately to suggest that 6 is not a factor of 132 because 32 is not 'in the 6 times table'.

Example 5: Factors and Multiples

Colin:

T:

Ch?:

T :

Karen:

T:

Karen:

T :

Sev.Ch.:

T:

Susan:

T:

Cos 32 comes in your four times table.

Well done. Yes?

Six

Six. Is six right? Anybody know? Is six a factor of 132? Karen?

Why do you think it isn't?

Because it doesn't come in your six times table.

We don't normally learn our six times table as far as that, do we? But if we did keep counting it would actually come into it. I'm trying to think what would be the easiest way to tell you to learn from./ Would 120 come in your six times table if you kept going for long enough?

[Yes. No

[Think. 120. Would that come in your six times table if you kept going long enough? Susan?

Yes

Yes it would. So if you count on from 120 in sixes, it would be 126, 132.

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In this extract, considerable use is made of pronouns by the teacher and the pupils and this may lead to some confusion between the participants. In particular, Karen seems to have used 'it' in her second response to refer to 32 whereas the teacher interprets her sentence to refer to 132.h the course of any classroom exchange much of the meaning remains implicit and the teacher will make judgements about when to explicate meanings. In making these decisions she will need to consider the individual participants and the group as a whole. In the above extract the teacher does not explicate the pronouns. However she does try to clarzfi the task which in this case is considering whether six is a factor of 132. She does this by taking the pupil's suggestion about the six times table as referring to the whole number and not just the last two digits and talking through ways to consider whether 132 is in the six times table. She refers back to ideas about 'times tables' going on infinitely and 'obvious' multiples which have been raised in earlier lessons.

INTERPRETATION

How then do these actions on the part of the teacher and her pupils help to induct the pupils into mathematical discourse? What effects do these practices on the part of the teacher have on the discourse as it develops? Using Mercer's (1995) suggestion that the talk of the classroom arises as a joint social construction by the pupils and the teacher, it is possible to explore the contribution made to the talk of the classroom of the features of clarzfiing the task, explicating pronouns and meaning and offering templates. The teacher is involved then in an on-going process of developing her pupils' ability to participate in mathematical discourse, which she does by refining their contributions to the talk in the classroom. Throughout the lesson she is involved in interactions with individual children and with the group as a whole, and she makes judgements about how to increase her pupils' understanding and their ability to communicate about the mathematics they are engaged with.

The many-to-one nature of .the interaction means that there are times when the teacher puts the needs of the group, as she perceives them, before the needs of the individual child and vice versa. In Example 5 above the teacher takes the pupil's contribution "it doesn't come in your six times table7' and uses it in a way that was probably very far removed from the pupil's idea when she spoke. The teacher appears to be doing this for the purpose of making some observations about the nature of times tables, factors and multiples to teach the whole group. This exchange is also interesting because of the implicit nature of much of the teacher's contribution. She uses pronouns to cover many different things over the space of a few sentences and seems to assume that the pupils can understand her line of argument. It is interesting to attempt to explicate these pronouns, in the following elaborated account.

Example 5: Factors and Multiples: Elaborated version

Colin: Cos 32 comes in your four times table.

T: Well done. Yes?

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Child?: Six

T : Six. Is six right? Anybody know? Is six a factor of 132? Karen?

Karen: No

T: Why do you think six isn't a factor of 132?

Karen: Because 32 doesn't come in your six times table. (or because 132 doesn't come in your six times table, which is the teacher's reading of this contribution)

We don't normally learn our six times table as far as 132, do we? But if we did keep counting 132 would actually come into our six times table. I'm trying to think what would be the easiest way to tell you to learn from. bause] 120 seems an obvious multiple of six to me. Would 120 come in your six times table if you kept going for long enough?

Several Ch.: [Yes. No

T: [Think. 120. Would 120 come in your six times table if you kept going long enough? Susan?

Susan: Yes

T : Yes 120 would. So if you count on from 120 in sixes, the pattern would be 126, 132 so 132 is in your six times table which means that six is a factor of 132

(Italics denote elaborated speech.)

From this elaboration it can be seen that the teacher takes for granted a common pool of assumed knowledge but that at the same time she is trying to check up, by asking questions of her pupils, the extent to which they do share that knowledge and understanding. In appealing to a trusted child, Susan, she is also engineering a continuing of the correct line of argument rather than immediately seeking to clarzfi or paying attention to the problems that the rest of the class may be having. Newman, Griffin and Cole (1989) describe the way in which teachers are involved in the dynamic assessment of their pupils in which they give the pupils a task and then observe how much and what kind of help the pupils need to complete the task successfully. The teacher and pupils work co-operatively to solve problems and in this context the teacher is dependent on the pupils' contributions to move the discourse forward. This phenomenon seems to be illustrated in the above example.

As well as the teacher re-expressing what the pupil says in order to clarify what has been said, the pupils also need to interpret the teacher's statements so that the lesson content is developed as a collaborative construction by the teachers and pupils. This interpretation is borne out in my examples by the ways in which pupils adopt the templates offered by the teacher and use them to express their own mathematical ideas and understanding of the context of the lesson in which they are involved.

In adopting the teacher's expressions and templates, the pupils seem to be adopting patterns of expression of their mathematical ideas which fit with the teacher's model.

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It seems as though the teacher is offering the pupils an 'inner voice' to frame their thinking in a mathematical form. This is illustrated by the pupils' adoption of the templates in the examples above and also in the teacher's posing of questions which suggest to the pupils what they should be thinking; for example when she asks in Example 4 "How do you know?" The process of clari@ing the task also serves to give the pupils a model for their reasoning in tackling the task they have been set.

The description of this behaviour as offering an 'inner voice' focuses on the psychological perspective, but also links with the description of patterns and routines in classroom interactions given by Voigt (1985) from a sociological perspective. This resonates with Vygotsky:

Every fimction in the child's development appears twice: first on the social level, and later, on the individual level; first between people (interpsychological) and then inside the child (intravsycholo~ical). (1 978, p.57, original emphasis)

During the course of their lessons, teachers interpret the offerings of their pupils and pupils interpret the offerings of their teachers, often re-phrasing each others' comments to fit their own frames of reference. In this context, the lesson content can be seen to be collaboratively constructed by the teacher and pupils using Mercer's (1995) cued elicitation and IRF exchanges as the building blocks in the discourse. My contention is that the teachers I have studied are engaged in developing the mathematical discourse of their pupils through the medium of the educational discourse of the classroom.

REFERENCES

Mercer, N.: 1995, The Guided Construction of Knowledge: Talk Amongst Teachers and Learners. Cleveden: Multilingual Matters.

Morgan, C.: 1998, Writing Mathematically. London: Falmer.

Newman, D., Griffin, P. and Cole, M.: 1989, The Construction Zone: Working for Cognitive Change in School. Cambridge: Cambridge University Press.

Northedge, A.: 1990, The Good Study Guide. Milton Keynes: Open University Press.

Potter, J. and Wetherell, M.: 1987, Discourse Analysis and Social Psychology: Beyond Attitudes and Behaviour. London: Sage.

Strauss, A. and Corbin, J.: 1990, Basics of Qualitative Research: Grounded Theory, Procedures and Techniques. London: Sage.

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