As there was no previous research which documented students' mathematical language systematically, a methodology needed to be developed for collecting and analysing the data. In setting up this probe study, one of our original research questions had been: What is an effective method for analysing children's talk about mathematics? This chapter, therefore, sets out our decision making process in regard to choosing a methodology. Some of the decisions which would affect what we could say about students' language choices were: how much data was to be collected; from what students; interacting in which situations. As well, how we chose to interrogate the data would have an impact on the type of model of student language that could be described.

Reviewing the literature suggested that we were likely to find differences between groups of students. Variables such as gender, age and socioeconomic background can be considered as socially constructed, with language use being one of the ways that individuals are positioned within a society (Wodak & Benke, 1997). In doing this research, it was important to recognise that specific features could not be considered as 'male' or 'female' but rather if there were differences these would occur along a continuum, as 'linguistic differences are very often a matter of probabilities and tendencies' (Laver & Trugill, 1979, p. 23). As we were uncertain how differences in linguistic choices would manifest themselves, the data had to be analysed flexibly enough so that interesting things could be identified. As a quantitative approach to research requires the researcher to know what they are investigating before they begin, it was felt that a qualitative approach would be more appropriate. Qualitative research has been described as having the following 5 characteristics (Bogdan & Biklen, 1982, p. 27-30):

In data collection, there were several issues which needed to be considered. These included: in what setting should the data be gathered; from whom should it be gathered; and by whom. The decisions about these would have an impact on the description of students' language that we would be able to produce. Qualitative research suggests the need for natural settings from which to gather data as context has an important effect on the production of linguistic data. Halliday (Halliday & Hasan, 1985) described the context of situation as being made up of what is going on, who is taking part and what role the language is playing. Changes to any of these affect perceptions of the context which then affect the language choices seen as appropriate. For example, how the teacher is perceived as interacting with students will influence students' language choices (see Khisty and Chval, 2002). For a robust model of student language to be developed from this research, it was important to keep the situation as similar as possible for all students. Yet there was a need for a variety of students to participate so that we were not relying on one or two students to provide a representative sample.

Studies into the language used by students in mathematics classrooms have, in general, only had a small number of participants. This has probably been because transcribing audiotapes of interactions takes large amounts of time (Swann, 1994) and produces huge amounts of data (Milroy, 1987 p. 22). Occasionally, studies on language in mathematics education have been done with larger numbers of participants (Rowland, 2000, Bills & Gray, 2001 and Bills, 2002). As part of his study, Rowland (2000) interviewed 230 students in one primary school to investigate their use of hedges. To do this he used a standardised set of questions and each interview took only five to ten minutes. The study by Bills (2002 and Bills & Gray, 2001) used 80 students who were interviewed at various times over two years. The transcribed interviews were then analysed to find the linguistic characteristics which accompanied correct calculations. Such studies constrained the language choices of students because they responded to a series of questions provided by an interviewer rather than being allowed more control over what was discussed (Rowland, 2000). However, the situations can be manipulated so that they are similar for every student and it was for this benefit that we decided to use data from the National Education Monitoring Project (NEMP). In NEMP, several hundred, randomly selected students in Year 4 and Year 8 from throughout New Zealand respond to the same set of tasks which are asked by about 100 teacher administrators (Flockton & Crooks, 1997 and Crooks & Flockton, 2001). The responses that the students give provide a snapshot each four years of what these students know in mathematics. Many of these tasks are video recorded and, therefore, can be transcribed relatively easily.

Interviewing students for NEMP is not the same as recording naturally occurring interactions in classrooms. However, the interactions between the students and the teacher administrators were similar to interactions that students would be expected to have with their own classroom teachers. Milroy (1987 p. 41), in discussing the collection of data for descriptive linguistic studies, stated that '[f]rom the interviewee's point of view, a co-operative response is often one which is maximally brief and relevant'. This could also describe the expected discourse patterns in teacher/child interactions in classrooms, except that the teacher administrators are told not to provide feedback on the correctness of the student's response, even though the provision of feedback is a typical part of classroom discourse (see Edwards & Mercer, 13 1987). With NEMP assessments, the students work with the same pair of teacher administrators over the course of a week and so have some opportunity to interact before doing the mathematics tasks. We were aware that the interviewer's age, gender, ethnicity and personality could affect the language choice of students (Bogdan & Biklen, 1982). On the whole, the teacher administrators -mostly female from middle-class backgrounds- would be similar to the teachers that students were likely to have in their own classrooms. Therefore, we hoped that many students would respond in the same way with the teacher administrators as they would with their classroom teachers and so would use language which closely resembled what they would use in their own classrooms. The students did the tasks in their own schools although not in their own classrooms. NEMP assessment is considered low stakes as it has no impact on the child's academic programme nor is it directly linked to school performance (Crooks & Flockton, 2001). Using the NEMP material was a compromise, as it allowed us to gather material from a large number of students where the style and set of questions were the same for all. Although it was not a classroom setting, the data was gathered in a context which was familiar to students. However, the decision to collect this data meant that we would be unable to comment on student-student interactions or even how students would use mathematics language when they had more control over the direction of the interview.

A main advantage of NEMP was that it was possible to choose students who fitted particular demographic descriptions such as gender, age (Year 4 or Year 8) and the decile ranking of the school attended. It is generally accepted in New Zealand that the decile ratings for schools relate to the socio-economic background of students (Bicknell, 1999). There are, of course, difficulties with such a categorisation as it is fraught with issues over who is making the decisions and what constitutes the factors which are relevant to such a decision (see Robinson, 1979). However, with few alternatives available, a decision was made to accept the common belief that children who came from high decile schools were from more affluent backgrounds. Tasks were also available whose responses could be related to the ethnicity of the students. These interested us as there had been studies to show that Pacific students living in New Zealand do not achieve as well as their European or Asian peers (Young- Loveridge, 2000) and so we wanted to know whether ethnicity was reflected in the language choices of students. As a result, videotapes of students were chosen based not only on gender and age, but also on their attendance at particular decile-rated schools and whether they were Pacific Islanders or not.

To produce a rich description of students' mathematical explanations and justifications, it was necessary to look at responses to more than one task. This would enable us to see how the task as part of the context affected the language choices of students and so give us more insight into the process of making those choices. We, therefore, transcribed videos of children responding to four different tasks. From tasks done in 1997, we selected 'Better Buy' and 'Weigh Up' (see Flockton and Crooks, 1997) and 'Motorway' and 'Bank Account' from 2001 (see Crooks and Flockton, 2001). These tasks are provided in the Figures below. Instructions for the teacher administrator are given in bold.

The two tasks from 1997 were done by the same set of students whereas the ones from 2001 were done by separate groups. By using the 1997 tasks, we were thus able to see how the actual questions affected the same students' responses. Bills and Grey's (2001) research had compared students' use of linguistic features between mathematical and non-mathematical explanations. This showed that students might use certain linguistic features such as logical connectives in one context but not in another one. This suggested that context had an influence on the linguistic choices that students felt were appropriate. By being able to compare the same students giving responses to two mathematical tasks we would be able to see how much was related to the student and how much was related to the task. Students attempting the 1997 tasks could be chosen based on age, gender and decile rating of school attended whereas students doing the 2001 tasks could also be grouped according to ethnicity.

As can be seen from the instructions for the tasks, Weigh Up and Motorway required students to provide explanations of what they did whilst Better Buy and Bank Account requested justifications. Better Buy required students to justify their choice of boxes. This was often done by students making reference to the calculation that they did. Bank Account requested students tell a story about the Bank Account and we had anticipated that students would justify why the amounts on the graphs changed during the week. As the correctness of responses varied for these tasks (Flockton & Crooks, 1997 and Crooks & Flockton, 2001), it can be considered that they were mathematically challenging.

In order to develop a robust model of the language that children used in giving mathematical explanations and justifications, we needed to ensure that we had a large enough sample size. Many descriptive linguistic studies used reasonably small samples. Labov's well-known generalisations of the speech of New Yorkers was based on a sample size of only 88 speakers (Labov, 1966). Hawkins (1977) investigated the nominal groups used by five-year olds in London. Having decided on his variables of social class, IQ, gender and communication index of mothers (CI), he formed a matrix of 2 x 3 x 2 x 2 cells and assigned 5 children to each cell. However, as he did not have middle-class children who had both low IQ and low CI, two of the cells remained empty (one for boys and one for girls). This meant that he had a total of 110 samples of five-year old children's talk to analyse. Hawkins (1977) recommended using such a matrix by stating that '[i]ts advantage is that the effects of each variable may be estimated independently' (p. 8). Increasing the number of samples results in data handling issues which could make the research impractical.

A decision was made to include those students who remained mute when asked a question as we felt that there may be patterns in their distribution. This sampling method ensured that any group, such as students from different decile schools, would contain at least 24 samples. This number increased when results across all four tasks were compared. As the students doing the two 1997 tasks were the same, our total sample size was 216. We anticipated that, with a sample size of 216 students and 288 transcripts to interrogate, patterns in linguistic choices would become apparent.

Although we would have access to the videos of students, we had to limit what we would analyse. Much information such as eye signals and gestures is lost when transcriptions are made (Swann, 1994), but, in order to keep the project manageable, it was decided that we would concentrate on the linguistic expressions used by students. As a result, only student responses were transcribed, as the NEMP procedure required the teacher administrators to ask set questions. It soon became apparent that this had not always happened, but by only having transcriptions of the student responses, we were forced to concentrate on their linguistic choices. However, it did mean that we missed opportunities to discuss the interactions between students and the teacher administrators.

One final advantage of using NEMP data was that ethics approval had already been granted. Obtaining consent from students and their parents and other interested parties such as the Ministry of Education is important in educational research to ensure that participants, especially children, are not exploited (Cameron, Frazer, Harvey, Rampton & Richardson, 1994). Yet 'going through the formal procedures that some educational systems require can be a long, laborious process' (Bogdan & Biklen, 1982, p. 122) so it was useful to have this ethical approval already obtained. Cameron et al. (1994) also raised the ethical issue of how the results would be used which might be contrary to interests of the participants. In doing research where differences between groups could be highlighted, we needed to ensure that any findings were seen as important only if they enabled teachers to better understand the mathematical learning of students. It was not possible to discover this, however, until we knew what some of these differences were.

This research set out to develop a robust model of student language in giving mathematical explanations and justifications. As a result, it was felt that a qualitative approach was the most suitable in order to achieve this. However, in determining how to implement such an approach, compromises had to be made between the aims of being as descriptive as possible, whilst also using naturalistic settings and keeping the study manageable.

Once the decisions had been made about what data to collect, we then had to consider how it was to be analysed. As we were unsure what aspects of students' language were most useful in producing a robust description, it was necessary to consider how others had approached their analysis of students' mathematical language. For example, Chapman (1997) examined her transcripts to discover the ways that the teacher rephrased students' utterances and how these rephrasing were then picked up by the students. In other studies such as that by Gooding and Stacey (1993), transcripts were coded so that specific types of responses (asking questions, responses to requests for clarification) were highlighted and their use related to common 18 attributes of the groups of students who used them. For our research, neither methods of analysis seemed appropriate as we were not focussing on the exchanges between students and the teacher administrators; rather we were comparing different students' responses.

We were also aware of the potential difficulties associated with a coding system which simply reflected what researchers expected to see (see Edwards, 1976) and how this did not support a qualitative approach to the research to be undertaken. Originally, we started by looking for particular features that we felt were more mathematical in student responses, such as the use of nominalisations or noun phrases as agents of actions rather than people, following the example of Hawkins (1977). These features were based on work described in Meaney (forthcoming).

As reported in Meaney and Irwin (2003), the results from this investigation were confusing (these are given at the beginning of each task chapter). We started again by classifying students' responses according to the clarity of their language and their accuracy. We then discussed what it was that contributed to some students' responses being classified as clear.

It was at this point that the work by Hasan (Halliday and Hasan, 1985) became valuable in our search for tools with which to do the analysis. This was because her beliefs about text structure enabled us to code students' responses but in ways that supported the illumination of patterns. It also enabled us to keep the context of the responses as an integral part of the description. Hasan's (Halliday and Hasan 1985, p. 56) described contextual configuration as the significant attributes of a social environment in which a text is constructed. Contextual configuration can be used to predict the text structure.

In this study, students' explanations were given in responding to questions on a mathematics task asked by a teacher administrator in a school environment. Hasan (Halliday and Hasan 1985) stated that the contextual configuration 'can predict the OBLIGATORY and the OPTIONAL elements of a text's structure as well as their SEQUENCE vis-á-vis each other and the possibility of their ITERATION' (p. 56 capitals and italics in original text). However, as Hasan also pointed out, the relationship between language and situation is bi-directional and some elements of the text structure will, in fact, help to construct the situation. For example, when a student provided a minimal response and the teacher administrator kept prompting, sometimes this probing became more about teaching the student than about assessing their current understanding. If these further questions become an obligatory element of the situation, then the situation changed from one of assessment to one of teaching.

By looking for the patterns of obligatory and optional elements and how they are sequenced and repeated across different students' responses, we could describe different children's perceptions of the situation. As there had been no previous work in this area, our coding was inductive rather than looking for expected elements, thus we hoped to limit the problems identified by Edwards (1976) in regard to coding.

In order to ensure reliability of results, all coding and counting was done with at least one person doing the majority of the work and another doing some checking. At times, the checking was done by research assistants who were trained in what was 19 being counted or coded. Occasionally, the Year 4 and Year 8 data were counted or coded separately after initial joint coding or counting. There was some cross checking. Discrepancies between counters and coders were discussed and clarified. A qualitative analysis needs analytical tools which do not constrain the data but which support the uncovering of patterns within it, so that a rich description of students' language choices can be made. Although we also began to look at the data through other lenses, Hasan's ideas about contextual configuration enabled us to reveal subtle differences in how groups of students constructed their justifications and explanations in mathematics through combining text elements. We are aware that this choice of analysis tools means that our description of students' language choices will be limited to the existence and ordering of elements within their texts.

Our research question was: What are the typical linguistic choices of different groups of students when giving mathematical explanations and justifications? In examining ways that it could be investigated, we were aware that these would have an impact on what we would ultimately be able to present as our findings. A quantitative approach might have produced a more systematic description showing how often terms and expressions were present in students' speech. Yet, it may not have enabled the context in which the students' responses were made to be an integral part of that description. We were aware that we had compromised the requirement of a natural setting for our qualitative data gathering. However, it would not have been possible to make any comparisons between the language used by different groups if the setting was not the same for all students. Our ultimate goal was to contribute to teachers' understanding of how their perception of students' knowledge about mathematics was affected by students' choice of language. It seemed valuable to produce a model which was as broad as possible and still placed importance on the context. Yet we were also aware that by looking at student-student or student-teacher interactions during learning experiences the model that we would have been able to produce would have been significantly different. To ensure that our study remained manageable we were forced to narrow our research question and then chose data gathering and data analysis methods which provided us with a robust model of students' language in giving mathematical explanations and justifications.

The following chapters provide information on each task. They begin by discussing the 1997 tasks and then provide information on the 2001 tasks. These chapters are then followed by further discussion on stories in mathematical tasks and text structures of explanations and justifications.