This report documents the language used by students in Year 4 and Year 8 in responding to mathematical assessment tasks which required them to do numerical or quantitative comparisons. It also describes some differences which were found between students of different ages, genders and the decile level of the schools they attended. This leads to a more extended discussion on story telling within
mathematics and the text structures of mathematics explanations and justifications.

In learning mathematics, language is the vehicle through which mathematical ideas are described by the teacher but it is also the medium that students use to discuss these ideas and make sense of them (Kaput, 1988). In order to evaluate students’understandings, teachers often listen and read students’ responses to tasks. For example, Moskal and Magone (2000) suggested that previous research had found that ‘students’ written explanations to well-designed tasks can provide robust accounts of their mathematical reasoning’ (p. 313) and so can be used by teachers to assess students’ knowledge. For students to make the most of language as a tool for their thinking and to adequately explain what they know, they need to learn how to express themselves mathematically (Chapman, 1997).

Although there has been much research on the mathematics register and teachers’ use of language, especially in regard to questions (Martino & Maher, 1994), little research has been done to document what is typical of the language used by students at different stages (see for example Ellerton & Clarkson, 1996). Yet curriculum documents such as the Mathematics in the New Zealand Curriculum (Learning Media, 1992) abound with statements that students should ‘develop the skills and confidence to use their own language, and the language of mathematics, to express mathematical ideas’ (p. 23). Being able ‘to make predictions, formulate generalizations, justify their thinking’ (Burns, 1985, p. 17) is believed to support students’ development of mathematical thinking. However, as Kristy and Chval’s (2002) research clearly demonstrated, it is not always apparent to teachers when everyday language is acceptable and when students should be encouraged to use mathematical language. In investigating students’ writing about mathematics, Morgan (1998) felt that there was a general lack of knowledge about language and language teaching. As a result she was unsure that students could adequately express themselves mathematically. This is supported by research by Bicknell (1999) in which New Zealand secondary teachers voiced their belief that the process of writing explanations and justifications should be explicitly taught to students. This research sets out to document the language used by Year 4 and Year 8 students in giving mathematical explanations and justifications of quantitative and numerical comparisons. By knowing what language students typically use at different ages to talk about mathematics, teachers would be able to develop their students’ fluency in as well as formal knowledge of mathematical language.

Students’ explanations and justifications is one area of mathematical language which has received attention as a way of improving their learning of mathematics (see Forman, Larreamendy-Joerns, Stein & Browns, 1998). In New Zealand, the advent of the National Certificate in Educational Achievement has emphasised the need for students to be able to write explanations and justifications in order to gain merits or excellences in their assessments (Irwin & Niederer, 2002; Meaney, 2002). These examples of mathematical genres are perceived as leading onto developing mathematical proofs (Bicknell, 1999). Bicknell (1999), using the work of Thomas (1973), gave these definitions:

[a]n explanation can be defined as making clear or telling why a state of affairs or an occurrence exists or happens, whereas a justification provides grounds, evidence or reasons to convince others (or persuade ourselves) that a claim or assertion is true.

Another who has done research on students’ explanations and justifications is Erna Yackel (2001), who using a symbolic interaction perspective, stated that:

[s]tudents and the teacher give mathematical explanations to clarify aspects of their mathematical thinking that they think might not be readily apparent to others. They give mathematical justifications in response to challenges to apparent violations of normative mathematical activity.

Using elements from both perspectives, we describe a mathematical explanation as the description of what was undertaken to solve a problem (the solution strategy) whereas a justification would be why a certain strategy was adopted and its consequent result being accepted as appropriate. Although we acknowledge the importance of the social interaction, we are primarily concerned in this report with the
explanations and justifications that students give in assessment situations. In these situations, there is limited interaction between the teacher and the student with the development of an explanation or justification not arising as a result of negotiation of meaning. Therefore our definitions imply that the explanations and justifications are given as a result of teacher questions in an informal or formal assessment situation.

Work by Krummheuer (1995) which described the components of students’mathematical arguments has been used by both Yackel (2001) and Forman et al. (1998). Using ideas from Toulmin (1969), Krummheuer (1995) proposed four components which were: claims; grounds; warrants; and backings. Claims are assertions of a point of view; in most cases these are the proposed solution to a task. Grounds are the unchallengeable facts from which the assertion is drawn. Warrants are the information which joins the grounds to the claims, while backings provide the context which make the warrants appropriate. Krummheuer (1995), Forman et al. (1998) and Yackel (2001) all used ideas about mathematical arguments in contexts of groups of students developing them in conjunction with their teachers. There appeared to be a need to investigate whether similar components could be identified in student responses where there was limited negotiation of meaning.

For students to successfully provide a mathematical explanation or justification as part of an assessment situation, they need an understanding of what is required mathematically and linguistically. All too often, students’ linguistic abilities in mathematics are ignored. For example, Moskal and Magone (2000) developed a framework for ‘the identification of response patterns across classrooms’ (p. 315) but they did so without reference to the language needed by students. There seems to be a sense that if students know the mathematics they will be able to respond appropriately to an assessment task. Bills and Grey’s (2001) and Bills’ (2002) research on the responses given by seven to nine year olds to a request about how they did mental calculations showed a relationship between the correctness of responses and some
linguistic features. These features included the use of personal pronouns (‘you’ and ‘I’), present tense and logical connectives such as ‘because’, ‘so’ and ‘if’ (Bills, 2002). Esty (1992) also stressed the importance of ‘five key logical connectives: “and”, “or”, “not”, “if … then” and “if and only if”’ which provided mathematics students with an understanding of when equations were true and therefore the limits
on them as generalisations. Bills’ (2002) felt that ‘you’ was a reflection of how the students’ teachers discussed mathematics. As a result, he felt that ‘I’ rather than ‘you’ used in a general or generic response showed that students had more ownership of a generalisable concept.

Bills and Grey (2001) categorised responses into three kinds: general, generic and particular. ‘General’ responses were those which invoked a rule with little mention of actual numbers. ‘Particular’ responses were those which revolved around the specific numbers in the calculation. ‘Generic’ responses were those between the other two categories in that they made use of specific numbers but as examples of a specific rule. Students who were incorrect in their calculation were most likely to give a ‘particular’ response whilst students who were correct were most likely to give ‘nonparticular’ responses (p. 153). The conclusion from both papers was that looking at the language that students used could be informative about students’ thinking. Their belief was that, as students were able to use these linguistic features in other types of explanations, the way that they expressed mathematical ideas reflected their conceptualised. For example, 80% of students who made a general response but used ‘I’ as the doer of the actions gave a correct response, suggesting that they had personalised their belief about how to do the calculation. However, this is based on the assumption that all students have the same linguistic resources to draw upon and that all students understand the value of being able to generalise in mathematics.

Figure 1 is presented to problematise the relationship between mathematics and language ability. It uses two continuums to illustrate how students’ language and mathematical knowledge interact as they respond. It is significantly harder to understand what students know in a situation where they give a minimal response or appear to answer a different question. In this situation, is it the mathematics or is it the
language which is problematic for them? Some students have the mathematics skills but the language of the task can cause them confusion. Clarke (1993) proposed that placing mathematical tasks in contexts increases the linguistic demands on students without requiring more of them mathematically. Teachers can interpret these responses as resulting from poor mathematical knowledge whereas it is, in fact, as a result of poor linguistic knowledge.

In considering to which of the quadrants in Figure 1 a particular student's response belongs, there is a need to remember that competence and performance are not equivalent. In the 1960s, Chomsky (1965) made a distinction in regard to language by stating that linguistic theory should be primarily concerned with the 'ideal speakerlistener' (p. 3). This would be similar to stating that mathematics learning theory should only concern itself with how an ideal student learns mathematics. Hymes (1972), in dismissing Chomsky's decision to ignore the outside factors which affect language performance, wrote:

The limitations of this perspective appear when the image of the unfolding, mastering, fluent child is set beside the real children in our schools. The theory must seem, if not irrelevant, then at best a doctrine of poignancy: Poignant because of the difference between what one imagines and what one sees; poignant too, because the theory, so powerful in its own realm, cannot on its terms cope with difference. To cope with the realities of children as communicating beings requires a theory within which socio-cultural factors have an explicit and constitutive role (p. 270)

Students' lack of performance may not be due to a lack of competence but rather due to a lack of knowledge of the type of response expected in a classroom situation. This was certainly the case for the Trackton students that were the subject of Shirley Brice Heath's (1982) research in the early 1970s. Recognition of socio-cultural factors which affect the perceptions of the context is equally important when considering how students exhibit their learning in assessment activities.

Students could fail to show their mathematical knowledge because they were tired or had no personal need to answer the question (Malcolm, 1982). Aboriginal children, for example, make the decision about when they will show competency in a skill. Adults in their society would not expect them to perform to someone else's time frame, nor would adults explicitly instruct them (Christie, 1985). Instead children are expected to watch and learn and decide when they are ready to show what they can do (Harris, 1980). There is a need to discover the cultural norms in which the student is operating. Even when students cooperate with the school requirements to perform, they can unintentionally mislead their teachers about their mathematical understanding because they have reacted to other aspects of the situation.

There has been significant research to show that children with different backgrounds do not make the same choices when using language. For example, Irwin and Ginsburg (2001) showed the differences in the choice that young children from different economic backgrounds made in whether or not to use language when engaging in tasks of a mathematical nature. This would have a marked effect on their teachers' understanding of their ability. Zevenbergen (2001) described the use in two different schools of the triadic dialogue sequence in which the teacher asks a question, the student or students respond and the teacher then provides feedback. Bernstein's (1990) work over many years consistently supported the idea that social class would have an effect on the language choices made by students because of their different perceptions of the context and of what were appropriate choices within that context. By drawing on this work and that by others to suggest that students from lower class backgrounds were less likely to be familiar with such a classroom interaction pattern, Zevenbergen hypothesised that these students would be at a disadvantage in learning mathematics. Research on teacher perceptions of students based on their speech showed that students who spoke non-standard versions of the language of instruction were more likely to be considered to have lower ability and behave inappropriately (Haig and Oliver, 2003). It is well known that 'the understanding that students are expected to explain their solution is a social norm, whereas the understanding of what counts as an acceptable mathematical explanation is a sociomathematical norm' (Yackel, 2001, p. 14). Yet this awareness of the role of the social background in the development of mathematical explanations has not resulted in research into the mathematical explanations provided by students from different socio-cultural groups. It, therefore, seemed valuable to investigate how students from different socioeconomic backgrounds used language when giving mathematical explanations and justifications.

There are also studies to show that there are differences in the language choices of boys and girls. Coates (2004), in summarising studies in this area, showed that working class girls were more likely to use standard forms as they grew older whereas boys were more likely to use a stable or increasing amount of non-standard forms. Other research has also shown that teachers nominated and interacted with boys more frequently (Wickett, 1997). It has been suggested that these interactions would have an effect on the scaffolding and modelling of language that boys and girls receive which might promote boys' ability to gain the academic language (Swann & Graddol, 10 1994). These studies suggest that it is worth investigating how boys and girls give mathematical explanations and justifications.

We were also aware that the acquisition of the mathematics register (or language to discuss mathematical ideas) occurs over time. Rowland (2000) found that the answers given by different-aged pupils to mathematics tasks showed differences in how uncertainty was expressed. For this research, it seemed important not only to investigate the effects of socio-economic background and gender on students' responses to tasks requiring mathematical explanations and justification, but also to look at the effect of age.

As well as looking for differences within variables such as whether younger children used different expressions to those of older children, it was also expected that combinations of variables would affect the choice of expressions. This is because no student is male or female without also being young or old and coming from a specific socio-economic background. For example, research has found that lower middle class women were more likely to imitate middle class language (Labov, 1990) and that young men were subject to more peer pressure to use a less standard form of language (Milroy, 1980 and Milroy, 1981 cited in Wodak & Benke, 1997).

It was quite clear that we were not so much interested in mathematical correctness of the explanations and justifications but how they were expressed (see Pimm & Wagner, 2003 for a discussion of the form of mathematical responses). There was a need to document how students typically structured explanations and justifications and to discover if different groups used different structures. It was then worth finding whether there was a relationship between these structures and their mathematical correctness. This was because it seemed that form could have a significant impact on knowledge being considered correct and that there was a need to have a more complete description of how groups expressed themselves. Our research question thus evolved into: What are the typical linguistic choices of different groups of students when giving mathematical explanations and justifications? In particular, we were interested in seeing if there were patterns of usage which were related to gender, socio-economic background, age and a combination of these variables.