Mathematics: She’ll be write!

1. Introduction – Mathematics: She’ll be write!

How students learn to speak, read, and write science and mathematics, and what is taking place in the classroom, laboratory, or informal learning context are critical areas for research. (Lerman, 2007, p. 756)

The focus of this Teaching and Learning Research Initiative (TLRI) project was to discover effective ways to develop students’ mathematical writing in te reo Māori. It was assumed that this would lead to better understanding of mathematics. The investigation was undertaken at Te Kura Kaupapa Māori o te Koutu which caters for students from Years 0–13, many of whom are second language users of te reo Māori. It involved all the teachers as well as two outside researchers considering a number of issues around the role of writing in mathematics.

Investigating the provision of professional development of effective ways to support students’ mathematical writing is an area in which very little research has been done previously (see Doerr & Chandler-Olcott, in press, for the exception). Consequently, this research report should not be seen as describing completed work, but rather as a snapshot of where we have travelled up to this point, in regard to the integrated, intricate nature of teacher professional development to improve student achievement. It was ethnographic research that was bounded by being conducted in one kura, where the language of instruction in mathematics was te reo Māori. The context as well as a detailed description of the research process undertaken by the teachers and other researchers is provided, so that our results and analysis can be better understood by others who may share some or none of the characteristics of the situation.

As a consequence of the research, a variety of different genres and text types were identified. The genres identified by the teachers were: whakaahua; whakamārama; and parahau. These have been organised in progressions showing how different layers of meaning are added to the text types. Using this information, we then explored how students could be supported to use writing to help their thinking processes and thus their mathematical learning. Various strategies were trialled by teachers and some of the results of these are discussed in this report. It was quite clear that both the amount and variety of writing done in all classes increased as a result of the teachers being involved in the project. The project used a Teacher Inquiry and Knowledge Building Cycle (Timperley, Wilson, Barrar, & Fung, 2007) approach that concentrated on developing programmes based on the identified learning needs of students and teachers. A fundamental component of this approach was the integration of reflection with action, which is also known as praxis.

This project built on an earlier TLRI project. In that project, Te Kura Kaupapa Māori o te Koutu’s teachers and the research team worked together in 2005 and 2006 to identify the teaching strategies that were effective for developing students’ use of te reo tātaitai or the mathematical terms and expressions within te reo Māori (Meaney, Fairhall, & Trinick, 2007). As a result of this project, teachers became aware of their need to concentrate more specifically on the teaching of mathematical writing. This is a need that has been recognised in many countries for some time (Ntenza, 2006). The results of this research, therefore, have implications not just for other kura kaupapa Māori but also for schools and mathematics teachers throughout the world.

An emphasis on mathematical communication is clearly indicated in the English- and Māori- medium curriculum statements (Ministry of Education, 1992, 1993). The document states that there will be opportunities provided “for students to develop the skills and confidence to use their own language, and the language of mathematics, to express mathematical ideas” (p. 9). The importance of language in helping children make sense of their world is supported by Campbell and Rowan (1997) who assert that “language has the power to help children organize and link their partial understandings as they integrate and develop mathematical concepts” (p. 64).

The teaching approaches recommended by the national curriculum statement (Ministry of Education, 1994) are, at heart, constructivist. The constructivist view is that people make “sense of the world in ways that are always historically and culturally specific” (Begg, 1999, p. 5). For students to be able to succeed in mathematics, they need to use mathematical language to help them make sense of the learning situation, rather than merely being expected to solve problems in the manner in which they have been shown. Developing a shared meaning of mathematical ideas is a key process within constructivist learning theory (Good & Brophy, 1990). This means that children should have the ability to verbalise someone else’s understandings to themselves so that they can reorganise external language into an “inner language” or “internalised thought”.

Within the field of mathematics education, there seem to be a number of issues relating to the forms of communication expected of students. Underlying these issues, though, is the expectation that children need to communicate effectively. Sfard, Nesher, Streefland, Cobb, and Mason (1998) stress the importance of developing children’s communication skills but question how this will be done and what should actually be taught, and comment that this is an issue that has not really been given much thought by the mathematics education community. They argue that children need to be taught how to communicate with their peers and teachers so that there is a base line of shared understanding.

A focus on how to support students’ writing in mathematics was chosen because it was believed that it would improve students’ reflection on their mathematical thinking (Southwell, 1993). The kura’s involvement in Poutama Tau, the New Zealand numeracy professional development project for teachers in Māori-medium classrooms, had meant that there was an awareness of the need to have students explain their thinking. Analyses of the Te Poutama Tau student data found that language proficiency was a significant factor in student achievement in the higher stages of the number framework (Christensen, 2003). In order for students to communicate mathematically in Māori medium, there is a need for the teaching community to understand what effective mathematical communication looks and sounds like in the classroom.

The effects of language learning on mathematics have been recognised for some time (Ellerton & Clarkson, 1996) but minimal research has been carried out on how students acquire the mathematical register. Although communication of all kinds is supposed to support students’ thinking mathematically (National Council of Teachers of Mathematics, 1989), writing, because it can be referred to again and again, supports the reflection process more easily than spoken language (Albert, 2000):

Writing is a valuable way of reflecting on and solidifying what one knows, and several kinds of exercises can serve this purpose. For example, teachers can ask students to write down what they have learned about a particular topic or to put together a study guide for a student who was absent and needs to know what is important about the topic. A student who has done a major project or worked on a substantial long-range problem can be asked to compare some of their early work with later work and explain how the later work reflects greater understanding. In these ways, teachers can help students develop skills in mathematical communication that will serve them well both inside and outside the classroom. Using these skills will in turn help students to develop deeper understandings of the mathematical ideas about which they speak, hear, read, and write. (National Council of Teachers of Mathematics, 2000, p. 352)

Research by Pugalee (2004) showed that students who wrote descriptions of their thinking were significantly better able to solve mathematical problems than those who verbalised their thinking processes. One of our underlying assumptions about increasing the quantity and quality of students’ writing was that it would lead to improvements in students’ understanding of mathematics.

The importance of students being able to explain their thinking process is also valued in New Zealand’s formal assessment processes. The National Certificate of Educational Achievement (NCEA) in mathematics requires students to be able to write explanations and justifications (Meaney, 2002a). The teachers at te Koutu believed that it was in the junior classes that students should start developing these skills and that there needed to be a coherent approach across the kura to the teaching of mathematical writing. This matched what Hipkins and Neill (2006) wrote about the impact of NCEA on high school mathematics teachers’ awareness about language:

both mathematics and science teachers give a relatively high priority to the need to develop language and literacy practices associated with each discipline. In at least two cases the teachers’ awareness of these issues has been sharpened by participation in school-wide literacy initiatives. (p. 63)

Having students write can also support teachers to understand better students’ mathematical misconceptions as it often provides the teachers with more information than what is gained from simply listening to students (Drake & Amspaugh, 1994). We wanted to investigate writing both as a way of supporting students’ mathematical thinking and also as a means by which teachers could better understand their students’ knowledge of mathematics.

The mathematics register in English makes use of the passive voice and logical connectives (Meaney, 2005a). As both of these are prominent features in traditional oral reo Māori, the teachers were also interested in exploring whether mathematics could be a vehicle for improving students’ reo Māori language skills in general. In the previous TLRI project (Meaney et al., 2007), we found that te reo Māori had many more logical connectives than English. Therefore, we also wanted to explore how providing students with information about using these logical connectives could support their mathematical thinking. Writing can more easily be used in explicit language learning as it can be referred to time and time again. However, there was concern about how a concentration on writing might result in a devaluation of speaking.

Orality and literacy

Te reo Māori has a long oral tradition and a concentration on writing must be considered within this context. Western beliefs about the value of writing in improving students’ thinking processes as propagated by researchers such as Vygotsky and Luria (Gee, 1989) cannot take precedent over issues relating to the how and why of teaching mathematical writing in the immediate situation of a kura kaupapa Māori. Although literacy is believed to have a role in the regeneration of a language (Hohepa, 2006), concern has also been raised about the possible imposition that writing can have on Indigenous communities’ ways of being (Cavalcanti, 2004; Street, 1995). Gee (1989) stated that:

Discourse practices are always embedded in the particular world view of a particular social group; they are tied to a set of values and norms. In learning new discourse practices, a student partakes of this set of values and norms, this world view. Furthermore, in acquiring a new set of discourse practices, a student may be acquiring a new identity, one that at various points may conflict with the student’s initial acculturation and socialization. (p. 59)

As a discursive practice, mathematical writing will have an impact on students’ identity. Whether this identity forming will be in conflict with the students’ Māori identity will depend on how the discursive practices are taught. In kura kaupapa Māori, “the pedagogy of these schools is based on, but not exclusively, Māori preferred teaching and learning methods” (Smith, 1990, pp. 147–148). It was, therefore, important that the role of writing in the teaching and learning of mathematics was not just accepted without question. With te reo Māori still in a process of regeneration after almost becoming extinct in the 1970s, there was a need to consider how the teaching and learning of writing in mathematics could be done in a culturally appropriate way. In discussing the role of reading in the home, Hohepa (2006) wrote:

A significant issue in the context of language regeneration concerns how language practices both reflect and construct cultural concepts and values. One way to address this issue is to ensure that ways of carrying out an activity such as book reading do not undervalue or replace existing cultural ways but are added to family repertoires (McNaughton, 1995). Also, ways of participating in the activity which are not inconsistent with the specific literacy goals, but which are consistent with culturally preferred ways of participating can be promoted. (p. 299)

Hohepa’s warning can also be related to mathematical writing. The advantages for students’ mathematical learning in being able to write mathematically had to be considered in relationship to other priorities that the kura and its whānau had for students. Issues of spoken mathematical language cannot be divorced from considerations of how to effectively teach writing in mathematics. As well, questions such as “What is writing in mathematics?” and “What constitutes an appropriate mathematical written text?” need to be situated in the wider context of this kura kaupapa Māori’s aspirations for its students.

What is writing in mathematics?

One of the first ideas that we had to determine was what it was that we, as teachers and researchers, meant when we referred to mathematical writing. In our original research applications, we had made reference to diagrams and models as also being examples of mathematical writing. Given that meaning is produced in mathematics through a variety of different written forms (O’Halloran, 2000), it was important to consider the relationship between the situations in which the writing was produced and the impact of the audience on the writing. The act of writing needed to be situated in the context that it was written in, thus acknowledging the impact of the social environment on the piece of writing that was produced (Gibbons, 1998). These were vital considerations given the situation in which we were investigating the teaching and learning of mathematical writing. Consequently, our beliefs about what constitutes writing in mathematics are discussed using Halliday’s ideas about context of culture and context of situation (Halliday & Hasan, 1985).

Using the ideas of the anthropologist Malinowski, Halliday described how a text or piece of language “that is functional” (Halliday & Hasan, 1985, p. 10) is both a process and a product that reflects the wider cultural background as well as the immediate situation in which it arises. The meaning that is conveyed by a text is influenced by both the context of culture and the context of situation. Linguistic choices are made by the producers of the text that illustrate their perceptions of the context of situation, often unconsciously (Meaney, 2005a). Figure 1 shows how Halliday (Halliday & Hasan, 1985) views the relationship between linguistic choices and the context of situation.

Figure 1 Relations of the text to context of situation (from Halliday & Hasan, 1985, p. 26)

Therefore, by looking at the linguistic choices used in a piece of mathematical writing it would be possible to see the producer’s perception of the context of situation. Another way of perceiving this set of relationships is shown in Figure 2.

However, viewing mathematics symbolism and other visual representations as language that can be analysed is a contentious issue. Halliday (2007) expressed doubts about whether mathematics could be analysed for meaning in the same way that texts that used words could be. He stated that “[m]athematics is not, of course, a form of visual semiotic, but it is expressed in symbols that look like, and in some cases are borrowings of, written symbols . . . We cannot read it, because it has no exact representation in wording” (p. 114). He went on to state that it could be verbalised in a range of equivalent mathematical forms but which were grammatically quite different and this would affect the meaning that was given to them. Halliday wrote how he would not describe these visual representations of information such as graphs as language: “they are semiotic systems whose texts can be translated into language, and that offer alternative resources for organizing and presenting information” (p. 115).

Figure 2 A model showing how language acts as a viaduct between culture and understanding (from Meaney, 2005a, p. 113)

As a result of the ambiguity that arises when interpreting mathematical symbols and other visual representations, Halliday would not refer to anything but written words as constituting a piece of writing. However, other researchers have not agreed with Halliday’s refusal to consider mathematics as something that can be analysed in the same way as language (see, for example, Kress & van Leeuwen, 2006). Several have used Halliday’s systemic functional grammar to describe how meaning is produced and interpreted in other forms than just words (Unsworth, 2001). O’Hallohan (2000) who applied systemic functional grammar to discourse in mathematics classrooms stated:

mathematics is construed through the use of the semiotic resources of mathematical symbolism, visual display in the form of graphs and diagrams, and language. In both written mathematical texts and classroom discourse, these codes alternate as the primary resource for meaning, and also interact with each other to construct meaning. (p. 360)

In this project, we have followed this tradition and have taken a broader definition of mathematical writing. Without acknowledging the multiple representations that could be used in mathematical writing, there would be very few mathematical pieces of writing that could be analysed. Nevertheless, it was important to note Halliday’s (2007) warning that there is a need to be aware of “meanings being lost, and what new meanings imposed, when there is translation between verbal and the non-verbal; and exploring the semiotic potential that lies at their intersection” (p. 116). Everyone will interpret a piece of writing in a variety of ways because they come to the piece of writing with different backgrounds. However, in interpreting mathematical symbols and visual displays, greater differences in interpretation may arise. It is therefore important to think carefully about how to support students developing a range of different visual information displays in mathematics so that shared meanings arise from this process.

The development of pieces of writing occurs within what has been defined as “literacy events”. Heath (1982) described these as “any occasion in which a piece of writing is integral to the nature of participants’ interactions and their interpretive processes” (p. 93). It is, therefore, important to consider how the context in which the writing is done will affect its form and appropriateness.

Appropriateness of pieces of writing

In considering effective ways to support students’ mathematical writing, it is important to have some understanding about what kinds of writing we are expecting students to use, as well as knowledge about how to support students to judge the quality of their own writing. In Chapter 5 of this report, we describe the teachers’ views about a good piece of writing. However, there is a need to consider what constitutes an appropriate piece of writing and how this may differ from good writing. This is an important distinction because appropriateness deals with whether the piece of writing achieved its purpose or function. On the other hand, discussions about quality or “good writing” assume that the function has been achieved, but distinctions can be made about the clarity or conciseness or some other criteria and the impact that this has on the quality of the piece of writing.

In considering the appropriateness of the writing, considerations revolve around whether the linguistic choices match the field, tenor and mode of the context of situation. In order to reflect the meanings available within the field, does the author of the piece of writing make clear the “kind of acts being carried out and their goal(s)” (Halliday & Hasan, 1985, p. 56)? How are the participants (including inanimate participants) labelled or highlighted and what are the types of verbs or processes that are used? The amount of detail as well as the choice of vocabulary will be affected by the situation. If the student is using writing to work something out, it is likely to be less explicit and much messier than writing that is for formal display (Ernest, 2007). There has been discussion about the role of formal mathematics terms in the learning of mathematics (Leung, 2005). Barnes (1976) identified distinct features of “exploratory talk” and “final draft talk” and a similar set of features could be identified with mathematical writing done as part of the process of thinking as opposed to writing done for formal presentation. Of course, the different audience, self as opposed to examiner, would have an impact on this writing. This is discussed in more detail in Chapter 3.

The interpersonal meanings realised through the tenor reflect the relationship between the author and the reader. Hasan described how the social distance between participants “affects styles of communication” (Halliday & Hasan, 1985, p. 57). When the participants are not familiar with each other, such as when a student is writing an exam paper, then there is a need for more explicitness than if the participants have a longstanding relationship such as when a student writes for their teacher. Morgan (1998) identified six audiences that students may write for in a mathematics lesson, including an indiscernible audience.

The mode shows how the type of writing, graphs, symbols and so on affects the textual meanings that are produced. Information presented in words as opposed to being presented in a graph will be interpreted differently by an audience. This is because the different types will highlight some aspects of the information while downplaying others. These aspects will differ depending upon the type of writing that is used.

For students to produce an appropriate piece of writing, they need to show an awareness of how changes in field, tenor and mode affect how they present their mathematical meaning. In Chapter 3, we describe what this may mean in regard to the pieces of writing that were collected during the project and how this relates to evaluations of the quality of individual students’ pieces of writing.

Report overview

This report consists of nine chapters and these are briefly described below. As the findings presented in Chapters 3 to 8 are quite distinct, rather than having one literature chapter, previous relevant research is described at the beginning of each of these chapters.

Chapter 2: Research methodology

This chapter outlines how we used a case study approach. It also describes the data that we collected.

Chapter 3: Genres

This chapter describes how students’ pieces of writing were classified according to their function: whakaahua; whakamārama; parahau. These categories were then related to Halliday’s ideas about field, tenor and mode within a context of situation.

Chapter 4: Whakaahua

The progressions for the different topics of descriptions are discussed. An initial explanation of how these types of descriptions are related to the year-level progressions is also given.

Chapter 5: Whakamārama me parahau

This chapter looks at the explanations and justifications in the set of writing samples and explores how different text types are often integrated to produce these genres. It also discusses the teachers’ perspectives on the benefits for writing as well as what constitutes a “good” piece of mathematical writing.

Chapter 6: Ways to improve mathematical writing

This chapter discusses how teachers facilitated students’ writing about mathematics. It describes acts of writing and the strategies that the teachers used to help students acquire mathematical writing. It provides a matrix to show how acts of writing, mathematics register acquisition (MRA) model strategies and genres are connected.

Chapter 7: Student writing

This chapter describes the typical writing done by students at the different year levels. We look specifically at the writing done around probability at the different year levels. This was connected to students’ beliefs about writing in mathematics, gathered through surveys and interviews. These beliefs, including the older students’ beliefs about completing bilingual exams, are described here.

Chapter 8: Teacher change

The effect of participating in this project on teachers’ teaching practices and their ability to reflect on their own learning are discussed in this chapter. It shows that all the teachers felt that their changes in practices had increased the quantity and/or the quality of students’ writing in mathematics.

Chapter 9: Conclusion

This chapter summarises the major findings and provides a discussion of the implications for this kura and for other mathematics classrooms. It also reflects on the limitations of the project and further research that this kura wishes to undertake.

2. Genres

One of the aims of the project was to describe the types of writing that students were currently using at the kura and then look at ways to improve both the quality and the quantity of the pieces of writing at all year levels. Little previous research had been done in this area and so there was little guidance on how to do this. Although curriculum documents emphasised the need for students to learn the language of mathematics, there were rarely specific examples of how to do this. For example, Doerr and ChandlerOlcott (in press) described how the National Council of Teachers of Mathematics (2000) Standards provided “limited discussion of different genres of mathematical writing that students might engage with and how they might learn to attend to audience and purpose in each of those genres and how these, in turn, might vary across mathematical tasks and grade level”.

At the research report meeting held in 2006 to set the parameters for what would be done in the She’ll be Write! project there was a sharing of anecdotal stories about students’ mathematical writing. For example, at the senior level of the kura it was felt that students did not want to elaborate on their explanations but rather kept them short, using simple language that they felt most comfortable with. Mathematical language was used only because students believed that teachers expected them to use it. This belief is similar to those given by students in Healy and Hoyles’s (2000) research on students’ development of proofs.

By systematically collecting and analysing students’ pieces of writing, it was felt that a better understanding could be gained about whether other, more appropriate types of writing could be introduced to students and how the ones that were currently being used could be improved so that students’ learning of mathematics and te reo Māori could be supported (Meeting 30 August 2006). One way of analysing the pieces of writing was to classify them into different genres and then to identify the features of each of the genres. This chapter provides a description of the different genres that were identified at the kura and their relationship to Halliday’s field, tenor and mode. Chapters 4 and 5 provide more details of the different genres.

What are genres in mathematics classrooms?

Any text, whether oral or written, is influenced by three components. These are: what is being discussed; who is involved in the text (producing it, interpreting it or within it); and the form the communication is taking (written, oral, gestures). Halliday described these as field, tenor and mode (see Meaney, 2005a). Changes to any of these will result in changes to the text that is produced (Halliday & Hasan, 1985). Consequently, types of texts will reflect the purposes that they serve in meeting the requirements of these three components. For example, demographic data are commonly presented in graphs, especially if they are going in reports designed for statistically literate adults.

If texts are continually produced to fulfil the same set of field, tenor and mode requirements, then their linguistic and other features will become stabilised over time. However, the inclusion of new features will continue to occur and so description of the features of genres should not be considered as rigidly fixed. Stabilisation occurs as a result of “negotiation among and between community members” (Wallace & Ellerton, 2004, p. 8). As a result of this stabilisation of features, sets of texts can be categorised as genres. To not structure the texts that are responding to the same set of field, tenor and mode in the conventional way can result in the meaning that is supposed to be produced being misinterpreted (Huang & Normandia, 2007). It is, therefore, important that students learn during their schooling experiences how to write the genres that are used in particular content areas (Unsworth, 2001). Knowing about genres is not just knowing what features to include but also knowing how and when genres are useful. As Pimm and Wagner (2003) wrote “[m]uch of this work (e.g. Martin, 1989; Halliday & Martin, 1993) is also rooted in questions of school systems developing greater equity by means of students gaining access to linguistic-cultural capital” (p. 162). For the students at the kura, it is valuable to understand the role of genres in mathematics. They can take this knowledge about genres with them if they go on to study further mathematics at university in English or another language. However, investigation of this would be for a future project and is not considered here. In this project, the purpose of identifying the genres was to have a starting point for teachers to consider how to improve the quality and quantity of students’ mathematical writing.

What is the relationship between the mathematics register and genres?

In discussing genres, there is a need to distinguish them from registers (Wallace & Ellerton, 2004). In this research report, we have used definitions more closely aligned with those of Halliday than those given in Wallace and Ellerton (2004). In every mathematics classroom, there would be a number of different genres. Some of these genres would be common across many classrooms. These would have a set of distinctive features that are organised into a specific structure and they all fulfil a certain function (Unsworth, 2001). However, if they were identified as mathematical genres then they would contain certain vocabulary and grammatical expressions that would support the mathematical meanings being presented. These texts would use the vocabulary and grammatical expressions of the mathematics register. Halliday described registers as:

the semantic configurations that are typically associated with particular social contexts (defined in terms of field, tenor, and mode). They may vary from ‘action-orientated’ (much action, little talk) to ‘talk-orientated’ (much talk, little action). (Halliday & Hasan, 1985, p. 43)

Consequently, aspects of the mathematical register would be apparent in all mathematical genres.

Mathematical genres

Although genres have received significant amounts of attention since the 1980s, especially in Australia and the United Kingdom (Unsworth, 2001), very little research has been done in regard to those typically found in mathematics classrooms. Like Morgan (1998), Marks and Mousley (1990), using the ideas of Martin (1985/1989), identified several genres that mathematicians would use and that, therefore, should be included in students’ repertoire of mathematical writing. These genres were:

Procedure: how something is done

Description: what some particular thing is like

Report: what an entire class of things is like Explanation: reason why a judgement has been made

Exposition: arguments why a thesis has been produced.

However, when they investigated the genres which were used in 11 classrooms (seven primary and four secondary), they found many instances of recounts, incorporating symbols and visual representations, but very few examples of other genres. Recounts described what a student had done during a mathematical activity and is generally expressed as a narrative. This would suggest that these students were not learning the conventions associated with mathematical writing because “mathematics cannot be narrative for it is structured around logical and not temporal relations” (Solomon & O’Neill, 1998, p. 217). In stories, cohesion is achieved by placing a series of events in a timeline. In mathematics, cohesion is achieved by relating separate ideas through logical connections. These relationships are timeless and therefore using time markers as is common in recounts is inappropriate in discussing mathematics.

Given that the purposes for writing are different for mathematicians than they are for students, it is perhaps not that surprising that students were being asked to write different kinds of genres. However, by the time students are in their final years of high school it could be expected that the students would have the skills to produce many of the genres used by mathematicians. It would be anticipated that some of these genres or early versions of these genres should be presented in all mathematics classrooms.

We were interested in considering what genres the students at the kura were using and how these related to those outlined by others such as Marks and Mousley (1990). Given that our purpose for categorising students’ pieces of writing was so that it could be a base for considering how to improve the quality and quantity of this writing, it was anticipated that our genres would be different from those of others.

What we did and what we found

The investigation of the students’ pieces of writing was done by the teachers with the researchers. It, therefore, must be considered as being at the second tier of the research model proposed by Doerr and Chandler-Olcott (in press) for investigating writing in mathematics classrooms that was described in the previous chapter.

At a meeting in March 2007, the teachers were provided with a variety of pieces of mathematical writing that had been collected during the previous year. These were classified by teachers, first in pairs and then as a whole group. Much of what is reported in the following paragraphs comes from the minutes of the meeting. The teachers classified the writing into three genres and an initial set of modes. Many of these categorised pieces of writing had been collected at the end of 2006. Over the course of 2007, other pieces of writing were collected. All writing samples were scanned, classified according to genre and mode and then named and filed.

Although the genres remained set, there were changes to how the modes were identified as more pieces of writing were added to the database. As Unsworth (2001) wrote “[g]enres are not fixed and invariant. They identify classes of texts with particular characteristics in common . . . Genres, as integral features of subject area learning and teaching, then, should not be considered as straitjackets but as starting points” (p. 127). It was important to our project to have relevant labels for the categories we identified. There was considerable discussion about these at the March 2007 meeting. The labels were: whakaahua (description); whakamārama (explanation); and parahau (justification).

In this section, we briefly discuss the genres and then the modes. The following section considers how changes to the field, tenor and mode would impact on the types of writing produced. The features of the genres are described in more detail in Chapters 4 and 5.

First, we need to acknowledge the limitations of our database of pieces of writing. Although more than 2,000 pieces of writing were collected, the database is still incomplete. The junior section of the kura runs a two-year cycle for their mathematics programme and so not all strands and topics were covered in 2007. In the senior part of the kura, the teachers taught multiple mathematics classes. For this project each teacher focused on only one of their classes. Consequently, few, if any, samples were collected from Years 9, 10, 12 and 13. As well, the primary researcher who collected the pieces of writing, did so during her once-a-term visits to the kura. During these visits, pieces of work were not always collected from the teachers, sometimes because a teacher was away or because writing had been part of a classroom display that was no longer available. Nevertheless, the database is extensive and does show a range of writing done in mathematics classrooms.

Three distinct genres were identified by the teachers and these can be seen in Table 2. The process used to do this classification was for each pair of teachers to look over a range of students’ pieces of writing. The samples came from students of different ages and were from a range of topics. The quality of the writing also varied. In pairs, the teachers classified the students’ writing according to categories that they felt were appropriate. Two pairs then shared their categories and decided on a joint set of categories. Then the whole group came together and had a more extended discussion about the genres. The teachers focused on grouping samples of writing that appeared to have the same function. They, therefore, identified the primary purpose of each piece of writing and then looked at the structure within each group. It was at this point that the genres were separated from the modes and Table 2 was developed.

Table 1 Writing genres and their modes

The modes of writing that students used in the genres were: pictures; iconic representations; graphs; geometric representations; symbols; and narratives. Other researchers had sometimes considered what we had labelled as modes to be genres. Solomon and O’Neill (1998) stated that “[i]n so far as genre shapes and constrains the nature of a text, then graphs, equations, proofs and algorithms can be considered as expressions of genre” (pp. 217–218). However, our definition of genre was based on the function that it performed and therefore the channel through which the function is delivered was considered to be the mode. Other researchers in semiotics used the term registers for different representations of mathematical ideas (Gagatsis, Elia, & Mousoulides, 2006). Duval (2002) identified four types of representation register: “natural language; geometric figures, notational systems and graphic representations” (cited in Gagatsis et al., 2006). These were similar to our modes of narratives, geometric representations, symbols and graphs in Table 2.

Ben-Chaim, Lappan, and Houang (1989) described three modes that were used by students in descriptions of an object made from cubes taped together. These modes were: verbal; graphic; and mixed mode. The verbal mode occurred when the student’s message was carried by words. A diagram might accompany the words but did not add any more meaning to what was stated in words. A graphic mode used drawings with, at the most, labels to accompany it. A mixed mode used both diagrams and words to convey meaning. As can be seen in Table 3, we collected a variety of pieces of mathematical writing and so had a larger number of modes. This larger number of modes also meant that we had to choose a greater number of labels and thus there was a need to move beyond those suggested by Ben-Chaim et al. (1989).

Although in Ben-Chaim et al.’s (1989) study, the graphic mode was believed to be the more successful at accurately conveying information about the object, the teachers felt that, commonly, genres would require a combination of modes rather than being exclusively one or the other. This belief is supported by O’Halloran (2000) who wrote:

Mathematics is not construed solely through linguistic means. Rather, mathematics is construed through the use of the semiotic resources of mathematical symbolism, visual display in the form of graphs and diagrams, and language. In both written mathematical texts and classroom discourse, these codes alternate as the primary resource for meaning, and also interact with each other to construct meaning. Thus, the analysis of ‘mathematical language’ must be undertaken within the context of which it occurs; that is, in relation to its codeployment with mathematical symbolism and visual display. (p. 360)

The genres were chosen because they fulfilled different functions. One was to describe, the next to explain and the final one was to justify. Although these categories were broader than those used by others in categorising mathematical genres (Marks & Mousley, 1990), it was still difficult at times to decide which of these genres a piece of writing should be categorised as. For example, a piece of writing could begin by describing something but finish by explaining something else. However, the component parts constitute the whole and could not be separated. An example of this can be seen in Figure 4. If this was the situation, then the piece of writing was categorised in the higher level, in this case the explanation.

Figure 3 ECKaL3 showing a combination of description and explanation

ECKaL3 starts with a description using symbols of the various combinations that different numbers and colours of blocks could form. It ends with an explanation of the equation relating the number of combinations to the number of blocks and the number of colours.

Whakaahua

Whakaahua was the label given to pieces of writing that described something. A range of these can be seen in Table 2. Writing descriptions provided students with opportunities to learn the conventions of mathematical writing. The descriptions could be in words, but could also be a numerical fact written in symbols. Descriptions of this kind have been classified as other things by different researchers. Marks and Mousley (1990) would separate out descriptions from reports whereas whakaahua contained both. Wallace and Ellerton (2004) stated that:

Description and report genres provide the nature of individual things and the nature of classes of things, respectively. Reports are characterised by generic participants, simple present-tense verbs, and a large percentage of ‘being’ and ‘having’ clauses (p. 9)

As more samples were collected and added into our classification system, it became clear that a defining characteristic for us of whakaahua was the way that new information was added. In systemic functional grammar, the way that new information is juxtaposed with given information is an important part of the construction of meanings in texts (Unsworth, 2000). If the addition of new information was cumulative (this is a triangle, it has three sides and it has three angles), or if new information could not be added without changing the function, then the piece of writing could be classified as whakaahua. Thus, equations such as 3 + 4 = 7 could be classified as a description because adding new information such as (3 x 1) + (2 x 2) = 9 – 2 actually changes the function of the equation from merely describing a simple fact to showing how each term has other ways of being described and that it is the combination of these descriptions that shows a deeper level relationship between numbers.

Whakamārama

Whakamārama, or explanation, used a series of steps to illustrate how something came to be. These were predominantly used when mathematics was employed to answer problems. Therefore, from our initial data set, as well as narratives about how to turn a net into a three-dimensional shape, multistep equations were classified as whakamārama. Other researchers describe this genre as procedural (Wallace & Ellerton, 2004). However, procedures have a sense of involving a lockstep process. For example, from earlier research on genres, Unsworth (2001) described the stages in a procedural text as goal, materials and steps (p. 123). The pieces of writing that we identified as whakamārama certainly explained how something had been done but did not always provide the steps in a set order. An example of such a piece of writing can be seen in Figure 5.

Figure 4 ECAwL4 showing an explanation provided in a nonlockstep manner

 

 

 

 

 

 

 

It was often the task that the student was responding to that determined whether a student provided an explanation in steps or in a nonlockstep manner. Figure 5 was a task where students were asked to explain the relationship between addition, subtraction and multiplication. The explanation begins with two related addition and subtraction equations written in symbols on the left-hand side of the page. The related multiplication equation is also written in symbols but underneath sentences state the multiplication in words. The explanation is not described explicitly, but the reader’s eye movement is channelled from 6 + 6 = 12 to 12 – 6 = 6 as 2 x 6 = 12 and on to 6 x 2 = 12. This is because readers of Western languages are taught to seek new information either below or to the left when reading texts. All the equations are related but it is up to the reader to draw this inference.

Parahau

Over the course of the project, far fewer parahau, or justifications, were collected. These were pieces of writing whose primary purpose was to provide information about why something was done. These pieces of writing were more reflective as students had to evaluate what options there were and to discuss why they chose a particular one to use.

Final genres and modes

Although the three basic genres did not change, the number of modes increased as more writing samples were collected. The final collection is given in the list below.

All genres included a range of different modes. However, whakamārama and parahau were more likely to use a combination of modes in the one piece of writing. In the samples that we collected, there were very few justifications that did not use a combination of diagrams, words and/or symbols. In the junior classes where justifications were just beginning to be taught, sometimes these were just narratives. More often, explanations and justifications contained a combination of modes as can be seen in Figure 6. In Chapters 4 and 5, more details are provided on the modes, both individually and how they were combined together.

Figure 5 JCUn

The relationship between field, tenor and mode, and the three genre types

It was quite clear that there was a relationship between the type of genre and the mode that was used. At the meeting in March 2007, the teachers also discussed how different audiences would have an impact on the writing. They felt that sometimes the writing would be exclusively for the student and therefore may convey limited, if any, meaning to others. This would be because of what Wallace and Ellerton (2004) described as “semantic discontinuity” whereby the reader must fill in gaps between the evidence and the conclusions described in order to make sense of what has been written.

Other researchers have connected the explicitness of a piece of communication, either written or spoken, with explicitness of the language used. In 1989, James Paul Gee wrote an article in which he contrasted different perceptions of orality and literacy. He believed that the claims for superiority in cognitive ability connected to literacy were in fact ways to privilege one group’s ways of thinking and thus were tests of participants’ ability to use language in particular ways. He equated this language use with explicitness:

Explicitness in language use can be placed on a continuum between two poles that we might label, following Givon (1970), ‘the pragmatic mode’ and ‘the syntactic mode’. At the syntactic-mode end of the continuum, speakers encode what they want to say using precise and varied lexical items and explicit syntactic structures (e.g., subordinating devices), leaving as little as possible to be signalled by prosody or inferred by the listener. The grammar takes on most of the burden for communication, and social interaction is downplayed. At the pragmatic-mode end of the continuum, speakers chain strings of clauses together fairly loosely through adjunction or coordination, use prosodic devices to signal meaning, and rely on the hearer to draw inferences on the basis of mutual knowledge. Social interaction and the participation of the hearer in a mutual negotiation of meaning are paramount. (p. 50)

Gee went on to link the distinctions based on explicitness to the differences that others had seen as those between oral and written language. What we found in classifying the pieces of writing into genres (and this is described in more detail in Chapter 5) is that all genres contained pieces of mathematical writing that require the reader to provide little or lots of details in order to follow the reasoning. Figure 5 was an example where the reader is expected to draw the appropriate inference. After investigating students’ classroom talk, Barnes (1976) distinguished between exploratory talk and final draft talk. The first type of talk occurs when students are thinking through what they are doing whereas the second is when there is a distant, more formal audience. One of the characteristics that he gave for exploratory talk was that of a low level of explicitness. It may be that writing for self or for an audience is also connected to whether the writing is to help thinking and thus is exploratory or whether it is for publication and therefore is final draft writing.

Predominantly, the teachers at the kura felt that most writing done in mathematics classes was done for the teacher so that they could assess the students’ learning. Similarly, students’ writing in external exams would be for examiners whom they would never meet. On occasions, pieces of writing might also be produced that could be displayed for other students. It may also be that some mathematical writing would be produced to be shown to parents or community members. They would all seem to be likely candidates for the written equivalent of final draft talk and therefore be very explicit as they are written for external audiences. Students’ beliefs about whom they were writing for is discussed in Chapter 7.

The pieces of writing that we collected and then classified into genres suggest that explicitness was less to do with being exploratory or writing for publication, but rather to do with background knowledge of the audience and whether or not one of the primary purposes of the writing was to fulfil an assessment requirement. When the audience had little or no knowledge of the mathematical activity being discussed, such as when the audience was family members, then more explicitness was needed. When the audience needed to know what the student is capable of, there was also a great need for the student to be explicit. Some students struggle with understanding the needs of an audience and being able to respond appropriately (Meaney, 2002a). If the audience for the writing—for example, the writer themselves or the teacher—had been involved in the mathematical task and the recording had no further use once the task had been completed, then the piece of writing was much less likely to be explicit. Therefore, there is a relationship between the explicitness embedded within a piece of writing through the grammatical structures or organisation of the diagrams and the audience. However, this relationship is not simple and relates to the context of situation in which the piece of writing was developed and the needs of the audience.

Given that the genre reflected the purpose for writing, it could be described as the reflection of the field of the context of situation. However, it was also common to find that there were particular audiences and modes that accompanied the fields for particular genres. Figure 7 is one representation of how this relationship could be perceived.

Figure 6 The context of situation for the use of the three genres of mathematical writing

Whakaahua are used to describe mathematical objects or facts. Complexity in these descriptions occurs with the amount of detail provided. Learning to write whakaahua meant that students were also involved in learning about mathematical writing conventions. At the very beginning levels of school, this involves just learning how to form the numbers or shapes. At later stages, students learn the appropriate way to write a number sentence or produce a graph (this is discussed in greater detail in Chapters 4 and 6). Whakaahua fit into what Unsworth (2001) described as “recognition literacy” in that it supports the “learning to recognize and produce the verbal, visual and electronic codes that are used to construct and communicate meaning” (p. 14). Without this literacy knowledge, whakamārama and parahau cannot be produced. However, where Figure 7 is different from the ideas of Unsworth is that he links recognition literacy to common experiences of everyday life whereas in the mathematics classroom, descriptions are very much about moving students into using the mathematics register. Whakamārama is about explaining a mathematical event or phenomenon; thus it is about using mathematics and recording what is done. Parahau, on the other hand, is about explaining why something has been done in a particular manner. This involves participating in reflection about what was done and why. Consequently, writing becomes part of the learning process in a much more conscious way than it is with whakaahua and whakamārama.

Figure 7 DGTrReUnL6

The model in Figure 7 should not be seen as rigid. It will not be the case that the field, tenor and mode are always combined in this way for each of the genres. Of the different parts of the model, the connections to the genres of the different tenor situations are the most contentious. Whakaahua and parahau are not always written by students for themselves or the teacher but can be written for others. For example, Figure 8 provides an example of a description that was written for public display. Nevertheless, it did seem that in our writing samples, whakaahua were mostly produced for the teacher as part of the learning about the conventions of mathematics writing, while whakamārama were mostly produced so that others, including the teacher, could follow the logic of what the student had done. Parahau, even when produced to fulfil the requirements of the teacher, would force a student to do some self-reflection. Therefore, Figure 4 should be considered as one interpretation of the relationship between field, tenor and mode of writing done in mathematics classrooms. It is, however, better to be considered a common combination, rather than the only interpretation of how the field, tenor and mode are related in each genre.

Teacher discussion about another genre

The students’ pieces of writing were all classified using the three genres described above. However, in the staff meeting held on 5 November 2007, there was a discussion about how to include students’ copying of the learning intentions for each lesson. Learning intentions are the teacher’s brief descriptions of what the students are expected to learn in a lesson or series of lessons. Some teachers had students write the learning intentions into their mathematics books. It was unclear how these learning intentions should be classified. There was, therefore, some discussion about whether another genre should be added to the classification:

They are all involved in the activity in some way. They are [providing] either a written representation of the activity which is sort of like the description or they are explaining what they are doing and then some of the seniors get to the point of justifying what they are doing . . . but everything happens within the activity. This sort of stuff happens before or after the activity doesn’t it? You’re setting [the goals for the activity] and the kids are involved in writing so they’re setting or reading with you . . . they are setting the parameters of what they are going to do. Then they are making some judgment about where they are afterward because I think maybe in this writing thing you need to put another category that’s in place for that sort of language. Because they will be using mathematical language whilst not exactly saying what they are doing. (T9, Meeting 5 November 2007)

One teacher (T1) also suggested that students could write the learning intentions in their own words rather than just copying what the teacher had written.

The project ended before this suggestion was followed up, but with research continuing in 2008 it will be interesting to see how this fourth genre is made use of.

Conclusion

One of the primary aims of the She’ll be Write! project was to document the types of writing that were undertaken by students at the kura. Previous research in this area had used genres from other subject areas and there had been little agreement about what would be typical in mathematics classrooms. The teachers at this kura decided to categorise the writing samples according to the functions that they fulfilled. Consequently, three mathematical genres were identified. These genres were whakaahua, whakamārama and parahau, and fulfilled the functions of describing, explaining and justifying.

Each genre was also considered in relationship to its possible main audience and to the modes used to convey the mathematical meanings. The modes used specific types of mathematical writing such as graphs, symbols and narratives. The relationship between the genres, the modes and their audiences is illustrated in Figure 7. The teachers appeared to use whakaahua to teach students about the conventions of mathematical writing. Whakamārama and parahau often used a combination of modes to express their meaning. These genres were related to supporting students to think mathematically and to reflect on this thinking.

3. Whakaahua (describing)

In Chapter 3, the genres and the different modes were described. It showed that ngā whakaahua, or descriptions, commonly only used one mode of mathematical writing whereas whakamārama (explanation) and parahau (justification) often combined a number of modes together. Our data suggest that it is necessary for students to master the different modes for descriptions, before they can use them in whakamārama or parahau.

Ernest (2007) described the conventions of mathematical writing as:

the rhetorical norms that tidy texts into modes of public address. These norms concern how mathematical texts must be written, styled, structured and presented in order to serve a social function, namely to persuade the intended audience that they represent the knowledge of the writer. (p. 66)

Certainly, the data supported the idea that students learn the conventions of mathematical language while learning to write descriptions. However, the pieces of writing in the whakamārama and parahau categories suggest that having control of the writing conventions supports students in their mathematical problem solving. We would suggest that even in the private workings of the messy texts of doing mathematics (Ernest, 2007), it is valuable to have control of the conventions so that students can make a choice over when to make use of them or everyday language in their problem-solving writing. As Meaney (2005a) showed, mathematicians fluctuate between formal mathematical writing conventions and everyday language in their joint problem-solving interactions. The data set included both messy and tidy texts. For us, having students produce whakaahua is a very important part of the process of learning to write mathematics.

Therefore, this chapter looks at how different layers of meaning are added to the modes of mathematically descriptive writing. It seems that adding different layers of meaning is often closely linked to learning different conventions of mathematical writing. Moving students to having control of the conventions of mathematical writing has been studied by a number of researchers. For example, Chapman (1997) was concerned with how students acquired more mathematical language through the interventions of the mathematics teacher. Student expressions such as “it was the same” were rephrased by the teacher as “the difference pattern is constant” (p. 161). Herbel-Eisenmann (2002) discussed the ways that students moved between contextual language, bridging language and official mathematical language. Her contention was that in order for students to be able to join the mathematical community they would need to be able to use official mathematical language. She stated: “‘Bridging’ helps students move from less to more mathematical language by encouraging multiple ways of talking about ideas” (p. 101). However, both these researchers were looking at spoken mathematical language. Although related, the issues surrounding written mathematical language are different and in need of investigation.

It is this relationship between the layering of meaning and the conventions of mathematical language that is explored in this chapter. Our focus is on the development of the modes and so refers very little to the tenor and only occasionally to the field. The field is important because the topic that is being covered has a direct impact on what mode is focused upon. Chapter 5 considers issues of quality of writing in regard to whakamārama and parahau. As later chapters draw on examples of transformational geometry and probability, this chapter mostly uses examples drawn from the modes—graphs, iconic diagrams, narratives, patterns, symbols, tallies and combinations.

In the 2005–6 TLRI project, one of the outcomes was the need to develop an across-the-school understanding about how topics and their accompanying mathematical language developed. There was little information about this for English-medium classrooms and there had been no research on this in regard to the development of mathematical writing in te reo Māori, except what was done in the previous TLRI project. As was noted in the last report (Meaney et al., 2007), triangles had been an area in which the teachers had begun to develop a progression showing how the ideas developed during the time the students were at the kura. It was felt that it would be useful if progressions for other areas were also developed. In the meeting in August 2006 on the possibilities for this new project, the following was noted:

Standards – at year groups

Language of triangles (this year’s project):

– who is responsible for teaching what language?

– when (at what year level) do they need to learn or know particular things?

– what language should be used at what year level?

– stick with simple constructions until such a time as they are ready for more complicated vocabulary? (when will that time be??)

Benefit for the young children if there is consistency throughout:

– learning new vocabulary

– able to condense

– foundation of words—database

– issue—even at year one—there is a limited amount of language to use (known language)

Suggestion of a word bank (familiar words)

– build on what is used at each year level

Writing formalises the mathematical terms and language used:

– same language throughout (consistency)

– consistency of language throughout the school and use by the teachers

– use of language as core content. (Meeting 30 August 2006)

In June 2007, the Tau (Year) 6 teacher, T7, restated this need in a discussion about his aims for the following two terms. He was an art teacher by training but was very reflective about his teaching of mathematics. He stated that his needs at that time were to know what his students would be doing in Tau 7 and beyond so that he knew better what they needed to learn in Tau 6. In the September 2007 meeting he again reiterated the need to know at what year levels different language should be introduced.

Another issue that was pertinent to other kura as well was that many teachers had not been taught mathematics in te reo Māori. This meant that they had to learn the vocabulary and grammatical structures of the mathematics register just before or at the same time as they were teaching it. The following extract from the staff meeting at the end of 2007 illustrates this difficulty for the teachers. The teacher had already discussed in her interview how she had started teaching the concepts of clockwise and anticlockwise turns, only to find that she was uncertain about the terminology to use. As T9 was walking past at the time, she had then asked him about the terms she should be using. In the final staff meeting, she again raised the issue:

For my own planning I need to be aware. You plan your unit but with all the reo that’s involved you can’t just go and copy everything . . . I chased my tail for my āhuahanga [transformation shape] unit. Because I came to you [T9] asking how to do the rotation. When I came upon it, I didn’t know how to say to them, clockwise, anticlockwise, and all that. So I should have had that at the beginning. If I had that structure, instead of running out of the classroom when I saw you coming past . . . So my classroom practice would mean me being a bit more onto it and going through things and knowing how to say this and this and having [it] ready, and since we are all doing the same kaupapa we should all be using them. Perhaps we could do it together—a bit more team planning. (T1, Meeting 5 November 2007)

Part of the reason for collecting writing samples was so that we could provide an overview across the kura that showed the development of the mathematics register. To some degree we did do this, but there is more work to be done and we found out other unexpected ideas while doing this exploration. This chapter provides details both of what we did find and what still needs to be discovered.

By examining students’ pieces of writing, we were able to produce both topic progressions and year-level progressions. Investigation of these writing samples was part of the second tier of research model proposed by Doerr and Chandler-Olcott (in press) that was described in Chapter 2. As had been the case in Chapter 3, it was a second-tier investigation because it was students’ pieces of writing that were being examined by teachers. Appendix A provides an overview of the progressions for some of the modes for descriptions. Appendix B provides the year-level progressions across various modes. It is this final set of progressions that could be used by teachers in developing their teaching programmes.

The progressions were built up from looking at the writing samples and identifying their purpose and the primary mode that was used. When more than one mode was carrying the majority of the transmission of information, then the mode was classified as a combination. As with the classifying of the genres, this process was not always simple and compromises were made. Writing of phrases or sentences often accompanied diagrams. In these cases, it was decided to keep the mode as that of the diagram. This was consistent with the classification of modes used by Ben-Chaim et al. (1989) that were described in the previous chapter. It was only when the words provided significantly different information than what was provided in the diagram that the whole piece of writing was classified as a combination. Figure 9 provides an example of two graphs. Although it also has some words and a numeral with it, the first one was classified under graphs, DGrSTeKL3a, while the second one was classified as a combination, DCUnL4. This was because the extra information provided in the sentences in DCUnL4 is new information that is not available from just reading the graph. The extra information provided in DGrSTeKL3a is information that is already given in the graph and so could not be considered new.

Figure 8 DGrSTeKL3a and DCUnL4

Progressions

Once the pieces of writing were categorised according to the genre and then the mode, it was clear that some of the features changed depending upon what was being described. This is consistent with research by Ernest (2007) who described the signs for the semiotic system of school algebra at the lower secondary school level. He outlined what they would look like at earlier year levels as well as at later year levels. Changes, therefore, do occur across the year levels and are well known. However, very little work apart from the geometrical drawing understandings investigated some time ago by Piaget and others (see Piaget & Inhelder, 1956) has been done on what these progressions may look like across a student’s learning of mathematical writing while at school. There are of course other types of mathematical progressions, such as the number knowledge progressions in the New Zealand Numeracy Framework (Ministry of Education, 2007), but these are not specifically to do with the development of mathematical writing.

The topic progressions outlined in Appendix A show how different layers of meaning are added to different modes. An example of these is given in Figure 10 which shows the stages for time. How these progressions relate to school year levels is shown in Appendix B.

Figure 9 The different kaupae (stages) in the progressions of time

Kaupae 1 is seen as the simplest because students are just adding the arms to the clock, whereas kaupae 2 requires the students to draw the complete clock from scratch. There is some sense that the layers are added on top of previous ones, but this is not always the case. Although drawing clock faces from scratch requires students to have more control over the conventions of clock faces, it is quite likely that students would be doing activities around both kaupae 1 and kaupae 2 at the same time. One day they might do one type of activity while the next day they might do the other. Therefore, it is not possible to see these progressions as being linear where one stage must be reached before another stage can be achieved. As with the van Hiele levels of geometric thinking (Nickson, 2000), depending on the activity, students will move between the stages in a fluid motion. Even if they show an understanding of the writing convention at a higher stage in one task, they may return to using the writing style from an earlier stage in responding to another task. A number of different issues, such as who they are writing for and the difficulty of the mathematics involved, will influence the students’ choice.

As was the case with the time progression, often the final kaupae was one that included a written description. In the piece of writing above, the sentences describe activities that are done at the times given on the clocks. A case could be made for categorising this piece of writing as a combination rather than as a time progression. However, as the student had repeated the time in words it was decided to leave it in the time progression. Although these could have been written by much younger children than those who produced the penultimate kaupae, their connection to explanations meant that it seemed more sensible to make them the final kaupae. Sometimes the adjustment of just a few words would have turned a description into an explanation. This was not the case with the example for kaupae 5 of the time progression. However, it was felt for consistency’s sake it would be best to make the final kaupae one that included sentences to support the description.

There were often divisions within the stages. Figure 11 shows the substages within kaupae 2 of the time progression. These are similar to those described by Pengelly (1985) and illustrate the various understandings that the students have to gain in order to successfully draw a clock face.

Figure 10 Divisions within kaupae 2

At kaupae 2a, the student has realised that a clock face is circular and something is written on it. By kaupae 2b, they realise that there is a need for some numbers to go around the outside of the clock face. Kaupae 2c shows that the student has understood that a clock face needs numbers to go all around the outside and that there should be arms on the face. These three substages would all be included in Pengelly’s (1985) first level. She had labelled this as an awareness of the numerals on the clock face. The next stage shows an understanding that the numbers should only go up to 12. This was the equivalent of Pengelly’s second level. It is not until the final kaupae that students are able to place them at appropriate positions on the clock face and also show that the arms are of different lengths. Although not all children would draw clock faces that illustrate each of these substages, it did seem that many children would show at least one of these earlier stages before being able to produce a conventional clock face. However, none of our examples showed the minute marks that Pengelly (1985) had described as the final level.

From looking at the stages and the substages for this progression, it is possible to see the need for students to have control over other aspects of mathematical writing in order to be able to present information about time appropriately. However, this may not correspond with when they gain the mathematical concept to match what they are capable of drawing. In order to be able to draw the arms on a clock, students will need to be able to draw straight lines and determine lengths of lines and the appropriate angles that the lines should be placed at. According to our year-level progressions, at this kura, students draw clock faces from Tau 2, yet drawing of angles is not formally taught until Tau 4 when the terms ä-karaka (clockwise) and huringa tua (anti-clockwise) were introduced with rotation. Actual labelling of angles does not appear until Tau 6. In our database, lines are not a focused area of writing until parallel lines are marked (<<) and labelled at Tau 7. It is possible that measuring the lengths of lines may occur at earlier year levels. However, we collected very few pieces of writing concerning measurement. Even with this limitation, the point is still interesting that students are expected to make distinctions between lines and angles at a stage when these themselves have not been formal areas of study. It would seem that students are expected to be aware of and make use of mathematical writing conventions before these are formal areas of mathematical study.

To draw a clock face, students must have those understandings but also be able to draw circles (if only roughly) and be able to place numbers in order around the circle at appropriate distances apart. Unlike rectangles, squares and triangles, circles never appear as an individual shape on a piece of writing in our database. The writing samples only have circles included as one of several shapes together and thus come under 2D shape progression. Writing numbers in sequential order is also important in drawing clock faces. This is something that is learnt in a similar time frame as that of learning to draw clock faces in Tau 2. However, most numeracy teaching in these early years is around base 10. Clock time uses 12 as its base unit for the hours and 60 as its base unit for the minutes. Although students are expected to draw clock faces at a relatively early age, they are not writing about the distinctions between am and pm time until Tau 5. Students respond to very simple questions involving minutes from Tau 3 but do not respond to complex questions involving understanding 60 minutes in an hour until the intermediate years (Tau 7 and Tau 8).

This would suggest that students are made aware of certain features such as angles and lengths of lines in other situations before they become the focus for a description in their own right. This is reinforced by the fact that shapes such as rectangles, squares and triangles also require an awareness of angle size and line length and are introduced from the first year at school.

The exception is that sequencing of number, that also occurs in clock faces, is a focus for descriptions at about the same time. This may be because of the pre-eminence given to mastery of symbolism in mathematics teaching because ultimately mathematical understanding tends to be associated with being able to provide a symbolic description. Therefore, whenever a possibility arose, the use of symbols was reinforced. O’Halloran (2000) suggested that:

the mathematical symbolism contains a complete description of the pattern of the relationship between entities, the visual display connects our physiological perceptions to this reality, and the linguistic discourse functions to provide contextual information for the situation described symbolically and visually. The major reason why mathematical symbolism is generally accorded the highest status by mathematicians is because this is the semiotic through which the solutions to problems are derived. In this respect, the visual display is not only limited in functionality, but also graphs and diagrams are usually only partial descriptions of the complete description encoded in the mathematical symbolic statements. (p. 363)

The sequencing in the progressions was determined from examining the writing samples. The number of samples that fell into each subcategory varied from one to more than 50. For example, in the time progression there is only one sample of kaupae 2a. However, there are 21 samples from kaupae 4b, from three different worksheets done by students in Tau 7 as part of a unit on time. The largest number of pieces of writing in one substage was 61, in narrative progression kaupae 3a. These samples are one-word or short-phrase answers to mathematical questions and came in response to many different worksheets, tasks or questions. Figure 12 provides some examples of these.

Figure 11 Pieces of writing that were classified as narrative kaupae 3a

Grouping stages across topic progressions

In Appendix A it can be seen that some large topics had progressions that are similar across several mathematical writing modes. For example, angles, lines, 2D shapes, rectangle, square, triangle, 3D shapes, rectangular prism, square pyramid, tetrahedra and triangular prism all seem to have the same set of stages in their progressions. There are also similarities in how symbol representations are developed. Whole numbers, fractions, decimals, integers and algebra seem to have a similar set of stages, although not quite as clearly the same as they are with the geometry modes. Table 3 provides an example of a table showing similarities in progressions. It shows the relationship between iconic and symbolic patterns.

Although the stages seem to be similar for both iconic and symbolic patterns, it would be rare for the stages of both types of patterns to be taught at the same year level. The stages for iconic patterns would be learnt at earlier ages than those for symbolic patterns. It may be that having students become used to the writing conventions of iconic patterns means that it is easier to introduce the writing conventions for symbolic patterns. Although examples are only given for either iconic or symbolic for the final two stages, this should not be taken as a distinction between the two patterns. It is more likely to be related to the collection of writing samples as it is easy to conceive that an algebraic equation could be connected to an iconic pattern and a word description connected to an iconic pattern.

Table 2 Patterns progressions

Year-level progressions

The year-level progressions were developed by the teachers. Once the stages of each of the topic progressions were stabilised, examples from each of the stages were provided so that teachers could place them on year-level charts. This was mostly done in the September and October staff meetings. During the principal researcher’s November visit, the year-level progressions were placed on a wall in the staff room and teachers were asked to place remaining samples from stages onto the chart and draw arrows to indicate when they would stop being used. They were also asked to check what had already been included.

Initially, in the September meeting, the teachers worked in three groups to determine which year level they thought that each geometric piece of writing should be focused on. At this time the geometric topic progressions were the most complete and extensive because of the kura’s focus on geometry for the project. The teachers were asked to think about where the ideas should be first taught and at what year level students would stop using them. They were to discuss in their groups where they felt the samples from each stage should go. The samples were then stuck on to a large chart under what they considered to be the most appropriate year level. At this stage, groups of teachers could have placed the same idea at different year levels. If this occurred all the teachers discussed together what they believed was the most appropriate year level or whether the writing began at one level but was revisited for several years.

After the initial placement of the writing samples, the teachers for each year level considered what the kura was expecting them to teach and decided whether they agreed with this. A considerable amount of discussion occurred among the teachers about the most appropriate years for the writing to be taught.

Findings from the year-level progressions

Examining the year-level progressions produces a number of interesting findings. Firstly it shows that the stages in the topic progressions are not linear, as discussed in the previous section. Usually, later stages are introduced at later year levels but this is not always the case and it is these exceptions that indicate that it is not a linear progression. Secondly, an examination shows that a related series of stages across modes would not always be introduced at the same year level. For example, being able to draw iconic patterns begins at Tau 1 while being able to produce symbolic patterns does not occur until Tau 3.

However, other findings also became evident with the year-level progressions acting like a fossil record of mathematical writing. This is not to say that stages record extinct remains but rather that there is a variation in how mathematical writing conventions develop. As a fossil record, what they do show is when mathematical writing conventions are introduced and when they are no longer used in mathematics classrooms. The reasons for the occurrence and disappearance of various stages of the modes are varied. One reason is due to the database being incomplete. However, it is also clear that there is a range of other reasons why a particular stage of a mode disappears. These are discussed in the following paragraphs.

Some mathematical writing conventions, such as tallies, are introduced in the first years of kura and remain unchanged at later year levels. DCUnL4 in Figure 9 shows a tally that was used by a student in Tau 11. This is not dissimilar to the tally used by a Tau 1 student (DSWNMaL2a) or one used by the teacher to keep class points that can be seen in Figure 13.

Figure 12 DSWNMaL2a (left) and classroom points tally (right)

Another reason for the disappearance of stages of modes in topic progressions is because writing conventions are transformed and appear at other stages. For example, once students master clock faces (kaupae 2) in the junior years of the primary school, they are not drawn again in later years. Instead, simplified versions are incorporated into summaries of different versions of giving time (kaupae 3) and from Tau 8 are not used even in this simplified form. This shows clearly that the recording of time information has been transformed into other forms, most commonly numerical either in 12- or 24-hour time. Time itself also stops being a focus of explicit teaching from the junior high school years because it is expected that students have mastered being able to solve problems involving time (kaupae 4) by this stage.

Similar examples can be found in other topic progressions. Kaupae 1a of the symbolic—whole number progression—involves students tracing numbers. Examples of this type of writing were drawn only by students in Tau 0 to 2. Kaupae 1b involves the students drawing an appropriate amount of objects to match the numeral that is given. This would only be expected of students in Tau 0 to 1. At the same time students were also expected to independently produce numerals. The information that is attached to these numerals becomes more complex as students move through the substages of kaupae 2. Table 4 shows each of the substages and provides information about the year levels that they occur in.

Table 3 Substages for whole number kaupae 2 progressions

Each substage provides an extra layer of meaning about numerals and how they can be used. At kaupae 2e, the numerals are still recognisable as being the same as those produced at kaupae 2a, even though they now have more than one digit. However, the descriptions that they provide are very different at each of the substages. Numerals do not disappear over the years of schooling nor are layers of meaning lost by moving through the substages. Rather, it should be considered that the numerals accumulate layers of meaning as students progress through their schooling. Not all of the layers may be actively used by a student when doing mathematics but they are available if needed. Therefore, the various substages and stages of different topic progressions can appear and disappear over the year levels.

As well, some progressions disappear altogether from the year levels because their purpose was to introduce another progression. Iconic pattern progressions become extinct or at least appear less frequently in the high school years. As their primary function may have been to introduce symbolic patterns, once students are competently using them the purpose of drawing patterns becomes redundant. This may be why this progression disappears from the year-level progressions. Iconic pattern progressions, therefore, could be seen as examples of what Herbel-Eisenmann (2002) called “bridging” language. It enables students to talk about patterns whose features are visually more noticeable than those of many symbolic patterns. In this way, students gain ways of describing patterns in writing that then can be transferred to when they describe symbolic patterns.

One of the aims of working with patterns is to be able to provide algebraic equations to describe various patterns, but in order to be able to do this students need to understand the distinct features of patterns. This is done through using writing to focus on small numbers of these features at one time. The various stages of the progressions show the change in focus on the features. Introducing this focusing process by first working with iconic patterns can be thought of as a way of easing students into understanding the importance of these features.

Much harder to understand is the appearance of short-lived progressions that do not seem to be related to other mathematical writing convention progressions. The progression that we have labelled as Technical Drawing would be one example of these. This progression has only two stages, examples of which are given in Figure 14. Both appear (at Tau 7) and disappear (at Tau 11) at the same time.

Figure 13 DGTDRaL1 (left) and DGTDMaL2 (right)

It would seem that they have a short life and connect with nothing else. Although they could be related to perspective drawings of 3D shapes such as rectangular prisms, the pieces of writing that we have do not make such a connection. It is, therefore, unclear why students are expected to learn these mathematical writing conventions.

It may be that it involves students in visualisation that is necessary in other fields of mathematical learning. However, there appears to be some contention about whether focused teaching increases spatial visualisation skills (Ben-Chaim, Lappan, & Houang, 1985). Being able to draw different faces of an irregular 3D shape and to represent it on isometric paper from other 2D representations of these objects requires students to be able to produce and manipulate mental images (Ben-Chaim et al., 1985). Being able to visualise flat, 2D images as having the depth of a 3D object requires students to learn how to make use of the conventions of the depth cues in the representations (Lowrie, 2002). These are Western conventions that take time for students to develop and involve considerable interaction with real-life objects (BenChaim et al., 1985). By the time students reach intermediate school level, it could be assumed that students would have had this interaction and could then take advantage of writing lessons that specifically focused on drawing 3D shapes on paper. The ability to produce mental images becomes essential in working with many mathematical ideas in high school. Like learning about time, it may also have some relevance to being able to deal with outside-school mathematical problems. Being able to read architectural plans is not something that is formally taught in schools but is something that many adults at one time or other have to be able to do. However, the connections are not clear and it may be difficult for some students to understand the point of learning these skills.

Conclusion

The development of progressions was done so that the teachers had an understanding across the kura of how the language attached to mathematical ideas developed. The stages in the topic progressions provide details of how layers of meaning are attached to the different modes. The substages show some of the features of each stage are developed so that students become proficient in describing the particular layer of meaning of the stage. Examples were provided from the time progression to illustrate these points. The similarities between stages of iconic and symbolic patterns illustrated how related modes could also have the same development of layers of meaning. This was not always the case because the three types of graphs each had very different progressions.

The development of the year-level progressions was done by the teachers after discussion with each other. These progressions will be of use to teachers in their planning. However, they also provided information about the topic progressions. For example, they reinforced that the topic progressions should not be considered as linear and that the stages of related progressions are rarely introduced in the same year level. By considering the year-level progressions as a fossil record, it was also possible to understand why some progressions appeared and disappeared at different year levels. Tallies, for example, could be considered as tuatara in that they appeared in the earliest years of school and were still appearing in the same form at the end of high school. Other progressions appeared for a few years but then transformed into other modes of mathematical writing. For example, iconic patterns disappear and are replaced by symbolic patterns, although there are a few years where examples of each would be written. The third type of “fossil” would be those like technical drawing that make a fleeting appearance in the year progressions but do not appear to be related to other progressions either before or after their appearance.

4. Whakamārama (explaining) and parahau (justifying)

Learning how to write whakaahua involved learning the conventions of mathematical modes and this was described in the previous chapter. This chapter looks at how these mathematical writing conventions are integrated in explanations and justifications. Students’ explanations and justifications have received attention by researchers because of their potential to improve the students’ learning of mathematics (see Forman, Larreamendy-Joerns, Stein, & Browns, 1998). The definitions that the teachers had given to whakamārama and parahau were:

	whakamārama (explanation)	Explain the process	How do you get there?
	parahau (justification)	Justify the result	Why is this best?

These have similarities with the definitions given by others. Bicknell (1999), using the work of Thomas (1973), suggested:

[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. (p. 76).

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. (page ref)

All these definitions suggest that whakamārama and parahau are both kinds of mathematical arguments. In regard to solving problems, Meaney (2007) wrote “[a] convincing argument makes a clear connection, using reasoning, between what is known about a problem and the suggested solution” (p. 683). Thus, both whakamārama and parahau are providing mathematical arguments because they outline their reasoning for solving problems, although their focuses are different. From analysing discussion in mathematics classrooms, Krummheuer (1995), using ideas from Toulmin (1969), proposed four components of argumentation: claims; grounds; warrants; and backings. These are outlined in Figure 15.

Claims are assertions of a point of view; in most cases these are the proposed solutions. Grounds are the unchallengeable facts from which the assertions are drawn. Warrants are the pieces of information which join the grounds to the claims, while backings provide the contexts for when the warrants are appropriate. A difference between whakamārama and parahau could be the inclusion of warrants and backings. An explanation would provide some warrants as they show the steps that the writer has gone through to reach their solutions. However, a justification would also need to provide the backings for why the warrants are valid.

Figure 14 Diagram showing the relationship between the components of an argument (adapted from Krummheuer, 1995, p. 245)

Benefits of writing

Both whakamārama and parahau were seen by the teachers in this project as being very valuable for students learning mathematics. In the June 2007 staff meeting, teachers played a place value game that was similar to a game that many of the teachers played with their own students. The game provided a focus for discussions about how to gain the most benefit from having students engage in mathematical activities. The teachers had to write down their strategy for winning and their reasons for why they felt the strategy worked before they played the game again. They could then modify their strategy. Following on from this activity, the teachers had a discussion based around the following questions:

• What benefits do you see for children in answering questions about their strategy use?
• What benefits do you see for teachers in having children answer questions about their strategy use?
• Are there any other benefits in having children write their answers rather than speak them?

The following is a summary of this discussion taken from the meeting notes (6 June 2007):

Benefits for children

• In junior classes, writing about what they are doing in mathematics gets children familiar with having to write about their thinking. Having children write about a strategy rather than just play the game means that some of them become aware that a strategy is required in order to win the game. Although teachers implement games because they can see the mathematical learning that is possible, children may think that it is just luck that helps them win. Making children explicitly aware of their own and others’ strategies can mean that the mathematical learning opportunities are made more accessible to the students.
• The process of having children write about their strategies in mathematics means that children cannot hide but must meet the expectation that they will have a strategy. Having students write about how they got their answers promotes metacognitive evaluation of what they did. They can use these reflections to refine their strategies and increase their learning opportunities. By writing about their strategy, children will have more time to reflect on what they did and why.

Benefits for the teacher

• Having children write about their strategies means that teachers become aware very quickly of who does not have a strategy. This may not always be possible when the strategies are discussed orally.
• Being able to explain and justify their answers requires students to be able to both do the mathematics and also to use the appropriate linguistic resources. Teachers can determine from students’ writing and their participation in the activity whether it is the mathematics or the language that causes students’ problems.

Specific benefits of writing

• Concentrating on writing means that the question can be asked: What do you want to do with language, rather than just maths language?
• Writing enables an activity to be spread over time as the results can be referred to again and again. This is not possible when an oral discussion occurs. Poutama Tau encourages students to talk through their responses. This can be built on so that there is explicit teaching about writing genres. The teacher models answers and strategies. Children can write to express what they are doing and this can be developed into explicit writing of different genres. By having students write about what they have been doing, they can consolidate their learning. Quiet children who may not have participated in a discussion will be encouraged to have a strategy. However, it is recognised that there is a relationship between speaking and writing. Children are more likely to write well if they have had opportunities to talk about what they are doing. This leads to the question of how to ensure that every child has an opportunity to talk about what they are doing.
• It is useful in later years if children have got used to writing in the junior classes.
• Children can be encouraged not just to answer problems, but also to write their own problems.

From this discussion it can be seen that the teachers felt that having students explain their reasoning through writing explanations and justification was important, even though this is not explicitly mentioned. This supports what other researchers have suggested. For example, Moskal and Magone (2000) stated 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. However, there is a need for teachers to expect students to use these genres as well as supporting students to produce them. It is only within the social milieu of the mathematics classrooms that students will be channelled into responding to these expectations. As Yackel (2001) wrote, “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” (p. 14).

Although we had collected some samples of explanations and justifications from the senior classes before this June meeting, it was clear that other teachers took up the challenge of expecting students to write whakamārama and to a lesser degree parahau during the final two terms.

However, having students write whakamārama and parahau was something that teachers were still grappling with over the course of the whole project. By the November 2007 staff meeting, some teachers, such as T7 and T3, were able to describe the strategies they had recently adopted for having their students write explanations and justifications. For other teachers this was still something they wanted to work on in 2008 as the fourth term of the year was not seen as being a good time to start something new. This is discussed in more detail in Chapter 8.

Identifying whakamārama and parahau

The next two sections describe the distinctive features of whakamārama and parahau. We used a linguistic analysis approach to identify these features. There are a number of ways that the features could have been identified. Lankshear and Knobel (2004) suggested a range of questions that could be answered when doing a critical linguistic analysis. These were:

What is the subject matter or topic of the text?

Why might the author have written this text?

Who is the intended audience? How do I know?

What kind of person would find this text unproblematic in terms of their values, beliefs, world views, etc.?

What world view and values does the author hold or appear to hold? How do I know?

What knowledge does the reader need to bring to this text in order to understand it?

Who would feel ‘left out’ in this text, but should logically be included? Is this exclusion a problem? Are there important ‘gaps’ or ‘silences’ or over-generalisations in this text? For example, are different groups talked about as though they constitute one homogeneous group?

Does the author write about a group without including their perspectives, values, beliefs in relation to the things or events being reported?

Who would find that the claims made in this text clash with their own world view or experiences? (p. 342)

Our interest was in the ways that students presented their mathematical ideas so not all of Lankshear and Knobel’s (2004) questions were relevant to the analysis that we wanted to do. These questions raised our awareness of the need to consider how students expressed their Māori identities through their mathematical writing. Mathematics is a Western construct that consequently is overlaid with Western assumptions about knowledge in general. This is manifested in the mathematical register that developed concurrently with the development of the mathematical ideas (Meaney, 2005a; Roberts, 1998). Nonetheless, Burton and Morgan (2000) found that it was possible for mathematicians to individualise their writing to reflect something of their personality and beliefs about mathematics. Meaney (2006a) also found that there were differences in how primary school students expressed themselves that were related to age, gender and the decile level of the schools they attended. However, there were also differences in students’ oral responses according to how the task was expressed (Meaney, 2007). Discussions about who the author was writing for and their purposes for writing are discussed in more detail in Chapter 7. An interesting area for further research will be to investigate the linguistic choices that students make as a way of investigating how they express their Māori identities.

The linguistic analysis used ideas around multiple literacies and how different modes are connected to present mathematical ideas. It therefore drew on the ideas of O’Hallohan (2000), Kress and van Leeuwen (2006), Lemke (2000) and Unsworth (2001). All of these researchers used understandings from Halliday’s (1985) systemic functional grammar to discuss mathematical symbolism, images, scientific multiple literacies and other multiple literacies respectively. Although none was directly relevant to the analysis undertaken with the samples we had, they did provide insights into what may be important features. In particular, these researchers highlighted the importance of the positioning of the visual images, the way the main participants (including mathematical objects) are emphasised and how coherence is maintained. For students in Johanning’s (2000) study, drawing pictures, graphs or a table was the first thing they did when they began solving a problem. However, they rarely saw this as a problem-solving strategy, but, rather, as a way of organising information. It was therefore useful to concentrate on explanations and justifications that used combinations of modes. In the discussion of the following kaupae, or stages, these ideas will be discussed in regard to systemic functional grammar’s ideas about field, tenor and mode in relationship to the context of situation in which the pieces of writing were developed.

Whakamārama

The primary purpose of whakamārama was to provide the reasoning for getting a particular answer, usually in such a way that another person would be able to reproduce it or that the author could remember what they had done. Explanations most often used a combination of modes. However, there were also 21 examples of explanations that only used symbols. There were also two pieces of writing that were classified as whakamārama and only used sentences. One of these can be seen in Figure 16.

Figure 15 ENArL3

The samples of whakamārama that were classified as combinations used a range of different modes. Most often they used some symbols and/or some words. However, there was also a large number of explanations that focused on geometrical ideas and thus also contained diagrams of shapes. As other chapters use examples from geometry and probability, in this chapter the examples will be drawn from other topics.

The following sections look at the simplest to the most complex explanations in our database. Differences in complexity are to do with how the different elements are combined to fulfil particular needs. We have not used whether or not the piece of writing was transparent or opaque in its explanation in the way that we determined the stages. This is because we saw the transparency as being related to the audience requirements rather than to the complexity of the writing.

It is not expected that students should be aiming to write the most complex explanation every time and so a simpler explanation should not be considered to be of a lesser quality. A discussion about judging the quality of an explanation or justification comes after the sections on whakamārama and parahau.

Figure 16 ECUnL1

Kaupae 1

I used eighteen 2 times

I worked out 5 lots of 16

I worked out 3 lots of 10 and 3 lots of 4 I worked out 4 lots of 10 and 4 lots of 3

I worked out 2 lots of 19

I worked out 5 lots of 20.

Figure 17 provides an explanation of some multiplications. Single-step multiplications would normally be classified as whakaahua as they just describe a basic fact. However, in this example, the student gives a brief explanation of how they worked out the answers but does not provide the particular strategy they used. Given that the Poutama Tau professional development project suggests that teachers have students explain how they worked out their answer, it is quite likely this is what the teacher had expected the student to provide. Instead, this student has mostly simply stated in words that they multiplied the two numbers together.

1. I whakamahi au i ngā teka[u] mā waru e 2 ngā wā (I used eighteen 2 times)

However, answers 3 and 4 are somewhat different in that they break down the multiplication into two parts: first the tens and then the ones. This shows a two-step process and moves the piece of writing from being classified as whakaahua to whakamārama. There are no warrants that would justify why it is useful to break 14 or 13 into their composite tens and ones.

	3. I mahi au 3 ngā tekau me 3 ngā 4	(I worked out 3 lots of 10 and 3 lots of 4)
	4. I mahi au e 4 ngā 10 ara i mahi ara 4 ngā 3	(I worked out 4 lots of 10 as well as worked out 4 lots of 3)

Explanations at this level are the simplest because the arrangement of information does not require the reader to integrate different representations simultaneously. Although mathematics should not always be read from left to right (Meaney, 2005a), this is the case for the piece of writing in Figure 17. Each answer is numbered and written down the page, then the multiplication operation with its constituent parts (number, multiplication sign, number) is provided, followed by the main relationship indicator as a process (=) and then the result. Each of these multiplications is then followed by its explanation written mostly in words but also with at least one numeral. The length of the word explanation is shortest for the final two statements and provides a repetition of the symbolic multiplication without any elaboration.

Using a systemic functional grammar approach to analyse the pieces of writing is to look for equivalents of field, tenor and mode of this piece of writing. The field is clearly that of doing simple multiplications on paper and this is recognised by the placement of the equation on the left-hand side of the paper. It was common in our samples to find the main focus of mathematical writing to be on the left with elaborations on the right. Lankshear and Knobel (2004) suggested that items on the left of the page tend to be more salient in terms of attention and importance than items placed on the right. This is because our writing system generally uses a left-to-right movement so that items placed on the left would be considered first.

The tenor of a text is reflected in the use of personal pronouns and the mood of the verb used. The responses in Figure 17 follow closely the symbolic calculations rather than how they would be said out loud in te reo Māori. Thus the mood of the verbs could be said to be “mathematicised” and to have removed a recognition of the author.

On the other hand there are repeated references to the author, au (I), in the second part of the response. The numerical equation, as is typical of mathematics, makes no reference to humans and in so doing represents itself as an unchanging fact (Burton & Morgan, 2000). However, the use of au suggests that this should be considered a personal piece of writing. It was the writer who was working out the answer and was responsible for its accuracy. Yet the clarity of the piece suggests that it may have been written with an expectation that it would be read by someone else or at least would be read by the author again at a later date. A similar example of writing is provided in Figure 18, but in this case the lack of transparency in displaying the meaning makes it unlikely that it was written for someone else to read.

Figure 17 ESUnL1

In the example given in Figure 18, the student is showing that doubling a multiplier gives double the product. Although the way the information is set out in Figure 18 is similar to that in Figure 17, there are fewer words and, from answer 5, making opaque the explanations of how the responses are gained. The reader must look across the page to locate the relevant information. This is much less a tidy text (Ernest, 2007) and is more likely to have been written for the writer than for an external reader, unless the reader has enough background information to fill in the missing links.

The mode of Figure 17 involves using numerical equations followed by sentence explanations. Cohesion is achieved through the repetition of the numbers in the sentences. As well, the pattern of the sentences is similar for all the responses so that visually it clues the reader into not expecting any major differences in the types of explanations.

Figure 18 ECUnL3

Kaupae 2

The words in the middle of Figure 19 state “[f]ind the information for Gauss 1–16 boxes of 4”. The 4 x 4 grid under the word Gauss, at the top, is repeated with grid lines in the bottom left corner. This grid shows how the numbers are arranged so that each column and row adds up to 34, thus providing another method for determining the sum of the numbers. In the top right corner, is a 3 x 3 grid showing how the numbers 3 to 11 can be arranged in columns (or rows) to equal 21. It is a simpler example of using the grids.

In the bottom right corner, eight combinations of 17 are shown by using connecting lines between numbers. Underneath this is “= 136” which is the total sum of the numbers 1 to 16. However, another operation is then connected to 136 by the addition of ÷ 4. The result of this is given as = 34. There is no discussion about how this would be an effective method for addition. If there had been, it would have been classified as a parahau.

The piece of writing in Figure 19 is typical of kaupae 2 as it requires the reader to integrate a variety of different visual arrangements. It uses grids (3 x 3 and 4 x 4), sequences of numbers, connecting lines between numbers and mathematical operations using = and ÷ signs, as well as words. However, it is showing the same process in a number of different ways and thus is not as complicated as a piece of writing that integrates a variety of new information.

How a reader would make sense of pieces of writing at this stage would depend upon their previous understanding. It is not transparent and therefore it is unlikely that it was written for an audience unfamiliar with this topic. Although Gauss’ name is underlined, it is unlikely that someone who did not know Gauss’ strategy would have any entry points into what is being explained. Thirty-four is highlighted by the writer by surrounding it with a box. However, its connection to anything else, except 136 ÷ 4, on the page is not transparent. The sentences do not give sufficient detail to fill in the missing gaps in information.

Using ideas from systemic functional grammar, it is possible to see that the field has something to do with sequential numbers (although their arrangement in the grids does not make this particularly clear). Both the underlining of Gauss’ name and the surrounding of 34 in a box suggest that these are the main participants. The lack of transparency in the explanation suggests strongly that it was written by the author for him/herself or for someone else who was already familiar with Gauss. This lack of transparency can be considered as a way of understanding the tenor of the context of situation. The mode is represented by the use of sequences of numbers, grids containing numbers, connecting lines, mathematical operations and words. Although there is repetition of the numbers, the cohesion of the text is not clear. It is difficult to recognise the new information and how it builds on what was previously provided. In fact it is difficult to know whether the writer even expected the audience to read from top to bottom or from left to right.

Figure 19 ECTeWL3

Kaupae 3

With the problem given at the top of the page, it is clear that Figure 20 provides the workings about this problem. On the left-hand side of the page, going from top to bottom are diagrams representing the coloured blocks. The copying and scanning have removed the colour so the diagrams are not as clear as they might be. However, the first line of diagrams has been disregarded and the second line is the appropriate one. The last line of diagrams shows a more complex set of combinations of blocks. The sentences written on the right-hand side provide an explanation of the pattern. This is the same sort of arrangement as those seen in Figures 17 and 18 where the main focus is on the left side of the page and the written explanation is on the right. However, underneath this pattern is a sequence of numbers 4, 8, 16, 30 written both horizontally and vertically on various places on the page. The additions given in a numbered list also refer to this sequence and appear to continue it further. However, there is not enough detail to know where it comes from and indeed what its relationship is to the list of additions.

The field is one of explaining patterns. This is realised through the inclusion of a visual representation of the first pattern and a repetition of the sequence of numbers forming the second pattern. Repetition appears to be a common way in our mathematical samples to emphasise what is being done, as well as providing cohesion. In this piece of writing, the tenor seems to change halfway through. The clarity of the explanation of the first pattern would suggest that it may have been written for someone else. However, the explanation of the second pattern based on the sequence of numbers is impossible for a reader to follow unless they were present when the activity was undertaken. The mode shows that the student used three different mathematical modes of writing: diagrams; word explanations; and symbolic explanations. Cohesion is achieved by repetition. In the first part it is the drawing of and the writing about pouaka (box). In the second part it is through the repetition of the sequence of numbers. New information appears to the left of or below previous information.

Kaupae 4

Figure 21 is really a series of explanations that are related to simplifying algebraic expressions and equations. These are the bulk of what appears on the page. Each one is separated by a blank space and begins on the left-hand side of the page, except for (10) that is written on the right-hand side of the page. Some algebraic equations are dealt with in one line, whilst others are explained over several lines. They may or may not have accompanying word explanations. As well, there is a general explanation of some of the rules for doing algebraic equations in the right-hand margin. Each equation could be thought of as belonging to a lower stage. However, their close proximity, the addition of repetitive features such as word explanations and arrows, as well as the set of general rules in the left-hand margin, suggest that this should be considered as one explanation. Consequently, it is up to the reader to make sense of all the parts and how they are related. This is what makes this such a complex explanation.

Figure 20 ECMa1L4

Therefore, the field is that of doing algebra, but it is also clear that this student is only at an early stage of learning how to make use of it, hence the inclusion of the arrows and written explanations. The clarity of the explanation suggests that it is for someone else to read or for the author to reread at a later date. With an audience for this writing, there is a need to be clear in providing details. The mode involves the students using algebra and words for providing their explanation. They have also used intermediate or bridging mathematics language (Herbel-Eisenmann, 2002) in their use of arrows to show how the expansion of the brackets is done. Cohesion is achieved through the repetition of the variables, although in different equations. This is one of the aspects that makes it a complex explanation in that there is a need to see both the similarities and the differences in how each equation is solved.

Summary

Whakamārama provide some insight into the students’ thinking as they give the information, or the warrants as described by Krummheuer (1995) that connect the problem with its solution. Explanations that use a combination of mathematical modes are often not simple pieces of writing to read. This is because they do not always set up the connections between the different formats of information clearly. Even at the higher stages of these explanations, it is not always easy to follow the reasoning unless there are word descriptions to accompany the explanations. These word descriptions become even more important in the production of parahau.

Parahau

Writing parahau involves the student justifying their answers. This involves the student providing a backing that contains details for the warrants. Knowing when the warrants are valid and why, would be the information that would convert an explanation to a justification. There were only eight of these in the database. The simplest of these was one written in words as a response to some questions about reflection.

The worksheet shown in Figure 22 required the student to draw both the object and its reflection. By having the heading at the top of the page, followed by the diagrams, it is clear that students are expected to use what they have done to answer the questions in words at the bottom. The questions could be interpreted as asking students to justify the placement of their reflections by mentioning the distance from the mirror line. Instead, the student has concentrated on making sure that the reflection is exactly the same as the original diagram. However, the mention of orite (same) would still provide a backing for the warrant, about using the mirror line as the focus for the reflection.

Figure 21 JCTeA

The field for this parahau is that of reflections. This is emphasised by the heading at the top of the page, followed the diagrams and the written questions, all of which repeat the theme of reflection. The tenor of the piece of writing is that of a student responding to a teacher’s request for them to show what they know about reflection, and as such is an assessment piece. The student, therefore, is not only answering questions, but also having to determine what the teacher wants to know from the questions. The student cannot simply respond in any way he or she likes; there are obligations when it is an assessment piece for them to tell to the best of their ability exactly what the teacher wants to know. The mode includes using diagrams and sentences. Cohesion is through the different representations of reflection and the repetition in the different forms enables the focus to be maintained.

The reasoning that is used to justify the way the reflections is done is procedural in that it refers to whether the process is correct rather than if the process is the appropriate one.

Figure 23 provides an example of an implicit justification that is based on reference to a set of mathematical facts. The two images in Figure 23 came from a student’s mathematics book and were on sequential pages. The first image is a description that provides labels for each side of a right-angled triangle. It also provides a visual description of the relationship for the sine, cosine and tangent rules. Below these sets of triangles, are equations given in words that state what the relationship is. In the second image, students are working out the length of the missing side and also the size of the angle. To work out the length the student has used Paitakōrahi (Pythagoras’) rule and to work out the size of the angle, the whenu (cosine) rule. An implicit justification for using the cosine rule is given by the repetition of the triangles showing the different rules from the previous page. This is the backing that supports the warrant for using the cosine rule. It has been written in the left-hand margin of the page. A reader must be able to read this combination. Without specific questions, they need to have appropriate knowledge to follow the explanations for solving the problems and to recognise the triangles on the left-hand side as the implicit justification.

Figure 22 DCUnL2 (left) and JCUn (right)

The field is one of answering questions about right-angle triangles, even when they are dressed up to be questions about a tree. The tenor is most likely to be the writer writing for themselves. The margin notes, as well as serving as a backing, are also a reminder of why a particular approach was taken for when the writer rereads it later. The mode involves a diagram of a tree which has been overlaid with a triangle. The use of the right-angle sign is supposed to indicate a right angle, even though the rough drawing does not suggest this. The drawings of the triangles are different from the right-angle triangle because they serve as information summaries.

A more explicit justification can be seen in Figure 24. This example was drawn as a result of students being challenged by their teacher to produce the dimensions for a cylinder that would hold one litre. The students had to justify what they did. A number of different approaches were taken.

Figure 23 JCKa1

The approach that this student took was to decide that the whitianga (diameter) was 10 cm and thus the pūtoro (radius) was 5 cm. She then determined the area of the circle at the top of the cylinder. Using this area and that one litre is equivalent to 1,000 cm³, she calculated the height of the cylinder. All the calculations are done using symbols, numbers and algebraic variables. The justifications for what was done are provided in sentences or phrases connected to the calculations with lines. Not every decision is provided with a justification, such as why the student began with a diameter of 10 cm. However, the majority of decisions are provided with a reason. Although the calculations proceed from the top to the bottom of the page in a logical order, the addition of justifications disturbs this flow as it requires the reader to follow the lines upwards and downwards as well as sideways. However, as was the case with many of the whakamārama, most of the justifications written in sentences appear on the left-hand side of the page.

The field is that of a student using their mathematical knowledge to solve the problem of working out the dimensions of a cylinder for a specific volume. The key aspects of the calculation are often labelled in the justifications. Whitianga and pūtoro are both labelled in the main part of the writing and pūtoro appears in the equation for the area of a circle as “p” in the following line. Although the piece of writing represents the working notes of the student, the addition of the justification suggests that the teacher will read it and expect the reasons to be made explicit about the decisions that were taken during the solving. The nonlinear addition of the justifications shows that this was not a polished piece of writing that would have a wider audience. The mathematical writing modes were words, sentences, algebraic calculations and a 2D representation of a cylinder. Cohesion is achieved by the repetition of the main participants in the calculations, either as labels or numbers in the calculations. The justifications also made quite explicit what the connection is between these components.

Summary

Justifications are different from explanations because they do not appear as part of “doing mathematics”. Although everyone would have reasons for what they do in mathematics, it is only when they make them explicit with their workings that they would become part of a piece of writing. Their addition to mathematical explanations means that they often appear on the margins (and in the case of Figure 23, literally in the margins) around the explanations. This is especially the case when they are added to workings rather than to a polished piece of writing that could be expected to have a wide audience. It may be possible that a polished piece of writing may find other ways to incorporate the justification, but this did not occur in the samples that we had collected. Generally, justifications are provided through providing words on the right-hand side of the page with the calculations on the left.

Judgements about the quality of mathematical writing

For teachers to improve the quantity and quality of students’ writing, then, there was a need not just to understand the features of explanations and justifications, but also to have a clear understanding of what constituted a “good piece of writing”. Identifying the features of an exceptional versus an average or unacceptable piece of mathematical writing is something that is rarely discussed in curriculum documents such as the National Council of Teachers of Mathematics’ (2000) Standards. Instead, there seemed to be an assumption that teachers and their students were already aware of the criteria for judging mathematical writing. However, this is rarely the case. As one of Doerr and Chandler-Olcott’s (in press) teachers stated, “there was no discussion at all about writing, what makes it good, what makes it acceptable, and what makes it mathematically correct”.

It is not a simple process of identifying those features that make up a “good” piece as opposed to the features of a “bad” piece. Many factors influenced what was being written as well as whether the pieces of writing conveyed their intended meaning adequately or extremely well. It is also known that demographic characteristics such as gender and ethnicity may influence how students choose to express themselves mathematically (Meaney, 2005b, 2006a). For example, Meaney (2005b) found that senior high school students embedded their algebraic responses within a narrative depending upon whether they answered correctly, their gender and the decile level of the school they attended, as well as on the actual question asked. There is therefore a need to identify some of the general issues that would be addressed in a piece of writing before considering how adequately they were addressed.

After analysing the texts written by students aged between seven and 13, Wilkinson, Barnsley, Hanna, and Swan (1980) proposed a model for assessing students’ development in writing. It had four components that all contributed to the quality of a piece of writing. These were:

• Cognitive: the writer’s awareness of the world: one’s ability to describe, interpret, generalise, and speculate
• Affective: the writer’s awareness of emotions and feelings of self and other people, including the reader and one’s environment and awareness of reality
• Moral: the writer’s awareness of a value system, attitudes, and judgements
• Stylistic: the writer’s awareness of syntax (the way words are organised), verbal competence, text organisation, cohesion, awareness of reader, and appropriateness of text. (Winch, Johnston, March, Ljungdahl, & Holliday, 2004, pp. 182–183)

The model is interesting because it places value not just on the stylistic features but also on what is being written about, as well as the writer’s awareness of their own and others’ responses to what is going on. However, all of these features would not be relevant to considering what constituted a “good piece of mathematical writing”. This is because writing in mathematics occurs to meet specific needs. As Burton and Morgan (2000) stated, “[t]he language used in mathematical practices, both in and out of school, shapes the ways of being a mathematician and the conceptions of the nature of mathematical knowledge and learning that are possible within those practices” (p. 445).

The work Doerr and Chandler-Olcott (in press) did with middle school teachers of mathematics also encompassed considering what were the features of good mathematical writing. The features these teachers identified are given in Table 5.

Table 4 Teacher-generated description of good mathematical writing (from Doerr & Chandler-Olcott, in press)

Characteristics of good math writing Contains examples/drawings Uses math vocabulary Restates the question Answers the question Is edited Responses are organized/sequential Explains examples Includes formulas where appropriate

Labels diagrams, examples, and numbers Addresses all parts of the question Addresses the key concepts Is clear and legible Has complete sentences and appropriate grammar

This list is quite explicit. However, it could well be that if the word “math” were replaced with another subject area such as social studies the list would not need to change in any particular way. There is little in this list that really seems to suggest that good mathematical writing has specific features connected to the writing being about mathematics.

Our teachers had also commented on students’ poor use of grammar when writing in mathematics. T5 had found it hard to let students write in his mathematics class because he was conscious of how much their reo Māori was influenced by English. He therefore felt that having them write might reinforce this inappropriate use of te reo Māori. T2 also described her students’ poor writing about probability:

It is something, the subject of probability, what they do is more or less explain what they saw from the data and what I had seen was the writing was erratic. They wanted to put every word they could think of on paper and who cares about grammar. (T2, Interview November 2007)

So the stylistic features that were identified in Table 5 were also issues for the teachers in our research. This is discussed in more detail in the next chapter.

What we did and what we found out

In the staff meeting in September 2007, the teachers discussed the question of what constituted good mathematical writing. Each teacher examined a group of students’ writing from their own class. They then shared their answers to the following questions:

• What is a good piece of writing?
• What is a poor piece of writing?
• What are the features that make it a good piece of writing?
• What strategies can be used to improve students’ writing?

T9 stated that he had difficulty determining a good piece. Sometimes the mathematics was better, sometimes the mathematical diagrams used for the explanation were better, sometimes the measurements were added (mm squared or not), and sometimes the written explanations were better although the maths was not as detailed as some of the others. In the samples he looked at, all of the students came up with the right answer. Four examples of this set are shown in Figure 25. The original activity had required the students to do the following:

• [Write] for someone who doesn’t know how to do it.
• Write why you should do it like that.
• At the end write anything else you know about Pythagoras.

Figure 24 Samples of T9’s class’s work on Pythagoras

 

T9 felt it was difficult deciding what was a quality piece of writing because there was a need to consider the student’s:

• written explanation
• diagrammatical representation
• mathematical accuracy using appropriate symbols and measurements.

Although this list is simpler than that provided by Doerr and Chandler-Olcott’s (in press) teachers, it is connected explicitly to the mathematical nature of the writing. It is possible to use these ideas to redesign the model of Wilkinson et al. (1980), so that it is more appropriate for mathematical writing. If this were done, then the components could be:

• mathematical: the writer’s awareness of the mathematical ideas and how they relate to what is being discussed
• integration of modes: the writer’s awareness of how to make use of a range of different modes to increase the clarity of the ideas they were presenting
• stylistic: the writer’s awareness of mathematical writing conventions in a similar way to the stylistic awareness described by Wilkinson et al. (1980).

Students who were good in one area of writing about mathematics were not necessarily good in another. To improve this group of students’ mathematical descriptions and explanations, then, all the different needs would have to be catered for. So some students would need support to ensure that they provided appropriate measurements and symbols. For others, it would be looking at the vocabulary and discussing more appropriate choices. For others, it would be discussing with them what had gone on in their head that had not been quite right. This is a mathematical issue that needed to be dealt with. So the features of a good piece of writing about Pythagoras’ theorem must have the mathematics correct, must use diagrams and text clearly and concisely and must be able to integrate these into a coherent whole.

Some of the teachers felt that sometimes there was a sense that a student knew more than they were able to put down on paper. However, if the students were to use mathematical concepts in the next stage of their learning their justifications of these concepts need to be clearer. One of the reasons for concentrating on writing is that it is easy to see when students are able to show their ideas clearly because it is permanent.

In the final interviews, the issue of a quality piece of writing and how to achieve it was raised. T8, who was in charge of the junior section of the school, felt that this question needed more discussion, especially in regard to how to improve all students’ writing in mathematics. She felt that on the whole the good things that happen in the writing lessons in the language classes were not necessarily being transferred over to the mathematics class. Teachers were too easily satisfied with what their students produced as mathematical writing. With the project continuing into 2008, she felt that this would be worth spending more time on.

Given that the discussion about what was a “good piece of mathematical writing” did not include assessing students’ awareness of their own or others’ cognitive, affective or moral understanding of the world as was included in the Wilkinson et al. (1980) model, then this may be an appropriate starting place for further considerations.

Conclusion

Whakamārama and parahau were seen by the teachers as being an important part of the writing that students should be doing in mathematics. They saw them as having benefits to the students in supporting them to reflect on what they were learning. They also benefited the teachers because these types of writing would help them to understand their students’ thinking. However, the samples we had of explanations and justifications that combined different mathematical modes would not necessarily be easy to teach. This is because the ways they arranged the different types of information were not linear. The easiest to follow were those that used lines or arrows to link the different components. The clarity in the writing was linked to the tenor of the context of situation. If the writing was produced for a wide audience, it was more likely to be clearer. The exception would be if the writer anticipated referring to their own writing at a later date. If the writer was writing for someone who was aware of the problem and already had insights into how the solution could be achieved, then there was often a lot of information left out.

Explanations were those that provided information about how the problem was linked to its solution. Justifications provided information about the reasons for why they used a particular solution strategy. Some justifications were very implicit and a reader had to have a considerable amount of detail filled in in order to understand why a particular solution was adopted.

The complexity of relating the purpose of the writing to the audience can be related to the teachers’ beliefs about what was a quality piece of writing. These teachers felt it was important for the students to be confident in all the mathematical writing modes. They also realised that difficulties with how students presented their ideas could be related to their lack of understanding of mathematical concepts rather than mathematical writing skills. However, ideas about what constitutes a “good piece of mathematical writing” need to be explored further.

Web editor’s note: The following is the second part of this report. The PDF version is available here

1. Ways to improve mathematical writing

The second focus for this project was to explore the ways that students were supported to produce appropriate mathematical genres. This was an essential part of the project because:

the teacher plays an important role in selecting writing tasks for students and in framing them in ways that attend to audience, purpose, and genre. The teacher also plays a role in responding to students’ work, especially that of students who are struggling with written expression, in ways that support students in achieving greater clarity and more coherence. (Doerr & Chandler-Olcott, in press)

In considering research on writing in general, Ivanic (2004) suggested that “[t]he ways in which people talk about writing and learning to write, and the actions they take as learners, teachers and assessors, are instantiations of discourses of writing and learning to write” (p. 220). She went on to state that these discourses revolved around beliefs about language, writing, learning to write, approaches to the teaching of writing and approaches to the assessment of writing. Given this complexity in approaches to writing, it is not surprising that it is not always clear to teachers what they themselves do to support writing and what other ways could be utilised that could be more effective.

In investigating students’ writing about mathematics, Morgan (1998) felt that there was a general lack of knowledge about language and language teaching. Consequently, 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.

In mathematical learning experiences, manipulating objects has been seen as a valuable way for students to gain understanding of mathematical concepts. This can also be related to research in art education. Pelland (1982) found that students who were able to handle an object (half an artichoke) were deemed by professional artists to produce better drawings than those students who were only able to look at an object.

Writing is often introduced to record experiences about the manipulation of ideas (see Burns, 2005) and in so doing supports the development of the ideas from the concrete to the abstract. Figure 26 suggests a development of the ideas about shapes that move from illustrating where shapes can be found in the environment to using diagrams to show the relationship between a net and its solid. The drawing of the triangle with its measurements may have been copied from using a concrete triangle, but was more likely to be constructed using written instructions. At every stage, the markings on the paper form an iconic representation that has some semblance of the actual object they are representing (Roth, 2001). However, as the ideas about shapes develop, the immediate relationship to a concrete item in front of a student becomes less important. This movement can be considered as another way of moving children from everyday language to official mathematics language (Herbel-Eisenmann, 2002).
Figure 1 Representations of shapes where the relationship to concrete materials becomes less obvious

As the forms of writing progress, the marks on the paper become more abstract and the relationship to actual manipulation of concrete objects less transparent. The final form in these progressions means that students are able to manipulate abstract concepts without the need for concrete materials at all.

One of the activities used in the kura that supported the movement from concrete to abstract symbolism was kanikani pāngarau (mathematical dancing) (Figure 27). This activity was taken from the New Zealand television programme, Toro Pikopiko E!, and initiated by T4. In this activity students learnt a series of movements for each of the numbers from 0 to 10. They also learn symbols for the four operations (+, –, x and ÷). Students were given problems through movement by the teacher or a student and asked to provided an answer also using movements. As students became better at this, they were asked to write the problem and solution before giving their physical response.
Figure 2 Kanikani pāngarau

We identified a number of teaching strategies that were used by teachers to increase the quantity and quality of students’ mathematical writing. This was done through analysing videos of teachers’ lessons as well as discussing with teachers what they had done. The data, therefore, included videos, interviews, meeting notes, students’ writing samples and photos from classrooms. Our results are not only useful to the teachers at this kura but could also be valuable to teachers at other kura as well as in mainstream schools anywhere in the world.

Our analysis has shown that it is impossible to separate writing from speaking, reading and listening. In our last project, the teachers had been most supportive of strategies that “encourage students to move between modes of expression such as speaking to writing” (Fairhall, Trinick, & Meaney, 2007, page ref). More often than not, the teachers used and expected the students to use all four language skills. Sometimes writing was used to support speaking, while at other times speaking and listening were used to support writing. At the meeting in August 2006, this relationship was described in this way:

1. you [the teacher] are talking, thinking, and writing
2. children are talking and thinking and you are writing
3. children use written language to present information.

The students were engaged in a mathematical activity while they were learning to use all four forms of communication fluently. Depending on the context and the amount of support that is provided to the students, the language activity could fit into any of the four stages of the mathematics register acquisition model (MRA). The following sections discuss the strategies according to the MRA model as well as “acts of writing”. Whereas the MRA model describes the strategies teachers use to support students acquiring mathematical writing, acts of writing refers to different types of writing processes that students had to integrate.

The mathematics register acquisition model

We analysed the teaching strategies by considering them in regard to the MRA model (Meaney, 2006b). This model was used in the previous TLRI project to identify the strategies that teachers were using to support mathematical register acquisition (Fairhall et al., 2007). It divides the acquisition of mathematical language, including written genres, into four stages, from Noticing to Output. These stages are shown in Table 6.

Table 1 Mathematics acquisition model

TAUMATA	WHAKAMĀRAMATANGA
KITENGA NOTICING	Ka kitekite i ngā kupu me ngā kīanga hōu me ako. Ka kitekite i ngā wā e kōrerotia ai. Taka huirangi ai te kōrero i ngā kupu me ngā kīanga hōu.	Students have to notice that there is new language to be learnt and when it is used by others. With prompting by others, students will use the new terms and expressions.
AKORANGA INTAKE	Ka kōrero i ngā kupu me ngā kīanga hōu i ngā āhuatanga rerekē kia akoako pai ai i ngā momo wā me kōrero.	Students start using the terms in a variety of situations. Feedback, both positive and negative, helps them to refine their understanding of when and how to use the terms and expressions.
TAUNGA INTEGRATION	Ka rite te kōrero i ngā kupu me ngā kīanga hōu.	Students will use these terms consistently except when the situation is challenging and they may revert back to simpler terms.
PUTANGA OUTPUT	He wāhanga pūmau ngā kupu me ngā kīanga o te reo tātaitai o te ākonga, ā, ka kōrerotia i ngā wā e tika ana.	Students are using the terms fluently even in the most demanding situations.

The four stages of the MRA model involve the teacher in gradually loosening control of the “what” and “how” in students’ use of mathematical language. In the initial stages, the teachers very much restrict students’ options in regard to terms and grammatical expressions as well the situations in which they are used. On the other hand, the final two stages provide students with increasing control over when and how they discuss their mathematical ideas. These stages have similarities with the three stages of the model for gradual release of responsibility that Doerr and Chandler-Olcott (in press) described for supporting students to become mathematical writers.

The 2005–6 TLRI project found that teaching strategies from each of the MRA stages were present in most lessons, although there did seem to be a relationship to the teaching of the mathematical concept. When a new topic was being introduced, teachers were more likely to use strategies related to the earlier stages of the model. At the end of a unit of work, teachers were more likely to use strategies from the last two stages of the model. Teachers used a range of strategies at each of these stages. Although all seemed to be useful to some degree in supporting students’ acquisition of the mathematics register, the teachers valued those that moved students towards being more reflective about their learning.

Acts of writing

In describing the strategies from the four stages, there is also a need to consider the acts of writing that the teachers engage students in. These are the processes that make up writing. We have labelled these acts of writing as physical, superficial and deep.

Walshe, March, and Jenson (1986) describe four parts to the physical act of writing that they felt supported the writing process. These were:

Handling—the physical manipulation of pen or pencil on a page; the computer keyboard and use of the mouse

Depicting—handwriting, spelling, punctuation

Scrutinising—the constant reading back before writing on

Restating—the so-called ‘shaping at the point of utterance’, which is really our earliest form of editing, the editing of inner speech. (p. 165)

Apart from these physical acts, Winch et al. (2004) also identified part of the writing process that was to do with revision, as a consequence of reflecting on a draft and then refining it. They stated that:

A sensitive teacher can lift the quality of thinking to higher levels during a writing activity through emphasising quality preparation and, once a draft is achieved, the limitless potential for pondering, cutting, extending, putting aside, returning, revising again, and so on until it is right. (p. 172)

However, our separation of the acts of writing was different. The actual physical control of writing implements and learning of the conventional mathematical terms and expressions were considered to be different from the editing and revision stages. The initial editing stage of checking the equivalent of spelling and punctuation in mathematics, such as learning more efficient ways of setting out working, was labelled as superficial acts of writing. This was not to suggest that they were unimportant, but rather to suggest that they did not lead to a revision of the thinking process that deeper acts of writing were able to do. Our final acts of writing were these deeper revision processes that contributed not just to improving students’ pieces of writing but actually to their thinking mathematically. For this to happen, Winch et al. (2004) stated that “time and opportunity are given to write without undue constraint” (p.171).

Physical acts of writing

These are the acts that reproduce conventional mathematical diagrams, symbols and so on. In the database there are countless examples of students who reversed numbers so that 3 was written backwards and 18 became 81. There were also other instances of students struggling to replicate conventional mathematical writing. For example, Figure 28 shows a student’s drawing of a square pyramid shape that accompanied a given diagram of a net.
Figure 3 Student drawing of a square pyramid

The diagram is not drawn conventionally as the middle edge is not lower than the two side edges. In looking at different ways a cube can be represented, Kress and van Leeuwen (2006) showed how different representations are all equally valid. However, they provide different information about both the object and how the drawer perceives the object. The conventional drawing of 3D shapes rarely shows what the drawer sees but rather shows what the drawer knows about the shape. The lack of perspective in Figure 28 has meant that it is difficult to know whether the diagram is supposed to be that of square pyramid or that of a tetrahedron, thus making the meaning the diagram is trying to convey difficult to interpret. Therefore, in mathematics, the ability to reproduce conventional mathematics is necessary if it is important that others are able to gain a specific intended meaning.

For students to be able to use writing in mathematics to support their thinking, then, it is valuable for them to have automated as far as possible the physical acts of mathematical writing. Cognitive approaches to writing have suggested that:

If young writers have to devote large amounts of working memory to the control of lower-level processes such as handwriting, they may have little working memory capacity left for higher-level processes such as the generation of ideas, vocabulary selection, monitoring the progress of mental plans and revising text against these plans. (Medwell & Wray, 2007, p. 12)

Learning how to replicate the conventional mathematical writing modes would come into the whakaahua genre lessons. Teachers did spend time providing students with activities that helped them recognise the essential features of the mathematical writing they were reproducing. For example, in the September staff meeting, T8 related how she and T1 and T3 had taught the students a series of sentence structures for mathematics. They all concentrated on these sentence structures for three weeks and the children were still using them in class. T1 shared resources with T8 and T8’s students were “blown away” that someone else in the kura was doing the same topic. T8 believed that the students often thought they were the only ones having to learn a particular mathematical topic.

Superficial acts of writing

Another act of writing that teachers provided lessons on was how to clarify meaning through editing. Sometimes students could reproduce the conventional mathematical writing but its meaning was not always easily interpretable because they failed to structure their writing clearly.
For example, T2 in the November staff meeting stated:

T2: The rest of Year 8/9, actually I don’t know if you’ve noticed it too . . . when kids think they have the right tau mahi (level of work) that’s what they do for [their lessons in] te reo writing, [but in their mathematical] writing they’ve omitted all grammatical things like full stops and capital letters and everything is just a word drrrrrrrrrr like this to explain what they mean . . . To put it on paper, everything, you might as well kiss all te reo grammar out of it because that is what they will show. Very erratic. And yet, there was something in there that was meaningful so I suppose what I’m saying is, is . . .

T9: Is it our job to fix it up?

T2: Is it our job to fix it up when it comes to something that uses common sense, you know.

T9: Definitely, we have no choice, we are a language school. We have no choice. If you are a language school you have got to expect that language is the vehicle, or the barrier, even stronger than it would be in a first-language school.

This had meant that the teachers not only had to teach the basic conventional mathematical writing but must also help the students improve the clarity of what they had written. This can be seen from Tau 2’s session on the problem of how many combinations of blocks of different colours could be made. Figure 29 shows different children working on this problem.
Figure 4 Children using blocks to find combinations

A systematic recording of their results was necessary for working out the answer. Figure 30 shows two examples of the recording of the different combinations. In the first one, the initial part of the recording is in the teacher’s handwriting indicating that they showed the student how to set out their results. The second example is in the child’s handwriting but follows the teacher’s example.
Figure 5 DPIAtL3a and DPIPaL3a

Although the children had the skills already to draw the blocks and to use the words and symbols to describe the combinations, the teacher’s intervention was needed to help them structure their recordings systematically. Students needed explicit instruction on how to reorder what they had previously produced.
Teacher support for these acts of writing occurred in regard to the teaching of all three genres. Superficial acts of writing are important as they focus on how to organise information in conventional ways that can lead towards deeper thinking processes.

Deep acts of writing

These acts of writing concentrate on improving writing by focusing on the planning, drafting, revising and polishing stages of the composing process (Ivanic, 2004). One of the approaches that had begun to be used in the kura during the project was an adaptation of an idea used in Helen Doerr’s project in the United States. At the New Zealand Mathematics Teachers Association (NZMTA) conference in September 2007, Helen had been one of the keynote speakers whom many of the teachers had heard. She mentioned the RAVE strategy that one of the teachers in her study had picked up during a language arts course and that the other teachers in the project had found valuable. RAVE was a mnemonic that stood for: Rewrite the question; Answer the question; use mathematics Vocabulary; and provide Examples. The teachers did not ever talk about using this mnemonic with their students but about developing their own one which reflected their needs and the language used at the kura. Many of the teachers at the kura felt that this was an approach to try. The following extract comes from the November staff meeting:

T7: I have been trying that RAVE.

TM: How did it go?

T7: It’s been going fine. I found . . . finally getting them to restate the question in a different way was a couple of lessons at least. To answer it as well, rather than just saying yes or no. And then to justify their answer and to attach an example onto that as well of what they actually did, if they were going to do some mihi hanga whatever. Getting them to answer the question, what they exactly did and why did they believe that the answer, to the question, they have given is right. Then they draw their tauira for me as well within the answer . . . And now they all have to stand up and give their answer.

TM: So they are orally presenting it? And then do they write?

T7: No, they are orally presenting their writing because if there are questionable answers that are produced and they’ve obviously heard everyone else’s, then they go maybe, maybe I have to redo mine. Plus I don’t mind telling them that’s not quite right, start again . . . I didn’t actually go to the hui, it was just a chance meeting with [T10].

. . .

T7: What I found interesting was that I had tried to tell the kids that mathematics is not just a series of numbers, it involves writing. We are fortunate enough that we are doing āhuahanga at the moment. I would like to try a times-table equation and see if they can give me that same sort of format answer to explain how they got their answer. We try to say to them that as they get further up towards where T9 is, you’ll notice that you do more explaining, more and more writing, so you need to justify your answer even if it is wrong, but you don’t know that. You justify as far as you can then the teacher will tell you what parts are wrong, once you’ve justified it pretty well. How you got to where you are, explaining what actual process they took. Because we looked at car logos in the car park, so they looked at the Mitsubishi and when they looked at it, straight away they thought the whole star series was what they were transforming but when they had to relook at it again and it was only one diamond that had been rotated three times, in a third of a circle so the writing about it they realised oh, yeah, you’re right. I’m not turning the whole star I am turning just one diamond around.

TM: So it was the process of writing that forced their thinking?

T7: Some of them got it wrong but they justified their answer. How come they ended up with it. Then at the end we added another bit where they look at it and say what they thought of it. I thought why not have a go and see how it ends up. Have a dive in and have a look. I was noticing if I asked them how they got their answer, the answer had been I just know, I just did it and it came out like this. Now they justify everything they’ve done, explain to me where they put the rawini tamariki (?)

Using the RAVE approach was seen by the teachers as something that could help students elaborate on their responses so that it got them thinking about the components that contributed to a quality response. There is more discussion about RAVE in Chapter 8. Figure 31 provides one student’s response to writing about the transformations that could be seen in the car logos.
Figure 6 JCUn

Integrating writing with genres and MRA model stages

The acts of writing are about the process of actually putting something on a page, while the MRA model considers the strategies the teachers used to support students acquiring mathematical writing. The next sections on the four stages of the MRA model explain these strategies more explicitly. However, Table 7 presents how acts of writing, genres and the MRA model can be integrated.
Table 2 Integration of acts of writing, genres and the stages of the MRA model

The relationship to genre is that learning how to physically produce mathematical writing conventions only occurs when learning to write whakaahua. Learning about the superficial aspects of writing occurs when learning about any of the three genres, while the deeper aspects of writing are learnt when learning to write whakamārama and parahau. The following sections outline the strategies teachers used to support students acquiring mathematical writing.

Kitenga

The kitenga stage is when the teachers introduce new terms or expressions or add extra meanings to ones that students are already familiar with. The function of this stage is to make students aware of new aspects of the mathematics register, whether these are new layers of meaning for already known terms or previously unheard terms or expressions. This stage is characterised by the teacher doing almost all of the cognitive work. They engineer the activity so that the new terms are needed. They ensure that the words are used frequently, mostly by themselves, but also by the students. At this level, students themselves rarely do any writing. If they do write, it is of a very limited kind that reinforces the physical aspects of the writing.

Modelling

At the kitenga stage, teachers model writing. As was the case in Doerr and Chandler-Olcott’s (in press) research, “students needed to have models of good writing before they could be expected to write such responses independently”. Our research showed that there were several different types of modelling done by teachers. These were: the writing of words, symbols or diagrams as a part of a focused discussion; the modelling by the teacher of the mode of writing that students would do as part of participating in an activity; and the modelling of an extended piece of writing that students would be then expected to copy into their books.
The first kind of modelling can be seen in Figure 32. As part of a teacher-controlled discussion, the teacher would emphasise words, symbols or diagrams by writing them on the board. The following is an extract from the video where the square is drawn on the board.
Figure 7  Writing as part of the discussion in the lesson

In this example, the teacher draws the shape on the board as part of a discussion about the features of a tapawhā rite (square). The students were sitting with the teacher in front of the board and had a worksheet on which they are colouring in different shapes. In order to do this, they needed to identify the different features of each shape. This worksheet can be seen in Figure 33. By drawing the square on the board, the teacher was able to channel the students into being able to describe and recognise the features of the square.

Student worksheet to accompany T1’s lesson

Teachers also modelled how they expected students to record information while doing an activity. Figure 34 shows a teacher setting out how to show the results from using a spinner. Students were not expected to copy these but were expected to produce their own tables. This part of the lesson belonged to the kitenga stage because it highlighted for students the features of a table. When the students use the tables themselves, in the next part of the lesson, they most likely would be operating at the integration stage. The teacher’s example would be there to remind them of how they should set out the information. However, the fluency students showed in producing their own table and the amount of intervention provided by the teacher would determine the stage the student was actually working at.
Figure 9. Teacher doing and recording

Another example of modelling is when the teacher writes something on the board that is then copied by students into their workbooks. Often these were extended pieces of writing. Figure 35 provides an example of a short piece of writing. These pieces of writing then become examples for students to use if they need to draw a similar example themselves. They also provide a model for explicitness in mathematical writing if the teacher expected students to refer to these pieces of writing later on.
Figure 10 Teacher writing on the board which is then copied into students’ books

In the November staff meeting, the teacher from Tau 2 described students’ modelling books. In these books, students, or the teacher, would write the learning intention for the day. The learning intention set out what it was that students were expected to learn. When they had completed the day’s worksheet, students would paste it under the learning intention. Students would then be able to refer back to this at a later stage. The focus for why students were doing the writing, then, is linked explicitly to the examples of the writing.

Providing examples of new writing

As well as modelling pieces of writing that students would be expected to master themselves in due course, teachers also started lessons by highlighting new material. In the following extract, T3 has āhuahanga (geometry) written on cardboard. She then had the children in her Tau 1 class read it with her. She finished by drawing different shapes that fitted into the category of āhuahanga and having the children name them.

Figure 11 Wall showing ine (measurement) and tauanga (probability) words

Kinaesthetic activities

As an introduction to the diagrams or symbols needed for writing, some teachers involved the students in physical activities to highlight features. Kanikani pāngarau described in the first section of this chapter was one example of this. Figure 37 shows a teacher with her students engaged in another activity around drawing shapes.
Figure 12 Making shapes with the body

In this lesson the teacher had previously had students manipulate concrete examples of the different shapes. Physical activities of making the shapes is one stage away from this manipulation, but students are not yet drawing anything on paper. This would be the next stage.

Restricted writing activities

At the kitenga stage, the only independent writing that students are engaged in is of a very restricted kind. Figure 38 provides an example of a student tracing numerals so that they are formed correctly. The tracing with arrows to show direction means that students will be able to draw the numerals conventionally. This activity is done up to Tau 2 because many students may still be reversing some numerals at this year level.
Figure 13 DSWNKaL1a

The activities in this stage of the MRA model were often concerned with the physical acts of writing as they were about ensuring that students are able to physically manipulate the writing objects as well as correctly produce conventional mathematical objects. However, modelling activities could also be about modelling both superficial and deeper acts of writing.

Akoranga

By this stage, some of the cognitive load has shifted to the students. They now need to give definitions and examples, rather than just being expected to notice and interpret those provided by the teacher. Nevertheless, the teacher is still very much in control and students’ contributions are usually short, thus providing them with little opportunity to provide inappropriate responses. The function of the akoranga stage is for students to form understandings of when and how new aspects of te reo tātaitai are to be used.

Worksheets

One way of ensuring that students were channelled into using correct mathematical writing structures was by providing them with worksheets where they had only limited ways to respond. These worksheets provided students with more opportunities to make mistakes than those in the kitenga stage. Figure 39 shows an example of a worksheet where a student could get the pattern incorrect but it is unlikely that this would occur. It may be that such a worksheet belongs to the kitenga stage. It is unclear whether such an activity would actually support students to learn about the sequential order of numbers.
Figure 14 DWNTaL3a

On the other hand, Figure 40 shows a worksheet that is mostly the teacher’s writing, but with spaces for the students to add in words or diagrams. The teacher has also written comments after the sheet was completed. In contrast to Figure 39, this worksheet provides opportunities for the students to show their understanding.
Figure 15 DGTrCArL3

Using students’ own words as a starting point for writing

Students contributed to the writing process by providing words either orally or by writing them. At the September staff meeting, T8 related how she transcribed some students’ contributions because they were too slow to write their ideas down and this impeded what else had to be done during that lesson. At the beginning of the next lesson, T8 asked them about their ideas from the previous lesson, whether they still agreed with them, and if they wanted to add anything to them. She found that doing the writing for these students meant that their ideas were valued. If they had to write it down they rarely got anything else done in the lesson. Having something written down meant that it was not just those students who could write whose ideas were appreciated.
T1 used various strategies around using the students’ own writing. In the first example in Figure 41, she began the student’s writing and then had them complete it with a sentence. In the second example, T1 had corrected the student’s narrative, and in the third example she had the student interpret what he had wanted to write and then rewrote it for him. The range of strategies employed that used the student’s own writing suggests that the teacher was actively monitoring students’ work while they were doing this writing.
Figure 16 DNAkL4 (left), DNHiL4 (middle) and DNTiL4 (right)

The akoranga stage mostly had students involved in superficial acts of writing. However, the strategies, such as the one of working with students to improve their writing, could also support students by being used to move students into doing deeper acts of writing.

Taunga

By the taunga stage, students have a good understanding of the new aspects of te reo tātaitai. The function of this stage is to have students use these new aspects but in a situation where the teacher is able to step in and provide support if necessary. Consequently, the teacher’s role has become one of reminding students of what they know and can do. The students are the ones who have the major responsibility for making use of the new language. If the student seems unable to operate at this level, the teacher is quickly able to supply more support, thus recognising that the student is still at the akoranga stage. If students do not need the teacher’s help then they would be operating at the putanga stage.

Correcting students’ writing

A very common strategy at this level is for teachers to collect in students’ work and check it for accuracy. Figure 42 shows an example of a piece of writing that has been checked by the teacher.
Figure 17 Students’ work that has been checked for correctness by the teacher

Sometimes, the students would write an initial draft and then check it, often by asking the teacher. In the junior grades this tended to be at a superficial rather than a deep act of writing. Once this had been done then a final version would be produced. Figure 43 shows an example of a sentence about rotation that has been written to accompany a set of diagrams. In Doerr and ChandlerOlcott’s (in press) research, the middle school teachers they had worked with had found that “the editing of student work began to yield improvements in the quality of students’ written responses”. This editing was done in a number of ways. Sometimes it was done in a whole-class situation where a sample piece of work was used. At other times students did the editing by themselves or occasionally with peers.
Figure 18 DGTrUnL5

In Figure 43, an earlier version of the sentence can be seen faintly underneath the final sentence. This is most obvious in the writing of kahuri. Writing that is displayed, such as this piece, often shows students’ growing fluency with new aspects of te reo tātaitai and, therefore, will come from the taunga stage. The teacher will closely supervise the work to ensure that students do produce an appropriate response for public display. However, the amount of support the teacher provides will depend on the students’ levels of fluency.
Public displays were not just flat posters, as Figure 44 shows.
Figure 19 Classroom display of 3D-shape posters

The problem with fragile displays such as those shown in Figure 44 is that they become damaged very easily. However, the folding of the 3D shapes would have given students immediate feedback about the appropriateness of the shapes. This takes the pressure off the teacher as always being the arbitrator of what is appropriate or correct. This, therefore, supports the students as having the responsibility for determining the accuracy or appropriateness of their own work.

Writing using computers

Another strategy one teacher used that provided students with immediate feedback was having students use MSWord drawing functions to produce tessellating patterns. Halliday (2007) described literacy as “a technological construct; it means using the current technology of writing to participate in social processes, including the new social processes that it brings into being” (p. 113). The use of computer technology to alleviate some of the demands of writing has been available in mathematics classrooms for some time. Winch et al. (2004) suggested that students find revision of narrative pieces of writing much easier if they can use word processing programs. It may be that students find being able to use computers to replace the tediousness of some parts of mathematics, such as tessellating patterns and drawing graphs, an incentive to engage with these topics. Brown, Jones, Taylor, and Hirst (2004) found that students were more able to engage with a problem about the diagonal properties of quadrilaterals using Geometers Sketch Pad whereas some had not been able to do so using a pencil and paper technique. However, the videoed lessons only showed one example of technology being used in this way and this was from a series of T2’s lessons.

Figure 45 shows the development of a pattern using a translated shape. Others in the class rotated their shapes to form their patterns. The software allows a very quick development of a complicated pattern that would have taken many hours to have drawn by hand.
Figure 20 Stages in developing a tessellating pattern using translation

The first picture shows the student choosing a shape. The next activity is to draw the original shape, copy it and then paste several examples onto the screen. The student slides (translates) the copies around the page to form a pattern. The final picture shows the student choosing colours to shade the shapes in the pattern.

Writing in public places
When students did mathematical writing on playgrounds or on whiteboards they were also displaying their fluency, but not in the same way as the static posters put up around the room, of which Figures 43 and 44 are examples. Public writing was done quickly and was only available for immediate scrutiny and discussion. If students had produced something that was not correct, then there were opportunities to discuss why this was the case. There were also opportunities to discuss well-presented pieces of work. The discussion had to be immediate as the work would be removed at the end of the lesson, if not earlier. These activities were part of the taunga stage because they allowed for instant feedback.

One of the junior classes used large pieces of chalk to draw 2D shapes on the concrete. This can be seen in Figure 46. There was a strong link to oral language in these activities where the students’ recording was just part of developing the students’ understanding of shapes. The teacher gave a description of the shape, and students had to draw it and jump into it when they had finished. This was followed by some students taking on the task of describing the shapes.
Figure 21 Drawing shapes on concrete

This activity had students concentrating on the features of shapes. It resembles those suggested by Juraschek (1990) for supporting students to move from van Hiele’s visualisation level where students are only aware of global features to analysis level where students are aware of specific features.

At the other end of the kura, students were regularly expected to present their ideas on the whiteboard. This was seen in all of T9’s lessons recorded since 2005. An example of this writing can be seen in Figure 47.
Figure 22 Presenting explanation of the length of the hypotenuse of a triangle

The scrutiny that accompanied these public writings meant that it was very easy for teachers or other students to highlight difficulties in understanding the meaning that the writer was trying to display. Consequently, the students themselves would clarify the meaning that they were trying to give. In asking students to display their knowledge, it is assumed that they have the skills to do so and that the classroom environment was supportive of them if they struggled in writing up a response. This supportive environment is used to remind the students of what they do already know and if they cannot resolve it themselves then the teacher can intervene by using strategies from the akoranga stage.

Putanga

The final stage of the MRA model allows students to show their fluency in using te reo tātaitai. Its function is to enable students to use their knowledge and skills without any support from the teacher. At this stage, there is not a series of strategies that teachers choose from. The teacher’s role is simply to provide opportunities for students to make use of the fluency they had acquired. Sometimes students who were engaged in learning a new topic would use other aspects of te reo tātaitai that they were fluent in. Often the work produced at this stage was for formal assessments.

Figure 48 shows a student completing a tally to record their results from using a spinner as part of a beginning activity on probability. This student had no difficulty with this part of the task. However, their fluency in being able to describe what they had done clearly was not so high.
Figure 23 Recording the results of a spinner using a tally

If the paper clip lands on the small number, perhaps to me it’s bigger, but if it lands on the same (equal) number, smaller number.

Assessment tasks also tested students’ fluency in being able to provide appropriate mathematical writing. Two teachers, T1 and T2, asked students to write about a topic both at the beginning and at the end of a unit of work. This enabled not only the teacher but also the student to be able to see what had been learnt and what improvements had been made in their writing. A teacher in Doerr and Chandler-Olcott’s (in press) research had used a similar approach in that she had given students the same writing prompt at the beginning, middle and end of a unit. This had provided her with insights into how students’ understanding had grown while completing the unit. In the second year of the project, the teacher had got the students themselves to look at the work and consider how to make it better.

In the September staff meeting, T2 described why she had students produce two examples of writing about transformations:

T2 mentioned that in her group she has some students who struggle with writing generally. She saw in the examples of writing about transformations (reflection, rotation and translation) that some students appeared to have played safe. For example, to show reflection a student chose the letter ‘T’. Although this was reflected, it does not actually change which would have been the case if she had chosen something like the letter ‘K’. A good piece of writing on this topic gave the explanation generally through diagrams. However, some students could have done better by providing a longer written text. Students need strong te reo Māori if they are to produce good narrative texts. Even with good mathematical vocabulary they also need good general writing skills.

Having students present their understanding of a topic means that they have to make some independent choices about what they are going to do. This shows you where they are at.

T2 had students complete a second piece of writing on this topic by having students choose a kōwhaiwhai pattern and then describe the transformations within it. In this case T2 felt that she provided more explanation about what she was wanting than she had with the earlier piece. The first piece was in some ways a diagnostic test to see what students knew about the topic.

Figure 49 provides the two pieces of writing from one student. It is possible to see a significant change in the type of transformation that is being discussed.

Figure 24 Transformation assignments by a Tau 8 student

Conclusion

The strategies teachers employed to support students improve their writing skills were varied in all four stages of the MRA model and all teachers used strategies from each stage. The acts of writing were also developed across the four stages. However, the focus for the physical acts occurred in the initial stage of the MRA model. Once students could independently recognise the features of the shapes or diagrams they had to reproduce they quickly moved onto being fluent so little was seen in the intermediate stages. Deeper acts of writing were more evenly spread across all four stages but were only connected to whakamārama and parahau genres.
The audience of the writing was also connected to the stage of the MRA model. For example, if the students were writing for themselves then they were generally fluent in the genre or mathematical writing mode they were using. If the teachers were slightly hesitant about whether the students were completely fluent, then either they would regularly check the writing themselves or set up activities where students would receive immediate feedback about the appropriateness of what they were doing. These activities were ones such as folding paper to produce a 3D shape or using the computer to produce a translated pattern. There were also opportunities through public displays of writing for other students to provide feedback if they could not follow what had been produced. This also gave the writers immediate feedback. The classroom environments at the kura were supportive and comments by teachers and other students were seen as helpful in the style of a tuakana–teina, older–younger sibling, relationship identified in the previous project (Meaney et al., 2007).

2. Student writing

This chapter outlines issues to do with writing in mathematics from the students’ perspective. Ultimately, the Mathematics: She’ll be Write! project was about improving students’ learning of mathematics. It is therefore valuable to hear from the students about their perceptions of writing in mathematics as well as to see what changes occurred during the project.

Little previous research has investigated students’ perspectives about writing in mathematics. In regard to students’ problem solving in mathematics, Albert (2000) investigated their use of spoken and written language. Students felt that “writing helped them keep track of their thinking and solutions” (p. 135). This was reinforced by Albert’s observations and interviews while students were engaged in problem solving. Her conclusion was that writing supported students’ self-talk and this contributed to their reflection on their problem solving. A similar conclusion was made by Meaney (2002b) after working with students in her junior high school class. However, she also found that unless students valued writing in mathematics, then it was unlikely that they would take advantage of the reflection that writing could provide (Meaney, 2002a).

For this project, we analysed survey results and interviews to discuss students’ beliefs about writing in mathematics and also described writing over the year from students at each year level. The amount of writing that students did varied across the year levels and was affected by the topic they were studying as well as their teachers’ engagement in the research project. As the teachers’ participation is discussed in Chapter 8, this chapter focuses on what students wrote over the year and their beliefs about this.

Writing over time

In order to discover the types of writing that students did and how these changed during the course of the project, we decided to document the writing done by two students from each of the classes. The students were usually the ones chosen by their teachers to be interviewed in September. When the teachers only had one student interviewed then a second student’s writing samples were also included. This student was chosen at random. The students who were interviewed were not always the ones we had the most samples from. However, by combining the samples of both students it was possible to get a sense of the writing that was done by all the students in the class over the course of the year. Occasionally, both students contributed the same piece of writing but these double-ups were less frequent than we had expected, suggesting that our collection of samples was not as rigorous as we had thought. Therefore, having two students’ samples was a more appropriate method of recording what each student was expected to write over the course of the year.

Some writing was collected in 2006, but it was not systematic and so has not been included in our analysis. The collection of material in 2007 was more systematic, but there were still some problems. Sometimes material was not collected from specific teachers during one of the researchers’ visits for a range of reasons.

Consequently, it is difficult to definitively say that students wrote more and at a higher quality as a result of their teachers being part of the project. Nevertheless, it did seem that the number of modes students were expected to use in each year level became more varied as the project progressed. There also seemed to be more of an emphasis on writing explanations and justifications later in the year. However, this may have been because the topics that were covered towards the end of the year lent themselves to being more appropriate for the writing of explanations and justifications.

Table 3 Writing in year levels across the year

	Tau 1	Tau 2	Tau 3	Tau 4	Tau 5	Tau 6	Tau 7	Tau 8	Tau 11
Feb	1 Time	1 Time	2 Time				2 Time 1 Calcul	1 Time	1 Probl solve 1 Meas 1 Calcul
March	1 Numb	2 Time 1 2D shape	2 Time 3 Numb		2 Numb		1 Stats graph 1 Word problem		4 Meas
April				1 Numb 1 2D shape				1 Calcul 2 Meas	1 Meas 1 Enlarge
May	1 Pattern		1 Patt 2 Rel graphs 4 Stats graphs	1 Rel graph 2 Stats graphs	1 Fract	1 2D shape	1 Word probl 1 Patt 2 Cart graphs	9 Meas	2 Algebra 2 Cart graphs
June	1 Tally	1 2D shape	3 Numb	4 Numb	1 Stats graph	3 Probl solve	3 Shape & angle 3 Fract & proport 2 Calcul 2 Patt 1 Time	1 3D shape 1 Geom	2 Algebra
July			3 Prob		4 Calcul		4 3D shape	1 Angle 3 Metric convers	1 Iso draw
Aug	3 Prob	2 Probl solve	2 Prob	6 Fract 1 Calc & meas	3 Prob 1 Fract	6 Prob	1 Meas 2 Calcul	1 Angle 4 Transf	2 Geo construct 1 Cart graph
Sept	1 Prob 1 Calc			1 2D shape	1 Prob		2 Meas 1 Fract	1 Transf 1 Prob	4 Pythag
Oct	1 2D shape	5 Fract 1 Transf 1 Patt	1 Transf	3Transf 2 Calcul 1 2D shape		13 Transf	6 Transf	1 Transf 1 Calcul	1 Pythag
Nov	2 Transf	1 Calcul 1 Transf	1 Transf	1 Transf		3 Transf			2 Angle

Calc = calculation, transf = transformation, probl solve = problem solving, numb = number, fract = fraction, meas = measure, patt = patterns, iso draw = isometric drawing, rel graph = relation graph, geo construct = geometric construction, Pythag = Pythagoras, prob = probability, geom = geometry

Table 8 sets out the topics and number of pages of writing done in each month between February and November 2007, for two students from each class. When there were two pieces of writing that were the same, then only one was included in the numbers in the table. For example, the next two pieces of writing were done by the two students in Year 7. However, as these were the same activity, they were only considered as one piece of writing in Table 8.

and

Table 8 shows that the type of writing students did varied over the course of the year. It also shows that, as could be expected, students are doing more writing in the later year levels. However, there are some exceptions with students in Year 3 doing more writing generally than their peers in the following couple of years.
Sometimes, the same worksheet would be completed by students at several year levels. For example, students in Tau 4 used a number of worksheets on fractions in August 2007. Some of these worksheets were then used by students in Tau 2 in October. As teachers planned their mathematics programmes together in the junior school, there were often possibilities of sharing resources. The lack of resources in te reo Māori is an ongoing issue in kura kaupapa Māori (Meaney, 2001) and so sharing of resources is useful for saving time in planning. The worksheet may be used as an introductory activity at one year level while in a later year level it may be used as a diagnostic test to check what students remembered before the topic was begun. Given that students were achieving at different levels in all classes, it was also possible that there were commonalities in the learning outcomes for some students in different classes. Figure 50 shows three similar worksheets on reflection from three different year levels.
Figure 25 Tau 2 (left), Tau 3 (middle) and Tau 4 (right)

The amount of writing did vary across the year but school holidays falling in April, July and September/October meant that the spread was not even. Although T1 had been at the initial discussions in 2006, she had not been at the kura for most of the first term and so there were no writing samples from her class for this period. As well, the last collection of samples was done in the first week in November; which meant that there were few samples for this month. However, it would seem that students were expected to do more writing towards the end of the year than had been expected of them at the beginning of the year.

The topic influenced the amount of writing. Number was an underlying theme across the year and was not tied to a particular time of the year as were other topics. However, written examples did not always appear in the data collection. It may be that with the focus of the Poutama Tau on mental strategies there were not many written examples. However, it may also be that these examples were not as valued as other pieces of writing and were not kept in the same way. An exception to this would be the situation in Tau 7 where students produced multiplication or addition tables throughout the year and these appeared regularly in the samples that were collected. The students would record the time it took them to complete the times or addition table. An example of this can be seen in Figure 51.
Figure 26 Times table

Some topics had students producing more writing than other topics. It was rarely expected that students would reproduce the same piece of writing in any topic, apart from the multiplication and addition tables seen in Tau 7. For example, there were five pages of writing on fractions done by students in Tau 2 in October. These can seen be in Figure 52.
Figure 27 Five pages of writing on fractions

In the pieces of writing, the students related iconic drawings to symbols, but each one required the students to integrate the two modes in different ways. The first one involved the students folding different shapes and recording how they produced two halves. As can be seen in the triangle, the students were not always able to recognise the two halves. On the next page, students drew different shapes and then showed how the first could be split into two halves and then into quarters. In the next sheet, the students represented fractions with different numerators. The final piece of writing showed how fractions of groups of items were determined. As they had been with the fraction of a whole, the students were channelled into being able to produce the conventional mathematical writing about the fractional amounts of a group of objects.

Students’ writing is affected by the topic they are writing about. Probability was one area that produced a significant amount of writing across the kura. It also involved students in integrating a variety of different modes. Consequently the sort of writing that students engaged with across different year levels is investigated in the next section.

Students’ writing about probability

Probability is an interesting topic to investigate in regard to writing. It has been suggested that the different facets of probability are difficult for students to grasp and have to develop over a number of years (Nickson, 2000). In a longitudinal study of how junior high school students developed ideas about probability, Green (1983) found that:

The concept of ratio is vital to children’s understanding of probability

The level of understanding of the language of probability is poor (e.g. words such as ‘certain’ and
‘least’)

A systematic approach to the teaching of probability and statistics in schools is necessary to overcome children’s misconceptions in connection with the subject. (Nickson, 2000, p. 94)
There is a need to integrate a variety of modes because probability concepts are built upon a range of different ideas. This is one of the reasons why students can find learning probability so difficult. Watson (2006) provided a diagram that showed the main ideas about chance, the precursor to probability, and how they were related. This can be seen in Figure 53.

Figure 28 Links between ideas and statistical elements related to chance understanding (from Watson, 2006, p. 130)

The pieces of writing collected from the kura showed that students were building on understanding developed in earlier years. The foundation for investigating the topic was that there were specific terms and expressions needed to discuss probability. Given that most of the students were second-language learners of te reo Māori, it is not surprising that the teachers focused on ensuring students had appropriate language as a beginning point for this topic. Interference from connotations from a student’s first language is known to affect their acquisition of second-language probability terms (Kazima, 2006). One of the reflections in the final meeting for the year was that probability was a particularly difficult area for students to grasp because:

Probability moves into language that hasn’t naturally been strongly supported in Māori. We have some words like puta noa and other things like tera pea but it is not as organised as possible . . . through to probable, to likely, to definite. Those people have worked on that a long time in English and have decided what a possible looks like, and this is what a probable looks like, and this is what a likely looks like and this is what a highly likely looks like, this is what a definitely looks like. (T9, Meeting November 2007)

For example, the word for probability in te reo Māori is tūponotanga which traditionally had the English connotation of “by accident” or “chance to hit”, that did not have a good outcome. Without explicit teaching, students may often be unaware that the same word had different meanings in the everyday context and the mathematical context. As well as concentrating on probability language, the students made use of previous work on statistical graphs in exploring probability concepts.

Probability over the year levels

Figures 54 and 55 show two of the pieces of writing that Tau 1 students did on probability. In the first piece, students were expected to draw reproductions of playing cards to show how three aces and two queens could be distributed. They were then asked to make predictions about the throwing of two coins. The students had to circle whether the coins would come up “māhunga, māhunga” (head/head) or “māhunga, whiore” (head/tail) or “whiore, whiore” (tail/tail). This worksheet had students produce drawings and use words connected to ideas to do with probability. They were related to actual activities of turning cards over and tossing coins, thus making the writing strongly connected to activities that students were engaged in.
Figure 29 Tau 2 student’s worksheet on probability

In the writing samples in Figure 55, students wrote sentences using the phrases written by the teacher above the box. The students also had to draw a picture to accompany their sentences.
Figure 30 Students’ definitions for specific probability terms

The writing students did about probability in Tau 1 required the students to integrate words with drawings. In Figure 54, the words and the pictures were connected to separate activities; in the examples in Figure 55, students had to match their sentences with their drawings. Although the students were not referring to actual activities as they had been in the previous example, they were expected to draw upon their own experiences. At the bottom of the worksheet in Figure 55, students had to find different probability expressions that had been introduced to students at the beginning of the lesson.

It is interesting to note that students in this year were already being introduced to the areas of language identified by Green (1983) as being problematic. This was not done in a simplistic way as students were writing about variation (Figure 54), one of the key ideas identified by Watson (2006) in regard to chance, through drawings and identifying combinations.

In Tau 3, students also were channelled into using the appropriate probability language. The first work they did was to place terms in a list from “definitely going to happen” to “definitely not going to happen” (Āe ka tino taea, Āe ka taea, E kore e taea). These terms were then used in a variety of different activities, including writing sentences using the expressions. An example of this is shown in Figure 56.
Figure 31 Using probability expressions in Tau 3

In Tau 5, students engaged in probability experiments using spinners. They used tables to record their results and then wrote about them in paragraphs. The activity involved students in thinking about ideas to do with proportion. Green (1983) had found these areas were not generally done well in school, making it difficult for students to understand probability. An example of a student’s writing can be seen in Figure 57.
Figure 32 A student’s description of their probability experiment from Tau 5

At this level of writing, students had to explain what they had done and what the results showed. This required a higher level of writing than just using the expressions to describe events. However, it was also clear that students were still grappling with expressing their ideas about what made a fair game. Some students were able to discuss how, even though the actual game had allowed one item to be more successful, over a longer run this would not be the case because it had a smaller proportion of the spinner. The teacher felt that some students were still developing an understanding of this idea and that many found orally explaining what had happened easier than having to write about it (T10, Meeting 5 September 2007).

Similar writing was expected of students in Tau 6. Students also worked with spinners and wrote about these experiences. The focus for these students was on looking at how the chances of winning were related to the proportions on the spinner. As was the case in Tau 5, some students were able to understand this while others struggled with seeing that two outcomes had an equal opportunity for occurring if the proportions on the spinner were the same (T7, Meeting 5 September 2007).

Figure 33 Tau 6 student’s explanation about the connection between the spinner proportions and the chances of winning

This writing involved students in integrating a range of different modes. They had to keep tables of results, draw graphs and explain what had occurred using diagrams of the spinners.

In Tau 8, students were also involved in playing a game. In this case, they had to choose a number between 1 and 100. They then had to throw two dice 10 times to try to get to a total that equalled their chosen number. They could use a calculator to keep track of the cumulative total. At the end of five games, they had to write about whether their chosen number was a “good” number for the total, whether they had a strategy for choosing the possible total and what they could see in the data that may have given them some idea about why the chosen number was a “good” number (Lesson notes from T2). Figure 59 provides an example of one student’s writing.
Figure 34 Tau 8 student’s recording of playing a dice game and his understanding about his strategy

This student was able to discuss how seven was a likely number to get from throwing two dice and therefore a total of around 10 lots of seven was a good total to aim for. However, this was a complicated set of ideas about probability, reflecting many of the interlocking ideas in Watson’s (2006) diagram and many students did not fully understand what was required of them.

In Tau 11, probability ideas were discussed as classical probability where theoretical outcomes were described (Nickson, 2000). An example of this is shown in Figure 60. This led on to describing possibilities as fractions.

Figure 35 Tau 11 student’s description of the results from tossing two dice

If the writing samples were typical of what occurred during the time a student was at the kura, then by the time students reached Tau 11 and the work with theoretical probabilities, they would have had many experiences of using probability language and participating in activities. Although students may not have grasped all of the ideas covered in particular years, they had many opportunities for meeting the ideas again in later years. Consequently, students also had opportunities for making extensive links between the different ideas that Watson (2006) perceived as being connected to ideas on chance. The kura focused their ideas about probability on students having a thorough control on the terms and expressions needed for discussing it. However, it was an area that the teachers still felt needed improvement.

The tamariki survey

Students from all year levels completed a survey about their beliefs about writing in mathematics. As the students were aged from five to 18, the survey used pictures and multiple choice predominantly. The survey was trialled with two senior students and five Tau 1 students. As a consequence, other students were asked to complete the survey. A blank survey is provided in Appendix C. One hundred and two students, or approximately half the total student population, completed the surveys. Some students did not complete each question so the totals rarely equalled 102. However, the students who failed to answer were different for each question.

The first question was about the types of writing and the amounts students felt they did of each. The results can be seen in Figure 61.
Figure 36 Graph of different types of writing

When the “lots” categories are combined with the “some” categories, it can be seen that students felt they often used many different modes when doing mathematical writing. More than half the students felt they were doing “lots” or “some” of each of the different types of writing. Of the different modes, students believed they did lots of calculations. Given that number is the underlying basis for much mathematics, especially with the strong emphasis on Poutama Tau at the kura, then this result is not surprising. However, it was not supported by the data used to construct Table 8.

The next two types of mathematical writing students felt they did much of was narratives and shapes. Just over one-third of students thought they wrote “lots” or “some” of these two modes. Few students felt they did “lots” of graphs or pictures.

Whakamārama and parahau (explanations and justifications) depend on narratives. So it is interesting to note that so many students felt they either did “lots” or “some” narrative writing. It may be that students did this survey in fourth term, just after doing the probability unit in Term 3 and the transformation unit in Term 4. Both these units involved students in doing more narrative mathematical writing than previous units and this may have swayed students’ beliefs about how much narrative writing they did.
Figure 37 Graph of frequency of writing in class

Figure 62 shows students’ beliefs about how often they wrote about mathematics. On the whole, the majority of students felt they wrote about mathematics at least two or three times a week. If this was the case, it suggests that the amount of writing collected for Table 8 grossly underestimates how much writing was done. It would be interesting to know whether the students felt the amount of writing or the type of writing had changed during the year. This may have given us a better indication of changes in classroom practices.
Figure 38 Places where mathematical writing is done

The majority of students believed they did most of their writing on paper. About half the students also felt they did some writing on the board. Slightly fewer students felt they sometimes wrote material to go up on the wall of their classrooms. If students were regularly, although not frequently, writing in these less permanent forms, this may explain why there were fewer writing samples contributing to Table 8 than had been suggested by Figure 62.
Figure 39 Audience for students’ mathematical writing

Figure 64 shows that students predominantly felt the writing they did was for themselves. This is a very interesting result as many of the teachers as well as the researchers had felt that students would see mathematical writing as something they did for the teacher. Morgan (1998), in considering the literature on school writing across the curriculum, stated that many studies suggested that “one of the roots of students’ difficulties and lack of motivation in their development as writers” (page ref.) was the fact that the teacher as examiner was the audience on most occasions. Although many students also felt they wrote sometimes for the teacher, mostly they believed they wrote for themselves.

Students’ beliefs that they were writing for themselves suggest that students could use this writing to reflect on their learning. However, without some guidance this may not eventuate. In research with a junior high school student, Meaney (2002a) found that a student who wrote predominantly for himself was unable to use his writing to check what he had done. He left out much of his reasoning because it had been self-evident when he had done the writing. This meant that he and others had difficulty in following what he had done when this writing was read later.

Students also described both their favourite as well as their least favourite modes of mathematical writing. The results for this are shown in Figures 65 and 66. It is quite clear that students felt that calculations were their most favourite mode of writing while narratives were their least favourite type of writing. Given that narratives are connected to the explanations and justifications, then it could indeed be a problem if students do not like to write them.
Figure 40 Favourite types of writing

Figure 41 Least favourite types of writing

As many students recorded graphs as their favourite as recorded them as their least favourite mode of writing. Students also enjoyed drawing shapes but did not enjoy producing patterns.

In Figure 67, the students described what their mathematical writing was for. The students felt that at least sometimes they wrote to fulfil all three purposes. More students believed they wrote so that they could learn mathematics rather than the other two purposes. The fewest students felt that they wrote in mathematics to help them solve problems.
Figure 42 Reasons for writing in mathematics

Figure 43 Difficulties in remembering the te reo Māori mathematical terms

By far the vast majority of students “sometimes” struggled to remember the te reo Māori mathematical words when describing what they were doing. As second-language learners of te reo Māori who would only encounter the mathematics register in classrooms, it is not surprising that students would “sometimes” find it difficult to recall the appropriate vocabulary. The approach the teachers adopted in focusing on ensuring the students had the appropriate vocabulary would seem to be meeting the needs of the students.

Conclusion

Students at each year level wrote a variety of different kinds of mathematical writing. As students progressed through the kura, they continued to integrate different modes but there was much more use of narratives. This was clear in the pieces of writing about probability that were analysed.

Students were also very clear about their beliefs about writing. They felt they did a lot of calculations and this was their favourite type of writing. They also felt they were expected to use words frequently but this was their least favourite type of writing. Given that explanations and justifications generally require students to use words at least in part, then it could be problematic if students are resistant to writing these. However, contrary to what other researchers have suggested, most students felt the writing they did was for themselves. This has great potential for supporting students to be reflective about their mathematics writing and also about their mathematical learning.

3. Teacher change

In Chapter 6 we described what the teachers did to improve students’ writing and in the previous chapter we discussed students’ views about writing in mathematics. In this chapter, we describe the impact of the project on teachers’ teaching practices and on their ability to reflect on their teaching. On the whole, the teachers felt that the project had resulted in their trying new practices and these had had some impact on students’ learning. In adopting these new practices, it was possible to see that the teachers were involved in a teacher inquiry model of professional development and this had supported them to reflect on their professional learning.

The teachers saw the project as an opportunity to improve students’ achievement but recognised that this was not always a simple process:

Now I am just for kids Tamsin and every child making progress, you know. Whether they are Māori or Pākehā I don’t care, but for them to do well in life and to be knowledgeable and to be able to impart their knowledge and to be able to share it, all those sort of things. If I can do any little bit to make a child move forward in their lives I am happy, yeah. But I know it is important how we do it. (T3, Interview November 2007)

Doing a professional learning project of this kind involves teachers changing in two ways. The first is in regard to their teaching practice, while the other is in regard to their ability to research their own practice. 2007 was the third year we had received TLRI funding and some teachers had been involved since the beginning. However, other teachers had only been at the kura since the beginning of 2007. This had made these teachers feel a little behind in their understanding of the project. For example, at the end of the year, T10 stated, “I just think that having come so late into the programme, having not really understood what’s happening, now I have a better understanding” (Interview November 2007).

In the initial project, Te Reo Tātaitai (Meaney et al., 2007), our primary purpose had been to document the teaching practices that were used to support students using te reo Māori to learn mathematics. Although the teachers did trial some new practices, the project had mainly been about sharing what was already occurring in the different classrooms in the kura. In this project, She’ll be Write!, there has been an explicit expectation that teachers would make changes to their teaching practices to enhance student learning. The uptake of the opportunity to change teaching practices was varied for the different teachers. One reason given for these differences was that “the penny drops more slowly if you are not a natural maths teacher” (T8, Interview November 2007). This was supported by a comment by T10 who said in her interview, “I enjoy teaching maths if I have the support.” The structure of the professional development opportunities, therefore, had to be accessible to the teachers who were all coming in with different backgrounds and expectations of the outcomes of the project.

This chapter documents how the teachers changed in order to increase the quantity of writing and quality of the writing that was done in their classrooms. It also describes how the teachers reflected on their own practice as part of the research process and how this contributed to the changes they made to their own teaching. In it, we describe the course of the project and the types of experiences that were provided to the teachers.

The teachers completed a survey and were interviewed in November 2006. As well, during staff meetings several teachers discussed ideas they had tried in their classrooms and these were recorded in the minutes or notes. Teachers also discussed with the main researcher their ideas about the project regularly during the year, as well as describing what was happening in their videoed lessons. This chapter draws on all of these sets of data to describe the changes that teachers made. A copy of the teacher survey is found in Appendix D.

Changes in teaching practices

In the survey, teachers were asked about their participation in the project as well as their ideas for a continuation of the project. The questions also asked about the range of modes that the teachers had taught in 2007, as well as about any changes they had made to their teaching practice to improve students’ writing. The results suggest that most teachers had increased the number of modes and/or genres as well as tried out different strategies throughout the year. For some teachers, the reasons why they had made changes were because an external force, the project, had channelled them into making these changes. For other teachers, the reasons for making changes had been internalised as they adopted the view that writing in mathematics would be beneficial to students’ learning. Sometimes it was a combination where initially the project had contributed to the teachers becoming aware of the issues but as the year progressed they internalised the beliefs about the benefits of writing for students’ mathematical learning.

The teachers were asked about the modes they had expected students to use and whether this was a different range from what they had taught previously. Table 9 sets out the modes and/or genres that teachers said they used in the survey, as well as the modes identified in Table 8 of Chapter 7. It also gives a summary of the reasons why teachers thought that they had increased, or not, the modes/genres they had expected students to use. It was decided to include the list of modes from two sources. Completing a survey at the end of a year, during a meeting, can mean that teachers may have not recalled all of the modes they had taught or used with students. The list of topics covered in the first column, that came from Table 8 in Chapter 7, often provides a richer understanding of the variety of modes covered by the teacher than their own recall of what they covered.

Table 4 Teachers’ beliefs about the modes and/or genres they expected students to use

Teacher and year level	Modes that students had used from Table 1 Chapter 6	Modes that teachers believed they had students use in 2007	Is this a different range than 2006?	Reasons for differences or not in range
T3 in Tau 1	Time, number, pattern, tally, probability, calculations, shapes, transformation	Pictures, symbols, writing, numbers	Yes	It’s been more focused—children are being made to attend, rather than take part and participate.
T6 in Tau 2	Time, shape, problem solving, fraction, transformation, calculations	Geometry, number, algebra, explanations, drawings, graphs	No	Not a different range but we did a lot more writing this year.
T8 in Tau 3	Time, number, patterns, relation graphs, stats graphs, probability, transformation	Graphs, symbols, explanations, diagrams	Yes	Made some attempt to think about writing. Develop some examples. Hadn’t focused on writing before.
T1 in Tau 4	Number, shape, relation graph, stats graph, fraction, calculations, shapes, transformation	Explanation, report, narrative	Yes	Attempting things a lot more and thinking of ways to get understanding out of the children (what genre would suit).
T10 in Tau 5	Number, fraction, stats graph, calculations, probability, fractions	Graphs, tally charts, word problems, justification, explanations/ descriptions	Yes	Due to my own raised awareness of the value of writing in maths— (because of this project).
T7 in Tau 6	Shape, problem solving, probability, transformation	Symbols, graphs, explanations	No	Ultimately all aspects are utilised on a yearly basis.
T5 in Tau 7	Time, calculations, stats graph, word problems, patterns, Cartesian graphs, shape, angles, fraction, proportion, measurement, fraction, transformation	Graphs, vectors, writing equations, Cartesian graphs, nets, 2D polygons, 3D polygons, transformations, cartoons, tables	Yes	Involvement with the research project.
T2 in Tau 8	Time, calculations, measurement, shape, geometry, angle, transformation, probability	Algorithms (graphs, diagrams, equations), questions to answer, explanations, survey questions	Yes/No, depends	The upper levels require “deep” or “explanations” to clarify solutions.
T9 in Tau 11	Problem solving, measurement, calculations, transformation, algebra, isometric drawing, geometric constructions, Cartesian graphs, Pythagoras, angle	Besides constructions and symbols used in equations, most writing has involved self-regulating instructions.	Yes	I had started to concentrate too much on symbols and relied too much on class conversation to develop vocabulary.

It is clear from Table 9 that all of the teachers had used a range of different modes. However, it was interesting to note the number of teachers who mentioned explanations. This suggests it was a genre that they felt was useful for students to master. This is confirmed in the reasons for making changes with what they had done the previous year. In the answers to the survey question on this issue, several teachers mentioned that they were interested in students describing their thinking. Another reason given was participating in the research project had contributed to them increasing the range of modes and/or genres that they expected students to use.

The first question in the survey had asked why the teachers had felt that students should write in mathematics. Many of these answers were directly connected to either having students provide an insight into their thought processes or that they helped the students clarify their thinking. For example, T8 wrote “so they can articulate their understanding and we can see their thought processes (whether correct or incorrect)”. T10 wrote “to consolidate understanding—writing requires justifying answers as well as further thinking so that they realise writing is also a significant part of maths”.
Questions 9, 10 and 11 were about the new practices that teachers had tried in 2007. The results for this are outlined in Table 10.

Table 5 New teaching practices in 2007

Teacher and year level	What were the new practices that were tried in 2007?	Why were these practices tried?	How do you know if they were effective?
T3 in Tau 1	Explained and clarified what was expected of them in more simple and easier to understand language.	Children weren’t really getting the gist of what was required from them.	Children gave and showed clear understanding or better understanding using pictures, words, numbers and symbols.
T6 in Tau 2	I try to explain things more clearly and I also write it out for them. We also write achievement objectives for the lesson for all of us to see.	I thought it would make things easier for the students as well as good modelling.	By this term (Term 4) the students are used to writing out their explanations with less help.
T8 in Tau 3	Presenting questions so that they would write their understanding. State what we are writing.	Was one way I thought may assist.	Not sure that it did other than they did start writing their ideas down.
T1 in Tau 4	Words around the classroom, new vocabulary in books, ideas of how to write things.	To see if by making sure that there was a build up of vocabulary. Then explanations would be easier.	As the year has gone on children can write more. Those that can only do a sentence, there is more depth in it.
T10 in Tau 5	Making them justify answers orally and then through writing. Making more displays and adding explanations.	First was done following suggestions. Second was adopted as a simple way to try to encourage writing.	Feedback Reo was being used in everyday situations following a unit. Their recall was stronger.
T7 in Tau 6	Creating a set writing plan to help students.	The need for explanations to be more specific, descriptive, and also to see inner strategies of students.	Seeing students being able to write at length and then being able to present.
T5 in Tau 7	Writing with words, nets.	Initially because of the project and later so they HAD to clarify their ideas.	Explanations they gave, and understanding gained.
T2 in Tau 8	Not much more.
T9 in Tau 11	Using models for instructions including numerals, arrows etc. to supplement words.	To ensure that the writing task didn’t drown the understanding required, i.e., in case the task got too big.	Students inform me that they return to their “instructions” or at least I direct them there.

Apart from T2 who had always used a range of writing activities with her mathematics classes, all of the other teachers had tried some new ways to support students’ writing in mathematics. These teachers felt that these new practices had been effective in either increasing the quantity and/or the quality of students’ writing. For example, T3 felt that her Tau 1 students were able to give a clearer description of their understanding using a range of modes. In the meeting in November, T8 described why she felt that the whole kura should introduce RAVE, as a way to support students’ writing:

I think we would be silly not to introduce the RAVE thing. It gives you a bit more direction and some of us have introduced it this year and it does make a hell of a lot of difference as to giving you a bit more direction, not so much to your teaching, but your end result of how much your kids produce. I have seen a great change in my kids and they are only Year 3s. I am amazed with some of them, the amount of words they learned just in a three-week period and the amount of writing they did. It might only be a sentence but it is rich. (Meeting November 2007)

However, this impact was not immediate, with many teachers suggesting that it had taken students some time to get to the point of being able to write explanations. In Chapter 7, students had suggested that writing in words was their least favourite type of mathematical writing. It may be that part of this reason was that it took some time before students had enough fluency in being able to write descriptions, explanations or justification to gain the benefits for their own thinking. Therefore, although the teachers saw this as valuable, the students did not see the same value until many months after they had been engaged in these activities, if at all.

The teachers believed participation in the project had resulted in their trying different approaches to the teaching of writing in mathematics. For some of them, the project had made them aware of the importance of writing in mathematics. T3 wrote in her survey, “I learnt how important children’s ability to express themselves clearly was through whatever genre they choose.” For others, being involved in the project had given them insights into their students’ language and/or their mathematical thinking. The process of gaining these insights is covered in the next sections.

Project outline

There were several strands of this project that provided experiences to teachers. All of these strands contributed to teachers learning something about their own teaching as well as about their students’ learning. T6 stated that the project had made her think about mathematics and the role of writing in it:

Thing is, it’s made me think about how much writing we do in maths. Because before, I’d never actually thought of it as writing. Golly gosh, it’s maths sort of thing, I’d never really thought about it as being a written language or anything. So for myself as a kaiako [teacher], it has made me think about it. (Interview November 2007)

This project was significantly different from that of professional development projects such as Te Poutama Tau in that there was no set content or pedagogy that teachers were expected to access or gain.
Professional development projects where there is set content to be covered have a process similar to that outlined in Figure 69. When there is knowledge or pedagogy that is seen as best practice, then it is imperative for teachers to be given this knowledge as soon as possible. Without it, teachers would have nothing to trial or reflect on. In our case, we had little previous research to draw upon to help us decide on what was best practice.

In a project such as She’ll be Write! the teachers and the researchers had identified a problem—that students were doing little writing in mathematics—but had no ready-made solution to implement. This then required all of us, teachers and researchers, to share current and past practices and to document implementation of any new practices. It also involved discussions about the contribution that writing could make to students’ mathematical learning.

Figure 44 Typical sequence of professional learning opportunities from Timperley et al. (2007, p. xxxviii)

Apart from one project in the United States (see Doerr & Chandler-Olcott, in press), this was an area where little work had been done previously although writing in mathematics was strongly supported by curriculum documents. As Doerr and Chandler-Olcott (in press) noted, the National Council of Teachers of Mathematics (2000) “Standards offer little sense of how writing activities might fit together or how students’ writing might develop across tasks and over time”. The Doerr and Chandler-Olcott project was ongoing at the same time as our own project and so there was little information about this available until Helen Doerr visited New Zealand in September 2007. Although her visit provided valuable input such as the writing strategy of RAVE, which was described in Chapter 6, our project was already well underway.

Our approach centred around a set of regular meetings and these formed the backbone for sharing ideas and planning for implementation. Usually each one had a theme and that became the focus for the discussions. Table 11 sets out the meetings with their focus and their outcomes.

Table 6 She’ll be Write! meetings

Date	Theme	Outcome
30/8/06	Setting up Discussing parameters for the project. Sharing some mathematical writing practices by TM.	Series of research questions developed. Timeline for the research.
13/3/07	Genres Sorting mathematical writing samples into genres by teachers. Discussion about the writing that students were currently doing at the kura.	Genres identified and named.
6/6/07	Benefits of writing in mathematics Strategy game used as a stimulus for discussion about having students write was beneficial to their learning. Sharing by teachers of writing activities that they had implemented in their classrooms.	Document that is now included in Chapter 6.
5/9/07	Progressions of writing samples Writing samples from initial topic progressions were placed by teachers into year-level progressions. Discussion about quality of students’ mathematical writing. Sharing by teachers of writing activities that they had implemented in their classrooms.	Initial placement of samples on year-level progressions. This process is described in Chapter 4. Summary of strategies teachers used for supporting writing collated and sent to teachers. These strategies are discussed in Chapter 6.
5/11/07	Summing up Sharing by teachers of writing activities that they had implemented in their classrooms. Discussion of future directions for the project.	Decision for all classes to implement RAVE in 2008 into their mathematical writing.

As well as these in-house sessions, the teachers also attended the New Zealand Mathematics Teachers Conference, NZAMT10, in September 2007. This allowed them several days to discuss not just the sessions they attended but also how the information they had heard related to She’ll be Write!. T10 stated:

I was quite excited when I got back. Two things happened for me. Understood what you are doing for the school here in trying to get us to write more. I came away thinking that Helen [Doerr]’s programme was what we needed to achieve, what you were asking us to do. It really made sense to me then. What we should have done was grabbed her and taken her to a classroom and thrashed it out. How do you start? What practical things do you do? As a team, we could have taken a better opportunity with her. I thought it was great opportunity having her there but also a great opportunity lost. (Interview November 2007)

An analysis of the teachers’ attendance at NZAMT10 is discussed in more detail in Meaney, Trinick, and Fairhall (in press).

It was clear from comments in the interviews that the teachers were also discussing the mathematics learning of the students in between these meetings. For example, the teachers who had not been able to attend NZAMT10 were provided with details about it from others who had attended. T1 told how T10 had described two ideas that she had gained from the conference. The first one was to use a passport to record students’ progress in learning their basic facts. The second one was the use of RAVE.

For the principal, T9, the main gain from having the teachers engage in this project had been that he now had a staff who were happy to discuss mathematics teaching. In his survey, in answer to the question “What had been the most interesting thing for you about being involved in the project?”, he wrote, “enjoying the development of staff, leading to more conversations about maths”. Until the original TLRI project had begun, this had not been the case. New staff who join the kura are quickly inducted into what is expected of them. This can be daunting especially for beginning teachers (two of whom started at the kura during the three-year project). However, it is now an established part of what being a teacher involves at this kura. Timperley et al. (2007) describe the necessity of having institutional support if a project is to be sustained beyond the initial intervention period.

Teacher inquiry and knowledge-building cycle

We therefore used many of the same sort of steps as were outlined in Figure 69 but did not have any front loading of new material. We had identified the issue of wanting to increase the amount of writing that students did in mathematics. This issue had been chosen because some of us initially, and all of us eventually, believed it would have an impact on the students’ ability to think mathematically. However, our approach was exploratory with much discussion based on the teachers’ reflection about their own practice. New practices were instigated and then reflected upon again. As T10 stated:

If we don’t hook on to some strategies that we believe will work and change classroom practice then we won’t do anything. You won’t see the change that we want. And that’s what you want to happen. You want the shift in classroom practice. (Interview November 2007)

It therefore seemed that rather than Figure 69 being typical of what was done during this professional development project, Figure 70 better describes the professional learning that occurred during the project. Figure 70 shows professional learning as a continual cycle of reflecting and implementing new practices. Timperley et al. (2007) described the underlying idea of the cycle as “co- and self-regulatory”, by which they meant “that teachers collectively and individually identify important issues, become the drivers for acquiring the knowledge they need to solve them, monitor the impact of their actions, and adjust their practice accordingly” (p. xlii). In order to do this, teachers need to identify students’ and teachers’ learning needs before moving on to considering designing and implementing teaching activities. The learning needs form the goals that teachers would aim to achieve as a result of participating in the inquiry cycle.

This model has significant correlation with ideas on praxis that were outlined by Freire (1996). Within praxis, action cannot occur without reflection. Reflection needs to happen simultaneously and continually with action and this will result not just in changes to action but in changes to the reflection itself and the knowledge base from which it was drawing (Carr & Kemmis, 1983). Action should be based on thinking about why it was needed and the consequences of following it through. As a result, some of the understanding about the initial situation and the new developing one will be changed and this would also result in further action being needed.
Figure 45 Teacher inquiry and knowledge-building cycle, from Timperley et al. (2007, inside front cover)

The following sections discuss three issues that arose during the course of the project. Each of the stages of the inquiry model is used as a starting point to analyse what occurred. However, these sections do not provide a chronological sequence of events because each of the stages was revisited several times and became important to individual teachers at different times. Consequently, although the kura worked on the project as a whole, each teacher was at different stages of this cycle at different times as they considered what was happening in their individual classrooms.

 Developing progressions

The topic and year-level progressions that were discussed in Chapter 4 had been developed as a result of an identified need from the last TLRI project, Te Reo Tātaitai (Meaney et al., 2007). However, although the teachers had been engaged in putting them together, few of them knew what to do with them.

Identifying student learning needs
Teachers felt that students often struggled with a topic when they had not covered it for several years. It was not just that students had to recall knowledge and vocabulary but sometimes what they had been taught was phrased in very different ways from how their current teacher was approaching the topic. In the following extract, T2 discusses the usefulness of having the year-level progressions. For her, they were necessary to ensure that there was a steady build up in ideas over the time a child was in the kura:

T: The research is supposed to feed back into this. We have the writing progressions which is quite useful when we put it into year levels and they had to think about who is introducing what.

T2: And we wouldn’t have known that if you hadn’t come into the picture. I don’t think anybody knows until somebody from the outside comes into the picture and says there is a flow on.

T: It was one of the things that came out of that previous project; people wanted to get a sense of how things developed and where they came from.

T2: And where they came from. Mmmm . . . think with writing it’s what you were expecting of the kids. You were doing a lesson. This is what you have to do and that is how they do it. Something like the probability strand. It’s where it’s common sense and you can’t see where their thinking is. It is very hard for many. It’s probably a process that needs to be worked up from the bottom . . . all of a sudden you have probability. The last time you had probability was in Year 5. You are in Year 10. It is quite a long way, you know, the way of thinking. (T2 Interview November 2007)

It was, therefore, important to be aware of students’ learning needs in this area and take advantage of the information provided in the year-level progressions so that students’ learning could be co-ordinated.

Identifying teacher learning needs
The teachers identified that they also did not necessarily have the information to know where the students would go with their mathematics learning:

We have a little bit of flow about where we want to get our kids to up to Year 6 so that we are covering specific things like, so, like, in Year 2 we cover this, so that they will know that by Year 3. But we, I don’t know where my kids need to go to get up to T5’s stages. I’ve no idea what they do up there. And they’re like ‘Oh, you were in high school’, yeah, like 10 years ago. So if we could set those things up. (T6, Interview November 2007)

T6 was very pleased to have the progressions as she saw them as one way to fill in her own lack of knowledge.

Designing tasks and experiences
Although she was yet to use the progressions in this way, T3 felt that they would be useful in helping to design programmes for the students:

We don’t want to compartmentalise anybody but at least we’ve got a benchmark to look at. Okay, this looks okay for my kids. I would like my tamariki [children] to get to this in terms of the writing in maths. Those that can’t, they can’t but you have got somewhere to go and have a look and get an idea as to where you should be at. But it’s not about boxing them, ae? You don’t want to be doing that either but I think it’s good to know where and if you’re progressing.

The progressions were therefore seen as a way to highlight what students needed to achieve.

Implementing teaching actions
There was discussion about how the progressions could be used. T7 had discussed how they could do the writing in mathematics across the year levels with his wife who worked at another school:

My wife she says it is very difficult at [her] school because they have this ultimatum, we think this is what they should have when the leave [primary school] and she was talking about the kura and she said it is a great opportunity that we have because we’ve got wharekura, Year 0 to 13. Now what do we want our stakeholders what do we want them to look at by the time they reach [T9]? Ultimately if we went backwards from what [T9] wants down each of the years what each person should implement and the steps increment as they go up so that ultimately he’s not having to try and do basically what should have been done in Year 7 or Year 5. So what does it look like at that end and go backwards from there then you know what Year 0 looks like.

However, T8 saw it differently. She felt that there were too many differences between the senior and junior sections of the kura and this made it difficult to co-ordinate what was done at all the year levels:

We are all in a unique situation where we can all link together but it can’t be wharekura dictated like they hoped because we have our structure in place. That gives the kids coverage. You have to teach the kids all those things. [T9, T2 and T5], this year have been more than happy to jump on board . . . We all did that triangle unit [in 2006] and it worked really well but that’s not always going to happen. It doesn’t have to happen every term because they are doing unit standards and we’re not. We are giving the coverage. We can’t have dictation from the wharekura and we can’t dictate to the wharekura. There might be just one unit that as a collective we can all do.

The progressions, therefore, had potential to support the kura in providing a co-ordinated coverage but more discussion would need to occur if this were to happen.

Reflecting on the impact of changed teaching actions
In the surveys, teachers mentioned that there was a need for benchmarks and exemplars to show the teachers and the students what were good pieces of writing in different topics. This means that the progressions would need to be developed further or reworked so that teachers could gain some more value from having them as reference material for their planning.

Explanations

One of the benefits that teachers saw in having students do more writing in mathematics was that it would encourage them to give better explanations of their thinking. T7 wrote in his survey that “[writing] helps them to explain their answers a lot clearer. It allows them in their own words to describe exactly what strategies they are using.” There was a lot of ongoing discussion about how to ensure that students did improve their ability to write clear explanations. This involved teachers concentrating on the deeper acts of writing that were discussed in Chapter 6. At the end of the year, the teachers were still grappling with helping students improve in this area.

Identifying student learning needs
Teachers felt this was an area their students needed to improve on. Since his original involvement in the project in Term 4, 2005, T7 had grappled with his students saying they “guessed” when he asked them how they got their answers. He had worked consistently on supporting students to give more appropriate explanations. Writing became one way of doing this, because it was permanent and could be referred to many times (Minutes September 2007).

T8 wrote in her survey that students needed to give “more explanations on how they know something is what it is or how to work something out” (T8, Survey November 2007). There was a recognition by some teachers that students were much better at giving oral descriptions than written ones. T8 felt that some students had an expectation that maths writing was about writing numbers or doing graphs and that they needed to be aware that writing was a part of learning mathematics (Minutes September 2007). This would correlate with the finding from the previous chapter that showed that writing narratives was students’ least favourite mathematical writing. It is likely that these students needed to see some value in what they were doing.

Identifying teacher learning needs
In the September staff meeting, T10 stated that as a consequence of watching the video of her mathematics lesson, she became aware that she never asked children to write down what they knew for Poutama Tau. She felt that if she were able to do this, it would help cement some of what they were learning. It would also mean that the next day they would have something to refer to if they forgot. This is especially true for multiplication strategies. Often there will be an oral discussion but she had not up to that point got them to write anything down.

However, T10 struggled with how to get students to write down their explanations. In November, she discussed why she wanted to implement RAVE as a consequence of attending NZAMT10. As was described in Chapter 6, RAVE provides guidelines for students to write explanations. She described her initial involvement in the project as something like being in a lolly shop:

actually I was plucking any out of the air that I thought might work, while I feel I am someone who has been to the lolly shop and I know exactly what it tastes like and I want to try it. (Meeting November 2007)

However, she was unsure that she was having an impact on students’ explanations:

I thought I was being a part of something, I didn’t know if I was contributing, but if we went with that [RAVE] I could see myself contributing. I could make a difference. I am not sure I was really making a difference. I was teaching the best I could but I didn’t have any new strategies that could help that writing component. (T10, Interview November 2007)

Identifying teachers’ learning needs to be connected with different possibilities for supporting this learning. This teacher recognised what she wanted to improve in her practice but this did not lead her immediately to adopting any different teaching practices, even when they were suggested by others. For her it was a matter of discovering a teaching approach that resonated with her current teaching practices. RAVE became the new approach that she wanted to implement.

Designing tasks and experiences
RAVE had been something that was promoted by some of the teachers. Although the teachers could see that RAVE had value, they understood that it was beneficial only if it was adapted to meet their students’ needs.
The following extract comes from T8’s interview:

T8: Helen had some great things to say. There was only a core bit of her korero [talk] that related to us or that could be used. RAVE is what we have taken from it, Getting used to justifying themselves.

T: That isn’t part of RAVE; that’s the movement here.

T8: We’ve not really taken the RAVE, we are making up our own. And we have to be aware of that. RAVE is a guideline. We must be careful not to get stuck into that.

The age of the students would also have an impact on how the RAVE equivalent would be implemented by the teachers. T6 felt that it would need to be modified if it were to be useful for her Tau 2 students:

It sounds like it would be good for the seniors, like even the Year 6s sound like they were doing really well with it and the Year 5s do well, I think. Yeah but I don’t think my babies would be able to, unless we simplify it. (Interview November 2007)

Young students who are just beginning to write could well find the demands of responding to RAVE too much. Therefore, a modified version would need to be put together that would support the writing of explanations for this year level.

Implementing teaching actions
Different teachers were implementing different strategies in order to support students’ writing of explanations. In order to support students moving from oral explanations to written explanations, some of the teachers in the junior section of the kura were writing down students’ explanations for them. For example, in the November staff meeting, T1 stated that, “You can write what they say in their own words too; asking them to explain something, you can do the writing for them.” This gave these students’ exemplars to follow while at the same time valuing the students’ own ideas.

T7 had been working with his students on improving their explanations for some time. In the last term, he had used a version of RAVE with his students. An example of his student’s explanation is provided in Figure 71. T7 described how he had implemented different activities to support students’ writing of explanations:

What I’ve been producing and role modelling and then getting them to design their own. Then they read all their stuff out and one boy read his out and his answer was quite good which was surprising for him. He said he took it home and his Mum helped him out . . . in terms of explaining what he was trying to say. Straight away in the question, it was a good example for everyone to follow. Read his out and I made everyone look back at their own to see similarities between theirs and his. The whole of them said no there were no similarities . . . Even they could see the value of it to work at their own level. (Interview November 2007)
Figure 46 T7’s student’s explanation of transformations

By having students compare their explanations with a good piece of writing, students were able to assess for themselves what features would improve their explanations. This is a very important component of making students responsible for their own learning. It was interesting that this piece of writing was produced after the student talked with his mother about it. Although students would be able to ask their parents for help with their mathematics homework, few parents would be able to talk about the mathematics ideas in te reo Māori. This is because many parents either do not speak te reo Māori or were never taught mathematics through it. Consequently they were only able to talk about mathematics in English.

Reflecting on the impact of changed teaching actions
Having students produce written explanations was seen as valuable by the teachers so there was a lot of reflection about the different strategies that various teachers had tried. T1 described how she felt her concentration on having students explain their answers had made a difference to their understanding of mathematical ideas:

But it has been more in my mind when I am trying to get them to explain things to me. Even today it was quite good. I was more aware and that’s one of the best things of the project. They need to give me things and explain what they are doing. We’ve done lots even for basic facts—How do you get it? Why do you get it? They are really good with their place values. They know they are taking a 10 from here and putting it [there]. They can say those things. (Interview November 2007)

The Tau 2 teacher, T6, had already begun having her class write out in words how they would say the symbolic number sentences. An example of this can be seen in Figure 72.
Figure 47 T3 student’s explanation of addition sums

In describing her impressions of the project, she made the following comment about what she had done to support students explain their thinking in te reo Māori:

They have to write out words. And I think that’s worked for my top group because they actually had to think about what they were doing and how they were going to explain it. Before, when we did the whole tāpiri (addition) thing, they’d just go, ‘four plus five equals nine’ but they’re, now, they’ve had to actually think about how they’re going to say it. And what I was getting my kids to do was to try to explain what they were doing, without giving the answer but explain it in a way that somebody else could follow just what they had written. It’s quite hard. (T6, Interview November 2007)

T7 had been the teacher who had been the most successful at trying RAVE in his classroom. It was in the process of trying it out that he had made modifications to it:

T: So it was the process of writing that forced their thinking?

T7: Some of them got it wrong but they justified their answer. How come they ended up with it. Then at the end we added another bit where they look at it and say what they thought of it. I thought, why not have a go and see how it ends up? Have a dive in and have a look. I was noticing if I asked them how they got their answer, the answer had been, I just know, I just did it and it came out like this. Now they justify everything they’ve done, explain to me where they put the rawini tamariki.(?) (Meeting November 2007)

Reflecting on what had happened when he had asked students to explain their answers, T7 found that students were able to justify what it was that they had done. Consequently, he had asked the students to evaluate their explanations and justifications. For this teacher, reflecting on his practice meant that he implemented new practices, not because he felt that the previous one had been inappropriate but because he felt that the students could be pushed further in their mathematical thinking.

Students’ use of te reo Māori in pieces of mathematical writing

Improving students’ grammar and vocabulary was a concentration on superficial acts of writing as discussed in Chapter 6. These improvements in themselves would not guarantee that students became better at thinking about mathematics. However, the teachers felt that without fluent control of the language the students would struggle with being able to express their thinking clearly and thus gain the advantages to their mathematical thinking.

As a kura kaupapa Māori, the focus was always on ensuring that students use appropriate reo Māori to express themselves. There was also a recognition that they needed to gain the vocabulary necessary for discussing a topic. In the previous chapter, the majority of students said that they at least sometimes struggled to find the appropriate term in te reo Māori when writing in mathematics. As most students were second-language learners, the teachers needed to ensure that students were given support to improve their fluency in the language even while learning mathematics.

Identifying student learning needs
There was much discussion about students having embedded errors in their use of te reo Māori. For example, T9 highlighted that many students continued to use “e” as a verb marker even when they used the passive tense.

T5 found it difficult to have students do writing in mathematics when he knew that their reo Māori was not grammatically correct. He felt that the students were fluent in te reo Māori but their thought patterns were in English and so their reo Māori reflected this. As second-language learners of te reo Māori, students can be expected to have interference from their first language, English. However, one of Doerr and Chandler-Olcott’s (in press) teachers in their research project also felt that she was not doing enough writing in her mathematics class because of students’ poor written communication skills.

T2 also felt that there were issues with students’ writing about mathematics in te reo Māori. In her survey, she commented on the students’ omission of grammatical markers when they were writing in mathematics. She went on to state, “even if they have a little to say, [grammar] is important to everything written”. She felt that students were not thinking about the best way to express their ideas. It was if students believed that it was sufficient to put down all their ideas in words without needing to ensure that someone else could follow it.

At the junior end of the kura, the teachers were aware of the students’ need for appropriate vocabulary:

The vocab is the be all and end all, for the Year 3s. If they haven’t got the vocab they are not going to get anything out. That has been my focus all year. If you look at my books for any new whenu [topic] that we do it has kupu hou [vocabulary list]. There might not have been anything that jumped out at them that they thought was kupu hou. If they thought this word was new it was up to them to write it so it’s not just you dictating, it was a natural process happening and realising it themselves. From the vocab you are going to get some writing. (T8, Interview November 2007)

The teachers had identified a number of learning needs of their students. In some cases, these learning needs were seen as a barrier to having students do any writing at all. However, by working with these students, teachers found that the writing became an opportunity for increasing students’ fluency in te reo Māori.

Identifying teacher learning needs
The teachers identified knowledge and pedagogy gaps in their own understandings about writing in mathematics. These then also became a focus for finding ways to fill those gaps.
Most of the teachers themselves had not been taught mathematics in te reo Māori. This meant that they often had to learn new vocabulary at the same time as they were expected to teach it. For example, T1 stated:

I chased my tail for my ähuahanga [geometry] unit. I came to you [T9] asking how to do the rotation. When I came upon it, I didn’t know how to say to them, clockwise, anti-clockwise, and all that. So I should have had that at the beginning.

Another issue was knowing what constituted a good piece of writing. T8 felt that this was an issue for many of the teachers. She felt that it was important for the teacher to ask themselves questions such as, “What were your expectations? Were you satisfied with a sentence? Were you hoping for a paragraph? If that was your expectation, what did you do? We have got to get out of just getting a sample and being satisfied with that” (T8, Interview November 2007). Therefore, teachers needed to be able to reflect on their expectations of students’ writing and be more explicit with these expectations with students.

T3 could identify that providing students with appropriate reo Māori was not a simple task for teachers because there was a risk that they could restrict rather than support students’ fluency in the language:

I am happy so long as it is not too false about putting the language into these kids. It is about their own experiences, their own knowledge and their own language . . . just putting it slowly into them so it fills their vocabulary so they have that ability and freedom to be able to use it when they need it. Otherwise I just don’t want to be teaching this language and the kids not knowing where they are. So it is up to us to make sure that we put it into them carefully and considerately because each one comes with different situations from home so we just have to be careful how we put that language to them. (Interview November 2007)

In having students discuss their emerging mathematical understandings, there is always a tension between encouraging students to use their everyday language so that they are fluent and encouraging them to use the appropriate aspects of the mathematics register that they are still acquiring (Meaney, 2005). Although te reo tātaitai, the mathematics register in te reo Māori, will provide students eventually with succinct, informative ways to describe ideas, being forced to use it before they have fully mastered it may limit their ability to discuss the mathematical understandings.

Designing tasks and experiences
At the November staff meeting, there was discussion about having students repetitively copy grammatical expressions that came up frequently. T9 had done this regularly in his previous school as a way of reducing a number of inherent errors in the students’ reo Māori. He stated that the first 20 pages of his high school students’ workbooks were filled with writing. This meant that passive voice was reinforced in describing mathematical sentences. By the time the students had completed all of this writing, their reo Māori was much improved. Although he had not done this in recent years, he feels that it is important to begin doing it again. T5 also decided to start the next year by having students write a lot so that the correct structures were drilled into them.

In the September staff meeting, T1 described how she had made more of an effort to put more work and words around the room. She recognised that for some students the transition into writing was quite difficult. So she listened to what students said and then wrote it down for them. Then she put the writing on the wall and reminded students that it was there in subsequent discussions. She also put up some of the things that she said so as to provide another model.

Implementing teaching actions
However, there was a recognition that ensuring a whole-kura approach to this issue needed a lot of organisation:

There’s a lot of correction that needs to be done, but that’s throughout their whole programme, not just in maths. And it’s like the lack of vocabulary that they’ve had up till now. But, they’re only six-year-olds, their second year of school, but they know the basics of täpiri [addition] and things like that. It’s more the grammar that they’re finding hard. I got what T9 was talking about though because it’s showing up in Year 2 now, the gaps and things like that. I think that while we were talking, well as everyone else was talking in the hui, they were saying ‘we have to work together’ and all this stuff and we always say that in every hui that we have but nothing ever gets done . . . And we’ve actually scheduled to meet but like maths meetings they always get shifted or they get cancelled. But it would be awesome if we actually sat down and did it, because I know that would help me a lot. I mean we sort of do that in junior school. (T6, Interview November 2007)

Several of the teachers had explicitly provided students with a vocabulary list to accompany units of work. For example, T7 provided a specific vocabulary list for the unit on probability because he felt that students needed explicit teaching of these terms (Minutes September 2007). On the other hand, as was mentioned earlier, T8 encouraged students to keep their own vocabulary lists where they had decided themselves what to include.
To support students’ acquisition of appropriate reo Māori expressions, T8 related how T3’s, T1’s and her own class all learnt a series of sentence structures for mathematics. They concentrated on these sentence structures for three weeks and the children were still using them (Minutes September 2007).

Several of the teachers also tried to make connections between the mathematics that students were learning and other experiences they had. For example, in T5’s video of a lesson on measurement he asked about different units of measures and the things that students had measured in them. At one stage he suggested that a book was a block of chocolate and asked how many were needed to cover a table. Although the students could give him an answer of 11, they also made it clear that blocks of chocolate were never that big, even at the Warehouse. T7 talked about how he had made use of an idea from another teacher in order to explain the concept of transformations:

The first week is learning the concept, learning what that actually means if you can utilise things from their own world like [T10] was saying last night, like transformers. I could explain how it would transform from a car into a human. Yeah. Transformers, yep. Utilising that sort of imagery they can understand. Ultimately getting them to look at something that has been translated or transformed or whatever. Look at the end product which has a separate name altogether, the transformed whatever. But the process is the transformation.

Implementing different teaching approaches was necessary to meet the diverse learning needs of the students. In the process, teachers also learnt about different pedagogical strategies as well as te reo tātaitai (the mathematics register).

Reflecting on the impact of changed teaching actions
Implementation of different strategies resulted in reflections on many of these strategies. T8 felt that one of the advantages of teaching mathematics in the junior part of the kura was that it was possible to use a thematic approach so that mathematics writing could be incorporated into the language lessons. However, T1 also thought that there was a need for more team planning as a consequence of her realising her lack of vocabulary when teaching the transformation unit:

So my classroom practice would mean me being a bit more onto it and going through things and knowing how to say this and this and having [the vocabulary] ready and since we are all doing the same kaupapa [knowledge base] we should all be using them. Perhaps we could do it together, a bit more team planning. (Meeting November 2007)

In the senior classes, T9 had implemented a policy of having students write rules in their own words and then having them discuss them with him. This allowed him to work on their general reo Māori but also improved the students’ language abilities:

You will see in those notebooks [students’ workbooks] what happened is, by increasing the writing, their vocal ability got a lot better. You can explain something to them and they will do it but I am not sure what’s going on in their heads. I know we don’t need to know exactly what’s going on in their heads but what I am talking about is what, the words they’re doing from one part to add on to another part. So when it comes to explaining it stays in the picture . . . so you’ll notice near the end that in the last month or so with that level ones [Tau 11 students], I’ll give them something then they’ve got to go down and write down what’s the rule. If it has been a show-andtell sort of thing, go down, write the rule, bring it back, lets discuss it, and I’ll say something such as, ‘That means to me that I start up here’ and they will say, ‘No, no, you are meant to start down here’. So, [I ask them] ‘Why use ‘Kei’? You should be using ‘e’. So I know to go from here to there. Little things like that.’ (Meeting November 2007)

Two examples of his students’ writing about probability are given in Figure 73.
Figure 48 Two examples of Tau 11 students’ writing about probability


T9 used the improvements that he could see in the students’ use of vocabulary and grammar as support for his continued use of a conferencing approach to students’ writing.

Conclusion

Teacher change was varied. Although all teachers noted some changes in how they engaged students in writing in mathematics, the impact was not the same. Some teachers used the vehicle of a project on writing in mathematics to find a way of dealing with the issue of students not being able to explain their mathematical thinking. This issue resulted in T7 trying and modifying a version of RAVE with his students. He could see significant improvement in both their thinking and their writing as a result. For other teachers, they had tried different ideas but nothing had really resonated with them until the end of the year. Being involved in the project had been the force that had kept them interested, but this alone was not sufficient to make them change their teaching practices. For other of the teachers, something happened along the way so that they had instigated some changes they felt were having a significant impact on their students’ mathematical writing.

It was clear that in making changes to their teaching practices, the teachers did engage both individually and collectively in a teacher-inquiry cycle. This chapter outlines three different issues that teachers engaged with over the course of the project. Being able to reflect on their own professional learning and having a supportive network of teachers within the kura to discuss ideas with meant that they were able to continually monitor the impact of their teaching practices on students’ learning needs.

4. Conclusion

The project was about how to improve students’ writing in mathematics as a way to help them improve their mathematical thinking. The research was based in a kura kaupapa Māori and involved 10 teachers of mathematics considering how to increase the quantity and quality of their students’ writing. The project arose from findings from a previous TLRI project (Meaney et al., 2007) that showed that the students at the kura were doing only limited amounts of writing in mathematics.

This new project, Mathematics: She’ll be write!, involved documenting the writing that was being done in the kura and organising it into genres and progressions. The research also identified the strategies that teachers were using to support mathematical writing. These strategies were ones teachers had used previously as well as new ones they had recently adopted. It was expected that teachers would try new strategies as part of participating in the project.

The theoretical framework for the whole project was that writing was a sociocultural activity that responds to changes in the activity that is being written about, the relationship between the writer and their audience and the method of presentation. These three premises are what Halliday (Halliday & Hasan, 1985) called the field, tenor and mode. Research done using his systemic functional grammar shows how different kinds of meaning are attached to these premises.

The research methodology was ethnographical in that it looked at a particular situation and reports just one set of teachers’ investigations of the mathematical writing that occurred in their classrooms. The research used a number of data gathering and data analysis methods to respond to the different research questions.

There are three parts to this conclusion. The first one is a summary of the findings. The second part is a description of the limitations of the research. The final section is about the directions the kura is now considering for further research.

Summary of findings

The research had two prongs in that it was about documenting what writing in mathematics might look like across the kura and also about identifying the strategies the teachers used in supporting writing. Chapters 3 to 5 discuss the classification of the writing samples and how the categories relate to students’ learning and doing mathematics. Chapter 6 describes the strategies the teachers used to support students’ writing, while Chapter 7 investigates students’ perceptions about writing in mathematics. The final chapter examines changes that teachers made as a result of being part of the project.

In order to investigate the first prong of the research, more than 2,000 pieces of writing were collected during the year. At a meeting in the first term the teachers decided to categorise the samples into three genres: whakaahua (description); whakamārama (explanation); and parahau (justification). As well as these genres, the different modes used in mathematical writing were also identified. The list of these from Chapter 3 is provided in Table 12.
Table 7 Mathematical modes used in different genres

There appeared to be a relationship between the genres, the modes and the audience for the writing. Learning to write whakaahua involves students in learning the conventions associated with mathematical writing and consequently their teachers are likely to be their audience. Frequently, the students only use one mode to describe mathematical objects. Writing whakamārama and parahau requires students to think about the mathematics that they are doing. This often involved a combination of modes and can be written either for others or for themselves as part of the reflection process in learning. How explicit students are in producing their pieces of writing seems to depend upon whether the audience shared in the activity that was the stimulus for the writing.

Whakaahua were the largest set of samples we collected. They were arranged in topic and year-level progressions. The topic progressions showed how extra layers of meaning were added to mathematical ideas. For example, ideas about patterns showed distinct stages from moving between producing simple patterns to providing an algebraic formula to explain how they were formed. Students began learning about patterns through iconic patterns before looking at symbolic patterns. Although the two kinds of patterns coexist for a while, symbolic patterns eventually became the only patterns that senior students engaged with. Some mathematical ideas such as tallies do not change once they have been introduced, while other ideas, such as isometric drawing, appear only briefly in students’ mathematical writing. Year-level progressions developed by the teachers suggest that, although the topic progressions are mostly linear, this is not always the case. Sometimes several stages of the topic progressions could be covered at the same time and occasionally later stages may be introduced earlier but continue on for more years.

Whakamārama and parahau were considered by the teachers to be more beneficial for students learning about mathematics. This is because they required students to think about what they were doing and thus be more reflective about their learning. The combinations of modes that students used suggested that, although mathematics does not have to be read from left to right, often the most salient pieces of information were on the left-hand side of the page. Issues about the quality of the writing were discussed by the teachers in regard to the mathematical accuracy of the writing, the ability to integrate different modes, and stylistic concerns.

Chapter 6 described the strategies teachers used to support students becoming writers of mathematics. These strategies linked to the four stages of the mathematics register acquisition (MRA) model but teachers also showed awareness of the need for students to make use of three different kinds of acts of writing. These acts of writing referred to being able to manipulate the writing instrument in physical acts of writing, through to revising the pieces of writing so that they became a tool for thinking mathematically in deeper acts of writing. These acts of writing were connected to the genres as well as to the four stages of the MRA model.

Students’ opinions on writing in mathematics were provided in Chapter 7, as well as a description of the writing by students in each year level. Students believed that they mostly wrote in their books. They mostly wrote for themselves and sometimes wrote for the teacher. Their favourite kind of writing was doing calculations and their least favourite was writing narratives. This is potentially problematic for the teachers’ programme of increasing the quality of students’ writing through supporting their writing of explanations. This is because explanations and justifications often require narrative input to be combined with other modes of mathematical writing.

The final chapter was about teachers’ change in teaching practice as well as in regard to their ability to be inquirers into their own professional learning. The teachers all believed that they had made some changes to their teaching practices although they were not always certain that these had resulted in improved student learning. However, several of the teachers could relate specific instances where they felt that students had increased the amount and variety of what they wrote. The teachers felt that this was making the students more aware of their mathematical learning processes. Three issues were also used to show how the teachers collectively and individually went through an inquiry and knowledge-generating cycle as a result of considering student and teacher learning needs.

The findings of the research were substantial, especially given that this is an area that has received very little investigation previously. As an ethnography, the results are not generalisable beyond this kura. However, teachers at other kura and also in mainstream schools could find that some of the findings have resonance with their own situations.

Limitations

An ethnographic case study can begin with research questions, but in the collection and ongoing analysis of data it is likely that these questions would change. This was the case with this research. However, the change in the research questions has meant that the data that were collected were not always sufficient to answer the new questions. Although over 2,000 pieces of writing were collected over the course of the year, it was clear that we had not collected enough to be able to say exactly what any student had written over the year or definitively how their writing had changed as a consequence of their teachers participating in the research. However, given the enormous time it took to tame the pieces of writing into electronic files that could be managed and classified, it is hard to imagine how we could have dealt with any more pieces of writing.

It was extremely interesting to talk with Helen Doerr from Syracuse University who had been involved in a similar project over the same period of time with fewer teachers. The amount of funding she received was substantially more and she was amazed at how much we had accomplished on our shoestring budget. The TLRI does a great job of funding research projects. However, as we worked across all year levels and had 10 teachers involved in the project, it would be more useful if we had been able to apply for a different level of funding so that we could have made use of the resources that we had in other ways.

One of our aims for the project had been to have an outside expert provide some professional development to the teachers on writing in te reo Māori or in mathematical writing. This was to widen the level of expertise that the teachers could draw upon. It soon became clear that there was no one whom we could call upon to provide this professional support. In the long term, given that the teachers were all developing their own understandings about the purposes and benefits of writing in mathematics, it was probably more appropriate that they worked through the issues themselves. It did, however, mean that being part of the project and having to turn up to meetings were the only reasons that teachers continued to experiment with writing in their mathematics classrooms for some time. The turning point for some was the attendance at NZAMT10 and listening to Helen Doerr. It is difficult to know whether, if such an event had been provided earlier in the programme, it would have had the same impact. By the time teachers attended NZAMT10, they had already been experimenting with different ways to support writing for three terms and been involved in many discussions about it. It may be that our original idea of providing outside professional development support early in the year may not have had the same impact.

Working in te reo Māori has its difficulties as people who are able to transcribe and translate for a report written in English are not readily available. As had been the case with our previous TLRI project, it was difficult to locate a transcriber of the video data that we had collected. This held up our data analysis and also resulted in our deciding not to try to undertake interviews with whānau that had been one of our original intentions.

We also continued to struggle to collect high-quality videos. This was partly due to the general difficulties of noise levels in classrooms but also of not being able to place the remote microphone in an appropriate place. We found out quite late in the research that it could not be placed in teachers’ pockets without it being knocked so that no sound was recorded. As each year of the project has progressed, we have learnt an enormous amount about the logistics of recording in classrooms.

There were limitations in the data collection for this research. However, given the funding and time we had for the research, our results are substantial and robust. They also lead us on to what further research needs to be done.

Ideas for future writing development

Although the teachers had all believed that they had made some changes to their teaching practice, in the last staff meeting and also in the final interviews there was a sense that more could be done to support students’ writing in mathematics. The following extract came from T10 and describes how she felt it was important for the kura to adopt RAVE in 2008:

Helen was my highlight because for me being new on the programme, I saw where we were wanting to go . . . the difficulty for me was I didn’t know I had the strategies to produce that, change my practice, or to change the practice of the children, and when I listened to Helen and saw the programme she had, that was what I thought was going to change my practice to achieve the outcome that we had. But unfortunately, I also went to one other really good forum. I decided to work on that instead which is on our Poutama Tau basic facts and decided that Term 4 is not a good idea really to try something new that you really wanted the children to really grasp, it went a bit airy fairy. But I think RAVE is the way for us to go but I do believe that we need to decide, do we want to do this? We have to do it as a whole school so that everyone starts together. If we make mistakes, we can share them and when we are incredibly successful then someone says, this really works, I think we should go this way. And we can iron out the wobbles together. It also means that you can talk together about what you might try and how you might try it. I also thought that what we could achieve today was a decision, yes or no, whether we go ahead or not but a small group will need to be put together, how we might do it and when we will achieve those things and what we will achieve in that time frame and they need to be realistic, knowing that everyone takes on everything else, but so we can tick those off as we go. This is really successful. We can call it our own name because it looks slightly different because we run it in the school. But it was the strategy that I lacked, it was the practice I lacked to try to get the children to write, whereas I thought that was an awesome strategy, it was the practice that I lacked, that I could try. I came away from talking to her saying, I think I can do that. This is exactly where we need to go. My disappointment about the conference was because Helen was there we should have captured her in a classroom and found a time when everyone had their other courses and sat down and said, what is the first thing you think that we should do? It was difficult to have those conversations over kai or when the band was playing that ghastly music. We yabbered to her as much as we could. But I thought to myself afterwards we should have taken that hour on whatever day it was and [said], let’s thrash it out now. Because I thought, this is what we need to get my children need to get them writing in maths. (T10, Meeting 5 November 2007)

At the end of 2007, the teachers in the primary section of the kura decided to investigate how to introduce RAVE into their classes. Consequently, T10 was going to approach Helen Doerr for advice via email.
2008 will also see two of the most experienced mathematics teachers take a year’s leave from July. This will result not only in new staff coming into the kura but also new structures being put into place. Consequently, one of the issues to be investigated is that of sustainability of the project, including the incorporation of the new teachers into existing activities. Without funding for 2008, it will be very much up to the teachers to move the project forward with only the resources provided by the kura. Sustainability is an issue for many projects once the funding stops. It is hoped that we can collect data to enable us to investigate this issue.

A note from a researcher

The project has been a fascinating one to work on. The enthusiasm of the teachers and the curiosity that comes from working in an area that no one had investigated previously has been immensely rewarding. It has also been fascinating to see the changes in teachers as they tried things with their own students. In 2004, mathematics was quite a scary subject for some teachers when we first began to discuss the possibilities of working together. Over the past three years, it has become something not only that a teacher teaches but something they discuss with other teachers. It has been wonderful to be part of that process, even if only as the “fly on the edge of the porridge bowl” (Meaney, 2004).
It would be wonderful to see some of the data that we have gathered over the past three years turned into resource materials for other kura. With permission from the parents it may be possible to combine videos with transcripts and pieces of writing to put together a multimedia professional development package that could be used in mathematics inservice days for Māori-immersion teachers of mathematics.

Publications

Fairhall, U., Trinick, T., Meaney, T. (2007). Grab that kite! Acquiring the mathematics register in te reo Māori. Curriculum Matters, 3, 48–66.

Maangi, H., & Meaney, T. (2009, forthcoming). It starts with the young ones: Mathematics, culture and language at the beginning of Māori immersion schooling. Australian Journal of Early Childhood.

Meaney, T., & Fairhall, U. (2009, forthcoming). Me āhua te āhua? Immersing students in Māori matters. In R. Averill & R. Harvey (Eds.), Secondary mathematics and statistics book (pp. 125–137). Wellington: New Zealand Council for Educational Research.

Meaney, T., Fairhall, U., & Trinick, T. (2007). Acquiring the mathematics register in te reo Māori. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice. Proceedings of 30th Mathematics Education Research Group of Australasia (pp. 493–502). Adelaide: MERGA.

Meaney, T., Fairhall, U., & Trinick, T. (2008). Genres in mathematical writing of Māori-immersion students. Paper presented at Topic Study Group 31, International Congress of Mathematics Education 11. Available from https://sandbox.tlri.org.nz//tsg.icme11.org/tsg/show/32

Meaney, T., Fairhall, U., & Trinick, T. (2008). The role of language in ethnomathematics. Journal of Mathematics and Culture, 3(1). Available from https://sandbox.tlri.org.nz//nasgem.rpi.edu/ index.php?siteid=37&pageid=543

Meaney, T., Trinick, T., & Fairhall, U. (2009, forthcoming). The conference awesome: Social justice and a mathematics teacher conference. Journal of Mathematics Teacher Education.

Valero, P., Meaney, T., Alrø, H., Fairhall, U., Skovsmose, O., & Trinick, T. (2008). School mathematical discourse in a learning landscape: Broadening the perspective for understanding mathematics education in multicultural settings. Paper presented at Topic Study Group 33, International Congress of Mathematics Education 11. Available from https://sandbox.tlri.org.nz//tsg.icme11.org/tsg/show/34

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Appendix A: Overview of the topic progressions

Te mahere tuhituhi pāngarau (writing in mathematics)

Kaupae (stage)	Geometric		Iconic	Symbolic		Narrative
	Shapes	Transformations		Whole numbers	Fractions
1	Recognise the outline of basic shapes √ Identify basic shapes	Use grid lines to draw simple reflected object Draw simple translations Use fold lines to draw reflected shape Draw reflected object without grid lines	a) Scaffolded into making iconic representations. b) Able to follow a model	a) Scaffolded to trace the outline of the numerals b) Draw an amount to match the numeral provided	a) Shade in the appropriate part of the diagram. At this level they are just working with a single whole b) Recognise the fraction and shade the appropriate amount. At this stage they are finding the fractional part of a group of things	Student’s not independent writing Recognise words and colours
2	Relate shape to what is seen in the environment Recognise that the shapes are found in the environment	Draw own objects and reflect accurately Draw own objects and translate accurately in one direction Draw simple rotations	a) Read and respond to pictures b) Respond with pictures to answer written questions	a) Recognise the sameness of the pictures and use this to answer questions using symbols b) Produce a numeral and connect it to the right number of objects. The objects are clearly linked to the numerals c) Connect the symbols with the words d) Recognise the arrangement of the digits, represents the value of the number e) Round numbers to the nearest 10	a) Supported to write fractions to describe diagramsb) Write own fractions and clearly relating them to diagrams c) Use diagrams to show fractional parts of amounts d) Independently produce equivalent fractions	Use single words or phrases to describe
3	Make rough drawing of shapes √ Draw sketches that contain all of the essential features of the shapes	Draw reflections with vertical and horizontal orientation Enlarge simple shapes in grid lines Simple enlargement shows scale factor Make own patterns using rotated shapes Draw own objects and translate accurately in one direction	a) Provide more complex representations to respond to questions b) Copy pictures drawn on the board to illustrate mathematical explanations c) Recognise and use perspective in a similar manner to when they are sketching 3D shapes d) Draw pictures representing real objects (roughly drawn still). Extra information is provided with the sketches e) Draw credible representations of real objects	a) Scaffolded into joining numbers together into a consecutive sequence b) Scaffolded to create a backward sequence of consecutive whole numbers c) Multi‐digit numbers are placed in an appropriate sentence d) Independently writing a series of numbers e) Students have to provide the information themselves f) Recognise the order of numbers. Use =, <, > signs to show the relationships between amounts g) Order numbers forward and backward to 100	The fractions are ordered	a) Choose to use words to answer word questions b) Although the answers are still words or phrases there is a need to group them
4	Draw using ruler √	Reflections and translations from a variety of orientations. Labels providedEnlargement shows scale factor a centre of enlargementRotate shapes around a point and label accurately	Arrows are used to describe movement on a hundreds board from one number to another. This is more abstract than the sketches used at earlier levels	a) Pictures are provided to support students to give numbers for answers b) Numbers are now used as answers to specific one‐step problems c) Whole numbers are used in equations but not just as answers to calculations. Numerals are used as answers in different contexts	a) Use diagrams to work with fractions to produce answers b) Fractions are used in questions that students need to answer	a) Short sentences are used to describe something in words b) Use more complex sentences to respond to questions c) More complex sentences are used to describe something
5	Draw and label diagrams with measurements √			a) Use symbols to describe the relationship between a set of objects or numbers √ b) Numerals used to describe simple addition relationships c) Create number story families d) The number stories have more numbers involved but are still  simple one‐step calculations e) The problems provide larger numbers but still just involve one‐step calculations f) Work with inequalities and produce a statement that is true	a) Simple addition of fractions	Use more complex sentences to describe
6	Various features of different shapes are labelled. Thus the similarities between shapes can be suggested √ Draw shapes that use right angles	Shapes reflected and described in words		a) Calculations arranged vertically. At this stage there is no advantage in arranging the calculations in this way b) The calculations are more complex because the numbers contain several digits. At this stage, there is a point in arranging the calculations this way to help get them correct c) The calculations are now without lines so that the students have less support to work out the answers		A series/sequence of sentences used to describe
7	Draw diagrams that contain compass construction marks			a) Calculate using other operations than addition. At this level students are supported with pictures b) The descriptions are involving other calculations than just addition. No scaffolding is provided c) The simple calculations are now arranged vertically d) The calculations are more complex	Use other operations than addition for their work with fractions
8	Draw diagrams that contain angles, measurements and construction marks
9	Further information is added to diagram
10	A lot of information is provided about the shapes √
11	Significant amounts of information are added to the diagrams

Appendix B: Year-level description progressions

Geometry

Years 0–2

Years 3–5

Years 6–8

Graphs

Years 0–2

Years 3–5

Years 6–8

Years 9–11

Iconic

Years 0–2

Years 3–5

Years 6–8

Narratives

Years 0–2

Years 3–5

Patterns

Years 0–2

Years 3–5

Years 6–8

Symbols

Years 0–2

Years 3–5

Years 6–8

Years 9‐11

Appendix C: Patapatai tamariki

Appendix D: Teacher survey

1. Why do you think students should do writing in mathematics?

___________________________________________________________________________ ___________________________________________________________________________

2. What kinds of writing (genres) have you had students do this year in maths lessons?

___________________________________________________________________________ ___________________________________________________________________________

3. Do you think that this is a different range to what you have done last year?

Yes/No

4. Why do you think that is?

___________________________________________________________________________ ___________________________________________________________________________

5. What other kinds of writing would you like to see students use?

___________________________________________________________________________

6. Why is this the case?

___________________________________________________________________________ ___________________________________________________________________________

7. What knowledge of te reo Māori do you think that students need to improve their writing in mathematics?

___________________________________________________________________________ ___________________________________________________________________________

8. Why is this the case?

___________________________________________________________________________ ___________________________________________________________________________

9. What have you tried that was different this year to help students in their writing in mathematics?

_______________________________________________________________________ _______________________________________________________________________

10. Why did you choose to try these things out?

_______________________________________________________________________ _______________________________________________________________________

11. How do you know if they were effective?

_______________________________________________________________________ _______________________________________________________________________

12. What has been the most interesting thing for you about being involved in the project?

_______________________________________________________________________ _______________________________________________________________________

13. What strategies would you like to use next year to help students in their writing in mathematics?

_______________________________________________________________________ _______________________________________________________________________

14. If the project was to continue, what support would you like?

_______________________________________________________________________ _______________________________________________________________________

Acknowledgements

This project would not have been possible without the active support of the kaiako of Te Kura Kaupapa Māori o Te Koutu: Aroha Fairhall, Tracy Best, Ngāwaiata Sellars, Ranara Leach, Heeni Maangi, Anahera Katipa, Maika Te Amo, Vianey Douglas and Horomona Horo. Hera Smith came very late into the project but was enthusiastic about all aspects. We also need to thank the tamariki and their whānau. Without their participation and active support, there would not be a project to report on.

The project also called on the skills of a number of people to help ensure its final completion. These people were: Taitimu Heke-Sellars, Tania Smith, Rawinia Treanor, Kelly Holtermann ten Hove, Chris Ketch, Patty Towl and Nancy Leslie who all provided much time and effort in trying to tame the data so that they could be analysed.

Professor Helen Doerr, from Syracuse University, met with the kaiako at NZAMT10 (New Zealand Mathematics Teachers Conference, September 2007) and provided an enormous number of ideas based on her experiences with middle school teachers in the United States. This has had a lasting impression not just on the teachers who attended but also on the future directions for the project. She also visited Tamsin Meaney in Dunedin and provided useful directions on how to analyse some of the enormous amount of data that was collected.

The Department of Education, Learning and Philosophy at Aalborg University, Denmark, also need to be thanked for providing office space and an Internet connection for the two months when the majority of the writing up of this project was being done. The books in the office were a great inspiration.

We would also like to thank the Teaching and Learning Research Initiative for providing the funding so that this research could be undertaken. A special thank you goes to Garrick Cooper, who gave active support and suggestions as the project progressed.