Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author. CULTURALLY RESPONSIVE TEACHER ACTIONS TO SUPPORT PĀSIFIKA STUDENTS IN MATHEMATICAL DISCOURSE A thesis presented in partial fulfilment of the requirements for the degree of Master of Education at Massey University, Palmerston North, New Zealand Ingrid Cheung 2015 ii ABSTRACT This study examines culturally responsive teaching to support a group of Pāsifika students aged 11-13 years old in mathematical discourse. It builds on previous work which has advocated culturally responsive practices where students learn mathematics through collaborative interaction that fosters greater student participation, engagement, and potentially better achievement in mathematics. In this study, the teacher’s actions drew on Pāsifika cultural practices and the value of the family, respect, and collectivism. This was significant in the establishment of social and mathematical behaviours which were important in supporting the development of productive mathematical discourse. In addition, the communicative and participation structures within the classroom that lead to mathematics learning are also considered. This study was situated in an inquiry classroom. A socio-cultural perspective provided the framework for analysing the classroom context. A case study approach drawing on a qualitative design was implemented. Data was collected through teacher and student interviews, classroom audio and video-recorded observations, and students’ written work. Detailed retrospective analysis of the data was undertaken to develop the findings of this classroom case study. Significant changes were revealed in the shifts of student discourse from long silences and hesitation to asking valid questions and developing mathematical justification with appropriate language and specific terms. The explicit instructional practices developed and implemented by the teacher fostered greater collaborative communication and interaction between group members and this was important in how they made mathematical meaning. The findings provide insights into the multi-dimensional ways that teachers can draw on students’ cultural strengths, values, and practices as invaluable resources which potentially will make a difference in students’ mathematical learning. iii ACKNOWLEDGEMENTS This study has been a collective endeavour. I would like to acknowledge and thank the many people who made this study possible. Most importantly, I wish to thank the teacher for his generosity and willingness to allow me to enter into his teaching world. His teaching beliefs and collaborative spirit contributed greatly to this project and were truly inspirational for me. I would also like to thank the students in his classroom for their amiable humour and keen participation in sharing their mathematics learning with me. I am extremely grateful for the assistance, advice and guidance of my three supervisors: Dr Jodie Hunter, Associate Professor Roberta Hunter and Professor Glenda Anthony. It has been a long journey for me and I would like to thank my main supervisor, Dr Jodie Hunter for her continuous support and encouragement with my research and for giving me the confidence to persevere with this study. I am especially indebted to the help of Associate Professor Roberta Hunter who so willingly gave her time and important professional insights and support. In particular, I wish to gratefully acknowledge the generosity of the following funding agencies for making this project possible: Ministry of Education, Graduate Women Manawatu Postgraduate Scholarship, and Graduate Research Fund at Massey University Institute of Education. Finally, I must acknowledge and thank my husband for his unwavering support and encouragement and my children for their forbearance throughout this long process. iv TABLE OF CONTENTS ABSTRACT ......................................................................................................... ii ACKNOWLEDGEMENTS .................................................................................. iii TABLE OF CONTENTS ..................................................................................... iv LIST OF TABLES .............................................................................................. vii LIST OF FIGURES ........................................................................................... vii CHAPTER ONE – INTRODUCTION ................................................................... 8 1.1 Introduction ............................................................................................ 8 1.2 Background to the study ........................................................................ 8 1.3 Research objectives ............................................................................ 11 1.4 Overview .............................................................................................. 11 CHAPTER TWO – LITERATURE REVIEW ...................................................... 12 2.1 Introduction .......................................................................................... 12 2.2 Discourse in the mathematics classroom ............................................. 13 2.3 Discourse practices within inquiry classrooms ..................................... 18 2.4 Engaging students in mathematical discourse ..................................... 24 2.5 Culturally responsive mathematics teaching ........................................ 27 2.6 Summary ............................................................................................. 33 CHAPTER THREE – METHODOLOGY ........................................................... 34 3.1 Introduction .......................................................................................... 34 3.2 Justification for methodolody ............................................................... 34 3.3 Researcher’s role ................................................................................. 37 3.4 The research setting, sample and schedule ........................................ 37 3.5 Data collection ..................................................................................... 41 3.6 Data analysis ....................................................................................... 44 3.7 Validity and reliability ........................................................................... 47 3.8 Ethical considerations .......................................................................... 48 v 3.9 Summary ............................................................................................. 50 CHAPTER FOUR – TEACHER ACTIONS TO SUPPORT PĀSIFIKA STUDENTS’ ENGAGEMENT IN MATHEMATICAL DISCOURSE .................... 51 4.1 Introduction ............................................................................................ 51 4.2 Culturally responsive teacher actions to develop productive mathematical discourse ................................................................................. 51 4.3 Building on cultural contexts and the home language to engage students in mathematical talk ....................................................................................... 58 4.4 Building group collaboration processes through using collectivism ....... 60 4.5 Summary ............................................................................................... 74 CHAPTER FIVE – DISCUSSION ...................................................................... 75 5.1 Introduction ............................................................................................ 75 5.2 The role of the teacher in creating a culturally responsive classroom to support collaborative discourse ..................................................................... 75 5.3 Building on cultural contexts and home languages to support discourse ...................................................................................................................... 80 5.4 Constructing a culturally safe learning environment to support mathematical discourse ................................................................................. 82 5.5 Enhancing mathematical discourse ....................................................... 86 5.6 Summary ............................................................................................... 93 CHAPTER SIX – CONCLUSION ...................................................................... 94 6.1 Introduction .......................................................................................... 94 6.2 The complex nature of teaching and the learning process................... 94 6.3 Drawing on cultural contexts and home language ............................... 95 6.4 Constructing a culturally safe learning environment ............................ 96 6.5 Using explicit mathematical language .................................................. 96 6.6 Teaching implications .......................................................................... 97 6.7 Opportunities for further research ........................................................ 98 vi 6.8 Concluding thoughts ............................................................................ 99 REFERENCES ............................................................................................... 100 APPENDIX A – INTERVIEW QUESTIONS..................................................... 108 APPENDIX B – MATHEMATICAL PROBLEM TASKS ................................... 109 APPENDIX C – SCHOOL CONSENT FORM ................................................. 112 APPENDIX D – TEACHER INFORMATION SHEET / CONSENT FORM CULTURALLY RESPONSIVE TEACHER ACTIONS TO SUPPORT PĀSIFIKA STUDENTS IN MATHEMATICAL DISCOURSE ............................................. 113 APPENDIX E – STUDENT AND PARENT INFORMATION SHEET / CONSENT FORM ............................................................................................................. 117 vii LIST OF TABLES Table 1. Summary timeline of research schedule ............................................. 38 Table 2. Initial coding categories and sub-categories ....................................... 46 LIST OF FIGURES Figure 1. Wooden place value block ................................................................. 67 Figure 2. Unifix coloured cubes......................................................................... 67 8 CHAPTER ONE – INTRODUCTION 1.1 INTRODUCTION This chapter provides the background context to the study. This context takes into account the international and national calls for changes to how mathematics is taught to students of diverse backgrounds (Bills & Hunter, 2015; Civil, 2014; Johnson, 2010). A focus in the study is on issues of equity in relation to the teaching and learning of mathematics for Pāsifika students. The continuing low mathematical achievement of Pāsifika students in mainstream schools in Aotearoa New Zealand is a challenge for educators and policy makers alike. Educational researchers in the 21st century have shown that a culturally responsive pedagogy could be a possible solution. The primary research objectives of this study are identified and an overview of the thesis is presented. 1.2 BACKGROUND TO THE STUDY 1.2.1 PĀSIFIKA STUDENTS This research project focuses on a group of Pāsifika students and their mathematics learning. The term Pāsifika refers to a heterogeneous group of people who originated from the island nations in the South Pacific. Pāsifika students as a group achieve significantly lower results in mathematics than their European New Zealand counterparts. Although this group obtained a 2.4% increase in achievement from 2012, they are still over-represented in terms of low mathematics achievement (Ministry of Education, 2014). According to the 2013 Mathematics results of National Standards, 60.8 % of Pāsifika students in years one to eight achieved at or above national standards which was still about 14% below the national average and 20% below the Pakeha/European cohort. The Pasifika Education Plan 2013-2017 (MOE, 2013), has called for a focus on lifting the school performance of Pāsifika students to 85% achieving National Standards and NCEA level 2 by 2017. This plan advocates that teachers draw 9 upon Pāsifika cultural values, languages, and identities to make links to curriculum areas and provide Pāsifika students with equal access to quality education. 1.2.2 CULTURALLY RESPONSIVE TEACHING Culturally responsive teaching is deliberate teaching to attend to the mismatch between a student’s home culture and the school culture (Ladson-Billings, 1992). It means that teachers proactively move beyond superficial, culturally appropriate, tokenistic efforts to meet the needs of their students to using evidence and research to inform their practice. Culturally responsive teaching is validating, comprehensive, multidimensional, and empowering (Gay, 2010). It is validating because it affirms and strengthens a student’s identity. It is comprehensive because it addresses the needs of the whole child. It is multidimensional because it encompasses the curriculum, learning environment, student-teacher relationships, instructional strategies, and formal assessments. It is empowering because it enables students to be successful learners and productive citizens (Gay, 2010). The ethic of caring (Noddings, 2008), is central to culturally responsive practices. It is related not only to Pāsifika students’ academic achievement, but also to students’ holistic growth as successful participants in societies that value their own cultures. When teachers truly care about their students, they have high esteem for them and view them as competent. Students, in turn, rise to the occasion by showing high levels of social, cultural, and intellectual behaviour. Teachers provide instructional support in order for students to move from what they know to what they need to know. They model the process, extend students’ thinking and abilities and possess in-depth knowledge of both the students and the subject matter (Gay, 2010). In this study it is of particular significance as to what teacher actions are responsive to engage Pāsifika students in mathematical discourse. Research shows that students of diverse backgrounds often have different ways of knowing, talking, and interacting and their background is not often acknowledged or supported by teachers from mainstream cultures (Delpit, 10 1988). In such cases, poor performance can be linked to inappropriate instructional practices that are insensitive to the social and cultural needs of the Pāsifika students (Tuafuti, 2010). Walshaw and Anthony (2008) contend that effective teachers use a range of organisational and instructional practices to enhance students’ mathematical thinking and ways of communicating. Allowing students opportunities to construct their own solution strategies to solve mathematics problems within their culture is motivating and encourages students to value multiple perspectives (Johnson, 2010). Culturally responsive teaching fits in with reformed mathematics education where the teaching and learning of mathematics emphasises problem solving and effective communication skills (e.g.,Bell & Pape, 2012; Chapin, O’Connor,& Anderson, 2013; Goos, Galbraith, & Renshaw, 2004). An essential notion of the inquiry classroom is one where teachers and students are actively working together to enhance mathematical understanding through effective mathematical practices. Drawing on Pāsifika values of respect, family, and collectivism enables student reasoning by way of explanation, justification, and validation in culturally appropriate ways (Bills & Hunter, 2015). A number of researchers (e.g.,Hunter & Anthony, 2011; Spiller, 2012) have called for further research focused on the development of culturally responsive teaching to foster Pāsifika students’ participation and engagement in mathematical discourse. If learning opportunities are to be created for all, it is necessary for teachers to find out about students’ cultural backgrounds, and what they know and think about while learning mathematics (Bills & Hunter, 2015).This is particularly important in a New Zealand context where Pasifika and Maori students’ underachievement continues to be noted. Furthermore, while international research has reported on culturally responsive teaching for students of various cultural backgrounds (Gay, 2010); research on Pāsifika and Maori students in primary school mathematics settings in New Zealand is relatively limited. It is against this background and for these reasons that this study was conducted on how teachers can support Pāsifika students’ mathematical discourse in culturally responsive ways. 11 1.3 RESEARCH OBJECTIVES The main objective of this study is to explore how a teacher draws on Pāsifika cultural practices and values to engage students in mathematical discourse. The study also seeks to examine the ways in which teachers support Pāsifika students to construct mathematical understanding. A related objective is to explore the classroom environment connecting the effects of specific classroom practices on the participants as they engage in mathematical reasoning. In particular, the following research question will be addressed: How can teachers support Pāsifika students to engage in mathematical discourse in culturally responsive ways? 1.4 OVERVIEW Chapter 2 reviews the literature from both a New Zealand and an international perspective, providing the background in which to situate the current study. The context and framework for the current study are provided through analysing and connecting relevant literature related to culturally responsive teaching that supports mathematical discourse in an inquiry classroom, collaborative interaction and communication, social and socio-mathematical norms, and the use of mathematical language. In Chapter 3, the methodology for the study is discussed. The research setting and sample, data collection, and data analysis are described and a timeline for the case study is presented. Chapter 4 and 5 present the findings of the study and the discussion of these findings. The culturally responsive teacher actions to support Pāsifika students in mathematical discourse are outlined. The teacher’s actions in drawing on Pāsifika values and cultural contexts to develop a safe learning environment and group collaboration to support mathematical discourse are illustrated. Finally, in chapter 6, the study’s conclusion is drawn and suggestions for further areas of research are described. 12 CHAPTER TWO – LITERATURE REVIEW 2.1 INTRODUCTION The previous chapter outlined the background context of the current study. This chapter reviews research literature both from a New Zealand and international context and provides the theoretical framework on which this study is based. In the western world, mathematics education reform has advocated a shift towards increased use of communication and problem solving activities within the mathematics lesson (Goos, 2004). Mathematical practices such as constructing arguments and critiquing the reasoning of others are central to learning and doing mathematics. For example, in the United States of America, (Common Core State Standards, 2012) advocate that teachers need to guide students to: justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose…also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which flawed, and – if there is a flaw in an argument – explain what it is. (p. 6-7) Similarly, within the New Zealand context, New Zealand Mathematics Curriculum (Ministry of Education, 2009), also emphasises problem solving, reasoning, and communicating mathematical ideas. However, there is limited guidance on how to successfully achieve this within New Zealand primary schools, particularly in culturally diverse classrooms. This review of the literature investigates teaching practices to support Pāsifika students to engage in mathematical discourse in culturally responsive ways. Section 2.2 examines literature on socio-cultural theory, in particular the place of the zone of proximal developmentin relation to constructing mathematical knowledge. Section 2.3 examines the critical role of the teacher in establishing the social and socio-mathematical norms which shape effective participation structures in the mathematics classroom. Section 2.4 examines the nature of 13 mathematical discourse and collective interaction in developing conceptual knowledge. The role of language and effective practices in using explicit mathematical language are discussed. Relevant literature is reviewed in Section 2.5 on culturally responsive pedagogical practices and associated outcomes for Pāsifika learners. 2.2 DISCOURSE IN THE MATHEMATICS CLASSROOM Traditionally, mathematics learning at school drew on a model of “whole class, teacher-dominated didactic instruction and individual seatwork” (Forman, 1996, p. 115) which valued memorization of number facts and students obtaining accurate answers through the flawless use of systematic procedures. Given that the teacher was the dominant voice in the transmission of knowledge, discourse interactions between the teacher and students typically an IRE (Initial- Response-Evaluate) model (Goos, Galbraith, & Renshaw, 2004). However, in recent times, mathematics education reform has promoted a shift in classroom practices to focus on communication, collaborative interaction, and understanding of deeper mathematical ideas (Anthony & Hunter, 2005; Kazemi & Hintz, 2014; Wells, 1999). The discourse is focused on generating meaning (dialogic discourse) where the teacher and students are in a more balanced partnership in the dialogue. In the words of Gee and Clinton (2000) discourses are described as: ways of talking, listening, reading and writing - that is using social languages - together with ways of acting, interacting, believing, valuing and using tools and objects, in particular settings at specific times, so as to display and recognise particular socially situated identities. (p. 118) Discourse in a mathematics classroom fosters a learning community as students and teachers interact with each other and engage in meaningful dialogue or talk to make sense of mathematical concepts and conjectures which are negotiated and developed (Cobb et al., 2011; McCrone, 2005). 14 Communication is central to learning mathematics and language is an integral part of discourse to communicate mathematical ideas (Moschkovich, 2012). Using mathematical language is important to develop deeper understanding of mathematical ideas. It involves both social language and specific mathematical terms (Johnson, 2010). As reported in Khisty and Chval’s study (2002) of two highly competent fifth grade teachers, both teachers created a positive learning environment and recognised that interaction among students and between students and teachers was important. However, one teacher neglected the consistent use of rich mathematical language and subsequently her students did not develop their fluency in the discourse of mathematics. In contrast, the other teacher in the study assisted her fifth grade Latino students to develop competent control of mathematical discourse. These students made significant gains in mathematics in explicitly using mathematical talk. For example, a student’s concise explanation of their group’s solution for the perimeter of a three-quarter circle was as follows: we multiply by pi to get the circumference of the circle, then we divide it by four to get the quarter circle. Then we multiply by three to get the curvy part of the three-quarter circle (p.163). The researchers concluded that the teacher’s consistent and explicit use of mathematical talk ensured that the student had access to the words necessary for such an explanation. Other studies (e.g., Johnson, 2010; Moschkovich; 2012; Selling, 2014) also support the notion that mathematical discourse requires explicit teaching and guidance. Moschkovich (2012) argues that teachers need to provide multiple opportunities for students to use mathematical language so that they can internalise the language and express mathematical ideas fluently. Engaging students in discourse fosters the development of mathematical language which enhances the conceptual meaning and understanding of mathematical ideas. Johnson (2010) supports this premise and contends that students need to be exposed to many contexts to give purpose for the language so that they can develop a meaningful grasp when applying a concept to real-life problems. A learning theory which supports discourse as a key way to learning mathematics is the socio-cultural perspective that draws on Vygotskian ideas of 15 cognitive development connecting the person with the setting, social, and cultural factors (Goos, 2004; Sfard & Cobb, 2006). 2.2.2 SOCIO-CULTURAL THEORY Vygotskian cognitive learning theory places emphasis on the social aspect of learning. Learning is seen as a social activity; the social origins of thinking and logical reasoning are created by social processes, inclusive of language and communication. Language has two functions: as a communicative or cultural tool we use for sharing and jointly developing the knowledge – “the culture” – which enables organised human social life to exist and continue… and as a psychological tool for organising our individual thoughts, for reasoning, planning and reviewing our actions.(Mercer, 2000, p. 10) Socio-cultural learning perspectives emphasise the importance of context. Learning is seen as contextualised, which is to view learning-in-activity within social, cultural, and institutional contexts. These social organisational processes are not considered merely as factors which may support or hinder learning, they are integral features of the learning itself (Foreman, 1996). Foreman argued that the three crucial constructs: activity setting, peripheral participation, and instructional conversation, play a central role in socio-cultural theory. The activity setting or learning environment can be understood as the relationship between thinking and learning, as well as the space and cultural tools within which thinking and learning occurs. Instead of viewing learning purely as residing within the individual, participation in the activities of a community is vital to learning (Lave & Wenger, 1991). Within a community, all participants are legitimate members with some members more knowledgeable than others (such as the teacher or older students), while other members (often the new students) are more peripheral. Mathematics classrooms as communities of practice have a united purpose through a common goal and collective social activity. Taking the contemporary participationist approach (Sfard & Cobb, 2006), learning mathematics is conceptualised as joint 16 participation in shared cultural activities that occurs in or outside of classrooms. Within this theoretical view, learning takes place when there are changes in the patterns of participation in discourse. Framing learning as a discursive practice, Goos and colleagues (2004) contend that knowledge is constructed through reasoning and argumentation. As part of the community of learners, the students and the teachers use dialogue as a means of communicating what they know and as a way to construct understanding of new concepts or ideas. The processes of learning and teaching are interactive both involving implicit and explicit negotiation of mathematical meaning. This instructional conversation between teachers and students is not static but it is in a state of construction and reconstruction (Mercer, 2000; Sfard & Cobb, 2006). Mutual accountability regulates the social participation between all participants whereby teachers and students work out who is responsible for what and to whom, what is important, what can be ignored and how to act or speak appropriately (Goos, 2014). Socio-cultural theorists believe that collaboration and dialogue are crucial for the transformation of external communication to internal thought. It is through the act of joint participation in activities that teacher and students are afforded opportunities to learn new knowledge and skills (Bell & Pape, 2012; Mercer & Littleton, 2007). However, it is important to note that participation in a community alone does not ensure significant mathematical learning takes place (Kazemi & Hintz, 2014; Lampert, 1998). Learning mathematics with understanding is a process that requires time for students to develop their wider mathematical practices within the support of a community (Goos, 2004; Yackel, 1995). Therefore, it is necessary to organise the learning environment so that it is socially and culturally safe for diverse learners to make conjectures and to practise explaining or justifying their mathematical ideas (Johnson, 2010; Spiller, 2012). 2.2.3 ZONE OF PROXIMAL DEVELOPMENT In Vygotsky’s original work of social learning, the zone of proximal development (ZPD) was described as being the difference between what a child is able to achieve individually and independently, and what a child can potentially do in 17 collaboration with the significant others (Mercer, 2000). The ZPD is traditionally linked to the notion of scaffolding used by Bruner (1990, 1996). In this arrangement, scaffolding supports the learner to achieve the learning goal. The learner is not a passive member; rather, active participation is required in the negotiation of meaning. In contrast to the traditional ZPD metaphor of an expert guiding an apprentice to learn within the zone, interthinking, a contemporary view of the ZPD perspectives offers a more empowering model of learning (Mercer, 2000). Mercer referred to student inquiry of each other’s reasoning in the ZPD as “interthinking” (p. 141). In this frame, learning is viewed as happening in a mutually unrestricted space which is known as the “intermental” (social) development zone where the shared knowledge and goals of all community members are created. The learning in this zone alters constantly as the students and teacher are required to consult and discuss their way through the activity together. It is a process in which participants in the discussion can see and think together and come to share a point of view or taken-as-shared- knowledge . The idea of participation within a mutual communicative space broadens the traditional view of the ZPD beyond scaffolding and guided participation to one where learning takes place through collective participation and active engagement in meaning making (Goos, 2004; Mercer, 2002). Through joint activity, the participants are able to negotiate each other’s meaning and endeavour to understand the diverse viewpoints of the community (Hunter, 2010). This “intermental” zone allows participants to work through partial mathematical knowledge, misconceptions, confusion, and uncertainty (Goos et al., 2004; Yackel, 2002). This requires the active engagement of all participants so that everyone shares responsibility in the collective inquiry of mathematical understanding. More recently, this “intermental” zone has been linked with the culturally responsive description of a “third space” of intellectual engagement by Lipka, Yanez, Andrew-Ihrke, and Adam (2009) where students’ views intermingle and cross cultural borders (Gay, 2010). 18 Another contemporary perspective of ZPD is the symmetrical model of reciprocal learning that advocated that teachers, and not just students, may be learning through classroom interactions. As Roth and Radford (2010) explain: the zone of proximal development is an interactional achievement that allows all participants to become teachers and learners. (p. 303) Roth and Radford described the reciprocal learning sequence of a geometry lesson. Twenty-two year two students were asked to classify three-dimensional shapes according to their geometrical properties (for example, cubes, spheres, rectangular prisms). The conversation illustrates how the teacher guided the students but in turn they guided her in relation to the assistance that they required. Through the exchange of questioning and the language use in communicating mathematical ideas, a clear model of reciprocal learning was shown as teacher and the students were learning from each other. 2.3 DISCOURSE PRACTICES WITHIN INQUIRY CLASSROOMS The term "Inquiry" is synonymous to reformed mathematics learning. Setting up inquiry classrooms is important in facilitating mathematical learning. According to Cobb and colleagues (2011) students need opportunities to jointly participate in mathematical practices through classroom interactions. They are expected to be actively engaged in thinking, doing, talking, and reasoning mathematically. Within such classrooms, the mathematical practices involve student questioning and participation in meaningful mathematical activity, collaborative work to construct understanding, and the creation of an environment where errors can be capitalised as learning opportunities (McCrone, 2005; White, 2003).The classroom discourse may include whole-class discussions, small groups collectively solving problems, discussion of solution strategies, sharing of conjectures, explanation or justification, and student reflection on their own work or the work of their group members (Lamberg, 2013; Manoucheri & St John, 2006; Kazemi & Hintz, 2014). Within inquiry classrooms, learning mathematics is a collective endeavour (Goos, 2004; McCrone, 2005). All participants, both students and teachers, 19 have responsibility to develop a social community of learners. All students are expected to explain and justify their mathematical ideas, listen and learn from others, and build on each other’s thinking. 2.3.1 SOCIAL AND SOCIO-MATHEMATICAL NORMS Collective participation in inquiry classrooms is shaped by social norms (Goos, 2014;Yackel & Cobb, 1996). Social norms are the common ways in which students take part in any classroom activities and in any curriculum subject. These include such activitiesas questioning, listening, turn taking, explaining, justifying, discussing different ideas, supporting each other within group activities, and making sense of others’ explanations (Goos, 2014). These social norms are also linked to culturally responsive teaching because it is imperative to create “a climate and ethos of valuing cooperation and community in the classroom” (Gay, 2010, p. 197) to promote equitable learning opportunities for diverse learners. Moving beyond social norms, socio-mathematical norms are related to explicit mathematical activities. They include analysing and talking about mathematical concepts, reasoning with a diverse range of tools, offering different strategies, and presenting mathematical arguments to reach a consensus. Additionally, they require the participants to judge what counts as an acceptable mathematical explanation or mathematically efficient solution (Yackel & Cobb, 1996). Students and teachers co-construct the social and socio-mathematical norms of the classroom to ensure equal participation of all students. Through participation in classroom communities, students learn classroom expectations and obligations on how to work on a mathematical activity. As students engage or participate in the negotiation of socio-mathematical norms, they develop mathematical beliefs and values. These help to increase students’ intellectual autonomy and enhance positive mathematical disposition. For example, in a study by Cobb et al. (2011), two first grade students used a foot-strip to measure the height of the cabinet. At first both students had different interpretations of what was to be measured, it only became taken-as-shared understanding when both students collectively agreed on the structured space 20 as a property of the object being measured. With the help of an adult, the students could explain the measurement of the height of the cabinet using a foot-strip. The development of socio-mathematical norms is essential to maintain the productive functioning of a learning community and to guide the quality of discourse within a classroom (Chapin, O’Connor,& Anderson, 2013). As Wood (2002) explained, the socio-mathematical norms regulate productive mathematical discussion or argumentation in the classroom. Makar, Bakker, and Ben-Zvi (2015) agree with this premise and maintain that both teachers and students are required to be explicit about discourse norms. In particular, students are expected to share not only their solution but also their thinking process in order to convince others of the validity of their solution. Furthermore, Kazemi and Stipek (2001) maintain that there are key socio- mathematical norms which are linked to a high press for conceptual thinking. These include: an explanation consists of a mathematical argument, not simply a procedural description or summary, mathematical thinking involves understanding relations among multiple strategies, errors provide opportunities to reconceptualise a problem, explore contradictions in solutions, or pursue alternative strategies, and collaborative work involves individual accountability and reaching consensus through mathematical argumentation. (p. 64) In their study, Kazemi and Stipek (2001) reported on four teachers in grade four and five classrooms, who all taught the same lesson on the addition of fractions. The researchers analysed conversations that created a higher or lower press for conceptual thinking. They found that in a low-press interaction, the class applauded the correct solution without analysis and the teacher glossed over inadequate or inaccurate solutions. However, in high-press exchanges, students explicitly linked their problem-solving strategies to mathematical reasons. 21 2.3.2 THE ROLE OF THE TEACHER IN THE INQUIRY CLASSROOM Teachers play a significant role in the development of an inquiry classroom. Goos and colleagues (2004) investigated the patterns of classroom social interactions that improved Year 11 and 12 students’ mathematical understanding. They demonstrated how the teacher facilitated a mathematical classroom inquiry community of practice. The teacher modelled the desirable mathematical thinking, discourse, and made explicit reference to mathematical language and symbols. Students were required to reflect and monitor their own thinking and reasoning. The teacher advanced student thinking by scaffolding inquiry practices and asking questions such as “how is this?” and “what is the reason for this?”. Furthermore, the teacher expected the students to take ownership in validating their own solutions; each student needed to develop clear explanations and justification of solutions in order to make personal sense of concepts. Across a range of research studies focused on developing inquiry with mathematics classrooms (e.g.,Hunter, 2007; Goos, 2004; Makar et al., 2015; White, 2003), researchers note that the teacher’s contribution to the discussion was to enrich the mathematical dialogue rather than to reduce the cognitive load of the students’ task. These studies show that discourse promoting conceptual thinking can be achieved through specific teacher actions. The teacher takes a key role in promoting students’ engagement in mathematical discourse (Makar et al., 2015; Yackel, 1995). In McCrone’s study (2005) of a year five classroom, the teacher specifically facilitated the development of specific behaviours in the classroom. In the first observation students did not actively listen to each other nor were they able to articulate their reasoning. As a facilitator and participant of the learning community, the teacher steered shifts in the discourse to ensure that students began to use mathematical reasoning and were conceptually focused on the collective task. As the term progressed, students began to listen, interpret, and respond to each other’s contributions. Supporting students’ active engagement is an important part of discourse rich classrooms. Effective teachers organise activities to encourage and support 22 students’ contributions to mathematical discourse, particularly for shy or less confident students. In Rittenhouse’s study (1998) of a year five classroom, the teacher responded to the needs of the students and made the conversation more comprehensible by providing the explicit words for students to participate in the discussion. When students reported back on behalf of a group in the study, they were contributing the group shared ideas rather than an individual idea. This exemplifies the important role of the teacher in guiding students on the peripheral to draw them into full participation in mathematical discourse (Rittenhouse, 1998). Teachers contribute important resources to a discussion by introducing mathematical language, signalling a new idea, connecting with previous learning and summarising key mathematical ideas (Khisty & Chval, 2002; McChesney, 2009). An example from the study by Khisty and Chval (2002) illustrates this. In this case, the teacher consistently made her mathematical talk explicit so that students could access the language they needed to participate in the discussion. In the classroom, students were expected to use correct mathematical terms and complete their mathematical explanation in full sentences. Connections were made between important ideas, for example, the teacher built both on the word opposite which the student knew to connect to a new word inverse and relational understanding that multiplication is the inverse of division. In this case, the explicit use of mathematical language enhanced the conceptual understanding of students. Another key role teachers take is using questions and prompts to develop students’ use of mathematical explanations. Franke and colleagues (2009) showed how teachers effectively used different types of questions and prompts to support students making complete and correct explanations when developing algebraic reasoning in elementary classrooms. Previous studies (e.g., Khisty & Chval, 2002; Moschkovich, 1999) have shown specific evidence that teacher- led questioning supported the development of mathematical language to engage in discourse which led to better conceptual understanding. 23 To foster student active engagement in discourse, teachers can also position students to take a specific stance to justify their thinking in mathematical discourse. In a New Zealand study led by Hunter (Alton-Lee, Hunter, Sinnema & Pulegatoa-Diggins, 2010), she reported that the teacher regularly halted students’ explanations and required students to take a stance: At some point, you are going to have an opinion about it. You are going to agree with it or disagree with it…Make sense of it. If you don’t agree, say so but say why. If there is anything you don’t agree with, or you would like them to explain further, or you would like to question, say so. But don’t forget that you have to have reasons. Remember it is up to you to understand. (p. 12) Through these actions, students become increasingly aware of the importance of validating their opinions with mathematical reasoning, not only to ensure they understand the mathematics themselves but also because they are accountable to the whole learning community in constructing new knowledge. Within the inquiry classroom, teachers use tasks to specifically facilitate students’ mathematical learning (Sfard & Cobb, 2006). Anthony and Walshaw (2007) in a synthesis of research studies that inform practice, commented that “in the mathematics classroom, it is through tasks, more than in any other way, that opportunities to learn are made available to the students” (p. 96). When designing tasks teachers need to consider the mathematical goals, as well as maintaining the level of cognitive demand. Contexts also need to be experientially real to foster student engagement in class discussion (Jackson, Shanhan, Gibbons, & Cobb, 2012). Jackson et al., (2012) argue that students engage in complex mathematical tasks when teachers discuss the contextual features and any unfamiliar language (specific terms or phrases) of the problem. These researchers explain that it is equally important for all participants to develop a common language to describe the key features of the task during discussion. 24 Challenging tasks foster student engagement in productive discourse with the careful support given by the teacher (Cobb et al., 2011, Rittenhouse, 1998). In enacting challenging tasks, the teacher takes a more active role than “not telling”. The teacher is required to actively listen to the students’ ideas in order to relate to the contributions made by various students about the tasks (Brodie, 2007). In addition, sometimes teachers need to provide additional information, missing links or an overall picture for the discussion to shape the mathematical ideas that are worth talking about (Brodie, 2007). Also, the teacher may need to provide additional support such as enabling or extending prompts to assist students to participate in the discussion (Sullivan, Mousley, & Jorgensen, 2009). It is the constant interactive support from the teacher or other members of the learning community that fosters students’ willingness to persevere in finding solutions for the tasks which in turn enhances conceptual understanding. The discussion above highlights the key role of teachers in influencing the students’ perception about their roles and their expectation of their peers in contributing towards productive mathematical discourse (Manoucheri & St John, 2006). We see from the research studies (Hunter, 2007; Khisty & Chval, 2002; Rittenhouse, 1998) that teachers have a complex role in an inquiry classroom as a task designer, participant, commentator, monitor, and facilitator, but primarily, their role is to promote the development of conceptual knowledge in students and to facilitate shared knowledge in the classroom community through mathematical discourse (Cobb et al., 2011). 2.4 ENGAGING STUDENTS IN MATHEMATICAL DISCOURSE Mathematical discourse requires mutual collaboration between students and students with the teacher. Effective collaborative interaction requires the students to be active listeners and critical participants. This socialisation process is not an easy task and it takes time to achieve the desired mathematical discourse (Chapin et al., 2013). Franke, Turrou, Webb, Ing, Wong, Shin, and Fernandez (2015) analysed various support moves (e.g. probing, scaffolding, positioning) that teachers used to engage students with 25 each other’s idea, they argued that it was the responsive-in-the-moment support move that allowed students learned how to listen actively to each other and build ideas together in mathematically detailed way. Some researchers (e.g., McCrone, 2005; Wegerif & Dawes, 2004; White, 2003) argue that an effective way to engage students in mathematical discourse is through the use of small groups. Small group interaction can lead to powerful learning. Wegerif and Dawes (2004) describe how: Children working in groups can offer one another chances to explore their conceptions, to employ their new vocabulary, and an audience for explanation, planning, suggestion and decision-making. In this way children learn to speak the language of maths. Challenges and explanations in groups, guided by teachers, can lead children to learn more expert ways of talking. (p. 102) Small group discussion offers many opportunities for students to engage in collaborative dialogue to support the development of mathematical thinking and the resolution of different points of view. The interaction between students helps them to develop new knowledge that makes sense to everyone in the group. Interactions with peers can empower learning. Goos (2004) explained the way in which one student viewed his interactions with peers in his senior mathematics class as an enriching learning experience: Adam helps me … see things in different ways. Because, like, if you have two people who think differently and you both work on the same problem you both see different areas of it, and so it helps a lot more. More than having twice the brain, it's like having ten times the brain, having two people working on a problem (p. 278). However, it is important that teachers provide a supportive group structure - a safe space for asking questions, clarifying ideas testing conjectures, giving and taking critical feedback, and building upon others’ strategies and solutions. For example, White (2003) found that English language learners with limited English were more comfortable to share their thinking with a friend rather than with the 26 whole class. Students clarified their thinking about the context of the problem and put forward a conjecture through peer discussion. "A lot of the time they won’t share something with the whole group. But they will share it with somebody sitting next to them, or they can sometimes get ideas from other kids who are sitting next to them." (p. 42) Students working collaboratively on solving challenging tasks show a greater level of cognitive engagement than those working independently (Walshaw & Anthony, 2008). 2.4.1 USING EXPLICIT MATHEMATICAL LANGUAGE IN DISCOURSE Research studies show that mathematical language is central to learning mathematics (Anthony & Walshaw, 2009; Brevik, Fosse, & Rødnes, 2014; Pimm, 1987; Schleppegrell, 2010). When students display fluency and accuracy in using mathematical language in discourse it furthers the development of mathematical reasoning. Developed through social interaction, discourse and language can be seen as a means for organising thinking for logical reasoning (Bruner, 1986; Mercer & Littleton, 2007). A range of pedagogical practices have been suggested to help students use mathematical language within classroom discourse. In the New Zealand context, Latu (2005) demonstrated that those Pāsifika students that were able to code switch between a first language and the language of mathematics (in English), performed better than those who had only restricted forms of English as their first language. The practice of “translanguaging” — where students receive task information in one language but discuss and record their thinking in another language of choice— has been found to be effective in supporting engagement in discourse in the studies by Garcia & Wei (2014). Other studies (e.g., Khisty & Chval, 2002; Moschkovich, 1999; Selling, 2014) have shown that English language learners are able to gain fluency and accuracy in mathematical discourse when teachers focus explicitly on the rich 27 use of mathematical language and specific terms. For example, Khisty and Chval (2002) documented how a teacher supported her 5th grade Latino students within collaborative problem solving activities. It was found to be important to give the students sufficient time to understand the language in the problem and provide multiple opportunities to practise explaining and justifying their mathematical reasoning using the correct mathematical language. The teacher frequently used mathematical words in her talk and capitalised on students’ cultural knowledge to make links between mathematical language, students’ understanding and their home language. In this study, the teacher capitalised on the students’ knowledge of Spanish to have them construct a meaning for quadrilateral (by connecting it to the Spanish word cuado). The teacher used her talk not only to extend students’ understanding but also to connect it to the meaning of a specialised mathematical term which led to the big mathematical ideas in the problem. Moschkovich (1999) outlined how a third grade classroom of English second language learners shifted from an informal use of terms to precise mathematical language. The teacher “did not focus primarily on vocabulary development but instead on mathematical content and arguments as he interpreted, clarified and rephrased what students were saying” (p. 18). The teacher listened carefully to what the students were saying, probing and revoicing what they said to maintain focus on the mathematical content of their contributions. As a result, the students gradually mastered both the use of the mathematical language and knowledge of how to participate in mathematical discourse. 2.5 CULTURALLY RESPONSIVE MATHEMATICS TEACHING Many studies have written about culturally responsive teaching approaches which resulted in successful learning outcomes for diverse students of different ethnicities (e.g., Au, 1993; Averill, Te Maro, Taiwhati & Anderson, 2009; Civil, 2014; Escalante & Dirmann, 1990; Gay, 2010; Johnson, 2010; MacFarlane, 2004). Culturally responsive teaching is a pedagogy that “empowers students intellectually, socially, emotionally and politically by using cultural references to impact knowledge, skills and attitudes” (Ladson-Billings, 1994, p. 17). 28 Culturally relevant teaching incorporates students’ culture into the curriculum to draw on history of students’ lives as well as unique ways of communicating, behaving and knowing while preparing students to effect change in society, not merely fit into it. (Ladson-Billings, 1994, p. 17) One way of developing culturally responsive teaching is for educators to focus on the strengths, that is what the students know instead of what they do not know. White (2003) analysed two third-grade teachers’ classroom discourse practices with African American and Hispanic students. In contrast to common practices where teachers often engaged students in repetitive and unchallenging tasks, these two teachers focused on developing students’ mathematical competence and creative thinking. They encouraged students to solve problems using their cultural knowledge and resources. They facilitated mathematical thinking by discussing students’ ideas and encouraging them to analyse answers to the questions being posed by others. They valued students’ ideas, allowed students to share their thoughts without judgement, and encouraged students to take risks to increase the variety of responses. Both teachers focused more on students’ thinking and their different solution strategies and less on the correct answer. By asking students to explain their answers, they not only learned how students thought about the problems but also provided the class with multiple ways to think about and solve problems. This study highlights that engaging students in mathematical discourse both maintains a focus on sense making and reasoning while also enabling teachers to reflect on students’ understanding and to stimulate mathematical thinking. Research studies on the use of successful culturally responsive practices offer invaluable insights into how teachers can capitalise on students’ culture to facilitate learning. A New Zealand study by Averill and colleagues (2009) looked at a bicultural framework of integrating English and Maori for culturally responsive teaching. The framework incorporated Maori concepts (such as harakia or prayer, kapa haka – performance, waiata – song and marae or meeting house) with teaching strategies (such as reciprocal learning – ako) (Averill et al., 2009). Similarly, Bills & Hunter (2015) and Johnson (2010) highlight the necessity of teachers incorporating students’ culture into classroom 29 practices by designing task problems that reflect students’ culture, using words in their home language, and building on cultural norms. English language learners, no matter how accomplished their English speaking becomes, still have their native language as a resource (Johnson, 2010). Students may think or reason mathematically to themselves in their native language or sometimes in English or in both languages. Schleppegrel (2010) advocated that teachers should utilise cultural tools of students (such as language, cultural nuances, logic, rhythm, gestures, drawing, materials) as invaluable learning resources in the mathematics classroom because they offer a different perspective in constructing mathematics knowledge. Using these cultural tools from the students’ world fosters greater student engagement in mathematical discourse. Moschkovich, (2010) states that teachers should make links with what students bring from home or communities to build on new knowledge and skills in the mathematics classrooms. Civil (2014) extends this notion and argues the need to broaden mathematical communication beyond the normal oral or written exchanges and in English language only. She stresses that the richness of students’ thinking in mathematics in their home language, their knowledge and experience should count towards mathematical development. Central to culturally responsive pedagogy is the caring perspective described by Noddings (2008). The teacher who genuinely cares for the students’ learning andorganises social and cultural conditions to establish a climate of mutual care and trust. According to Gay (2010) culturally responsive caring goes beyond feelings of empathy and kindness; the focus is turning the students’ personal interests and strengths or their cultural ways of doing things into opportunities for academic success. As discussed earlier, dialogue is a powerful learning tool and it is also a culturally effective way to learn mathematics (Gay, 2010). All participants learn from one another through dialogue, not trivial small talk but in the search for meaning and deeper understanding of the tasks. Noddings (2008) argues dialogue is important to learning and it shows how students care for each other’s learning. As they talk about their ideas the language they use will be expanded and polished. Progressively, they can develop a logical reasoning through individual contributions. 30 Teachers who truly care about the development of their students’ mathematical competency show interest in how students construct or express their ideas, no matter how unexpected or unconventional they seem. It is by modelling the practice of evaluating of each other’s ideas through dialogue that teachers encourage their students to make logical judgements about the ideas voiced by other class members (Anthony & Walshaw, 2009). Classroom routines should be in place so that caring can be encouraged and monitored. Noddings (2008) contends that organising for learning in groups provides opportunities to strengthen the ethos of care. However, for group work to be successful, teachers need to continually remind students that “they are engaged in this work to help one another – not simply to produce a better product or surpass another group” (Noddings, 2008, p. 171). In a socially and culturally safe classroom, everyone is encouraged to show their mutual understanding of care through respectful dialogue. A key component of respectful dialogue is the use of inclusive language such as “we want to know”, “what happens when we…”. Noddings advocates that teachers need to be socially and culturally aware of the unpleasant behaviour that may happen. For example, a classroom member can potentially pick on the more vulnerable students, changing the learning atmosphere from caring to competing. Hence, Noddings (2008) believes that it is necessary for teachers to have a strong grasp of interpersonal reasoning and to maintain caring relations during dialogue when a student is feeling distressed or uncomfortable with the direction the dialogue has taken. Teachers may need to interrupt the flow of discussion to assure the student that “he or she is thinking well” and assure students that it is acceptable to experience indecisions or frustration when working through complex tasks. It is important to note the conflict of beliefs and emotions that may emerge as Pāsifika learners socialise into the community of mathematical inquiry where practices such as questioning, disagreeing, and challenging have not been common experiences for the students in previous classrooms. Spiller (2012) explains the Pāsifika value of humility may influence some students to hold back from expressing their views. They may refrain from contradicting what has 31 been spoken because they did not want to look clever in front of their peers. Teachers therefore need to be aware of these values so they can work with students to address these issues sensitively. Pāsifika learners encompass a diverse group; however, Anae, Coxon, Mara, Wendt-Samu, and Finau (2001) highlight a common set of cultural values which are important to all Pāsifika people. These Pāsifika values include: respect, reciprocity, communalism, and collective responsibility. In Hunter’s (2008) study, the teacher incorporated Pāsifika values into the community of mathematical inquiry. The requirement that students worked collaboratively was framed within an appropriate cultural setting (preparing an umukai [village feast] and the collaborative roles all participants held). In a year seven and eight classroom, the teacher guided students’ attention toward Pāsifika concepts of reciprocity, collectivism and community as the students developed mathematical explanation, representations, and justification with their groups. He called on their concept of respect and reciprocity as prerequisites to actively listening, questioning, checking the understanding of all the members of the group and supporting each other when reporting back to the class (Hunter & Anthony, 2011). The relationships between teachers and students were socially caring and responsive which was central to their positive outcomes. Research studies (e.g., MacFarlane, 2004; Johnson, 2010; Tuafuti, 2010) show that when collaborative discourse is practised in culturally diverse classrooms students feel secure and empowered when their language, culture, and power is shared. Tuafuti (2010) contends that the culture of silence has relevance in learning for Pāsifika students. It is a sign of respect to people who are in position of authority such as teachers and elders and it is expected for students to listen attentively and learn from the teacher. It is viewed as disrespectful to argue or question teachers or peers. However, students need to be taught how to disagree and argue mathematically to learn mathematics (Spiller, 2012). Tuafuti (2010) advocates that classroom practices should value culture and empower active participation of students in discourse so that their voices are heard. Similarly, Fletcher and colleagues (2005) argue that maintaining the cultural identity of Pāsifika students is one of the important factors in helping 32 Pāsifika students to succeed in school. Gay (2010) agrees with this premise and suggests that teachers can use culturally appropriate ways to encourage students to contribute to mathematical discussion such as allowing longer thinking time, make the language accessible for them, story-telling, repeating instructions, or choral reading or any preferred way chosen by the students. Caring for the students’ learning is central to culturally responsive practices. Spiller (2012) claims that learning is more effective when teachers take full responsibility for their Pāsifika students. Pāsifika students prefer a learning environment where they have a space to think, and they are allowed to do work for themselves. The work should be interactive and challenging so that the students can respond to the learning opportunities that are purposeful for them. More importantly teachers should: …show them respect as a person, speak quietly to them, listen attentively to them when they have something they want to say and respond with respect to their ideas and questions…do not singled them out for help, they will ask for help when they need it and they want to be allowed to ask their friends first. (p. 65) In other words, it would make a difference to Pāsifika students’ achievement in mathematics if they were given dignity and opportunities to participate in their learning. 33 2.6 SUMMARY To meet the needs of diverse learners, mathematics education reforms advocate changes to teaching and learning practices that include a focus on personal, social, and cultural factors. Mathematics inquiry classrooms reflect the aims of reform education in that they provide opportunities for students to become active participants of an effective learning community, and construct mutual understanding through collaborative discourse. Within inquiry classrooms, the teacher takes on an important role in guiding the construction of the social and socio-mathematical norms associated with productive mathematical discourse. The literature also highlights the pivotal role the teacher takes in guiding the students’ roles as active listeners and participants to ensure productive mathematics discussion and collaborative interaction happens. Many research studies have shown how students’ mathematical reasoning is enhanced through participating in discourse with the appropriate use of mathematical language. Adapting culturally responsive pedagogical practices to enable all students to participate and contribute in mathematics classrooms is a key equity issue. Emphasis is placed on using the cultural capital of Pāsifika learners to improve their participation, engagement, and outcomes in mathematical learning. In the increasingly diverse classroom contexts, mathematics teachers are in fact language teachers of mathematics as well as cultural facilitators. Teachers’ actions are central to the orchestration of productive mathematical discourse in a culturally responsive and collaborative learning environment. When students are positioned to be held accountable for completing the mathematical activity, they are empowered to become academically competent in mathematics with the support of the learning community. 34 CHAPTER THREE – METHODOLOGY 3.1 INTRODUCTION The previous chapter outlined the theoretical framework for the current study. This chapter provides an overview of the research design. Section 3.2 presents justification for the selection of a qualitative approach for this research project and describes the use of a case study method. Section 3.3 outlines the role of the researcher. Section 3.4 provides details on the setting, the participants, and the research schedule. Section 3.5 discusses the data collection methods used in this study. Section 3.6 explains the process of data analysis. Section 3.7 considers the steps taken to preserve the validity and reliability of the findings of the study. Section 3.8 summarises the methods used to ensure that ethical standards were maintained at all times. 3.2 JUSTIFICATION FOR METHODOLOGY 3.2.1 QUALITATIVE RESEACH – AN INTERPRETIVE PARADIGM This study aims to investigate the key research question: How does a teacher support Pāsifika students to engage in mathematical discourse in culturally responsive ways? The study is guided by a qualitative interpretive research paradigm (model of inquiry). This view frames reality as a social construct in which the interpretation of the lived experiences needs to be understood from the views of the observed in context (Burton, Brundrett, & Jones, 2014). Often, there are “multiple realities or interpretations of a single event” rather than a single observable reality (Merriam, 2009, p. 9). Taking Patton’s description (1985, quoted in Merriam, 2009): Qualitative research is an effort to understand situations in their uniqueness as part of a particular context and the interactions there. This understanding is an end in itself, so that it is not attempting to predict 35 what may happen in the future necessarily but to understand the nature of that setting – what it means for participants to be in that setting, what their lives are like, what’s going on for them, what their meanings are, what the world looks like in that particular setting – and in the analysis to be able to communicate that faithfully to others who are interested in that setting… The analysis strives for depth of understanding. (p. 6) This study strives to provide an in-depth understanding of the ways in which a teacher supports Pāsifika students to participate in mathematical discourse. Qualitative research was chosen for this study because it allows a systematic investigation to take place in a social context. It focuses on the processes rather than the product. In terms of the current study, the qualitative research design aims to capture the different aspects of culturally responsive teaching which are drawn on to facilitate classroom discourse. A key emphasis is on making meaning from the perspectives of the teacher and the Pāsifika students in the natural setting—one primary mathematics inquiry classroom. Qualitative research employs an inductive approach to “find a theory that explains their data” (Goetz & LeCompte, 1984, cited in Merriam, 1998, p. 4). This contrasts with quantitative research which uses a deductive approach to find data to match a theory (Yin, 2012). Qualitative studies are undertaken because “there is a lack of theory or existing theory fails to adequately explain a phenomenon” (Merriam, 2009, p. 7). While there is a body of literature that documents the discourse patterns in mathematics inquiry classrooms (Chapin, O'Connor, & Anderson, 2009; Civil & Planas, 2004; Cobb, Stephan, McClain, & Gravemeijer, 2011; Hunter, 2007; Kazemi & Hintz, 2014), there is a lack of classroom-based research that specifically examines how teachers support Pāsifika students to engage in mathematical discourse in culturally responsive ways. 36 3.2.2 CASE STUDY After a critical review of various research methods, a qualitative case study approach was adopted for the research design. Case study research enables a detailed investigation of one site and one group in a naturalistic environment encompassing historical, social, and cultural contexts. According to Yin (2010), case study is a holistic form of research. It is characterised by multiple sources of data being collected to generate rich descriptions in order to support the theoretical assumptions, and to build on knowledge that supplements further research (Burton et al., 2014; Merriam, 1998). In this study the research builds on concepts, hypotheses or theories rather than testing existing theory. A case study of one classroom is used to describe a teacher’s culturally responsive actions to engage Pāsifika students to talk about their mathematical thinking and ideas. This study explores the construction of mathematical discourse within an authentic setting of a classroom. It also investigates the culturally responsive tasks and teacher practices which support Pāsifika students’ mathematical learning. There are many varied forms of case study designs; each with its own purpose, methods, and complexity (Berg & Lune, 2012; Merriam, 1998). The case study research can be solely descriptive in nature or interpretative where data is analysed to develop categories attributed to the task; or descriptive and evaluative which permits explanation and judgement to take place within data analysis (Merriam, 1998). An interpretive case study is appropriate for this project because the data provides holistic descriptions of the classroom interactions which are used to support theoretical conclusions. Rather than just describing what was observed or what the students’ responses were to their teacher’s instructional practices, taking the interpretative approach allows the researcher to analyse all the data and develop categories of culturally responsive practices. In this way the study aims to develop an in-depth understanding and expand the range of interpretation of the teacher’s actions. 37 3.3 RESEARCHER’S ROLE Merriam (2009) contends that the researcher is the main instrument in qualitative studies. The role of the researcher in this study was as the sole collector of data and as a participant observer. Initially, the researcher spent three days in the classroom getting to know the teacher’s mathematics programme and observing the students’ engagement in mathematical discourse. The students understood that the female researcher was a primary school teacher who had been teaching students of similar Pāsifika backgrounds. The relationship between the teacher and researcher was professional. As the primary instrument for gathering and analysing data, the researcher needs to collect, interpret, understand, and produce meaningful information (Berg & Lune, 2012). The researcher’s experiences in using an inquiry approach to teach mathematics with Pāsifika students meant that the researcher was familiar with expected classroom practices, learning objectives, and potential pitfalls or outcomes. The researcher shared her interpretation of events with the teacher during each visit and provided space for the teacher to either support or refute the researcher’sviewpoint. 3.4 THE RESEARCH SETTING, SAMPLE AND SCHEDULE This section outlines the setting for the study, information on the participants and the phases of the study. 3.4.1THE SETTING AND THE SAMPLE This research was conducted at a suburban full primary school during term one and term two of the 2015 school year. Kelly School (pseudonym) has a decile1 rating of one. The students attending this school are from low socio-economic 1 Each state and integrated school is ranked into deciles ranging from 1 to 10. Decile 1 schools draw students from low socio-economic communities. The lower the school’s decile, the more state funding it receives. 38 communities and predominantly of Pāsifika ethnicity. Each teacher at Kelly School has been involved in the professional development to support the development of mathematical inquiry communities in their classrooms for the past two years (see Hunter, 2007). The research took place in a Year Seven and Eight composite class. The teacher, Mr J, has taught for eight years and was in his third year of working to build mathematical inquiry communities. Following an invitation from the researcher, Mr J agreed to be involved in the research, viewing it as an opportunity to reflect on his own practice and provide insights towards his postgraduate study. There were 28 students in this class and 15 Pāsifika students agreed (with parental consent) to participate in the research. Mathematics in this classroom consisted of the students working collaboratively in heterogeneous groups to solve mathematical problems. The problems were set to cover Mathematical Curriculum Levels from Level Two to Four of the New Zealand Curriculum Document; equating to Numeracy Level Stages Five to Eight of the Numeracy Professional Development Projects (Ministry of Education, 2009). 3.4.2 THE RESEARCH STUDY SCHEDULE This study consisted of three phases of data collection over five months (February-June, 2015) and involved 15 lessons. The summary research timeline (see Table 1) provides an outline of the research schedule and further details of the activities and problems used in each three lesson block are provided in Appendix B. Table 1. Summary timeline of research schedule Date Field work details Phase one- 6 lessons Term 1 Week1 Initial Meeting Meeting with the school principal and teacher participant regarding research’s purpose and outline, discuss consent for school, teacher, and students Week 2 Researcher is introduced to class by the teacher 39 Three days of observations Classroom Task: Filling up the petrol (See Question 1 Appendix two) Two meetings with teacher participant to discuss about lesson plans and students backgrounds. Collate the consent forms from the students/ parents/teacher Week 7 Meeting with the teacher to discuss Term 1 week 2 data Three days of classroom observations Classroom Task: Sunday Feeds (See Question 2 Appendix two) Phase Two - 9 lessons Term 2 Week 1 Meeting with the teacher to discuss Term 1 week 7 data Three days of classroom observations Classroom Task: Estimating volume and measurement for Ta’ovala (Tongan’s traditional mat skirt- See question 3 Appendix two) Week 4 Meeting with the teacher to discuss Term 2 Week 1 data Three days of classroom observations Classroom Task: finding the area of a Tivaevae (Cook Island’s patterned quilt- See question 4 Appendix two) Week 7 Meeting with the teacher to discuss Term 2 week 4 data Three days of classroom observations Classroom Task: Finding fractions, percentages and decimals using the sharing of pizzas – See question 5 Appendix two) Individual interviews with fifteen students Post-interview with teacher participant Phase One: Preparation and data collection An initial meeting between the researcher and the teacher took place before commencing any data collection. Their purpose of the meetings was to discuss the research plan, the objective of the study, the timeframe for data collection, and the consent forms for the teacher, students, and the school. The teacher 40 introduced the researcher to his class and explained the purpose of the project. Students were invited to participate and consent forms were sent home. In the second visit, the consent forms from the teacher, parents, and students were collated. The researcher spent six lessons in the class with the video recorder positioned facing the teacher and the audio recorder by the teacher at the front of the class so that students could become familiar with the presence of the researcher and the data collecting tools. Phase Two: Data collection Data collected during the nine observed lessons included video and audio footage of the students’ collaborative interactions during problem-solving activities, written samples of group work, teacher developed lesson objectives, and researcher field notes. Individual interviews with 15 students were conducted at the end of the study. Additional data included reflective discussions between the researcher and the teacher following each lesson. These discussions were audio-recorded and transcribed as part of the lesson’s footage. Transcriptions of the lessons and data from interviews were analysed by the researcher to identify a range of actions the teacher utilised. The results were sent to the teacher participant and meetings were held with the teacher prior to the commencement of the next phrase to cross-check the validity of the data. Every lesson followed the same format. They were 50-60 minutes in length. They began with the teacher launching a contextual mathematical problem to the big group. The students in small groups of three or four worked collaboratively to solve the problem for approximately 15-20 minutes. During this time the teacher monitored student reasoning and interjected in group discussions as necessary. This included him seeking clarification on the written work presented, extending the mathematical discourse, and ensuring social and socio-mathematical norms were being enacted. In the final plenary, the small groups were called together to form one large group. The teacher then carefully sequenced the student discussion of the reasoning they used to solve the problem. 41 At the conclusion of the lesson observations, the researcher interviewed the teacher (See Appendix A). The questionsexplored how Mr J used culturally responsive ways to support Pāsifika students’ engagement in questioning, agreeing, disagreeing, mathematical explanations, and argumentation. Fifteen students were interviewed (See Appendix A for interview schedule) to explore their perceptions of the teacher’s actions and support for their mathematical learning. In a post-research interview the researcher and teacher reflected upon and verified the emerging data. 3.5 DATA COLLECTION Case study research provide flexibility and multiple sources of evidence (Yin, 2012). Employing multiple methods of collecting data allows rich and unique data to be surfaced (Lichtman, 2013; Merriam, 2009). Multiple sources of data are used because “no single source of information can be trusted to provide a comprehensive perspective” (Patton, 1990, p. 214 quoted in Merriam, 1998). Multiple data can help the researcher uncover meaning, develop understanding, and discover insights relevant to the research problem. The qualitative data collection for this study included observations, interviews, classroom artefacts, and detailed field notes (commentaries of the lessons observed and reflections on the interviews). Triangulation of data collected through multiple sources of evidence, enabled the establishment of the validity and reliability of the study. 3.5.1 OBSERVATION Observations, a primary data source in qualitative research (Yin, 2012), were made through video recording, with the aim of capturing the moment-to-moment detail of complex classroom interactions. Audio recordings were made simultaneously to ensure clear audio data. Video-recorded observation has become widely used in research to collect and archive large amounts of both 42 visual and audio data within the natural contexts of classrooms (Berg & Lune, 2012; Burton et al., 2014). Viewing the footage offers time for reflection on what has been observed (Sherin, Linsenmeiser, & Van Es, 2009). This study involved a sequence of video-recorded teaching episodes. In each lesson, the primary objective was to record the interaction between the teacher and the students. At the beginning of each lesson, the teacher normally outlined and discussed the targeted mathematical problem with the students. Following this, the camera was positioned to capture discussions between the teacher and one specific group. Finally, a sharing session was recorded where the small groups joined together for a large group discussion to share their strategies or ideas on the problem. The video-recorded data became a permanent record and was readily accessible for subsequent review or analysis. Viewing the video footage following the lesson provided the researcher and the teacher participant with opportunities to validate interpretations of students’ responses and the teacher’s actions made during instructional activities. Although video is a valuable vehicle for gathering and storing data, it is not flawless. The introduction of the video camera in a classroom may cause changes in the ways participants interact and behave. To minimise undesirable effects caused by the video recording, the teacher explained and discussed the purpose of taping and modelled the normal routines and behaviour expected when engaged in mathematical inquiry before formal observation commenced. Furthermore, observations were made during three lessons a week at the start of the project, followed by every fourth week until the end of the project and involved 15 lesson recordings. This supported the gathering of more representative data as described by Yin (2012). Transcribing the video-recorded lessons supported the researcher to reflect retrospectively on what had occurred during the observation in relation to the research questions. The emerging themes and patterns from the data were then matched against the theoretical framework. 43 Detailed field notes are used in qualitative research to supplement as much information about the complex interactions in the classroom as possible (Yin, 2012). Written commentary on teacher actions, student actions, board work and materials used, were recorded promptly during and after each observation by the researcher. These field notes were incorporated in the lesson transcripts in brackets to document an accurate account of events from all angles that were not captured by the video-recorded observations. 3.5.2 INTERVIEWS In qualitative research, interviewing is a common means of collecting information in the participants’ own words (Berg & Lune, 2012). Interviewing is necessary to understand what is on “someone else’s mind” (Merriam, 1998, p. 76) and enter the world of the participant (Merriam, 1998; Yin, 2012). In the current study, interviewing was used to clarify the reasons for the instructional activities, teacher actions or student responses, and to investigate a potential explanation for a specific comment or behaviour. In the study, discussions with the teacher were conducted immediately prior and post each lesson. The goal was to explore the content of the lessons and also theculturally responsive practices the teacher used to support the Pāsifika students to meet these goals over time. The data from the interviews provides an important description of both the teacher and the students’ perspectives about mathematics learning in the classroom. The interviews with the fifteen students were conducted after the study to explore how they interpreted the role of the teacher in the learning process. The interviews with the students were about 10 minutes in duration for each student and were audiorecorded to allow a less intrusive method of recording data. Field notes were taken during the interviews to supplement recordings. The interviews followed a semi-structured format (Appendix, A) which allowed the researcher to respond to the “emergent worldview of the respondent, and to new ideas on the topic” (Merriam, 1998, p. 76). Data gathered from interviews was triangulated with evidence from fieldnotes and the classroom observations. This strengthened the validity of the data generated from the interviews. 44 3.5.3 CLASSROOM ARTEFACTS As part of the data collection in this study, student’s written work and digital photos of mathematical representations on the whiteboard using diagrams or materials were collected. This collection of artefacts complemented other methods of data collection. A research field log was maintained by the researcher to record reflections during each stage of the study such as entering any potential emerging biases, assumptions and interpretations of events or interesting events that unfolded that wererelevant to the study. These were supplementary to the focus of the study and were another means of triangulating the data. 3.6 DATA ANALYSIS The purpose of analysing data is to make sense of it (Merriam, 1998). Analysis of data in the currentstudy meant making sense of the teacher’s culturally responsivepractices to supportPāsifika students to engage in mathematical discourse. Data analysis began with organisation into manageable units with codes that match emerging patterns (Merriam, 1998; Newby, 2014; Yin, 2012). Both the coding used in Hunter’s study (2007) to characterise teacher actions and the questioning techniques described by Boaler and Brodie (2004) in their study were drawn upon during the initial data analysis. For example, the coding categories such as “provides wait time for other children to ask questions”, “emphasises the value of working together” (Hunter, 2007, p.164) and “exploring mathematical meanings and/or relationships” and “inserting terminology” (Boaler & Brodie, 2004, p. 4) were used to form descriptive codes for the initial data analysis. Data collection and analysis evolved continuously through the three phases of data collection that shaped this study. Data retrospectively analysed outside of the classroom involved the identification of categories and themes. The transcripts from video and audio recordings were read and re-read to develop potential broad themes which formed the initial categories (See Table 2). These broad themes became the headings of the findings chapter. Further coding was refined and reduced to 45 sub-categories which narrowed the coding and analysis for close description of the teacher’s actions. Each theme was examined and matched against the whole range of data collected including transcripts, field notes, written work samples, and artefacts. 46 Table 2. Initial coding categories and sub-categories Initial coding categories of teacher’s action Sub-categories of teacher’s action Gives individual thinking space/time to structure thoughts Respect silences, give time/space to compose self/understand the tasks Elicit prior knowledge/connect cultural experience Connect with home, church or community experience Rehearse ideas with peers/small group in a safe way. Encourage think-pair-share, swap student roles in scribing/contributing Orient to what is understood Model active listening, invite alternatives / extend potential ideas Foster safe to take risk learning environment in a culturally sensitive way Endorsing mistakes are part of learning/discovery of new knowledge Model an action of what to say in a culturally responsive way Model polite exchange of words eg. “You can say, I’m not sure, can you ask someone else please?” Give voice to a quiet member to be culturally inclusive Ask students to repeat a statement or read the problem Striving for mathematical language/insert terms Encourage the repeat use of mathematical terms in student voice Incorporate cultural belief of collective responsibility Emphasise the responsibility for themselves and the family Using body cultural tools/language Use drawings, students’ words, facial expression and hand gestures Questioning and prompting for mathematical understanding Press for clear, logical mathematical explanation and justification Revoicing/seeking a response to agree/disagree Encourage students to revoice ideas in own their words Connecting to big mathematical ideas Reflect on key mathematical ideas 47 3.7 VALIDITY AND RELIABILITY Qualitative research needs to ensure the validity and reliability of the findings. Validity is concerned with the honesty, richness, and scope of the data. It is gained through detailed descriptions and systematic measurement (Newby, 2014). Internal validity looks at the credibility of findings and whether the interpretation of an event can be sustained by the data. In the current study, internal validity was achieved by the prolonged period of observations, comprehensive field-notes, and triangulation of multiple sources of data. External validity concerns the transferability of data, that is, the degree to which results can be generalised to other settings (Newby, 2014). The provision of rich and in-depth descriptions of the teacher’s culturally responsive practices that support Pāsifika learners in mathematical discourse established external validity. Reliability is concerned with the accuracy and comprehensiveness of the research. Complex and diverse classrooms pose difficulties in fulfilling the traditional notion of reliability because there will never be any two classrooms where the conditions of learning and teaching are identical (Burton et al., 2014). Lichtman (2013) claimed that it is more appropriate to view reliability as the trustworthiness of the data, the fit between what is recorded as data and what occurs in the setting under study. In the current study, the trustworthiness of the findings can be established by the use of triangulation of multiple sources of data because the wider and deeper perspectives help to safeguard against bias from collecting only one form of data. 3.7.1 Limitations of Case Study Some of the criticisms about case study research are that it is too particular; the sample size is too small and that the results cannot be generalised (Berg & Lune, 2012; Merriam, 1998). In contrast, Flyvberg (2006, in Lichtman, 2013) argued that conducting case studies is valuable because it is a systematic way to produce exemplars in any discipline and “a discipline without exemplars is an ineffective one” (p. 94). Ruddin (2006, cited in Lichtman, 2013) noted that in 48 some instances case studies can be generalised (p. 797). Furthermore, Yin (2012) claimed that findings from case studies can be generalised but on “analytic rather than statistical grounds” (p. 177). This means case studies can be generalised to other situations on the basis of analytic claims to inform the relationship among a particular set of concepts, theoretical constructs or sequence of events. This study alone is not sufficient to generalise a theoretical construct on how teachers support Pāsifika students to engage in mathematical discourse in culturally diverse ways. However, the findings of the rich dense data in this classroom may be transferable across to similar settings for future studies. Another concern about case study research is that the researcher may let personal beliefs and biases affect the interpretation of findings (Berg & Lune, 2012; Yin, 2012). Some critics maintain that rather than accounting for or eliminating sources of bias, it is more appropriate to identify and acknowledge factors which may impact on the researcher’s interpretation of data (Burton et al., 2014; Lichtman, 2013). In order to maintain objectivity in this study, detailed field notes and constant reflective monitoring of the researcher’s own assumptions, beliefs and biases were used throughout the study. Objectivity was further enhanced through discussion with participants to clarify the interpretation of findings, the triangulation of multiple sources of evidence over a period of time and by asking the teacher and research supervisors to give critical feedback on emerging findings. 3.8 ETHICAL CONSIDERATIONS This study followed Massey University’s code of ethical conduct for research, involving human participants (Massey University, 2000). The ethics of social research focus on the need to protect all participants from possible harm. It also includes showing respect to participants with informed and voluntary consent, confidentiality, truthfulness, and social and cultural sensitivity. Ethical approval was obtained prior to data collection. All participants involved in the study were provided with the relevant information to give their informed consent (See Appendix, C,D,E). This research involved children under the age of fifteen years 49 old, therefore consent from their parents or guardians was also sought and obtained. It is important that anonymity and confidentiality is guaranteed and that privacy is neither invaded during a study or once the research is complete (Berg & Lune, 2012).Regarding anonymity, the researcher was cautious during filming to ensure that unintentional filming of students who had not consented to participate in the study did not occur; if it occurred, however, the researcher was ethically bound to ensure that none of the footage was used as part of the study and was destroyed. Anonymity and confidentiality are key ethical issues (Newby, 2014). Although, all participants in this study were allocated pseudonyms, assuring anonymity and confidentiality could not be guaranteed given that the staff and students in the school community knew who the participants (teacher and students) in the study were. However, particular care was taken to exclude any identifying information about the teacher, students or school within any written reports. Harm to the teacher participant was observed and minimised through open and honest discussion. Potential harm to students was reduced given that the research was undertaken during the normal classroom programme and practice. Potential harm to the school was curtailed by the absence of any identifiable information in reporting. Sensitivity to social and cultural issues was observed at all ti