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. 
 



 i 

 

 

 

 

 

“It Makes Me Feel Proud Of Who I Am”: Developing Functional 

Thinking Through Culturally Located Tasks 

 

 

 

 

 

A thesis presented in partial fulfilment of the  

requirements for the degree of  

Master of Education 

in  

Mathematics Education  

at Massey University, Manawatū, New Zealand  

 

 

 

 

 

 

Bronwyn Elizabeth Gibbs 

2020   

 
 

 



 ii 

Abstract 

Success in algebra plays a major role in equity and lifelong opportunities well beyond the 

mathematics classroom. Nationally, and internationally, high failure rates in algebra see 

many non-dominant students excluded from equitable higher education, career, and economic 

opportunities. There appears to be limited research focused on non-dominant students and the 

development of algebraic thinking through culturally located tasks. This study examines the 

representations Māori and Pāsifika students use when engaging with contextual functional 

tasks, and the ways Māori and Pāsifika students generalise culturally located tasks involving 

functions.  

A design based research intervention and qualitative research methods, drawing on a Pāsifika 

research methodology, were selected as most appropriate for the study. Twelve 10-12 year 

old Māori and Pāsifika students from a low socio-economic, urban school in New Zealand 

participated. Students engaged in an intervention of eight lessons focused on developing 

functional thinking with growing patterns drawn from Māori and Pāsifika cultures. A range 

of data were collected and analysed, including interviews, field notes, video recorded 

classroom observations, and photographs of student work.  

 

Findings revealed that when Māori and Pāsifika students were given opportunities to draw on 

their cultures to make sense of functional relationships, they constructed increasingly 

sophisticated and abstract representations to identify, communicate, and justify 

generalisations. There was significant growth in their conceptual understanding of both 

contextualised and decontextualised growing patterns. Additionally, aligning tasks with non-

dominant students’ traditions and experiences strengthened students’ mathematical and 

cultural identities.   

 

This study offers a contribution to the literature regarding how culturally contextualised tasks 

support non-dominant students to engage in early algebra, in particular, the representation 

and generalisation of functions. To address disparities and structural inequities in 

mathematics education, educators must acknowledge that students bring their own cultural 

knowledge and strengths to the classroom, and provide opportunities for all students to learn 

mathematics in ways they see as relevant to their cultural identities and communities. 

Recognising that mathematics is inherently cultural is a key lever for equity.  



 iii 

Acknowledgements 

I would like to acknowledge the support I have received from a number of people throughout 

my studies.  

 

First of all, I want to sincerely thank my two supervisors, Doctor Jodie Hunter and Professor 

Roberta Hunter. I would like to thank Jodie and Bobbie for so willingly sharing their 

extensive knowledge, invaluable advice, and guidance throughout this research project. Over 

the last five years they have consistently encouraged me to work towards a vision for 

equitable mathematics education, to take up opportunities to extend my practice and 

understanding, and supported me to take on further study. It has been a privilege to work 

through this journey with you. Meitaki ma'ata.  

 

I would like to thank the student participants for being so open in sharing their experiences, 

perspectives, and enthusiasm for learning mathematics throughout the study. I learnt a great 

deal from all of you, and hope that the learning you engage with treasures your world views 

and expects nothing less than for you to reach your fullest potential.  

 

I would like to express my gratitude to the teacher participant for opening up her classroom 

and teaching practice, and committing her valuable time and resources to this study. Thank 

you for your belief in this project and in your students. Thank you to the principal and board 

of trustees for allowing me to undertake the research within your school setting, and to 

whanau who gave consent for their children to participate in the study.  

 

Finally, I would like to thank my friends and family. This study would not have been possible 

without you encouraging me to achieve my goals. Thank you for believing in me and my 

ability to complete this research. I am especially grateful to my daughters Poppy and Ruby 

for your interest, encouragement and motivation.  

 

E kore e mutu ngā mihi aroha ki a koutou. 

 

 

 



 iv 

Table of Contents 

Abstract ii 

Acknowledgements iii 

Table of Contents iv 

List of Tables and Figures vii 

Chapter One: Introduction 1 

1.1 Introduction 1 

1.2 Background to the study 1 

1.3 Rationale 2 

1.4 Objectives 4 

1.5 Overview 4 

Chapter Two: Literature Review 6 

2.1 Introduction 6 

2.2 Sociocultural framework 6 

2.2.1 Concept of culture 7 

2.2.2 Māori and Pāsifika cultures 7 

2.2.3 Culture and mathematics 8 

2.3 Contextualisation of tasks 9 

2.4 Functional thinking 11 

2.4.1 Pathways of student thinking related to functions 12 

2.4.2 Layers of generalisation 14 

2.4.3 Representations to support reasoning and generalising 15 

2.4.4 Forms of representation 16 

2.4.5 Difficulties with functions: from patterns to generalisation 19 

2.5 Growing patterns and indigenous cultures 21 

2.6 Summary 22 

Chapter Three: Methodology 24 

3.1 Introduction 24 

3.2 Justification for methodology 24 

3.2.1 Pāsifika framework 25 

3.3 Researcher Role 27 

3.4 Data Collection 28 



 v 

3.4.1 Interviews 28 

3.4.2 Observations 29 

3.4.3 Video recording of lessons 30 

3.4.4 Student work 31 

3.5 Participants and research setting 31 

3.6 The research study schedule 32 

3.7 Data Analysis 35 

3.7.1 Coding and developing themes 36 

3.7.2 Validity and reliability 37 

3.8 Ethical considerations 38 

3.9 Summary 39 

Chapter Four: Findings and Discussion 40 

4.1 Introduction 40 

4.2 Students initial understanding of functional relationships 40 

4.3 Representing and generalising functional relationships 42 

4.3.1 Using T-charts to support functional thinking 42 

4.3.2 Visual representations to support generalisation 49 

4.3.3 Natural language to express generalisations 54 

4.3.4 Symbolic representation 55 

4.3.5 Multiple representations 59 

4.3.6 Challenges with functions 62 

4.4 Culturally located learners engaging with contextual tasks 63 

4.5 Phase three: students final understanding of functional relationships 68 

4.6 Summary 69 

Chapter Five: Conclusion 71 

5.1  Introduction 71 

5.2  Summary of research questions 71 

5.2.1   What representations do Māori and Pāsifika students use when engaging with 

contextual functional tasks? 71 

5.2.2 How do Māori and Pāsifika students generalise culturally located tasks involving 

functions? 72 

5.3  Key findings, implications and recommendations 73 

5.4  Limitations of the study 75 

5.5  Suggested areas for further research 75 

5.6  Final thoughts 76 



 vi 

References 77 

Appendices 90 

Appendix A1: Semi-structured initial and final interview 90 

Appendix A2: Task based interview 91 

Appendix B: Record sheet used for classroom observations 92 

Appendix C: Assessment tasks 93 

Appendix D1: Task one: sāsā 94 

Appendix D2: Task two: ngatu 95 

Appendix D3: Task three: vaka 96 

Appendix D4: Task four: titi 97 

Appendix D5: Task five: tukutuku panel 98 

Appendix D6: Task six: tivaevae 99 

Appendix D7: Task seven: fala 100 

Appendix D8: Task eight: kapa haka 101 

Appendix E1: Thematic analysis table used to group codes into themes 102 

Appendix E2: Thematic analysis 103 

Appendix F1: Principal consent form 104 

Appendix F2: Board of Trustees consent form 105 

Appendix F3: Student participant consent form 106 

Appendix F4: Parents of student participants consent form 107 

Appendix F5: Teacher participant consent form 108 

Appendix G1: Student and parent research information sheet 109 

Appendix G2: Board of Trustees research information sheet 111 

Appendix G3: Teacher research information sheet 113 

 

 

 

 

 

 

 

 



 vii 

List of Tables and Figures 

Summary of Tables  
 

Table 1 Cultural context, pattern and function type of tasks used   33 

in the lesson sequence       

  

Table 2  Summary of research activities and data gathering strategies 35 

implemented during each phase of the current study 

Table 3 Participants functional thinking and representations pre-teaching 41 

intervention 

Table 4 Participants functional thinking and representations post-teaching 68 

intervention 

Summary of Figures  

 
Figure 1 A covariation and correspondence approach for a growing pattern  14 

(as cited in Wilkie, 2012, p. 4) 

 

Figure 2 Pattern generalisation as objectification, showing layers of   15 

generality and different types of thinking (as cited in Wilkie, 2016, p. 

338) 

 

Figure 3 A typical growing pattern and recursive approach to finding the  20 

next position (as cited in Beatty, 2014, p.1) 

 

Figure 4 The Ula Model for collaborative engagement (as cited in Sauni,  26 

2011, p. 57) 

 

Figure 5 T-chart showing recursive relationship in the dependent variable 43 

Figure 6 The three columned t-chart      46 

Figure 7 T-chart representing the quadratic pattern    49 

Figure 8 Visual representation of the multiplicative structure of the   50 

growing pattern 

 

Figure 9  Using the visual representation to justify the function  52 

 𝛾 = 6 𝑥 +  3 

 

 Figure 10 Visual representation of triangular numbers to explore the   53 

function  y =
𝑥(𝑥+1)

2
  

 



 viii 

 Figure 11  Variable notation characterising the rule as an equation  58 

 Figure 12 Multiple representations and generalisation of the ngatu pattern  60 

 Figure 13 Multiple representations in group workbooks   61



 1 

Chapter One: Introduction 

1.1 Introduction 

This chapter provides the background to the study and establishes the context of the research. 

Section 1.2 highlights mathematical achievement challenges for Māori and Pāsifika students 

in New Zealand schools, and the need for educators to recognise mathematics as a cultural 

endeavour if equitable outcomes are to be achieved. Section 1.3 explains the rationale of the 

study, and section 1.4 presents the specific research questions this study investigates. Finally, 

an overview of the chapters is provided in section 1.5.  

 

1.2 Background to the study 

New Zealand communities, like those around the world, are becoming increasingly culturally 

diverse. Within New Zealand, Māori and Pāsifika peoples are two of the fastest growing 

groups compared to the overall population (Statistics New Zealand, 2018). Current 

projections indicate that Māori and Pāsifika children will make up the majority of New 

Zealand primary school students by 2040. The implications for the education system are far-

reaching, as these changing demographics require a greater capacity to respond to diverse 

cultures, and optimise mathematical learning opportunities for all students (Alton-Lee, 2003; 

Hunter & Hunter, 2018; Seah & Andersson, 2015).  

 

In New Zealand, national and international data (e.g. Educational Assessment Research Unit 

and New Zealand Council for Educational Research, 2018; Ministry of Education, n.d.) show 

that Māori and Pāsifika students are underachieving in mathematics compared to all other 

cultural groups. For a number of years, New Zealand governments have sought to reform and 

improve educational outcomes for Māori and Pāsifika students. Education policies and 

Ministry of Education initiatives, such as the Ka Hikitea Māori education strategy (Ministry 

of Education, 2013a) and Pāsifika Education Plan (Ministry of Education, 2013b), promote 

an equitable education system, focused on raising academic success and reducing disparities 

in outcomes. However, this vision is yet to become a reality. 

 

Evidence from the research (e.g. Hunter & Hunter, 2018; Louie & Adiredja, 2020; Rubel, 

2017) suggests that the persistent underachievement of non-dominant students within 

mathematics education can be attributed to three interrelated factors: deficit theorising; 



 2 

prevailing beliefs that mathematics is culture free; and structural inequities within the 

education system. A common thread between the three components, is that they all 

problematise non-dominant student’s cultures. Deficit theories assign blame for disparities in 

educational outcomes to students themselves, and perceived deficiencies in their family, 

community, and cultural backgrounds (Hunter & Hunter, 2018; Louie & Adiredja, 2020; 

Rubel, 2017). Related to deficit theories are pervasive Eurocentric beliefs that mathematics is 

culture and value free, and involves universally accepted facts, rules and procedures (Hunter 

& Hunter, 2018; Louie, 2017; Nasir, 2016). Rather than questioning assumptions about the 

education system non-dominant students are positioned within – for example, which students 

and their language, values, knowledge and ways of being are privileged while other groups of 

students are disadvantaged - the structures of the education system are framed as neutral and 

normal, and students and their cultures as the problem that needs to be fixed (Bills & Hunter, 

2015; Davis, 2018; Louie & Adiredja, 2020).  

 

In contrast to focusing on gaps in achievement, a significant body of research (e.g. Bills & 

Hunter, 2015; Hunter & Hunter, 2018; Louie, 2017; Wager, 2012) questions the gaps in 

opportunities for all students to learn mathematics in ways they see as relevant to their 

cultural practices and cultural identities. These research studies show that privileging non-

dominant students’ cultural capital in the mathematics classroom has a powerful influence on 

equitable educational outcomes. When learning is viewed as inherently cultural, non-

dominant students’ values, languages, traditions, and worldviews, are not deficits to be 

overcome but assets which can propel students to academic success (Si’ilata et al., 2018).  

 

1.3 Rationale  

Over the past two decades the teaching and learning of algebra has been a prominent research 

area in mathematics education (e.g. Blanton et al, 2015; Kaput, 2017; Rivera & Becker, 

2011). Algebra is considered critical to students’ mathematical development, and has been 

labelled as a gateway to academic and professional success (e.g. Blanton et al., 2018; Knuth 

et al., 2016; Morton & Riegle-Crumb, 2019). However, for many non-dominant students 

algebra is not a gateway to opportunity, it is a gatekeeper. Nationally, and internationally, 

high failure rates in algebra exclude non-dominant students from equitable economic, 

citizenship, higher education and career opportunities - particularly those related to the 

twenty first century STEM subjects of science, technology and engineering (Hunter & Miller, 



 3 

2018; Knuth et al., 2016; Morton & Riegle-Crumb, 2019). Research into the teaching and 

learning of algebraic concepts is important, because success in algebra plays a major role in 

equity and lifelong opportunities well beyond the mathematics classroom for non-dominant 

students.  

 

A key aspect of learning to think algebraically is exploring functional thinking, including 

generalising the relationship between two variables, and representing, justifying, and 

reasoning with these generalisations (e.g. Blanton et al., 2018; Wilkie, 2016). Teachers 

commonly use visual growth patterns to develop functional thinking, because growing 

patterns provide a means for students to focus on the underlying mathematical structure of a 

pattern and explore generalisation (Beatty, 2014; Markworth, 2012; Miller & Warren, 2012; 

Wilkie, 2016). However, the types of growing patterns presented and discussed in the 

mathematics classroom make a significant difference to the ways in which all students are 

provided with opportunities to develop functional understandings (Beatty, 2014; Markworth, 

2012).  

 

The majority of studies into the development of students’ functional thinking through 

growing patterns are with students from majority backgrounds, or, if they feature non-

dominant students, patterns are generally decontextualized rather than culturally located 

(Hunter & Miller, 2018). There appear to be few international studies which have considered 

how young non-dominant students’ engage in functional reasoning, and the role of culture in 

this process.  

 

Within the New Zealand context both Māori and Pāsifika cultures have a long and rich 

mathematical heritage with a strong emphasis on geometric patterning used, for example, 

within craft design and architecture (Finau & Stillman, 1995; Meaney et al., 2013). There is 

some evidence that drawing on the mathematics embedded within Pāsifika patterning 

provides a powerful means of supporting young culturally diverse students to develop 

understanding of growing patterns and engage in functional thinking (Hunter & Miller, 

2018). However little is known about Māori and Pāsifika students’ representations of 

growing patterns, and how they move through the stages of mathematical generalisation. The 

current study aims to extend upon the limited research available regarding how culturally 

contextualised patterning tasks support non-dominant students to develop their understanding 



 4 

of growing patterns, represent their functional thinking, and engage in the generalisation 

process. 

 

1.4 Objectives  

The purpose of the study is to explore how 10-12 year old Māori and Pāsifika students’ draw 

upon culturally embedded mathematics to develop algebraic understandings and make sense 

of functional relationships.  

 

In particular, the study addresses the following research questions:  

  

1) What representations do Māori and Pāsifika students use when engaging with 

contextual functional tasks? 

  

2)  How do Māori and Pāsifika students generalise culturally located tasks involving 

functions?  

 

1.5 Overview  

Chapter Two summarises national and international literature that is focussed on the role of 

culture in the teaching and learning of mathematics. It also defines the role of functional 

thinking in constructing early algebraic understanding, looking particularly at the research on 

representing and generalising functions. The literature review describes how culturally 

contextualised tasks provide opportunities to improve non-dominant students outcomes in 

mathematics education, as well as strengthening cultural knowledge and understanding.  

 

Chapter Three presents the research design and methods used in the study, and explains the 

data collection and analysis. The participants and research setting are introduced, and the 

timeframe for the study is outlined. Considerations regarding ethics, the role of the 

researcher, and the validity and reliability of the research are also addressed.  

 

Chapter Four integrates the findings, and discussion of the findings, to present the results of 

the study. The chapter examines the shifts that occurred in how non-dominant students were 

able to articulate and symbolise generalisations, and use multiple representational forms to 

support generalisation. The key developments in student thinking are identified and analysed, 

supported by evidence from the data and the theory discussed in the literature review.  



 5 

 

Finally, Chapter Five concludes the research by addressing the research questions, presenting 

a summary of the key themes, and providing recommendations for educators. It addresses the 

limitations of the current study, and provides suggestions for areas of future research.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 6 

Chapter Two: Literature Review 

2.1 Introduction 

The previous chapter introduced the background to the study and established the context of 

the research. This chapter summarises relevant literature that highlights the links between 

mathematics and culture. This is followed by a review of research studies investigating the 

development of functional thinking in early algebra. Section 2.2 introduces the sociocultural 

research framework and discusses mathematics as a cultural endeavour. It draws attention to 

the disconnect between mathematics education and the cultural worlds of Māori and Pāsifika 

learners. Section 2.3 examines relevant literature focused on the use of contextualised tasks 

aligned with non-dominant students’ cultural practices. It considers how culturally located 

tasks offer opportunities to both improve educational outcomes in mathematics, and build 

cultural knowledge and understanding. Section 2.4 defines the role of functional thinking in 

developing early algebraic understanding, looking particularly at the literature on 

generalising and representing functions. This section also outlines difficulties with functions, 

and the limitations on students’ awareness and ability to successfully generalise the 

functional relationship between two varying quantities, depending on the task. Finally, 

section 2.5 highlights research investigating growing patterns and the development of 

algebraic thinking in indigenous cultures. 

 

2.2 Sociocultural framework 

The theoretical framework underpinning this research is a sociocultural view of learning 

mathematics grounded in the work of Vygotsky (1986), who saw the social environment as 

instrumental to children’s learning. A sociocultural framework emphasises relationships and 

culture as essential facets of learning and development (Nasir & de Royston, 2013; Planas & 

Valero, 2016). From this perspective, understanding mathematical learning requires a focus 

on how students participate in mathematical activities with other students, and how they draw 

on cultural artefacts and practices, to solve problems in a local context (Nasir & Hand, 2006). 

A central concept of sociocultural theory is that children construct and modify their 

mathematical capabilities and competencies through the cultural practices they engage in 

their everyday lives, because they are culturally and socially situated learners (Planas & 

Valero, 2016). Therefore a sociocultural framework draws our attention to the consideration 

of relationships between cultural and content knowledge in mathematics and provides a lens 

to understand and support students using knowledge from their cultural backgrounds to 



 7 

construct new mathematical knowledge and understanding in the classroom (Nasir et al., 

2008).  

 

Sociocultural theory highlights that learning is inherently cultural, and mathematics 

classrooms are cultural and social spaces. Different forms of knowing and being are 

validated, based on historical, economic, political, and social power structures that serve to 

perpetuate inequities in both schools and society (Davis, 2018; Nasir et al., 2008; Nieto, 

2010). Rather than focusing on gaps in achievement researchers (e.g. Louie, 2017; Wager, 

2012) question the gaps in opportunities for all students to learn mathematics in ways they 

see as relevant to their cultural identities and communities.   

 

2.2.1 Concept of culture 

Nieto (2010) defines culture as:   

 

The ever-changing values, traditions, social and political relationships, 

and worldview created, shared, and transformed by a group of people 

bound together by a combination of factors that can include a common 

history, geographic location, language, social class, and religion (p. 

136). 

 

Researchers (e.g. Nasir et al., 2008; Nieto, 2010) regard culture as a social construct because 

culture cannot exist outside of social interaction and culture is produced, as well as 

reproduced, between people in local contexts. Closely related to the concept that culture is 

socially constructed is the concept that culture is learned (Fecho, 2016; Nasir et al., 2014). 

This differentiates culture from ethnicity, as culture is not a passive legacy passed down 

through our genes or inherited, but is dynamic and learned through interactions with families, 

communities and other social groups (Nieto, 2010).  

 

2.2.2 Māori and Pāsifika cultures  

Māori are indigenous to New Zealand, and are a dynamic, heterogenous population made up 

of divergent groups and cultural identities (Greaves et al., 2015). Pāsifika is an umbrella term 

used to describe a diverse, heterogeneous group of people who originate, or identify in terms 

of ancestry or heritage, from the Pacific Islands of Samoa, Tonga, Niue, Cook Islands, 



 8 

Tokelau, Tuvalu and Fiji (Coxon et al., 2002; Samu, 2015). Each group is unique in terms of 

how they identify with particular ways of knowing, being, and viewing the world (Mahuika 

et al., 2011). There are, however, a set of common values which Māori and Pāsifika peoples 

share. For example, family and collective responsibility are an integral part of life for both 

Māori and Pāsifika peoples (Hunter et al., 2019). Other core cultural values include 

reciprocity, respect, service, inclusion, relationships, spirituality, leadership, love, and 

belonging (Ministry of Education, 2013b).  

 

2.2.3 Culture and mathematics   

Mathematics is traditionally thought of as a discipline that is objective and independent of 

values and culture (Davis, 2018;  d’Entremont, 2015). The perception that mathematics is 

culture-free, and consists of neutral, universal truths, is perpetuated by assumptions such as 

mathematical properties remain the same regardless of who is doing the mathematics, for 

example, five plus five is always 10 (Nasir, 2016; Parker et al., 2017). Research studies (e.g. 

Hunter & Hunter, 2018; Louie, 2017; Nasir, 2016; Parker et al., 2017) show both students 

and teachers frequently consider mathematics learning and their cultures as distinct entities. 

For example, Parker et al. (2017) describe the challenges involved in developing mathematics 

teachers’ cultural responsiveness when the belief that mathematics is culture-free has been 

normalised within mathematics education. Louie (2017), and Nasir (2016), show that 

mathematics educators often frame mathematics as a culturally neutral, fixed body of 

knowledge to be received and focus on developing universal approaches to instruction, rather 

than culturally specific ones. In the New Zealand context, Hunter and Hunter (2018) report 

statements from middle years Pāsifika students that indicate a strong belief their cultural 

background is non-mathematical.  

 

Every culture has mathematics and the mathematics students know, understand, and come to 

school with is linked to the particular cultural practices and cultural identity of the learner 

(Nasir et al. 2008; Wager, 2012). Students from non-dominant cultures enter school with rich 

mathematical experiences, backgrounds, and knowledge (Nasir et al. 2008; Wager, 2012). 

For example, Barton and Fairhall (1995) identify mathematical aspects of  Māori culture 

implicit in geometric patterns in raranga (weaving), kowhaiwhai (painting) and whakairo 

(carving). Hunter et al. (2019) provide examples of the mathematics embedded within 



 9 

Pāsifika homes and communities such as food preparation, making traditional clothing, and 

patterns in drumming.   

 

However, educators frequently under-utilise or dismiss the cultural assets of diverse students 

(Aguirre et al., 2017; Wager, 2012). When all children are expected to conform to the 

dominant groups’ practices emphasised in school mathematics, there is a disconnect between 

diverse students and what they perceive to be another group’s mathematics which is alien to 

their reality (Aguirre et al., 2017; Wager, 2012). Many non-dominant students believe that 

mathematics has been developed and is owned by a community they are not part of  (Barta et 

al., 2012). Louie (2017) argues that a key product of the dominant culture characterising 

mathematics education, (i.e. the model of everyday instruction and classroom practices), is 

exclusion. Students who do not conform to the dominant stereotypes about what people who 

are good at math are like, typically White or Asian males from economically privileged 

backgrounds, are excluded from developing positive mathematical identities through narrow 

definitions of what counts as mathematical activity and mathematical ability. Indigenous 

students in Matthews et al.’s (2005) study on indigenous perspectives of mathematics 

education in Australia, perceived mathematics as an assimilation process and a subject where 

they had to become white in order to succeed. Similarly, students interviewed by Hunter and 

Hunter (2018), examining practices which have marginalised Pāsifika students in 

mathematics classrooms, expressed “a belief that to be successful in mathematics you must 

enter a “white-space” (p. 5).  A “white-space” is a pervasive space in the education system 

representing dominant white values, knowledge, culture and language as the culturally 

neutral norm, thereby marginalising or alienating non-dominant learners from their own 

cultural and mathematical identities (Battey & Leyva, 2016; Davis & Martin, 2018; Hunter & 

Hunter, 2018). 

 

2.3 Contextualisation of tasks  

A mathematical task is defined as a set of problems or a single complex problem, that focuses 

students' attention on a particular mathematical idea (Stein et al., 1996). Mathematical tasks 

directly determine what learning opportunities are made available to students and are 

significant in establishing how students come to “view, develop, use, and make sense of 

mathematics” (Anthony & Walshaw, 2009, p. 13).  

 



 10 

Tasks embedded within cultural contexts provide opportunities for non-dominant students to 

see their experiences reflected in school mathematics, and to recognise that the activities they 

engage in at home and in their communities involve mathematics which is meaningful and 

valued (Lipka et al., 2005; Nasir et al., 2008; Wager, 2012). Meaney and Lange (2013) report 

on students transitioning between home and school contexts. They highlight the potential 

limitations on learning caused by discontinuities when the two contexts have significant 

differences in what counts as valued mathematical content and knowledge. These researchers 

contend that having to mediate between in and out of school knowledge systems results in 

students more likely to be unable to perform well at school in mathematics topics they master 

easily at home. On the other hand, drawing on connections between the mathematics 

culturally diverse students learn in the classroom and the mathematics embedded in their 

everyday lives, repositions students as having valid cultural capital in their own mathematics 

classrooms (Hunter & Hunter, 2018; Wager, 2012). Matthews et al. (2005) emphasise that 

contextualising mathematics has the potential to change the educational environment so that 

indigenous students believe they can perform well in the education system. Both Leonard et 

al. (2010), and Martin (2009) undertook research on opportunity and access in mathematics 

for underrepresented minority students in the USA. They report that when non-dominant 

students mathematics knowledge is valued alongside school mathematics knowledge students 

develop their mathematical identity and disposition, and feel empowered as learners and 

doers of mathematics. 

 

Children create meanings for mathematical concepts when they work within contexts that 

already have meaning for them. There are rich opportunities for students to develop new 

mathematical knowledge, and construct progressively more abstract understandings, as they 

generate formal mathematics from cultural ideas (Lipka et al., 2005; Nasir et al., 2008; 

Wager, 2012). As part of a long term project on mathematics in cultural contexts, Lipka et al. 

(2005) undertook a case study in rural, predominantly Yup'ik villages where students 

simulated building a fish rack structure used to dry salmon. The cultural context enabled rich 

explorations into how area can change while perimeter stays the same, and students explored 

properties of a rectangle, developed geometric proof that their bases were in fact rectangles, 

and investigated how quadrilaterals are related. The students participated in challenging  

mathematics based on a practical problem that community members of their Yup’ik villages 

solve every time they build a fish rack.  

 



 11 

Researchers (e.g. Beatty & Blair, 2015; Davis & Martin, 2018) report that aligning tasks with 

non-dominant students’ cultural practices not only improves educational outcomes in 

mathematics, but strengthens diverse students’ cultural identities. Davis and Martin’s (2018) 

research focused on teaching practices and assessment that stigmatises African American 

students. They argue that for mathematics education to be liberatory for African American 

students, they must receive an education that produces growth not only in their mathematical 

skills, but builds on their cultural knowledge base. Beatty & Blair (2015) explored 

mathematical content knowledge based on the Ontario curriculum expectations and the 

mathematics inherent in indigenous cultural practices through Algonquin loom beading. The 

research focus was both the students’ mathematical thinking and cultural connections. 

Lessons in the study emphasised the cultural importance of looming, providing an 

opportunity for students to connect to their own cultural heritage. Community members, 

educators and students expressed pride in the mathematical thinking the children were 

demonstrating, such as patterning, multiplicative thinking and spatial reasoning, alongside the 

development of deeper cultural knowledge and understanding. 

 

2.4 Functional thinking 

Functional thinking entails generalising relationships between two or more varying 

quantities, representing and justifying these relationships in multiple ways (such as words, 

symbols, tables, or graphs), and reasoning with these generalised representations to 

understand and predict function behaviour (Blanton et al., 2015; Stephens et al., 2017; Wilkie 

& Clarke, 2015). Functional thinking provides an important entry point for developing 

algebraic understanding, and is regarded by many mathematicians as a powerful, unifying 

strand, because functional thinking is threaded through all of mathematics, is a crucial part of 

mathematical development, and leads to a deeper understanding of the structural form and 

generality of mathematics (Blanton et al., 2018; Kaput, 2017; Kieran et al., 2016).  

Researchers have found that elementary school children are capable of deeper functional 

analysis than previously thought, and that functional thinking begins at grades earlier than 

typically expected (e.g. Blanton & Kaput, 2011; Brizuela et al., 2015; Kieran et al., 2016). 

Blanton & Kaput’s (2011) examination of children’s capacity for functional thinking found, 

for example, the types of representations students use, the ways students organise and track 

data, and how they express functional relationships, can be scaffolded in instruction 

beginning with the very youngest students, at the start of formal schooling.  



 12 

2.4.1 Pathways of student thinking related to functions  

Learning trajectories are used in mathematics education to provide a research based 

representation of the ways student thinking in a particular domain develops over time (Fonger 

et al., 2018; Sarama & Clements, 2019). The levels of thinking in the progressions of a 

learning trajectory are not intended to be interpreted as stages students progress through in a 

linear sequence, but as a depiction of the growing sophistication shown in their reasoning 

(Stephens et al., 2017).  

 

Several researchers (e.g. Blanton et al., 2015; Markworth, 2010; Stephens et al., 2017; 

Wilkie, 2014) have developed learning trajectories specifically for functional thinking, 

describing the typical development of progressions of student thinking in generalising 

functional relationships. Generalisation is at the core of functions, and involves noticing a 

commonality in terms of a sequence, and deliberately extending the range of reasoning from 

specific situations, to more general ideas and conclusions identifying patterns, structures and 

relationships (Blanton et al., 2017; Kaput, 2017; Wilkie, 2016). Three types of functional 

thinking are evident from the learning trajectories, and serve as a framework to plan for, 

interpret, analyse and assess the kinds of functional reasoning found in mathematics 

classrooms: recursive, covariational, and correspondence thinking.  

 

Recursive thinking involves looking for a relationship in a single sequence of values, and 

indicating how to obtain a number in a sequence given the previous number or numbers 

(Stephens et al., 2017). When students display recursive thinking, they consider the 

relationship between successive terms in a pattern by referring to a sequence of distinct, 

particular instances only, and add the constant from term to term to extend the pattern 

(Blanton et al., 2015; Miller, 2016; Stephens et al., 2017). For example, Stephens et al. (2017, 

p.154) asked students to describe patterns they noticed in a task involving a growing pattern 

of seats and tables being joined for a birthday party. “The number of people is going up by 

2s” was evidence of recursive thinking, because the response of adding two each time 

suggests the student was attending to only one variable, the people, and therefore only the 

recursive structure of the pattern.  

 

Students engaging in covariational thinking analyse how two quantities are coordinated and 

vary in relation to each other (Blanton & Kaput, 2011; Blanton et al., 2015; Stephens et al., 

2017). For instance, Blanton and Kaput (2011) report on a cutting string task, where children 



 13 

could describe the covarying relationship between the number of cuts on a piece of string and 

the resulting number of pieces of string when the string is folded in a single loop. Students 

were not looking for a recursive pattern such as adding two every time, but a relationship 

between the two variables: “Every time you make one more snip it’s two more” (p. 51). A 

critical marker of this level is that, similar to the recursive level, children could describe a 

relationship within specific cases but not as a generalised functional relationship over a series 

of instances (Blanton et al., 2015; Carraher & Scheilman, 2016).  

 

With a correspondence approach the focus is on identifying the correlation between the 

variables in one set and the related variables in another set (Wilkie & Clarke, 2015). Students 

must pay attention to the correspondence between the two variables and identify an explicit 

rule so that they can calculate a variable no matter which term they are looking at (Wilkie, 

2014). For example, Wilkie (2014) reports on students giving a rule for an upside-down T 

plant growing pattern in words: “if you multiply the day number by 3 and add 1 more, you 

will be able to find the total number of leaves for the plant on any day” and symbols: “‘t’ is 

the total number of leaves on the nth day. The rule for the ‘upside-down T’ plant is t = 3n + 

1.” (p. 25). A distinction of children’s thinking at this level is their focus on the structure of 

the pattern, their awareness of what constitutes a functional relationship, that is, a 

correspondence between two variables, and explicitly stating a function rule which describes 

a generalised relationship (Rivera & Becker, 2011; Stephens et al., 2017; Wilkie, 2014). 

Figure 1 provides an example of a covariation and correspondence approach to understanding 

functional relationships in a sequence.  

 

 

 

 

 

 

 

 

 

 



 14 

Figure 1 

A Covariation and Correspondence Approach for a Growing Pattern (as cited in Wilkie, 

2012, p. 4) 

 

2.4.2 Layers of generalisation  

As students’ functional reasoning develops, so does their capability to articulate the 

underlying structure of growing patterns algebraically (Cooper & Warren, 2008; Radford, 

2010). Radford (2010) undertook a six-year long program of research with students as they 

moved from Grade 2 to Grade 6, to determine their transition from non-symbolic to symbolic 

expressions of algebraic thinking in pattern generalising activities. From his research, 

Radford (2010) theorised that the ways students express relationships and convey 

generalisations develops through three levels of algebraic generality—factual, contextual, and 

symbolic generalisation. As students notice pattern structures as recursive, covariational or 

correspondence relationships, they express these structures in factual, contextual, or symbolic 

forms (Warren et al., 2016).  

 

Radford (2010) views the process of generalisation as ‘objectification’, where students 

progress from arithmetic thinking to algebraic thinking, and recognise functional 

relationships in more abstract and analytic ways (see Figure 2). At an early level, factual 

generalisation is in the form of a concrete rule which allows children to calculate a numerical 

value for particular instances. For example, “there will be 10 triangles, and since there are 

three matches for each triangle, there will be 30 matches altogether” (Kanbir et al., 2018, 

p.100).  Contextual generalisation focuses on more descriptive terms such as “you add the 

next figure from the top row”, and is language driven to explain the generalisation (Radford, 

2018). The generalisation still refers to material objects in the sequence, but has moved to a 

new layer of generality where students’ attention has shifted from specific numbers to the 

variables and their relationship (Radford, 2010, 2018; Twohill, 2018). Symbolic 

generalisation requires a different perspective on the mathematical objects involved, and 



 15 

students use algebraic notation, including letters, symbols or signs, to express the 

generalisation (Cooper & Warren, 2008; Kieran et al., 2016; Radford, 2010). The focus is 

now relational, and children represent their thinking symbolically, without reference to 

specific or situated instances (Twohill, 2018; Wilkie, 2019).  

 

Figure 2 

Pattern Generalisation as Objectification, Showing Layers of Generality and Different Types 

of Thinking (as cited in Wilkie, 2016, p. 338) 

 
 

2.4.3 Representations to support reasoning and generalising 

Researchers (e.g. Blanton, 2008; Blanton et al., 2018; Cañadas et al., 2016), provide evidence 

that elementary school children can develop and use a variety of representational tools to help 

them reason with functions, describe recursive, covarying, and correspondence relationships, 

symbolise relationships, and express generalisations.  

 

Individually, students use representations as a tool to make sense of ideas and explore the 

problem in their own way. Socially, representations are a way for children to communicate 

their reasoning with others as they represent, explain and justify generalised relationships in 

diverse ways (Blanton, 2008; Stephens et al., 2017; Tripathi, 2008). Investigating student-

created representations also gives educators and researchers insights into students’ functional 

thinking and emerging conceptual understandings (Blanton, 2008; Stephens et al., 2017; 

Tripathi, 2008).  

 

Representation is a dynamic process, and students move through different phases in their 

choice of representation, showing various levels of complexity in the ways they represent the 

relationship between two terms in a growing pattern (Blanton et al., 2015; Bobis & Way, 



 16 

2018; Stephens et al., 2017). Blanton (2008) discusses representation as referring to both the 

process and the product of expressing an idea. For instance, children may represent their 

thinking with pictures or tables showing the process of their developing awareness of the 

functional relationship. As students develop their understanding of the relationship between 

the variables they represent a generalisation of the relationship - the product of their thinking 

- using representations such as words or symbols. Blanton (2008) describes children’s 

representations transitioning from direct modelling, to mathematising, to mathematical 

understanding, as children’s representations develop from the more concrete to more abstract.  

 

2.4.4 Forms of representation 

Different forms of representation can be helpful to support the development of algebraic 

reasoning. These representations include concrete, verbal, numerical, graphical, contextual, 

pictorial, or symbolic components (Tripathi, 2008).  

Through drawing, students may identify the underlying functional relationship between two 

variables, and express a generalisation. Cañadas et al. (2016) report on children using 

drawing to make the specifics of a problem explicit, for example, students drew desks and 

people to represent the context of a growing pattern. They used both drawings and natural 

language to express and justify the relationships between the two varying quantities: “I draw 

desks first and then I draw the people and then I counted by 2’s.” (p. 95). Conversely, Moss 

and McNab (2011) designed a teaching intervention to support second grade students 

understanding of linear functions, through geometric and numeric representations of growing 

patterns. All of the students had worked with repeating patterns, but none with growing 

patterns. Results showed that when students used visual representations of geometric and 

numeric patterns they were able to notice the constant in visual arrays, represent 

generalisations in their natural language, and identify and express two part rules for growing 

patterns (e.g.  y=ax+b). Moss and McNab (2011) concluded that when visual representations 

were prioritised students were better able to find, express, and justify functional rules.  

Using concrete materials to manipulate both variables in a growing pattern gives students the 

opportunity to model the growing pattern, examine the pattern structure and construct 

generalisations (Cooper & Warren, 2008). Twohill (2018) investigated the strategies that nine 

and ten year old students used when constructing general terms for shape patterns. Prior to 

constructing the pattern terms of tiles, two students tended strongly towards recursive 



 17 

thinking with comments such as “each time you’re adding two”. After using materials, one 

student identified the pattern as a top and bottom row containing n + 1, and n, tiles 

respectively. The other student identified the terms as containing “t diagonal pairs, with an 

additional tile on the top right corner” (p. 224).  

A common representation for solving functions problems is creating a t-chart, or function 

table, where students make two columns of data and record corresponding entries for the 

independent and dependent variables (Blanton & Kaput, 2011). Blanton et al. (2015) report 

on children in the early grades initially creating function tables as a means to organise 

covarying data, but by Grade 2, they were beginning to use tables as a tool for thinking about 

the data and reflecting on the relationships and patterns they could see in the table. Similarly 

Blanton and Kaput (2011), discuss the shifts from students in the early grades using t-charts 

as a place to record numbers, to students in the middle grades using t-charts as a tool that can 

be used to determine relationships in data and an important structure in mathematical 

reasoning. For instance, a student in a task where students were to find the number of body 

parts a growing snake would have on day ten and day n stated: “I know that on day 10 the 

snake will have 101 body parts and I know that on day n the snake will have n x n +1.  I 

know this because I used my t-chart and I looked for the relationship between n and body 

parts” (p. 11). This suggests that the t-chart’s structure helped the student use it as a tool to 

compare data and find relationships.  

As students are plotting a graph they can directly see how relations change, rather than, for 

example, reading a table showing a limited number of discrete input and output values. 

Caddle and Brizuela’s (2011) exploration of fifth grade students discussing linear graphs 

found that using a graph prompted different reasoning than other representations, and gave a 

more complete view of the function. Similarly, Brizuela and Earnest (2017) examined how 

young, multi-ethnic, students worked with multiple representations to choose the best deal 

between a grandmother either doubling a child’s money, or tripling his money and then 

taking away $7. Brizuela and Earnest (2017) found students questioned their understanding 

about the nature of the relationship between the variables as they constructed a graph and 

interpreted the best deal, and the functions became more explicit in a graph than with other 

representations.  

 



 18 

Students’ natural language descriptions are another form of generalising and representing 

mathematical concepts. Cañadas et al. (2016) investigated second grade students articulating 

their ideas about functional relationships. They found the use of natural language is helpful 

for students in the early grades because of their familiarity with it, and students used natural 

language to express relationships as a recursive pattern (counting by twos) or a functional 

relationship (doubling). Interestingly, Stephens et al. (2017) and Blanton et al. (2015) both 

unexpectedly found that students were generally more successful representing generalisations 

using variable notation than using words. For example, in Stephens et al. (2017) 66% of 

students provided a correct function rule for a growing stars pattern in variables: x · x = y, and 

only 32% in words: “The picture number times itself equals the number of stars” (p. 157). 

This clearly challenges the notion that variables as a varying quantity should not be 

introduced until secondary school. 

 

Researchers (e.g. Blanton et al., 2015; Brizuela et al., 2015; Stephens et al., 2017; Wilkie, 

2014) underline the importance of students using letters or symbols to represent variables and 

generalise functional relationships through explicit rules. Brizuela et al. (2015) provides 

evidence that children as young as six can use variable notation in meaningful ways to 

express relationships between quantities and represent generalisations. They argue that 

allowing variable notation to become part of mathematical language as children represent 

their algebraic ideas gives them opportunities to develop a deep and powerful means of 

representing generalised relationships. Wilkie (2014) reports on upper primary students being 

keen to explore and experiment with letters to represent variables in algebraic equations. This 

tended to flow naturally from students’ own attempts to create a number sentence for their 

rule which often contained a mixture of numbers and words, for example: “The answer is the 

day × 3 + 1” (p. 26). Both Wilkie (2014) and Brizuela et al. (2015), consider students should 

be given opportunities to use variables and variable notation early in their formal schooling, 

in order to give children multiple opportunities to fully explore and develop their 

understandings over time.  

 

Researchers (e.g. Cañadas et al., 2016; Neilsen & Bostic, 2018; Stephens et al., 2017; 

Tripathi, 2008) provide evidence of the benefits of using multiple representations in order to 

develop a deeper, richer and more flexible understanding of functional relationships. When 

children have multiple ways to represent an idea they can choose representations that are 

intrinsically meaningful to them (Blanton, 2008; Daryaee et al., 2018). Using multiple 



 19 

representations enables students to develop a deeper understanding of underlying 

mathematical structures, because different mathematical representations highlight different 

aspects of mathematical relationships (Neilsen & Bostic, 2018; Stephens et al., 2017; 

Tripathi, 2008). Each representation provides a different way for students to examine and 

compare the relationships, and students learn about and deepen their understanding of 

functions when they are explored while making connections across diverse representations 

(Carraher & Scheilman, 2016; Daryaee et al., 2018).   

 

2.4.5 Difficulties with functions: from patterns to generalisation  

Although young students are capable of sophisticated functional thinking, there are a number 

of common difficulties related to the development of functional thinking reported in the 

research literature.  

 

Students often have difficulty transitioning from looking at the relationship between 

successive terms in a pattern as recursive, to focusing on the relationship between variables 

and viewing patterns as functions (e.g. Blanton et al., 2015; Isler et al., 2015; Stephens et al., 

2017; Wilkie, 2014). Research identifies numerical and visual geometric growing patterns as 

a common vehicle for supporting the development of functional thinking (Blanton & Kaput, 

2011; Cooper & Warren, 2011). However, the growing patterns typically used, and the ways 

in which the pattern structures are presented, can limit students’ awareness and ability to 

successfully generalise the functional relationship between variables and across instances 

(Beatty, 2014; Miller, 2016).  Beatty (2014) shows when students work with growing 

patterns (such as Figure 3), they are able to describe the pattern and extend it to the next 

position based on additive reasoning, but have difficulty predicting values for terms further 

down the sequence (i.e., the 10th position, 25th position, 100th position, nth position). 

 

 

 

 

 

 

 



 20 

Figure 3 

A Typical Growing Pattern and Recursive Approach to Finding the Next Position (as cited in 

Beatty, 2014, p. 1) 

 

Many students over-generalise the applicability of proportional methods to growing pattern 

contexts, believing that proportional reasoning can be applied to all linear relations because 

they increase (or decrease) at a constant rate (e.g. Ayalon & Wilkie, 2019; Lannin, 2003). 

Lannin (2003) provides an example of joined cubes in a row and “smiley” stickers on rods of 

ten cubes. A student claimed that because a rod of ten cubes has 42 stickers, you could 

multiply 42 by two for a rod length of 20. The student was unaware that when she used this 

method she was overcounting the extra stickers where the two rods joined. In this case the 

student was focused on the numeric relationships (examining the increase in the number of 

stickers in a table), without connecting her reasoning to the context and considering the 

functional relationship between the variables.  

 

Moving beyond particular cases and expressing generality can create a challenge for students. 

Lannin’s (2005) research shows students relying on empirical evidence to support general 

statements. This difficulty appears to be due to the traditional focus in mathematics on 

calculating only a particular instance of a situation, rather than determining a general relation. 

Lannin (2005) describes students using guess-and-check strategies, and experimenting with 

various operations and numbers provided in the problem situation, to construct a 

generalisation. This trial and error approach led to students guessing a rule without 

considering why the rule might work, and attempting to find a formula to fit a particular 

instance of the pattern. Lannin (2005) emphasises the importance of students constructing 

and justifying a general relation in the problem context, and developing arguments which are 

independent of particular instances.  

 



 21 

The role of variable as a varying quantity is an essential tool to develop and express 

functional relationships, but is one with which students may struggle (e.g. Blanton et al., 

2017; Isler et al., 2015). Students’ typical difficulties with variables include believing that 

variables stand for a fixed unknown quantity, label, or attribute (e.g., l stands for leaves, d 

stands for day, or x is always three), rather than as a symbol that can stand for any real 

number in a functional relationship (Blanton et al., 2015; Blanton et al., 2017; Isler et al., 

2015; Wilkie, 2014). Other conceptions regarding variables include thinking that two 

different variables (e.g. x, y) in the same equation cannot represent the same value, and 

believing that the value of a variable has something to do with its position in the alphabet 

(Brizuela et al., 2015; Wilkie, 2014).  

 

2.5 Growing patterns and indigenous cultures  

The majority of studies into the development of students’ functional thinking are with 

students from majority backgrounds, or if they feature indigenous students then tasks are 

generally decontextualized or shared context rather than drawing on culturally located tasks 

(Hunter & Miller, 2018). It appears that there is limited research that focuses on indigenous 

students and the development of algebraic thinking through engaging with their culture 

within the mathematics curriculum. 

 

Miller’s (2014) study with Year 2 and 3 students in an urban indigenous school in North 

Queensland investigated the role of culture in young indigenous students mathematical 

generalisation of growing patterns. Results indicated that the type and context of the pattern 

impacts on indigenous students’ abilities to access the structure and relationship between the 

variables. Young Australian indigenous students were more successful extending and 

generalising growing patterns that came from the natural environment (e.g. identifying the 

relationship between possum tails and eyes) than they were extending and generalising 

growing patterns represented by decontextualized geometric shapes. Miller selected growing 

patterns for tasks deliberately to ensure that the functional relationship was transparent and 

explicitly represented. This was achieved by using materials where the variables were explicit 

(i.e. pattern term cards and coloured tiles) or could not be physically separated (i.e. plastic toy 

kangaroos and crocodiles). Miller (2016) conjectured that representing growing patterns in 

this manner assisted students to attend to both variables in the pattern, and potentially pushed 

them towards functional thinking rather than recursive thinking.  



 22 

 

Further evidence demonstrates the importance of educators and researchers understanding 

students’ cultural representations, such as gesture in indigenous contexts, and providing 

opportunities for students to engage in and express their own mathematical understandings in 

culturally appropriate ways. Miller’s (2014) research explored how young indigenous 

students use gesture to generalise growing patterns, and brings attention to the importance of 

cultural interpretations of gesture and actions within the classroom. Miller provides two case 

studies of students using gesture to support their explanations, articulate their understanding 

of the structure of growing patterns, and generalise the rule of growing patterns. Miller 

argues that these students are demonstrating contextual and symbolic generalisation within 

Radford’s (2010) three layers of generality, and these young indigenous students are using 

gesture to represent generalisations of geometric growing patterns, rather than alphanumeric 

symbolism. If these cultural signs are missed or misinterpreted the true understanding of 

students’ knowledge will potentially be unseen.  

 

In the New Zealand context there is some evidence that drawing upon the mathematics 

implicit in Pāsifika and Māori patterning provides a powerful means of developing culturally 

diverse students’ early algebraic reasoning and understanding of functional patterns (Hunter 

& Miller, 2018).  Hunter and Miller’s (2018) research concentrates on the use of culturally 

located patterns from Pāsifika and Māori cultures to develop young culturally diverse 

students’ understanding of functional patterns and support generalisation. Hunter and Miller 

focused on a pattern from a Cook Island tivaevae (a communally sewn traditional quilt) 

which grew in multiple directions. Their evidence showed that when mathematics is 

embedded in a cultural context, in this case the structure of a tivaevae pattern, young 

culturally diverse students are able to make a meaningful connection to the mathematics, 

begin to see covariation, develop their understanding of growing patterns, and articulate 

generalisations as they see the structure of the pattern growing in multiple ways. 

 

2.6 Summary 

Despite mathematics being positioned as a value and culture free subject area, researchers 

(e.g. Hunter & Hunter, 2018; Nasir et al. 2008; Wager, 2012) have shown that mathematics is 

a cultural product, and is closely tied to the cultural identity of the learner. As this literature 

review depicts, setting mathematical tasks in contexts centred on non-dominant students 



 23 

traditions, experiences, and cultures, gives more equitable opportunities to participate in, and 

develop, higher level mathematical thinking (Anthony & Walshaw, 2009; Hunter & Miller, 

2018). All children can be successful in mathematics when their understanding is linked to 

meaningful cultural referents (Ladson-Billings, 1997). 

 

Evidence from research studies highlights that young children are capable of sophisticated 

sense making of functional relationships, and generalising and representing these 

relationships in diverse ways (Blanton et al., 2015). However, in contrast to the research, 

Māori and Pāsifika students are well behind their peers in national and international measures 

of mathematics achievement (e.g. Educational Assessment Research Unit and New Zealand 

Council for Educational Research, 2018; Ministry of Education, n.d.). There appear to have 

been limited studies which have investigated young non-dominant students’ understanding of 

growing patterns, and little is known about Māori and Pāsifika students’ representations of 

growing patterns, and how they move through the stages of mathematical generalisation.  

 

The objective of this study is to provide insight into the representations Māori and Pāsifika 

students use when engaging with contextual functional tasks, and how Māori and Pāsifika 

students generalise culturally located tasks involving functions.   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 24 

Chapter Three: Methodology 

3.1 Introduction 

The previous chapter discussed the literature related to the current study. This chapter 

outlines the research design and methods used in the study. Section 3.2 provides a 

justification for the selection of design based research and qualitative methods used. This 

section also details the Ula model, and how the current study aligns with this Pāsifika 

theoretical framework. Section 3.3 describes the role of the researcher, and data collection 

methods are explained in section 3.4. The participants and research setting are introduced in 

section 3.5. Section 3.6 outlines the research project and the instructional sequence that forms 

the basis of the study. Section 3.7 describes the data analysis, and discusses the validity and 

reliability of the findings of the research. Finally, section 3.8 elaborates ethical 

considerations.   

 

3.2 Justification for methodology 

The choice of methodology was influenced by the aim of this study. This was to provide 

insight into how Māori and Pāsifika students generalise culturally located tasks involving 

functions, and the representations Māori and Pāsifika students use when engaging with 

contextual functional tasks.     

 

Design based research is a prominent methodology in mathematics education research, and 

is appropriate for developing research based solutions to complex problems in educational 

practice (Plomp, 2013; Prediger et al., 2015). Several key features define design based 

research, namely “it is interventionist, theory driven, context-specific, collaborative and 

contains a … focus on local impact and theory generation” (Crippen & Brown, 2018, p. 

490). Design based research was selected as the most appropriate methodology for the 

current study for the reasons outlined in the following paragraph.  

 

Firstly, design based research aligns with the sociocultural perspective that underpins the 

current study. Both design based research and sociocultural theories emphasise the social 

aspects of learning, such as collaboration, the active construction of knowledge, and the 

integration of cultural experiences into the learning process (Prediger et al., 2015; Steffe & 

Thompson, 2000). Secondly, design based research is well suited to studying real-life issues 

in naturalistic environments (Anderson & Shattuck, 2012).  The current study is classroom 



 25 

based because there are complexities and social interactions in natural settings that would not 

be present if the study took place out of context (Anderson & Shattuck, 2012). Thirdly, a goal 

of design based research is to investigate the possibilities for educational improvement by 

supporting and studying the development of new or different forms of practice (Cobb et al., 

2015). The goal of the current study is to understand how students develop the generalisation 

and representation of functional thinking through culturally located tasks. An interventionist 

methodology such as design research that aims to bring about the intended developments in 

order to study them is therefore appropriate.  

 

Design based research is descriptive and explanatory by nature, so design based researchers 

often make use of qualitative methods to study learning in the design intervention (Reimann, 

2016). The essence of nearly all qualitative studies is a focus on people, and narrative 

information about their perceptions, actions, beliefs or behaviours (Merriam & Tisdall, 2016; 

Yin, 2016). A qualitative approach to data collection and analysis was appropriate for this 

study, in order to provide an insight into Māori and Pāsifika students lived experiences in an 

upper primary school mathematics classroom. A qualitative approach allowed the researcher 

to draw on student voices which not only provided insights into participants points of view, 

but also privileged the voices and experiences of  Māori and Pāsifika students.  

 

3.2.1 Pāsifika framework 

In order to reflect a culturally appropriate theoretical framework for the research a Pāsifika 

research methodology, the Ula model, was drawn upon (Sauni, 2011). The Ula model (see 

Figure 4) is a metaphor for collaborative engagement based on fa’asamoa: Samoan cultural 

principles, values and beliefs (Sauni, 2011).  

 

 

 

 

 

 

 

 

 

 



 26 

Figure 4  

The Ula Model for Collaborative Engagement (as cited in Sauni, 2011, p. 57) 

 

 

The circle of flowers represents the use of cultural values throughout the research study 

(Sauni, 2011). The space inside the ula represents the va (the space between), which is not an 

empty space but holds the ula together through relationships (Anae, 2010; Sauni, 2011).  

 

The Ula model also supports the current study to be responsive to Māori participants through 

integrating core cultural values such as whanaungatanga (whanau or family type 

relationships), manākitanga (caring) and kotahitanga (unity). To teu le va - cherish, nurture 

and care for the relational spaces - is a way of affirming the importance of relationships, and 

engaging with both Māori and Pāsifika participants within the same research project (Naepi, 

2015).  

 

In the current study building trusting relationships, and creating a comfortable atmosphere for 

the participants and the researcher to engage in meaningful dialogue, were prioritised. 

Reciprocity was encouraged by the researcher sharing their personal identity with 

participants, listening carefully and respectfully, and utilising a strength based approach – 

participants were viewed as experts when sharing their experiences, views and 

understandings. Being open and approachable, and sharing a sense of humour when 



 27 

appropriate, also helped students feel at ease during interviews and classroom observations 

(Sauni, 2011; Vaioleti, 2006). 

 

3.3 Researcher Role  

In qualitative research studies the researcher is “the primary instrument for data collection 

and analysis” (Merriam & Tisdell, 2015, p. 15). Accordingly, the researcher in the current 

study was the sole collector of data. This enabled the researcher to maximise the efficiency of 

data collection and the quality of the data collected (Merriam & Tisdell, 2015). For example, 

immediately responding to and adapting questions in an interview, or checking with 

participants to validate the researcher’s interpretation of events.  

 

During classroom observations the researcher’s role was observer as participant, primarily 

observing and gathering information, but having some level of interaction with participants 

(Merriam & Tisdell, 2015; Mertler, 2019). This meant the researcher sat in with the class and 

participated in small ways, as a means for generating a more complete understanding of the 

groups’ activities, but was not integrally involved in the lesson.   

 

The relationship between the researcher and the teacher was professional and collaborative. 

Unlike traditional studies in education that pose the researcher as expert, design based 

research and the Ula model allow for a reciprocal relationship as the teacher and researcher 

share knowledge and resources (Jung & Brady, 2015; Sauni, 2011). During the research 

lessons the teacher had primary responsibility for teaching the lesson, but the researcher was 

available to assume a tuakana (older sibling) role if required, and provide support, advice, or 

further information in the moment. For example, helping the teacher decide how she might 

select and sequence students solution strategies in order to highlight important mathematical 

ideas.  

 

Participants understood the researcher’s role as an observer and collector of data, and data 

was openly collected (Merriam & Tisdell, 2015). The students knew the researcher as a 

mentor in mathematics, and were used to the researcher working in classrooms supporting 

teachers for pedagogical change in mathematics. This meant that participants felt comfortable 

when observations took place. However, participants who know they are being observed can 

behave in ways differently than they normally would (Merriam & Tisdell, 2015; Yin, 2016). 



 28 

For example, the participants in the current study may have tried to present themselves in a 

favourable manner or regulate their behaviour in reaction to the presence of the researcher 

(Merriam & Tisdell, 2015).  

 

The researcher’s experiences in using an inquiry approach to teach mathematics with Māori 

and Pāsifika students meant familiarity with expected classroom practices and potential 

outcomes of the lessons. However, this strength can also be seen as a weakness, due to bias 

or over-familiarity with the research context leading to critical data being missed (Merriam & 

Tisdell, 2015). When the researcher is the primary instrument for data collection and 

analysis, it is important to be aware of biases and assumptions that might impact the study, 

and monitor how they may be shaping the collection and interpretation of data (Merriam & 

Tisdell, 2015). 

 

3.4 Data Collection  

Collecting data from different sources in the learning environment is consistent with both 

design based and qualitative research methodologies (Merriam & Tisdell, 2015; Prediger et 

al., 2015). Using multiple methods for data collection produces rich and detailed information, 

and enhances the validity of findings (Merriam & Tisdell, 2015).  

 

Data collection tools utilised in the current study were interviews, field notes, video recorded 

classroom observations, and photographs of student work.  

 

3.4.1 Interviews 

Interviewing is a valuable way of gaining in-depth information related to participants’ 

experiences, perspectives, and constructions of reality (Flick, 2018; Seldman, 2019).  

Interviews are used to find out from people things which cannot be directly observed, such as 

thoughts, feelings and intentions (Merriam & Tisdell, 2015). 

 

Conducting interviews in culturally appropriate ways is essential within Māori and Pāsifika 

contexts. Prior to interviews in the current project participants were given a brief explanation 

about the purpose of the study, the conduct of interviews, confidentiality, and consent. 

During interviews a talanoa (conversation) format was employed (Vaioleti, 2006). The 

interview questions were asked in a conversational rather than inquisitorial manner, allowing 



 29 

respectful, reciprocal interactions. Each interview lasted approximately 15 minutes and took 

place in a quiet breakout space within the participants’ classroom. All interviews were video 

recorded and wholly transcribed for coding and analysis.  

 

Two types of interviews were used in the current study: semi-structured individual interviews 

and task-based interviews.  

 

3.4.1.1 Semi-structured interviews. Semi-structured interviews were undertaken 

before and after the intervention (see Appendix A1). Semi-structured interviews sit in the 

middle of a continuum between structured and unstructured interviews (Merriam & Tisdell, 

2015).  The questions in semi-structured interviews are open-ended  and flexibly worded 

(Merriam & Tisdell, 2015). In the current study, interview questions exploring participant’s 

perceptions of connections between their cultural identities and mathematics were developed 

from the literature review. The interview protocol provided a framework of ideas to 

investigate with participants, while also leaving room for participants to share their 

experiences beyond the expectations of the researcher. Semi-structured interviews provided 

space for reciprocity, as the researcher could respond to participant’s ideas in the moment, 

and participants could elaborate on the points of each question that were meaningful to them.  

 

3.4.1.2 Task based interviews. Task-based interviews have been used by researchers 

in qualitative research in mathematics education to gain insights into participants developing 

knowledge, and ways of explaining, reasoning, justifying and representing mathematical 

ideas (Assad, 2015; Pepper et al., 2018). Task based interviews were utilised for the 

assessment tasks, and additionally, subsequent to lessons in the teaching sequence (see 

Appendix A2). During task-based interviews participants were asked to think aloud while 

reflecting on a previously completed task, in order to gain an in-depth perspective into shifts 

in student thinking which might not have been obtained from observations in the classroom 

setting. A semi-structured interview protocol allowed for open-ended prompting if required, 

depending on the judgement of the researcher in response to participants descriptions.  

 

3.4.2 Observations  

Observations are important in qualitative research because they provide researchers with a 

first-hand account of the activity being studied while it is occurring (Cohen et al., 2018; 



 30 

Merriam & Tisdell, 2015). Researchers have the opportunity to notice behaviours that might 

otherwise be taken for granted or go unnoticed by participants, which leads to richer, and 

more valid and authentic data (Cohen et al., 2018; Merriam & Tisdell, 2015).  

As with other forms of data collection, observational data must enable the research questions 

to be answered (Cohen et al., 2018). In the current study eight semi-structured observations 

were made. Semi-structured observations are conducted when the researcher has considered 

what will be observed and recorded beforehand, and developed loose categories of data to 

gather (Cohen et al., 2018). Key themes that emerged from the literature review served as 

prompts to guide the observations (see Appendix B). This meant the researcher had a lens for 

the observations, but could also adapt the focus in the moment based on what emerged during 

the course of the lesson (Punch & Omacea, 2014).  

Observations were recorded in the form of field notes and video recordings. Field notes were 

written up as soon after the observation as possible in order to add details and provide a 

summary of what had been observed. Alongside factual descriptions of what happened, the 

field notes had a reflective component. Reflective comments were based on, for example, 

initial interpretations, and tentative themes and ideas that emerged during the lesson. In this 

respect the researcher was engaging in preliminary data analysis alongside data collection, 

and could note things to ask, observe, or look for in the next round of data collection 

(Merriam & Tisdell, 2015).  

3.4.3 Video recording of lessons  

Video recording is a qualitative research method that captures complexities of social activity, 

which is not possible through observation alone (Cohen et al., 2018; Wang & Lien, 2013).    

Three features of video-recording underpin it’s strengths in qualitative research: video 

recording provides a real-time sequential record; captures authentic behaviours in their 

situational context; and is a permanent record that can be viewed multiple times (Cohen et al., 

2018; Wang & Lien, 2013).    

In order to gather a rich picture of classroom learning, and to provide a broad scope for data 

collection, all lessons in the current study were video recorded. Two video cameras were 

used during observations. While participants worked on the problem in small groups, a video 

camera was focused on two groups. The students included in these groups varied in different 

lessons. The purpose was to investigate the ways participants were engaging with the task, 



 31 

building conceptual understanding, and reasoning about the structure of the patterns and 

relationships. When the teacher was facilitating a larger group discussion there was one 

camera recording the teacher and participants, in order to document the collaborative 

discourse and shifts in thinking that were occurring.  

Video recording the lessons allowed many details to be captured which would otherwise have 

been missed (Basil, 2011; Wang & Lien, 2013). For example, video recording captured non-

verbal cues, reactions and gestures when participants were working in their small groups. The 

video records allowed the data to be analysed more than once, which helped to overcome the 

“fleeting nature” of observation (Basil, 2011, p.251). This allowed for reflection on what had 

been observed, and the possibility of reinterpretation, because the researcher wasn’t restricted 

by memory of the lesson (Wang & Lien, 2013).  

All video recordings were downloaded at the conclusion of the lesson, and wholly 

transcribed.   

3.4.4 Student work  

Digital photographs of student’s responses to tasks were taken during lessons, as a means of 

capturing participants varied forms of representation and development of generalisations. 

This provided concrete evidence of how students were engaging with the pattern structures 

and the types of representations students were using.  Names and faces were not 

photographed to assure anonymity.  

 

3.5 Participants and research setting  

The research was conducted with a group of twelve students from one Year Six to Eight class 

(10-12 year olds), in a low socio-economic, high poverty, urban school in New Zealand. 

Ethnicities of the twelve students were Māori (40%), and Pāsifika (60%). The study 

participants were selected from a larger group of possible participants by purposive sampling. 

Qualitative research methods are likely to produce large amounts of data, and therefore the 

number of participants was limited to keep the study manageable (Merriam & Tisdell, 2015; 

Punch & Omacea, 2014). The teacher and the researcher selected the group of twelve based 

on who was likely to be available and willing to participate, and able to clearly communicate 

their experiences and opinions (Merriam & Tisdell, 2015; Punch & Omacea, 2014).  

 



 32 

The school was selected because they are part of a longitudinal mathematical research project 

called Developing Mathematical Inquiry Communities (DMIC), designed to address the 

persistent underachievement of Māori and Pāsifika students in mathematics (Hunter & 

Hunter, 2018). The teacher was an experienced educator who has long-term involvement 

with DMIC professional development, and core Māori and Pāsifika values underpin her 

pedagogy and classroom practices. She regarded her active participation in this research as 

professional development and a means to reflectively inform her instructional practice. 

 

3.6 The research study schedule   

Phase one  

The first phase of the study began with individual semi-structured interviews with 

participants and the teacher. Additionally, a pre-intervention assessment was administered to 

ascertain what students already knew about representing and generalising functional 

relationships. Participants showed their algebraic thinking through parallel assessment tasks, 

one contextualised and one decontextualised, representing the same function. Questions were 

intended to ascertain students’ understanding of growing patterns, including their ability to 

predict further positions in the pattern and describe, in general terms, the relationship 

between the pattern and its position. The assessment tasks mirrored the types of tasks and 

discussions that occurred during the teaching phase, and aligned with Level 4 learning 

outcomes of the New Zealand Curriculum (see Appendix C).  

The teacher and the researcher analysed participant responses to the assessment tasks, and 

collaboratively built an understanding of the progressions of thinking that could be seen, and 

the shifts in thinking to be developed through the intervention. Data from the pre-intervention 

assessment and interviews, evidence from the academic research discussed in the literature 

review, and the teacher and researcher’s personal and professional experience, were used to 

construct a series of contextual tasks (see Appendix D1-8).  The tasks were designed to build 

on current student understandings, and increase in complexity over a sequence of lessons.  

The mathematics in all the tasks was embedded in a cultural context relevant to the students’ 

backgrounds. Patterns were drawn from Māori and Pāsifika cultures and chosen both for their 

context, and the ways they lent themselves to multiple ways of seeing and representing the 

pattern’s growth. For example, it was likely that some students would see the pattern growing 

recursively, while others would view the pattern in a way that lent itself to finding an explicit 



 33 

rule (Markworth, 2010). Table 1 provides an overview of the cultural context, pattern and 

function tasks used in the teaching intervention.  

 

Table 1 

Cultural Context, Pattern and Function Type of Tasks Used in the Lesson Sequence 

 

 

After the tasks were drafted, a subset of the tasks were piloted in a class with similar aged 

students at the same school. The teacher and students participating in the pilot work were not 

participants in the study. The pilot work provided the researcher with an understanding of 

how students made sense of the task components and helped with testing the appropriateness 

of the tasks, for example, the language used, the mathematical content, and the cultural 

context. The results of the pilot work were discussed with the teacher.  

 

 

 Context Pattern (visual) Possible function type 

Task 1 Sāsā 

 

𝛾 = 3 𝑥 + 1 

Task 2 Ngatu  

 

𝛾 = 8 𝑥 + 4 

Task 3 Vaka 

 

𝛾 = 6 𝑥 − 1 

Task 4 Titi  

 

𝛾 = 6 𝑥 +  3 

Task 5 Tukutuku panel  

 

𝛾 = 4 𝑥 − 6 

Task 6 Tivaevae 

 

𝛾 = 24𝑥 + 4 

Task 7 Fala 

 

𝑦 = 𝑥2  + 𝑥 + 4 

Task 8 Kapa haka  

 

y =
𝑥(𝑥 + 1)

2
 



 34 

Phase two 

The second phase of the project involved teaching the series of problems. The instructional 

sequence that formed the basis of the research was comprised of eight tasks taught over eight 

one-hour lessons. The teaching intervention took place over a five-week period in Term 4 

(October / November). The number and length of lessons was intended to give participants 

enough opportunities to engage with the concepts, and to gain traction toward algebraic 

understanding.  

 

Each lesson followed the same structure. First, the teacher launched the problem in a whole 

group setting, then participants worked in small groups to discuss the problem, look for 

relationships, and represent the relationships in multiple ways including words, symbols, or 

variable notation. After participants had enough time to explore the problem in small groups, 

the teacher would facilitate a larger group discussion of students’ approaches, connecting 

students thinking to the big mathematical idea which underpinned the lesson. 

 

The collaborative and iterative nature of design-based research allowed the teacher and the 

researcher repeated opportunities to discuss and reflect on the students’ responses to each 

task in the sequence. After each lesson the teacher and the researcher reviewed the lesson, 

analysed classroom events and student responses, and made any modifications required for 

the next lesson based on, for example, what was noteworthy about how children were talking 

about or representing the functional relationship.  

 

Additional data collected at this stage included audio recordings of the collaborative planning 

meetings and field notes regarding reflective discussions between the teacher and researcher.  

 

Phase three 

The final phase of the data collection consisted of participants completing a post-intervention 

assessment, and a concluding semi-structured interview.  

 

Table 2 summarises the research activities and data gathering strategies implemented during 

each phase of the study. 



 35 

Table 2 

Summary of Research Activities and Data Gathering Strategies Implemented During Each 

Phase of the Current Study 

 Research activity Data gathering strategy  

Phase 

One 

 

Individual semi-structured 

interviews  

 

Two pre-intervention assessment 

tasks 

 

Collaborative planning of teaching 

unit and series of tasks with teacher 

 

Pilot tasks 

Semi-structured interviews video recorded and 

transcribed   

 

Think-aloud video recorded and transcribed  

Photographs of student work 

Phase 

Two 

Algebra intervention (8 lessons)  

 

 

 

 

 

Task based interviews with 

participants  

Reflective discussion and 

collaborative planning after each 

lesson 

Semi-structured observations  

 

Obervational field notes: descriptive notes and 

reflective comments  

 

Group work video recorded and transcribed 

 

Large group discussion video recorded and 

transcribed 

 

Photographs of student work 

 

Task based interviews video recorded and 

transcribed  

 

Audio recordings  

 

Researcher notes 

Phase 

Three 

Post-assessment tasks 

 

 

 

Final student semi-structured 

interviews 

Think-aloud video recorded and transcribed  

Photographs of student work 

Semi-structured interviews video recorded and 

transcribed   

 

3.7 Data Analysis 

Data analysis is the process of making meaning out of the data (Merriam & Tisdell, 2015). 

Analysing the data in this study meant making sense of the nature of students’ generalisations 

and representations, when patterns and relationships they know from their cultural 

experiences were the basis for functional thinking.  



 36 

In both qualitative and design based research, two phases of data analysis are carried out: 

ongoing and retrospective analysis (Merrian & Tisdell, 2015). 

In the current study ongoing analysis was an interactive and iterative process throughout the 

data collection phase. For instance, analysis of the pre-intervention assessment informed 

design of the tasks. After lessons, the researcher and teacher’s collaborative analysis 

informed the teaching of the following task, and the next stage of data collection. This 

analysis continued in a cycle throughout the series of tasks. Additionally, data was collected 

and analysed concurrently. Transcribing interviews and video-footage throughout the lesson 

sequence allowed the researcher to highlight items of potential interest and begin to 

informally identify themes from the data.   

 

Once all the data from assessments, transcripts from interviews and video observations, field 

notes, and student work were collected, retrospective and formal analysis began.  

 

3.7.1 Coding and developing themes  

Thematic analysis was the method used for “systematically identifying, organizing, and 

offering insight into patterns of meaning” across the data set (Braun & Clarke, 2012, p. 57). 

Thematic analysis is a flexible method of qualitative data analysis and allows the researcher 

to derive meaning and understanding from the data, in order to answer the research questions 

being addressed (Braun & Clarke, 2012; Merriam & Tisdell, 2015).  

 

The systematic analysis of the data began with coding. Initially the data were coded with 

respect to functional thinking, Māori and Pāsifika cultures, and culturally located tasks. 

Codes were a combination of  descriptive and interpretative. In the next phase analysis 

shifted from codes to themes in a manner consistent with the thematic analysis proposed by 

Braun and Clarke (2012). The basic process of generating themes involved gathering the 

codes into possible themes, then “collapsing or clustering codes that shared some unifying 

feature together, so that they reflected and described a coherent and meaningful pattern in the 

data” (Braun & Clarke, 2012, p. 63). The subsequent phase involved reviewing the 

developing themes in relation to the coded data set to determine that they did in fact address 

the research questions. Eventually the following themes were identified: generalisations to 

support functional thinking; representations to support functional thinking; contextual tasks; 

and culturally located Māori and Pāsifika learners.  See Appendix E1 for the thematic 



 37 

analysis table used to group codes into themes, and Appendix E2 for illustrative excerpts of 

coded transcripts removed and grouped on to a table. 

 

3.7.2 Validity and reliability  

All research involves analysing data in a reliable way, to produce conclusions that are valid 

(Merriam & Tisdell, 2015). Reliability, in terms of replication, is challenging in qualitative 

research because human behaviour is dynamic and the social environment being studied 

relies on context (Merriam & Tisdell, 2015). Replication of a study will not produce the same 

results. However, qualitative studies can convey consistency, transferability, and 

trustworthiness (Flick, 2018; Merriam & Tisdell, 2015). 

 

Transferability can be achieved through thick descriptions, detailing the setting, participants, 

and themes. This technique allows readers to experience the events being described, and 

judge for themselves how the findings could be transferred to other contexts. Additionally, 

thick descriptions provide enough evidence to prove that the results are consistent with the 

data collected, and the conclusions are trustworthy and make sense (Merriam & Tisdell, 

2015).  

 

Using multiple sources of data helps to eliminate bias that can result from relying on one 

research method. In the current study many sources of data collection were used and the data 

collection method and analysis was systematic. An audit trail detailing how the study was 

carried out, and how the findings were drawn from the data, strengthens reliability (Merriam 

& Tisdell, 2015). Triangulation of multiple data sources also supports confidence in the data 

(Merriam & Tisdell, 2015). In the current study, data has been triangulated wherever 

possible. For example, data from interviews and classroom observations were cross checked 

and compared, in order to verify that both sets of data were telling the same story (Atkins & 

Wallace, 2012).  

  

Additionally, the iterative nature of design based research strengthened the credibility of the 

research findings. In the current study, the ongoing discussion, reflection, and modification of 

lessons, made it possible to test developing theories from earlier lessons in later lessons. 

Retrospective analyses were contrasted with the informal analyses conducted while the study 

was in progress to strengthen the credibility of the research claims.  



 38 

 

Diversity amongst participants allows for a greater range of application of the findings 

(Merriam & Tisdell, 2015). In the current study participants were of Samoan, Tokelauan, 

Cook Islands and Māori descent. Some were New Zealand born, and others Pacific born. 

They varied in terms of home language, gender, age, and time at school in New Zealand. 

Identifying patterns that exist across a range of people increases the chance that patterns can 

be applied to another example, compared to collecting a narrower range of data (Holley & 

Harris, 2019). 

 

Member checks are the procedure where a study’s findings are shared with participants, who 

can give feedback to correct or otherwise improve the accuracy of the study (Merriam & 

Tisdell, 2015; Yin, 2016). In the current study tentative interpretations of interviews and 

observations were checked with participants, who were asked if they accurately represented 

their realities. Collaboration between the researcher and the teacher enabled facts to be 

checked and interpretations to be corroborated or revised. Similarly, the researcher discussed 

themes and emerging findings with colleagues and supervisors for peer review. Asking them 

to comment on whether the findings were plausible based on the data adds another layer to 

the study’s validity.   

 

3.8 Ethical considerations  

The research was designed and conducted in a responsible manner, in accordance with the 

Massey University Code of Ethical Conduct for Research, Teaching and Evaluations 

Involving Human Participants (Massey University, 2015). The project was reviewed and 

approved by the Massey University Human Ethics Committee prior to data collection. Ethical 

considerations taken into account included respect for participants with informed consent, 

respect for privacy and confidentiality, and social and cultural sensitivity.  

 

Written consent was obtained from all participants, including the school Principal, Board of 

Trustees, students, parents or guardians, and the teacher (see Appendices F1-F5). This 

research involved children under the age of fifteen years old, therefore consent from their 

parents or guardians was sought and obtained. A detailed information sheet was provided 

alongside the consent form to ensure that participants had clear details of why the research 

was being conducted, what was involved throughout the research, and what they were 



 39 

consenting to (see Appendices G1-3). All participants in this study were allocated 

pseudonyms, and care was taken to exclude any identifying information about the teacher, 

students or school within any written reports. Participants also had the right to withdraw from 

the research at any time.  

 

Sensitivity to social and cultural issues was observed at all times by the researcher, for 

example, maintaining the daily opening and closing karakia, the classroom routines, the 

social groupings selected by the teacher, and respecting any silences during interviews.  

 

Risks that required consideration included the time commitment of the teacher and her 

students to the project. Disruptions were avoided as lessons under study took place as part of 

the normal classroom programme. Meetings were held at times and locations that suited the 

teacher so as not to burden her with an increased workload.  

 

3.9 Summary 

This chapter has outlined the research design and methods used in the study, including the 

rationale for selecting design based research and qualitative methods of data collection and 

analysis. The Ula model for Pāsifika engagement provided a culturally appropriate 

framework to inform the research. A variety of methods to collect data were used, including 

interviews and classroom observations. Data was analysed using thematic analysis, 

identifying codes and generating themes. Triangulating data supported the credibility of the 

interpretations, and conducting the research in a thoroughly documented and ethical manner 

ensured the reliability and validity of results. The findings and discussion of the study are 

presented in Chapter Four.    

 

 

 

 

 

 

 

 

 

 

 



 40 

Chapter Four: Findings and Discussion 

4.1 Introduction 

The previous chapter provided an overview of the research design and methods used in the 

study. This chapter presents the findings and discussion in relation to how Māori and Pāsifika 

students generalise culturally located tasks involving functions, and the representations Māori 

and Pāsifika students use when engaging with contextual functional tasks. Section 4.2 

outlines the initial functional understandings of the twelve Māori and Pāsifika students 

involved in the study. Section 4.3 draws on the data analysis from the series of eight 

contextual tasks to describe student learning and provide insight into the development of 

students’ capabilities in representing and generalising functional relationships. Section 4.4 

highlights connections between the cultural contexts of the tasks, and the cultural worlds of 

Māori and Pāsifika learners. Finally, section 4.5 presents student understandings of functional 

relationships at the conclusion of the teaching intervention.  

 

4.2 Students’ initial understanding of functional relationships  

Prior to commencing the teaching intervention participants compl