Vernacular Numeracy Practices - an Exploration of the Numeracy Resources Young People Bring to Their Learning

Vernacular numeracy practices - an exploration of the numeracy resources young people bring to their learning

A thesis submitted to The University of Manchester for the degree of Doctor in Education EdD

in the Faculty of Humanities

2017

Nuala Broderick

School of Environment, Education and Development

Table of Contents

Table of Contents ...... 2 List of Tables ...... 5 List of Figures ...... 6 Glossary of terms and abbreviations ...... 7 Abstract ...... 8 Declaration ...... 9 Copyright ...... 9 Acknowledgements ...... 10 Chapter 1. Introduction...... 12 1.1. Introduction...... 12 1.2. Background and Context ...... 12 1.3. Statement of purpose ...... 14 1.4. Personal motivation for the research ...... 16 1.5. Scope and limitation of the research ...... 17 1.6. Organisation of the thesis ...... 17 1.7. Conclusion...... 18 Chapter 2. Literature Review ...... 19 2.1. Introduction...... 19 2.2. Review Methodology ...... 19 2.2.1 Numeracy in the Further Education and Skills Sector ...... 21 2.2.2 The importance of numeracy ...... 24 2.3. Conceptualising adult numeracy ...... 25 2.3.1 The relationship between numeracy and mathematics ...... 28 2.4. Theoretical perspectives ...... 30 2.4.1 The turn to the social in numeracy and mathematics research ...... 31 2.4.2 Funds of Knowledge ...... 37 2.4.3 The New Literacy Studies ...... 40 2.5. Conclusions from the literature review ...... 42 2.5.1 Research questions ...... 44 Chapter 3. Methodology ...... 45 3.1. Introduction...... 45 3.2. Developing the research questions ...... 45 3.3. Explanation of key concepts used in the research ...... 46 3.4. Research methodology and design ...... 48 3.4.1 Quality criteria and research process ...... 52 3.4.2 Establishing trustworthiness and supporting transferability and generalisability ...... 53 3.5. Sampling ...... 58 3.5.1 Research site ...... 58 3.5.2 Participant sample ...... 60 3.6. Data generation methods ...... 62 3.6.1 Data generation with students ...... 64 3.6.2 Data generation with teachers ...... 72 3.7. Data analysis ...... 74 3.8. Ethical considerations ...... 77 3.9. Conclusion...... 80

Chapter 4. Findings (1) ...... 81 4.1. Introduction...... 81 4.2. The students in the study ...... 82 4.3. Students’ numeracy events ...... 83 4.3.1 Students’ numeracy events outside college ...... 84 4.3.2 Vignettes ...... 89 4.3.3 Students’ numeracy practices evidenced in events and translated into the language of the ANCC ...... 97 4.3.4 Students’ numeracy events inside college ...... 101 4.3.5 Classroom observation for hair and beauty – renting a flat ...... 103 4.3.6 Classroom observation for construction students - budgeting ...... 105 4.4. Conclusion...... 106 Chapter 5. Findings (2) ...... 107 5.1. Introduction...... 107 5.2. From numeracy events to numeracy practices ...... 107 5.3. Exploring students’ numeracy practices ...... 108 5.4. Street, Baker and Tomlin’s analytical model of numeracy practices ...... 109 5.5. Conclusion...... 116 Chapter 6. Conclusions and discussion ...... 118 6.1. Introduction...... 118 6.2. Research purpose and answering the research questions ...... 118 6.2.1 RQ1: What numeracy events and practices do learners engage in, in their everyday lives? ...... 118 6.2.2 RQ2: What numeracy events and practices do learners engage in on their vocational and functional skills programmes? ...... 121 6.2.3 RQ3: How are learners’ everyday numeracy practices, conceptualised as their vernacular numeracy practices, used in their vocational and functional mathematics learning? . 124 6.3. Conclusions ...... 127 6.4. Implications for practice ...... 127 6.5. Addressing the aim of the research ...... 128 6.6. Limitations of the research ...... 131 6.6.1 Sample size ...... 131 6.6.2 Self-reporting data ...... 132 6.6.3 Ideological ...... 132 6.7. Contribution to knowledge...... 133 6.8. Final reflection ...... 134 References ...... 136

APPENDICES ...... 158 Appendix 1. Original research questions - September 2013 ...... 158 Appendix 2. Student profiles ...... 159 Appendix 3. Lily’s clock face ...... 160 Appendix 4. Paul’s collage ...... 161 Appendix 5. Daniel’s list ...... 162 Appendix 6. Dom’s list ...... 163 Appendix 7. Jack’s list ...... 164 Appendix 8. Data generation and reduction - numeracy events ...... 165 Appendix 9. Lily’s numeracy events ...... 166 Appendix 10. Emma’s numeracy events ...... 167 Appendix 11. Paul’s numeracy events ...... 168 Appendix 12. Daniel’s numeracy events ...... 169 Appendix 13. Students’ vernacular numeracy practices using social practice analytic framework ...... 170 Appendix 14. Students’ college numeracy practices using social practice analytic framework ... 171 Appendix 15. Comparing and contrasting students’ vernacular numeracy practices and college numeracy practices using social practice analytic framework ...... 172 Appendix 16. Lily’s timetabling task sheet ...... 173 3

Appendix 17. Student collages (content) transcribed into list of activities (step 1) and use in interviews to stimulate discussion ...... 174 Appendix 18: Prompts used with students for semi-structured 1:1 interviews and how they relate to analytical framework ...... 175 Appendix 19. Students’ numeracy events and practices in college ...... 176 Appendix 20. Headings for use in observation of teaching in vocational setting and functional skills class to inform general notes taken during observation ...... 177 Appendix 21. Pen portrait used in focus group with teachers...... 178 Appendix 22. Questions used in the semi-structured interviews with teachers ...... 179 Appendix 23. Numeracy event 1: timetabling appointments in college ...... 180 Appendix 24. Numeracy event 2: online shopping at home ...... 182

Word count: 50,354

List of Tables

Table 3.1. Explanation of concepts used in the study adapted from Navigating Numeracies ...... 48 Table 3.2. Questions-methods matrix ...... 51 Table 3.3. Coding and classifying to support data analysis ...... 76 Table 4.1. Student Profiles ...... 83 Table 4.2. Emma’s on-line shopping ...... 99 Table 4.3. Lily’s working part-time in a hair salon ...... 100 Table 4.4. Paul’s buying parts for his own motorbike and checking cost of parts in work ...... 101

List of Figures

Figure 2.1. Adult Numeracy Concept Continuum of Development ...... 27 Figure 3.1. Lily’s clock-face ...... 68 Figure 3.2. Paul’s collage ...... 68 Figure 4.1. Categories of students’ numeracy events ...... 85 Figure 4.2. Student’ numeracy events embedded in technology ...... 85 Figure 4.3. Students’ part-time work ...... 86 Figure 4.4. Students numeracy events embedded in managing money ...... 87 Figure 4.5. Students numeracy events embedded in sport and leisure activities ...... 87 Figure 4.6. Students’ numeracy events embedded in household responsibilities ...... 88 Figure 4.7. Students’ numeracy events embedded in self-management ...... 89

Glossary of terms and abbreviations

ABE Adult Basic Education ALCC Adult Literacy Core Curriculum ALM Adults Learning Mathematics ANCC Adult Numeracy Core Curriculum BIS Business Innovation and Skills BKSB Basic and Key Skills Builder BSA Basic Skills Agency DfE Department for Education and Skills DIUS Department for Innovation, Universities and Skills E1; E2; E3; NQF five-level system to categorise skill levels in relation to literacy and L1; L2 numeracy. Starting at the lowest level, these are: Entry Level 1 (E1), Entry Level 2 (E2), Entry Level 3 (E3), Level 1 (L1) and Level 2 (L2). E&TF Education and Training Foundation FE Further Education FS Functional Skills FSM Functional Skills Mathematics GCSE General Certificate of Secondary Education HD Handling data: HD1 Data; HD2 Probability LfLFE Literacy for learning in further education LLN Literacy Language and Numeracy LSS Learning and Skills Sector MSS Measures, shapes and space: MSS1 Common measures; MSS2 Shape and Space N Number: N1 Whole numbers; N2 Fractions, decimals and percentages NAO National Audit Office NFER National Foundation for Educational Research NIACE National Institute of Adult Continuing Education NLS New Literacy Studies NQF National Qualifications Framework NRDC National Research and Development Centre OECD Organisation for Economic Co-operation and Development Ofsted Office for standards in education PIAAC Programme for the International Assessment of Adult Competencies QCA Qualifications and Curriculum Authority QCF Qualifications and Curriculum Framework SfL Skills for Life VLP Vernacular Literacy Practices VNP Vernacular Numeracy Practices

Nuala Broderick: University of Manchester

Degree: Doctor in Education (EdD)

Thesis title: Vernacular numeracy practices– an exploration of the numeracy resources young people bring to their learning

Abstract

Current narrow definitions of numeracy and mathematics within the functional mathematics agenda, conceptualised as the autonomous model (Street 1984,1995; Street, Baker and Tomlin 2008) of numeracy, offer students entering further education limited and deficit subject positions and may contribute to a continued lack of success in college mathematics and numeracy.

Drawing on New Literacy Studies theory (Barton and Hamilton 1998; Baker and Street 2004; Heath 1983; Gee 1996; Ivanic et al 2009; Street 1984, 1995, 1998, 2000) and a funds of knowledge perspective, (Moll, Amanti, Neff, & Gonzalez 1992), I suggest that adopting a social practice approach to numeracy teaching and learning provides affordances to reframe these deficit positions through an examination of students’ everyday numeracy practices. A social practice theory of numeracy constructs it as numeracy embedded in social purposes, context dependent and reflective of students’ everyday social practices. It also recognises the importance of power in structuring relations between people and ideas within practices. In an effort to avoid reifying the distinction between home/school, formal/informal, I propose the term vernacular numeracy practices as a way of conceptualising students’ out of college numeracy practices.

Using focus groups, classroom observations and interviews, the research explores students’ vernacular numeracy practices embedded in a range of purposes and across different settings of students’ lives. The invisibility of students’ socially situated, vernacular numeracy practices, both to themselves and their teachers is revealed. However, students were willing to engage with the research practice which provided opportunities for them to access and share their vernacular numeracy practices. These vernacular numeracy practices are not readily understood across learning contexts and the challenge for tutors and students is how to access and use them to challenge the largely deficit subject positionings available and taken up by students.

Within the research, students’ vernacular numeracy practices were ‘translated’ into the language of the adult numeracy core curriculum as a way of supporting students and tutors to ‘see through to the maths’ (Coben 2003). The purpose of extracting the mathematics which was embedded in students’ out of college practices, was to support students to engage in the process of ‘repairing their mathematical identity’ which Gee (2007) suggests, needs to take place before students can move on to become successful learners of mathematics. The research concludes that valuing and using students’ vernacular numeracy practices may provide opportunities for students with low or no qualifications in mathematics to recruit ‘identities of participation’ (Solomon 2009) in place of identities of exclusion.

Declaration

No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning;

Copyright i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trademarks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/

Acknowledgements

Completing my EdD thesis has been both challenging and satisfying. I have received support, guidance and inspiration, from many people on this journey.

Firstly, I would like to thank all the participants who took part in the study. I am very grateful for their generosity in participating and giving so freely of their time.

I would like to thank all the staff at the School of Environment, Education and Development at The University of Manchester. Providing the opportunity to undertake research alongside my ‘day job’ in such a supportive and encouraging environment has been a dream come true for me. My experience throughout the EdD programme has been overwhelmingly positive and affirming.

I owe an enormous debt of gratitude to my supervisors, Professor Alan Dyson and Professor Julian Williams. Their encouragement, insight and support have been crucial to my development. Alan kept me focused on structure and coherence and kept me grounded when I veered towards flights of fancy in my written work. Julian supported me in developing not only my writing and thinking, but also a relationship with theory. I am grateful to him for introducing me to the Social Theories of Learning Group and the Bourdieu Reading Group, at the University. Both provided me with opportunities to read, think and develop a critical awareness of my own and others work, in a wonderfully collegiate environment. Both my supervisors have read and provided feedback on many versions of my thesis. Ultimately, however, the flaws are my own.

I would also like to give a special mention to Professor Helen Gunther who provided words of encouragement and support on my work.

A very special thanks to my friends and family for their sustained and constant support on my journey. A special mention to Loretta, Barbara, Rachael, Sue, Jenny, Iona and Graham, for always asking about my progress and showing an interest in the research.

To my sisters for always believing that I could see it through to the end. My sister Geraldine, who not only provided emotional support but who also challenged me to explain and justify my research and in so doing, did much to aid and clarify my thinking.

And finally, to my husband Gonie and my children Stephanie, Daniel and Jennifer. They have been my most constant supporters. Their love, support, patience and belief have kept me going during those times when it almost seemed easier to give up. I shall be forever grateful to you all.

Dedication

I would like to dedicate this thesis to my growing family: Gonie, Stephanie, Daniel, Jennifer, Angela, Lucas, Cora and Amber. Each of you is a source of inspiration to me. My love always.

Chapter 1. Introduction

1.1. Introduction

My research is concerned with challenging the labelling of some young people at age sixteen, entering further education, as innumerate. It seeks to achieve this through an examination of young people’s numeracy practices as they are enacted in their everyday lives, using the lens of a social practice theoretical perspective. It does not seek to glamourize or dichotomize (Street 2012; Swanson and Williams 2014) informal practices but rather to legitimize the vernacular as a starting point, and use it to establish a way into repairing students’ fractured mathematical identities. These fractured mathematical identities are revealed in the research students’ stories of their experiences of learning school mathematics and are operationalized for many of them not only as a ‘fear and loathing’ of mathematics, but also as a lack of recognition of their own mathematizing.

This introductory chapter, then, is divided into two main sections. The purpose of section one is to situate my research within the field of post 16 numeracy; to establish the background and context to the research and identify some of the key, relevant literatures. It identifies the purpose of the research and the research questions to be answered. It provides insight into my motivation for undertaking the research as well as addressing the research scope and limitations. Section two provides an overview of the structure of the thesis.

1.2. Background and Context

Mathematics is considered an important subject (Esmonde 2009; Hodgen and Marks 2013; Legner 2013; Norris 2012) however, it is fair to say that there are problems with mathematics education (Andersson et al 2015; Baker 2005; Baker and Rhodes 2007; Brown et al 2008; OECD 2004). There is a significant ‘tail of underachievement’ for mathematics (BIS 2012; Whitehouse and Burdett 2013; Sturman et al 2012) in England across compulsory school age children as well as adults. Both industry and higher education (CBI 2008; Institute of Physics 2011) have also expressed concerns about the skill levels of entrants to mathematics rich courses. Underachievement affects children at the higher end of the ability scale and research (Whitehouse and Burdett 2013) shows they are not being sufficiently challenged. Important also are the ways that lack of success in successfully engaging in mathematics may be seen to contribute to issues of social and economic exclusion (OECD, 2004).

The focus of this research, however, is on those young people who do not achieve a grade C or above in mathematics, at GCSE and subsequently enter further education. Lack of success is compounded for students from low income families (Ofsted 2011) and research shows that of those who do not gain a level 2 qualification in maths, by age 16, few go on to achieve them through further education (NIACE 2013; Ofsted 2011; House of Commons Public Accounts Committee 2009; Wolf 2011). Students may then go on to work and adult life and many experience disadvantage across a range of life domains (Moser 1999; DfES 2001; Bynner and Parsons 2007).

Important to students’ success in mathematics, is the approach taken by individual colleges to identify and assess young people’s mathematical or numeracy capability on entry to further education. Research shows that ‘more of the same’ (Moser 1999; Ofsted 2011) in terms of how young people with previous negative experiences of school mathematics experience the subject in further education, is not an option. Also, important to their success are the discourses which circulate about falling standards in literacy and numeracy and the reasons for underachievement.

These powerful public discourses (Ivanic et al 2009) about and around young people and adults’ literacy and numeracy levels are important in shaping not only public opinion, but also policy and practice (Papen 2005). The public discourses define the problem and consequently, according to Papen (2005) they determine the causes and resultant actions. Acknowledging the importance of discourses about literacy and numeracy situates the debate around definitions, and what emerges is recognition that the ‘non- neutral’ definition does not exist. Important to this research project is the definition of numeracy adopted. The social practice definition of numeracy arises from discourses of empowerment and recognition that there are a range of literacy and numeracy practices going on in people’s everyday lives. This view stands in contrast to those definitions of numeracy which emanate from discourses of deficit. Policy documents in the area of literacy and numeracy (Moser 1999; BSA Adult Numeracy Core Curriculum; DfES 2001; DIUS 2007; BIS 2012) focus on target groups and subsequently construct the groups’ identities in terms of lack and being problematic. The students in this research, labelled as they and others are within the Skills for Life strategy (DfES 2001) as ‘functionally illiterate’ and having serious problems with numeracy, do not see themselves as such. The institutions within which much of the Further Education and Skills Sector policy is enacted can often be complicit with the deficit discourses (Howard 2006; Ivanic et al 2009) and focus on the skills model - the commodification of literacy and numeracy - and the value of people’s skills to the economy. Those with the requisite skills at the ‘right’ level (Leitch 2006) are useful to the economy and those without are seen as inadequate and lacking. For young people attending further education colleges these discourses

13 can influence what takes place within the classroom and highlight the issue of power in relation to literacy and numeracy teaching and learning.

Some form of participation in maths classes is now compulsory for students, and for those without a level 2 qualification, it is mandatory they acquire one while at college. Students experienced a curriculum that is defined for them rather than negotiated on the basis of their needs (Tett, Hamilton and Hillier 2006). Teachers in this research believed they had little power over what was taught in the functional mathematics classroom as well as the vocational classroom. However, what this research has shown is that there is space, if teachers are supported to recognise it, which could accommodate a more empowering pedagogy without compromising the need to meet institutional and global obligations.

Researcher attention in mathematics educational research has tended to focus on developing compulsory school age children’s mathematical skills as well as those of undergraduates on mathematics rich courses. There has also been some attention paid to the impact of teachers’ mathematical skills on student learning. The development of mathematics in adults and young people who have completed their compulsory schooling and who are either in further education or community education, has not always been an important area for research (Coben 2005). This has been the case both in terms of a focus on the learner as well as the context where learning takes place.

1.3. Statement of purpose

The purpose of this research therefore, is to challenge the dominant view in the literature, in public, policy and institutional discourses, which focus on the individual deficits of young people without a mathematics qualification and which characterises them as innumerate, a burden to the economy and the health service, as well as a barrier to social cohesion (Leitch 2006, Education and Training Foundation 2014). Drawing on the theoretical perspective of the New Literacy Studies (Street 1984,1995, Barton and Hamilton 1998, Baker and Street 2004, Ivanic et al 2009) and a funds of knowledge perspective, (Moll, Amanti, Neff, & Gonzalez, 1992), the everyday numeracy events and practices, what I have called, the vernacular numeracy practices (after Barton and Hamilton 1998) of two groups of young people are explored. I attempt to reframe the crisis narrative and dominant deficit view, in a more positive way, using the findings from this research.

There is a research history which has paved the way for a focus on social practices in relation to literacy and numeracy learning and learners. This ‘turn to the social’ (Lerman

2000) or ‘the social turn’ (Street, Baker and Tomlin 2008) in numeracy research, encompasses much more than the social as Lerman (2000) described. More anchored in social literacies research than mathematics education research, the social in numeracy, on which this research seeks to build, draws on the concepts of literacy events and practices and adopts the terms numeracy events and practices (Street, Baker and Tomlin 2008) as important concepts for research. Literacy events were identified initially by Heath (1983) as ‘any occasion in which a piece of writing is integral to the nature of the participants’ interactions and their interpretative processes’ (Heath, 1983, p. 93, quoted in Street, Baker and Tomlin 2008, p 18), however Street (2000) and other literacy researchers recognised the limitations of literacy events and identified the term literacy practices (Street 1984; 1998; 2000; Barton and Hamilton 1998; Hamilton 2000). A focus on literacy practices links literacy events ‘to something broader of a cultural and social kind’ (Street, Baker and Tomlin 2008, p 19). Street, Baker and Tomlin’s development of the concepts of numeracy events and practices provided opportunities, in their research, to focus on ‘occasions when numeracy activity is integral to the nature of the participants’ interactions and their interpretative processes’ (Baker, 1998 quoted in Street, Baker and Tomlin 2008, p 20). In adopting the term numeracy practices, they have provided ‘a nuanced language of description and a lens through which to view practices in different contexts’ (Street, Baker and Tomlin 2008, p 21).

This focus on numeracy events and practices allows for a ‘pluralisation’ of numeracy practices and the ‘possibilities they have for pedagogic practices’ (Ivanic et al 2009, p. 21). Before an understanding of pedagogic practices can be identified, a thorough analysis of numeracy events and practices needs to take place. Street, Baker and Tomlin (2008) provide the framework for analysing the research students’ out of college numeracy events and practices. They propose the four dimensions of content, context, values and beliefs, social and institutional relations. This four dimension framework, the ideological model of literacy and numeracy, (Street 1984) allows for an analysis of the relationships between in college and out of college numeracy practices. This research builds on the ethnomathematical and situated research which seeks to understand how groups use numeracy in their lives and how their numeracy practices have evolved from their cultural needs. An often quoted example of this type of situated, social mathematics is taken from Lave’s (1988) research on observing people making ‘best buy’ decisions in a supermarket. Making calculations formed only one aspect of their decision making. Participants were mostly concerned with providing meals for their families and used trial and error to reach a solution to a problem. Lave’s participants were correct in their calculations 93% of the time, in contrast with pen and paper best buy calculations, where only 44% (Capon and Kuhn 1979) of answers were correct.

1.4. Personal motivation for the research

My motivation for undertaking this research has grown from both personal and professional experiences. I started my working life as a volunteer numeracy and literacy tutor in Durban, South Africa, in 1984, where inequalities in education where part of education policy. Black children received no state support towards their education over the age of eight, unlike the children from other population groups. The adult students I taught, had all left school at eight or nine years when state funding ceased. They were working in menial jobs, mainly in white households, as servants, and were saving to pay for completing their education. Their desperation to learn in the face of state sanctioned inequalities was my motivation to work in adult numeracy and literacy on returning to England in 1985.

My work in the field of adult numeracy and literacy has spanned thirty years. I have lived and worked through the effects of successive governments’ pre and post 16 education policies. Some years ago, while interviewing young, male prisoners as part of the evaluation of the pilot for adult functional skills tests, one young man, when asked what mathematics he would find useful to study, replied,

“I want to be able to do the maths I couldn’t do in school. I want to do maths. I don’t want to do this everyday stuff. I want the hard sums I couldn’t do at school and I want my bit of paper that says I can do it.”

More recently my work has been with vocational teachers and supporting them to embed literacy and numeracy into their vocational programmes. I also work with adult literacy and numeracy trainee teachers, many of whom entered teaching via non- traditional routes and who have much in common with their students. Many of the people I work with are labelled as ‘target groups’ in government policies. They are usually positioned as having a lack or deficit. This also includes literacy, numeracy and vocational teachers, who themselves have been the subject of a series of government reviews and targets. Trying to support learners as well as trainee teachers to deal with the effects of this labelling has been a significant part of my work. While I have been part of the government’s Skills for Life Strategy through my work, I had not recognised that I was contributing to a deficit discourse of numeracy and literacy learning and learners until I started to engage in research activity which had not been available to me while working in further education. Engaging with a literacy, language and numeracy research community, all of whose research was from a social practice perspective, led to a growing awareness of my own complicity in contributing to the sedimented moral panics around literacy and numeracy. This questioning of how a deficit view of learners and

16 learning is maintained and its effects on learners, led me to want to research the social practice perspective on numeracy teaching and learning.

1.5. Scope and limitation of the research

This section introduces the scope and limitations of the research. Specific methodological limitations are discussed in chapter 3 and all the limitations are summarised in chapter 6. This research is concerned with a small group of fifteen students and six tutors in a particular further education college in the North West of England. Time spent with the students and their teachers was over one academic year and to an extent followed the cycle of life of the college, as it takes place over one college year. Engaging with this ‘college life cycle’ is important to research (Heath and Street 2008, p 60) in terms of ‘grounding data collection with the life rhythms of the group’ under study. So, although the group is small and the time relatively short compared to other broadly ethnographic type studies, it is grounded in the life cycle of the college. The study offers, then, a particular perspective of students’ vernacular numeracy events and practices. However partial it might be, it does attempt to provide a detailed analysis of these vernacular practices using a social practice analytical framework which reveals the relationship between practices in and out of college. It also offers an analysis of students’ experiences of numeracy within their vocational and functional skills classes.

The study does not claim to reflect the vernacular numeracy practices of all vocational students nor their experiences of prior mathematics learning or vocational and functional skills learning. However, it does seek to reveal the hidden nature of the embedded mathematical knowledge which students draw on in several activities which are important in their lives. This knowledge is hidden both to the students and their teachers. Further, an analysis of the students’ narratives of their experiences of learning mathematics in school, as well as an analysis of the discourses circulating in and outwith the college, reveal how the students are positioned in relation to knowledge and power and how this affects them in terms of how they experience the learning of mathematics.

1.6. Organisation of the thesis

This chapter has aimed to introduce and provide some contextual background on post- 16 numeracy as it applies to those students with low or no qualifications in mathematics on entry to further education. It has also provided an account of my interest and motivation for this research. Chapter 2 presents a review of the literatures which are relevant to my research as well as an account of how the review was conducted. These

17 important literatures include the theoretical perspectives and frameworks as well as the significant empirical studies which have informed and influenced this research. This chapter also identifies the gaps in the literature and leads to the formulation of the research questions which this study attempts to answer. The research methodology, how the research was designed and carried out, to answer the research questions, is the subject of Chapter 3. Details of the traditions which the study seeks to build on are set out, along with data generation methods used as well as reasons for the methodological choices made within the research. The approach to data analysis, its trustworthiness and the effects of researcher positioning are also discussed. The limitations of the research methodology are also outlined in this chapter. Chapters 4 and 5 present the findings from the analysis. The findings begin to answer the research questions. They include an analysis of the students’ vernacular numeracy events and practices, using the theoretical framework which has guided the research. Not only are students’ vernacular numeracy practices explained, but how they experienced functional skills mathematics is also presented and discussed. These chapters also draw on vignettes of numeracy events and practices, to show, among other things, how students’ previous experiences of learning schooled mathematics influenced their subsequent learning. How tutors in the study viewed their students’ numeracy practices is also revealed. Chapter 6 attempts to answer the research questions and discusses the findings and their implications in light of the wider literature. The limitations of the research are summarised and ways of moving forward from this study are suggested. How the research contributes to knowledge is included in this chapter along with some final reflections on the study.

1.7. Conclusion

This chapter sets the scene for the research. It establishes a rationale for the research as well as providing some background and context and why this research is potentially important. The personal motivation for the research is also explained.

Chapter 2. Literature Review

2.1. Introduction

This literature review begins with an outline of how the review is structured, and is followed by a section on the methodology used to undertake the review. It is then divided into four sections each of which addresses a key question(s) which guided the review. The first section sets the context for the literature review. It provides a brief overview of the Further Education and Skills Sector and the broad policy initiatives which have led to the spotlight being focused on the importance of numeracy. It asks what is known about the FE and Skills Sector that is pertinent to numeracy teaching and learning in FE. There is also a discussion of the importance of numeracy for adults and the way in which deficit models are used to explain low attainment in young people and adults entering further education. This leads into an exploration of the concept of numeracy, the contested nature of numeracy and its relationship to mathematics. The third section of the review traces the turn to the social in mathematics education. It includes an examination of those theoretical positions, for example, situated cognition, the New Literacy Studies and funds of knowledge, which seek to not only reframe notions of what constitutes legitimate mathematical knowledge and numeracy activity, but also to disrupt the deficit discourses as they relate to numeracy and mathematics learners and learning.

The literature review concludes with an attempt to justify adopting a social practice approach to numeracy for this research as well as setting out the research questions.

2.2. Review Methodology

Work undertaken for a previous research paper provided an initial focus for this review. That work investigated a national Literacy, language and numeracy (LLN) policy implementation in a further education college. However, during focus group activity with vocational and LLN teachers it emerged that these teachers viewed the majority of their learners as deficient in both literacy and numeracy, on entry to college. Even though many of the students entering college had GCSE English and maths at grades G-C, broadly equivalent to Levels 1 or 2 on the Qualifications and Credit Framework (QCF), up to 80% of students, when initially assessed at induction to their vocational programme, achieved a level, particularly in numeracy, which was below Entry level 3. The teachers in the study very much located the fault of low student attainment with the individual student, their home environment or their previous schooling, rather than with the on-line assessment instruments used at induction. This deficit model view, prevalent in public and government discourses seemed to fit with the recurring moral panic

19 concerning education standards in general and the teaching and learning of numeracy in particular, which has persisted for many years. Their views were, however, at odds with a body of research (Baker and Street 1996; Barton and Hamilton 1998; Barton, Hamilton and Ivanic 2000; Ivanic et al 2009) showing that although many young people and adults did not have formal qualifications in English at entry to college, they nevertheless engaged in a sophisticated set of literacy practices outside of their formal learning environments which could be used or mobilised to support their learning. Notwithstanding that this research was specifically literacy focused, I could see close parallels between the concept of literacy as a social practice and numeracy as a social practice given the ways literacy and numeracy are intertwined in everyday life (Johnson 1994; O’Donoghue 2003). This was the starting point for the literature review. I drew on academic literature; practitioner-focused publications; government reports; large-scale representative surveys and case studies. There is not a vast research literature on adult (post 16, up to and including level 2) numeracy as practised in further education in England, however, it has grown over the past twenty five years. Prior to the setting up of The National Research and Development Centre (NRDC) on-line publications bank as part of the Moser Report (DfEE 1999) recommendations, the majority of UK adult basic skills research focused on adult literacy and was carried out by the Lancaster University Literacy Research Centre with all research from Lancaster underpinned by a social practice perspective. The emergence of the Adults Learning Mathematics International Research Forum (ALM) in the early 1990’s, provided a dedicated focus to researching adult numeracy and mathematics in adult and community settings as well as in Further Education and higher education. Alongside this growing research on adult numeracy, is an established body of mathematics education research which has as its focus mathematics teaching and learning in schools. This initial investigation into the ‘deeply contested’ concept (Coben, 2003 p 9) of adult numeracy, was important to the research in order to have a workable framework for identifying numeracy events and practices outside of formal learning.

The many competing and overlapping definitions of adult numeracy which I examined are detailed below in section 2.4. This examination resulted in a rejection of the autonomous model of numeracy (Street 1995, Baker 1998) in favour of the ideological model (Baker and Street 1996, Street Baker and Tomlin 2008). This ideological model of numeracy foregrounds ‘the social’ in numeracy, not only in terms of the content of what is taught, but also the context, values and beliefs and social and institutional relations. This focus on the ‘social turn’ (Lerman 2000) in numeracy opened up the possibility of locating numeracy practices within a wider context in the sense of them being connected to issues of power; authority relationships; the nature of knowledge; knowledge production and the legitimisation of certain forms of knowledge. This moved the review towards

20 examining theories which could illuminate the unequal access to numeracy knowledge in educational settings as well as investigating more critical approaches to the teaching and learning of numeracy or mathematics.

This led me to use the following set of questions to guide the scope and analysis of the relevant literatures.

 What are the historical and institutional FE contextual factors that have proved important in shaping the development of numeracy teaching and learning to date?

 Why is numeracy important? These questions are answered in section 2.3

 What is numeracy? What are the implications of adopting and rejecting competing concepts or discourses of adult numeracy for the research?

 What is the relationship between numeracy and mathematics and how will this affect the research? These questions are answered in section 2.4

 How have particular theoretical approaches to numeracy and mathematics influenced the teaching and learning of numeracy and how do these contribute to the research? What role does the problematized concept of context play in the transfer of learning? These questions are answered in section 2.5

2.2.1 Numeracy in the Further Education and Skills Sector

In this section I trace the development of the FE and skills sector as a site of growing importance and concern for post 16 numeracy development. There is also a discussion on the importance of numeracy for adults.

Historically, the FE and skills sector has been under resourced and largely neglected by governments until the election of New Labour in 1997 (Lucas 2004, p 35). Post 16 numeracy has been described as the ‘Cinderella’ of the trinity of adult basic skills of literacy, language and numeracy (Griffiths and Stone 2013, p 1) with some commentators suggesting that adult basic education is itself the Cinderella of further education and the FE sector the ‘Cinderella Service’ of education (Moser 1999, Bradley 2000) sandwiched as it is between schools and universities. This metaphor of marginalisation serves to highlight, historically, not only the lack of attention and funding for the further education sector as a whole, but also the lack of a body of theory and research into adult numeracy learning and teaching in particular (Coben 2003, p22). Publication of the Moser Report, A Fresh Start (1999) was a landmark for adult literacy

21 and numeracy in England. Prior to 1999, adult numeracy provision, in particular, was very much a ‘marginal backwater of educational provision’ (Coben 2003, p 55). What provision existed had emerged from the adult literacy campaigns of the 1970s (Oughton 2013). Classes were predominantly staffed by non-specialist volunteers, but provision was usually learner-centred, responding to individual learner need and interest rather than a laid-down curriculum very specifically linked to employability (Tett, Hamilton and Hillier 2006). With the Further and Higher Education Act, 1992, adult numeracy became more formalised and integrated within a wider system of vocational education (Hillier 2006). After the OECD (1997) assessed the literacy and numeracy skills of adults in 12 developed countries, with only Ireland and Poland scoring lower than Britain, the way was clear for the greatest changes to adult basic education to begin. The government’s response to the Moser Report (1999), the Skills for Life strategy, (DfES 2001) included plans to professionalise the teaching workforce; reform of the curriculum in line with the curriculum reforms in schools; make national qualifications (tests) in literacy and numeracy available; set up the National Research and Development Centre (NRDC) to undertake research into literacy, language and numeracy and finally, plans to alter the types of learning opportunities available to adults, for example, a greater emphasis on workplace literacy and numeracy and more intensive programmes of study. The clear message from Moser (1999) and The Skills for Life Strategy (DfES 2001) was that there should not be ‘more of the same’ (Leitch 2006; Moser 1999; Ofsted 2011) in terms of scope and range of provision as well as who was being taught and who was doing the teaching. The underlying perspective of the Strategy was one of increased skills to improve the economy and promote social cohesion although there has been greater emphasis placed on the creation of a skilled workforce than the informed citizen (Tett, Hamilton and Hillier 2006). So along with the increased funding for adult numeracy came ‘tightly drawn boundaries’ (Tett, Hamilton and Hillier 2006, p 7) for adult numeracy teachers who now had to follow a curriculum ‘designed for learners rather than with them or by them’ (Tett, Hamilton and Hillier 2006, p7). Critics of the Skills for Life Strategy (DfES 2001) have framed their criticisms in terms of a rejection of the skills model (Herrington and Kendall 2005, Howard 2006) which perpetuates social and educational inequalities through a commodification of literacy and numeracy (Bathmaker 2005), driven by a market ideology and a ‘vision of the needs of global economic competitiveness’ (Tett, Hamilton and Hillier 2006, p7). There has also been criticism (Hodgson et al 2007) that policy in the FE and skills sector, affecting numeracy teaching and learning, is based on ‘half right’ assumptions. Hodgson et al (2007) suggest that currently the FE and Skills Sector providing as it does, ‘vocational opportunities, including different learning styles and places of learning” (DfES, 2005:7) for students whom the compulsory education sector has failed, does not address,

“Uncomfortable questions about the roots of alienation within the school system. Moreover, there is an under-estimation by policymakers of the effects of GCSE ‘failure’ on learner identities and the cost of repairing these identities in FE. Policy appears to compound the problem facing these learners by expecting FE to act as the ‘zone of repair’ without adequate resources for concerted remediation.”

Hodgson et al 2007 p326

This theme of repairing learner identity is further highlighted (Education and Training Foundation 2014) as crucial for those learners who have had previous negative experiences of formal learning. These learners, it is argued, ‘need to be supported to develop an identity through which they can view themselves as being competent’ (The Research Base 2014, p 5). Given that 75% of school leavers (The Research Base 2014) who have not achieved a grade A* - C in mathematics, enter the FE and skills sector, its importance in providing a positive second chance environment for learners in significant. Yet, many learners, according to Brown et al (2007) ‘would rather die than do maths’ after leaving school.

There is also some concern that learners who undertake initial and diagnostic assessment on entry to FE are having their mathematical skills ‘mis-diagnosed’ (Lydon 2013). The results of these initial, ICT based assessments, appear not to reconcile with learners’ on-entry qualifications. This is resulting in learners being placed on courses, in some instances, lower in level than previously attained. This provision does not stretch them, nor does it lead to progression to higher level provision or employment (Watson 2004, Hayward et al 2004).

Mathematics teaching and learning remains a key area of FE and skills sector provision (Ofsted 2014) above and below level 2, with foundation mathematics (up to and including Level 2 Functional Mathematics) deemed to be one of the weakest subjects in FE colleges (Ofsted 2011, 2014). However, the teaching of mathematics has now become a condition of funding in FE. The achievement of a GCSE mathematics qualifications or equivalent, for those learners without one, has become mandatory, post Wolf (2011). The achievement of Functional Skills mathematics at Levels 1 and 2 will now constitute an ‘interim qualification’ on the road to GCSE mathematics. That GCSE mathematics may not be the most suitable qualification for employment (Hodgen et al 2013) has been ignored. One maths specialist (Norris 2012) suggests that ‘GCSE is not a good assessment of functional numeracy’, as has been demonstrated through BIS (2012) and Skills for Life surveys. The BIS survey (2011) noted that GCSE pass rates stand at approximately 60%. This survey which gathered statistics on competence in ‘everyday maths’, however, has shown that just 24% of 16 to 24 year olds are working at level 2.

Norris (2012) noted the potential disconnect between GCSE pass rates and adult numeracy skills. This, combined with what has been called ‘the absence of a mature and integrated culture of theory and research in adult numeracy’ (Coben 2003 p55), leaves practitioners in the Sector, guided by initiatives, good practice tick boxes and checklists, rather than research informed practice. This situation, however, makes the FE and Skills Sector an important site for research into numeracy teaching and learning.

2.2.2 The importance of numeracy

There is, within education research literature, a strong case made for numeracy being important not only in ensuring that the UK has a highly skilled workforce at all levels (Noss 1997; Hoyles et al 2002; Wake and Williams 2007; Wolf 2011), but a lack of confidence as well as competence in numeracy disadvantages not just the economy but also the individual (Bynner and Parsons 2005, 2006; Wolf 2011; OECD 2013). While the Cockcroft Report (1982) underlined the importance of mathematics education for work, with the need for out-of-school calculation methods and a statement that much of the mathematical needs of employment could be summarised as a ‘feeling for measurement’ (DES/WO 1982:24, para 8 5), more recent research (Hoyles et al 2002; Coben 2008) has demonstrated the real complexity of workplace numeracy. Low levels of both literacy and numeracy have a continuing adverse effect on people’s lives and there is some evidence that poor numeracy may be a bigger indicator of disadvantage than poor literacy (Bynner and Parsons 2005). Problems with numeracy may lead to the greatest disadvantages for the individual in the labour market and in terms of general social exclusion (Smith 2004, Bynner and Parsons 2006).

In establishing the importance of numeracy in society The NIACE (2011) Committee of Enquiry into Adult Numeracy, went so far as to assert that having adequate numeracy skills was a basic human right for UK citizens, while at the same time noting that the possession of inadequate numeracy skills is very often worn as ‘a badge of honour’ (p3) in a way that poor reading and writing skills would never be. In other research (ProBonoEconomics 2014) almost 50% of people surveyed felt it was socially acceptable to be poor at maths as opposed to just 30% who felt the same for English. In its Manifesto for a Numerate UK, National Numeracy (2014) stated that being numerate is an ‘entitlement for everyone in the UK’ (p2). Yet, despite the recognition of the importance of sound numeracy skills for both the economy, health and social inclusion, only around half of young people (16-18 year olds) entered for Level 2 numeracy qualifications (approximately GCSE equivalents) are currently successful, (Ofsted 2011). The picture for adult (16 years to 65) numeracy skills is that approximately 17 million

24 adults in England are stated (Moser 1999; DfES 2001; Leitch 2006) to be working at entry level on the National Qualifications Framework.

This section highlights both the importance of numeracy and being numerate in people’s lives as well as the importance of the FE and Skills Sector as a destination for many young people and adults who have been failed by their previous schooling. The Sector is expected to engage and motivate learners, develop their literacy and numeracy, English and maths, address issues of alienation and inequality as well as the culture of acceptance of failure at mathematics. All of this is to be achieved within a volatile sector subject to competing policy imperatives and whims of government.

2.3. Conceptualising adult numeracy

In this section I explore a range of definitions of numeracy which populate the literature and conclude with the definition of numeracy which informs and is used throughout this research project. Crucial to this research is a definition of numeracy which reflects not only the complexity of the concept, but supports a methodological approach in the research.

Within the field of adult mathematics education, numeracy as a term is both important and highly contentious (O’Donoghue 2003; Coben 2000, 2003; Gal 2000; Evans 2000; Wedege 2010). Coben considers numeracy, ‘a highly slippery concept’ (2003, p9) and Gal (2005) states, ‘The construct “numeracy” does not have a universally accepted definition, nor agreement about how it differs from “mathematics”.

As Coben puts it,

There is no shortage of definitions but there is, crucially, a shortage of consensus, with the term meaning different things in different educational and political contexts and in different surveys of need.

(Coben 2000, p 35)

At one end of a continuum is the rich interpretation of numeracy as the ability to recognise when mathematics may be useful, in which contexts, to apply these and interpret the results. This definition was put forward in the Crowther Report (DES 1959) and acknowledged by Cockcroft (DES 1982) as the source of the concept and the term numeracy. Crowther stated that numeracy was, ‘as the mirror image of literacy’ (par. 398) and explained that both literacy and numeracy were needed by all people over the age of fifteen (the then earliest age at which young people could leave school), in order to communicate between the two cultures of the scientific and the literary. Crowther’s

25 view that being numerate meant the ability to reason quantitatively and have an understanding of science, implied that numeracy was seen as a ‘high level skill’ (Roper 2005). However, this view of numeracy did not last and by the time of publication of the Cockcroft Report (DES 1982), the ‘limited proficiency’ view of numeracy had taken hold. Cockcroft’s foundation list of mathematical topics would underpin the teaching and learning of mathematics to ‘lower attaining students’. (Paragraph 458, page 136, DES 1982). It comprised the following: number; money; percentages; use of a calculator; graphs and pictorial representation; spatial concepts; ratio and proportion; and statistical ideas. This foundation list formed the basis for the National Curriculum in schools and subsequently was adopted by the Moser Report (1999) and operationalised in the government’s Skills for Life strategy to improve adult basic skills in England (DfES 2001). In the Skills for Life strategy (DfES 2001) numeracy is defined as the ability "to use mathematics at a level necessary to function at work and in society in general" (DfES, 2001). Coben (2003) suggests that this limited proficiency approach has established the overall opinion of numeracy as:

“A relatively limited set of low-level (by comparison with the Crowther conceptualisation) uncontextualised mathematical skills, systematised in the Standards for Adult Literacy and Numeracy (QCA, 2000) and operationalised in the Adult Numeracy Core Curriculum (2001) and the associated Subject Specifications for adult numeracy teacher training (DfES/FENTO, 2002) and teaching/learning materials”

(DfES Readwriteplus, 2002)

One way of making sense of this ‘conceptual confusion’ (Condelli et al 2007) is to use Maguire and O’Donoghue’s (2004) numeracy organizing framework. This framework allows for the grouping together of definitions along a continuum of sophistication, which traces development in the concept of numeracy. In the initial formative phase, numeracy is considered to be basic arithmetic skills. This is representative of Baker and Street’s (1996) ‘autonomous’ model of numeracy which they categorise as abstract, decontextualised and (thought to be) value free; in the mathematical phase, numeracy is “in context,” with explicit recognition of its importance in everyday life. The Cockcroft (DES 1982) definition of numeracy and being numerate would fit here alongside the Adult Numeracy Core Curriculum (2001) and the Functional Skills Standards - Mathematics (Ofqual 2007; 2012) currently in use in further education colleges in England. The third phase, the integrative phase, views numeracy as a multifaceted, sophisticated construct incorporating the mathematics, communication, cultural, social, emotional, and personal aspects of each individual in context. It is within this third phase that numeracy as a social practice and Street’s (1995) ideological model sits.

Figure 2.1. Adult Numeracy Concept Continuum of Development

Phase 1 Phase 2 Phase 3

Increasing levels of sophistication

FORMATIVE MATHEMATICAL INTEGRATIVE (basic arithmetic skills) (mathematics in context of (mathematics integrated everyday life) with the cultural, social, personal, and emotional)

(Maguire and O’Donoghue’s (2004) organizing framework for numeracy)

For the purpose of this research I adopt the definition of numeracy put forward by Street, Baker and Tomlin (2008), which sees numeracy as a social practice or the ‘ideological model of numeracy’. Their concept of numeracy as a social practice comprises the four dimensions of: content, context, values and beliefs, social and institutional relations. If each component is examined in turn, then the content relates to the activities, techniques, procedures and processes of numeracy with which students engage. The abstract, de-contextualised aspect of numeracy, e.g. four rules of number, decimals, percentages, fractions, data handling, proportion and ratio, measurement, spatial awareness. Content would be the content of the Adult Numeracy Core Curriculum (BSA 2001) and Functional Skills standards up to Level 2 of the National Qualifications Framework. The context of the model includes the framing of the occasions when numeracy is done. For example, using ratio to increase quantities in recipes in a college catering kitchen or deciding how many pizzas to buy to share with friends. Or as in the case of one of the research students, estimating distances between buildings in his free running activities. Values and beliefs, the third component of the model, identifies how individual beliefs and values affect the numeracy practices learners adopt. For example, ‘I’m no good at maths’, or the invisible numeracy – the numeracy one can do which is dismissed by students as ‘just common sense’. That which they cannot do, is ‘proper maths.’ It might also include the absolutist view of numeracy – there is only one right way of doing a procedure or only clever people can do maths. The final component of the model is social and institutional relations. This acknowledges the issues of power and control as they relate to content, context, the curriculum, what is taught and how. It is power and control exercised by different institutions and roles within the institutions. It is student/tutor relations and the power involved in this. The role of Ofsted in exerting power over what occurs in classrooms in terms of the curriculum and pedagogy. It is the individual experiences of learners in learning numeracy either at home or in

27 school/college and the effect their experiences have on their relationship with numeracy. This view of numeracy as outlined above affords the opportunity to identify, analyse and compare numeracy events and practices in different settings, e.g. what happens in relation to numeracy inside and outside formal learning settings.

Despite the lack of consensus over the years about what exactly the term numeracy means, I claim the important element is its functionality and use outside academe. This leads one to consider what functionality students have outside academe, and how academic study might be made to serve this functionality.

2.3.1 The relationship between numeracy and mathematics

In this section I seek to explain what is relevant to this research in terms of the relationship between numeracy and mathematics. The distinction between numeracy and mathematics is subject to some debate within the research literature (Coben 2003). The literature acknowledges the problematic relationship between numeracy and school mathematics, mainly because being numerate does not appear to be an automatic consequence of participation in school mathematics (Coben 2003, O’Donoghue 2003). Much of the research literature is concerned with defining numeracy rather than mathematics, given perhaps that the term numeracy is recognised as originating in 1959 (DES 1959) and has been around a very short time compared to mathematics. The lack of an operational definition, however, need not be seen as a disadvantage. The important point in not having this agreed definition, ‘is that numeracy is defined and re- conceptualised according to the contexts within which it is used and by whom’, (Withnall 1995). This fluidity of meaning can be seen as a strength (Kaye 2013). However, when definitions of numeracy are examined, it is clear that this raises questions about mathematics (Kaye 2013). Cockcroft (1982) talks about numeracy being ‘an ability to make use of mathematical skills’ and ‘to cope with the practical mathematical demands of everyday life’, as well as to have ‘an understanding of ‘information presented in mathematical terms’. (Paragraph 39, page 11, DES 1982).

And in Coben’s (2000) influential definition:

“To be numerate means to be competent, confident, and comfortable with one’s judgements on whether to use mathematics in a particular situation and if so, what mathematics to use, how to do it, what degree of accuracy is appropriate, and what the answer means in relation to the context.”

(Coben, 2000 p 35; emphasis in the original)

Evans (2000) proposes a ‘reconstituted idea of numeracy’.

“Numeracy is the ability to process, interpret and communicate numerical, quantitative, spatial, statistical, even mathematical, information, in ways that are appropriate for a variety of contexts, and that will enable a typical member of the culture or subculture to participate effectively in activities that they value.”

(Evans, 2000, p 236)

It would seem therefore that there cannot be numeracy without mathematics, although Evans’ definition suggests there is a difference between numerical, quantitative, spatial and statistical information and mathematical information. Evans’ definition, as Coben (2003) further points out, then leads one to ask who is a ‘typical member of the culture and what are the activities they value? The definition proposed is influenced by the perspective of whoever is making it. The relationship between numeracy and mathematics is either portrayed as mathematics encompassing numeracy as in the case of Adults Learning Mathematics - A Research Forum (ALM), or conversely as in the Australian adult numeracy teacher education pack, Adult Numeracy Teaching – Making Meaning in Mathematics (ANT), where numeracy is seen as "not less than maths but more" (Johnston and Tout, 1995; Yasukawa, Johnston and Yates, 1995). As Tout explains:

“We believe that numeracy is about making meaning in mathematics and being critical about maths. This view of numeracy is very different from numeracy just being about numbers….. It is about using mathematics in all its guises – space and shape, measurement, data and statistics, algebra and of course, number – to make sense of the real world and using maths critically and being critical of maths. It acknowledges that numeracy is a social activity. That is why we can say that numeracy is not less than maths but more.”

Tout 1997, p 13.

What these views have in common is that numeracy and mathematics are not interchangeable terms. O’Donoghue (2003) agrees with this non-interchangeability.

“Mathematics and numeracy are not congruent. Nor is numeracy an accidental or automatic by-product of a mathematics education at any level. When the goal

is numeracy some mathematics will be involved but mathematical skills alone do not constitute numeracy.”

(O'Donoghue, 2003 p8)

It seems that rather than the conceptual nature of the relationship between numeracy and mathematics being at the heart of the matter, it is the use of each term to further a particular agenda that is. Being clear about what is involved in the teaching of each, is, O’Donoghue (2003) argues, crucial, given that both overlap. The lack of clarity in the use of both terms, especially in policy literature, (Kaye 2013; O’Donoghue 2003) is of concern, mainly because there is the potential for numeracy to be seen more as an economic construct than a pedagogical one. In the spirit of clarity of definition, for the purposes of this research I adopt the ‘not less than but more’ view of numeracy. This means that being numerate involves being able to use those ‘autonomous’ mathematical skills (Street, Baker and Tomlin 2008) in a range of contexts, while at the same time recognising the importance of the changing social, political and economic context ‘in ways which imply a critical notion of numeracy and a broader range and perhaps higher - or deeper - level of mathematical capabilities on the part of the individual’ (Coben 2003 p 17).

One implication, put forward by Coben (2003) of adopting this view is in the recognition that the fluidity of the term numeracy can lead to the idea of a pluralistic concept of ‘numeracies’ as has happened with the term literacies. This pluralistic view of literacy and numeracy is further examined in the next section which traces the influence of various theoretical perspectives on literacy and numeracy research. Often referred to as the social turn in mathematics or the turn to the social, this includes the New Literacy Studies, situated cognition and funds of knowledge.

2.4. Theoretical perspectives

In this section I identify how social theories of learning in general, as well as theories of learning relating to literacy and numeracy in particular, have influenced and continue to influence, the field of numeracy teaching and learning. I am concerned that so many young people and adults are alienated from participating in formal mathematics environments, therefore, I ask what theoretical tools are available to explain and address inequalities in terms of student positioning, student identities and student experiences of teaching and learning numeracy.

2.4.1 The turn to the social in numeracy and mathematics research

I intend to show how changes in the influences on mathematics education research across a range of fields, has led to an increased focus on and acceptance of the social in mathematics. This social turn signifies the recognition that ‘meaning, thinking and reasoning are products of social activity’ (Lerman 2000, p 23). I also intend to show that the ‘social’ in terms of Lerman’s (2000) social turn, is, like numeracy, something of a slippery concept itself, with ‘the social’ taking on different meanings for different researchers.

Psychological paradigms, methodologies and research questions are generally accepted (Atweh et al 2001) to have dominated mathematics education research in the 1960’s and 1970’s. Lerman (2001) suggests that traditional psychology has done very little to improve society, in particular, the poorest in society. Lerman, agreeing with Apple, (1995) that psychology’s influence on mathematics education led to an ‘individualizing view of students’ and consequently a lack of recognition of the influence of social structures and social relations which form those individual students. Therefore, from this perspective, psychology is then ‘unable to situate areas such as mathematics education in a wider, social context that includes larger programs for democratic education and a more democratic society. (Apple 1995 p 331) Three fields outside of mathematics are credited by Lerman (2000) for initiating the turn to the social in mathematics education. These are, anthropology (e.g. Lave), sociology (e.g. Walkerdine), and cultural psychology (e.g. Nuñes; Cole). For these researchers, knowledge and identity are intricately linked and situated in specific practices. Lerman (2000) suggests that the most significant moment in the turn to the social, was the publication of Jean Lave’s (1988), Cognition in Practice.

In her research of people doing their grocery shopping, attending a slimming club and interviewing participants on how family money is managed Lave (1988) provided early research into the mathematics people use in their everyday lives and is considered ground breaking in terms of ‘a socio-cultural model of learning as interaction between the learner and their environment’ (Colwell in Coben 2003, p 40) or context. Lave put forward the notion that there exists, in different settings, an infinite number of different types of arithmetics (1988, p 63). Mathematics taught at school is one type of arithmetic practice. Lave further argued against the view that people’s cognitive abilities are stable and constant across contexts. As Lave puts it:

Several years of exploration of arithmetic as cognitive practice in everyday contexts has led to a kernel observation….[that] the same people differ in their arithmetic activities in different settings in ways that challenge theoretical boundaries between activity and its settings, between cognitive, bodily, and

social forms of activity, between information and value, between problems and solutions.

Lave (1988, p 3)

Therefore, according to Lave, everyday mathematical problems are interconnected with issues that are not primarily about mathematics. This accounts for the many forms of maths in practice (1988 p 101). These other issues or activities, conceptualised as structuring resources, give structure to activities or practices.

Lave’s work also challenged notions of learning transfer, ‘the use in one context, of ideas and knowledge learned in another’ (Evans 2000 p 74) and the rationale for providing compulsory mathematics education for schoolchildren, in that school mathematics provided an easily transportable toolkit which could be used in any situation. Lave (1988), in her comparison of school mathematics problems and out of school problems, showed that problems outside school and their solutions are part of a larger context of activity. The shopping problems were not only about calculations and best buys, but rather were related to other things, like feeding the family and space in the fridge. The school mathematics problem, on the other hand, is an end in itself: ‘these kinds of problems are specialised cultural products which belong to particular social practices, those of mathematics education.’ (Colwell in Coben 2003 p 41). Lave proposed a model of learning, a socio-cultural model, ‘where cognition is constituted in dialectical relations among people acting, the contexts of their activity and the activity itself’ (Lave 1988 p 148).

Walkerdine is credited (Lerman 2000) with bringing a focus on subjectivity and mathematics education; something which Walkerdine (1997) suggests is missing in Lave’s work. Both Lave and Walkerdine however share a view of mathematics not as an abstract cognitive task but as something deeply bound up in socially organised activities and systems of meaning. Walkerdine (1988, 1997), like Lave (1988), presents mathematical reasoning as bound up with the larger activity within which it takes place. Numbers for her are signifiers that signify forms of activity. Mathematical activity, then, is activity whose forms are orchestrated so as to exhibit specifically mathematical signs. Walkerdine demonstrates that a mathematical signifier such as "more" can be embedded in wholly different signs in different settings. In the homes of the people she studied, for example, children and adults employ the signifier ‘more’ in the context of regulation over the consumption of food and other things. (As in, ‘A: Can I have more? B: No, you cannot have more.’) This ‘more’ has quite a different discursive logic than the ‘more’ of classroom mathematics assignments, and it is no wonder, Walkerdine concluded, if children stumble in their acquisition of the pairing of signifiers and signifieds that is

32 specific to mathematical activity. Walkerdine, like Lave, sees cognition as inseparable from its context. The context of any social action is constituted by the practices in play and the related discourses.

Evans (1999) links the notion of subjectivity and the mathematics classroom as a site of complex practices. He argues that the idea of giving real world contexts for mathematical concepts does not provide meaning for students. He, similar to Lave and Walkerdine, sees school mathematics practices as one of a range of practices that an individual might call upon to use in everyday situations. Depending on whether the individual was successful or unsuccessful in the school mathematics context, she/he might focus on the mathematical calculation required or more likely, the identity called up, would be one of low confidence and lack of success. Evans maintains that thinking and emotion are inseparable so that human mathematical activity is also always emotional, rather than only cognitive. Evans (2000) proposes a complex concept of context whereby mathematical thinking is constituted by the discursive practices in which subjects have their positioning. A discourse, Evans suggests, is a system of signs that organises and regulates specific social and institutional practices; it provides resources for participants to construct meanings and identities, experience emotions, and account for actions. Discourses specify what objects and concepts are significant and what positions are available to participants in the practice – the various roles that may be adopted, together with their possibilities for action and relationships with other participants. They also provide standards of evaluation. These form the basis of social relations of power which regulate how the positionings of participants come about – how individuals come to take up particular discursive positions from those available (Evans, 2000). Positioning is particularly relevant to understanding emotion as it affects how individuals' identities are constructed within a power structure of social relationships. Positioning is not permanent; neither is it completely determined, nor freely chosen: participants are constrained and enabled by their personal histories and the discursive resources available to them. These resources may be drawn from discourses other than those underlying the practice(s) in which they are immediately involved. Using problem- solving interviews, Evans makes judgements about which practices provide the position within which the subject responds to a particular problem. His analysis shows how ideas, emotions and actions of participants are shaped by the dynamic of interactional practices, and how positions available in discourse can be realised as positionings in practice.

Saxe’s (1991) study of child sweet-sellers in Brazil also sought to examine how mathematics is used in everyday life and the role of culture in the development of

33 cognition. Saxe found that the children’s cognition was linked to their cultural and social relationships. He proposed that,

“Culture and cognition are constitutive of one other. Social conventions, artifacts and social interactions are cognitive constructions and cannot be understood adequately without reference to cognizing individuals. At the same time, individuals’ cognizing activities are interwoven with conventions, artifacts and other people in accomplishing of everyday life.”

(Saxe, 1991, p 184)

Nunes, Schliemann and Carraher’s (1993) research in Brazil with children and adults using mathematics in a range of work situations, compared these with calculation practices used by students with similar or more schooling. Their study showed that the workers, in most cases, performed better than the students. They used mainly oral methods with the context for the calculations kept constantly in mind. Notwithstanding their lack of education, the workers were able to calculate ratios which were not met in their working situations and were also able to solve problems which were unfamiliar. The research also showed some level of learning transfer of mathematical knowledge, if the contexts were meaningful to the workers. For example, the fishermen were able to calculate ratios in an agricultural context, about the yield of ground cassava from fresh cassava.

Student centred learning, described by Stech (2008) as ‘active and reformist conceptions of teaching/learning’ are strongly nurtured by this (situated learning) conception. Steen and Turner (2002) highlight the problems of teaching a situated numeracy. They acknowledge the difficulties of taking numeracy outside of the mathematics classroom and further state that in schools, students would learn numeracy through all subjects.

Thus mathematics teachers should not, and cannot, bear the entire burden of helping students become numerate. Like literacy, mathematical literacy is everyone's responsibility.

(Steen and Turner 2002, p293)

High quality mathematical tasks are authentic, intricate, interesting, and powerful.

(Steen and Turner 2002, p290)

This is not to say that the everyday skills and knowledge cannot be harnessed and used in more formal settings. However, the focus needs to be on ‘meaningfulness and the learner’s own resources and approaches to problem solving’. Schliemann stresses that:

Everyday mathematics research has documented how people represent and solve problems through their own invented methods commonly used in specific situations. Schools can and must engage students in situations that are part of their everyday experiences as well as in situations that are new for them. (…) By explicitly recognising these alternative methods of representing and solving problems teachers can understand more clearly how students think and better design situations to help them to advance and to cope with new situations and problems.

(Schliemann, 1999, p 29)

The situated cognition theorists (Lave 1988; Saxe 1991; Nunes, Carraher and Schliemann 1993) highlight the ‘unexplained discontinuity of understanding’ (Boaler 2000) in a subject seen by many as fixed and immutable. Demonstrating the inconsistency between mathematical performance carried out in different contexts, for example, the school and everyday situations, they challenged the widely held view that mathematics learned in school could be unproblematically applied to any real world situation. Lave, in her Adult Maths Project, demonstrated the inappropriateness of assuming that students can learn something, retrieve it from memory and apply it to a new situation, independent of the activity itself, the context or the process of socialisation. This rejection of unproblematic learning transfer on the one hand, does not necessarily lead to an equally polarised view that all learning is situation specific and that context, as a concept, has no place in the formal learning environment. Lave (1988), in particular, rejected the use of context within mathematics lessons and cited the ‘shopping context used in the classroom’ as having no value to students and little effect on performance in the supermarket, and serving only to ‘disguise mathematical relations’, (Lave 1988 p 99).

Boaler (1993) proposes a third way within the polarised views of learning transfer theory and situation specific learning, and suggests that these ‘inconsistencies’ across contexts may mean that the context in which maths is learned dictates the mathematical procedures selected, rather than the problem to which it is applied. She advocates recognition that context is a powerful determinant in terms of performance, but rejects, like Lave, much of the use of contextualised learning in mathematics classroom, proposing that this artificial contextualisation, is potentially as alien to young people as abstract maths. Boaler rejects using a ‘real world veneer’ (Maier 1991) when presenting classroom mathematical tasks to students. Instead she proposes an approach to mathematics teaching and learning based on the folk maths or ethnomathematics of D’Ambrosio (1985, 1991) which draws on the maths used by people outside of academic institutions. D’Ambrosio (1991) and Maier (1991) advocate using students’ own real life ‘ethnomathematics’ as a basis for mathematics learning in order to reduce the problem of

35 learning transfer. In this way the gap which students perceive as existing between ‘formal’ maths and ‘real’ maths is subsequently reduced.

Gutierrez (2013) acknowledges the role that Lerman (2000) has played in providing the social lens through which mathematics education practices are viewed. However, she argues that there is a clear line between the social as Lerman defines it and the social in the socio-political. Gutierrez explains her understanding of the sociopolitical turn in mathematics education by referencing a growing body of research by practitioners and researchers who ‘seek to foreground the political and to engage in the tensions that surround that work’ (Gutierriez 2013, p 14). The ‘that work’ that Gutierriez signals is the teaching and learning of mathematics and the political nature of this practice is deemed so by virtue of the fact that teaching and learning mathematics is a social process and therefore inevitably is a political practice. This sociopolitical turn signals, she argues, the shift in theoretical perspectives which,

Acknowledge that knowledge, power, and identity as interwoven and arising from (and constituted within) social discourses. Adopting such a stance means uncovering the taken-for-granted rules and ways of operating that privilege some individuals and exclude others.

(Gutierriez 2013, p 14)

While Gutierrez cites a variety of perspectives which can be considered part of the sociopolitical turn in mathematics education, she maintains that the three most important theoretical perspectives are, critical mathematics, critical race theory and latcrit theory and post-structuralism. For the purposes of this research, I focus on critical mathematics education and its influence on critical numeracy and a social practice approach to numeracy teaching and learning. Rooted in the critical theory of the Frankfurt School and the critical pedagogy of Freire, two of the main goals of critical mathematics are ‘to develop within learners “conscientizacao” (a kind of political awareness) that allows an individual to recognize her or his position in society and as a part of history (Freire and Macedo 1987) and to motivate individuals to action’ (Gutierrez 2013).

Frankenstein (1989, 1995, 2009) and Skovmose (2011) are perhaps the best known advocates of ‘a criticalmathematical literacy’ and critical mathematics and mathemacy, respectively. Frankenstein (2009) claims the main goal of a criticalmathematical literacy is not to understand mathematical concepts better, although that is needed, but rather it is, ‘to understand how to use mathematical ideas in struggles to make the world better. In other words, the question to be investigated about the criticalmathematical literacy curriculum is not “Do the real real-life mathematical word problems make the mathematics more clear?” The key research questions are “Do the real real-life

36 mathematical word problems make the social justice issues more clear?” (Frankenstein, 2009, p 2).

Skovmose (2011) operationalises critical mathematics as ‘mathematics in action’ and ‘the variety of forms in which mathematics is brought into effect’, including technological, economic and business settings. Mathemacy, Skovmose (2011) argues, embraces critical mathematics as well as a critical mathematics curriculum for active citizenship. Kerka (1995) proposes that a critical numeracy means that learners empowered with functional skills can participate fully in civic life. They can sceptically interpret advertising and government statistics, and take political and social action. Shore et al (1993) suggest that teachers can use numeracy as well as literacy to counteract the perspective which blames people with numeracy problems for their own difficulties, as well as helping students to examine how society positions them and treats them differently.

While anthropology and ethnomathematics research, outlined above, influenced the recognition that mathematics may be embedded in a range of activities and practices, anthropology was also influencing research into literacy studies which in turn has influenced classroom numeracy practices for adults. In the following section I explore two important theoretical perspectives, Funds of Knowledge and the New Literacy Studies (NLS) and explain their relevance and importance for this research.

2.4.2 Funds of Knowledge

The concept of funds of knowledge has been imported from anthropology (Wolf 1966, Velez-Ibanez and Greenberg 1990) to education (Moll, Amanti, Neff and Gonzales 1992, Moll 2002) in order to explore how insights gained from a knowledge of students’ lives and practices outside of the classroom (the social) can be brought into the classroom and used to enhance learning. It is presented as “historically accumulated and culturally developed bodies of knowledge and skills essential for household or individual functioning and well-being” (Moll, Amanti, Neff, and Gonzalez, 1992, p. 133).

In their anthropological study of households in the US-Mexican borderlands, Velez- Ibanez and Greenberg (1990) whose research purpose was to instigate possible reforms in US public education policy in schools which served US-Mexican school children in the south west USA, described the formation of, 'strategic and cultural resources, which we have termed funds of knowledge, that households contain' (1990, p 313). They relate their use of the term to Wolf's (1966) categorisation of economy in peasant households into several funds, which may take the form of labour, produce or currency. Velez-Ibanez and Greenberg (1990) suggest that 'entailed in these are wider sets of activities requiring specific strategic bodies of essential information that 37 households need to maintain their well-being' (p.314), and define these bodies of information as 'funds of knowledge'.

Since Moll’s original work, the term ‘funds of knowledge’ has entered the discourse of both researchers (Baker 2005, Oughton 2010) and policy-makers. For example, a DfES document for teachers of pupils from minority ethnic backgrounds stated that:

Schools have much to gain from the experiences and understanding of pupils, their families and communities. Drawing on their funds of knowledge enriches a school in a range of valuable ways.

(DfES, 2004, p8, emphasis added)

Although the focus of much of the funds of knowledge literature is primary and secondary language and literacy studies, there is a study which explores the relationship between learners’ numeracy funds of knowledge and numeracy teaching and learning in a post 16 setting. This study, ‘Making use of Learners’ Funds of Knowledge for Mathematics and Numeracy: Improving Teaching and Learning of mathematics and numeracy in Adult Education’ (Baker and Rhodes 2007), set within the Maths4Life Project sought to investigate ways that teaching and learning adult numeracy can build more effectively on what learners know – their funds of knowledge for mathematics and numeracy. The report is aimed at helping professionals involved in adult numeracy to improve the teaching and learning of numeracy to adults. A significant outcome of their study is the distinguishing between ‘narrow’ and ‘broad’ funds of knowledge. They explain the narrow and the broad as follows:

In most numeracy classes teachers plan and design their own lessons around the mathematics and numeracy topic or content they are scheduled to teach. For example, when teaching measuring, calculating, or telling the time their objectives are closely tied to the skills and understandings learners need to master those topics. Their plans for these lessons account of what learners’ skills or knowledge are of this mathematics and numeracy content. In this project we see this as a narrow view of the learners’ funds of knowledge for mathematics and numeracy. What teachers do not often do is to extend this view to take positive account of the learners’ broader knowledge, practices, backgrounds, or experiences. For example, learners come to classes with different histories, identities, dispositions, beliefs, personal attributes, expectations, aspirations, experiences, relationships to learning and to mathematics, practices, knowledge and motivations. These are what we call in this project the learners’ broader funds of knowledge for mathematics and numeracy.

(Baker and Rhodes 2007, p3)

One of the key findings from the Baker and Rhodes (2007) study is that not only is utilising learners’ funds of knowledge much more than recognising the formal facts and

38 skills learners have acquired in their previous educational or formal settings, it is also about ‘developing a reciprocal relationship where teachers come to understand that learners’ experiences of numeracy practices within informal settings are valid and valued.’ (Baker and Rhodes 2007, p13). It suggests that more effective teaching can arise if teachers are sensitive to and take positive account of learners’ broader funds of knowledge for numeracy. Through greater effective teaching and learning of numeracy, Baker and Rhodes (2007) suggest there would be a lessening of the perception held by many learners that numeracy is irrelevant and boring. While this research recognised the value of learners’ broad funds of knowledge, it did not research any methods of identifying those broad funds of knowledge. By focusing on the whole learner rather than their deficit, teachers and learners together can ‘attempt to reconstruct knowledge, attitudes and understandings’ (Baker 2005, p 17) as opposed to trying to ‘paste new skills over old’ (ibid. p 17) with little effect. However, this approach is not without tensions and conflicts, not least for teachers and their practice. As Baker and Rhodes (2007) conclude in their study,

There are substantial and troubling conflicts between the formal numeracy curriculum and the learners’ experiences beyond the classroom. Dominant approaches to the teaching of numeracy tend to be about finding out what learners cannot do; that is seeing learners in deficit and then working on that.

(Baker and Rhodes 2007 p14)

This literature was influential in focusing my research gaze because it provided affordances for new ways of looking at students’ out of school or college numeracy and mathematics practices in order to understand, not only why some students are ‘left behind’ (Civil 2014) but also in thinking about the implications for students of the discontinuities between home and school numeracy practices. While disrupting the discourse of deficit, it also built on that tradition which, Baker (1996) calls, ‘challenging ways of knowing’ and this led me to question the potential benefit to students of challenging the hegemony of belief around the legitimacy of knowledge acquired and used outside of formal learning and what role or purpose this knowledge could have in college. There are, however, limitations in the funds of knowledge literature in that it does not appear to provide a focus for students and teachers to reflect on and expose the more political aspects of practices both inside and outside formal learning. Historically, learning has been conceived of as an acquisition of knowledge, (Oughton 2013) with the human mind as a container to be filled with knowledge, and the learner as gaining ownership of that learning. Sfard (1998, p. 8) points out that ‘if people are valued and segregated according to what they have, the metaphor of intellectual property is more likely to feed rivalry than collaboration’. Sfard suggests that the acquisition metaphor, which funds of knowledge appears to espouse, has become so natural to us that we

39 would probably never become aware of its existence if another, alternative metaphor was not available. The alternative metaphor is that of learning as participation, within which the learning of a subject is regarded as the process of becoming a member of a certain community (Lave and Wenger, 1991). Within this metaphor, there is a shift from learning as acquisition or having (Oughton 2013) to learning as ‘doing’, which then invites discussion around situatedness, context and culture. Therefore, funds of knowledge, while initially influential, did not totally fit with my purpose and led me to explore different theoretical perspectives. This exploration is the subject of the next section.

2.4.3 The New Literacy Studies

The new literacy studies (NLS) emerged as an alternative theoretical perspective for conceptualising literacy and numeracy. The NLS brings together writings about literacy and numeracy theory and practice which sees literacy and numeracy as social practices rather than technical skills to be learned in formal education (Street 2012). Underpinning the perspective is the belief that literacy and numeracy need to be researched as they occur in social life, taking the context into account and the different cultural meanings for different groups. As it relates to practice, the NLS advocates that the learner is central to the learning process, and the literacy and numeracy which learners bring from outside formal education to the context of the formal classroom, is taken into account by all those who hold power over the formal learning environment, for example, teachers, those who design the curriculum and those who assess it. Exponents of the NLS, (Baker and Street 1996; Barton and Hamilton 1998; Gee 1996; Heath 1983; Heath and Mangiola 1991; Street 1984; 2001; 2004; Villegas 1991) stress the importance of building upon students’ own knowledge and skills. Because there are many contexts in which literacy and numeracy are practised, there are, it is argued, many versions of literacy and numeracy and these are constantly evolving. This ideological model, stands in contrast to the ‘autonomous’ model (Street 2001) which underpins the dominant approach to literacy and numeracy. It pretends to be neutral but is overladen with beliefs and values about literacy and numeracy, which are never acknowledged. The ideological model allows for multiple literacies (and numeracies). If these were acknowledged within the formal educational setting, then equality would be enhanced. As Street states, ‘It is in this sense that literacy is always ideological – it involves contests over meanings, definitions and boundaries and struggles for control over the literacy (and numeracy) agenda’, (Street in Crowther et al, 2001, p 18).

The importance of identifying the ideological and the autonomous models allows for the recognition of the pervading nature of the autonomous model within public and policy discourses. These discourses give voice to judgements about people in terms of lack

40 and deficit in relation to literacy and numeracy. Following on from the judgements are the policies that are implemented to address the lack and deficit. The consequences of these ‘half-right policy assumptions’ (Hodgson et al 2007) are far reaching for students and teachers in the further education and skills sector.

Rethinking literacies across the curriculum, (Ivanic et al 2009) an empirical study, which draws on theory and research in literacy studies which suggests that, students who appear to have low levels of literacy in educational settings can be highly literate in other domains of life: in their work, domestic, community and leisure activities (Barton and Hamilton 1998; Hamilton and Hillier 2006; Ivanic et al 2009). This research has been influential in shaping my thinking about, and focus on, students in further education who are reported (Casey et al 2006; Ofsted 2011) as having low levels of numeracy skills in their college setting. While Ivanic et al (2009) focus on literacy in further education, Navigating Numeracies, Home/School Numeracy Practices, (Street, Baker and Tomlin 2008) referred to in previous parts of the chapter, explores the usefulness of understanding children’s home and school numeracy events and practices using a social practice approach. Notwithstanding that the focus of the study is primary school aged children’s numeracy, it provides useful insights into the relationship between home and schooled numeracy practices, in particular, the disjunctions between home and schooled numeracies and the resulting disengagement and underachievement of primary school age children with formal numeracy in the classroom. This research also introduces into the literature the concept of numeracy events and practices which are key concepts in my research.

Literacy events and practices are important concepts to the NLS. Street, Baker and Tomlin (2008) outline the history of the development of the terms, emanating as they do from Heath (1983) who described a ‘literacy event’ as ‘any occasion in which a piece of writing is integral to the nature of the participants’ interactions and their interpretive processes’ (Heath 1983, p 93, quoted in Street Baker and Tomlin, 2008, p 18). Street (2000) identified limitations with the concept and a move towards literacy practices emerged. Street (1984; 1998) uses the term literacy practices to, ‘focus on social practices and conceptions of reading and writing’. Street, Baker and Tomlin (2008) explain that using the term literacy practices enables the linking of literacy events to ‘something broader, of a cultural and social kind’ (Street, Baker and Tomlin 2008, p19). Literacy practices begin to explain those invisible things which people bring to a literacy event. Practices attempt to understand or explain the events and the patterns around literacy to link them to ‘broader occurrences of a cultural and social kind’, Bailey (2007).

Adopting the terms numeracy events and practices, Street, Baker and Tomlin (2008) suggest, is their contribution to the ‘social turn’ in numeracy. They see numeracy events and practices as ‘analogous’ to literacy events and practices. This linking, they propose, is useful for researchers carrying out social practice research in the field. In order to operationalise the concept of numeracy practices, Street, Baker and Tomlin (2008) break it up into four components, always at play together. These components are: (i) content, (ii) context, (iii) values and beliefs and (iv) social and institutional relations. This is the ideological model of numeracy, drawing as it does on the ideological model of literacy. Street, Baker and Tomlin (2008) argue that such an understanding of the components of numeracy practices supports the identification of the relationships between ‘numeracy practices observed in homes and schools, which are central to our research’, (Street, Baker and Tomlin 2008, p22).

2.5. Conclusions from the literature review

The research literature confirms that further education is an important site for post 16 students in terms of providing access to numeracy and mathematics provision for many students. The literature reviewed has provided an understanding of the contested nature of adult numeracy. It highlights the complexity of mathematics and numeracy as concepts and it calls for the need to examine the implications of adopting particular definitions of numeracy both in research and practice. The concept of numeracy and what it means to be numerate is, however, presented in sector policy and curriculum documents, as ‘uncontestable’ or straightforward. The literature further points to a complexity in the relationship between mathematics inside and outside the classroom, which is also absent from further education policy and curriculum documents. This relationship, conceptualised as a ‘disjunction’ within the literature, is shown to have some impact on student achievement in primary and secondary school settings.

Students’ confidence in their own numeracy knowledge and competence is not necessarily the automatic consequence of formal learning. School leavers’ negative experiences of learning schooled mathematics are shown (Evans 2000, Brown et al 2008) to be potentially more ‘portable’ than school mathematics. These negative experiences, operationalised in a ‘damaged identity’ (Gee 2007) may adversely affect students when participating in subsequent mathematical learning, particularly when these damaged identities are invoked through a ‘more of the same’ approach, rather than deliberately ‘repaired’ (Gee 2007).

There is evidence of research which examines the transfer of school mathematics to out of school contexts and there is some research evidence of mobilising out of college

42 literacy practices to support in-college literacy learning. However, there is little research evidence of how the relationship between in and out of college numeracy practices is enacted in the functional skills classroom and the resulting effect it may have on learners’ experiences and achievement. I have identified this as a significant gap in the literature.

The literature further highlights how particular theoretical perspectives invade public discourses, government policy and political imperatives and yet these are presented as ‘natural’ and often remain unchallenged. These deficit positionings of numeracy students and teachers, leave many students entering further education with little choice in the identity positions they take up. Given this public positioning of learners entering further education as a deficient community in respect of their numeracy skills and knowledge, (Moser 1999; DfES 2001; Smith 2004; CBI 2008; DIUS 2007; Wolf 2011; Ofsted 2011; Vordeman et al 2011; NIACE 2015) adopting a social practice approach to numeracy is useful in conceptualising the resources which learners might bring from the everyday to their formal learning settings. Ultimately, I am concerned to disrupt this notion of deficit as it is applied to students from disadvantaged backgrounds on leaving compulsory schooling and entering further education. Studies in literacy in further education and in numeracy in primary schools, have shown that students have greater capabilities – their vernacular numeracy practices (after Barton and Hamilton 1998) – than are realised or acknowledged by themselves and the formal environments they inhabit. However, what the studies I have cited do not tell us is which are the crucial events and practices or funds of knowledge that can be recognised and used in college numeracy work by teachers and students, nor how these might be identified. Therefore, I aim to explore what numeracy practices students engage in, in their everyday lives. I further aim to explore how these numeracy practices are recognised by teachers and students and finally if they are recognised, how are they used by teachers in vocational and functional skills classes. I see the recognition that students engage in numerate practices in their everyday lives as a starting point for building further learning. This recognition of students’ vernacular numeracy practices is conceptualised as the ideological model of numeracy and is operationalised in the social practice approach to teaching and learning mathematics.

The literature review has also shown the relationship between ‘conceptual frames’ (Street, Baker and Tomlin 2008) and the very particular use of terminology as part of those conceptual frames. This is most evident in the multiplicity of uses of the terms practice and practices. Many of the literatures examined (Lave; Walkerdine; Evans; Baker; Street) use the terms practice and practices yet these terms mean something different depending on the theoretical perspective adopted. This calls for conceptual clarity within the research.

The review also points to the gaps in research in relation to numeracy teaching and learning up to and including level 2, in further education in England. In particular, it highlighted that there are gaps in the literature investigating numeracy as a social practice among young people and adults attending colleges of further education, in England; specifically, that current practices in further education teaching and learning numeracy, focus on what students do not know, rather than what they do. The research questions attempt to address this gap in the literature.

2.5.1 Research questions

I have identified the following research questions as a result of undertaking the literature review. The development of these research questions over the course of the early stages of the research is detailed in chapter 3. In chapter 3, I also explain how my theoretical perspective and position in relation to adult numeracy, has influenced the study’s research methodology. I justify the research methodology adopted in relation to ontology, epistemology and theoretical perspectives and aim to provide the conceptual clarity needed in relation to all terminology used. I explain the data collection methods used and how the data generated was analysed.

1. What numeracy events and practices do learners engage in, in their everyday lives?

2. What numeracy events and practices do learners engage in on their vocational and functional skills programmes?

3. How are learners’ everyday numeracy practices, conceptualised as their vernacular numeracy practices, used in their vocational and functional mathematics learning?

Chapter 3. Methodology

3.1. Introduction

In the previous chapter I outlined the significant literatures, key developments and relevant conceptualisations as well as the important theoretical perspectives which have influenced the study’s focus, aims and questions. The main purpose of this chapter then is to provide the details of my research methodology and design, the rationale for my choice of methods and to describe how I have implemented them. The chapter begins with an explanation of how the research questions developed as these influenced and in turn have been influenced by the study’s research methodology and design. There is an explanation of the criteria used to guide the quality of the research. There is also a discussion on how the ‘criteria for quality’ support the trustworthiness and credibility of the research and to what extent generalisation is possible from qualitative research. This leads to an explanation of the assumptions made about important conceptual and theoretical issues relating to the research. There is a description and explanation of the research setting and sample; the data generation methods used and the approach taken to data analysis. Ethical issues are discussed as well as the positioning of the researcher and how the process of reflexive engagement was enacted in this study. The chapter concludes with a summary of the strengths and limitations of the research methodology and the impact these have on the findings from the research.

3.2. Developing the research questions

The research literature acknowledges that generating research questions is ‘not a single act or decision’ (Campbell et al. 1982, cited in Robson, 2002: 55) and research questions are likely to be modified as the research progresses and the researcher gains greater understanding of the important issues relating to the research topic (Bazeley 2013). In fact, ‘it is highly likely that […] research questions will be initially underdeveloped and tentative’, being clarified and changed during the research process (Robson, 2002: 165). Campbell et al. (1982, cited in Robson, 2002: 55) view the process for selecting research questions ‘as often non-linear and involving considerable uncertainty and intuition’.

In identifying my initial research questions, I was mindful that they would undergo subsequent changes and that such alterations are an inherent part of an interpretative, qualitative research design. Bazeley (2013) refers to this as, ‘a funnelling process’ which allows for the narrowing down of a broad area of interest which when mediated through a conceptual framework allows for the formulation of specific research questions. My research questions developed in this way. A Funds of Knowledge perspective resonated

45 with me as it challenged the deficit view put forward in the Skills for Life (DfES 2001) strategy for adult literacy and numeracy which compared the literacy and numeracy skills of many adults to those of primary school children. I framed my initial set of research questions (Appendix 1 page 159) to incorporate the term ‘resources’ which is central to ‘funds of knowledge’. The term resources in this context refers to households and communities containing ‘ample cultural and cognitive resources with great potential utility for classroom instruction’ (Moll et al 1992 p.134). However, my engagement with the New Literacy Studies literature and its application in post 16 settings, led me to recognise a conceptual framework which afforded a more specific focus on individual students’ engagement with numeracy or mathematics inside and outside college. This opened up the possibility for examining participants’ numeracy practices across both sites. More specifically, it offered a wider lens with which to view students’ numeracy practices. This lens starts from the premise that students engage in and use numeracy or mathematics in their everyday lives and these social uses of numeracy are important as units of enquiry in and of themselves as well as in relation to students’ use of numeracy in college. The NLS framework, or wider lens, also provides affordances to explore students’ uses of numeracy as more than just the mathematical or numeracy content of the problems they may solve. It points to an examination of the ideological nature of numeracy (Street 1984) in the sense that content, context, values and beliefs and social and institutional relations - those key aspects of the NLS conceptual framework – are infused with relations of power. Consequently, I rejected using the ‘funds of knowledge’/’resources’ framework, not only for the reason cited above, but also on the grounds that in aiming to reject the deficit positioning of students within the current discourses surrounding numeracy, there was a seeming contradiction of deploying an economic metaphor to empower disenfranchised populations (Hinton, 2015; Oughton, 2010). Further the concept of ‘funds of knowledge’ and ‘resources’ seemed to support the reification of the teacher’s privileged position as arbiter of which student resources “count” as acceptable in the school space (Zipin, Sellar and Hattam, 2012; Rodriguez, 2013). I decided to link the research questions directly to the key concepts in the conceptual and analytic framework derived from Street, Baker and Tomlin (2008) in order to provide a clear focus for data generation and data analysis, or questions that are ‘analysable and answerable’ (Bazeley 2013, p 46). The final research questions provide direction and focus on what is important and relevant to the research.

3.3. Explanation of key concepts used in the research

Education is a multidisciplinary field (Grenfell 2012, p 50) and one of the features of educational research can be a ‘crossing of boundaries between disciplines’ (Adams et al, 2012, p 5) linking education with other fields including sociology, anthropology,

46 philosophy, history, linguistics and psychology. Not only does this boundary crossing potentially provide ‘creative imaginings’ and ‘new understandings’ (Adams et al 2012, p 5) but it can also give rise to new terms which are potentially confusing when used in a different context.

As stated above, this study draws on research in the New Literacy Studies which in turn, draws on potentially ambiguous conceptual terms. Terms used include but are not limited to: numeracy as social practice; numeracy events and practices; ideological model of numeracy; autonomous model of numeracy; vernacular numeracy practices, funds of knowledge. In table 3.1 below, I present these key concepts to show how they are understood and used within this research. Importantly, in the sections which follow, I explain and account for the decisions I have made in relation to the research design and methodology, and attempt to show an alignment between the underpinning philosophical tradition of the research and the methods which support this.

Summary of concepts used in the study

Table 3.1. Explanation of concepts used in the study adapted from Navigating Numeracies (Street, Baker and Tomlin 2008, p. 20 – 21.)

Concept How concept is used and interpreted in this study

Numeracy as a Coming out of literacy theory, this approach emphasises the social practice uses, meanings and values of numeracy in everyday activities and the social relationships within which numeracy is embedded.

Numeracy events An observable moment when numeracy activity is integral to the nature of the participants’ interactions and their interpretative processes. (Baker 1998) For example, shopping, reading a nutrition chart on food labels, weighing ingredients.

Numeracy More than the behaviours that occur when people do maths or practices numeracy (numeracy events). Includes the context and settings in which numeracy events are located. College numeracy practices may be characterised as having an educational purpose with the teacher in control of what is learned as well as when and how it is learned. Out of college numeracy practices may be characterised as more authentic, student controlled, multimodal.

Autonomous Sees numeracy and mathematics as abstract, model of numeracy decontextualized and value free. Often privileged over less formal forms of mathematics.

Ideological model Sees numeracy practices as having four dimensions: content, of numeracy context, values and beliefs, social and institutional relations and all are permeated by power.

Funds of Uses insights gained from a knowledge of students’ lives and knowledge practices outside of the classroom (the social) inside the classroom to enhance learning

Vernacular The sum total of the resources students call on when engaging numeracy with numeracy activities outside of college. practices

3.4. Research methodology and design

The aim of this research is to challenge the dominant view in the literature, in public, policy and institutional discourses, which focuses on the individual deficits of young people without a mathematics qualification and which characterises them as innumerate. The study seeks to challenge the above view through adopting the view of numeracy as a social practice (NSP). It sets out to explore and represent the numeracy events and practices which a particular group of young people engage in, both inside and outside 48 college, with a view to challenging the positioning of those students as innumerate. The implication of this theoretical perspective for the research design and methodology is significant.

Choosing a research methodology is contingent upon the research aims, epistemology, and the research questions (Jackson 2013). As outlined above, this study adopts a view that numeracy is a social practice. Students’ numeracy events and practices are significant concepts within this perspective. The research questions set out at the end of chapter 2, focus on students’ numeracy events and practices both outside and inside college, as units of inquiry. Therefore, developing a research methodology consistent with accessing, exploring, analysing and reporting these events and practices is important for the study. In order to find out what numeracy events and practices the students in the study engage in, I adopted what Ivanic et al (2009, p 193) refer to as ‘a broadly ethnographic’ research methodology. This ‘broadly ethnographic’ methodology is similar to ‘adopting ethnographic tools’ (Street 2012, p39). It is ethnographic in approach in that it uses everyday practices as a focus for interpretation, yet it is not ethnography in the anthropological sense, that is, the long-term, in-depth study of a social group (Street 2012, p 39). It uses the ‘tools of observation to comprehend a certain situation, but without the general theory of culture and society’ (Street 2012, p39). Applying these ‘ethnographic tools’ enables the researcher to work alongside research participants, thereby supporting the gathering of the participants’ perceptions and perspectives while trying to ensure an equitable relationship between researcher and participants. An ethnographic approach emphasises the importance of understanding things from the point of view of participants (Denscombe 2002) and encourages the collection of a range of perspectives and experiences to assist in the analysis of the complexity of situational life (Jeffrey and Troman 2004).

In deciding on a case study design I have been guided by the methodological literature (Yin 2009; Stake 1995; Hitchcock and Hughes 1995), and also by two significant studies (Ivanic et al 2009; Street, Baker and Tomlin 2008). Both studies adopt a social practice perspective towards literacy and numeracy as their theoretical position, and use case studies as their preferred design. While different writers provide their own descriptions, explanations and justifications for using a case study design there are some consistent key messages on using this approach. These are, the nature of the phenomenon to be investigated and the implication that there is no single truth to be uncovered (Stake 1995; Yin 2009) but rather a view of the social construction of reality (Baxter and Jack 2008) wherein the researcher enables participants to tell their stories. Through the telling of the stories, the researcher can better understand participants’ views of reality and their actions. Another common factor across proponents of the case study is the opportunity

49 provided to explore a phenomenon within its own context using a variety of data sources. While the use of many data sources is advocated to ensure that many facets are explored, there is also a caution about defining the case and placing boundaries around it either by time and place (Creswell 2003; Yin 2009); time and activity (Stake 1995); by definition and context (Miles and Huberman, 1994). This ‘binding’ (Stake 1995) of the case, it is argued will ensure that the study remains reasonable in scope.

The research aims and questions of this study point to an exploratory case study design (Yin 2009). Yin’s (2009) definition of a case study as, ‘an empirical inquiry that investigates a contemporary phenomenon within its real-life context’ (Yin 2009, p 18) resonated with the aims of this research. Further, his position that a case study design should be considered when: (a) the focus of the study is to answer “how” and “why” questions; (b) you cannot manipulate the behaviour of those involved in the study; (c) you want to cover contextual conditions because you believe they are relevant to the phenomenon under study; or (d) the boundaries are not clear between the phenomenon and context, is useful as I wanted to capture what Geertz (1973) calls, ’the up close reality…of participants’ lived experience of, thoughts about and feelings for a situation.’ The case, in this instance, is the collection of the numeracy events and practices of a small group of vocational students, in and out of a college of further education, in the North West of England.

Two significant (Brown 2008; Jarvis 2007; Pollard 2009) empirical studies Rethinking literacies across the curriculum, (Ivanic et all 2009) and Navigating Numeracies, Home/School Numeracy Practices, are concerned with examining students’ everyday literacy or numeracy practices as well as college or school practices. Rethinking literacies across the curriculum, (Ivanic et al 2009) drew on theory and research in literacy studies which shows that, students who appear to have low levels of literacy in educational settings can be highly literate in other domains of life: in their work, domestic, community and leisure activities (Barton and Hamilton 1998; Ivanic et al 2009; Tett, Hamilton and Hillier 2006). This research had been influential in shaping my thinking about, and focus on, students in further education who are reported (Casey et al 2006; Ofsted 2011) as having low levels of numeracy skills in their college setting. While Ivanic et al (2009) focus on literacy in further education, Navigating Numeracies, Home/School Numeracy Practices, (Street, Baker and Tomlin 2008) explores the usefulness of understanding children’s numeracy events and practices using a social practice approach. Even though each of the two studies has differing research aims, they nevertheless share similarities in terms of methods used to identify students’ literacy or numeracy events and practices both in and out of the classroom. Methods to achieve

50 these significant insights, in both studies, include, interviews; observations; focus groups; as well as document analysis.

I chose to use a set of data generation methods consistent with a broadly ethnographic approach and which, the literature suggested would support the examination of students’ numeracy events and practices both inside and outside college. These include: focus groups; interviews; classroom observations; researcher reflections and relevant document analysis. The link between each method and the research questions is summarised in table 3.2 below.

Table 3.2. Questions-methods matrix (Wellington 2000)

Research Question Research Methods Researcher Reflections

1. What numeracy Focus groups: Interviews: Reflections on events and practices focus group Clock-face activity 1:1 semi-structured do learners engage activity and interviews with in, in their everyday Reviewing data generated interviews with students lives? with students students

2. What numeracy Observations: Document analysis: Reflections on events and all 8 classroom observations Adult numeracy core practices do observations across construction, hair curriculum learners engage and beauty and functional in on their Functional skills maths skills vocational and standards: E1 – L2 functional skills programmes? Vocational programme units of work Course materials

3. How are learners’ Observations: Interviews: Reflections as everyday numeracy part of 8 classroom observations 1:1 semi structured practices used in observation across construction, hair interviews with their vocational and process and beauty and functional teachers functional skills mathematics learning? Focus groups X 1 with teachers using student

pen portraits as stimulus for discussion

3.4.1 Quality criteria and research process

There is some debate in the methodological literature (Arthur et al 2012; Bazeley 2013; Guba and Lincoln 1994; Hammersley 2007; Lincoln and Guba 2000; Mason 1996; Northcote 2012; Spencer et al 2003) on if, and how, qualitative research should be judged. The debate seems to take place on at least two levels. At a macro level researchers engage in debates about epistemological and ontological differences both in and between qualitative and quantitative research and the implications of these differences for evaluating quality. In quantitative research, the ‘holy trinity’ of objectivity, reliability and validity are important criteria in establishing and assessing the quality of research undertaken (Spencer et al 2003, p. 59). Mason (1996) argues for the assimilation of reliability, validity and generalizability into qualitative research with little change of meaning from their use in quantitative research where they were established. Hammersley (2007) argues for guidelines or explicit quality criteria, to be used by researchers when carrying out their own research, as well as evaluating wider qualitative research. Only through use of these guidelines, according to Hammersley, (2007) will a consensus emerge, across educational researchers, on judging and evaluating qualitative research and staving off the criticism that much qualitative research is of poor or questionable quality.

The issue of quality in qualitative research is therefore framed as part of a larger and contested debate about the nature of the knowledge produced by qualitative research, whether its quality can legitimately be judged, and, if so, how. Smith and Deemer (2000) for example, claimed that attempts to apply criteria to qualitative research would inevitably result in confusion and inconsistency, because criteria are incompatible with the basic philosophical assumptions of this type of enquiry. Advocates of the antirealist position (Guba and Lincoln 1994) argue that qualitative research represents a distinctive paradigm (Kuhn 1970) and as such it cannot and should not be judged by conventional measures of validity, generalisability, and reliability. At its core, this position rejects the belief that there is a single, unequivocal social reality or truth which is entirely independent of the researcher and of the research process; instead there are multiple perspectives of the world that are created and constructed in the research process. Guba and Lincoln (1994) suggest the need for alternatives to these concepts (validity, generalisability, and reliability) when establishing the quality of qualitative research. Trustworthiness (credibility; transferability; dependability and confirmability) and authenticity have been put forward by Guba and Lincoln (1994) as two ‘primary criteria’ for assessing a qualitative study. Bryman (2008) acknowledges that authenticity as a criterion has been controversial given its emphasis on the wider impact of research and has not been influential or popular in evaluating the quality of qualitative research.

Northcote (2012) acknowledges the problems experienced researchers encounter in evaluating the quality of qualitative research and suggests it is unsurprising that post- graduate students experience difficulty in deciding how to evaluate their own qualitative research. In terms of this study, then, I set out what has guided my thinking in terms of quality. I stated above that the debate about quality in qualitative research is carried out on two levels. On the one hand researchers engage in philosophical debates about the nature of knowledge and on the other hand, these ideas are translated into lists, checklists and guidelines designed to support the evaluation of quality in qualitative research. I have examined a range of approaches to ensuring quality in qualitative research (Bazeley 2013; Guba and Lincoln 1994; Hammersley 2007; Lincoln and Guba 2000; Mason 1996; Northcote 2012; Spencer et al 2003) and although these are not presented in any uniform way, there are common themes which unite and which relate to the overarching concepts of credibility, transferability, dependability and confirmability (Guba and Lincoln 1994) as the means of establishing trustworthiness in qualitative research. In the following section I set out how I attempted to ensure trustworthiness in my research. I also explain how I have approached the issue of generalisability in my research. For this purpose I draw on Williams (2000, 2002) and Yin (2009, 2014) to better understand how to realistically and cautiously address this issue. In particular, Williams’ (2000) concept of moderatum generalisation establishes an argument in support of limited generalisation from qualitative research and Yin’s (2014) generalising to theory or analytic generalisation serves as an appropriate logic for generalising for a case study. I present below an outline of how these concepts were operationalised in the research and how decisions made over the course of the study were taken to ensure the overall trustworthiness of the research and to support a view that there is potentially a case to be made for limited generalising.

3.4.2 Establishing trustworthiness and supporting transferability and generalisability

I now set out the strategies used in this research to guide and support establishing credibility and ultimately trustworthiness in the research presented. There is also a discussion on transferability and generalisability and how methodological decisions taken during the research influence each.

Credibility

Credibility establishes whether or not the research findings represent plausible information drawn from the participants’ original data and is a correct interpretation of the participants’ original views (Graneheim and Lundman, 2004; Lincoln and Guba, 1985). That is, do the research findings capture what is really happening in the research context

53 and whether the researcher learned what they intended to learn. The view put forward by the antirealists, for example Guba and Lincoln, that there are multiple accounts of social reality, places a responsibility on the qualitative researcher to ensure that a rigorous process of data generation has been adhered to. Bryman (2008) advocates not only carrying out the research in accordance with accepted research practices but specifically to submit data generated for ‘respondent validation’ (Bryman, 2008 p 377) or ‘member checks’ (Bazeley 2013; Lincoln and Guba 1985; Onwuegbuzie and Leech, 2007). Asking participants to verify their contribution to the data is according to Guba and Lincoln (1989) important to trustworthiness of qualitative research. In my research, both students and teachers were provided with transcripts of focus group activity and interviews to verify their contributions. It was important to have participants check the accuracy of the focus group activity in particular as this formed the basis for identifying their engagement with numeracy outside of college. I presented students with data from their focus group in a table and held a second focus group to examine and verify the data. This proved easy for students to read and follow. Unexpectedly, presenting students with my ‘translation’ of their vernacular numeracy practices into the language of the Adult Numeracy Core Curriculum, led to a discussion with students on their views of what is mathematics and what is valuable knowledge in relation to numeracy and mathematics. I gave teachers typed transcripts and summaries, where relevant, for verification. However, none of the research participants challenged any of the data generated.

Triangulation is another strategy suggested for ensuring credibility. Triangulation can involve the use of different methods, especially observation, focus groups and individual interviews, which form the major data collection strategies for much qualitative research (Shenton 2004). It involves testing hypotheses across the different sources of data as well as cross-checking the accuracy of data from one source with data obtained from other sources, (LeCompte and Schensul 2010). I employed focus groups, interviews and observations as data generation methods in my research. While focus groups and interviews could be seen to suffer from some similar methodological shortcomings, (Shenton 2004), Guba (1981) suggests that using these methods in concert, compensates for their individual limitations. The use of observations was a way of triangulating what teachers had said in their interviews about how they use students outside numeracy practices in college. Similarly, the use of the interviews with students was useful in exploring data generated in the focus groups on their out of college numeracy activity.

The use of well-established research methods (Yin 2009) is an important strategy in supporting credibility in research. Methods used for data generation and analysis should

54 draw on those that have been successfully utilised in previous studies. I have detailed earlier in this chapter the influence of two significant empirical studies in terms of making methodological decisions about my research. Furthermore, I developed a research methodology consistent with the nature of the topic under investigation. There is a history within qualitative research of using a case study approach and the methods to support data generation which I used, are also well established within similar studies. I tried to replicate, in part, the data generation methods used in these studies. However, I adapted some data generating methods to meet local need. This is detailed in the relevant section in this chapter.

Lincoln and Guba (1985) and Erlandson et al. (1993) are among those who recommend researcher familiarity of the culture of participating organisations and ‘prolonged engagement’ between the researcher and the participants in order both for the former to gain an adequate understanding of an organisation and to establish a relationship of trust between the parties. The danger emerges, however, that if too many demands are made on staff, gatekeepers responsible for allowing researcher access to the organisation may be deterred from cooperating. I had an existing relationship with the College, however, with the exception of the assistant principal, who was the gatekeeper, I did not have any prior knowledge of the research participants. I spent nine months in data generation activities at the College. Prior to that, I visited the College to speak to teachers and students about the research. I also read college reports to familiarise myself with more detailed college background. The College, also, is similar to other colleges I have worked in and with for many years. I am aware of the issues affecting staff and this supported developing a professional relationship with participants. However, familiarity with the culture can also be a threat to trustworthiness. I felt sympathy for the pressure teachers in the college were under. I recognised how difficult it was for vocational teachers to grapple with having to support students’ numeracy, when they themselves in some instances felt insecure in their own numeracy. Trying to maintain a professional distance while attempting to build rapport and trust so you can observe classes, requires a constant awareness of one’s position.

Dependability

Lincoln and Guba (1985) stress the close ties between credibility and dependability, arguing that, in practice, a demonstration of the former goes some distance in ensuring the latter. For example, the use of “overlapping methods”, such as using focus groups and individual interviews support dependability. However, in order to address the dependability issue directly, the processes within the study should be reported in detail, thereby, in theory, enabling a future researcher to repeat the work, if not necessarily to

55 gain the same results. This entails adopting an ‘auditing approach’ (Bryman 2008, p 378) to record keeping for each phase of the research. This would include keeping details of problem formation, selection of participants, fieldwork notes, interview transcripts and data analysis decisions. There are built in milestones or opportunities as part of the EdD programme to share and defend data gathering and analysis on the above processes. I participated in supervision sessions as well as regular peer meetings where decisions on data analysis were discussed. The process of writing analytic memos in the early stages of data generation and using these as a basis for discussion with colleagues supported data analysis decisions. The formal thesis writing process required the detailed recording of both strategic and operational data generation.

Confirmability

Shenton (2004) argues that the concept of confirmability is concerned with ensuring as far as possible that the research findings are the result of the experiences and ideas of the informants, rather than the characteristics and preferences of the researcher. Miles and Huberman (1994) consider that a key criterion for confirmability is the extent to which the researcher admits their own predispositions. To this end, beliefs underpinning decisions made and methods adopted should be acknowledged within the research report, the reasons for favouring one approach when others could have been taken explained and weaknesses in the techniques actually employed admitted. In terms of results, preliminary theories that ultimately were not borne out by the data should also be discussed. Earlier in this chapter as well as in chapter 2, I set out the ‘theoretical inclinations’ (Bryman 2008) which have influenced my research.

Transferability and Generalisability

Transferability refers to the degree to which the results of qualitative research can be transferred to other contexts with other respondents (Bitsch 2005; Bryman 2008). Lincoln and Guba and Guba and Lincoln (1985, 1994) have argued that ‘the only generalisation is: there is no generalisation’ and they embrace the alternative term of transferability for qualitative research. They further suggest that whether or not findings ‘hold in some other context, or even in the same context at some other time’ is an empirical issue rather than an epistemological one. Transferability, they suggest, is in the eyes of the reader of the research as it is they who will decide whether there are enough similarities between the reader’s own context and the research context under investigation. However, this is not to imply that methodological issues are not important in supporting transferability. Bitsch (2005) argues, that transferability is facilitated by research which adopts thick description and purposeful sampling. Thick description or

‘rich accounts of the details of a culture’ (Geertz 1973) provide what Lincoln and Guba (1985) refer to as ‘a database for making judgements about the possible transferability of findings to other milieux’. The importance of thick description of context and methodology is emphasised by Li (2004) who advocates thick description about the research process itself along with the context. This ‘rich and extensive set of details concerning methodology and context, should be included in the research report’ with a view to ‘enabling judgments about how well the research context fits other context’s (Li, 2004, p. 305). I have attempted in this research study to adopt methods which provide ‘rich accounts’ of the vernacular numeracy practices of a group of young people who have entered a further education college in the North West of England and who do not hold a level 2 qualification in mathematics. These accounts have, in part, been realised through the use of vignettes of students’ numeracy events and practices. The reasons for using these vignettes are explained in a later section of this chapter, but they are designed to provide those ‘rich accounts’.

Although Lincoln and Guba reject outright that generalization is possible, Williams (2000) argues that qualitative researchers are, in many cases, able to produce what he calls moderatum generalisations, that is, ones in which aspects of the focus of the research ‘can be seen to be instances of a broader set of recognizable features. This is the form of generalization made in interpretive research, either knowingly or unknowingly’ (Williams 2002 p 131). That qualitative researchers can make these types of generalisations is not only the case, but Williams (2000, 2002) suggests they already do, without always acknowledging this. He refers to this as the ‘do it’ yet ‘deny it’ issue. Williams (2000, 2002) makes a compelling case for the hypocrisy of qualitative researchers, justifying this view with evidence that almost all interpretative researchers make generalising statements about their research findings yet without ‘commenting upon the basis upon which such generalisations might be justified’. Moderatum generalizations are therefore more limited and tentative than those associated with statistical generalisations (Williams 2000) but they help to challenge the view that generalization is impossible beyond the immediate evidence in the case.

If Williams invites limited generalizing in qualitative research, Yin (2009, 2010) offers the possibility of analytic generalizing from case study research. This is where a previously developed theory, for example, social practice theory, conceptualised in this research as numeracy as a social practice, and operationalised into Street, Baker and Tomlin’s (2008) analytical framework, has the potential to provide useful insights into the problem being researched as well as ‘pertain to newer situations other than the case in the original case study’, (Yin 2010 p 21). Yin (2010) argues for the theory to be stated early on in the research; grounded in the literature, most importantly if similar results have

57 been found with other case studies, and the findings from the research need to show how the empirical results supported or challenged the theory. If it supported it, then, the researcher needs to show how the findings can be used to ‘pertain to newer situations’.

Crucial to ‘transferability’ and generalization, both moderatum and analytical is the issue of sampling in the research because ‘the sample is the bearer of those characteristics that it is wished to infer to a wider population’, (Williams 2000, p 216). Details of decisions taken in relation to sampling are detailed in the next section.

3.5. Sampling

Sampling, as it relates to qualitative research, refers to the selection of individuals, units, and/or settings to be studied and is a complex issue. (Fletcher and Plakoyiannaki 2010, p 837). Rapley (2014) makes the case for the importance of sampling decisions in qualitative research being made for sound analytical reasons. His reasoning is that qualitative researchers generally work with relatively small numbers of people, interactions or situations, and yet, notwithstanding this, they may wish to make claims about their research. Consequently, sampling must ‘not be ad-hoc or left to chance’, (Rapley 2014, p 49). The primary purpose, then, of sampling for a qualitative researcher, is to collect specific cases, events, or actions that can clarify or deepen the researcher’s understanding about the phenomenon under study. Sample suitability, according to Cohen, Mannion and Morrison (2007) is important to the overall quality of a piece of research. The methodological literature advocates an approach to sampling which is guided by the aims of the research and the researcher being clear in defining the population with which the research is concerned as this will have a bearing on the potential usefulness of the research findings and claims made (Cohen, Mannion and Morrison 2007; Denzin and Lincoln 1994; Silverman 2010; Yin 2009). Further, the literature also suggests that, purposive sampling, often used in qualitative research, is useful when ‘information-rich’ cases for in-depth study is the desired outcome of sampling (Patton 2002). Keeping in mind the evidence from the literature above, the decisions taken in relation to site and participant sample for my research are detailed below.

3.5.1 Research site

The process of selecting sites and/or participants is influenced by the researcher understanding and taking into consideration the unique characteristics of specific research participants and the settings in which they are located (Devers and Frankel, 2000). However, there is also a need to ensure that the choices made around sampling support any attempt to suggest that the findings from the research could be of use to 58 other researchers. The focus of this particular study is post-16 numeracy and mathematics teaching and learning for students who have left school without achieving a GCSE grade C or above, in mathematics. The majority of students with low or no qualifications in maths who progress to other learning opportunities, do so in further education colleges, (Moser 1999; Casey et al. 2006; Ofsted 2011, 2012). Therefore, the research focus and questions pointed to a research site which provided access to this cohort of students. Over the last twenty years further education colleges have been the main setting for my work. They have also been a site of significant government funding and scrutiny for literacy and numeracy. Consequently, in seeking a research site, I chose a setting which not only provided access to the relevant cohort of learners, but also one where I had an understanding of the context, issues and concerns of the organisation. Recognising the importance of pragmatism and flexibility, (Hammersley and Atkinson 2007) in choosing a research setting, I decided to approach a site with which I had an existing relationship. In my role as a literacy and numeracy advisor to colleges, I have attended regional forums (north west literacy, language and numeracy research forum; dialogue north west) with college staff where numeracy was a focus. Staff from one college, in particular, showed an interest in my research topic. The college is a large, further education college, in the North West of England. It has 3,000 full-time 16-19 year olds studying on a range of vocational programmes. The college is located within an area of high economic and social deprivation and the attainment of school leavers at GCSE is below the national average (Ofsted 2013).

I applied, in writing, to the principal to undertake my research. Applying to the head of an organisation is advised (Festinger and Katz 1966) where the organisation is hierarchical and where lower levels of staff (class teachers) are dependent on their superiors. In the application letter, I briefly outlined the aims of the research, research sample, broad methods to be used and the practical implications of these for participants and the college. Providing this general overview serves two purposes (Cohen, Mannion and Morrison 2007). It clarifies, at an early stage, the potential shape of the research for the researcher as well as the organisation. It also provides insight into potential problems which may occur. However, it is difficult at the beginning of the research to know precisely how it will affect those involved (Denscombe 2007). Consequently, negotiating access is an ongoing ‘balancing act’ (Hammersley and Atkinson 2007) between ethical and strategic considerations. Although not employed at the College, I had previously undertaken some advisory work in a different department. While I did not know any of the teachers potentially participating in the research, I had a professional relationship with the Assistant Principal and Head of Functional Skills, who were, in effect, my sponsors, in the college.

Much of the research literature highlights the effects of researcher position in terms of insider, outsider. My existing relationship with the Assistant Principal and Head of Functional Skills, was useful in gaining access but it also concerned me as I approached tutors to obtain informed consent. On the one hand, these relationships did not quite make me an insider researcher (Adler and Adler 1994) but they did give me some advantages of the insider researcher, for example, having a greater understanding of the culture being studied and a general understanding of the politics of the institution, not only the formal hierarchy but also how it “really works”, (Bonner and Tolhurst 2002). In general, insider researchers have a great deal of knowledge, which takes an outsider a long time to acquire (Smyth and Holian, 2008). This ‘semi-insider’ relationship did give rise to misunderstandings as to my purpose in undertaking the research. For example, as part of the early discussions with the assistant principal and functional skills staff, on the focus of the research, the Head of Functional Skills asked, when I carried out the classroom observations, that I report to him on whether functional skills teachers were carrying out their lessons in accordance with college guidelines and the Ofsted framework for inspection. Being able to use the University’s ethical guidelines and rules was helpful in the meetings with the Head of Functional Skills. I was able to stress that I was bound by these rules which included confidentiality and anonymity for all participants.

Threats to the trustworthiness of the research alongside ethical considerations are discussed, more fully, in the relevant sections of this chapter.

3.5.2 Participant sample

The approach to participant sampling taken for this study was one of purposive sampling (Cohen, Manion and Morrison 2007; Bryman 2008). Palys (2008) suggests that engaging in purposive sampling involves making a series of strategic decisions about ‘with whom, where and how’ (Palys 2008, p697) the research is to be carried out. In the previous section, I set out my reasoning for deciding to conduct my research in a particular further education college. Keeping in mind that participant sampling decisions need to align with ‘what researchers want to accomplish and what they want to know’, (Palys 2008, p 697) I reasoned that I needed a rationale for sampling which would help me to realise my research aims, which have as their focus an exploration of young people’s numeracy events and practices. In particular, young people without a level 2 qualification in mathematics who are entering further education. Therefore, thinking critically about the research sample (Silverman 2013) and ensuring a rationale for inclusion (Cleary, Horsfall and Hayter, 2014) are crucial in making decisions about the research sample. I devised the following inclusion criteria: two groups of students, without a level 2 qualification,

60 participating in a vocational programme, at or below level 2, with a functional skills component. For staff participation, I decided to interview two members of staff from each vocational programme as well as the functional skills teacher from each vocational programme. I decided on the two members of vocational staff from each vocational area who taught each group for the greatest number of hours each week. Two tutors was an attempt to guard against eliciting purely idiosyncratic views as well as providing opportunities to involve the tutors who potentially knew the students best. While more vocational tutors would have provided a greater breadth of interviewees, in practical terms I did not have the capacity to interview larger numbers. I was also aware, from the progression panel discussion, of the need not to collect an excessive amount of data and to keep the process manageable.

Access to staff and students was initially facilitated by a senior manager within the College, often referred to in the literature as gatekeepers (Becker 1970; Cohen, Manion and Morrison, 2007). Because the gatekeeper in this instance was head of Foundation Learning (FL) and Assistant Principal, my access to staff and students was restricted to those in this particular programme area. FL is vocational provision for students with low or no qualifications including in English and maths. While my existing relationship with the gatekeeper expedited access to a particular programme area of the College, I became aware that it could potentially compromise some teacher participation in the research.

I had two meetings with vocational and functional skills staff, and two meetings with students, where I provided both oral and written information on the proposed research. Students and tutors agreed to participate in the research, after these meetings. My research sample, therefore consisted of six teachers and fifteen students. All fifteen of the students had left compulsory schooling without any qualification in mathematics. There were four vocational teachers and two functional skills teachers. The students comprised two groups drawn from the Foundation Learning (FL) curriculum area. Group, in this instance, is defined as students on a particular programme, e.g. Level 1 Construction and Level 1 Hair and Beauty. The numbers making up the groups differed between vocational areas. However, in total there were fifteen students at the start of the research.

Introduction to Construction group: the students taking the Level 1 Introductory Diploma in Construction included: eight young men, aged 17-19, without any formal qualifications, including in numeracy or literacy. At the beginning of their course all the students had an initial assessment of their literacy and numeracy skills using an on-line

61 initial assessment tool, Basic and Key Skills builder (BKSB) Seven of the students were recorded at Entry 2 and one at Entry 3 for numeracy.

Introduction to Hair and Beauty group: the students in this group comprised seven young women and one young man aged 16-18. None had a qualification in literacy or numeracy and all had been assessed using BKSB. Five of the learners were assessed at Entry 2 with one assessed at Entry 3 and one at E1.

Introduction to the teachers: the six teachers participating in the study, three male and three female, stated they came into teaching in further education with a range of prior experience. In the case of the four vocational teachers, they worked in industry, on average for 8 years, prior to beginning part-time teaching in the College. They subsequently all worked full time. The two remaining functional skills teachers both taught other subjects, accountancy and business, prior to teaching functional skills. One was considered a maths specialist, with a degree in accounting and the other did statistics as part of a business degree.

Appendix 2, page 160 provides a table with student profiles.

3.6. Data generation methods

In order to provide a brief overview of the approach taken to the research design I present below, a summary of activity which occurred in the data generation phase of the research. Following this summary details of the data generation methods used and how each data set was analysed is then set out. Finally, how the analyses were synthesised is then explained.

1. Student focus group activity was carried out and written up as student numeracy events and practices. This data set was analysed in order to identify underpinning mathematical skills linked to level descriptors in the adult numeracy core curriculum. Numeracy events were grouped under headings: part-time work; travel; gambling; using computer games; shopping; cooking; sporting activities; managing mobile phone; money; helping at home.

2. Student interviews were undertaken and transcribed and written up and analysed to identify underpinning mathematical skills linked to level descriptors in adult numeracy core curriculum. Numeracy event headings, used in 1 above were further populated and refined.

3. Data from 1 and 2 above were examined together to begin to populate ideological model of numeracy framework. Concepts in the framework provide the lens to produce tables, showing each student’s everyday numeracy events and this meant I was able to begin to say something about numeracy practices. General headings formed. Framework provided for identification of headings and supported emerging themes in relation to student numeracy practices. (Research question 1)

4. Using data from 1 and 2 above, pen portraits were written to use with teachers in focus group to begin to identify what mathematical skills students draw on in their out of college activities as well as joint activity to identify the underpinning mathematics in vocational subjects.

5. Student data linking numeracy events to adult numeracy core curriculum shared with students. Student reflections on the data were recorded and analysed using themes identified from 3 above.

6. Teacher interviews transcribed and summarised. Data then presented question by question in table. Similar answers were grouped together. Data from interview questions analysed in terms of research questions.

7. Interview transcripts and interview summaries shared with teachers to check for accuracy.

8. Classroom observation pro-formas and researcher notes analysed using ideological model of numeracy conceptual framework to identify emerging themes. Research questions 2 and 3.

9. Using data from 8 above, table produced comparing student out of college numeracy events and practices with their in-college numeracy events and practices.

10. There were two phases to data generation. Each group of students participated in two focus groups; one semi-structured, 1:1 interview and were observed in their vocational and functional skills classes. In phase 1 (October 2013 – January 2014) I carried out two focus groups with students and one with teachers. I undertook interviews with students and teachers and carried out eight classroom observations. In phase 2 (February – May 2014), I completed the remaining student and teacher interviews and carried out the second round of focus groups with students. Throughout both phases I kept a researcher diary for on-going reflections on the research activity process as well as recording particular incidents. I also started

data analysis from early on in phase 1 and continued throughout both phases, with more intensive data analysis taking place in phase 2. Details of activities in each of these phases are provided below.

3.6.1 Data generation with students

Focus groups

I carried out five focus groups in total with both students and tutors. The methodological literature reports on overlaps between focus groups and group interviews (Bryman 2008; Silverman 2010). Bryman (2008) defines the focus groups method as, ‘a form of the group interview’ and ‘an interview with several people on a specific topic or issue’, (Bryman 2008, p 473). This differs from a group interview, according to Bryman (2008) in that the focus group typically emphasises a specific topic or theme while a group interview may ‘span more widely’. Bryman (2008) further states that group interviews are sometimes used as a means of avoiding individual interviews and saving time and money, whereas this does not apply to focus groups. Gibbs (1997) suggests a focus on ‘features’ may provide a more useful sense of differentiating between the two. She suggests that focus groups are organised, involve collective activity, and promote discussion and interaction, while group interviews give the interviewer or mediator a more prominent or central role in proceedings, asking participants specific questions. One key characteristic which both Bryman (2008) and Gibbs (1997) highlight, is that in the focus group, the participants are believed to have particular experience of an issue which the focus group will draw on. This was important in my decision to use focus groups as a method.

I recognise that my focus group activity could fit into a definition of either group interview or focus group, depending on the literature used. In the case of my focus groups, perhaps what was more important was the purpose and process rather than the name. Alongside these (purpose and process), being aware of the strengths and limitations of the activity as a data generation method was important in deciding on the method. I used a group activity as my initial engagement with students because this part of the data generation process was early on in the research and I was concerned that students may feel vulnerable and intimidated if I started with 1:1 interviews (Kvale and Brinkman 2009).

In trying to answer my first research question, ‘What numeracy events and practices do students engage in in their everyday lives’, I carried out a focus group with each group of construction and hair and beauty students. Each focus group lasted one hour, and all students attended. The purpose of the focus group was to find out what activities the 64 participants engaged in, in their free time, in order to identify their numeracy events and practices. In order to stimulate data generation (Cohen et al 2007) in the focus groups, I used the ‘clock-face activity’ based on Ivanic et al’s (2009) research. This activity had been used successfully in previous research (Ivanic et al 2009) to gather information on learners’ literacy events and practices. Students in that study were specifically asked to record all activities in a day which involved literacy. I modified this approach with my participants. Rather than ask them to record activities in a day which involved numeracy, I asked them to capture all the activities they engaged in, over a weekend. I did this because it has been noted in the literature that students, when thinking about numeracy, unlike literacy, often do not ‘see through to the numeracy’ (Coben 2000) in their activities. I further modified the activity by providing picture prompts, representations of everyday activities from young people’s lives. I did this because, in discussions with teachers prior to the focus groups, concerns were raised about students’ writing, in particular that this would get in the way of the data generation. Students were invited to use these pictures to populate the clock face or to generate a collage of activities which they felt represented their daily activities. They also had the option of writing a list of activities or using any combination of the three. Two scribes were also available to support students in writing, although these were not used. I provided students with large sheets of paper, pens and pencils, as well as the prompt pictures, and students produced data in a variety of ways, including, writing on or around a clock face, and using the prompts to show experiences meaningful to themselves. Figures 3.3 and 3.4 are two examples of students’ representations of their out of college activities. Further examples of what students produced are in Appendices 3-7, pages 161-165.

An important strength in using the focus group method is that it can generate a substantial amount of data within a relatively short space of time (Wellington 2000). I found this to be the case with both the student focus groups. Students initially struggled to represent their activities. They did not see the importance or worthiness of recording their out of college activities and some students made fun of the activity. However, I provided further explanations, and all the students, either individually or in pairs, completed some form of representation of their out of college activities. The main strengths of the activity was that it allowed students to take ownership of their contribution to the research, firstly by representing their activities in their own way and secondly by choosing what to include. It also led to some recognition by the students of the amount and range of activities they engaged in. By reflecting on their out of college activities, students were stimulated to discuss their numeracy events during the follow up interviews.

This initial focus group was successful in supporting all students to participate and to produce a representation of their activities. There was some interaction between participants in the groups. They informally, looked at each other’s collages and commented on the activities. I noted down these, ‘unsolicited oral accounts’ (Hammersley and Atkinson 2007, p112) at the time of the focus groups, though I was not sure how I would use them. Overall, what was extremely useful, was the focus group provided an opportunity to begin to develop a professional relationship with the students as well as identify issues for consideration in the 1:1 interviews which followed.

Second focus group with students

The purpose of the second focus group was to share with students a summary of the data which had been generated in the initial focus groups and the subsequent interviews. Each student’s record of activities from focus group 1 was transcribed into a table. See Appendices 9,10 and 12 pages 167, 168 and 170 for a sample of these. Each transcribed record was brought to the 1:1 interviews and formed the basis for a discussion. Students identified the most important activities on their list and these became the focus for further discussion in the interview, linked as they were to the concepts of the ideological model of numeracy. Furthermore, there was a need to check for accuracy in transcribing and also this provided an opportunity to discuss with students the links between their out of college numeracy events and the abstracted mathematics in their formal learning in college. I knew from the literature review and previous experience that many students dismissed their use of maths outside the classroom as common sense. It was becoming clear from my time spent in college that there was little recognition of the value of students’ vernacular numeracy practices. Creating opportunities to discuss with students their use of potentially embedded mathematics outside of the college began to take on a greater significance, and consequently I used all the opportunities that arose to have these discussions with students.

Each student who attended the second focus group had a transcribed handout showing their out of college numeracy events and expressing the content of the events in terms of the language of the adult numeracy core curriculum and the functional skills mathematics standards. This translating (Evans 2000) of their numeracy events into more formal statements provided opportunities for the students to reflect on and discuss their everyday numeracy events and how they could be interpreted in the language of college mathematics. See Appendices 9-12, pages 167-170.

Both focus groups also provided opportunities for the students to discuss their views of their own numeracy skills and their experiences of learning maths at school. These

66 emerged in focus group one, were included in the 1:1 interviews and presented in focus group 2.

Relevant here to the discussion on strengths and weaknesses of the data generation method, is that no matter which method (focus group or group interview) was used, ultimately the data generated in the focus groups was self- reporting, rather than directly observed. Silverman (2010) has suggested that self-reporting data has potential validity issues but acknowledges, along with Bryman (2008) and Hammersley and Atkinson (2007) that the reality is more complicated. They suggest that while some qualitative research asks research participants to reconstruct events by thinking back over time, there is no other way to get at this information. However, a significant advantage of the methods used to investigate students’ ‘vernacular numeracy practices’ is that it examines them through the lens of what is possible from the reflections of the students in the College setting. This is important in that the students can be brought to recognise their own vernacular numeracy practices.

I was aware that my age, gender, nationality, social class and ethnicity positioned me as an outsider in terms of the research participants’ world, (Kellett 2013), and following students around their out of college environments was not practicable or possible. Consequently, I adopted a view put forward by Hammersley and Atkinson (2007) which views the stories people recall and tell ‘as relevant to the phenomena to which they refer as well as indicative of the perspective they imply’, (Hammersley and Atkinson 2007 p. 97). As Hammersley and Atkinson (2007) point out, there is no reason to disbelieve or affirm the validity of individual accounts on the grounds that they are subjective. However, in trying to maximize trustworthiness of the data, I compared the activities generated by the students in the focus group work with other studies with participants of similar age and circumstance, (Fowler et al 2002; Ivanic et al 2009) and there were similarities across the activities.

Figure 3.1. Lily’s clock-face

Figure 3.2. Paul’s collage

Interviews with students

I carried out 1: 1, semi-structured interviews with 10 learners in total across both vocational areas. I intended to interview all the students, but some of the students were unavailable for interview on the days when I attended the College. Six construction and four hair and beauty students were interviewed. These are the students with an asterix (*) beside their name in Appendix 2, page 160. The purpose of interviewing for educational research has been stated variously as (i) to find a way in to understand the lived experience of other people and the meaning they make of those experiences (Kvale and Brinkman 2009); (ii) to provide rich and detailed answers from the interviewees’ perspective (Bryman 2008) (iii) to enable interviewees to discuss their interpretations of the world in which they live (Cohen, Manion and Morrison 2007). What these views share is the ‘importance of interaction for knowledge construction’, (Kvale and Brinkman 2009). My view of interviews supports the stance that interviews are useful for gathering information while at the same time they are practical sites of interaction in which knowledge and meaning is co-constructed between interviewer and interviewee. Blommaert (2006) argues that asking is often the worst way of trying to find out something, as people are not actively aware of what they are doing or what they believe about a situation. However, I think there is a place for interviews, as regardless of whether one bears in mind that asking may not be the best way of finding out the answer, some sort of answer will always be forthcoming. The purpose of the interviews, then, was to explore more deeply, learners’ numeracy events and practices, using data generated from the focus group. Using the headings of Street, Baker and Tomlin’s (2008) analytic framework, I devised prompts which would allow the student to discuss their numeracy events in terms of content, context, values and beliefs, social and institutional relations. See Appendix 18, page 176 for interview prompts used with students. The process of interviewing was not as clear cut as the prompts might suggest. The areas of the framework inevitably overlap and we moved between events, in most cases, rather than pursue a linear approach. Each interview lasted for approximately thirty minutes.

In interviewing the students, I was aware that, with the exception of interviewing a group of young offenders, these students were younger than any previous interviewees. I prepared for this by reviewing the purpose of the interviews to confirm the appropriateness of this method of data generation. I was also aware that observing students in their natural environment and recording numeracy events as they happen (LeCompte and Schensul 2010) would have been preferable to accessing these events second hand through interviews, especially as a significant amount of time was spent in the student interviews asking students to explain their numeracy events. However, carrying out the interviews as well as the focus group activity provided an opportunity to

69 dig deeper into the participants’ experiences. I viewed these young people as agents in their own right with valuable knowledge to share. The recognition of young people’s knowledge can go some way to equalising power relations in the interviewing process (Devine 2000, Kellett 2011).

At the beginning of each interview I acknowledged the importance of the interviewee’s contribution to the research and asked each participant if they were willing to proceed with the interview, in terms of location and whether they were comfortable being in a 1:1 situation and the interview being recorded. Six of the ten interviews were recorded as the other four did not want to be recorded. In one instance, one of the students was nervous at being interviewed alone and he asked another student to accompany him. I tried to provide opportunities for the students to ask questions and make comments during the interview. Giving students time to think and answer questions and engaging in active listening helped to generate an environment of respect for participants. I made notes during each unrecorded interview and at the end of each interview session I read through each set of notes and rewrote any illegible excerpts while the content was fresh in my mind. I included in these annotations, any observations during the interview. For example, Graham started out his interview by giving one word or very short answers, but after a short period visibly relaxed and removed the hood from his jacket and sat forward in his chair. This coincided with greater fluency in his answers.

I listened to each recorded interview several times and read and re-read accounts of those that were not recorded as soon as possible after carrying out the interviews. This process provides a sense of the whole and a context for specific meanings which emerge later (Hycner 1985). I decided to fully transcribe four of the interviews and selectively transcribe the remaining two. The fully transcribed interviews were used in writing up the vignettes and were also used for the pen portraits which were shared with teachers in a separate focus group, which is explained in the next section. I used the process of selective transcription (Bryman 2008), for example, in Graham’s interview, as the initial part of the interview, the one word answers part, did not yield anything new to the data already transcribed.

The interviews with the students were an important part of the overall research. They generated data which was crucial in terms of applying the analytical framework and ultimately, addressing the research questions.

Classroom observations

The purpose of the classroom observations was to observe each student in their vocational and functional skills class in order to (i) observe what types of numeracy/mathematical events and practices students engaged in on their learning programmes and (ii) to observe whether and how vocational and functional skills teachers drew on their learners’ vernacular numeracy practices in the classroom. The observations were also important to begin to identify the in-college numeracy practices learners engaged in and how these related to students’ vernacular numeracy practices. Observational techniques are used extensively to acquire data in naturalistic settings (LeCompte and Preissle 1993, p 3.) In order to ensure that all the students were observed in both a practical and theory class as well as a minimum of two functional skills classes, I carried out eight classroom observations, each lasting approximately 45 minutes to one hour. At the beginning of the study, teachers understood that participation in the research required participating in classroom observations. However, given that further education has been subjected to a ‘continuous stream of policies and initiatives aimed at improving the quality of teaching and learning and raising standards’, (O’Leary 2014) classroom observation has led to ‘a typology of resistance’ (Cockburn 2005, p 376) on the part of some teachers in further education. Furthermore, with Ofsted, proclaiming that a satisfactory grade in classroom observation, is now unsatisfactory, the ‘stakes have been raised higher still’ (O’Leary 2014, p 25). So, while classroom observation has become a ‘prominent feature of the professional life of teachers’ (O’Leary, 2014.) there is evidence of an ‘artificiality’ (Wragg et al 1996; Cockburn 2005) in observed lessons, with some teachers ‘teaching to a formula’ and ‘playing the observation game’ (O’Leary 2011) to ensure an acceptable grade. In order to play down the role of observation in the research, I tried not to use the term observation in conversations with class teachers. Instead, I referred to classroom observations as my ‘just sitting in on classes to see what numeracy is involved’. In attempting to ‘equalise the unequal power relationship’ (Cockburn 2005, p 384) between the observed and observer, I developed and shared with teachers an observation pro-forma to record classroom activity. There is a copy of this in Appendix 19, page 177. I explained before each observation what I was looking for: (i) identify and record any numeracy activity which students engaged in whether obvious or embedded, and (ii) note any reference made by the teacher or student to out of college numeracy. The eight observations allowed all the students in the research to be observed in their vocational and functional skills classes, along with the teachers taking part in the research. In their vocational classes, students were observed participating in both a practical setting as well as in a theory class, which takes place outside of the practical workshop or training salon. Three observations were carried out in hair and beauty; one in the training salon and two

71 in theory sessions. Two observations were carried out in construction; both were in practical sessions. Three observations were carried out in functional skills classes; two in functional skills mathematics classes, taught by a specialist maths teacher and one observation in a combined English and maths functional skills class.

The observation pro-forma while useful, did not always allow for capturing other useful data during the observation. Therefore, alongside the pro-forma, I wrote detailed notes using a set of prompts. See Appendix 20, p178. Data from both of these was then used to support the presentation of data for analysis of classroom activity using Street, Baker and Tomlin’s (2008) organising framework for an ideological model of numeracy. See Appendices 13-15 pages 171-173. I also noted those instances which did not always fit into the pro-forma. For example, when Kurt decided not to participate in the class activity and reconfigured it according to his own desire not to do subtraction. Having a framework, from the outset, did help in managing the data generated from the observations. Over the period of carrying out the observations, teachers became accustomed to my presence in the college as these were taking place alongside student interviews, focus group work and informal discussions.

3.6.2 Data generation with teachers

Interviews with teachers

Understanding teachers’ beliefs about numeracy and mathematics is important to this research in that there is evidence that teachers’ views about the nature of mathematical knowledge is connected to how they teach it. Teachers who tend towards a belief in the autonomous model (Street, Baker and Tomlin 2008) or the absolutist view of mathematical knowledge are more likely to teach in a ‘transmission’ style (Lerman 1998) where mathematical information is presented to students as fact. Conversely where teachers felt that mathematics was a more socially constructed process, they tended to use a more learner-centred, interactive approach in their teaching. The ‘affective’ component of numeracy includes the beliefs, attitudes and emotions that contribute to an individual’s ability and willingness to engage, use and persevere in mathematical thinking and learning or in activities involving numeracy (Ginsburg et al 2006). I interviewed each teacher once, for approximately 45 minutes. These semi-structured interviews were designed to find out teacher views on: the importance of numeracy in their vocational subject; their views on the numeracy skills and knowledge their learners bring with them to college; how they find out about their learners’ numeracy skills and knowledge; their perception of the numeracy practices their learners engage in outside of formal learning; how they support and motivate learners to engage with numeracy and their own

72 experiences of numeracy. The use of interviews was an appropriate method in engaging teachers with the topic because it gave access to their own stories as well as their insights, experiences and maths biographies. Teachers were generous with their time and displayed no sense of unease in the 1:1 interviews. By the time the interviews were carried out with the teachers, I had been in their classrooms and had had both formal and informal discussions with them. The interviews were carried out over a three week period. Each interview lasted for approximately 45 minutes and I provided each teacher with the broad headings, prior to the interview. The interviews were recorded using a digital recorder. I listened to the interviews a number of times to get an overall sense of the content. I fully transcribed three of the interviews, one from each vocational teacher and one from the functional skills maths teacher. I selectively transcribed (Bryman 2008) two, choosing those parts which added something new to the data. One interview was partially lost due to a fault with the digital recorder. Please see Appendix 22 page 180, for interview questions for teachers.

Focus group with teachers

I undertook a focus group with teachers which lasted 45 minutes and all six teachers attended. Preliminary data from the classroom observations showed that there were few connections made between students’ out of college activities and in college learning, in either the vocational or functional skills classrooms. The purpose of the focus group was therefore: (i) to present to the teachers an account of aspects of students’ out of college lives; (ii) to find out from the teachers how relevant they believed their students’ vernacular numeracy practices were to in-college mathematical practices in functional skills and vocational classes; (iii) to explore one approach to identifying the mathematical skills students used outside college as they went about their lives.

The stimulus material for the teacher focus group was three student pen portraits which I had compiled using data from the student ‘clock face activities’ and student interviews. Student pseudonyms were used. Although there is a tradition in traditional ethnography of using life histories or narratives to understand the experiences of individuals (LeCompte and Schensul 2010), I decided against using a ‘composite picture’ (LeCompte and Schensul 2010) of the group. I wanted to give a broad sense of each student’s numeracy events outside college rather than focus on one particular issue or problem. There was also a sense of wanting to do justice to the students who had engaged with the data generation process. The timing of the first focus group with teachers was important as it had to come after students had taken part in both of their focus groups and some student interviews had taken place. See Appendix 21, page 179 for an example of one pen portrait used in the focus group.

Teachers spent 10-15 minutes reading the pen portraits. They identified one activity from a pen portrait and discussed which mathematical skills were needed by students to carry out the activity. They estimated the level of these mathematical skills, using the adult numeracy core curriculum. At the end of the first focus group, teachers were using their ‘maths eyes’ (Maguire 2011) and had begun to identify the underpinning maths skills in aspects of their vocational work.

3.7. Data analysis

The following data were analysed for this study:

1. Fifteen individual student collages/clock-face sheets/lists, transcribed into list form from focus group activity - 148 activities listed

2. 60 numeracy events

3. 18 numeracy events analysed and translated into language of the Adult Numeracy Core Curriculum.

4. Transcripts and written up notes from semi structured student and tutor interviews

5. Field notes

6. Research diary

An account of how each data set was analysed has been presented earlier in this chapter. This section attempts to show how the various strands of data have been brought together and synthesised in order to provide answers to the research questions. An important stage in data analysis is data reduction.

One of the challenges of qualitative research which uses a range of data generation methods to explore practices, is how to manage the large amounts of data in preparation for analysis. Using a combination of the research questions and data generation methods helped to ensure a coherence in the data organisation (Cohen, Mannion and Morrison, 2007, p 468). Linking data generation to research questions in the early stages of the research was helpful when it came to data analysis. Focusing on each research question in turn, pointed to the data generation instrument used. Data from each instrument was analysed separately, then brought together for closer analysis. For example, research question 1 required the focus group data and student interview data. By organising and analysing the data from each instrument separately then combining the analyses, all the relevant data was collated. All the data from the student focus groups was analysed together and tabulated together. Similarly, so too was data from student interviews, teacher interviews and classroom observations. This organisation and analysis is supported through the use of the conceptual framework, in this case, the ideological model of numeracy. Using the conceptual framework from the beginning provided a lens for seeing through the data to begin to identify answers to the research questions.

The analytical framework was crucial to data analysis in the study. This led to the development of a series of supporting frameworks for use across all data generating activities. For example, the use of the framework below helps the reduction of the data from teacher interviews.

Table 3.3. Coding and classifying to support data analysis

Teacher Interview Response Category question/topic

Teacher understanding of learners’ numeracy/ attitude to numeracy

How have you identified There isn’t that much Numeracy equated with the underpinning involved, I don’t think on the school mathematics. numeracy skills and level 1 course. I’m not Invisible unless it’s knowledge your students really sure because I now identified as related to need to succeed on their think there might be maths school mathematics. course? (vocational in areas I haven’t thought Hidden numeracy not teacher) of. recognised.

How would you describe Most of them have very low Deficit positioning of your learners’ numeracy skills. They’re in FL and students. Lack of skills? (all teachers) have lots of issues. numeracy linked with other ‘problems’.

How are your learners’ All students take BKSB Narrow notion of numeracy skills online assessment. We get numeracy assessed at entry to a handout of their level, but college? (all teachers) not much more than that.

Would you have a sense No – not at all. We have a Deficit view of students of how much numeracy hard enough time trying to Students positioned as activity your learners keep them at college and lacking; vulnerable and engage in outside get them through what they needing support. college? (all teachers) need to do for their course. Judgements about literacy I can’t imagine they do very and numeracy ability in much because they’re at a students used to make low level. broader judgements about students

How confident do you I don’t feel I have the skills. Lack of confidence in own feel in developing your I can do the maths I need maths skills. learners’ numeracy for my own subject but I skills? (vocational tutors don’t know how to teach and embedded maths. functional skills teacher) I don’t really see it as my job to teach maths. I do know though that the kids would probably enjoy maths more in their vocational area rather than doing in on its own. But, you know, one of them, the other day said that she would like to get a GCSE in maths and English.

Teacher Interview Response Category question/topic

How do you engage and We tend to concentrate on Numeracy only available motivate your learners in the students doing the level to students on higher level learning numeracy as 2 programmes. At level 2 courses. part of their students need to know vocational/functional about proportion and skills curriculum? fractions. We also do a (vocational and fs tutors) module on setting up your own business and there’s

some maths in that.

How did you get on with I wasn’t in the bottom set, Negative experience of maths at school? (all but I wasn’t in the top set maths at school. Effect on teachers) either. own role? I didn’t like it at all. I felt I never wanted to do any maths again once I left school. Once I left school and started work I could see that maths was used in various areas of everyday life that wasn’t like it was at school.

3.8. Ethical considerations

In this section I attempt to draw together issues relating to ethical considerations and researcher positioning and how these affected and were dealt with in this study.

Throughout the research process there is the potential for the researcher to indulge in what Bourdieu called, ‘a blinkered self- interest’ (Bourdieu 1990, p 28) that is, to fail to recognise the existence of the ‘perspectiveless view’ and the ‘partisan partiality of individual viewpoints’, (Bourdieu 1990, p 29). In order to avoid these ‘errors’, Bourdieu called for researchers to remain reflexive in terms of their habitus (gender, social class) as well as their position in the field which they are researching (Bourdieu and Wacquant 1992). I have interpreted this reflexive approach, in this study, as having a critical awareness of how my position, not only as a researcher, but how my identity more widely, influences the research at each stage of the process. This includes not only the choice of subject, research design and methods, but also my position within the field I am researching, the research setting, and my relationships with research participants.

As a student of the School of Education at the University of Manchester I am bound by the ethical rules of the University. The School has published an ethical practice policy

77 and guidance for research (School of Education, 2011). The focus of the guidance is on individual rights and fundamental freedoms. There are two main implications of this focus, firstly, any study undertaken should be designed in a way that takes account of the rights of participants and secondly as a researcher, I have a duty to meet both these objectives (p 2). The ethical practice policy and guidance also outlines an ethical protocol which identifies principles of ethical practice for research in the School of Education.

This research was designed to minimize and anticipate any risk to subjects, make use of voluntary participation, and assure the anonymity of participants and the confidentiality of the information they provided. In demonstrating research integrity, the ethical practice policy and guidance for research outlines the need to establish the level of risk associated with the research. The level of risk associated with this research was established in consultation with my supervisor and was considered to be of low risk. This was recorded in the Research Risk and Ethics Assessment form.

Hammersley and Atkinson (2007) point to five aspects of ethics as they apply to social research generally and ethnographic type research in particular. These are, informed consent; privacy; harm; exploitation and consequences for future research. Informed consent, privacy and harm are the three aspects which most directly affected my research and I deal with these now.

Informed consent: I followed a process to gain informed consent from all the research participants. I provided oral and written information to all the participants and held two meetings with staff and students to ensure that potential participants had as much information as was possible about their involvement in the research project. However, as Hammersley and Atkinson (2007) note, ‘it is not always possible to tell all of the people being studied, everything about the research’, (Hammersley and Atkinson 2007, p210). I was not aware during the early stages of the research what exactly all the consequences of involvement would be for either the students or the teachers, in the research. While I did not deliberately withhold information from the participants, it was not always possible to keep updating participants. An example of this would have been my research questions. These evolved over time and the initial questions which I gave to the teachers appear somewhat benign compared to the final set. I was particularly concerned that the teachers would interpret, negatively, research questions 2 and 3 which could be seen as leading to negatively judging them and this may have had an impact on their behaviour especially during the classroom observations. I did share the observation pro-form with the teachers, but I did not share my field notes where I made

78 notes or research diary which recorded ‘instances’ which occurred outside those being observed according to the observation pro-forma.

Privacy: in the context of this study I will discuss confidentiality and anonymity which I see as aspects of privacy. I promised all the participants confidentiality and anonymity at all times. While the functional skills manager, in the early stages of the research, wanted me to report back to him what was happening during the observations, it was relatively easy to deny this request as it was such a violation of the confidentiality and anonymity commitment. However, there were other occasions when because of the small numbers of participants, teachers could potentially ‘guess’ who various participants were in a focus group. When presenting the pen portraits to teachers, their initial action was to guess who the students were. I had to state that students were entitled to confidentiality and would not disclose the real names of the students in the pen portraits.

Harm: Hammersley and Atkinson (2007) note the potential, unintentional harm to research participants. As stated earlier in this chapter, I built up a good working relationship with the teachers and felt some empathy for them in their roles. I was conscious that I was adding to their stress levels by carrying out observations in their classes. No matter how much I tried to reassure them I was not there to judge them, but observe particular activities as they occurred, I believe that for some of the teachers, observations were stressful. Two of the teachers in particular tried to prepare what they thought I wanted to see. However, discussing this after the observation, and continuing to stress the purpose of the observations seemed to have an effect.

I realised that there was also the potential for harm in relation to the teachers who knew I had a close working relationship with the assistant principal and head of department. I decided to deal with this head on and in the early meetings with staff I affirmed that although I had worked with departmental colleagues previously, I was bound by a code of ethics and would not disclose any information.

In trying to challenge notions of what it means to be numerate, I’m aware that the voice of those being researched needs to be heard, because I don’t know personally what it means to be considered innumerate. There is a responsibility then not to perpetuate the positioning of the research participants, both students and teachers, as other, and to try to ensure that the focus is not on deficit but provides opportunities for participants to show what they do with numeracy and so help to transcend objectifying representations of the participants. Viewed from this perspective, researcher positioning is closely linked to the ethical nature of the research process (Curtis, Murphy and Shields, 2014). As stated above, this research was designed to minimize and anticipate any risk to subjects,

79 make use of voluntary participation, and assure the anonymity of participants and the confidentiality of the information they provided.

3.9. Conclusion

This chapter outlined the methodological traditions which I have drawn on for this study and locates it within a qualitative research paradigm. It sets out the data generation methods used and attempts to reconcile these with that tradition as well as the research aims. I aimed to use a ‘broadly ethnographic’ approach to guide data generation and this reflected the importance of capturing and representing the numeracy events and practices of the students in the study. While students’ numeracy events and practices which took place in college were observable in the classroom, a challenge for the research has been accessing students’ out of college numeracy events and practices. There was no direct observation of students participating in their everyday activities. However, the range of everyday activities which students recorded, aligned closely with the everyday activities of participants in the pilot research study and are consistent with those reported in other research from similar geographic, context and age-similar studies. (Fowler et al 2002, Ivanic et al 2009). This chapter and the preceding chapter on the literature form the foundations of the research and inform and guide the analysis and presentation of findings which are considered next in Chapters 4 and 5 and subsequently the discussion and conclusions in Chapter 6.

Chapter 4. Findings (1)

4.1. Introduction

The previous chapter set out the methodology, research design and data analysis methods used in this research. This chapter, and the one that follows, set out the research findings. Firstly, the research purpose and questions are restated and there is a brief discussion about the gap in the literature which this research seeks to fill. Following on from this there is an account of the research participants, including three vignettes which aim to bring to life students’ vernacular numeracy practices which include not only the mathematics content of those practices, but also the context, their values and beliefs and social and institutional relations. In wanting to bring students’ vernacular numeracy practices to life, and provide a ‘rich account’, they are presented in a broad sense and the details of these vernacular numeracy practices are presented in chapter 5.

There is a short recap on the two important concepts of numeracy events and numeracy practices and how these are operationalised in the research. Next, findings relating to students’ numeracy events outside college are summarised. Data is then presented on the numeracy events students reported participating in outside college. Following on from this, three vignettes are included and a selection of student numeracy events are translated into the language of the Adult Numeracy Core Curriculum (ANCC). Findings from students’ numeracy events inside college are summarised. This data was generated in the observed functional skills mathematics classes and vocational classes. Three classroom observations are presented which give a sense of the numeracy events students participated in inside college. This completes chapter 4 and provides the focus for the analysis and discussion, in chapter 5, which moves the research on from describing students’ numeracy events to analysing students’ numeracy practices both inside and outside college.

The purpose of this research then was to identify and examine the vernacular numeracy practices of a group of further education students with a view to challenging, at a conceptual level, what it means to be numerate. In order to realise the aims of the research the following research questions were formulated:

1. What numeracy events and practices do learners engage in, in their everyday lives?

2. What numeracy events and practices do learners engage in on their vocational and functional skills programmes?

3. How are learners’ everyday numeracy practices used in their vocational and functional mathematics learning?

A substantial cohort of young people leave compulsory schooling and enter further education labelled innumerate. This research seeks to present an alternative view which challenges this position through a focus on young people’s social uses of numeracy, what I have called their vernacular numeracy practices. This research also seeks to explore aspects, other than ‘skills’, which may influence students’ ability to engage in learning mathematics inside college. These other aspects include an examination of not only the content of numeracy but also the context, values and beliefs and social and institutional relations.

4.2. The students in the study

All the students in the study live within a five-mile radius of the college which serves an area ranked the sixth most deprived in England (Dept. of Communities and Local Government 2015). On average, 48% of pupils, in the area, achieve 5+ GCSEs at grades A*-C (incl. Maths & English) compared to 59% nationally (DfE 2013). Nine of the fifteen students in the group completed their statutory schooling. All the research participants are referred to by their pseudonyms. Table 4.1 below profiles the students in terms of their course, gender, age and results of initial assessment in numeracy at entry to college. The results of their initial assessment in numeracy places them all at entry level (E) on the national qualifications framework (NQF). The NQF sets out the level at which all qualifications in England are recognised. Levels range from entry level to level 8. Within the NQF, the further education sector uses a five-level system to categorise skill levels in relation to literacy and numeracy. Starting at the lowest levels, these are: Entry Level 1 (E1), Entry Level 2 (E2), Entry Level 3 (E3), Level 1 (L1) and Level 2 (L2). This positioning of the students in relation to level on the NQF is important for their learning journeys in college. For post 16 students entering further education, achieving an entry level numeracy assessment result positions them as having ‘severe numeracy difficulties’ (Cara and De Coulon 2008) or ‘poor numeracy’ (Carpentieri et al 2009) and this has consequences for students accessing learning programmes in college. In government publications (ANCC 2002; ALCC 2002; DfES 2001; DIUS 2007) the levels on the NQF are interpreted in terms of levels of attainment for children in primary and secondary school. (See the ANCC 2002, p 4). I note this practice here as evidence of the deficit positioning of further education students within policy and practice documents. The legacy of this ‘insulting’ (Swain 2006) practice is still current as the interviews with vocational tutors will show in the next chapter. Students entering further education

82 without a Level 2 qualification in English and maths, (broadly equivalent to GCSE grades C and above) are required to continue to study both subjects until this is achieved.

Table 4.1. Student Profiles

Male/ Numeracy initial Student Age Vocational programme female assessment result

Graham M 17 Introduction to Construction Skills E2

Paul M 17 Introduction to Construction Skills E2

Kurt M 18 Introduction to Construction Skills E2

Dom M 17 Introduction to Construction Skills E2

Daniel M 17 Introduction to Construction Skills E2

Jack M 18 Introduction to Construction Skills E3

Liam M 17 Introduction to Construction Skills E2

Mike M 17 Introduction to Construction Skills E2

Andy M 17 Introduction to Hair and Beauty E2

Angela F 17 Introduction to Hair and Beauty E2

Emma F 17 Introduction to Hair and Beauty E2

Lily F 17 Introduction to Hair and Beauty E2

Petra F 18 Introduction to Hair and Beauty E2

Sophie F 17 Introduction to Hair and Beauty E3

Tasha F 17 Introduction to Hair and Beauty E1

4.3. Students’ numeracy events

In Street, Baker and Tomlin’s (2008) framework for analysing home and school numeracy practices, numeracy events and numeracy practices are identified as two key concepts. They are also important for this research. The focus initially in this chapter is on students’ numeracy events. Numeracy events in this research are, ‘those occasions in which a numeracy activity is integral to the nature of the participants’ interactions and their interpretive processes, (Baker 1998; Street, Baker and Tomlin 2008). Numeracy events are ‘observable episodes’ (Ivanic et al 2009) always situated in students’ lives, highlighting the importance of context in relation to the value and meaning of that event. Understanding how students use mathematics, through their own accounts of their 83 numeracy events, is a different focus to measuring their numeracy skills at entry to college. Finding out about the numeracy events in students’ lives, outside of formal learning, is the first step in understanding how students use mathematics and in making the move from describing numeracy events to analysing numeracy practices. Numeracy practices, at their most simple are, ‘what people regularly or habitually do with numeracy, whereas a numeracy event is just one instantiation of a practice’, (Ivanic et al 2009 p 48). That analytical move from numeracy events to numeracy practices is the subject of chapter 5.

4.3.1 Students’ numeracy events outside college

This section presents the data which was generated with students in order to identify the numeracy events they engaged in outside college. The data generation methods used for this purpose are set out, in detail, in Chapter 3. However, in summary, the data presented in this section draws on methods outlined in table 3.2, page 51, which include focus groups and one to one interviews with students. Both of these were important in the data generation process which contributed to students being able to identify their social uses of numeracy and ultimately contributed to an analysis of their vernacular numeracy practices. Details of students’ identification of their social uses of numeracy are set out on pages 65 – 70 and Appendices 3-12, pages 160-169. This data was then examined and coded using the lens of the Adult Numeracy Core Curriculum, see Tables 4.4, 4.3 and 4.4, pages 98-100.

Figures 4.1 – 4.7 represent students’ out-of-college numeracy events.

Figure 4.1 provides graphical representation of the categories and numbers of students in the group (15) who reported they engaged in each category of event. For example, all fifteen of the students use technology every day; engage in leisure activities; manage their money; undertake household responsibilities; socialise with friends; organise their lives (self- management). Twelve of the fifteen students undertake some form of part- time work.

Figure 4.1. Categories of students’ numeracy events

Figures 4.2 - 4.7 take each of the categories and show the number of students who reported engaging in numeracy events which contribute to the category. For example, under the heading of using technology, all fifteen students manage their mobile phones and use an ATM. Thirteen of the students who use social media use facebook with fewer, five, using Pinterest. Twelve of the fifteen students use technology to shop. This inlcudes browsing and searching for products online although not always purchasing them online. Nine of the fifteen students play computer games and using a computer at work was reported by five students.

Figure 4.2. Student’ numeracy events embedded in technology

Figure 4.3 shows the range of part-time work in which students are employed. The majority of students (12) had some form of part-time work. Only three students did not work outside college. Students did bar and café work as well as working in a salon. Working hours ranged from four hours per week, for dog walking, through to twelve hours per week for bar work and those students working in a salon on Saturdays, averaging eight hours. Work, in particular, generated many mathematically mediated opportunities for students including using technology.

Figure 4.3. Students’ part-time work

Figure 4.4 below shows the events students reported on around managing their money. Students reported that while they did not have to budget for paying rent or utility bills, all of them shopped for clothes and food and almost all paid for their own cosmetics and mobile phones, mostly, but not always, on a pay as you go basis. Three students reported getting into difficulties with paying for their phones on a contract basis and switched to using pay as you go. This made them feel more in control. They were all able to report on the ‘phone deal’ they had which inlcuded, numbers of free texts, free phone calls and the amount of data they could use. Accessing the internet on their phones was very important to them as the majority of students did not have access to a

86 computer at home. Internet use at home was mainly for satellite television or for some students, to play on their Xbox or Xbox Live. Almost all of the students in the construction group engaged in gambling of some kind either placing bets or buying scratch cards. Students explained how to calculate the odds when placing a bet, although as Paul explained, he relied on a friend of his father’s to provide racing tips which almost always won. Jack no longer used his Xbox and sold it at a Cash Generator shop to buy new trainers.

Figure 4.4. Students numeracy events embedded in managing money

Playing computer games

Figure 4.5. Students numeracy events embedded in sport and leisure activities

Although students were quick to deny doing any numeracy or maths outside of college, it became clear that their sport and leisure activities drew on a range of numeracy skills. Whether it was paying busfares or reading tram times from electronic displays, through to keeping score in their sporting activities, all the students participated in some form of sport and leisure. Some of them moved between imperial and metric measures in the gym for example, where weights were in kilos and yet they stated they used miles on the running machines and expressed their weight in stones and pounds.

Being part of a family and the responsibilities this entailed was reported as being important for all the students. It also provided opportunities for students to use their numeracy skills along with taking responsibility for self management. Students reported cooking for the family in order to eat healthily, being part of a slimming club and checking labels on food as well as weighing themselves. Some students looked after siblings and regularly took children in the family on days out.

Figure 4.6. Students’ numeracy events embedded in household responsibilities

Figure 4.7. Students’ numeracy events embedded in self-management

4.3.2 Vignettes

I now present, through the use of vignettes, examples of students’ experiences of engaging with numeracy events and practices. The examples drawn on here came from three students, two from hair and beauty and one from construction. They were chosen because they were illustrative of all 15 students. The vignettes draw on college, at-home and in-work events and practices. They are significant in supporting the contribution to knowledge within the thesis and were chosen because they were valuable in explaining the contribution to knowledge. I could have done this for all 15 of the students, but space (number of words) was limited and it would also have been repetitive. The students were all assessed at Entry level 2 at the beginning of the college term, yet they seemed to engage in mathematics practices outside college, at a higher level than their in-college initial assessment suggests. The vignettes are presented at the beginning of the chapter firstly, to provide a sense of who the students are, in part, through their own words and to avoid the impression of research participants who are ‘bleached of biography and background’ (Grenfell et al 2012). Secondly, I wanted to foreground their lived experiences, their small stories, (Bamberg and Georgakopoulou 2008) including those experiences of engaging with mathematics and numeracy inside and outside college as well as their accounts of learning mathematics during their compulsory schooling. In constructing the vignettes, I used data generated in the focus groups, interviews with students and classroom observations, using a common framework. They are constructed to provide a sense of: (i) the events which students identified as important to them outside college, many of which, but not all, involved numeracy; (ii) their beliefs

89 about numeracy and mathematics and (iii) their reported experiences of learning mathematics in school.

Vignette 1: surfing and shopping event and practices

In the hair and beauty focus group Emma completed a list (Appendix 17 page 175) which showed she engaged in a wide range of activities outside college. Emma, subsequently, in interview, identified surfing the internet, shopping and her part-time job as an important part of her life.

“It’s great having the internet on my phone. I use it all the time. I’ll get a text from a friend about something that’s been in a magazine, you know, and I can check it on Top Shop’s website or whatever. I have my own private boards (on Pinterest) but I share some with friends and we follow each other. I mostly pin clothes, make up and sometimes music too and food. I pin things in the morning and during the day and then before I go to sleep I look at them. It’s relaxing reading my boards and it gives me ideas for styling. I want to do well in hair and move on to do the higher courses and then further on I want to do styling.”

Emma also bought make up on line, but only after she tried it out in a shop first.

“It’s way cheaper. We’d never just buy make up on line. We’d go to Boots in town and have a look and try them out. I’ve been caught out before with that on line and it turned out it wasn’t what I thought it was. So, we’ve learned now what to do.”

She had favourite sites she checked daily and then compared products and prices across sites to see where the best value was. Sometimes she just bought an item of clothing, like a pair of leggings, from a site, but at other times she bought clothes and some household items through an on-line catalogue.

“There aren’t many clothes shops around here and there’s no big stores. And because I work at weekends it’s not easy getting to (nearest city). But buying online is great. And I can buy what I need and then pay it off each week. You need to watch your sizes when you buy from different sites. And so a size 12 from Top Shop might be smaller than a size 12 from Next. We love it when we buy from some of the discount sites that have things in American sizes. My mam says it’s the only time she gets to be a size 12.”

Emma explained,

“An American size 12 is really a size 14-16 in our sizes depending, so when I buy something from eBay that’s coming from America, I can order a size 8.”

But mostly Emma orders from UK sites as the postage and packing is cheaper and sometimes free and there are times when payments can be spread over 3 equal payments or over a number of weeks.

Emma and her mother share an account as Emma cannot get credit.

“I shop with (friend) quite a lot and also my mam. It’s just like going out shopping together, but we can stay at home. Sometimes, if it’s for a special occasion, and it’s the weekend, we might have a cocktail. We did that when we bought our outfits for the Christmas party.”

In this instance, the purpose of shopping this way was to pool their order because there was an offer of no postage and 5% discount on orders over £50.

“The new Summer tops and things are out already now in the catalogues and we like to get sorted early. But there can be sales of stuff from last year too and we always try and get a discount. Otherwise there’s no point. We’re going with (friend’s) family to Spain and we want to make sure we get what we need.”

They chose to buy from a catalogue because they could pay weekly and choose the number of weeks for payment. Because Emma does not have access to credit, she used her mother’s account and Emma paid her back each month. Emma’s friend, in turn paid her.

“There’s this catalogue on line at Littlewoods and it’s really good. It’s mam’s account but we always share. She knows I’m good at paying her each month. I never miss because she would just stop. (Friend) pays me and she always does as well. She’s the only one I do it with. I always check first with mam, that it’s ok.”

Emma explains how she approaches the task and does her calculations for shopping.

“I knew that I wanted to buy a bikini, two tops – one with long sleeves and one sleeveless. Also, I wanted a pair of shorts and a sun dress. I reckoned, from looking at the sites, I look at them all the time to check prices, that I could get all these for under £40 and if it was any better than that then I might get a pair of beach sandals. If they cost more then it’s just not worth it. I could get them from Top Shop and just buy them one at a time.”

Emma went on to explain,

“But with (friend) wanting to buy as well, we could get them all now and save a bit too.”

“You have to search for what you want. We always start with the sale stuff. If I have time I search one time and put them in a basket and then come back when you’re ready to buy. It can take time because I’m on my phone and the screen is small. If I keep things in the basket then the price is there and I can keep track. I try to add up as I go along, not exactly, but the rough price. You know, like £6.99, then I add £7, or even £5.50 I always add £6, like that. I always cost things more, so I know I can really afford them and I won’t be disappointed then. I do check the prices and work out how much it will really cost.”

Emma and her friend used tables of data to work out sizes, colours, catalogue number and price. Once they knew roughly what they want to buy, they worked out how to pay by instalments. Emma explained:

“You need to choose the size and colour of what you want from a table that you click on. I know that a size 12 from ‘Very’ is too big for me, so I always get a 10 from there. Sizes are not the same everywhere and this catalogue has more sites together. I write down what each thing costs and add it all up on the calculator, before I put them into my basket. I do it twice to make sure I get the same answer. For the holiday things, our cost altogether came to £89. Mine was £49 and (friend’s) £40. We got free delivery and we spread the cost over 20 weeks. There’s no interest for 20 weeks. They take the money out on the same date every month.”

When asked how she calculated how much to pay her mother back each week, Emma explained,

“We work out how much I have to pay mam.”

Emma keyed in £89 into her mobile phone calculator and divided it by 20. She showed me £4.45. When I ask her if she pays her mother weekly, she says,

“No. I pay her every month because the catalogue will take it out from her bank every month. It all has to be paid off in 20 weeks or else you get charged interest. You can pay it off earlier through your account online. We always pay it on time.”

Emma multiplies the £4.45 by 4 and gets £17.80.

“I pay mam £20 every month. That’s easier to keep track of. (Friend) will pay me £10 a month for four months. That’s straight. So I just add another £10 for the four months and then I’ll pay mam the £9, so it’s all clear in 5 months.

Mam knows that I will pay her back from my own money from my job at the salon. It works out ok for us. She lets me pay her each week or month instead of giving it in one amount. I know I’m lucky but I don’t like shopping online with mam as she takes too long to look at sites and I like to move between a few sites at the same time.

I can work out the cost of an order as I’m going along, because it’s easy to get carried away and before you know, there’s too many things in your basket, and then you have to add postage and packaging too.

Table 4.2 below summarises Emma’s online shopping event and maps the steps in the process to the ANCC descriptors.”

When stimulated to explain how she works out her online shopping costs and payments Emma can do so with ease and fluency. Her account, explanation and demonstration of shopping calculations suggests there is a mis-match between her assessed numeracy skills at entry to College and those she reported using outside. The level descriptors from the ANCC show that Emma, in a single numeracy event, is using mathematics not just at Entry 2, but also at Entry 3, Level 1 and in one instance Level 2. Please see Appendix 10 for Emma’s numeracy events.

Vignette 2: working in a salon and timetabling appointments

Lily’s ‘clock-face’ (Appendix 3, p 161) showed that she enjoyed going out with friends; looked after her nieces; took turns to cook at home; had a part time job in a local hair salon; liked shopping and was a member of a slimming club. Like all the students in the study Lily spent a lot of time on her phone each day. She estimated she spent around two hours a day, when in college, using her phone to access the internet and longer at the weekends. She used her phone to access Facebook and had favourite sites that she used for looking at clothes and hair and beauty products. While she did not shop online she used the internet for window shopping, to see what products were available as well as looking at the latest trends in hair styles and hair products. She communicated with friends mostly by texting and messaging via Facebook. She has two young nieces whom she spent time with, usually at the weekend after work. At least once a month Lily took her nieces out to give her sister a break from childcare. Lily identified her part-time job in a salon as well as taking her nieces out as the activities that were most important to her.

If we look at content in relation to Lily working part-time in a salon, she has a range of duties in the course of the working day. The duties Lily reported on are listed in Table 4.3 below.

Lily, like Emma, was assessed at Entry 2 for her numeracy and yet she too appeared to use maths skills at a higher level in her part-time job.

In her 1:1 interview Lily said she left school with a D in GCSE Art. It was the only GCSE she turned up for. When asked about her experiences of school and maths in particular, she said:

L: “I never got on with school, but I liked art. I didn’t feel stupid in it. In maths I was in the bottom set and I couldn’t keep up with anything the teacher said. I started to feel, you know, like I couldn’t breathe when I went in. I tried to keep up, but it was awful. I was sick. It was maths made me feel sick. The school kept ringing our mam and saying I wasn’t there, but I just couldn’t go back.”

In particular, Lily said she has a real block with telling the time. This was the third occasion Lily highlighted time as problematic.

L: I can’t tell the time like that (Lily pointed to the analogue clock on the wall). I can a bit, but I use my mobile phone to say what time it is and I can read the time on my phone, but don’t ask me to add minutes to the time on the clock like that one. I just remember sitting and staring at pages and pages of clock faces with numbers and hands pointing…. and you’re told to add maybe 28 minutes to the time that was written on the clock. I felt so stupid that I couldn’t do something so easy. My niece who’s seven can tell the time. It’s one of the things that everybody just knows, isn’t it? I just never got on with it.

R: How do you get to college and work on time, and meet up with your friends?

L: If I’m out late on a Friday, I set the alarm on my phone to get up (for work). I don’t like being late for work. I know my bus for college comes at 8.20 and it takes about 20 minutes and I’m always on time. I have time to get a drink in the cafeteria and be in class before it starts. When I was at school mam would get me up but she works shifts now and I can get myself up and out now.

It’s no problem meeting up with friends. We all meet at someone’s house and get ready together, so we’re never going to be late going out because we’re all together.

Vignette 3: working in a garage

Paul enrolled on the Introduction to Construction Skills in September, at the beginning of the college year, but later transferred to Motor Vehicle. He has a part-time job in a local garage where he works from 9.00 until 3.30 each Saturday. The garage is owned by a friend of Paul’s dad who ‘gave him a chance’ because he knew Paul and didn’t mind that he wasn’t ‘good’ at English and maths. Paul started out at the garage helping on

94 reception but didn’t like it when it was busy and having to talk to customers and answer the phone at the same time.

My dad’s friend took me on. He knew I had some problems with English and maths from school. They put me on reception, I think because I like to look smart, but I got very flustered when I had to answer the phone and deal with somebody in front of me, together. The woman that worked reception didn’t come in on a ~Saturday. It just didn’t work out.

He moved to working in the ‘back’ and helps with checking the stock as well as valeting the cars after they’ve been serviced.

I like it when I’m asked to phone a company about ordering parts. I use Auto Data Direct to search for information on the cars and parts. It’s an online data base for car repair companies. Sometimes I look up parts on the computer to see how much they cost. I’ve also compared the price of parts, online, for a motorbike I’m fixing at home with my dad. I asked the boss first and I can get the prices at trade (price).

Paul also identified cutting the grass for his neighbour who cannot manage to do this anymore as important for him. The neighbour paid Paul for doing this. Paul thought of asking other people in the local area if they needed their grass cutting too.

I’m not sure what to charge. My neighbour gives me £5, but his garden is not very big. I thinks it usually takes about 30 – 40 minutes to cut his (neighbour’s) lawn and tidy things up, but I’m not sure. I could time it, so I can work out how much to charge. I knows Graham (another student) did that last year, and I might ask him how much to charge.

Paul also identified his mobile phone was very important to him.

I have a Blackberry but I want an iPhone. I’m on a contract and my phone is in my dad’s name. I pay £16 a month for my plan. I know I have 2000 free minutes every month as well as 300 blackberry messages and 3000 free texts. I don’t trust the company. You need to really check your bill because the phone company makes mistakes on the charges. I always check my bill, because I never go over my limits. I check how many messages and minutes I’ve used on my phone. It’s all on the website on my Blackberry. I know I just need to give my dad £16 in a month and I budget for that each week. I just know that’s £4 a week and I give it to him then.

When I asked Paul if he had a bank account, he explained,

Yeah, I do, but I don’t take any more than I need in a week out unless I tell my dad. I take £20 a week out but not all together. I do £10 on a Friday for the weekend and my gran gives me a few quid every week too when she comes. I give dad the phone money on Friday. Me and Dan and sometimes Graham or Jack buy some pints but we’re watching how much we drink. Too much is not

good, is it? We did about that in college. A lot of people around here drink too much.

Paul had chosen betting on horses and football as something he enjoyed. I asked him how he decided which horses and teams to bet on and where he placed his bets.

We go to the local betting shop and place our bets there, me and Dan do. For the horses, my dad’s friend at the Garage (where Paul works) sometimes gives me tips on the horses. If he gives me a tip I try to use it ‘cos he just seems to get it right ‘cos they win. I’ve won £20 and £15. I never put more than £3 on a horse or the football. If United are on a good run, which they’re not now, I might put £2 on Rooney to score first in the game, if we’re playing at home or against one of the lower sides that are a bit crap.

Paul hated school and left when he was about fourteen. His parents tried to get him to go back but he would not and for a while he stayed away from home and slept at his friend’s house. Paul says he couldn’t get on with school and didn’t like the teachers.

I were a naughty boy at school. One time, one of the teachers told me I was dyslexic but I didn’t know what that meant.

Paul started to go to a gym. Because he’s small in build he said he wanted to work out. He liked it that some of the other boys in his class also went to the gym. They had competitions to see who could do the most reps and lift the heaviest weights. Daniel always beat him. Paul wanted to take up boxing because he thought this would be good to build up his muscles.

Paul said that coming to college had helped him. He made friends with other lads in the class, even though they fought a bit. When I asked what was different about learning maths at college compared to school, he stated,

We don’t do any maths in motor vehicle, but if we asked the teacher he would definitely teach them. He’s dead cool. I know there is maths you need to know for working in a garage and doing the tyre pressures an’ all that. You need them (maths) to get a job in a garage.

Paul explained his plans for maths in college,

I know I’m not one of the kids who’s good at it (maths) and English. But I want to get my Level 1 exam. I’m going to try in January, Annie (the FS maths teacher) says. I want my Level 2 qualification in motor vehicle because I won’t get a proper job in a garage without it. What I really want is to go on the motor vehicle apprenticeship here, but I need to pass my maths and English at Level 1 to get onto it and then you have to pass Level 2 to get your apprenticeship.

4.3.3 Students’ numeracy practices evidenced in events and translated into the language of the ANCC

Street, Baker and Tomlin’s (2008) framework for analysing numeracy events and practices, comprises four components. These are: content, context, values and beliefs and social and institutional relations. These framework dimensions are not seen as distinct and separate from each other but always at interplay together and permeated by power (Street, Baker and Tomlin 2008, p 21). Tables 4.2 – 4.4 set out the content aspect of a numeracy event. The content is the abstracted mathematics that students used when carrying out their ‘everyday’ events. Street, Baker and Tomlin (2008) place less emphasis on the content aspect of their model when analysing primary school children’s numeracy events and practices. For this study, however, the mathematical content of the numeracy event is important. It is important because it shows the extent to which students engage in numerate behaviour outside of college, across a range of levels, higher at times, than the level students were assessed at in college at the beginning of their course. Consequently, it supported the central aim of the research which was to challenge the labelling of the students as innumerate. This step in the process of examining students’ social uses of numeracy was important in stimulating students to discuss their views about what is mathematics. I suggest that it is through the ‘translation’ (Evans 2000, p101) of the content of students’ out of college numeracy events, first into the generalised numeracy practices they evidence, and then into the abstracted, decontextualised mathematics content of these that belong to the ANCC, that both students and tutors were challenged to review their fixed views of what constitutes mathematics and, in the case of tutors, what it means to be numerate.

It is possible to compare, in terms of content, the numeracy events inside and outside college and which in chapter 5, is useful in illustrating the discontinuity between in- college and out of college practices.

My reasons for using the ANCC are twofold. The lesson observations showed that the ANCC was used in functional skills and vocational classes. Both functional skills and vocational teachers were required in the College to ‘map’ their lessons, where appropriate, to both the adult literacy and numeracy core curricula. Therefore, I reasoned, the discourse of the ANCC was one that was familiar to tutors in the study.

Table 4.2. Emma’s on-line shopping

Numeracy event Summary of student Translated into decontextualised Identified from explanation of what the language of adult numeracy core focus group event involves curriculum and interview

Shop on-line for Use smartphone to access MSS1/E3 Add and subtract sums of clothes and make Pinterest and Facebook to money using decimal notation. Know how up get information about fashion to enter sums of money in a calculator. sites and fashion ideas. Round sums of money to nearest £ and

Check sites daily and ‘pin’ 10p and make approximate calculations. ideas.

When shopping online – use MSS1/L1 Add, subtract, multiply and favourite sites with on-line divide sums of money and record. catalogues. Understand place value of whole numbers and decimals. Know that, for column Get permission from mother addition, decimals should be aligned by to use her account. the decimal point. Sometimes shop with friend and buy clothes together to N2/L1 recognise equivalences between share cost of postage. common fractions, percentages and decimals, for example, 50% = ½.

Read tables of data to check HD1/E3 Extract information from lists, sizes and styles are tables and simple diagrams. Understand available. that tables are arranged in rows and columns. This is the same skill descriptor for both levels but E3 context will involve more complicated data.

Use calculator to work out Rounds up or down to estimate rough cost of a list of items MSS1/L1 Add, subtract, multiply and before completing on-line divide sums of money and record. purchase.

Complete online payment MSS1/L2 Understand dates and times with mother’s debit card. written in different formats

Work-out schedule of N2/E3 Read, write and understand repayments with friend to decimals up to two decimal places in mother. practical contexts. N2/E3 Use a calculator to calculate using whole numbers and decimals to solve problems in context, and to check calculations.

Table 4.3. Lily’s working part-time in a hair salon

Numeracy event - Summary of student Translating underpinning numeracy identified from explanation of what into language of adult numeracy core focus group and the event involves curriculum interview

Working part-time Take bookings for MSS1/E2 Read and understand time in hair salon appointments. displayed on analogue and 12 hour digital clocks in hours, half hours and

quarter hours. MSS1/L1 Understand and use common date formats

Set timing on MSS1/E3 Read, measure and record equipment. time. Understand and use a.m. and p.m. Read analogue and 12 hour digital clock to the nearest five minutes

Take payments in MSS1/E3 Add and subtract sums of cash from clients. money

Shop for lunch for staff MSS1/E3 Add and subtract sums of in salon money

Ensure containers are MSS1/E3 Read, estimate, measure and kept filled with compare capacity using standard units shampoo and conditioner.

Check stock at end of N1/E3 count, read, write order and the day and record. compare numbers up to 1000

Table 4.4. Paul’s buying parts for his own motorbike and checking cost of parts in work

Numeracy Summary of event student Translated into decontextualised Identified from explanation of language of adult numeracy core focus group what the event curriculum and interview involves

Checking cost Read tables of data HD1/E3 Extract information from lists, tables of motorbike to locate specific and simple diagrams. Understand that parts and parts; check tables are arranged in rows and columns. buying on-line catalogue numbers; This is the same skill descriptor for both costs and levels (E2 and E3) but E3 context will availability. involve more complicated data.

Use calculator to MSS1/E3.2 Rounds up or down to estimate work out rough cost MSS1/L1 Add, subtract, multiply and divide of items before sums of money and record. completing on-line purchase for parts both in work and for own motorbike parts.

Complete online MSS1/L2 Understand dates and times payment for spare written in different formats parts with dad’s N2/E3 Read, write and understand decimals debit card. up to two decimal places in practical contexts.

Work-out schedule N2/E3 Use a calculator to calculate using of repayments whole numbers and decimals to solve problems in context, and to check calculations.

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4.3.4 Students’ numeracy events inside college

I undertook eight classroom observations across students’ functional skills mathematics and vocational programmes. All of the students participated in some form of functional skills mathematics classes alongside their vocational programme and all the vocational classes observed included some mathematics. While none of the vocational tutors were observed teaching mathematics to their students, they were observed using mathematical terms and demonstrating their application to the vocational subject. For example, in a hair and beauty session on styling techniques, tutors demonstrated the importance of holding brushes at 45, 90 and 180 degree angles from the model’s head, in order to achieve a particular aesthetic look when styling hair. They encouraged students to check the symmetry of a finished haircut and compared the characteristics and function of styling brushes based on their diameter measured in centimetres. They also explained quantities of solutions used in sterilising equipment. In order to provide a sense of the numeracy events students participated in, within college, three different numeracy events, which were part of the classroom observations, are set out below.

Classroom observation in hair and beauty – timetabling appointments numeracy event.

The classroom session which I observed Lily in was a one hour, hairdressing theory class. Previous theory sessions I observed included students learning about hair shafts; managing equipment in the salon; compiling colour boards and working with head models. This observed session was about booking appointments in a salon as part of reception duty. The tutor (G) had explained previously that he was ‘acting up’ as course tutor for ‘the Level 1s’ while a colleague was off ill. His previous role was as an assessor in the Hair and Beauty department. He asked me, as I approached the classroom, to help Lily during the class. ‘I know she won’t cope with what we’re going to do, it’s about time’. I had explained previously to G that I would not take part in the lesson as I was observing. However, I felt unable to say no to him as I knew from our 1:1 interview that G had concerns about his own ability in mathematics. G, as personal tutor to the Hair and Beauty students, was present at one of the information sharing meetings, when Lucy told the group she had difficulty telling the time using an analogue clock.

It took five minutes or so for all the students to arrive. There were seven students in the observed session, including Lily. The tutor, G. handed out a pro-forma Appendix 16, page 174, which represented a daily appointment sheet in a salon. G explained the task to the students.

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G: “OK, guys. We’re looking at timetabling in the salon today and I want you to work out the timings for the list of appointments, yeah, and put them in the appointments sheet. You’ll need to look at who wants what in the appointments list and what time they can come in at, and calculate what time to book each one in. You need to work out if the stylist has enough time for all these people. Has everybody got the sheets? OK. I need you to read the list of appointments that’s written out on the sheet and I want you to decide what time you can slot them in on the appointment one. Make sure you don’t forget to put in your 45 minute lunch and the stock taking time. You’ve got 20 minutes to complete and then we’ll come back together.”

In the class Lily sat slightly apart from the group, looking quite tense. I sat near her and after approximately five minutes where she just sat and looked at the sheet, I asked her how things were going. I knew from the focus group that Lily worked part-time in a salon.

R: How are you getting on with the task, Lily?

L: I really hate this stuff.

R: Which bit is…..

Lily: I just hate the exercises on time. I can’t do it.

R: OK. But how do you sort out your appointments in your salon, the bookings?

L: It’s just not like this. At ours, a lot of the appointments work in half hour slots, but on the appointment book at reception, the slots are in 15 minutes. Four of the slots make up an hour. If a client is in for a wash and blow dry, it’s nearly always half an hour, I know that’s two slots. If it’s a cut and blow dry for short hair it’s about an hour, so that’s four of the slots. Then we have regulars and we get to know how long their hair takes. Some are forty five minutes, so I ‘cross-out’ three slots. Then for long hair, about your length (pointing to my hair) it’s about an hour for a wash and blow dry, isn’t it? Also, I can ask the stylist if I’m not sure how long an appointment will take.

R: Is this sheet like the pages in your appointment book?

L: It’s in 15 minute slots, isn’t it? I didn’t see that.

R: Do you have any thoughts on how you might decide where to book them in?

Sophie, another student, made a comment to the students sitting near her, which Lily heard.

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S: This would never happen in real life. You wouldn’t get all these appointments at once and you need to make them overlap. None of our stylists ever take 45 minutes for lunch on a busy day. It’s just not right.

Lily looks at the task sheet and appointment sheet again and says to me:

L: There aren’t enough separate slots, are there? You just overlap the appointments. Why didn’t he say we could do that from the beginning? It didn’t say that on the sheet. You can get the cut and colour started, then while she’s waiting for her colour to take, the stylist just starts on someone else. That way everyone gets sorted. It’s what happens in the salon all the time. But I’d just ask the stylist when the customer rings up or is in the salon, if she can fit in the client and we do. If it’s a regular, the stylist always tries to fit them in.

4.3.5 Classroom observation for hair and beauty – renting a flat

I observed the following two budgeting sessions as part of functional skills for Hair and Beauty students and construction students. The first session outlined below was in a FS English session where the teacher aimed to embed numeracy in the lesson. The task set for the students was on budgeting to rent a flat. Students were asked to imagine they were employed on the minimum wage and to work out how many hours per week they would need to work to be able to afford to rent a flat in the local area.

Students worked in pairs, and used computers to search for flats, discussing with each other which area of the town they wanted to live in; what type of flat they wanted; whether they would wish to share with a friend. The six learners in the class struggled to complete the activity. They struggled to understand the calculations for working out an imagined wage, based on the minimum wage, times the number of hours worked in a week, times the number of weeks in a year.

The teacher wrote out the algorithm for calculating their potential annual earnings, on the board (£4.98x40x 52 =£10,358.40) because the students said they could not work it out. Many of the students were more focused on the reason for and value of, doing the task. They were trying to make sense of it. One student, Angela commented to the group,

It just doesn’t make sense to do it that way. Why don’t we search for a job we think we will get or will be qualified to do and see how much it pays and then we can find a flat we can afford and work out the budget?

Another student, Petra, said,

I’m not doing this. My dad says he’ll buy me a flat. He’s a builder.

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Petra and another student, Sophie, began talking about what they were going to do after college and were disengaged from the task.

I noted in my observation notes that I felt an air of dissatisfaction and frustration in the class that day. I thought the class seemed to feel they had little control over the activity set for them. The functional skills tutor, sat apart from the group doing some work on a computer. About ten minutes from the end of the session the tutor explained to the group that the real purpose of the activity was (not to develop any meaningful numeracy skills) to provide a source of information for the learners to subsequently use to take part in a group discussion to fulfil the assessment needs for FS English speaking and listening.

I asked the tutor after the class, why he had chosen this activity, he stated,

Well, what I really wanted them to know was that they, especially, need to work hard and pass their qualifications otherwise they’re going to end up unemployed or in dead end jobs on minimum wage that they just can’t live on. I wanted them to know that college is important and they can’t mess around like they did at school and leave here with nothing. (My italics)

R. So, it was to motivate them to engage with……

T. Yeah, that’s right.

R. Do you think they understood what you were trying to do in the class?

T. Probably not. They’re not very interested in functional skills. It’s just the way… They like their practical sessions.

R. Might it have helped with their interest if Angela’s suggestion had been considered?

T. Eh, when was that? What?

R. When Angela said about finding a job first…..

T. Oh, I didn’t catch that.

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4.3.6 Classroom observation for construction students - budgeting

The second budgeting session I observed was with the construction students in a functional skills maths class. Students were asked to give examples of things that needed to be budgeted for each month. They suggested items such as rent, electricity, gas, loans, food, mobile phone bills and transport, which the tutor recorded on the whiteboard. Students were asked to approximate how much they might spend on each item. Students then totalled up the imagined amounts and checked their answers on the calculator. They then subtracted this total from an imagined monthly salary total. Students participated enthusiastically in this activity, shouting out contributions to the list of expenditure.

On the surface this activity appeared as an authentic activity, attempting to relate real life expenditure, an everyday practice, to the functional skills requirements for students to be able to add, subtract amounts of money in familiar contexts. This is what Boaler (1993) calls ‘inauthentic, authentic activities’. None of the students in the group worked out a monthly budget in real life. Neither did any of them have responsibility for paying rent or many of the other items in the classroom budget. Yet, they responded enthusiastically to the task and worked together to complete the calculations. Students do not always expect their functional skills maths classes to draw on their real life experiences, unlike their vocational classes, where they seemed to expect it to reflect what happens in work. This was the case for Lily and Sophie in hairdressing.

In their one- to- one interviews, some of the students spoke about the reality of budgeting on a daily basis, with their money coming from either benefits paid through college bursaries, minimum wage or casual part-time work or money borrowed from friends and family. One student sold his old Xbox console at a Cash Generator store to pay for new trainers.

I wanted a new pair of trainers, but I didn’t want to ask my step-father to buy them and my mam, I know, can’t afford the ones I want. She has my step-brother and sister to buy for too. It was Dan that told me I could get some money for my Xbox even though it’s a couple of years old. He said the ‘Cash Generator’ shop in (town) will give a fair price. I didn’t want to go on my own with it so I asked my mam to come with me. I was a bit…, not sure what to do.

Another student sold cigarettes to classmates at 50p each to make money in college.

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4.4. Conclusion

The numeracy events presented in this chapter are illustrative of those which students engaged in both inside and outside college. They are however, of limited value unless they are examined to generate an understanding of students’ vernacular numeracy practices. The first part of this analysis involved translating a selection of students’ out of college numeracy events into the language of the ANCC. This focus on the content aspect of a numeracy event, demonstrates how the numeracy event can generate a numeracy practice(s) through an understanding of the particular event and the generalised practice involved. For example, when Emma shops on-line, (the event) she rounds up, never down, and approximates the cost. She uses her calculator to both estimate and accurately calculate the cost of her items of clothing, prior to purchasing them. The maths content requires one to understand both the particular event and the generalised practice involved. Further analysis of students’ numeracy events and practices is the focus of discussion for the next chapter.

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Chapter 5. Findings (2)

5.1. Introduction

The previous chapter set out the numeracy events students engaged in, inside and outside college. This chapter continues to present findings from the data and has as its particular focus the move from reporting students’ numeracy events to presenting an analysis of students’ numeracy practices both inside and outside college. The analysis is carried out using the lens of Street, Baker and Tomlin’s (2008) social practice analytical framework.

5.2. From numeracy events to numeracy practices

Numeracy events and numeracy practices are key methodological and conceptual tools in the New Literacy Studies (Street 2000) as well as for this research. Consequently, the relationship between them is also important for my analysis and findings. Baker (1996) and Street, Baker and Tomlin (2008) define numeracy events as ‘occasions in which a numeracy activity is integral to the nature of the participants’ interactions and their interpretative processes’, (Baker 1996 p 3; Street, Baker and Tomlin 2008, p 20.) Another way of thinking of a numeracy event, according to Papen (2005) is ‘that which people do with mathematics or numeracy: numeracy events are the uses of numeracy and mathematics which can be observed and described’ (Papen 2005 p 31). Examples from the previous chapter of out- of- college numeracy events included, shopping on-line; paying for items in a shop; weighing ingredients in a recipe; calculating discounts for an amusement park using on-line vouchers. In college numeracy events included, learning to hold different sized hair brushes at 45, 90 and 180-degree angles from the head, when styling; arranging tools in size order when clearing up in the workshop or learning how to calculate the mean and range of a series of numbers in a FS maths class in preparation for an exam. ‘Numeracy practices’ however, is the more over-arching term, which encompasses both the uses and the meanings of numeracy in a particular numeracy event. These uses and meanings have to do with learned ways of using mathematics with social norms, with values, attitudes and feelings, as well as with social and institutional relationships. While ‘numeracy event’ is primarily a descriptive concept, the term ‘numeracy practices’ moves into the realm of analysis, trying to understand the meanings of events observed, looking for patterns across events, similarities and differences between them and trying to understand their relationship with other elements of the world. One of the tasks in analysing the data generated from students’ reporting of their many numeracy events, is to see those regularities and patterns in that data and this is how the move takes place from describing numeracy events to analysing and

107 understanding students’ numeracy practices. The concept of numeracy practices therefore, operates at a greater degree of generality and abstraction than the concept of numeracy events. At its most simple, numeracy practices are ‘what people regularly or habitually do with numeracy whereas, a numeracy event is just one instantiation of a practice’, (Ivanic et al 2009 p 48). While there is a value in examining students’ numeracy events, there is a danger in ‘simply piling up more descriptions of local (numeracy) events without addressing the underlying question of practices’, (Street 2012, p 43). So, paying attention to students’ numeracy practices requires the researcher to take account of students’ awareness, framings and discourses of numeracy, demonstrated in how they talk about and make sense of it. Alongside this, numeracy practices also represent ‘recurring patterns in the values, attitudes, feelings and social relationships which are associated with students’ uses of numeracy in a particular context’, (Street Baker and Tomlin 2008, p 20). In Street’s words, ‘participants not only ‘do’ mathematics, they also have ideas about what they are doing’ (Street 2012 p 37). It is in this analysis that it is possible to make the analytical move from studying numeracy events to identifying students’ vernacular numeracy practices, that is, their numeracy practices as they go about their everyday lives – contrasted with formal schooling. Using students’ numeracy events as ‘units of enquiry’ (Street, Baker and Tomlin 2008, p 18), they are examined using Street, Baker and Tomlin’s (2008) social practice analytical framework to move the analysis forward from numeracy events to numeracy practices. It is through this analysis of students’ numeracy practices, in and out of college, that the characteristics of each, their similarities and differences, can be examined. This analysis is the focus for the rest of this chapter.

5.3. Exploring students’ numeracy practices

Keeping in mind the explanation above of the relationship between numeracy events and numeracy practices, this section, draws on the generated data relating to students’ numeracy events, to examine their numeracy practices in and out of college, to see whether and how an understanding of these practices can provide some useful insights for tutors and students to further develop successful learning of mathematics in college. The concept of numeracy practices, can be operationalised for analysis into the following four components: content, context, values and beliefs and social and institutional relations, (Street, Baker and Tomlin, 2008). I intend to focus the analysis, mainly, on one of Lily’s in-college numeracy events which she participated in as part of a hairdressing theory class and Emma’s at home, on-line numeracy shopping event. These were referred to in Chapter 4 and are set out in Appendix 23, page 181 and Appendix 24, page 183 respectively. The next section sets out an analysis of each of these events using Street, Baker and Tomlin’s (2008) analytical model.

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5.4. Street, Baker and Tomlin’s analytical model of numeracy practices

Numeracy events 1 and 2: timetabling appointments in college and online shopping at home

Using the four components of the analytical model: (i) content (ii) context (iii) values and beliefs (iv) social and institutional relations, I now explore numeracy event 1, outlined in Appendix 23, page 181: timetabling appointments in college and numeracy event 2, online shopping at home, outlined in Appendix 24, page 183. This analysis seeks to identify the important characteristics of both in-college numeracy practices and vernacular numeracy practices and the implications of these for students’ learning experiences in college.

Content: Street, Baker and Tomlin (2008) describe this aspect of the model as ‘the activities, techniques, procedures and processes that individuals engage in.’ (p 21). It is the mathematics aspect of the numeracy practice. In the observed timetabling appointments event described in Appendix 23, page 181, doing the calculations (content) and getting the (right) answer, was the aspect which was most stressed in college rather than using mathematics as a tool to solve a particular problem. When G. set out the activity for students he framed it more in terms of calculating time, rather than understanding how to book appointments. Even though the session was called, timetabling appointments, it was explained by the tutor in terms of the mathematics content - ‘work out’, ‘calculate’. This was further reinforced by how the work was set up. In the college numeracy event, the timings for clients were not expressed in the ‘real salon’ way, as Sophie highlighted. A fringe trim was allocated 10 minutes and two children’s haircuts allocated 25 minutes in total. A colour and cut combined was said to take one hour and thirty five minutes and so on. Students were not asked to round these up or down, as in a salon, and consequently, they interpreted timetabling the appointments as calculating the time and choosing a slot in which to fit the client. This seemed to reinforce the idea that calculating time rather than understanding how to book appointments efficiently in a salon was more important. Even though the artefacts used to record the appointments attempted to replicate a salon appointment-book page, divided into 15-minute slots, (Appendix 16, page 174), it did not appear to support Lily’s understanding or successful completion of the activity. This content aspect was also observed as framed in terms of ‘filling a gap’ in students’ knowledge, rather than building on what they already knew. The balance of focus in the classroom was very much on what students could not do, rather than areas of strength. It seemed to reinforce the ‘skills model’ or ‘autonomous model’ of maths or numeracy which appeared to be the dominant view of mathematics and numeracy observed in the College.

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In Emma’s reported on-line shopping event, set out in Appendix 24, page 183, the content aspect while important to the numeracy practice, was not foregrounded to the same extent as in college. In her 1:1 interview when Emma was asked to explain how she did her calculations for her on-line shopping, she chose ‘an example’ from the many on-line shopping transactions she has performed. The example she gave was the most recent and fresh in her mind. However, all she had in front of her, in the interview, was her phone and she demonstrated her calculations using her phone calculator. Emma at all times was aware of the relationship between the numbers she keyed in and what they represented. Her on-line shopping involved estimating calculations in her head as well as written calculations. She stated that she always ‘rounds up’ the cost of clothing she intends to buy to the nearest pound to make sure she can afford them. She then checks the calculation, twice, on her calculator, using the exact amounts. I have interpreted Emma’s use of her calculator as the use of written methods. Emma’s calculations, addition, subtraction, multiplication and division were representations of the problems she encounters when buying on-line. When Emma said, ‘What I do is’, I interpreted this as what is usual in her on-line buying activities, what she habitually or generally does, which is inferred in her use of the simple present. I suggest that what this shows in Emma’s case, is that although each on-line buying activity comprises context specific or particular information, Emma’s understanding is not restricted to the specific situation.

Emma explained her choice to disregard, rather than use, the precise answer given for her calculation of £89 (cost of clothes) ÷ 20 (number of weeks over which cost is spread) x 4 (weeks in a month) = 17.80, in favour of paying her mother £20 per month as, ‘much easier’. She can afford to ignore the ‘right’ answer in favour of a ‘better or more convenient’ answer, because getting the right answer means something different to her in terms of her vernacular numeracy practices. What seemed to be more relevant, and one could suggest, culturally relevant, to Emma, rather than a ‘right’ answer, was one that satisfied her own values of always paying back her mother in a timely fashion and thus maintaining access to this form of credit. Emma’s choice of algorithm was however, also suggestive of school, as she ‘used’ all the numbers that appeared to be part of the problem. The cost= 89; number of weeks that are interest free =20; number of weeks in a month =4. Rather than just efficiently dividing 89 by 5, she included the 20 weeks even though she did not intend to pay her mother back weekly. Instead, because the money is taken from her mother’s account monthly, Emma paid her back monthly. Emma seemed to demonstrate that she used both formal and informal ways of calculating and ultimately solving problems, depending on her purpose. Buying stylish clothes, which not everyone would be wearing, cost-effectively and conveniently, was her goal, rather than performing arithmetic calculations for either practice or assessment. And yet, performing the calculations correctly, both mentally and using written methods, was also part of that

110 goal. In contrast to Lily, Emma’s calculations appeared not to be an end in themselves, but a tool to solving a problem and achieving a specific goal.

Context: This component of the model is defined as, ‘the framing of those occasions when numeracy is done and the purpose for that use of mathematics’. (Street, Baker and Tomlin 2008 p 22). While many of the numeracy events students participated in outside college could not easily be imported wholesale into the vocational or the functional skills maths classroom, timetabling appointments in a salon would appear to share similarities across college and salon contexts. Boaler (1993, 2000) suggests that learning mathematics in context is motivating for students as it provides not only a richer curriculum but also it can enhance transfer through showing the links between school mathematics and real life problems. Contextualising ‘time’ did not appear to help Lily achieve the goal of the session. Calculating time was presented in such a way that what seemed like a useful context on which to draw from students’ lives, became in reality, a decontextualised skills activity. There was a ‘veneer of relevance’, (Maier 1991) or what Boaler (1993, 2000) has referred to as the ‘the inauthentic authentic’ which did not support Lily’s understanding of the problem she had to solve, instead it only served to disguise mathematical relations rather than clarify them. Even though she works part time in a salon, Lily’s knowledge and experience from outside the classroom were not experienced by her or seen by the tutor, to be of value. She was positioned as unable to invoke her knowledge and experience of numeracy practices in a salon, her vernacular numeracy practices, and consequently, was unable to engage with the problem. However, when Lily was stimulated to engage directly and explicitly with her knowledge of vernacular numeracy practices, she could see how the problem of timetabling appointments could be solved, and she demonstrated how she, habitually or generally, timetabled appointments in the salon where she works.

In contrast to Lily’s college numeracy practices, Emma’s online shopping took place at home and was framed within the context of what a young person needs for going on holiday. Emma’s purpose was also about getting value for money when shopping.

Values and beliefs: This component is concerned with ‘the ways individuals’ beliefs, values and epistemologies affect the numeracy practices they adopt.’ (Street, Baker and Tomlin 2008, p 22). In a discussion with the tutor after the session on timetabling appointments, he stated that the main purpose or motivation of the session, which was not obvious to the students, was to introduce and ensure that students understood concurrent appointments and using time efficiently in the salon. However, the tutor assessed Lily’s competence in relation to this based on his belief of her as someone unable to calculate time rather than as someone who works part-time in a salon and

111 understands appointments. This was a belief Lily herself had expressed in a previous group meeting where the tutor was present, and that belief remained unchallenged. Lily’s own self- assessment in relation to timetabling appointments took place before she saw what problem had to be solved. It was based on the teacher’s exposition before the work sheet was handed to students. In the classroom Lily equated ‘booking appointments’ with manipulating or calculating time. She invoked memories and emotions from school, ‘I really hate this stuff’, ‘I just hate the exercises on time’, where ‘time’ meant sitting and looking at pages of calculations –devoid of reference or context. When Lily was prompted to explain how, in the real context of her work, she booked appointments in the salon, she suggested two possible methods. One involved drawing on her knowledge of salon practices of manipulating the variables of hair length, hair treatment and usual time taken to complete the treatment, to the nearest 15 minutes, and then matching that to time available as laid out in the appointment book, for a particular stylist. She also reported that she had the option of asking the stylist if they were prepared to ‘fit in’ a particular client, a regular, even though there may not be sufficient time available in the appointment book. But none of this knowledge was initially available to Lily, it was overshadowed by her feelings of incompetence and powerlessness in the face of ‘calculating time’.

Sophie, interpreted the problem to be solved as timetabling appointments, yet rejected it as vocationally meaningful or authentic, because it did not reflect her experiences of work practices in the hair salon she is familiar with. This discontinuity between her experience in the salon and the college classroom, seemed to be a factor in her alienation or disengagement. Prior to understanding the teacher’s less than clear purpose, Lily seemed agitated and when she did work out what the focus was, her fear and agitation were replaced with anger, directed at the tutor. ‘Why didn’t he (the tutor) say we could do that’? Lily understood about making concurrent appointments, but because this was not made explicit in the class, she was not able to draw on what she knew. She needed permission from the tutor (why didn’t he say we could do that) to deviate from the college numeracy practices about time and draw on her own experience of making appointments. Lily’s previous negative experiences of learning maths in school may have influenced her in a situation which she interpreted to be about school maths, even though the tutor may present it as vocationally relevant. Lily described feeling stupid in school maths classes, in her one- to- one interview, so much so she stopped attending school maths classes. Lily expressed an inability ‘to do time’ on three occasions, yet, she demonstrated her use of time both inside and outside college in ways that did not evoke these negative feelings. Lily’s anger may suggest that in some instances, those feelings of alienation at school are continuing in college, even though teachers explicitly expressed the view of wanting college to be different from school for their students. Lily’s view of her own inability to

112 calculate time does not stand up to scrutiny. Time is recognised as a complex and multifaceted concept that has been shown to be difficult for learners to understand due to its abstract nature and the absence of concrete representations (Siegler and McGilly, 1989; Burny et al., 2009). According to Block and Zakay (1997) the complexity to estimate durations of time requires a person to encode temporal properties of events, construct cognitive representations and use those representations for actions. Lily, as she explained in her one to one interview, and in the numeracy event above, appeared to have no trouble estimating time either in her part-time work or in her day to day life. Nor did she seem to have any problems with arriving at work or college on time, nor when she planned trips out with friends or with her nieces. Although she assessed herself as having problems ‘with time’, what she may have struggled with at school, was measuring time. She described, in her one to one interview, sitting in class at secondary school and staring at multiple clock faces and having to add on and take away minutes and hours from the time displayed on the clock faces. Most of the students in their discussions used metric measurements. For example, Lily spoke about doubling up in recipes and used grams, although she did talk about her weight and height at the slimming club, in stones and pounds and feet and inches. Time, however, is one aspect of measurement that has not gone metric, so the relationships between the units, for example, 24 hours in one day; 60 minutes in an hour, and 60 seconds in a minute, can make adding and subtracting units challenging. When faced with calculating differences on the clock faces, Lily, from what she reported, appeared not to have had a strategy for undertaking these calculations. Her experience in the numeracy event above seemed to take her back to that feeling of not having a strategy to solve the problem. Nor was any opportunity for Lily to address this in college observed, which suggested that although the College stressed the need to provide students with learning opportunities which did not repeat negative experiences from school, it did not always realise this for learners. Lily did not complete her work sheet and defaced it. (Appendix 16, page 174). Sophie also chose not to complete her worksheet.

Social and institutional relations is that part of the model which is concerned with control of content, management of context and ‘the ideology exercised by different institutions and roles’. (Street, Baker and Tomlin 2008 p 22). In the tutor-student relationship in the classroom, there are a number of possible explanations for tutors not drawing on students’ experiences and knowledge in out of college contexts as a stimulus for engaging them in class. There was some evidence, from the tutor focus group, that when presented with the extent of the numeracy events students engage in outside college, tutors were surprised and took some time to reconcile their view of the students they ‘knew’ in college with the pen portraits presented to them. There was also evidence from the one to one interviews with tutors that their perceptions of what students do

113 outside college, especially those enrolled on Foundation level programmes, could not be of value in college. S., a hair and beauty tutor, expressed her view that students who had part-time jobs in local salons might pick up ‘bad practices’ and so students’ out of college experiences, even those most closely related to the vocational programme, were not always seen as valuable or valued for use in college. S. stated:

I know that some of the Level 1s have part time jobs in salons and I like to ask them, when it’s appropriate, how things are done where they work. Some of the small salons around here though have ways of working that we wouldn’t teach in the salon. I feel responsible, as an assessor for the College, that when students say they’ve been trained here, that they’re really up to scratch.

M. who teaches construction, highlighted the institutional power and pressure exerted over teachers which may influence their ability to engage with students’ vernacular numeracy practices:

I’m just so used to having to justify what and how I’m teaching in terms of getting them ready for work and making it relevant to that, not just with these guys but all the time…that I don’t think about what students actually bring to the class, like what they do and know already from outside. I suppose I just assume that because they’re in foundation learning and have problems, you know, that they don’t really ……you know, bring very much with them. I have a syllabus to get through and I need to make sure the lads stay on programme and pass, otherwise I have to justify my course running again next year. Getting them through Level 1 means they will probably stay on for Level 2 and 3 and that’s important.

There was also some evidence that vocational tutors were unsure how to support their students with their numeracy in vocational classes, even though this had become a key part of their job.

G. from hair and beauty also raised the issue of getting students through their course while at the same time supporting them to develop individually. He had had some negative experiences at school generally and in relation to maths, in particular, and wanted to be inclusive in his practice.

I’m being honest here, but I just don’t really know where to start with maths. Some of us staff have just passed our own level 2 (in maths) since working at the college. That’s why being involved in this is good. Mainly, I want the students who come from poor backgrounds and that’s a lot of our students, almost all foundation learning certainly, to get a fair deal and be able to get a job and I know their maths will help them in that in general.

I think this kind of thing should be done by the whole team. I think we need to sit down and agree how we do it. I know (head of department) wants us to look

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at it. It’s a weakness on our self-assessment report, so it will happen and I want to do it right.

When teachers were asked how they found out about their students’ knowledge and understanding of numeracy they identified the initial assessment carried out at induction. However, when asked what this told them about their students, most of the vocational teachers acknowledged that they couldn’t interpret the results. J. said,

To be blunt, I just put it (the printout of levels for each student) in my course file and I keep it there in case I get observed. I’m happy helping the kids with any numeracy that comes up. I can tell them how I do it in construction, but I can’t really show them different ways.

Both hair dressing tutors stated they didn’t understand the print out from the on-line initial assessment. G. explained,

We’re not exactly sure what the levels mean. We know that entry level 2 means very low numeracy, but I wouldn’t know exactly what a student could or couldn’t do. Sometimes, I don’t think I should have to know. It’s usually when we’re all stressed and there’s going to be an observation by someone on the quality team. Then I think the maths teacher should teach them their maths and I can stick to hairdressing. But most of the time I really want to help them. Our foundation learning students have had some bad experiences in school and at home.

In taking the stand that they did, of not recognising that students might have valuable knowledge and experiences, the teachers closed off the potential gain for students which could be realised through understanding more about their students’ social uses of mathematical/numeracy practices. Teachers could potentially better understand the overlap and divergence in the ways their students are engaging with mathematical activity in the multiple contexts of home, leisure and part-time work. This insight may offer the possibility to help teachers build links or bridges between out of school practices and the kind of mathematics necessary for vocational and functional skills mathematics success. A stance that includes a wider lens for legitimate mathematical activity, and that gives credibility to students’ vernacular numeracy practices, makes visible the importance of a more active and collaborative relationship between teacher and student and would begin to put into practice the ‘not more of the same’ approach which teachers said they wanted for their students. However, what the teachers’ views suggested was that the functional skills model of mathematics was the dominant model in the College and remained very much closely aligned with the ‘skills’ model of mathematics and numeracy. Rassool (1996) suggests that the concept of functional mathematics and numeracy matches skills with quantiﬁable educational outcomes (which can be measured through adequate testing procedures) and with ‘economic needs’ (Rassool

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1999 p 6). Central to the concept is the assumed correlation between individual skills and the overall performance of society in terms of modernisation and economic productivity (OECD 1997; OECD 2013). A functional view of numeracy can be found in current deﬁnitions of adult basic education in the UK. The Moser Report for example (DfEE 1999) refers to it as ‘. . . the ability to read, write and speak in English, and to use mathematics at a level necessary to function at work and in society in general’ (DfEE 1999 p 2). Moser’s influential definition is at the heart of the ANCC and functional skills standards, both of which drive mathematics provision in the College for students on vocational programmes up to and including level 2. Within this framework, functional mathematics is linked to the concept of human resource development and the debate about ‘basic’ skills that occupies a central place in current employment policies. Mathematics is seen to have high economic value and it serves as an indicator for economic and societal development.

5.5. Conclusion

This chapter and the previous one presented accounts of students’ numeracy events from inside and outside college which demonstrated the extent of mathematically mediated practices which occurred across all domains of students’ lives including, family, leisure, part-time work and learning in college. Students’ out-of-college numeracy events were analysed and using the ANCC, the mathematics content of events and the generalised numeracy practices students engaged in, demonstrated their uses of mathematics above the levels they were assessed at on entry to college. In the case of Emma, a hair and beauty student, whose initial assessment placed her at Entry level 2 in terms of the National Qualifications Framework (NQF), her vernacular numeracy practices when translated into the language of the ANCC, showed her to draw on numeracy practices from across Entry levels 2, 3, level 1 and level 2. Further analysis, using the four components of Street, Baker and Tomlin’s (2008) analytical model, revealed further discontinuities between vernacular numeracy practices and college numeracy practices. In particular, how students are positioned as being in deficit in college even though this was not the case outside college. Tutors’ perceptions of students’ use of mathematics outside college assumed a basic level of use and equated this basic level with basic contexts. They did not recognise that students operated and engaged in complex contexts as part of their daily lives. With few references or connections directly made to students’ lives, outside college, in a way which expressed positive value or use in relation to in-college work, tutors were unaware of the extent of students’ vernacular numeracy practices and consequently, few opportunities were available for students to reflect with their tutors on their social uses of numeracy. Therefore, students had little opportunity to challenge their deficit positioning in college.

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Although there was an explicit college aim of not wanting to replicate students’ previous negative experiences of school mathematics, this was not always realised for students. Tutors, under pressure of accountability, a strong college belief in the importance of assessment-driven practices and as a result of their own beliefs and experiences of learning mathematics, were not always in a position to understand and draw on students’ vernacular numeracy practices for classroom use.

Students initially resisted seeing a value in their social uses of mathematics, conceptualising them as not ‘real maths’. However, when they were encouraged and stimulated to re-view their numeracy events translated into more formal language - the language of the Adult Numeracy Core Curriculum - they recognised them as ‘school maths’ and engaged in a re-consideration of their views. This also stimulated them to express their views about numeracy and mathematics and their previous and current experiences of learning. They viewed functional skills mathematics in college as numeracy and not real maths. They equated maths with school maths and they did not see school maths as the same as college maths. This may be because their functional skills assessments are all contextualised, and they associated contextualised learning with ‘not real learning’. Students encountered and valued a wide range of technology enabled, numeracy-related artefacts which mediated their vernacular numeracy practices, in their everyday lives. These included mobile phones, computers, TVs, gaming devices, ATMs, digital screen displays in a range of contexts. Technology was seen as a normalised part of students’ lives and students could not conceive of an identity without it, in particular, their mobile phones. Students were not observed using technology in college, in the same way.

There were differences evident in students’ vernacular numeracy practices and in- college numeracy practices. However, these may not be inherently different and consequently differences between in and out of college practices could potentially be reduced.

Tutors’ prevailing discourses about students were of people in deficit and lack and not just in terms of numeracy but also in terms of ‘moral values’. Tutor views about the purpose of college were framed as much in terms of inculcating particular values and beliefs in students, as they were about learning. Tutors expressed views about their students’ lack, e.g. a work ethic and the knowledge of the impact of ‘poor skills’ on their future lives.

College numeracy practices were seen, at times, to disempower students.

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Chapter 6. Conclusions and discussion

6.1. Introduction

This chapter serves a number of purposes. Firstly, it provides a reminder of the purpose of the study; summarises the findings from the research in order to answer each of the research questions; it explores the findings through a dialogue with the literature to make some sense of them for the purpose of arriving at a set of conclusions and to understand their implications for practice and further research. Through this dialogue I hope to establish where and how my research resonates with previous research; where it diverges or takes the discussion on post 16 numeracy forward and what cautious claims to a contribution to knowledge can be made. The limitations of the research, introduced in chapter 1 and expanded on in chapter 3 are discussed. Finally, the chapter concludes with some implications for practice and my final reflections on the research.

6.2. Research purpose and answering the research questions

This section restates the purpose of the research and summarises the research findings as well as answering the research questions. It also engages in a wider discussion and exploration of the relationship between the findings and previous research.

The purpose of this research then, is to explore students’ social uses of numeracy conceptualised as students’ vernacular numeracy practices. Through this exploration of students’ vernacular numeracy practices, the research seeks to challenge, at a conceptual level, what it means to be numerate. In particular, the study aims to challenge the public labelling of those young people who enter further education without a mathematics qualification, as innumerate. The following research questions were devised to support the purpose of the research.

6.2.1 RQ1: What numeracy events and practices do learners engage in, in their everyday lives?

The research showed that all the students reported many numeracy events as part of their daily lives which were contextualised across their life domains including, family, leisure and part-time work. These numeracy events were shown to generate opportunities for mathematically mediated practices. Events within the domains of students’ lives included: on-line shopping; managing their mobile phones; using a computer at work to price and order car parts; timetabling appointments in hair salons; ‘free running’; placing bets on horses and football and playing computer games;

118 shopping to a budget; looking for discounts to take children on days out and cooking to recipes. Details of all these events are reported on in chapter 4.

The numeracy events reported on by students are consistent with out of college interests of other young people of similar age and circumstance, (Fowler, Hodgson and Spours 2002; Ivanic et al 2009) and share similarities with Bishop’s (1988) six categories of fundamental activities that “are both universal, in that they appear to be carried out by every cultural group ever studied, and also necessary and sufficient for the development of mathematical knowledge” (p. 182). Mathematics, as cultural knowledge, “derives from humans engaging in these six universal activities in a sustained, and conscious manner” (p.183). Bishop’s six fundamental mathematical activities are: counting, locating, measuring, designing, playing, and explaining. What is significant about Bishop’s (1988) list is that the students displayed greater engagement with these foundational mathematical activities outside college rather than inside college.

Students’ numeracy events were translated into the language of the ANCC. This afforded a way of examining the content of the numeracy event and expressing that content in the re-contextualised language (Wake and Williams 2007) of abstracted skills. This analysis showed that habitually, students demonstrated numeracy skills above the level they were assessed at on entry to college. This suggests that students’ initial assessment at entry to college is limited and their vernacular numeracy practices go unrecognised and unacknowledged. Students’ everyday numeracy practices, conceptualised as their vernacular numeracy practices - tended to make use of mathematics as a tool to solve problems rather than as an end in itself.

The Funds of Knowledge literature (Moll, Amanti, Neff and Gonzales 1992; Moll, Amanti and Gonzales 2005; Moll 2002) reviewed in chapter 2 demonstrated how insights gained from a knowledge of students’ lives and practices outside of the classroom (the social) can be brought into the classroom and used to enhance teaching and learning. Funds of knowledge has expanded from meaning family and community resources of disadvantaged groups to the individual resources students bring to their learning (Baker and Rhodes 2007, Oughton 2010). Baker and Rhodes (2007) differentiate between what they call ‘narrow’ and ‘broad’ funds of knowledge. The broad funds of knowledge learners bring to their learning are more than recognising the formal facts (narrow) and skills learners have acquired in their previous educational or formal settings, it is also about ‘developing a reciprocal relationship where teachers come to understand that learners’ experiences of numeracy practices within informal settings are valid and valued (broad)’ (Baker and Rhodes 2007: p13). ). Baker and Rhodes (2007) not only stop short of offering a comprehensive framework for identifying and analysing learners’ numeracy

119 practices, they also resist suggesting ways for finding out about learners’ funds of knowledge. I think my research offers a practice to begin to identify these vernacular numeracy practices with students.

Within the literature examined in chapter 2, there are several examples of studies which aim to explore participants’ contextual uses of numeracy or mathematics without explicitly referring to these in the language of numeracy as a social practice or students’ numeracy practices. These include Rogoff and Lave (1984); Lave (1988); Saxe (1988; 1991); Nunes, Carraher and Schliemann (1993); Scribner (1984) and Walkerdine (1988, 1997) and these provide early examples of research which examined the mathematics (practices) of the groups studied in their specific contexts. These ‘street maths’ (Nunes, Schliemann and Carraher, 1993) studies examined the maths learning that takes place in work and in more formal settings. (Lave 1988) challenged the notion of unproblematic learning transfer of school maths to the workplace and suggested that maths needed for out of school practices is best learned in those contexts. This has been seen as the basis for the argument for ‘embedded learning’ (Brookes 2005) in vocational programmes in the Further Education and Skills Sector in England.

My research and many of the studies above attempt to capture people’s uses of numeracy and mathematics in their out of school setting. However, where there is some divergence between my research and those above, is in how school numeracy practices are examined. I observed students in their vocational and functional skills mathematics classes, the majority of these other studies tested students in the laboratory or outside with traditional school maths problems, rather than observing students having to deal with college or school maths as part of their curriculum. However, the findings from my research share some similarities with the studies above in so far as the students’ vernacular numeracy practices are located in a real-life need to solve authentic problems, usually initiated by the students themselves and considered important for managing a particular aspect of their lives. Other similarities between these studies and my research are the demonstrable inconsistency between mathematical performance carried out in different contexts, for example, the school and everyday situations. Where we may differ is in the explanations for the inconsistency. I have used a theoretical framework to explain these inconsistencies, predominantly through social factors with the other research sometimes emphasising cognitive reasons for the differences. Masingila et al (1996) had similar findings when looking at people’s everyday maths practices in their work. They reported that people have more than sufficient mathematical knowledge to deal with problems; their mathematics practice is nearly always correct; and problems can be changed, transformed, abandoned and/or solved since the problem has been generated by the problem solver.

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The New Literacy Studies, (NLS) as applied to numeracy by Street, Baker and Tomlin (2008) is an attempt to adopt a more social perspective on numeracy and mathematics education, conceptualised by Lerman, (2000) as the ‘turn to the social’. The NLS offers a way of seeing what students can do, in terms of content, context, values and beliefs and social and institutional relations. I shifted in focus from Street Baker and Tomlin’s (2008) approach in that I chose to foreground the content aspect of their analytic framework as it was in this re-looking at the mathematics embedded in students’ vernacular practices that two important results emerged. Firstly, my assessment of the maths content of students’ vernacular numeracy practices suggested it was at a higher level than the content identified through in-college assessment practices. Secondly, it provided a focus and stimulus to engage both students and tutors in beginning to recognise the potential curricular value of students’ vernacular numeracy practices. By objectifying the mathematical basis of students’ vernacular numeracy practices, through the process of translation, I have provided some evidence of acquisition of knowledge by students in the study.

In getting students to report on their out of college activities, it became very clear the extent to which the students used technology to function. Students encountered and valued a wide range of technology enabled, numeracy-related artefacts in their everyday lives. These artefacts included mobile phones, computers, TVs, gaming devices, ATMs, digital screen displays in a range of contexts. Technology was a normalised part of students’ lives and students could not conceive of an identity without it, in particular, their mobile phones.

6.2.2 RQ2: What numeracy events and practices do learners engage in on their vocational and functional skills programmes?

In college, in the observed vocational and functional skills classes, students’ numeracy practices tended to be assessment driven. This is referenced in particular, in functional skills classes, when tasks were introduced in the lesson as relevant to or needed for ‘the test’. For example, Jack’s learning of mean and range for the Level 1 FS exam and hair and beauty students carrying out a task involving calculating rent costs, because it was needed for a presentation as part of a formal functional skills assessment. In construction, some numeracy tasks were also linked to ‘good work practices’. Paul was asked, by the tutor, to arrange a set of spanners, measured in mm, in size, on a work bench, because this was part of ‘keeping a tidy work space’. Findings from the research showed that while students’ use of mathematics as part of their vernacular numeracy practices tended to be in pursuit of a goal linked to a clear purpose, with the student in control of the task, in-college maths practices more routinely had the teacher firmly in control, were curriculum driven and tended not to be collaborative. The role of context in

121 the college classroom is dealt with more fully as part of RQ3, however, I just want to note here that when some tasks were contextualised, for example, looking at budgeting or timetabling appointments the goal still tended to be computational, for example, to ensure that students could add and subtract decimals, rather than exploring ways of budgeting on a low income or generating concurrent appointments in the salon. My research confirmed Boaler’s (1993) conclusions that ‘inauthentic authentic’ practices which give a veneer (Maier 1991) of the real world, may actually impede students’ mathematical understanding. This was evident in the timetabling appointments session when Lily was unable to solve a salon timetabling problem. Masingila et al (2010) go so far as to suggest that school maths practices are often grown from a transmissionist paradigm of instruction and are largely irrelevant to students’ everyday interests, usually with the goal specifically school related and alienated from students’ vernacular interests. I would want to add that in my research, class goals were not always acknowledged and shared with students. Moreover, in half of the observed classes, in post observation conversations, the tutors identified other goals for the session that were not made explicit or shared with students. These related more to ‘values and beliefs’ rather than mathematics.

Teachers’ prevailing discourses about students were of people in deficit and lack. The emphasis was always on level of skill and their level characterised as ‘basic’, ‘modest’, ‘low-level’. This label was attached from when the students entered college and achieved Entry level in their initial assessment of literacy and numeracy. Tutors frequently referred to students by their level, ‘the level 1s’ or ‘the foundation students’. This notion of lack and deficit is not just in terms of numeracy but also in terms of ‘moral values’. In one example the covert goal for a functional skills mathematics task was framed in terms of inculcating particular values and beliefs in students which the tutor believed the students lacked, e.g. a work ethic and the knowledge of the impact of their ‘poor skills’ on their future lives. Big institutions like further education colleges are charged with implementing government policy on literacy and numeracy. Current policy and practice around numeracy in further education has been heavily influenced by the first Skills for Life policy (Tett, Hamilton and Hillier 2006) which in turn was influenced by the Moser Report (Moser 1999). Hodgson et al (2007) have noted the way people in these documents are positioned. Skills for Life policy focuses on people over the age of 16 with literacy, language and numeracy needs. In the 2001 version of The Skills for Life Strategy, learners are always seen as ‘them’ and not ‘us’. They are viewed with sympathy, on the grounds that they risk being cut off from “the advantages of a world increasingly linked through information and technology” (DfES 2001 p. 1) they are taken to “have stressful lives” (DfES, 2001 p. 8) and “pressing personal and social problems” (p. 17) which need to be addressed. This apparent sympathy is accompanied by the

122 implication that people with basic skills needs are a drain on the economy and on society. This “has disastrous consequences for the individuals concerned, weakens the country’s ability to compete in the global economy and places a huge burden on society” (p. 6). The Skills for Life Strategy (DfES 2001) presents a highly dysfunctional picture of learners with basic skills needs and goes as far as to suggest that some people suffer from “inertia and fatalism” (p. 7) and that “there is evidence that some of those with a need still have a deep-seated reluctance to address their literacy and numeracy skills needs” (p. 16). The tutors’ discourse of deficit seemed to be an extension of this same discourse of deficit which pervades public policy documents and frames the discussion about literacy and numeracy.

Students’ in-college numeracy practices were disconnected from their vernacular numeracy practices. There were few references or connections directly made to students’ lives, outside college, and consequently their vernacular numeracy practices, in a way which expressed positive value or use in relation to in-college work. This is not surprising given that there was no formal mechanism for establishing how students use mathematics in their everyday lives. Tutors were also unsure as to the value and relevance of students’ numeracy practices to their work in-college. Even where some students’ part-time work was vocationally relevant, tutors expressed concern and reservation as to the quality of the practices used in students’ part-time work places. While much of the literature acknowledges the benefits of making connections between students’ lives outside college and their mathematics learning in college or school, Resnick (1987) has argued that 'the process of schooling seems to encourage the idea that ... there is not supposed to be much continuity between what one knows outside school and what one learns in school' (1987, p. 185). Observation and analysis of students’ ‘in college’ numeracy practices showed that in-college, vocational numeracy activity, had aspects in common with both in-college functional skills numeracy practices as well as vernacular numeracy practices. The vocational sessions, in some instances, were problem or goal directed (Masingila et al 1996) requiring contextualized reasoning and situation specific competencies (Resnick 1987). These were similar to the students’ vernacular numeracy practices. However, the vocational numeracy practices were also similar to functional skills mathematics practices in that they were largely transmissionist in approach. The tutors told the students what to do and how to do it. There were times when the vocational sessions were as curriculum driven as they were in functional skills. As one tutor, put it, ‘I have a scheme of work I need to get through to make sure the lads pass otherwise my course may not run next year’. The students were still positioned in deficit, even though some of them had part-time jobs in the vocational area of work.

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Students looked to make sense of their numeracy events both in college and out of college. Sense making and what happened in ‘real life’ were important to students and when their experience in college conflicted with what they perceived as ‘making sense’ or inauthentic, this seemed to act as a barrier to learning. This was evident in three of the classes observed, where students invoked a discourse of sense-making and became frustrated when their view of sense-making was not enacted as part of the activities. I concluded that students’ in-college numeracy practices were sometimes disempowering for them as learners.

6.2.3 RQ3: How are learners’ everyday numeracy practices, conceptualised as their vernacular numeracy practices, used in their vocational and functional mathematics learning?

In college, students seemed to have little or no opportunity to reflect on their vernacular numeracy practices as valuable and valued knowledge and ways of working. In fact, students’ perceptions of their vernacular numeracy practices as ‘just common sense’ and ‘not real maths’ could not be challenged, as there did not appear to be a formal mechanism in college for students to reflect on and share views either with each other or their tutors. This is not surprising, given that in the interviews with teachers, they generally did not see the mathematical value of students’ vernacular numeracy practices. However, they could be brought to do so when the ‘content’ was translated or abstracted in the research. The vocational teachers recognised some of these practices insofar as they addressed vocational contexts, for example, in the hair dressing salon and where the maths content was also made explicit. To some extent also it seemed the vocational teachers recognized the significance of the ‘social and institutional relations’ to numeracy practice when these too were made explicit. In exploring students’ vernacular numeracy practices, this research was interested in understanding the role of everyday context in supporting the successful ‘understanding and doing’ of mathematics. In Street Baker and Tomlin’s (2008) analytical framework for understanding students’ social uses of numeracy, context is a key component.

In their empirical research, Street, Baker and Tomlin (2008) seem to interpret context primarily as location, but do also reference the specificity of context as including the relationship between the individual’s (child’s) cultural resources in terms of numeracy practices and how this either advantages or disadvantages the child depending on the location. For example, school might teach counting through asking children to count the number of place settings needed when setting a table for dinner at home. This does not take into account those children for whom individual place settings for meals is outside their cultural norms. What is missing from this definition and interpretation is the recognition of the problematized nature of the term. I suggest that an understanding of

124 the contested nature of context is important in identifying how best to support students in developing their mathematical knowledge and understanding. Theories of context seem to polarise into representations of context put forward by the proponents of learning transfer on the one hand and on the other, those who advocate, what Evans (1999) calls, the strong form of situated cognition. Traditional learning transfer theory suggests that students will be able to demonstrate their knowledge and understanding of mathematics in situations outside the classroom so long as they recognise that their previous learning applies to the new context, they can retrieve that learning, and finally that the previous knowledge can be transformed to fit the demands of the new situation. This view, Evans (1999, 2000) argues, proposes a model of teaching and learning which involves the transmission and internalisation of a body of knowledge (and skills). A problem or 'task', and the mathematical thinking involved in addressing (or producing) it, are seen as able to be described adequately in abstract terms, with little or no reference to the context - e.g. simply as 'proportional reasoning' or ‘calculating time’. Hence it is claimed to be possible to talk about 'the same mathematical task' occurring across several different contexts. Therefore, traditional views expect that the transfer of learning, e.g. from school to everyday situations, should be relatively unproblematical - at least, in principle, for those who have been properly taught.

Boaler (1993, 1998) has suggested a middle ground in relation to context and mathematics teaching in the classroom. She suggested that, the argument for learning mathematics in context, can be summarised under two broad headings. Firstly, context is used to motivate students through a richer curriculum and secondly, learning in context will enhance transfer through showing the links between school mathematics and real life problems. Boaler however, dismisses much of the learning mathematics in context work that has taken place over the past twenty to thirty years, relegating it to what she describes as ‘distracters or even barriers to understanding’, (Boaler 1998). In my research there was some evidence of tutors attempting to use everyday contexts to engage and motivate students as in timetabling appointments and budgeting, in particular. However, these attempts resonate with Boaler’s ‘distracters and barriers’ as they were not successful in engaging and motivating students in class but rather appeared to alienate them. Nor was it just students having to contend with contextualising as distracting, their own views about contextualised mathematics or as they said, ‘numeracy’ and not mathematics, also served as a barrier to engaging in college mathematics practices.

What the literatures above generally agree on, although they express it in different ways, is making use of students’ vernacular numeracy practices either as a starting point from which to build further knowledge and understanding, or as a means of demonstrating

125 respect for what students already know and do with mathematics and as a means of engaging and motivating them to progress to further learning. What is either implied or explicitly stated in these views is the need to find out what students know and how they use mathematics in their lives. Or as Wedege (2010) puts it, ‘understanding (mathematical) knowledge acquired in everyday practice as well as knowledge wanted or needed in everyday practice’, (Wedege 2010, p 34). A starting point for building these bridges between contexts, Evans states, is in discerning those areas of overlap between students’ everyday lives and their school mathematics. The discourses surrounding practices in and out of school or college need to be examined for similarities and differences as well as the tasks involved in each. The importance of making connections is also highlighted by Masingila et al (2010). They suggest this connection making, while essential for constructing mathematical knowledge, is often absent in classrooms.

Current policy and practice around functional skills mathematics suggests a belief that a likelihood of situation specificity is reduced if mathematics is learned in real life contexts and the links between school and real life requirements are made explicit, (ANCC 2000, Ofsted 2011, Ofqual 2012). In all of the observed sessions both ends of the learning transfer spectrum were evident. In the case of Lily’s timetabling appointments session, what appeared to be an appropriate ‘area of overlap’, ended unsuccessfully for Lily as well as Sophie. I would argue that this lack of success stems from tutors not knowing what their students know and do outside college, including their values and beliefs. For Lily in particular, it could be seen that the discourse of ‘calculating time’ was very strongly emotionally charged and because the lead in to the task was neither discussed or presented in a way which might de-escalate the emotional aspect, Lily could neither recognise nor draw on her vernacular numeracy practices and harness them for in- college use. Nor was there an attempt to give Lily’s vernacular numeracy practices an enhanced mathematical recognition which might also have enabled her to recognise and draw on them to complete the task in college. There appeared to be no mechanisms available to the tutor to gain an understanding of Lily’s or other students’ social or cultural values, which could then be used in the classroom to support the development of timetabling concurrent appointments in the salon.

I think this is where my research has the potential to break down some of the barriers to engagement. Through the process of reflecting and engaging on their own vernacular numeracy practices, translated into abstracted skills, students were open to re-evaluating their views on the nature of mathematics and numeracy and their own abilities to successfully engage with each or both. This was also the case with tutors.

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6.3. Conclusions

Students labelled as innumerate or having very low numeracy skills, by their initial assessment at entry to college, have demonstrated significant engagement with mathematics as part of their vernacular numeracy practices. Very often this engagement took place above the level at which they were formally assessed. This negative labelling is sedimented in both public and private discourses, in public policy documents, college policies and tutor conversations and actions in the classroom.

The use of formal assessment, at entry to college, as the single metric against which students’ personal numeracy competence is assessed, I suggest, oversimplifies and misrecognises what it means to be numerate. The social view of numeracy and consequently understanding students’ vernacular numeracy practices, is potentially useful in supporting teachers in further education, to better understand the role of mathematics in everyday life and what this says about learners’ numeracy-related demands and interests outside college.

There are discontinuities and inconsistencies between students’ vernacular numeracy practices and in-college numeracy practices, however, many of these are not inherent differences and could be reduced. A point of departure for reducing the differences would be to recognise and acknowledge students’ vernacular numeracy practices as a basis for developing more generalisable schemas. This was most apparent in relation to the use of technology in college to support mathematics learning.

6.4. Implications for practice

There is general agreement within the literature that drawing on students’ out- of -school or college mathematical knowledge is beneficial to in-school or college learning. Masingila et al (2010) go so far as to state that students gain ‘mathematical power’ when their in-school mathematics experience builds on and formalises their knowledge gained in out-of-school situations and when their out-of-school mathematical experiences ‘apply and concretize their knowledge gained in the classroom’. However, it is also acknowledged in the literature that understanding students’ out of college mathematics practices is not straightforward given the number of students in a class and the number of classes one teacher may teach. One of the implications of this research is in providing a connection between home and college numeracy practices.

The findings and conclusions from the research present some possibilities for numeracy in the Further Education and Skills Sector in England, for students with low or no

127 qualifications in mathematics, and their teachers. There is some evidence in the literature to suggest that current models of initial assessment reinforce deficit positionings for those students with previous negative experiences of learning mathematics (Looney, 2007). Adopting a social practice approach to numeracy or mathematics assessment at entry to college, could highlight what students can do rather than only what they do not know. Supporting students to represent their everyday activities, their vernacular numeracy practices using the ‘clock-face’ activity or collage, would encourage students to explore their social uses of numeracy and provide a point of entry into discussing not only their social uses of mathematics, but also those other aspects of the model which can support students to explore their previous experiences of learning mathematics. The aim is not to slavishly match each event to a set of level descriptors in the ANCC, but to start the discussion about the legitimacy of students’ vernacular knowledge including their vernacular numeracy practices.

Finally, encouraging critical reflection on students’ vernacular practices does not mean a descent into romanticism and a privileging of the vernacular over the formal. The critical approach implies that all practices – including one’s own – need to be examined. Finally, and perhaps most importantly, the above criticism emanates from the view that access to the dominant mathematical practices of formal institutions and workplaces is the only goal for functional skills mathematics students in further education.

6.5. Addressing the aim of the research

There is some evidence, based on the findings, that there is a case to be made for ceasing to label young people as innumerate because they do not have a qualification on entry to further education. Relying on such a narrow definition of numeracy and mathematics reinforces many young people’s negative emotions about mathematics and sediments the circulating discourses about their lack and deficit, in public discourses as well as policy documents.

The stated broad aim of this research was to challenge, at a conceptual level, what it means to be numerate. This means to confront head on those limiting and deficit views of the students who have all been classified as having very low numeracy skills and are described as innumerate by their teachers and the College. This labelling is the result of the students not having a qualification in mathematics at Grade C or above and is further compounded by their initial assessment of literacy and numeracy at entry to college, which then places them on a level, which in many cases, compares their skills to those of primary school children. This ‘insulting’ practice (Swain 2006) has been identified as contributing to students’ poor self- image and identity in relation to mathematics in further

128 education. The findings in chapters 4 and 5 present a picture which conflicts with and contradicts not only the College but public and policy perceptions of what it means to be numerate. While there are many definitions of numeracy and what it means to be numerate being used by organisations who have influence within the Further Education and Skills Sector, in practice, students in this study have been judged on whether they are numerate or not, solely on the basis of their lack of qualifications in formal or school mathematics. However, if the lens is widened to include definitions of numeracy which move beyond qualifications only, then there is evidence from the research which allows students to challenge the positions open to them, in terms of being labelled innumerate.

As stated earlier there is concern that learners who undertake initial and diagnostic assessment on entry to FE are having their mathematical skills ‘mis-diagnosed’ (The Research Base 2013). This is resulting in learners being placed on courses, lower in level than previously attained. This provision does not stretch them, nor does it lead to progression to higher level provision or employment (Watson 2004; Hayward, Priestley and Young 2004). It is not only the initial assessment tool that labels them as innumerate, many policy documents agree.

The majority of students in the research were all assessed to be at Entry level 2, with two assessed at Entry level 3 and one at Entry level 1 on the national qualifications framework. Given these labels, students were expected to know number concepts including the following: how to count up to 20 items; recall addition and subtraction facts to 10; multiply using single digit whole numbers and other competencies from the numeracy core curriculum. They did know these, but they were able to do so much more, which the range of events in the findings show. Despite continuing debates and the lack of consensus about what exactly the term numeracy means, the students in the research, as evidenced in chapters 4 and 5, demonstrated their functionality and successful numerate behaviour outside college.

Over and above providing possibilities for students to demonstrate their functionality outside of the college, the social practice model of numeracy widened the lens to include, in a definition of numeracy, the recognition of those other factors which influence students’ learning, these are - values and beliefs and social and institutional relations.

There are critics of a social practice approach to numeracy, who are willing to accept a social practice approach to literacy. Barwell (2003) for example, while embracing the ‘social’ in mathematics education and literacy as a social practice, has questioned the validity of numeracy as a social practice, mainly because he contests the lack of a tradition in the use of the term numeracy within the field of mathematics education.

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Barwell’s argument goes like this: literacy as a social practice, which sees literacy as more than the decontextualised skills of reading and writing, has a long and rich research tradition. Ethnographic researchers (like Heath, Street and Barton and Hamilton) have historically researched literacy practices and have subsequently decided to draw attention to numeracy in their research. They are, he suggests, ‘tripping over’ numeracy in their research and are confusing mathematics and numeracy. There may well be, he offers, mathematizing going on in activities such as cooking, shopping, reading tables and charts, food labels as well as calculating budgets, but, he suggests, this is not mathematics, as there is no mathematical abstraction taking place, which for him is at the heart of doing maths. He does not explain, however, what is for him, the difference between doing maths and mathematizing. Numeracy and numeracy practices are, he offers, a subset of literacy, rather than related to mathematics and mathematics education. Barwell’s argument, could be seen to exemplify the difference between the ‘autonomous model’ and the ‘ideological model’ of numeracy and mathematics. Regardless of what term Barwell wishes to use, it does not change the outcomes for the students who were failed by their previous learning experiences. Coben (2006) on the other hand, while not endorsing the notion of numeracy as a social practice does not take issue with the use of the term numeracy in the way Barwell does. Her concern, seems to be that a social practices conception of numeracy implies that adult practices are ‘so rich in mathematical knowledge and understanding that no teaching – or even further learning – is necessary.’ (Coben 2006, in Tett, Hamilton and Hillier 2006, p 101). While she appears to accept, fallaciously, in my view that this ‘no more teaching is necessary’ approach applies to literacy as a social practice, she argues that this is just not acceptable for numeracy or mathematics. This view signifies a misunderstanding of what a social practice approach to numeracy or mathematics really is, as well as to literacy. Coben (2006) goes on to suggest that ‘pluralising’ (allowing for a recognition of many numeracy practices) atomises numeracy and mathematics teaching and learning. She links this to the problem of learning transfer. I would suggest that the current functional skills standards and the adult numeracy core curriculum already atomise numeracy and mathematics and support the transmissionist approach and what has been called, in primary school mathematics classes as well as in adult numeracy classes, ‘death by 1000 worksheets’ (Askew 2011; Ofsted 2011). Coben acknowledges that in the literature on adult numeracy ‘there is recognition of the view that numeracy and mathematics are culturally determined and socially formed practices’, (quoted in Tett Hamilton and Hillier 2006, p 102). What may be at the heart of some critics of a social practice approach to numeracy, is not a de facto unwillingness to accept the theoretical perspective and what it has to offer to students, but rather that the theory has its roots in anthropology and literacy theory.

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In summary, the findings have shown that students’ vernacular numeracy practices when examined, included some recognizable mathematical content, related to a sense of context, for example, the numbers are attached to specific quantities of money, time and distance, that have particular sense in the contexts; are attached to values and beliefs, for example, the need to budget within one’s limits and relate to social and institutional relations, for example, the ‘rules and norms’ of being a receptionist, hair stylist, mechanic, that set the social expectations which frame all of these.

6.6. Limitations of the research

The limitations of this study have been highlighted within relevant chapters (chapter 1 and 3) of the thesis as the research has progressed. This section provides a summary of the main limitations and addresses what actions have been attempted to address the limitations and also to recognise the effects the limitations may have on interpretation of the research findings.

The limitations in this research can be categorised as methodological and ideological. The main methodological limitations are the sample size and the use of self-reporting data for part of the study.

6.6.1 Sample size

The sample size of this study is fifteen students and six teachers. I recognise that fifteen students and six teachers do not represent the student or tutor population within further education. The fifteen students do however, comprise two groups of vocational students with no qualifications in mathematics, which was a key criteria for choosing students to be included in the study. Although the numbers are small, the sampling was purposive, often used in qualitative research, when ‘information-rich’ cases for in-depth study is the desired outcome (Patton 2002). I reasoned that I needed a rationale for sampling which would help me to realise my research aims, which had as their focus an exploration of young people’s numeracy events and practices. In particular, young people without a level 2 qualification in mathematics on entry to a college of further education. I devised a set of inclusion criteria to support addressing the aims of the research.

In order to address the limitation of sample size, I attempted to generate multiple data sources which included generating data using collages, clock faces, lists and discussions. Alongside this there were individual interviews with students and observations of students in their vocational and functional skills classes.

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6.6.2 Self-reporting data

The second methodological limitation is the use of self-reporting data in the study. Although adopting a broadly ethnographic approach, I had to rely on students self- reporting their out of college numeracy events as well as trusting students to tell their own stories as accurately as possible in interviews. It was not possible nor practical for me to observe students outside of college. However, I recognise that self-reported data can contain certain potential and actual biases, which may prove limiting to the study. These biases include selective memory, exaggeration or embellishing events or experiences and telescoping. In terms of addressing this issue I recorded the self- reported data shared with participants. This was then followed up in a one to one interview with each participant who spoke to the data. I asked each participant to explain aspects of what they had self- reported, to give them an opportunity to amend the data and to explain or clarify. I also checked the data for incongruence with similar studies. None of the participants reported activities which were out of the ordinary when compared to the pilot study and other studies examined as part of the literature review.

6.6.3 Ideological

I am including as an ideological limitation my own bias in favour of the theoretical perspective used within the study. In adopting a social practice approach to numeracy and mathematics I am aware that I am seeking to find certain phenomena in the data generated. Alongside this, as a longstanding teacher in further education I had a certain sympathy for both student and teacher participants in the study. I made regular visits to the college to undertake the research and developed professional relationships with the participants. In attempting to address the bias, recognising its presence through reflexive practice was an initial activity. Planning for the bias, ensured that pro-formas used in classroom observations, in particular, were used as a basis for recording data that was then used in the analysis and findings. Recognising that there are inherent problems for the practitioner- researcher (Carmichael and Miller 2006) who undertakes research as part of gaining a qualification, rather than the professional who defines their identity as a researcher is useful when reflecting on one’s research limitations. A further limitation of the research is in that in my position as a researcher I was enabled to conduct conversations with students that might not be so easy for their regular tutors to carry out.

Maintaining reflexivity within the research process provides opportunities to focus upon the assumptions, the implicit value-judgements that often affect and direct the ways in which the research operates. I have now learned to adopt a distance between myself and my research, what I term a ‘symbolic detachment’ after Bourdieu (1990), however,

132 this has taken years to learn. This symbolic detachment, encouraged by the doctoral supervision process, provides a space for I as a practitioner-researcher, to loosen the grip on what can be tightly held tenets of faith around the research topic undertaken.

6.7. Contribution to knowledge

This research is cautious in its claims to contribution to knowledge. Firstly, it seeks to foreground the context of the FE and Skills sector and in particular, those students who enter college without a level 2 qualification in mathematics. These students, as shown by the students in the study, had limited ways to challenge or contradict their negative and deficit positionings in not only public policy documents and debates, but also in the private or local discourses constructed by their college tutors. An analysis of the ‘circulating discourses’ has shown how tutors’ ‘sedimented’ (Davies 2008, Rogers and Mosley Wetzel 2014) beliefs constructed students as vulnerable, lacking, needing support and their numeracy practices as without value.

Secondly, while seeking to contribute to the field of post 16 numeracy studies, this study also builds on a tradition of exploring and recognising ‘different ways of knowing’ (Baker et al 1996). This tradition of challenging ways of knowing has been more visible in research within the field of literacy studies than in numeracy studies (Baker 1996, 1998; Coben 2006; Lerman 2000; Street 2005, 2012). I would further suggest that in challenging what is recognised as legitimate and valued mathematical knowledge in the FE college classroom, I want also to ‘know something different’. Firstly, to construe mathematics and numeracy as social, resists the dominant view which sets up mathematics and numeracy as autonomous skills, independent of social context and universally true across space and time. Viewing maths and numeracy as social starts from knowing, understanding and respecting how students engaged with and used their mathematical knowledge in their own lives rather than only knowing how they performed in formal assessments. This partial perspective of formal assessment, limits what a teacher knows about their students’ knowledge and understanding of mathematics. I have called this, knowledge of students’ vernacular numeracy practices. I propose the use of the term, vernacular numeracy practices, to more accurately describe students’ social uses of numeracy that occurred repeatedly across situated events, and in which some mathematical knowledge and content could be discerned. Currently, the terms used to describe these social uses of numeracy include, but are not restricted to: ‘everyday’; ‘real-life’; ‘informal’; ‘folk’; ‘concrete’; ‘contextualised’; ‘home mathematics’; ‘street mathematics’; ‘natural’ and ‘out of school/college’. These terms, some of which are used to describe context, are also used as metaphors of competence and imply a level of proficiency in terms of the mathematics used. Consequently, I suggest, they are

133 always linked to student ability – a limited proficiency view of both the mathematics and the students and reinforce notions of deficit. I further suggest that the semantic connotations of the current terms used to describe numeracy and mathematics used outside formal learning will always point to a lack because of the perception that the terms describe knowledge which is rooted in and connected to student experiences and beliefs and their subjective or tacit knowledge, and will always be viewed in opposition to and therefore inferior to objective knowledge. Some of the terms used to describe this objective knowledge in relation to mathematics, are, ‘school mathematics’; ‘formal mathematics’; ‘abstract mathematics’; ‘decontextualised’ and ‘generalised’. In proposing the use of the term vernacular numeracy practices I challenge this limited proficiency view of the everyday, the informal, by showing that some students can, in sufficiently stimulating interview conditions, identify significant vernacular numeracy practices that embed mathematical knowledge that was seen to have high potential academic curricular value and apparently at a higher level than the mathematics shown through traditional academic assessment. However, to focus on level alone in relation to value would be to misrepresent these vernacular numeracy practices. As well as the dissonance in relation to levels of student performance between vernacular and college numeracy practices, there is also the need to highlight the complexity of the contexts wherein students’ vernacular numeracy practices are enacted. I propose that there is some formality in students’ vernacular numeracy practices that is linked to the context in which they are operating and not just the arithmetic or mathematics they are using. I suggest this complexity is not captured or referenced in any of the current terms used to describe students’ social engagement with mathematics and numeracy.

Expressing students’ vernacular numeracy events in the language of the Adult Numeracy Core Curriculum is helpful in attesting to both the non-linear nature of their vernacular numeracy practices and to challenge the limited proficiency view of the everyday, the informal. It may be helpful to think in terms of the usefulness of using vernacular numeracy practices in terms of a ‘transfer of practices’ rather than a ‘transfer of learning’. This transfer of practices starts with recognising that students are numerate in their own lives.

6.8. Final reflection

The particular focus of this research has been to explore numeracy as a social practice – how students use numeracy in their everyday lives rather than focusing on numeracy as an autonomous skill. The intention was to move away from thinking about the ‘sums on the page’ and explore those other aspects of learning which frame the social uses of numeracy. The research has foregrounded the importance of acknowledging the

134 context, values and beliefs and social and institutional relations, as much as the content aspect of numeracy. And yet, the sums on the page remain important. Students enjoyed having their vernacular numeracy practices discussed in the language of the formal register of the curriculum and standards. Supporting students to reflect on the value of their own knowledge may be a starting point to support greater movement of practices from outside to inside and back again. That there are still so many young people leaving compulsory schooling without feeling confident and competent in using their schooled mathematics suggests there is still some way to go in getting it right. This research is an attempt to highlight the practices that we cannot afford to continue to ignore. Allowing them to remain unacknowledged and under used for learning is, I would suggest, unsustainable, if students’ second chance opportunities for learning are to be successful.

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APPENDICES

Appendix 1. Original research questions - September 2013

1. What resources do students bring to college from everyday life which could support numeracy learning?

2. How do vocational teachers and functional skills teachers view these resources?

3. What evidence is there that these resources are used in either vocational or functional skills classes?

4. How could these resources be mobilised to support numeracy teaching and learning?

158

Appendix 2. Student profiles

Male/ Numeracy initial Student Age Vocational programme assessment female result

Graham* M 17 Introduction to Construction Skills E2

Paul* M 17 Introduction to Construction Skills E2

Kurt* M 18 Introduction to Construction Skills E2

Dom* M 17 Introduction to Construction Skills E2

Daniel M 17 Introduction to Construction Skills E2

Jack* M 18 Introduction to Construction Skills E3

Liam* M 17 Introduction to Construction Skills E2

Mike M 17 Introduction to Construction Skills E2

Andy M 17 Introduction to Hair and Beauty E2

Angela F 17 Introduction to Hair and Beauty E2

Emma* F 17 Introduction to Hair and Beauty E2

Lily* F 17 Introduction to Hair and Beauty E2

Petra F 18 Introduction to Hair and Beauty E2

Sophie* F 17 Introduction to Hair and Beauty E3

Tasha* F 17 Introduction to Hair and Beauty E1

159

Appendix 3. Lily’s clock face

160

Appendix 4. Paul’s collage

161

Appendix 5. Daniel’s list

162

Appendix 6. Dom’s list

163

Appendix 7. Jack’s list

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Appendix 8. Data generation and reduction - numeracy events

Data Data generated Where? What? generation method

Focus group 148 events Student Decision to separate generated from 15 collage/clock- numeracy events from non- students face/list numeracy events.

90 numeracy Table generated Provisional categories used events identified for each student in order to group/organise from 15 student and shared at numeracy events collage/clock- interview with face/lists student

Interview 50 numeracy Individual Final categories decided to with events chosen by student table classify numeracy events students 10 students generated after based on data and literature. interview

Classroom Numeracy events Observation pro Budgeting; area; planning a observations generated from 8 forma party; timetabling classroom appointments; buying a flat; observations annual wage; organising tools in size order; reading gauges for tyre pressure; 180, 90 and 45 degree angles when blow-drying hair; different sizes of brushes and their uses; measuring liquids in salon.

165

Appendix 9. Lily’s numeracy events

Translating Numeracy events underpinning - unit of activity in Summary of student explanation of numeracy into student’s what the event involves language of adult everyday life numeracy core curriculum

Cook family meals Decide what to cook based on recipes from slimming club. Check to see what

to buy. Shop for food. Read food labels to ensure that fat content is below 5%. Try to double up recipe to freeze for another meal.

Days out with Decide where to take niece and nephew nieces based on weather, preference and whether discount vouchers are available, from Groupon or Wowcher, for example. Also just searching for discount online sometimes means you find a code you put in on the site. Using bus – paying fares, calculating cost of entry to soft play area, looking for discount deals.

Go to slimming Weigh self before going to slimming club club to check if any weight lost. Arrive on time for weigh in. Record weight in log book. Work out how much weight lost or put on. Weight is recorded in stones and lbs. Log exercise or time in gym on exercise log book. Rewards are given in club for the more exercise you do. Buy 10p raffle ticket in club. You can win a basket of healthy food each week.

Go out with friends Browse social media to see what friends to the cinema suggest

Work part time in Take bookings for appointments. Date and time salon Take payments in cash from clients. Blow-dry hair. Set timing on equipment. Shop for lunch for staff in salon Ensure containers are kept filled with shampoo and conditioner. Check stock at end of the day and record on stock-taking sheet.

Shop for clothes Decide what I want to buy and then work and make up out how much I can afford to spend.

166

Appendix 10. Emma’s numeracy events

Numeracy events - unit of activity Summary of student Translated into decontextualised Identified explanation of what the language of adult numeracy core from focus event involves curriculum group and interview

Shop on-line Use smartphone to MSS1/E3 Add and subtract sums of for clothes and access Pinterest and money using decimal notation. Know make up Facebook to get how to enter sums of money in a information about fashion calculator. Round sums of money to

sites and fashion ideas. nearest £ and 10p and make Check sites daily and ‘pin’ approximate calculations. ideas. MSS1/L1 Add, subtract, multiply and When shopping online – divide sums of money and record. use favourite sites with Understand place value of whole on-line catalogues. numbers and decimals. Know that, for column addition, decimals should be Get permission from aligned by the decimal point. mother to use her account. Sometimes shop with friend and buy clothes together to share cost of postage.

Read tables of data to FSS/E3 Extract, use and compare check sizes and styles are information from lists, tables, simple available. charts and simple graphs

Use calculator on phone Rounds up or down to estimate to work out rough cost of

items before completing on-line purchase.

Complete online payment Reading numbers with mother’s debit card.

Work-out schedule of Calculations repayments with friend to mother.

Work in hair salon

Shop for food

Go to gym

Budget for day/week

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Appendix 11. Paul’s numeracy events

Numeracy events - Summary of student Translated into decontextualised unit of activity explanation of what language of adult numeracy core the event involves curriculum Identified from focus group and interview

Work part time in Search database of garage parts used in garages for particular part, usually for a service or repair Match part number to catalogue number to make sure it’s the same one. Check prices Check availability Check with manager Ring supplier and order

Follow football team Check score of my progress on mobile team and check phone scores of competition. Look at points table each week to see position of my club in premiership. Try to work out best finish in table near the end of the football season.

Place bets on horses and football matches

Use my motor bike and carry out repairs at home

Cut grass for family and neighbours

Go to the pub and Check what money I clubs with friends or have to spend and drink at home decide whether to go out or stay in. share cost of beer between mates, from off licence.

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Appendix 12. Daniel’s numeracy events

Numeracy events - Summary of student Translating underpinning unit of activity in explanation of what the numeracy into language of student’s everyday event involves adult numeracy core life curriculum

Going out on his Reading tables MSS1/L1 Read and interpret scooter. distance in everyday situations.

Buys petrol. Using ATM to get cash Read, estimate and measure and compare length using non- Read gauge on petrol standard units. pump. Make sure not to go over my limit of what to spend on petrol.

Places a bet in the Calculating possible betting shop winnings or losses Working out odds

Free running Estimating length and MSS1/L1 Read, estimate, height; comparing lengths measure and compare distance

Working in cafe Using a till FSS/E3 Begin to develop own strategies for solving simple problems.

Taking payments Understand practical problems in familiar contexts and situations.

Select mathematics to obtain answers to simple given practical problems that are clear and routine.

Giving correct change Solve practical problems involving multiplication and division by 2, 3,

4, 5 and 10

Checking stock Add and subtract using three-digit numbers

Shopping for clothes Budgeting Using ATM Counting number of reps on exercises

Manages own bank Use ATM account

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Appendix 13. Students’ vernacular numeracy practices using social practice analytic framework

Components of Area of activity - What can we say about vernacular numeracy managing money reported vernacular numeracy practices practice

Content Estimating; add; subtract; Sometimes right answer not as multiply and divide. important. Rounding up and down Spikey profile of levels. Time and date formats

Context Shopping online & face to Certain; confident; participation; can face; mobile phone; explain what was going on; use selling Xbox for trainers; technology all the time. working in café; handling cash in work; buying lunch for stylists.

Values, beliefs and Need to shop and buy Framed within broader cultural values emotions clothes that are value for money. Problem solving Need to pay back money Student driven/in control borrowed – Emma. Seen as common sense – not maths

Social and institutional Entering into contract Want qualifications. relations with online seller.

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Appendix 14. Students’ college numeracy practices using social practice analytic framework

Components of college College numeracy category numeracy practices practices

Content Mean; median; mode; place value; angle of brushes; ratio of sterilising solution; 3,4,5 rule. Ordering spanners Calculating time Budgeting

Context

Values, beliefs and emotions

Social and institutional relations

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Appendix 15. Comparing and contrasting students’ vernacular numeracy practices and college numeracy practices using social practice analytic framework

Components of Vernacular College numeracy Category vernacular numeracy numeracy practices practices practices

Content

Context

Values, beliefs and emotions

Social and institutional relations

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Appendix 16. Lily’s timetabling task sheet

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Appendix 17. Student collages (content) transcribed into list of activities (step 1) and used in interviews to stimulate discussion

Student Activity – numeracy event

Emma Work part time in salon

Shopping on-line

Shopping face to face

Cooking meals

Going out with friends to the cinema

Using mobile phone

Using social media and Pinterest

Using an ATM

Budgeting (daily and weekly)

Travelling on the bus

Using catalogues virtual and real to check pricing of clothes; styles and sizes available

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Appendix 18: Prompts used with students for semi-structured 1:1 interviews and how they relate to analytical framework

Question/prompt to stimulate Street Baker and Tomlin’s Document discussion around numeracy analytic framework for events numeracy as social practice

1. If I asked you to choose which Content Collage/clock- activities on your list are the most face/list important for you, which ones would they be? Which of the activities on your clock-face/list are most important to you.

2. If we look at this one in particular, Context tell me about where it takes

place/happens/ what you’re doing during the activity?

3. Is there anybody else involved? Context Values and beliefs

4. Do you ask for help?

5. Who would you ask to help you if you got stuck?

6. Are you able to tell me how you do Content that? Talk me through how you do that. (any calculations/ number activity

7. Do you ever need to write things down or do you do it in your head?

8. Why do you need to carry out the Context/Values and beliefs activity?

9. Why is it important to you? Values and beliefs

10. What would happen if you didn’t Social and institutional relations complete the task?

11. In the focus group you chatted Values and beliefs about what it was like learning Social and institutional relations maths at school, before you came to college. Could you tell me what Feelings associated with your memories of that are? How it previous experiences of learning made you feel? maths

12. Is college different to school? In Student perceptions of whether what way? experience in college is ‘more of the same’

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Appendix 19. Students’ numeracy events and practices in college

Class Topic Numeracy Links made to out content of college numeracy

Hair and Beauty

Construction

Functional skills – maths

Functional skills - combined

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Appendix 20. Headings for use in observation of teaching in vocational setting and functional skills class to inform general notes taken during observation

 Are explicit links made between numeracy and topic being taught in session?

If yes, describe how does this take place?

 Are connections made between numeracy related topics in the session?

If yes, describe how is this done?

 Are connections made between learners’ prior knowledge and learning and topics developed in session?

If yes, describe how is this done?

 Was there discussion between learners and between teacher and learners around numeracy?

If yes, how was this facilitated?

 Outline of underpinning numeracy skills mapped to ANCC and FSS – outlined separately in detail

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Appendix 21. Pen portrait used in focus group with teachers

Paul’s story

Paul loves cars. He has a part-time job in a local garage where he works from 9.00 until 3.30 each Saturday. The garage is owned by a friend of his dad’s who ‘gave him a chance’ because he knew Paul and didn’t mind that he wasn’t ‘good at English and maths’. Paul didn’t like working on reception in the garage as it was very busy and the phone would ring when customers were there and he wasn’t sure what to do. Now he works in the back and helps with checking the stock as well as helping the lads with the cars. He is sometimes asked to clean the cars. He likes it when he’s asked to phone about ordering parts. Sometimes Paul looks up parts on the computer to see how much they cost. He has also compared the price of parts for a motorbike he has at home. He’s been late a couple of times and has been given a warning by the boss. He’s usually late because he’s been out on a Friday night and had too much to drink.

Sometimes Paul cuts the grass for his neighbour who can’t manage to do this. The neighbour pays Paul for doing this. Paul is thinking of asking other people in the area if they need their grass cutting too. He’s not sure what to charge. His neighbour gives him £5, but his garden is not very big. He thinks it usually takes about 30 – 40 minutes to cut the neighbour’s lawn, but he’s not sure. He’s going to time it, so he can work out how much to charge.

Paul hated school and left when he was about fourteen. His parents tried to get him to go back but he wouldn’t and for a bit he was homeless. Paul says he couldn’t get on with school and didn’t like the teachers. He says he was a ‘naughty boy’ at school. One teacher told him he was dyslexic but he didn’t know what that meant. Paul has started to go to a gym. Because he’s small in build he wants to work out. He likes it that some of the other lads in his class also go to the gym. They have competitions to see who can do the most reps and lift the heaviest weights. Dean beats him every time. Paul wants to take up boxing because he thinks this would be good to build up his muscles.

Paul said that coming to college has helped him. He’s made friends with other lads in the class, even though they fight a bit.

He knows he’s not good at maths but he’s going to take his functional skills exam in January

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Appendix 22. Questions used in the semi-structured interviews with teachers

1. How have you identified the underpinning numeracy skills and knowledge your students need to succeed on their course? (vocational teacher)

2. How would you describe your learners’ numeracy skills? (all teachers)

3. How are your learner’ numeracy skills assessed at entry to college? (all teachers)

4. Would you have a sense of how much numeracy activity your learners engage in outside college? (all teachers)

5. How confident do you feel in developing your learners’ numeracy skills? (vocational tutors and embedded functional skills teacher)

6. How do you engage and motivate your learners in learning numeracy as part of their vocational/functional skills curriculum? (vocational and fs tutors)

7. How did you get on with maths at school? (all teachers)

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Appendix 23. Numeracy event 1: timetabling appointments in college

The Hair and Beauty tutor, G. asked me, as I approached the classroom, to help Lily during the class. ‘I know she won’t cope with what we’re going to do, it’s about time’. G, as personal tutor to the Hair and Beauty students, was present at one of the information sharing meetings, when Lily told the group she had difficulty telling the time using an analogue clock. Lily also shared this again with the focus group when she compiled her ‘clock-face’.

It took five minutes or so for all the students to arrive. There were seven students in the observed session, including Lily. Two students, G explained, were from another group but joining in the session. He handed out a work-sheet and a pro-forma (Appendix 16, page174) which represented a daily appointment sheet in a salon. G explained the task to the students.

G: “OK, guys. We’re looking at timetabling in the salon today and I want you to work together to work out the timings for the list of appointments, yeah, and put them in the appointments sheet. You’ll need to look at who wants what in the appointments list and what time they can come in at, and calculate what time to book each one in. You need to work out if the stylist has enough time for all these people. Has everybody got the sheets? OK. I need you to read the list of appointments that’s written out on the work- sheet and I want you to decide what time you can slot them in on the appointment one. Make sure you don’t forget to put in your 45 minute lunch and the stock taking time. You’ve got 20 minutes to complete and then we’ll come back together.”

In the class Lily sat slightly apart from the group, looking quite tense. I sat near her and after approximately five minutes where she just sat and looked at the work-sheet, I asked her how things were going. I knew from the focus group that Lily worked part-time in a salon.

R: “How are you getting on, Lily?”

L: “I really hate this stuff.”

R: “Which bit is…..”

Lily: “I just hate the exercises on time. I can’t do it.”

R: “OK. But how do you sort out your appointments in your salon, the bookings?”

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L: “It’s just not like this. At ours, a lot of the appointments work in half hour slots, but on the appointment book at reception, the slots are in 15 minutes. Four of the slots are an hour for the stylist. If a client is in for a wash and blow dry, it’s nearly always half an hour, I know that’s two slots. If it’s a cut and blow dry for short hair it’s about an hour, so that’s four of the slots. Then we have regulars and we get to know how long their hair takes. Some are forty five minutes, so I ‘cross-out’ three slots. Then for long hair, about your length (pointing to my hair) it’s about an hour for a wash and blow dry, isn’t it? Also, I can ask the stylist if I’m not sure how long an appointment will take. Colour is different.”

R: “Is this sheet like the pages in your appointment book?”

L: “It’s in 15 minute slots, isn’t it? I didn’t see that.”

R: “Do you have any thoughts on how you might decide where to book them in?”

Sophie, another student, makes a comment to the students sitting near her, which Lily hears.

S: “This would never happen in real life. You wouldn’t get all these appointments at once and you need to make them overlap. None of our stylists ever take 45 minutes for lunch on a busy day. It’s just not right.”

Lily looks at the task sheet and appointment sheet again and says to me:

L: “There aren’t enough separate slots, are there? You just overlap the appointments. Why didn’t he say we could do that from the beginning? It didn’t say that on the sheet. Colour is different. You can get the ‘cut and colour’ started, then while she’s waiting for her colour to take, the stylist just starts on someone else then comes back to the colour and does the cut. That way everyone gets sorted. It’s what happens in the salon all the time. But I’d just ask the stylist when the customer rings up or is in the salon, if she can fit in the client and we do. If it’s a regular, the stylist always tries to fit them in.”

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Appendix 24. Numeracy event 2: online shopping at home

Emma and her friend buy clothes from a catalogue because they can pay weekly and choose the number of weeks for payment. Emma does not have access to credit and so she uses her mother’s account and Emma, pays her back. Emma’s friend, in turn pays her.

“There’s this catalogue on line at Littlewoods and it’s really good. It’s mam’s account but we always share. She knows I’m good at paying her back on time. I never miss because she would just stop. (Friend) pays me and she always does as well. She’s the only one I do it with. I always check first with mam, that it’s ok.”

Emma explains how she approaches the task and does her calculations for shopping.

Emma went on to explain,

“But with (friend) wanting to buy as well, we could get them all now and save a bit too.”

“You have to search for what you want. We always start with the sale stuff. If I have time I search one time and put them in a basket and then come back when you’re ready to buy. It can take time because I’m on my phone and the screen is small. If I keep things in the basket then the price is there and I can keep track. I try to add up as I go along, not exactly, but the rough price. You know, like £6.99, then I add £7, or even £5.50 I always add £6, like that. I always cost things more, so I know I can really afford them and I won’t be disappointed then. I always check the prices and work out how much it will really cost.”

Emma and her friend used tables of data to work out sizes, colours, catalogue number and price. Once they know roughly what they want to buy, they work out how to pay by instalments. Emma explained:

“You need to choose the size and colour of what you want from a table that you click on. I know that a size 12 from ‘Very’ is too big for me, so I always get a 10 from there. Sizes are not the same everywhere and this catalogue has more sites together. I write down what each thing costs and add it all up on the calculator, before I put them into my basket. I do it twice to make sure I get the same answer. For the holiday things, our cost altogether came to £89. It was a little under the £89 but not much, so we rounded it up for working out the

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payments. It’s just easier. Mine was £49 and (friend’s) £40. We got free delivery and we spread the cost over 20 weeks. There’s no interest for 20 weeks. They take the money out on the same date every month.”

When asked how she calculated how much to pay her mother back, Emma explained,

“We work out how much I have to pay mam.”

Emma keys in £89 into her mobile phone calculator and divides it by 20. She shows me £4.45. When I ask her if she pays her mother weekly, she says,

Emma multiplies the £4.45 by 4 and gets £17.80.

“Mam knows that I will pay her back from my own money from my job at the salon. It works out ok for us. She lets me pay her each week or month instead of giving it in one amount. I know I’m lucky but I don’t like shopping online with mam as she takes too long to look at sites and I like to move between a few sites at the same time. I can work out the cost of an order as I’m going along, because it’s easy to get carried away and before you know, there’s too many things in your basket, and then you have to add postage and packaging too.”

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