Kees Hoogland

Beth Kelly

Javier Díez-Palomar

Introduction to TWG07 Adult Mathematics Education

Keywords: Adult Mathematics Education, Numeracy, PIAAC, Language, Agency

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Introduction to TWG07 Adult Mathematics Education

Kees Hoogland 1 , Beth Kelly 2 and Javier Díez-Palomar 3 1 HU University of Applied Sciences Utrecht, The Netherlands; kees.hoogland@hu.nl 2 University College London, United Kingdom; beth.kelly@ucl.ac.uk 3 University of Barcelona, Spain; jdiezpalomar@ub.edu

Introduction

Adult mathematics education is a fast growing tree in the mathematics education forest. Many countries have in recent decades adopted policies on lifelong learning and, in most cases, mathematics and numeracy are included in these policies (see for instance the Lifelong Learning Platform, lllplatform.eu). This is a consequence of the rising awareness that the current society is rapidly changing due to technological, digital and environmental developments, and that citizens need a broad repertoire of mathematical knowledge, skills, and confidence to deal with those changes, both in work situations as in daily life. This idea is not completely new. An increasing interest in the relevance of mathematics needed for a well-functioning citizen started in the late 1950s in the UK with the Crowther report (DES Department of Education, and Science, 1959) but really blossomed two decades later with the Cockcroft report (DES Department of Education and Science, 1982), the foundation of Adults Learning Mathematicsan international research forum (ALM) established in 1993 with a strong European input (see e.g., Safford-Ramus, Keogh, O'Donoghue & Maguire, 2018), and the start of the Adults Numeracy Network (ANN) established in 1994 with a main focus on the USA. Numeracy was at that time defined in the Crowther report (1959) as scientific understanding and thinking quantitatively and in the Cockroft report (1982) as the capacity to understand and use basic principles of mathematics and science presented in mathematical terms.

With the rapid growth of adult education in the 1970s (Milana & Nesbit, 2015) we witnessed a rise in formal mathematics education for adults, as a result of growing opportunities for adults to take another chance in getting mathematics certificates for further study or new careers. During this time, a large part of adult numeracy and mathematics education took place in informal and nonformal education environments, for instance in community centres or in the workplace, or was hidden in vocational or literacy courses for adults, which made it hard to get a complete overview of the field. Therefore, the mathematics referred to in AME has many appearances. It ranges from formal mathematics for advanced technological careers, to typical school mathematics content supporting the attainment of formal certificates and diplomas, and to the multifaceted concepts of mathematical literacy and numeracy, which deal with the use of numbers, patterns and structures in daily activities. This last appearance can also be described as Numeracy as a social practice [START_REF] Yasukawa | Numeracy as social practice : global and local perspectives[END_REF]. This makes the concept of numeracy relevant to large segments of the populations in many countries; Indeed, the results of the Programme for the International Assessment of Adult Competencies (PIAAC), discussed in the next section, found that many adults have their optimal functioning in daily life hampered by numeracy problems.

In this paper we show the subthemes raised by the participants in the TWG on AME. We consider this TWG as new platform for scientific discussion to push AME towards a more recognized position within the scientific international community. We present a brief summary of each of the contributions presented in the TWG during the conference. Finally, we conclude with some remarks on new avenues and challenges for the research in AME in the forthcoming years.

Emerging themes in Adult Mathematics Education

In TWG07 on AME the breadth and depth of the emerging field of AME is mirrored. Contributions come from various backgrounds, cultures, and experiences. In the first working group discussion the various themes of the contributions were made explicit and put into categories.

These are the categories covered by the contributions of the participants in this TWG:  The definition, scope, and assessment of numeracy;  The role of language and dialogue;  Affect, including motivation;  Special groups and vulnerable learners;  Teaching and learning activities, including vocational education;  The role of power structures, inclusion, agency and self-efficacy. These themes more than anything else make clear that AME is a multi-layered phenomenon and much more than just mathematics education for a special target group. Mathematics education for adults intertwines the learning of mathematics with language and dialogue, affect, and dealing with learners' reality, but also with societal power structures and sociological concepts like equity, inclusion, and agency.

Regarding the definition, scope and assessment of numeracy "Adults' numeracy and literacy practices matter" (p. 75) is the opening line of the chapter by [START_REF] Coben | Scoping the Development of a Measure of Adults' Numeracy (and Literacy) Practices[END_REF] in the ICME13-monograph series title Contemporary Research in Adult and Lifelong Learning of Mathematics -International perspectives (Safford-Ramus, Maaß, & Süss-Stepancik, 2018). This brief sentence summarizes in the shortest possible way the importance of the theme numeracy in the field of AME.

The results of the last PIAAC survey show that in all but one participating country, at least 10% of the adults are proficient below level 1 of the 6-point scale in literacy or numeracy (OECD, 2013;PIAAC Numeracy Expert Group, 2009). In other words, significant numbers of adults do not possess the knowledge, skills and confidence to succeed in today's society regarding quantitative matters. These results on numeracy give rise to serious concern for the economic development of Europe. This is an even more pressing issue since the amount of numerical data that needs to be interpreted and used is rapidly rising and all pervasive due to technological developments and the prevalence of (big) data. To gain a better insight in these trends a second cycle of PIAAC will be starting in 2021, which is now in preparation [START_REF] Tout | Review of the PIAAC numeracy assessment framework: Final report[END_REF]. David Kaye, chair of ALM, made a contribution to the discussion in the group on the development of the definition and historical development of the concept of numeracy, based on his earlier publications in an ICME13 monograph [START_REF] Kaye | Defining Adult and Numeracy: An Academic and Political Investigation[END_REF]. In the Crowther report (1959), numeracy was closely related to understanding science and developing technologies in society. In the next twenty years, the meaning changed to basic operations, whereby the association with science was no longer present and the level of mathematical understanding to which numeracy refers became much lower and not much more than "able to perform basic arithmetic operations". The persistence of the latter definition makes it hard to develop, disseminate, and implement a concept of numeracy which better equates to an ability to cope confidently with the mathematical demands of adult life. For that purpose this latter definition is much too restricted because it refers only to more mechanical and lower-order skills like performing arithmetic operations and not 'mathematizing', which is the skill of translating an encountered problem into an adequate operation so that the mathematics can be used with confidence to cope with practical everyday situations. What is so significant about these shifting definitions is that the debate about what numeracy might include was already twenty years old in 1982, and that the tension between numeracy being little more than arithmetic versus being a significant part of everyone's social practice is still being repeated in debates on numeracy policy development. The continuing tension can most likely be explained by the fact that it is not just an academic and educational debate, but rather a political one. In many countries we see a reoccurring 'back-to-the-basics'-movement as a simple answer to the complex issue of how to prepare children and adults to deal with situations in everyday life which involve numbers, patterns, and structures.

Two contributions especially address this issue too. Diez-Palomar, Hoogland, and Geiger are members of the PIAAC Numeracy Expert Group, which is responsible for the development of an assessment framework and the items in the next PIAAC international comparative assessment, which is happening from 2021 onwards. They performed a limited literature review to explore which concepts from mathematics education theories could enrich the numeracy framework to make it more appropriate to adults functioning adequately in our number-drenched society [START_REF] Tout | Review of the PIAAC numeracy assessment framework: Final report[END_REF]. The most relevant concepts taken into account are number sense, big ideas in mathematics, and embeddedness and authenticity. Common in these concepts is that they all acknowledge the richness and multifaceted aspects of adults' cognitive processes in dealing with numeracy situations. Furthermore, Hoogland, Auer, O'Meara, Diez-Palomar and Van Groenestijn report in their contribution on the first steps towards creating a Common European Numeracy Framework (CENF) for adults. These steps are made in an Erasmus+ project with the same name, which started in 2018, and is aims to broaden the perspective on numeracy, foremost by considering numeracy as a social practice, focusing on numerate behaviour and numeracy practices. For a much more sophisticated discussion on seeing numeracy as a social practice, we further refer to the work of [START_REF] Coben | Scoping the Development of a Measure of Adults' Numeracy (and Literacy) Practices[END_REF] and [START_REF] Yasukawa | Numeracy as social practice : global and local perspectives[END_REF].

Regarding the role of language and dialogue.

The importance of the role of language and dialogue in AME was emphasised by no less than three contributions by Griffiths, Wessel, and Diez-Palomar & Anagnostopoulou respectively. Griffiths report on his investigations into discussions that took place during and following a particular small group activity in which a scene is read aloud and a mathematical task is undertaken. In his research, the concept of 'thinking as communication' is utilised and employs an overarching framework for analysis. To understand what happens when the 'reading aloud' activity is undertaken, it is important to investigate what happens when learners undertake related activities. Griffiths describes the key elements to the research with some initial analysis drawing on data from an exploratory phase along with observations from the main study.

Wessel gives an overview of an Erasmus+ project that aims to empirically identify the potentials and challenges of language-responsive teaching designs for low-achieving students in vocational contexts. Using topic-specific designs the research delivers first insights into developing and experimenting with two teaching units (percentages and proportional reasoning) and summarizes the questions that come up when adapting design principles for language-responsive mathematics to the teaching and learning in vocational education. The empirical investigations are carried out with young adult pre-apprentices in lower level vocational education in Germany in the technical sector. These young adults often struggle with mathematics and have only limited proficiency in the language of instruction.

Diez-Palomar and Anagnostopoulou discuss the effectiveness of dialogic learning as an adults' learning theory. Their paper specifically focuses on how so-called Mathematics and Physics Dialogic Gatherings can enhance the critical thinking of adult learners and hence develop their learning in both topics. Two studies were considered where the participants engaged in dialogic reading activity reading classics in mathematics or physics. The former study took place in Barcelona, Spain and involved adult women of above 40-years-old with low literacy skills engaging in mathematics. The latter took place in Kendal, UK and involved two groups of adults. Although the samples were extremely diverse, the results indicate that using classical readings can significantly improve critical thinking and dialogic talk in all groups and provide the potential to create further learning opportunities.

Regarding affect, including motivation

In various contributions, attention is given to the role of affect as an important factor influencing the outcomes of AME in the classroom. It acknowledges that learning mathematics involves not only cognitive processes, but also, to a substantial level, an array of psychological and affective factors.

Kelly undertook extensive doctoral research on the role of motivation in AME. Her research is based on interviews with adults about their motivation to study mathematics in the workplace, through classes organised and funded by trade unions in the UK. The findings point to motivation as a function of individual's emotions and cognition that are influenced by social experiences. The research identifies a difference between the initial motivation to re-engage with learning and that required to continue, or persist. Continuing learning relies on a socially and emotionally supportive learning environment which adults identify as 'different' from previous learning experiences. The term 'Affective Mathematical Journey' is developed to describe the positive emotional changes experienced by adults, which helps them overcome negative memories. The adults' resultant increase in confidence and motivation encourages them to successfully develop and use their mathematical skills both inside and outside the classroom.

Regarding special groups and vulnerable learners

Engaging in mathematics education as well as trying to improve the management of various numeracy situations, such as finances, is a big step for many adults, especially when their past experiences with education are negative.

Redmer used PIAAC data to get a better insight into the numerate behaviour of older people. Demographic changes render basic numeracy skills increasingly important in older age. Among them, skills in financial matters are increasingly relevant. The skills measured in PIAAC and Competencies in Later Life (CiLL) represent comprehensive cognitive abilities that can be seen as the basis for successful participation in social and economic life. In a secondary analysis, Redmer shows that financial practices continue to be important for older people, although their numeracy skills proficiency may be lower than those of younger people. The analysis clearly indicated a gender difference in dealing with financial matters: men handle bills and bank statements, whereas women manage the household, although there was some discussion about whether this distinction would continue in following generations.

Byrne did research on a very special group. Her exploratory study is part of a larger study on assessment and teaching in mathematics classrooms in Irish prison education centres. Her purpose is to enhance knowledge in this field, to enhance practice and to enable better learning and teaching through research. A so-called Mathematics Teaching Triad is used as a theoretical construct for the study. This triad looks at management of learning, sensitivity to students, and mathematical challenges. When finished, the research project will present data derived from a wider investigation, including data from teachers in the secure sector across Ireland and data from prisoners attending mathematics classes. In this paper Byrne analyses the data gathered from a survey on small sample of teachers, covering the time available for mathematics classes, class composition, teacher satisfaction and teacher's identities through metaphor.

Regarding teaching and learning activities

Keogh reports on developing and validating an instrument to let people recognise their own numerate behaviour. His reasoning is that adults know more about mathematics than they think and use it more often than they realise. Despite their apparent self-perception of being a 'non-maths' person, they behave in mathematics-informed ways, which are dismissed, routinely, as common sense or something other than mathematics. The doctoral research found that given the opportunity to reflect on their own behaviour at work, through a mathematics-sensitive lens, people readily recognise their numerate behaviour even if they talk about it in different ways. This work introduces a set of tools and a methodology to harvest real tangible benefits for the learner. The tools explore the locus of a person's expertise in increasing granularity, building a platform from on which to capture a learner's mathematics knowledge, skills and competence, to help revise possible mistaken self-perceptions and to inculcate confidence in his/her learning of mathematics.

Hoogland, Heinsman and Drijvers report on conducting a literature review on how numeracy is used in vocational education and whether there are effective practices identified in peer-reviewed literature of the last five years. The rationale for this study is that numeracy and mathematics education in vocational education is under pressure to keep up with the rapid changes in the workplace due to developments in workplace mathematics and the ubiquitous availability of technological tools. Vocational education is a large component of education for 12-to 20-years-olds in the Netherlands and the numeracy and mathematics curriculum is on the brink of a reform. Preliminary results show that in most of the articles the concept of numeracy was not clearly explained, however when specified the PIAAC definition is used. In the search for articles on numeracy education practices which also reported on effects of those practices, only four such articles could be found out of more than 600 publications on numeracy and vocational education. Discussion in the group suggests a wider search of publications which includes practices involving problem solving, motivation, and math-anxiety.

Regarding the role of power structures, inclusion, agency and self-efficacy

Last but certainly not least is the notion of the use of numeracy in society, including the importance of being numerate, equity in access to relevant numeracy education and the perception of gender issues around mathematics and numeracy, all of which are never value-free. In her research Heilmann uses adults' numeracy skills in the PIAAC data to analyse the issue of power. She states that theoretical discussions increasingly view numeracy as embedded in power relations, and if this is the case, these power relations should be visible in the results. She gives an overview of the underlying theories, focusing on three elements of numeracy skill proficiency: the distribution of skills in society, the process of defining a hegemonic view of numeracy skills and finally, the value of numeracy skills in the context of power relations. The value of numeracy skills is analysed using the example of gender relations in the labour market. The analyses explores how gendered hegemony might influence the way numeracy skills matter in terms of monthly wages and the probability of getting into leadership positions.

Forwarding the field

Although there is a broad acknowledgment that an array of psychological and sociological factors are important in (the results of) adult education, there is not yet a well-researched set of examples how in practice this can be taken into account in a more systematic and effective way. The practice of adult numeracy education is still a plethora of different content descriptions and goals that vary from very back-to-the-basics to very sophisticated higher-order skills. Reported teaching practices and lesson materials for adults are quite often not very different from those used in primary schools. Research-validated practices explicitly addressing the psychological and sociological issues are still very rare. The twin goals of trying to establish a firm definition of numeracy which acknowledges the multifaceted nature of adult's mathematical practices while identifying consistent teaching approaches should be of concern to the global AME community.

Currently in Europe there are some developments which are promising to come to a more common definition of numeracy and identify effective teaching approaches. The ALM organisation is strongly anchored in Europe and functions through its yearly conference in July as a community of practice for researchers in the field of AME in Europe and beyond. The European Basic Skills Network (EBSN) in combination with the EPALE-website of the European Union provides a platform where practitioners, policy makers and researchers can share their ideas, thoughts and results. Furthermore, in 2018 an Erasmus+ project started, which originated in The Netherlands, Austria, Ireland, and Spain, and is aiming to involve all European countries in an attempt learn from each other experiences to find some common ground for terminology, perspectives, development and practice, and policy making .

In the future, it could be helpful if national and regional level policy documents explicitly define and openly discuss the chosen definitions of numeracy in relation to the European or world-wide used definitions, like the one by OECD in the PIAAC programme. Especially when societies become globalised, as Safford-Ramus, Misra and Maguire (2016) advocate, numeracy must remain a dynamic concept that recognises and responds to changes in society.

Conclusion

AME plays a vital role in empowering individuals to fulfil their full potential, enrichening their repertoire to cope with situations and problems in daily life and the demands made by a globalised world, increasing reliant on technology. In the second cycle of PIAAC, which will be conducted from 2021 onwards, a large body of data will become available, which will for the first time provide a longitudinal comparison with the results of the first cycle of PIAAC. To some extent, this will also point to trends in the development of give aspects of the field of AME Despite the relevance of numeracy in our society, the role that AME plays in the field of research into mathematics education is still rather small. There are international research associations in AME, but the main educational mathematics conferences that are referents in the field do not usually include AME as working group. We have over seventy years of "numeracy" as research topic devoted in national and international reports and studies. In the last decade, results of important large-scale surveys such as PIAAC have becoming available. However, there are still numerous gaps in terms of research understanding how adults learn and use mathematics, develop their quantitative understanding of everyday facts or situations, understand particular mathematical objects (proportions, functions, big numbers, shapes, etc.), and their properties, relationships and representations. The themes identified in this CERME working group layout a possible research agenda. The discussion will continue on the scope and definition of numeracy. Furthermore, sociological aspects will remain in the foreground of discussions on AME, such as its relationship with societal power structures and promoting individual inclusion, agency and self-efficacy for special groups and vulnerable learners. Another important aspect in AME is that adults bring their own life experiences to the learning environment, which implies that teaching and learning is influenced by a broad range of factors including the recognition that motivation to learn is developed through social and emotional experiences, while teaching activities need to be sensitive to the role of language and dialogue as well as relevant to the context of learners' lives.


Catherine Byrne
email: catherine.byrne8@mydit.ie

Brian Bowe
email: brian.bowe@dit.ie

Michael Carr
email: michael.carr@dit.ie

Doing the time in the mathematics class 1

Keywords: Assessment, mathematics, prison time, metaphor

This exploratory study is part of a larger study on assessment and teaching in mathematics classrooms in Irish prison education centers. Our purpose is to enhance knowledge in this field, to enhance practice and to enable better learning and teaching through research. We are using the Mathematics Teaching Triad (TT) theoretical construct for the study. This triad looks at Management of Learning (ML), Sensitivity to Students (SS) and Mathematical Challenges (MC). When finished, this research project will present data derived from a wider investigation, including data from teachers nationwide and data from prisoners attending mathematics classes. This paper analyses the data gathered from a survey on small sample of teachers, covering the time available for mathematics classes, class composition, teacher satisfaction and teacher's identities through metaphor.

Introduction and context

Irish Prison Education

The Council of Europe document Education in Prison (1990) states that education of prisoners must "be brought as close as possible to the best Adult Education in society outside" in philosophy and content and that it must "link prisoners to the outside community" (p.14).

Prison education in Ireland is a partnership between the Irish Prison Service and the Education Training Boards (Irish Prison Service 2016). The Joint Strategy Statement states that its aims are to assist persons in custody to cope with their sentences, to prepare for life after release, to develop an appetite and capacity for lifelong learning and to achieve their full potential as learners. It also aims to promote the importance of prison education. (IPS, 2016, p.6) So prisoner learners can access similar education and accreditation to learners in adult education elsewhere, through accredited (leading to state certification) and non-accredited courses (not leading to state certification). An adult education approach is followed in prison education which looks at the whole person rather than deficit models, which see the student's weaknesses or mainly as an offender [START_REF] Warner | Against the narrowing of perspectives: How do we see learning, prisons and prisoners[END_REF].

Teaching in Prisons

Students entering prison education may have low levels of education (O'Mahoney, 1977;[START_REF] Morgan | The Prison Adult Literacy Survey: Results and implications[END_REF] and a history of negative or disrupted educational experiences [START_REF] Muth | Radical conversations: Part one social-constructivist methods in the ABE classroom[END_REF].

People are motivated to start education in prison for many reasons, including personal development, family and work preparation (Costelloe, 2003). School in prison represents freedom, an escape, a distraction and stops them from sinking [START_REF] Behan | Learning to escape: Prison education, rehabilitation and the potential for transformation[END_REF]. They come to have a laugh, make the most of their time, and to learn how to learn [START_REF] Carrigan | Never really had a good education you know, until we came in here': educational life histories of young adult male prisoner learners[END_REF]. In prison education they feel trusted [START_REF] Behan | Learning to escape: Prison education, rehabilitation and the potential for transformation[END_REF], feel normal and their true selves [START_REF] Carrigan | Never really had a good education you know, until we came in here': educational life histories of young adult male prisoner learners[END_REF]. The positive feelings of the students seem to have an impact on the teachers. Prison teachers interviewed by [START_REF] Michals | Prison teachers and their students: A circle of satisfaction and gain[END_REF] were positive about their work, felt satisfaction and felt that they had learnt from their students. [START_REF] Muth | Radical conversations: Part one social-constructivist methods in the ABE classroom[END_REF] and [START_REF] Byrne | Maths in prison[END_REF] state that teachers in prison have noted the transformative nature of prison education, and the growth of their students. This may lead to teachers feeling empowered in the classroom, as [START_REF] Messemer | The level of influence that six dimensions have on teacher decision-making in the correctional education classroom[END_REF] states that teachers interviewed in US correctional education centers felt they had a strong level of influence to make the necessary instructional decisions in their classrooms. We decided to investigate if there were other reasons for mathematics teacher feeling satisfied in their work. As classes in prison are generally longer than in schools outside, we adapted a study on this topic with mathematics teachers in secondary schools. This found that longer more frequent classes increased teacher satisfaction, which led to increased innovation in methodology and more teaching for understanding [START_REF] O'meara | Time allocated to mathematics in post-primary schools in Ireland: are we in double trouble[END_REF].

Why I am doing research?

Our study is based on the Jaworski Teaching Triad [START_REF] Jaworski | Mathematics teaching: What is it[END_REF] which has three components, the Management of Learning (ML), Sensitivity to Students (SS) and Mathematical Challenges (MC). We chose this because it offers a framework for all the proposed data from this study, both qualitative and quantitative. ML refers to the role of the teacher in the classroom learning environment. In this context this refers to groupings, timetable, resources, interactions. Sensitivity to Students (SS) describes the teacher's knowledge of the learners, attention to their needs, affective, cognitive and social. In this context it describes the engagement with students, from the point of entry. Mathematical Challenges (MC) describe the challenges offered to the learners to help them develop mathematical thinking through tasks, questions and reflections, in accredited and non-accredited tasks.

The domains are interlinked and interdependent. A balance between SS and MC is needed to enable teaching to take place. In prison education there could be a risk of too much SS, due to vulnerability of learners, and too little MC which could mean strong relationships in class but little mathematical progression. Too much MC and too little SS could cause stress to an already traumatized and stressed group. Harmony leads to achievement when there is a balance between challenge, support and stimulation. The results from the teachers (see below) show their SS, ML MC. The prison education center is where is two cultures collide, the culture of the classroom and the prison [START_REF] Wright | Going to teach in prisons: Culture shock[END_REF]. There has been very little research on the everyday experience of prison education internationally, so we decided to gather data on the practices and rationale. We built this study on previous studies of mathematics teachers, in Ireland, Finland and Switzerland. O'Meara & Prendergast (2018) investigated the length of mathematics classes and the impact on teacher satisfaction. [START_REF] Oksanen | Finnish mathematics teachers' beliefs about their profession expressed through metaphors[END_REF] used metaphor to explore mathematics teachers' beliefs about their profession and Beeli-Zimmermann (2014) explored adult numeracy teachers' beliefs.

We distributed the survey by email after a discussion with each teacher. During the call, the rationale for the study was given and the invitation was issued to take part. All who were contacted responded. We will extend the study nationwide in future.

The questions are divided into two types; quantitative information on the class structure and allocation and qualitative questions on the teachers' identity using metaphor (Table 1 The O'Meara & Prendergast (2018) study is interesting because it showed that mathematics teachers in secondary schools were more satisfied with longer and more frequent mathematics classes. Classes in prison are longer so we wanted to investigate the impact of the longer class time on the satisfaction of prison mathematics teachers. Teachers' beliefs can shape their actions, and this could have an effect in prison education, where teachers' values may differ from the prison culture [START_REF] Wright | Going to teach in prisons: Culture shock[END_REF]. So, we decided to evaluate how mathematics teachers in prison felt about their work as this can impact on their teaching [START_REF] Beswick | Knowledge/beliefs and their relationship to emotion[END_REF]. We are interested in doing this research because there is a lack of research in the field of prison mathematics education. After teaching for many years in this sector, I felt it was timely to investigate it.

What am I doing?

So, we developed a survey for teachers of mathematics in prison center, based on other studies and issued it to a pilot group of mathematics in prison. We intend to involve a wider group in future.

The survey has quantitative and qualitative parts: the quantitative part investigates the timetabling, the composition and the access to mathematics classes. The qualitative part used metaphor to assess teachers' identity and feelings about their work through metaphor. We will analyze this data using a framework such as Linguistic Metaphor Identification [START_REF] Steen | A method for linguistic metaphor identification: From MIP to MIPVU[END_REF]. This was originally planned as an online survey, but these can be problematic in the prison context, so we simplified it and used email. Five emails were sent and five responded.

There is a small number of responses for several reasons. There is a small number of mathematics teachers in the Irish prison education system. There are 13 centers in total and all have at least one teacher, either part time or full time. This data will be part of the next stage in this study. It is a subject with relatively low participation rates, which will be investigated in the next stage of the research. This is the first stage of the research and in the next stage, further data will be gathered from the remaining mathematics teachers in Irish prisons. The final stage of the project will consist of surveying the students attending mathematics classes in prison.

There is often conflict over mathematics, mathematics or numeracy [START_REF] Kaye | Adults learning mathematics: Inside and outside the classroom[END_REF] and the discussion is beyond the scope of this paper. This paper uses the term mathematics, as that is the term used widely in prison education by teachers and students.

The research questions in this study are:

(1) Are long mathematics classes common practice in Irish prison education centers and how often can students attend? (2) Is there a link between teachers' level of satisfaction to the availability of mathematics classes? (3) How do Irish mathematics teachers in prison see their identity?

Results

The quantitative data was analyzed using Excel. Teachers surveyed stated that their mathematics classes varied between 1.5 and 2 hours long (Figure 2). They also stated that students could enrol in as many classes as they wished each week. All teachers stated that they were satisfied with this.

Figure 2: Hours available for Mathematics classes in prison education centers daily

We found that mathematics classes in prison were longer, at least one hour (Figure 2). Students could come more often if they wished; all levels and abilities attend at the same time, so students could join classes with people they choose to learn with.

Figure 3: Maximum numbers attending mathematics classes in prisons surveyed

In Figure 3, the numbers attending range from 1 to 10 and one respondent was unhappy with the low numbers. 10 is the maximum number in any class according to the respondents.

Students worked individually within the group as all were at different levels and the teachers respected and protected their privacy. This mixture in the class was a challenge at times and took a toll on teachers, in terms of emotional and physical energy, according to some respondents.

A challenge that teachers felt in this context was getting more students to class, as they knew there were many who were not coming to school or who were in other classes but had not enrolled in mathematics yet. They also felt that the students were anxious at the beginning so had to work individually rather than in groups as at the start, they were afraid to let others see where they were at. Some teachers were the only mathematics teacher in their center, some were part of a team in their center.

In response to the question "Are students of mixed ability?" all teachers stated that all students are mixed level. Teachers were asked: "Do you like teaching mathematics in prison?" Those surveyed all said they enjoyed teaching mathematics in prison and seeing their students grow in confidence and in skills. Teachers stated: "I enjoy it" "I love it" "I like it" "Yes, I love teaching mathematics …to see a student's absolute pleasure at reaching their target."

Responses to the question "What metaphor describes you as a mathematics teacher?" ranged from "bit of everything" (2 responses) to "facilitator", "member of a band" and "jazz musician" and "juggler" and "bridge builder" and "breaking down barriers". Teachers stated that they wanted to ensure students had a positive experience and would return to mathematics class the next day, so adapted their behavior according to the needs of the students during each class. These teachers see themselves in different ways, as facilitators of the group, as led by the group and as a mixture of both. [START_REF] Oksanen | Finnish mathematics teachers' beliefs about their profession expressed through metaphors[END_REF] stated that in their findings teachers saw themselves in five categories: subject specialist, pedagogue, didactics expert, self-referential and contextual metaphors. We will develop our coding into categories in the future wider study.

As part of my own reflective practice, I picked a metaphor which I felt represented my work as a teacher of mathematics in prison. I feel that I am a tightrope walker, balancing on a fine line in class between challenging my students with mathematics, being sensitive to them and managing the learning while keeping an eye out for risks.

Discussion

There were many findings from this initial stage of gathering the research. Looking at the data from teachers, they had long mathematics classes of and mixed levels and were generally satisfied with the amount of time students could access classes. They were concerned that so few came to mathematics class. Teachers showed flexibility of approach in their use of words related to their management of learning, such as juggle, rotate, flexibility. Their actions showed a tolerance and acceptance of where the students were at. The irregular attendance, mixed ability groups, different groups, different class lengths, and frequent breaks during class allowed the students to feel comfortable and enjoy being mathematics class. These factors presented a challenge to teachers at times. Teachers showed an awareness of the role of emotions in mathematics class and noted that they allowed respect and privacy to students, offered individualized teaching within the group. They saw their role as breaking down barriers around learning mathematics and acknowledged that learning is challenging. Teachers expressed their pleasure at seeing student reach their target and acknowledged the difficulties they had in their mathematics learning histories.

Figure 1 :

 1 Figure 1: Teaching Triad, Jaworski, B. (1992)

Table 1 : The survey for prison mathematics teachers

 1 ).

	1	How long are mathematics classes in your center?
	2	How often can people attend mathematics?
	3	How satisfied are you with this?
	4	How many students are in your mathematics classes
	5	Are students of similar or mixed ability?
	6	Do you like teaching mathematics?
	7	What metaphor describes you as a mathematics teacher?
	8	Please explain


Javier Díez-Palomar
email: jdiezpalomar@ub.edu

Kees Hoogland
email: kees.hoogland@hu.nl

Vincent Geiger
email: vincent.geiger@acu.edu.au

Numeracy in adult education: discussing related concepts to enrich the Numeracy Assessment Framework

Keywords: Numeracy, assessment, adult learning

. In 2018 and 2019 the framework for the second cycle of PIAAC will be developed. This second cycle of the PIAAC survey aims to update the data about the numeracy skills of adults in different countries around the World (Hoogland, Díez-Palomar, Maguire, 2019). The objective of this paper is to highlight some relevant findings from literature on the concept numeracy in order to discuss a potential enrichment of the PIAAC Numeracy Assessment Framework (NAF).

Introduction

We are now well into the 21 st century, and lifelong learning is becoming a crucial feature in adults' lives, especially in terms of numeracy, because of the need to live in an increasingly globalized world characterized by rapid technological and economic change. There are major societal and policy pressures on education to prepare citizens for a complex and technologized society [START_REF] Hoogland | Towards a Common European Numeracy Framework[END_REF][START_REF] Voogt | A comparative analysis of international frameworks for 21 st century competences: Implications for national curriculum policies[END_REF]. What is expected from numeracy education in the current situation? New means of communication and types of services have changed the way individuals interact with governments, institutions, services and each other, and social and economic transformations have, in turn, changed the nature of the demand for skills as well. Globally, too many adults and young people lack the necessary numeracy competencies to participate autonomously and effectively in our technologized and number-drenched society. As a consequence, many people are disadvantaged in terms of employment and face preventable challenges in relation to social well-being and financial security. The results of the last PIAAC survey (OECD, 2016) show that a quarter of the participating countries in PIAAC have numeracy outcomes below level 2 of the 6-point scale. These outcomes give rise to serious cause for concern for the future economic development for many nations. This is an even more pressing issue since the amount of mathematical data that needs to be interpreted and used is increasing rapidly due to technological developments and the emergence of (big) data. However, numeracy is a complex notion that entails different components [START_REF] Geiger | A rich interpretation of numeracy for the 21 st Century: A survey of the state of the field[END_REF]. More than a decade ago (in 2003), several scholars meeting in Strobl discussed, during an ALM (Adults Learning Mathematics) conference, the meaning of numeracy versus mathematical literacy (Maasz & Schloeglmann, 2003). It was not clear whether numeracy was just referring to the ability to use mathematics in different situations, or something wider that we can call "mathematics literacy." In this paper we adopt the definition of numeracy developed for the first cycle of the PIAAC survey as starting point, to discuss the concept of "numeracy" seeking to identify what numeracy concepts are, so a meaningful assessment framework can be developed to recognize these concepts. We will discuss the notion numeracy in order to enrich the basis to develop the NAF that will be used in the second cycle of the PIAAC survey. We draw in a literature review exploring different concepts of numeracy, putting the emphasis on its components. In the next section we present the methodology used to conduct the review discussed in this paper. Then, we introduce the main findings.

Methodology

In order to answer the research question mentioned above, we conducted a literature review on the concept numeracy. Typically, a literature review is used to frame the problem presented in the introduction of the study [START_REF] Creswell | Research design. Qualitative, quantitative and mixed methods approaches[END_REF]. In our case, we are using the literature review as a research instrument to identify relevant literature in order to discuss the different definitions about numeracy. According to [START_REF] Creswell | Research design. Qualitative, quantitative and mixed methods approaches[END_REF], the first step of doing a literature review is identifying "key works useful in locating materials" (p. 34) that may be relevant for the purposes of the study. In our review we elaborated on past finding by cross-referencing the concept numeracy with a number of reoccurring concepts in the discussions on numeracy. These concepts are: big mathematics ideas, number sense, embeddedness, and authenticity. We decided using those categories since they emerge from the first cycle of PIAAC. We have carried out a documental analysis of articles published in scientific journals included in the main databases and repositories, including the Web of Science, Scopus, for relevant journal articles, and ERIC Database for scholar documents and reports. Additionally, Google Scholar was used to identify potentially relevant literature, that we have subsequently contrasted with the Web of Science and, especially, with the Journal Citations Report database. We used the mentioned categories above as keywords to conduct the searches.

Results

Following [START_REF] Creswell | Research design. Qualitative, quantitative and mixed methods approaches[END_REF] recommendation, we present the global results of the review for each of the selected concepts. We also use what Creswell calls a literature map to illustrate the different components regarding to numeracy found in the literature review. Drawing mainly on the documents produced by the OECD (PIAAC, etc.), we created a first approach to the concept of numeracy, using a web-based software to count the number to times (frequency) that each word is used. This provided a first view of the concept of numeracy and how is it being used (Figure 1). We can see that the word "numeracy" is usually connected to concepts such as mathematics, skills, literacy, education, teaching, workplace, use, work, and data (see Figure 1). However, this first approach is based in a limited exploration. For this reason, we conducted a more fine-grained analysis, using a purposeful procedure [START_REF] Creswell | Research design. Qualitative, quantitative and mixed methods approaches[END_REF] using the three key words, as cited above, in the methodological section. As we can see in Figure 2, those three keywords provided a conceptual network in which aspects such as competence, access, use and interpret mathematics, embeddedness and social practice, invisibility, authenticity, workplace, powerful mathematical ideas, appears interconnected.

Numeracy and big ideas in mathematics

An entry point to identify relevant schemes, models or instruments to establish the basis of the new numeracy components is the so-called "big ideas in mathematics." This is a well-known research domain in mathematics education [START_REF] Jones | Elementary students' access to powerful mathematical ideas[END_REF]. There is a general agreement that mathematics proficiency means noticing connections among different mathematics' concepts and competence in using them. After many decades of mathematics education as a sort of utilitarian discipline looking for ways and strategies for children to perform calculations and solve problems, the vast majority of the mathematics education researchers defend a more relational focused approach to teach and learn mathematics. Jones and his colleagues (2002) provide a great summary of the main contributions of the research to what they call "powerful mathematical ideas", including the following domains: whole number and operations, rational numbers, geometry, probability, data exploration and algebraic thinking and other underrepresented domains. It could be argued that being numerated means using the contents of all these domains not just as procedures (instrumental understanding in Skemp's terms) but in a critical / meaningful manner.

In a more recent article, [START_REF] Hurst | Developing the big ideas of number[END_REF], quoting [START_REF] Charles | Big ideas and understandings as the foundation for elementary and middle school mathematics[END_REF], state that "big ideas" allow us to see mathematics as a coherent set of ideas, encouraging a deep understanding of mathematics. It could be suggested that being numerate as defined within the PIAAC NAF may link to the idea of being able to access, use, interpret and communicate mathematical information around what the international scientific community calls "big ideas in mathematics." Although it seems that everyone might understand what "big ideas in mathematics" encompasses, the reality is that the construct remains contentious. Kuntze and his colleagues (2011) mention a plethora of different terms referring to the area of big ideas, e.g. fundamental ideas [START_REF] Schweiger | Fundamental ideas: A bridge between mathematics and mathematical educa on[END_REF], central ideas or universal ideas [START_REF] Schreiber | Bemerkungen zur Rolle universeller Ideen im mathematischen Denken[END_REF], core ideas [START_REF] Gallin | Sprache und Mathematik in der Schule. Ein Bericht aus der Praxis[END_REF], leading ideas [START_REF] Vollrath | Rettet die Ideen![END_REF], basic ideas and basic conceptions [START_REF] Hofe | Grundvorstellungen mathematischer Inhalte[END_REF]. [START_REF] Charles | Big ideas and understandings as the foundation for elementary and middle school mathematics[END_REF] define "big idea" as "a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole." This definition is also shared by other authors such as [START_REF] Hurst | Developing the big ideas of number[END_REF]. In their article, they track the notion of "big idea" back to the work of [START_REF] Bruner | The process of education[END_REF], who inspired Clark's (2011) definition of big idea as a "cognitive file folder" that we can file with "an almost limitless amount of information." (Clark, 2011, p. 32). Big ideas became conceptual structures (schema in Skemp's terms) that we can use to provide a NAF where content might be characterized by multiple connections. As [START_REF] Bruner | The process of education[END_REF], [START_REF] Hurst | Developing the big ideas of number[END_REF], [START_REF] Clark | Concepts as organizing frameworks[END_REF] and other authors claim, big ideas may become bridges for the transfer of learning. Drawing on their thoughts, we suggest here using big ideas in mathematics as a skeleton for developing PIAAC NAF.

Numeracy and number sense

Number sense appears to be one of the main components of "numeracy." Being numerate means having a certain sense of numbers and how we use them to represent, inform, predict, estimate the reality.

McIntosh, Reys and Reys (1992) develop a framework for number sense including three components: numbers, operations and computational settings, which are interconnected. According to them, number sense involves being able to use numbers, operations and their applications in different computational settings. They talk about the meaningful understanding the Hindu-Arabic number system, the development of a sense of orderliness of the number, the multiple representations for numbers (including the idea of composition / decomposition), the understanding of mathematical properties, and the relationship between operations. For them, having "number sense" means being able to solve problems in the real world, providing suitable answers, using (or creating) effective strategies to compute, count, etc. It is not just reproducing instrumentally a certain algorithm but being able to use the mathematical knowledge and components in a flexible manner. [START_REF] Yang | Number sense strategies used by pre-service teachers in Taiwan[END_REF] define number sense as "a person's general understanding of numbers and operations and the ability to handle daily life situations that include numbers. This ability is used to develop flexible and efficient strategies (including mental computation and estimation) to handle numerical problems." (Yang, Reys & Reys, 2009, p. 384). Regarding the components of number sense, these authors argue "Number sense is a complex process involving many different components of numbers, operations, and their relationships." (Yang, Reys & Reys, 2009, p. 384). Among these processes, they highlight two aspects, (1) the use of benchmarks in recognizing the magnitude of numbers, and (2) the knowledge on the relative effects of an operation on various numbers. [START_REF] Faulkner | The components of number sense: An instructional model for teachers[END_REF] claim that "the characteristics of good number sense include: (a) fluency in estimating and judging magnitude, (b) ability to recognize unreasonable results, (c) flexibility when mentally computing, (d) ability to move among different representations and to use the most appropriate representations" (p. 25). The main components of their approach to number sense are: quantity and magnitude, numeration, equality, base ten, form of a number, proportional reasoning, algebraic and geometric thinking.

Numeracy, embeddedness and authenticity

The concept of embedded mathematics emerges primarily from studies related to mathematics in the workplace but also has significance in the broader notion of numeracy. The embeddedness of mathematics refers to a deep connection to the context in which it is utilized. This can mean that the way mathematics is used to operate on a task is fundamentally shaped by the context in which it is employed. This includes socio-cultural influences that afford or constrain action in school, civic, personal or workplace environments. In this view there is a clear separation between school mathematical knowledge, how it is taught, learnt and practiced, and the use of this knowledge outside of schooling. As [START_REF] Harris | Mathematics in a cultural context: Aboriginal perspectives on space, time and money[END_REF] notes:

In work [. . .] mathematical activity arises from within practical tasks, often from the spoken instruction of a supervisor and always for an obvious purpose which has nothing to do with the numbers working out well. Thus, students taught to react to isolated, abstract and written commands in the specialist language and carefully controlled figures of a school mathematics class, find themselves confronted with the urgent spoken, if not shouted, instructions in a completely different context and code (p. 138). [START_REF] Yasukawa | Production workers' literacy and numeracy practices: using cultural-historical activity theory (CHAT) as an analytical tool[END_REF] make a clear connection between embeddedness and social practice arguing that numeracy practices cannot be understood independently of the social, cultural, historical and political contexts. They illustrate this point, they compare students completing calculations individually, using paper and pen and perhaps a calculator against the use of mathematics in the supermarket, in which the same calculations completed at a checkout counter by the shop assistant using a cash register. In this situation the shopper might perform an estimation to avoid being overcharged. However, the shop assistant is equally concerned with charging the customer the correct price and recording accurate record of the items sold via the cash register. The calculations are the same but the purpose -which is related to context -is different.

Embeddedness has led some researchers to talk about the invisibility of mathematics within work or social contexts. This means that mathematics can be fundamental to activities that are not obviously mathematical [START_REF] Fitzsimons | Adult numeracy for work and life: Curriculum and teaching implications of recent research[END_REF]. This is most clearly apparent in the use of technology in the workplace where digital tools used to complete tasks often obscure underpinning mathematical activity. As [START_REF] Kent | Characterizing the use of mathematical knowledge in boundary-crossing situations at work[END_REF] argue, within technomathematical situations in workplaces there is a shift from 'fluency in doing explicit pen and paper mathematical procedures to a fluency with using and interpreting output from IT systems and software, and the mathematical models deployed within them' (p. 2-3).

Building on this point, [START_REF] Wedege | People's mathematics in working life: Why is it invisible? Adults Learning Mathematics[END_REF] defines two forms of invisible mathematics as (a) subjectively invisible mathematics where people do not recognize the mathematics that they do as mathematics and (b) objectively invisible mathematics in which mathematics is hidden in technology.

Discussion

Drawing on the contributions coming from the literature review, some considerations emerge.

First, big ideas in mathematics, number sense, embeddedness, and authenticity are important concepts in trying to define the notion of numeracy in the context of the 21 st century. The Numeracy Expert Group working at the PIAAC survey defined numeracy as "ability to access, use, interpret and communicate mathematical information and ideas" (PIAAC Numeracy Expert Group, 2009, p. 21). However, this definition does not provide clues about what "mathematical information and ideas" means. This is a relevant topic, since in the 21 st century the "mathematics" that may be relevant for adults probably include some of the components highlighted by the scholars working around the notion of big ideas of mathematics. This is also interlinked to the notion of authenticity, since relevant mathematics, perhaps, must to be also authentic (or they are relevant because they are also authentic). If we want to create a new framework for numeracy assessment, then probably we need to look for the mathematics embedded in real situations and draw on them in order to be able to measure adults' numeracy.

Second, more research is required to identify other important elements of numeracy, especially in the current context of the 21 st century and the "new" skills that adults must to develop. In fact, numeracy capability is increasingly vital in a world characterized by rapid technological and economic change.

Third, numeracy is vital for social well-being, financial security and informed citizenship. Hence, a framework to assess numeracy must explore how this notion is embedded in authentic practices related with those societal dimensions mentioned above.

Fourth, the critical aspect of numeracy, (not discussed here) related to making evidenced based judgements and decisions, is an aspect of numeracy that has often been underplayed for adults but is an essential element for informed participation in personal, civic and work life. More work is needed in identifying how to promote this critical capability.

Fifth, evidence suggest that adult numeracy has been limited to studies in context (workplace, personal settings or activities, such as shopping, etc.). However, there is a lack of research in terms of cognitive, epistemological considerations of how adults learn/ use mathematics (numeracy skills), which means that additional research is needed to cover those aspects.

Figure 1 :

 1 Figure 1: Word cloud of terms related to numeracy (based on OECD documents)

Figure 2 :

 2 Figure 2: Literature map about the notion of numeracy

Acknowledgment

This work is supported under the OECD Programme for the International Assessment of Adult Competencies (PIAAC) survey.


Javier Díez-Palomar
email: jdiezpalomar@ub.edu

Evgenia Anagnostopoulou

Mathematics and Physics Dialogic Gatherings: Fostering Critical Thinking Among Adult Learners

Keywords: Dialogic learning, adult scientific literacy, gatherings

This paper discusses the effectiveness of dialogic learning as an adults' learning theory. It specifically focuses on how Mathematics and Physics Dialogic Gatherings (M&PDG) can enhance the critical thinking of adult learners and hence develop their learning in both topics. Two studies were set where the participants engaged in dialogic reading activity reading classics in mathematics or physics. The former study took place in Barcelona, Spain and involved adult women of above 40-years-old with low literacy skills engaging in mathematics. The latter took place in Kendal, UK and involved two groups of adult A Level students, one studying Psychology and the other Physics. Although the samples were extremely diverse, the results demonstrated that classical readings can significantly improve critical thinking and dialogic talk in all groups and provide the potential to create further learning opportunities.

Introduction

Dialogic learning has been one of the main approaches to adults' learning theories in the last decades [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF]. This approach suggests that adults are self-responsible of their own learning. They become active agents, creating social spaces for learning, drawing on solidarity and social interactions. In order for adults to be active in their learning and express themselves, they must use processes that promote their way of thinking, understanding and their ability to reason; in other words, they must develop their critical thinking (as defined by the Foundation for Critical Thinking, 2017). The aim of this paper is to discuss Mathematics and Physics Dialogic Gatherings (M&PDG) as learning spaces for adults to develop their learning in both topics. Dialogic gatherings were originally implemented by Flecha in La Verneda Adult School, in Barcelona, in the late 1970s. In 1978, a group of people asked for popular education for adults and occupied a public building to offer literacy courses for adults who never had the opportunity to attend school. Drawing on Freire's work (who also visited the school several times), the adults participating in La Verneda Adult School created a democratic popular movement of education based on the principles of dialogic learning [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF]. M&PDG are part of this initiative [START_REF] Díez-Palomar | Using dialogic talk to teach mathematics: The case of interactive groups[END_REF][START_REF] Díez-Palomar | Mathematics dialogic gatherings: A way to create new possibilities to learn mathematics[END_REF]. In the following sections, we will introduce the theoretical framework, the methods used in the research reported here, and the main findings, for further discussion.

Theoretical framework

According to [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF], learning is framed by seven principles: egalitarian dialogue, cultural intelligence, solidarity, transformation, the creation of meaning, instrumental dimension of learning, and equality of the differences. Dialogic gatherings are spaces where all these seven principles are evident. Flecha coined the term "dialogic literary gatherings" (DLG) in the late 1970s. He created the first DLG in Barcelona, with a group of non-academic women, most of them barely literate, drawing on debates between some of the most outstanding scholars in adult education and social sciences at that moment, such as Freire or Habermas. Freire (1970) believed that learning matters because individuals discover that through learning they can make and remake themselves because they realize that they are beings capable of knowing. Learning for individuals is a practice of freedom. We can resist oppression and transform situations of domination, such as the lack of schooling, for example. [START_REF] Habermas | On the pragmatics of social interaction: Preliminary studies in the theory of communicative action[END_REF] also assumed understanding as a universal ability of human beings. In his Pragmatics of Social Interaction, he claims that meaning and understanding are two categories eminently human. Individuals develop their understanding through social interaction, sharing repertoires of meaning embedded in particular actions denoted by linguistic or semiotic signs [START_REF] Habermas | On the pragmatics of social interaction: Preliminary studies in the theory of communicative action[END_REF]. Drawing on this idea, [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF] states that all individuals may be able to develop understanding (learn) using social interaction as a form of scaffolding. The DLG is defined as learning spaces where a) adult learners with low literacy skills participate, b) the readings are universal literature classics, and c) the process is based on dialogic learning [START_REF] Puigvert | La Tertulia: A dialogic model of adult basic education in the new information age[END_REF]. The DLG become spaces where individuals participate, interact, and share meanings around particular readings / words/concepts. The DLG became one of the successful educational actions validated in the research project INCLUD-ED. Strategies for inclusion and social cohesion from education in Europe (2006Europe ( -2011)).

Drawing on this background, La Verneda Adult School developed the Mathematics Dialogic Gatherings (MDG) using classic readings in mathematics to conduct the gatherings [START_REF] Díez-Palomar | Mathematics dialogic gatherings: A way to create new possibilities to learn mathematics[END_REF]. The research discussed in this paper starts with this experience and extends the MDG to science (Physics). The second author of this paper used this approach to create PDG in the UK, drawing on the principles of DLG [START_REF] Soler | Reading to share: Accounting for others in dialogic literary gatherings[END_REF][START_REF] Soler | People understand everything, and the better it is written, the better it is understood[END_REF].

Methodology The research question

In this paper, we discuss some of the main findings from two independent studies focused in the same research question: "Can mathematics and physics classical readings promote the development of critical thinking of adult learners?".

The setting and participants

The first study reported in this paper was held in Barcelona (Spain). The setting for the study was an adult school placed in a working-class neighbourhood in Barcelona, Spain. Data was collected from May to June 2016 (seven sessions). Participants included six women between 40-years-old and over seventy-years-old, with low literacy skills. The second study discussed in this paper was conducted in Kendal College, Cumbria, in the United Kingdom. Data was collected during the last semester of the academic year 2016-2017. It involved two groups of A Level students, male and female: group PSY, who studied psychology and group PHY, who studied physics.

Research design

In both studies, we created Dialogic Gatherings. These gatherings are spaces in which participants engage in dialogic reading activity involving classic texts in mathematics or science. Participants are the ones choosing the readings. The book selected for the first study was Historia de las matemáticas, the Spanish translation Jean-Paul Collette (1979) book Histoire des Mathématiques.

For the second study, the reading selected was Dialogue Concerning the Two Chief World Systems, by Galileo Galilei (1632). In both cases, we used the methodology of the DLG created by Flecha (see Figure 1).

Results

The participants in the Mathematics Dialogic Gatherings (MDG) engaged from the very beginning in mathematical discussions around the history of mathematics. The six women seemed comfortable talking about the first traces of mathematics in the Palaeolithic Era, the mathematics used by the Babylonians, the contributions made by the people living in ancient Egypt, the mathematics formalized by the Greeks, etc. Drawing on the description of how humankind developed grouping and the idea of number as the strategy to keep track of cattle, crop, the six women discussed such notions as number system, algorithm, unit of measure, value, equivalence, and so forth. They related some of these concepts with their everyday life. For instance, when discussing the concept of the base of a number system, they rooted their arguments in their previous knowledge about using different coins whose value is defined by groupings in base 1, 5 or 10. Their knowledge about duros (a coin equivalent to 5 pesetas in the old Spanish currency) facilitated their understanding about ancient number systems which a base different from the Hindu Arabic numbers that we use nowadays. Jean-Paul Collette mentions in the first chapter of his book that a bone was found in the Czech Republic with some tally marks carved in groups of approximately 30. According to the experts, it could be a moon calendar. Interestingly, one of the women in the group highlighted this excerpt of the book because she stated: "I remember that I read somewhere that some people think that those marks correspond to the menstruation cycle." Then, a further discussion about mathematics and human biology started, connecting mathematics to gender issues. During the sessions, many other issues arose: Why the Romans didn't have the zero in their number system?, How to translate quantity from one unit of measure to another?, and What strategies are best in problem solving?

For every topic, the participants in the MDG used their own personal background to, in [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF] terms, "create meaning with the mathematical content. The book presented many symbols and codes (numerals, formulas, graphical and visual representations), that usually are abstract representations of real phenomena. Data suggest that women were able to create a zone of proximal development where everyone was free to contribute with their piece of knowledge drawing on their own experiences. According to [START_REF] Hutchins | Distributed cognition[END_REF] this can be defined as episodes of distributed cognition. That was the case when developing an understanding an ancient algorithm for multiplication presented by Jean-Paul [START_REF] Collette | Historia de las matemáticas[END_REF] in his book.

In the same vein, the Physics Dialogic Gatherings project further examined participants' progress with respect to their background knowledge, in a different setting.

During the first sessions, the psychology group got involved in discussions in a descriptive and philosophical manner. They felt more confident to discuss philosophical ideas, such as the definition of perfection in the text, and they tend to avoid or ignore any scientific or mathematical part of the book. An example from a participant: "If I were in this conversation I would question the definition of perfect and what is exactly they are trying to achieve by defining these principles? What are they going to achieve if they finally define that the Earth is perfect? What does perfect mean?" They often exchanged information directly quoting from the text without showing any understanding of the ideas transmitted. For example: "Pythagoras says it's defined by beginning, middle and end, but others are saying perfect is how you form the body; the length, and thickness."

They did not challenge the obvious. They preferred to agree with that. Direct quotation from the text:

"They say about the lines that the straight line is the shorter one and obviously you can have many other lines. I agree with that."

During the later sessions, they felt confident to discuss science too, starting with simple arguments and poor terminology. Their dialogic skills evolved with time when they felt the need to use definitions and self-created terminology. They created their own terms when the lack of scientific knowledge did not help.

"I think it depends what fast is. In our situation our fast is like steady and we have air-resistance and when it reaches terminal velocity, it will be like balanced. Whereas if you are in a car and it speeds up and in fact you are in control of the fastness, you will just accelerate until you reach where you want to go and then you will put another force acting into it because you want to slow it down. So it depends on which fast it is acting on." They could formulate arguments where critical thinking was evident. As they were feeling more comfortable, they progressed to using scientific terminology, which emerged either from the text or from prior knowledge that they had not used for a long time, but they recovered it for the sake of their argument. "So, say you have a forward motion, velocity and then you have the air resistance and gravity. They would balance out in a way that the one motion is still more than the others because it still moving forward, because if the others weren't there it wouldn't have to speed up."

Their critical thinking was developing, and they were more involved in discussing their ideas in a dialogic manner, even if these ideas could be wrong.

"If the plane is tilted, the ball starts moving and speeds up until it reaches towards the end. And then reaches a flat surface again. So yes, I agree with that. There has to be some kind of limit, because… I don't know… It can't just speed up, speed up, speed up. It has to be some kind of limit."

Towards the final sessions, they were able to combine physics and philosophy into their arguments, which triggered further discussions and carried forward their learning skills. "It's got to have some purpose to move. A body which is not moving, will move if it has a purpose."

The physics group seemed more open to disputes and reasoning than the psychology group, possibly because they considered the text to be more on their area of expertise. They approached the ideas presented with scepticism and doubt, rather than with unquestioning acceptance, which already indicates a level of critical thinking [START_REF] Scriven | A statement by Michael Scriven and Richard Paul[END_REF]. In the early sessions, the group was dealing with the texts purely scientifically, ignoring any philosophical quotations in contrast to the psychology group. The physics group was seeking for proofs and they even performed their own inclass short experiments to prove a point.

"Salviati is saying that he doesn't believe that 3 is necessarily more perfect than 4 or 2, and gives the example of legs on a chair or table. And I was wondering if it is harder to knock over a table with 3 legs or 4 legs, because with 2 legs obviously it is not stable." They employed their scientific background knowledge to explain the ideas presented in the reading.

"Triangle is the strongest shape in nature so I guess it may be stronger."

"…because there are more ways where you can pass the center of gravity." At later sessions, the group was involved in philosophical critical thinking too. The engagement with the philosophical terms expanded their discussion away from the actual text and they were able to connect science with philosophy. Their critical thinking and dialogic talk development evolved to a point where they were disputing scientific methods that they had never disputed before. They questioned science and mathematics and they concluded that they had more trust in their senses than some scientific proofs.

"Does mathematics support or help science, do you believe things because you see them to be mathematically true or do you believe things because you see them to be physically true? What creates new knowledge?"

Discussion

In 1976 Wood and his colleagues coined the term "scaffolding" as a metaphor to analyse how adults assist children when solving problems. They defined it as an action that "enables a child or novice to solve a problem, carry out a task or achieve a goal which would be beyond his unassisted efforts" (Wood et al, 1976, p. 90). This concept moved forward the old Vygotskian notion of ZPD [START_REF] Bruner | Actual mind, possible worlds[END_REF]. [START_REF] Cazden | Peekaboo as an instructional model: Discourse development at home and at school[END_REF] extended the concept from its original use in the context of dyadic adult-child interactions, to a study of teacher-student interactions in classroom settings. More recently, a special issue published in 2015 in the journal ZDM Mathematics Education (47,7) explored the interlink between this metaphor and the idea of dialogic teaching. Evidence suggest that dialogic teaching pushes forward to notion of scaffolding, since dialogue creates opportunities for participants to exchange and share knowledge from which create their own understandings. In this paper, we found evidence suggesting that this approach can also be used in the context of adults learning in both mathematics and physics.

Overall, the Mathematics and Physics Dialogic Gatherings project proved to be an enjoyable method for the teaching and learning of physics and mathematics. Data collected suggest that egalitarian dialogue can create opportunities for learning. The participants, drawing on their dialogic talk, created [START_REF] Vygotsky | Mind in society: The development of higher psychological processes[END_REF] zones of proximal development where all participants contribute with their cognitive potential. Our results also confirm Hutchins' (2000) notion of distributed cognition, since many times the different participants ended with a complete explanation of a particular notion, such as the uses of the zero in a number system, after sharing different pieces among them. The egalitarian dialogue, as defined by [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF], means that everyone has the same opportunity to share his/her previous knowledge throughout a dialogic process in which every participant shares arguments based on valid claims. The participants of the psychology and the physics groups as well as the mathematics group were able to make valid claims on the understanding of mathematical, scientific and philosophical concepts. This suggests that the critical thinking of both groups was significantly improved.

In addition, the results also reveal that using an egalitarian dialogue based on personal previous knowledge, adults were, in Freire's terms, able to read (and re-read) critically the world. The different notions discussed during the sessions in both settings gained from this personal way to present them to the audience in the M&PDG. Linking them to personal situations in the everyday life appears to be a successful strategy to create meaning around the concepts discussed in the group. As [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF] states: "the creation of meaning is one of the more seminal principles of the Dialogic Learning theory since learning is strongly connected to motivation, as previous studies have largely demonstrated [START_REF] Mehler | Cognitive capacity of very young children[END_REF]". Freire narrates the case of a woman who learned to read and write by sending love letters to her husband. For this woman, the words acquired full meaning because they transmitted her feelings to her husband, who was far away. In a similar vein, for the women participating in the MDG recalling their memories about duros and pesetas was a way to make meaning to the idea of base in a number system. In this situation, grouping, which is an important component of numbers in number systems (as well as in the development of number sense), was fully understood by the six women participating in the MDG. They not only used the notion of grouping (base); they were also able to explain different algorithms to do mental calculation drawing on the use of grouping as a cognitive strategy to solve particular cases of calculations (mainly addition and subtraction, and sometimes multiplication as well).

The research demonstrated that background knowledge is important to establish an egalitarian dialogue, in the sense that everyone uses their own repertoire of knowledge to participate in the dialogue. Although the two groups were very diverse in terms of background knowledge, they ended up in parallel ideas when combining mathematics, philosophy, and science. They approached the notions from a different perspective but somehow, they concluded in similar ways. Combining mathematics, science and philosophy opened new perspectives into their learning of physics and mathematics. In fact, results suggest that "heterogeneity" may be a successful component of M&PDGs. Without that heterogeneity, participants in the gatherings would not be able to introduce different examples to illustrate particular scientific or mathematical notions, making learning more depth and meaningful for everyone in the group. The Dialogic Gatherings method can, therefore, benefit heterogeneous groups, where people can learn from each other, taking advantage of the knowledge that everyone shares within the discussion. This is consistent with previous research in the educational arena [START_REF] Flecha | Sharing words: Theory and practice of dialogic learning[END_REF][START_REF] Vygotsky | Mind in society: The development of higher psychological processes[END_REF][START_REF] Wertsch | The voice of rationality in a sociocultural approach to mind[END_REF].

Summarising, evidence suggests that classical readings and egalitarian dialogue could be an excellent method to improve critical thinking in mathematics and physics and have the potential to create further learning opportunities in classrooms.

Conclusions and further research

In this paper, we have discussed how adults in two different settings (Barcelona and Cumbria) draw on their participation in the M&PDGs to develop their critical thinking around mathematics, science, and philosophy. Results suggest that adults use their own personal background to make meaning to the academic concepts in the books selected. Egalitarian dialogue mediates learning through a participatory process in which adults share their personal understanding of the topics discussed in the gathering. They use practical examples, drawing on their memories and looking for plausible explanations. In doing so, they acquire proficiency in using expert scientific and mathematical jargon. However, evidence reported here are limited since they belong to two single case studies. The sample is neither representative nor random. The examples introduced in this paper illustrate some intuitions about how adults, using classic readings in mathematics and science, can further learn and create a critical thinking aligned to what some authorities claim as being critical citizens in the rise of the 21 st century. However, due to the limitation of the research design, more research is needed in order to generalize the benefits of the MP&DGs in supporting adult learners' critical thinking, in other settings.

Figure 1 .

 1 Figure 1. Scheme of how a M&PDG works (Díez-Palomar, 2017)

Acknowledgment

The authors want to express their gratitude and acknowledgment to all the adult people participating in the M&PDGs.