Graham Griffiths
email: g.griffiths@ucl.ac.uk

Using dialogue scenes with adult mathematics learners: Research questions and methods

Keywords: Collaborative discussion, discourse analysis, dialogue, adult education, mathematics

. To understand what happens when the 'reading aloud' activity is undertaken, it is important to investigate what happens when learners undertake related activities.

Introduction

This work started with an interest in teaching practices and a wish to investigate, in some detail, the collaborative discussions that occur between learners when undertaking a particular type of activity [START_REF] Griffiths | Using dialogue scenes with adult learners: some exploratory data[END_REF]. From this starting point, my work has become more focused on generating data to inform a theoretical perspective on learner-learner discussions. I am interested in the interactions that occur between learners during the implementation of a form of classroom activity, namely the reading aloud of a scene of dialogue concerning mathematics and a linked, collaborative mathematical task.

Theoretical perspectives on learning

Dialogue (and conversation) has been seen as a useful focus of attention in the educational world and utilised in a variety of ways. For example, classroom dialogue has been investigated within the framework of dialogic teaching by [START_REF] Alexander | Towards Dialogic Teaching: rethinking classroom talk (5th edition)[END_REF] and has resulted in a special edition of ZDM (see [START_REF] Bakker | Scaffolding and dialogic teaching in mathematics education: Introduction and review[END_REF]). Much of this work has been interested in the dialogue that occurs between teacher and learner whereas I am interested in learner-learner interactions with minimal input from teachers when undertaking the specified task. Having said this, there are important ideas that are in common to such work and my research. For example, notions such as the importance of sense making and socio-cultural influences on learning involved in such discussions are also important in my work. Such sense making and socio-cultural perspectives draw upon literature that stretches back to writers such as Bakhtin (published in English in Bakhtin 1986), and Vygotsky (published in English in [START_REF] Vygotsky | Mind in society: The development of higher psychological processes[END_REF]). These writers have also influenced more recent work that I am utilising (in particular [START_REF] Sfard | Thinking as communicating[END_REF].

Central to my work is the question of whether reading aloud has an effect on the resulting collaborative discussions compared to reading silently. This was initially motivated by my own experience of observing such activity. Nonetheless, there is a question as to what the literature says about the effects of reading aloud and silently. Much work has already been undertaken into studying the effects of reading aloud compared to reading silently when related to language and the effect on subsequent educational performance. For example, [START_REF] García-Rodicio | A comparison of reading aloud, silent reading and follower reading. Which is best for comprehension? / Una comparación de lectura en voz alta, lectura en silencio y lectura de seguimiento. ¿Cuál es mejor para la comprensión?[END_REF] provide evidence that suggests there is no significant difference in performance between reading aloud and reading silently when dealing with comprehension exercises (in fact, they note that individuals following others reading aloud can result in significantly worse performance on tests). It is worth noting that this type of study takes an acquisitional perspective (in the sense of [START_REF] Sfard | On two metaphors for learning and the dangers of choosing just one[END_REF]) and is concerned with performance on some standardised test. My study has an interest in participation in a given activity type and the evidence in the transcriptions of discussion related to that rather than the effect on some standardised test. As such the perspective taken by Garcia-Rodicio, Melero & Izquierdo (2018) and similar work is not of interest to me. Other work, for example [START_REF] Duncan | Reading aloud in Lewisham: an exploration of adult reading-aloud practices[END_REF], has a focus on the perceptions of individuals and their practices related to reading aloud. This work does suggest positive aspects to the practice of reading aloud. Although Duncan's work is not interested in mathematical development, I am more interested in the practice focused work of this type of work.

The theoretical perspective that I intend to use, that of thinking as communication, has been developed in recent years and been given a detailed rationale by [START_REF] Sfard | Thinking as communicating[END_REF]. My research takes the participatory approach of seeing thinking as communication that Sfard argues for. Within this perspective, learning is not seen as concerned with changes to some internal schema, that is responding to experiences, but rather the communication itself is viewed as thinking and learning. Sfard argues that the job of the researcher is to investigate communication. The discussions that happen during and following an activity would provide an example of communication which could be investigated and [START_REF] Sfard | There is more to discourse than meets the ears: Learning from mathematical communication things that we have not known before[END_REF] proposes a form of discourse analysis and suggest some aspects to look for in such work. In particular, she identifies for analysis the mediating tools involved in discourse and a set of meta-rules that govern discourse.

For example, the text in the scenes provides an example of a mediating tool. When discussing the scenes and engaging in classroom discourse related to mathematics, it is not surprising to hear direct extracts from a text being used by participants along some paraphrasing of sections. In addition, the mathematical examples used within the text are mirrored back in many cases. Indeed, one of the reasons that [START_REF] Swan | Collaborative learning in mathematics: A challenge to our beliefs and practices[END_REF] has constructed the scenes (DfES 2005) that initiated my work is for discussion to focus attention on key elements of mathematics. It seemed a reasonable working assumption that by asking learners to read text aloud, it is likely that, at least for some groups and individuals, focus will be put on those words and phrases. Part of my research is investigating this aspect, to see the ways in which specific parts of the text are employed in discussion of the scenes.

The term 'meta-rules' might suggest a strict set of guiding principles while, in reality, this might be less rigid. Indeed, Sfard suggests that there could be a wide range of possibilities for a hypothesis to govern a discussion and such a meta-rule will need to be interpreted from the discourse. As an example, in a discussion of mathematics between two boys, she proposes that one of the contributors is focused on mathematical meaning while the other appears to avoid discussing mathematical ideas while employing, for example, face saving devices (such as humour) which suggest a concern for social position. This can be seen as another, different form of, socio-cultural issue raised by [START_REF] Lerman | Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics[END_REF] who argued that the role that individuals play and aspect of power and position in society will have an impact on discourse and need to be considered.

Within the discussions that I am concerned with, such meta-rules may be related to resources that the participants utilise other than those in the text (identified as mediating tools). Such additional resources include the use of mathematical discourse that draws upon prior experiences (either within their present educational setting or from an earlier situation) as well as non-mathematical discourse. In the case of our task, which involves a discussion around the price of rail tickets, it is not a surprise to note that participants have contributed with discourse concerning their own use of the rail system (thus introducing an element of socio-cultural influence).

In my research, I use an overarching framework developed by Sfard, Morgan and others (in particular [START_REF] Morgan | Investigating changes in high-stakes mathematics examinations: a discursive approach[END_REF]. The framework structures the characteristics of discussions into two main categories for analysis: (a) the mathematising aspects of discourse and (b) the subjectifying aspects of discourse. Further elaboration on this, with examples, is provided below. The reason for selecting this form of analysis is that it is designed to be of interest to mathematics education rather than utilising more general categories that occur with other forms of discourse analysis. The analysis of the collaborative discussions consists of investigating a range of factors and I have been interested in the following sub questions under the two identified categories.

(a) The mathematising aspects of discourse:

(i)
What resources do the individual participants use in the discussions (please note that resources here might mean physical resources such as a calculator but also the examples that are chosen by the participants and the proposed calculations that are employed)?

(ii)

To what extent do these resources relate to the text in the activity (a mediating tool) or draw upon other experiences (and therefore suggest meta-rules are being employed)? (b) The subjectifying aspects of discourse:

(i)
How do the individual participants contribute to decision making within the group?

(ii) How is the discussion regulated by the individual participants involved?

(iii) How do the participants see themselves in relation to mathematics (in general and specific to the activity)?

The specific activity that my work is concerned with consists of a mathematical task along with a scene in which four characters discuss the task (see Figure 1 and Figure 2 for details of the task and activity). The task asks participants to agree or disagree with a given statement related to a percentage problem. The activity of interest involves the learners taking and reading aloud the parts of four characters who argue different positions related to the task. This is then followed by a discussion in which the participants are intended to come to their own conclusion concerning the task.

Task

Rail Prices

In January, fares went up by 20%.

In August, they went down by 20%.

Sue claims that:

"The fares are now back to what they were before the January increase" Do you agree?

If not, what has she done wrong?

Scene 1 (as in DfES 2005)

Harriet: That's wrong, because … they went up by 20%, say you had £100 that's 5, no 10.

Andy: Yes, £10 so its 90 quid, no 20% so that's £80. 20% of 100 is 80, … 20.

Harriet: Five twenties are in a hundred.

Dan: Say the fare was 100 and it went up by 20%, that's 120.

Sara: Then it went back down, so that's the same.

Harriet: No, because 20% of £120 is more than 20% of £100. It will go down by more so it will be less. Are you with me?

Andy: Would it go down by more?

Harriet: Yes because 20% of 120 is more than 20% of 100.

Andy: What is 20% of 120?

Dan: 96…

Harriet: It will go down more so it will be less than 100.

Dan: It will go down to 96.

Scene 2 (amended features)

Harriet: That's wrong, because … they went up by 20%, say you had £100 that's 5, no 10.

Andy: Yes, £10 so its 90 quid, no 20% so that's £80. 20% of 100 is 80, … no, 20.

Harriet: Five twenties are in a hundred.

Dan: Say the fare was 100 and it went up by 20%, that's 120.

Harriet: And, because 20% of £120 is more than 20% of £100. It will go down by more so it will be less.

Andy: What is 20% of 120?

Dan: 96…

Harriet: It will go down more so it will be less than 100.

Dan: It will go down to 96.

Sara: No, it has gone up £20 to £120 and then it went back down the same, so that's £100.

Activity RAS1&2

In groups of 4:

Choose to read out the parts of:

Activity RSS1&2

Read the scene below

In groups of 4:

Discuss what you think about Sue's claim that "the fares are now back to what they were".

Activity NS

In groups of 4:

Read the statement from Sue below.

Discuss what you think about Sue's claim that "the fares are now back to what they were". Activity RAS1 is then followed by the Task and Scene 1 from Figure 2 (and similarly for RAS2), Activity RSS1 is then followed by the Task and Scene 1 from Figure 2 (and similarly for RSS2) and Activity NS is followed by the Task only from Figure 2.

Figure 2: The three activity types

During an exploratory phase, examples of the activities described in Figure 1 and Figure 2 were undertaken and reported on in [START_REF] Griffiths | Using dialogue scenes with adult learners: some exploratory data[END_REF]. Examples of discussions were included which showed some different responses to the different activity types. Data taken from this exploratory phase along with early observations from the main phase will be used to illustrate issues in this text.

Methodology

My study takes a qualitative, inductive approach, using a series of case studies and utilising the theoretical frameworks noted earlier for analysis. The work explores the data that is gathered to see how it fits the proposed framework with an aim of suggesting variables for further study.

The data collected consists of audio recordings and observational notes of learners undertaking the activities along with audio recordings of short focus group discussions (one per cohort) about the process immediately following the activity.

The research design of the main phase

To identify some characteristics in the response of learners to the activity involving reading aloud, it is helpful to identify what happens when the activity involves reading silently. In addition, the use of the mathematical task without a dialogue scene is used as a way of identifying the extent to which elements of analysis relate to the use of mathematical scenes whether read silently or aloud.

A final issue concerns the nature of the scene itself. It is possible that responses may be produced as a result of the particular elements of the text. For example, characters within a text may have more dialogue and thus be given priority, or the first and last statements might have more dominance. In order to take this set of issues into account, alternative versions of the text have been developed and used.

The research design chosen considers 5 activities (see Figure 2) which are used as interventions with additional (to the exploratory phase learners) groups of (four) learners across 4 institutions (giving a total of 20 groups and 80 learners):

RAS1activity which involves reading aloud the scene as described by Swan (in DfES 2005) RAS2activity which involves reading aloud an alternative version of the scene RSS1activity which involves reading silently the scene as described by Swan (in DfES 2005) RSS2activity which involves reading silently an alternative version of the scene

NSactivity which involves the task without the scene

To distinguish which characteristics are related to reading aloud, the overarching research question is then refined to become the following question:

What are the similarities and differences in the characteristics of learner discussions during and following the identified activities?

The analysis is considered in relation to contextual information, such as participant backgrounds and experiences, that becomes known as the research progresses including information noted when agreeing the groups to be studied with centres, the recording of the discussions that occur while undertaking the full activity and focus group discussions following the activity.

Observation and discussion

The following text describes some initial thoughts utilising data collected from the exploratory phase along with initial observations from the main data collection without detailed analysis. The first category concerns the nature of the mathematical tasks involved in the research, thereby addressing the mathematising aspects of the activities (and resulting discourse). The second two categories are concerned with the responses of participants to the activities, and with others in a group, thereby addressing issues related to subjectifying aspects of the activities (and discourse).

The relationship between mathematics, the context of the activities and the activity types

The three activity types (RAS, RSS, NS) have one thing in common in that the notion of percentages is linked to rail prices and as such is making links to the 'everyday'. The task, deciding whether an individual is correct or not, is not necessarily what might be understood as a traditional task, that is, one in which a calculation is undertaken and presented with an answer. In addition, the task, while relating to aspects of real world activity -the buying of tickets, the notion of sales and discountinvolves the combination of information that can be seen as somewhat artificial (and perhaps encourage responses involving humour for example in the exploratory phase a contributor suggested that a passenger will be "ripped off").

In the main phase of data collection, while participants most often use the example of the price given within the scene (that of £100), other examples of mathematising are clear such as the strategies used to calculate 20% which were not evident in the text. From early observations of the main data collection phase, it is not yet clear whether reading aloud or not is significant in relation to this.

The relationship between participants and the activity types

A key aspect to consider is how participants in the activities relate to the information and roles presented in the scene. One example is the scene use of the proposed £100 ticket price. This would suggest the use of rail travel between towns and cities rather than more local commuting. Another example concerns the (assumed) genders of the participants in the scene and the selection of roles for reading aloud. I have seen the choice of character being led by gender on a number of occasions (and discussed in the focus groups).

In the exploratory phase, one contributor talks about 'they' ("what they're saying…") and another specifically mentions the character Harriet by name. In the main phase, some individuals undertaking RSS noted that they only read the first few lines of the scene. One of the groups selected to read aloud had identified, and agreed, mathematical statements which would lead to them saying Sue was wrong but did not want to commit to this judgement. In another discussion following the non-scene task, participants suggested £5 as a rail ticket. These are examples of subjectifying aspects of the discussion while on some occasions being linked to mathematising aspects such as related calculations.

The regulation of discussion

The utterances that are involved in a discussion will follow some, usually unspoken, set of rules (meta-rules in Sfard's terms). I have already noted that [START_REF] Sfard | There is more to discourse than meets the ears: Learning from mathematical communication things that we have not known before[END_REF] argues that some participants will have a focus on mathematical meaning while others may focus on alternative, and potentially face saving, aspects of discussion. I draw upon two aspects of discussion that have been written about which can be interpreted as involving subjectifying aspects of discourse, one concerning school discussion, uncertainty and hedging, and another that has been noted in relation to adults and the use of humour.

The significance of uncertainty and signs of hedging in discussions has been noted (in particular [START_REF] Lakoff | Hedges: A study in meaning criteria and the logic of fuzzy concepts[END_REF]). In my exploratory investigation, one participant starts by suggesting that she does not understand what is happening and yet then follows with the mathematically expected responses.

Oughton (2010) has noted the importance of humour in the interactions between learners. She suggests that adults use positive, affiliative humour in group working in order to deal with anxieties that they feel undertaking mathematics. Interestingly, within the exploratory data we see the use of humour to end discussion ("So... you still get ripped off whichever way"). The comment seems to be a mechanism to end the discussion and may be an attempt to disguise uncertainty about meaning.

Conclusion

Having noted that the study is anchored in practice, the study itself is likely to be of interest to researchers interested in forms of discourse analysis within mathematics education. The study has generated data that is intended to inform a theoretical understanding of the effect of reading aloud on classroom discussion with such mathematical scenes. This detailed analysis is still to be undertaken although observations so far suggest some interesting indicators of what to focus on.

While of primary interest in the research field, it is expected that the study will help to inform classroom teachers of (adult) mathematics classrooms. It is not intended to identify a toolkit that enables teachers to produce their own scenes but rather to identify a potential use of the existing scenes in two main ways: a) To propose and describe (including key features) the classroom activity of reading aloud dialogue scenes involving mathematics (not an expectation of the original Swan scene) b) To outline what might be expected to happen in the discussions between learners when undertaking the activity This will then allow teachers to make their own decisions about whether to employ these activities (or perhaps similar ones) with their own learners.

Figure 1 :

 1 Figure 1: The task and scenes 1 and 2.

Acknowledgment

The author thanks the many individuals who have supported this project so far including the staff and learners involved and particularly supervisors Prof. Candia Morgan and Dr. Samantha Duncan.


Lisanne Heilmann
email: lisanne.heilmann@uni-hamburg.de

Reproducing social difference in the concept of 'numeracy skills'

Keywords: Basic skills, gender bias, labour market, adult education, PIAAC

This paper is concerned with adult numeracy skills as these are presented within the Programme for the International Assessment of Adult Competencies (PIAAC). It will focus on what we can learn from PIAAC results about adult skills, about the assessment and about societal structures. Theoretical discussions increasingly view numeracy as embedded in power relations. If so, these power relations should be visible in the results. This paper will provide a brief overview of underlying theory in the field, 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. This value of numeracy skills can be analysed by the example of gender relations in the labour market. The analyses will use PIAAC-data to indicate how gendered hegemony might influence the way numeracy skills matter in terms of monthly wages and the probability of getting into leadership positions.

Introduction

Educational research often looks at skills as something to acquire. Once acquired, it is anticipated that skills bring benefit to their possessor by transforming skills into success. Following that line of argument, a common aim is to make education available and suitable for more people; assuming, that if education would only find a way to convey knowledge more efficiently, this would bring direct benefit to people in terms of happiness, health, or success in the labour market [START_REF] Grotlüschen | Adults with low proficiency in literacy or numeracy: OECD education working papers[END_REF][START_REF] Kittel | Mathematische Grundbildung im Erwachsenenalter[END_REF]OECD, 2016a). [START_REF] Craig | The promises of numeracy[END_REF] arguedin a paper called 'the promises of numeracy' -that numeracy education is largely associated with empowerment and seen as relevant for social participation, while innumeracy is connected to "personal, social, and cultural costs" (Craig, 2018, p. 64). This paper is a reflection on PIAAC results with this perspective in mind. The OECD assessed via PIAAC the literacy and numeracy skills of 16-65 year-olds. In 2011/12, 24 countries participated in the first round of assessment. Nine other countries followed in a second round in 2014/15 (OECD, 2016a).

There is a variety of definitions and concepts around numeracy, mathematics, quantitative literacy used in international contexts. (e.g. [START_REF] Karaali | What's in a name? A critical review of definitions of quantitative literacy, numeracy, and quantitative reasoning[END_REF][START_REF] Vacher | Looking at the multiple meanings of numeracy, quantitative literacy, and quantitative reasoning[END_REF]. For this paper, the terms numeracy and numeracy skills refer to PIAAC's definition: PIAAC defines numeracy as the ability to access, use, interpret and communicate mathematical information and ideas, in order to engage in and manage the mathematical demands of a range of situations in adult life. (OECD, 2012, p. 34) An alternative view on skills focusses more on their embeddedness in society and in power relations. Being embedded in social constructs means being greatly influenced by power structures. Thus, it is worth asking, whether Craig's 'promises' are equally true for those who acquire them.

Using the example of gender to look at a concrete representation of societal power relations, PIAAC results offer the potential to analyse the intersection of gender and numeracy empirically and with quantitative methods. In this paper, I choose a regression analysis to test the value of numeracy skills in terms of income and the chance of getting into a leadership position. The scope of difference between these values for men and women will also be explored.

Numeracy in relations of power

Distributing numeracy

How do people acquire numeracy skills and why do some attain higher skill proficiencies than others? Focusing on gender differences, the achievement of young boys and girls has received much attention for the past decades and continues to be of high importance (OECD, 2016b). Surveys such as PISA1 and TIMMS2 test students at the ages of 10 (TIMSS) and 14/15 (TIMSS, PISA). These assessments show that boys tend to perform higher on numerical and science-related tests. The difference is small, but nevertheless significant [START_REF] Mullis | Literacy, numeracy and problem solving in technology-rich environments: Framework for the OECD survey of adult skills[END_REF]OECD, 2016b). A comparison between these and further assessments of numeracy and mathematical competencies from the years 1994 to 2007 show that gender differences increase with age and grade. The performance of boy's on numeracy and mathematical proficiency increases from the seventh grade on. The difference between boy's and girl's performance on such assessments can be related to stereotypes and expectancies. Not only do teacher seem to expect and (unconsciously) support different skills and interests of boys and girls, but also, students expectations of themselves have been shown to align with their interests and their self-concept to what they think is expected from them [START_REF] Budde | Mathematikunterricht und Geschlecht: Empirische Ergebnisse und pädagogische Ansätze[END_REF].

Regarding the adult population, results from PIAAC show men scoring higher on the numeracy scale than women with the gender gap greater in older age cohorts (OECD, 2016a). The presented findings on young students and adults show differences in how numeracy skills are distributed among genders.

Determining numeracy

In a comprehensive look at the social aspects of numeracy, [START_REF] Street | Navigating numeracies: Home/school numeracy practices[END_REF] transferred insights from the New Literacy Studies to the field of numeracy and argued that numeracy is a social practice. This idea of a multiple of skill interpretations originates in the New Literacy Studies [START_REF] Street | Introduction: The new literacy studies[END_REF]. Contemplating the great variety of numerate practicesnumeracy itself has to be seen as multiple numeracies. The relativity of numeracy is not only evident in the social distribution of skills but also in determining what constitutes as mathematics or as numeracy. As such, all numeracies are embedded in power relations and subject to the societal hegemony. A variety of studies have demonstrated how numeracy skills and practices are embedded in economic outcomes [START_REF] Grotlüschen | Adults with low proficiency in literacy or numeracy: OECD education working papers[END_REF] or how values, beliefs and power relations determine the idea and the relevance of numeracy and mathematical education (c.f. [START_REF] Ares | Mathematics and numeracy as social and spatial practice[END_REF]. The relativity of numeracy is not only evident in the social distribution of skills but also in determining what constitutes as mathematics and as numeracy. Different studies showed how skills and practices are embedded in economic outcomes [START_REF] Grotlüschen | Adults with low proficiency in literacy or numeracy: OECD education working papers[END_REF] or how values, beliefs and power relations determine the construct of numeracy and mathematical education [START_REF] Ares | Mathematics and numeracy as social and spatial practice[END_REF]. Following postmodern theories, some researchers view mathematics and numeracy to as constructed in discourse in every aspect [START_REF] Ernest | Postmodernity and social research in mathematics education[END_REF]. Velero (2004) suggests this perspective allows us "to perform a very fine grained analysis of how mathematics and mathematics education are used by people in particular discourses and of the effects of those discourses on social practices and, consequently, on people's lives" (p. 11).

Rewarding numeracy

Continuing from the perspective of that power relations shape numeracy conceptualizations and acquisition, the following analyses will show empirically that gender relations affect the value of numeracy skills on the example of labour market outcomes. Regression analyses of PIAAC data will be used to show how numeracy proficiency does not have the same value for everybody. Looking at two labour market outcomes (wages and leading positions), the analyses that follow will reveal how the effect of numeracy skills changes with gender.

Method: Data and sample

The presented research was carried out with PIAAC datasets for 13 European countries. The analysis disregarded those European countries whose PIAAC data did not disclose all variables employed. It therefore includes the following thirteen European countries: Belgium, Czech Republic, Denmark, Spain, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Norway and Poland.

PIAAC scales its numeracy test items based on item response theory (IRT) to account for differences in item difficulties. Based on the IRT scaling of the items and a population model, plausible values are drawn. This approach increased the measurement accuracy [START_REF] Yamamoto | Scaling PIAAC Cognitive Data[END_REF]. PIAAC views numeracy skills on a scale from 0 to 500, which it sections into five levels3 (OECD, 2013).

To exclude a possible bias arising from any gendered occupational segregation, different areas of work were included as control variables in the regression analyses. To exclude any effects of women working part-time, those participants who are unemployed, out of the workforce or work part-time 4 were not incorporated in the analysed data.

Method: Variables and analyses

Variables on monthly income and leadership positions 5 serve as indicators of labour market outcomes. The income variable indicates the monthly income exchanged into US-Dollars and transformed as purchase power parity (PPP). This way, all outcomes are given on the same scale. Both income and leadership variables include employees as well as self-employed workers.

The effect on monthly income was regressed with a multinomial linear model. Logistic regressions were used for the leadership positioning. On both occasions, three models were employed: Model 1 regressed against male gender and numeracy; model 2 added background variables (b.v.) 6 and model 3 regressed against the interaction term between numeracy and male gender. For the analysis on monthly income, the average work hours per week (h/w) were also controlled for in the first model.

Result: Same numeracy, wages

Table 1 shows the result of the first two models on the monthly wages; showing significant results 7 .

Table 1: Difference of monthly income of men and women with the same numeracy proficiency

The results indicate how much higher men's monthly wages are when having the same average numeracy proficiency as women. A comparison of the two models suggests that structural differences cannot explain men's higher wages, as in most countries the differences in income rise in the further adjusted model. In Norway, for example, women on average earn almost 1.000 USD (PPP) less than men despite having the same level of skill proficiency.

6 Background variables include: Age (AGE_C), educational qualifications (EDLEVEL3), the parent's educational background (PARED), native language (NATIVELANG), different fields of work (ISCO1C) and the average hours of work per week (D_Q10) The last column in Table 1 shows how many points on the numeracy scale are the equivalent of the gender difference in income. Men in these European countries seem to score from 30 (Germany, Ireland) to 60 (Finland, Norway, Greece, Belgium) to 88 (Italy) points lower on the 500 scale when earning a similar income to women. There are no significant differences in Denmark, Spain, Netherlands, and Poland.

Result: Exchanging numeracy into wages at different rates

Table 2 presents the third model of regression on the monthly income. The analysis tested for the increase of income against numeracy proficiencyagain, not for individuals but structurally. The table indicates how much additional skill points relate to higher income for men in comparison to women. For all genders, higher numeracy scores relate to higher income. However, for men higher numeracy points relate to a higher increase in income. The difference is smallest in Ireland (with Germany as a close second). Here, a man's increase equals 117 percent (118 percent) of women's.

In Italy, men's income increase equals more than 140 percent of the average woman's increase in income.

Country

Table 2: Average value of men's numeracy in relation to women's

There are no significant differences in Denmark, Spain, Netherlands, Poland and the Czech Republic. For all thirteen European countries, higher numeracy scores relate to an increase of income by 123 percent of women with equivalent scores.

Result: The probability of leadership

Regarding the probability of men and women of getting into a leading position, we found similar results, which are shown in Table 3. These results show the effect coefficient of the binomial logistic regression analyses. The effect coefficient is the exponent of the regression results. In both models, men show a higher probability of being in a leadership position. In the unadjusted model, men in Italy and Germany are between 14 and 20 percent more likely to be in a leadership position when having equal numeracy skills as women. In Finland and Belgium, men are 81 and 89 percent more likely to be in a leadership position.

In the adjusted model, men in Spain and Finland are almost twice as likely to be in a leadership position as women with similar numeracy proficiency, educational background, in the same field of work, and with further background variables controlled. In the Czech Republic they show a more than two and a half times higher probability.

Women in Finland or Spain are almost half as likely to get into a leading position with the same numeracy proficiency, education and further background variables. There are no significant results in Ireland.

Averaged for all analysed European countries, men's probability to be in a leadership position is 148 percent of that of a woman with equivalent numeracy skills. When controlled for background variables, the overall average for men is 59 percent higher in relation to women.

Country

Discussion

As a consequence of the use of IRT, plausible values and the nature of our calculations, these results cannot be interpreted at the level individual case. In addition, countries' results can't be compared because of the different economic positions and related purchasing power within countries. However, the strong and significant correlation in at least nine of the 13 analysed countries indicate strong gender differences in the way (numeracy) skills can be transformed into labour market outcomes.

Furthermore, it seems numeracy does not correlate (strongly) with all job-specific skills. In contrast, there is no reason to assume that the gender difference in numeracy is bigger than in other jobspecific skills.

In this paper I have argued that numeracy is embedded in hegemonic structures of society in (at least) three specific ways: (1) influencing what is recognised in definitions of numeracy, (2) selecting who acquires numeracy, and (3) determining the societal value of numeracy skills. The analyses of PIAAC data could show women's numeracy skills relate to a lower income and to lower probabilities of being in a leadership position. On a scale of 500, men score between 32 and 88 point lower than women while having a similar income. With the same numeracy proficiency, men on average earn between 300 and 750 USD more than women. Similarly, men's probability of getting into a leadership position is around one and a half times that of women with the same numeracy proficiency.

In some cases, the adjustment of the model for various background variables increased the respective coefficients. This might indicate, that part of the gender differences in the labour market are intensified by (some of) those background variables, like different fields of work or educational attainments. This result, however, cannot be explained by the first model.

In conclusion, this study provides evidence that numeracy skills are entwined in power relations. A multitude of factors, which -in some way or anotherappear to be influencing women's opportunities along gender lines, accumulate to different outcomes in the labour market. My analysis indicates that higher levels of numeracy skills relate to an increase in labour market outcomesbut it is not an equal increase for everybody. This might suggest a deeper reflection on our concepts of numeracy and skills and our willingness to use real-life outcomes as promises for numeracy education.

Men's increase in income compared to women's increase

	Belgium	138 %
	Finland	130 %
	France	122 %
	Germany	118 %
	Greece	122 %
	Ireland	117 %
	Italy	141 %
	Norway	129 %

Table 3 : Odds ratios of logistic regression results: average probability of men getting into a leading position compared to women with equal numeracy skills

 3

		not adjusted	Adjusted for b.v.
	Belgium	1.89 ***	1.62 ***
	Czech Republic	1.75 ***	2.60 ***
	Denmark	1.77 ***	1.52 ***
	Spain	1.75 ***	1.93 ***
	Finland	1.81 ***	1.95 ***
	France	1.68 ***	1.65 ***
	Germany	1.21 *	n.s.
	Greece	1.52 **	n.s.
	Italy	1.14 ***	1.61 ***
	Netherlands	1.57 ***	1.37 **
	Norway	1.55 ***	1.26 *
	Poland	1.40 **	1.72 ***

Programme for International Student Assessment (by OECD)

Trends in International Mathematics and Science Study (by IEA, International Association for the Evaluation of Educational Achievement)

The score boundaries for numeracy in PIAAC are: 0-175 points (below level 1); 176-225 points (level 1); 226-275 points (level 2); 276-325 points (level 3); 326-375 points (level

4); 376-500 points (level 5).4 Excluding all participants who work less than 31 hours in an average week (D_Q10).5 Variables names are EARNMTHALLPPP for monthly wages and J_Q07a and J_Q08a for leading positions.


Kees Hoogland
email: kees.hoogland@hu.nl

Karen Heinsman
email: karen.heinsman@hu.nl

Paul Drijvers
email: paul.drijvers@hu.nl

Numeracy and mathematics education in vocational education: a literature review, preliminary results

Keywords: Numeracy, mathematics, vocational education, VET

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 stream in education for 12-to 20-years-olds in the Netherlands and the numeracy and mathematics curriculum is on the brink of a reform. To assess what is known from research on numeracy in vocational education, we are in the process of conducting a systematic review of the international scientific literature of the past five years to get an overview of the recent developments and to answer research questions on the developments in vocational educational practices. The work is still in progress. We will present preliminary and global results. We see vocational education from the perspective of (young) adults learning mathematics.

Introduction Practical background / Our aim

Vocational education in the main stream of the educational system in the Netherlands (preparatory vocational for 12-to 16-year-olds and specific vocational for 17-to 21-year-olds) is on the brink of bringing the curriculum up-to-date to the needs of the current society and the contemporary workplace. In the 1990s the curriculum was built around four domains: numeracy, algebra, geometry & measurement, and information processing & statistics. In that timeframe the curriculum was considered quite modern and ahead of developments in other countries, mostly caused by the Realistic Mathematics Education movement which was inspired by the "mathematics is a human activity" -approach by Hans Freudenthal [START_REF] Gravemeijer | Developing realistic mathematics education[END_REF].

In the 2010s, however, from an alleged low level of arithmetic skills of the students, a strong topdown ministerial policy was introduced by adding to the existing mathematics curriculum a separate numeracy/arithmetic programme with a digital nation-wide summative examination. After 10 years of piloting and political quarrels, this path was abandoned again. Now new paths are explored to see whether the vast body of knowledge on mathematics for the workplace and mathematics for the future citizens can provide valuable clues for a modern, relevant and effective numeracy or mathematics curriculum for students who will work with mathematics in very practical situations and who will have to deal with the quantitative world around them. From this perspective the students in vocational education are seen as young adults who can benefit from the body of knowledge collected around adult mathematics education.

From this perspective we started a review of the most recent literature on numeracy in vocational education.

Theoretical background

For a theoretical background, we address the following issues:

 Societal changes  Shifts in mathematics education  Previous reviews
To start with, there are major societal and policy pressures on (mathematics) education to prepare young learners and school leavers for a complex and technologised society. In literature, these new educational goals are referred to as 21 st century skills or 21 st century competences [START_REF] Voogt | A comparative analysis of international frameworks for 21st century competences: Implications for national curriculum policies[END_REF]), global competences (OECD, 2016) or the 4 th industrial revolution [START_REF] Schwab | The fourth industrial revolution: What it means, how to respond[END_REF]. Common among the concepts mentioned above is the acknowledgement that across education, government, and business, the skills and knowledge needed to succeed in work, life and citizenship have significantly changed in the 21 st century. As workplaces are being transformed, the skills and knowledge workers now need are intertwined with technology. 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 demand for skills as well.

These major societal and policy pressures translate also into pressures on mathematics education to prepare young learners and school leavers for a complex and technologised society. For instance, PriceWaterhouseCoopers (2015) reported recently that businesses "competing in a global economy driven by data, digital technologies and innovation will need more employees trained in science, technology, engineering and mathematics (STEM). Research indicates that 75 per cent of the fastest growing occupations now require these skills" (PWC, 2015, p. 4). Similarly, in their recent 2017 review, the National Council of Teachers of Mathematics (NCTM) (2017), stated that "Today mathematics is at the heart of most innovations in the information economy. Mathematics serves as the foundation for STEM careers and, increasingly, careers outside STEM, and mathematical and statistical literacy are needed more than ever to filter, understand and act on the enormous amount of data and information that we encounter every day" (p. 6). Many of the mentioned changes are related to technological developments, particularly information and communications technologies (ICT), and have profoundly altered what are considered to be the key knowledge and skills that individuals need as economies and society continue to evolve.

There is a growing body of knowledge on the desired numeracy and mathematics competences in workplaces (D. [START_REF] Coben | Benchmark assessment of numeracy for nursing: Medication dosage calculation at point of registration[END_REF][START_REF] Geiger | A rich interpretation of numeracy for the 21st century: A survey of the state of the field[END_REF][START_REF] Hoyles | Global competency for an inclusive world[END_REF][START_REF] Hoyles | Improving mathematics at work the need for techno-mathematical literacies[END_REF][START_REF] Straesser | Numeracy at work": A discussion of terms and results from empirical studies[END_REF][START_REF] Wake | Preparing for workplace numeracy: A modelling perspective[END_REF]. Consistent in the reported results are conclusions which boil down to the observation that the mathematics tasks which people undertake at work in the 21 st century involve more than plain calculation skills or straightforward procedural proficiency. Nowadays, workplace practices involve sophisticated mathematical problem-solving skills and the ability to recognise and engage with the mathematics embedded within multifaceted, real-life workplace settings. Many 21 st century workplace mathematics requirements and practices are integrated with technology. [START_REF] Hoyles | Improving mathematics at work the need for techno-mathematical literacies[END_REF] argue that this requirement for mathematical capabilities is driven by the need to improve production processes and productivity, and that there will be greater demand for what they call 'techno-mathematical literacies'. This involves "a language that is not mathematical but 'technomathematical', where the mathematics is expressed through technological artefacts" (Hoyles et al., 2010, p. 14). Furthermore, the pressures for moving toward the teaching and learning (and assessment) of 21 st century skills include a consistent demand for students to be digitally and technologically competent, to be able to engage with technology in all its guises, and to be highly ICT literate [START_REF] Voogt | A comparative analysis of international frameworks for 21st century competences: Implications for national curriculum policies[END_REF].

There have been a few recent reviews on related topics. At the start of this century [START_REF] Coben | Adult numeracy: Review of research and related literature[END_REF] did a comprehensive review on adult numeracy focusing on research and related literature. The results of this review are relevant to our research questions as far as vocational education can be considered as (young) adults who are learning mathematics. This is indeed the perspective we chose.

The main conclusions from Coben's review were:

 Numerate practices are diverse and deeply embedded in the contexts in which they occur.  The transfer of learning between contexts is problematic.  The impact of adult numeracy tuition is under-researched and more detailed studies are required.  Adult numeracy teacher education needs continuing professional development.

At the same time, [START_REF] Falk | Literacy and numeracy in vocational education and training -review of research[END_REF] conducted a review of research on literacy and numeracy in vocational education and teaching from an Australian perspective. Their main conclusions were: Significant and multiple implications for literacy practice have emerged from forces for change such as globalisation and technological advancement. (…) Adult and Community Education (ACE) and VET sectors have been merged to a large extent. These changes have had important consequences for the way practitioners carry out their work. (Falk & Millar, 2001, p.3) The overall picture of adult education and vocational education and training (VET) is that it is a rapidly developing field with a strong focus on connecting mathematics education to the world around us. Effective practices to incorporate these ideas in courses and educational programmes are still under-researched.

Research questions

In the Netherlands a broad curriculum reform has started which includes redefining the mathematics curriculum in vocational education. This review is also intended to collect scientific results of research on vocational policies and practices to support this process and make it more evidenceinformed.

To get a better grip on how the concepts of numeracy and mathematics are used in the international scientific community our first research question is:

1. Which concepts of numeracy (or mathematics) are used in research in a vocational context? Furthermore, we are interested in well-researched practices in mathematics or numeracy education in vocational tracks or streams. Hence our second and third research questions: 2. Which numeracy teaching practices are described in research in a vocational context? 3. Which effects of numeracy teacher practices are reported in research in a vocational context?

The review of literature is still ongoing. The results we report in the results section are preliminary and still quite general.

Method Search terms

We conducted a systematic review [START_REF] Gough | An introduction to systematic reviews[END_REF][START_REF] Petticrew | Systematic reviews in the social sciences: A practical guide[END_REF] of the research literature through the definition of search terms derived from the research questions.

In various countries the concept numeracy is used when referring to the mathematical activities in the vocational classroom, for instance in Australia. In The Netherlands two words are used to distinguish the more basic numeracy as is taught in primary education (Dutch: rekenen) and the more advanced use of mathematics as is taught in secondary education (Dutch: wiskunde). In most countries, however, mathematics is used to refer to all the kinds of activities which deal with the numbers, patterns and structures in the vocational education or in the workplace.

Regarding the specific part of the educational infrastructure we are focusing on, two labels are used most common to refer to this specific kind of education which prepares students for the workplace. These labels are "vocational education" and VET, which is short for vocational education and training. So our main logical string of search terms is:

Search string = (Numeracy OR mathematics) AND ("vocational education" OR VET).

Search engines and databases

We used our search string in a variety of educational databases which together assemble most international educational research: WorldCat, ERIC, PsycInfo, Web of Science, Scopus, and Education Research Complete.

Scope

We narrowed down our search using the following limitations:

 Scientific literature  Published in peer-reviewed journals  English language  Recent, last five years, date of publishing: 2014-2018 or 2014+.  Search terms present in title, abstract, or keywords.

Further narrowing down

The search-string led to 611 hits in the selected databases. There were several ways in which we narrowed down the amount of initial hits (see Figure 1).

First, the articles in this 'technological catch' were brought together, deduplicated and checked for obvious mishits, such as articles on veterans or veterinarian issues. This resulted in a set of 477 articles to be considered by experts.

Figure 1: Search strategy and remaining articles after each step

Second, the first expert screening followed, based on the titles of the articles. Three experts scored independently whether the article was supposed to be relevant for the three respective research questions. For each a score 1 (relevant), 2 (possibly relevant), 3 (maybe relevant) or 9 (not relevant) was given. After this, articles with three 9-scores by each experts (i.e., not relevant for any of the research questions) were discarded from the selection. This reduced the set to 93 articles remaining. Third, the abstracts of the remaining articles were scored by three experts, again independently. The experts decided from the content of the abstract whether they supposed the article to be relevant for either of the three research questions. After this, again articles with three 9-scores by each expert (i.e., not relevant for any of the research questions) were discarded from the selection. This reduced the set to 78 articles remaining.

Fourth, the 78 articles were linked to the respective research questions to be examined further: 65 articles were judged to be relevant for research question 1 about the definition of numeracy, 12 articles seemed relevant to answer research question 2 about teaching practices and 4 articles were scored as relevant for research question 3 on the effects of teaching practices.

Fifth, the full text of the remaining articles are being analysed for relevant sections which can be used to answer the actual research questions. This analysis is ongoing.

Cross-referencing and snowballing

In the next phase of our research, we will cross-reference our set of relevant sources with Google Scholar, which is nowadays considered to be the search engine which uses the most comprehensive database of products from the scientific community. However, the ways to refine searches in Google Scholar are still rather limited.

In a preliminary analysis of the remaining articles we found multiple references to specific books and journal specials (for instance, ZDM 2015, 47-4, and Educational Studies in Mathematics 2014, 86-2) of particular interest for our research questions. In a later stage of our research we will add some of these sources to the set of sources we use as corpus to answer the research questions.

Results

The review is ongoing. The results and conclusions in this paper must be read as preliminary findings.

Regarding research question 1 -on the used definition of numeracy -we found that a large part of the articles address policy matters, like comparative studies (for instance PIAAC), country policies, regional and school policies, most of the cases assuming the definition of numeracy from the PIAAC framework as the working definition. In articles, which were not on policy matters, in roughly half the cases the definition of numeracy is not explicitly defined, making it difficult to value the inferences made from the gathered data. This problem will be elaborated in a later article.

Regarding research questions 2 and 3on actual practises -, the major conclusion is that from the found hits and relevant articles only a very few are addressing actual educational practices on numeracy in vocational education (12 of 477) and even fewer (4 of 477) report on effects of educational practices or interventions. Most of the 12 articles addressing educational practices describe observations in classrooms, providing conclusions and implications for education based on these observations. These articles tend not to be on describing current teaching practice and their effects, but in most cases report on how the authors would like to see the teaching practice change.

The importance of real context in education for transfer of the numeracy-skills and motivation of the students is emphasized.

Some themes are reoccurring in the selected articles. First, all articles describe the importance of connecting numeracy to work-related situations. This transfer or boundary crossing is important since students can solve mathematical problems in class, but often do not recognize these problems in work-related knowledge; the integration of mathematics and statistics learnt in school with work related knowledge is a problem [START_REF] Bakker | A boundary-crossing approach to support students' integration of statistical and work-related knowledge[END_REF]. [START_REF] Wake | Making sense of and with mathematics: The interface between academic mathematics and mathematics in practice[END_REF] (2015) show in their research how in a math computer game proportional reasoning increases, but analyses of transfer problems showed that there was no measurable transfer. They conclude students in vocational education have trouble applying school knowledge in another context. Van Schaik, Terwel and Van Oers (2014) describe a science workplace in which students have to build a trike. This article shows how numeracy is predominantly hidden in other subjects, like science. And also in science transfer is difficult for teachers to achieve. When given a practical assignment, students seem to forget the need to apply the knowledge they learned before, but also teachers do not always teach their students the missing school-math when they need it and connect the classroom knowledge with the practical assignment.

Only four articles provide experimental research in which a specific teaching practice, differing from the regular approach is being implemented. These articles are too small in number and too divers to find trend-setting answers to our third research question.

Conclusion and Discussion

When we used the search string (Numeracy OR mathematics) AND ("vocational education" OR VET) is used in Google Scholar for products since 2014, we found 16,500 hits. There is no lack of output around these themes. Refining the sources to a corpus of relevant literature from which some conclusions could be drawn or trends could be discerned is not a straightforward task. The sources are multifaceted and of very different quality, scope and choice of themes.

The articles are from dozens different journals, which reflect the multifaceted character of both the concepts numeracy and vocational education. It also reflects that the research area is young, not well-established and literally 'all over the place'.

The definition of numeracy used in the articles is influenced heavily by the work around the PIAAC Numeracy Framework. In that sense the PIAAC programme has a formatting power on the development of the concept of numeracy.

  speaks about 'hidden mathematics'; workers (and students) do not recognize mathematical situations as such and therefore do not use school knowledge in work-situations.[START_REF] Dalby | Connecting mathematics teaching with vocational learning[END_REF] as well as[START_REF] Coben | Meeting the mathematical demands of the safety-critical workplace: Medication dosage calculation problem-solving for nursing[END_REF] stress the use of 'real work' problems in which students can use mathematics as a tool, instead of the mathematics they know as a school subject. Contextual math provides motivation since students experience the need of it for their own work. According to[START_REF] Bakker | A boundary-crossing approach to support students' integration of statistical and work-related knowledge[END_REF] students themselves report to appreciate a work-related approach and claim they learned a lot in only five one-hour meetings. Second, a reoccurring theme is problematic transfer from numeracy interventions to practice. Ter Vrugte et al.


Kees Hoogland
email: kees.hoogland@hu.nl

Marlies Auer
email: marlies.auer@bfi-ooe.at

Javier Díez-Palomar
email: jdiezpalomar@ub.edu

Niamh O'meara
email: niamh.omeara@ul.ie

Mieke Van Groenestijn
email: mieke.vangroenestijn@kpnmail.nl

Mieke Van

Groenestijn Ini- Tiating

Initiating a Common European Numeracy Framework

Keywords: Numeracy framework, professional development, pilot courses, Erasmus+

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In 2017 four educational institutes from The Netherlands, Austria, Spain, and Ireland respectively, took the initiative to develop a numeracy framework for adults. A grant was awarded by the European Union in an Erasmus+ project to work towards such a framework in the years 2019-2021.

As result of this project a tentative Common European Numeracy Framework (CENF) will be established and based on this framework a set of professional development modules (PDM) for adult numeracy educators will be developed. The CENF will incorporate the latest insights into numeracy skills and competencies which are relevant for our technologised and numbers-drenched society.

The suite of professional development modules developed for teachers and volunteers who teach on numeracy courses will be based on the CENF. The PDMsvwill foster the capability of teachers and volunteers to effectively develop basic skills and competences among adult learners in order to strengthen their employment prospects, develop their socio-educational and personal skills, and counteract social exclusion. It will also equip adult learners with the skills highly needed to participate in civic and social life.

In short, the objective of this project is to first develop a Common European Numeracy Framework and secondly to expand and to improve the number and the quality of numeracy courses (or other relevant educational activities) offered to adults to improve their life chances of prospering in society. To reach this objective, we will use on a multi-level approach for addressing teachers and volunteers that are involved in the delivery of numeracy courses, the teacher educators who deliver courses to those teachers and volunteers, and policy makers and other stakeholders who are responsible for creating opportunities for such courses. This multi-level approach already has started by connecting the associated partners to our goals in this application.

The CENF will comprise of information on the effective teaching and learning of numeracy, especially in relation to adults who have left school early and those from disadvantaged backgrounds. Such a framework can also be used as a mechanism for systemically tracking and monitoring the progress of adult learners in the field of numeracy. The professional development modules will be made available on-line in an attempt to actively contribute to extending and developing educators' competences and maximize the chance of European wide use of the project outputs.

Background

Too many European citizens lack the necessary numeracy competencies to participate autonomously and effectively in our technologized and number-drenched society. Consequently many citizens are overlooked for certain jobs and have problems in their daily life, dealing with the abundance of number-related issues. In literature, these competencies are referred to as 21 st century skills' or '21 st century competences [START_REF] Voogt | A comparative analysis of international frameworks for 21st century competences: Implications for national curriculum policies[END_REF], global competences (OECD, 2016a) or skills for "the 4th industrial revolution" [START_REF] Schwab | The Fourth Industrial Revolution: what it means, how to respond[END_REF].

The results of the last PIAAC survey (OECD, 2013(OECD, , 2016b;;PIAAC Numeracy Expert Group, 2009) show that a quarter of the participating countries in PIAAC have results below level 2 of the 6-point scale. Scores below level 2 are associated with having problems in daily activities due to low numeracy skills. These results on numeracy give rise to serious cause for concern for the future 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 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 2020 (Hoogland, Diez-Palomar, & Maguire, 2019;[START_REF] Kirsch | PIAAC: a new design for a new era[END_REF][START_REF] Tout | Review of the PIAAC numeracy assessment framework: Final report[END_REF].

On average, school leavers between the age of 16 and 24, perform at a lower level than people between the ages of 25 and 44 (OECD, 2016b). The lack of numeracy skills among these people increases the risk of unemployment and may influence family life and social inclusion. In particular, low-skilled workers are at risk in the labour market, particularly when faced with adaptations in the labour market. The Council of the European Union emphasizes that adult learning is a means for upskilling or reskilling those affected by unemployment, restructuring and career transition, while simultaneously it makes an important contribution towards social inclusion, active citizenship and personal development. The European Council recommends the enhancement of basic skills including literacy, numeracy, problem solving and digital skills as part of the Europe 2020 Strategy (European Commission, 2017).

In the frame of lifelong learning, therefore, adult numeracy education has an important role in the development of good programs for basic skills for the future.

Evidence-informed adult numeracy education may be the key for attaining the goals set out in the Europe 2020 strategy. In most European countries, adult numeracy education is a locally based endeavour with a plethora of practices, some efficient, some less so. Furthermore, there is a variety of underlying assumptions on what constitutes good adult numeracy education. The availability of a good collection of (piloted) professional development modules, based on the CENF (which contains relevant content and insights for adults learning numeracy and mathematics), will generate a quality adult numeracy education across Europe, and thereby contribute to policies and activities which address the low-numeracy levels in many European countries. Effective numeracy education in European countries, based on a common framework, may lead to a higher level of societal participation and inclusion of adults, and thereby to the improvement of the European economy.

Project methodology as starting point for workgroup discussion

In this section we present the methodology which will be used in the Erasmus+ project (Hoogland, Diez-Palomar, & Vliegenthart, 2019). In the working group of CERME11 we will use these headings as a guide for the discussion. The aim of this discussion paper is to inform the participants of TWG07 and to tap into their knowledge, experiences, and networks to strengthen the impact of the project.

The project methodology can be laid out as follows.

The context

The number of European citizens lacking the necessary numeracy competences to participate autonomously and effectively in our technologised and number-drenched society is a serious concern for the economic development of Europe. (OECD, 2012(OECD, , 2013(OECD, , 2016b;;PIAAC Numeracy Expert Group, 2009). The European Council recommends enhancement of the basic skills -literacy, numeracy, problem solving and digital skills -as part of the Europe 2020 Strategy (European Commission, 2017).

Evidence informed adult numeracy education is the key for reaching the goals as mentioned in the Europe 2020 strategy. In most European countries adult numeracy education is a locally based endeavour with a plethora of practices, some efficient, some less so and with a great variety of underlying assumptions on what constitutes good adult numeracy education. A good set of (triedout) professional development modules based on a common European numeracy framework (which contains relevant content ánd insights in adults learning numeracy and mathematics) will generate a qualitative and quantitative impulse in adult numeracy education across Europe, and thereby contribute to policies and activities which address the low-numeracy levels in many European countries.

The objectives

The ultimate objective of the project is to enlarge the number and to improve the quality of numeracy courses (or other relevant educational settings) offered to adults to improve their chances in society. To reach this objective we develop a Common European Numeracy Framework (CENF) and set of professional development modules (PDMs)

The target group and the participants in the project.

The ultimate target group of the project are the participants in adult numeracy courses. The aim is to provide them with numeracy competences which will benefit them in participating adequately in daily life and work-related situations in our number-drenched society. Numeracy to this regard is considered to be a social practice (Yasukawa, Rogers, Jackson, & Street, 2018). However, the availability and quality of adult numeracy courses around Europe are dependent on many factors, of which many are outside the scope and influence of this project. Therefore, we choose a multi-level approach of addressing teachers and volunteers who are actually involved in the numeracy courses, the teacher educators who provide courses to those teachers and volunteers, and policy makers and other stakeholders who are responsible to create opportunities for such courses (both by teachers and by teacher educators). In doing so, the project will involve the local, regional, national and international communities of practitioners, researchers, and policy makers around adult numeracy education by information, surveys, presentations, and presenting a CENF.

The activities

On the basis of literature studies and broad European surveys on used frameworks, good practices, and theoretical insights we will develop a draft framework and draft modules. This will be followed by try-outs in the partner countries to test the drafts empirically and tp collect feedback from various stakeholders from the adult numeracy community. The framework and modules will be revised with the gathered experiences, evaluations and comments. The end result will be made available as a blended on-line course, while the availability as cMooc will be investigated.

The methodology

The ways of gathering the information are literature reviews, surveys and peer reviewed development of products, according to academic standards for research methodology.

The methodology for dissemination, implementation and sustainability is based on the multilevel approach of educational changes as for instance propagated by Hargreaves, Lieberman, Fullan, and Hopkins (2010) and [START_REF] Fullan | The six secrets of change: What the best leaders do to help their organizations survive and thrive[END_REF]; [START_REF] Fullan | Breakthrough[END_REF].

The result

The result will be an increase in the quality and quantity of numeracy courses and professional development courses on adult numeracy education. The concrete results of the project are a CENF and a set of PDMs for teachers and volunteers who work in adult numeracy education. Both output we consider as essential in reaching the ultimate objective.

The impact

The existing European networks on adult numeracy education support the project as associate partners and express their willingness to advocate the results through their network communications and in their network meetings. The CENF will have an impact across Europe.

Long term benefits

A common approach and a reinforcement to create high quality national professional development courses based on a common framework can have a long-lasting effect on the development of high quality adult numeracy education around Europe.

Discussion

The discussion points on the starting project followed the project methodology. Below, to each discussion point a short outcome of the discussion in the Thematic Working Group is mentioned

The context and the objectives Q: What should be added to the context to make the picture more complete or richer in examples?

A: Special attention is needed for participants who are vulnerable learners, often with negative educational experiences.

Q: Can a Common European Numeracy Framework be helpful in the further development of adult numeracy education?

A; Yes, as long as it is developed bottom-up and connects to existing practices and frameworks.

The target group and the participants in the project.

Q: Should the framework aim at participants in adult numeracy courses, at practitioners in adult numeracy education in the European countries, at adult numeracy course designers, or at policy makers?

A: Both practitioners and policy makers must be able to read and understand the framework. For participants materials can be designed which are closely relate to the framework, for instance rubrics or descriptions of typical behaviour.

The activities, the methodology, the results of the project Q: Which other activities are paramount to reach the envisioned goals?

A: The project would strengthen as the participants' voices were heard in the developmental stages of the framework. The participants are adults who have an opinion on how learning goals could and would look like.

Q: Which activities should be added to make the results stronger and more widely-spread?

A: The design and the implementation of the framework would benefit by international parallel research and validation of parts of the framework

The impact and the long term benefits Q: Which actions might increase the chance to reach a really common European Numeracy Framework on adult numeracy education? Who are the most important stakeholders?

A: Both policy makers and networks of practitioners and researchers are important. Translating the framework in as many European languages as possible can help the acceptance and use.

The results of this discussion will be input for the further development of the Common European Numeracy Framework..

Acknowledgement

This project is carried out in an European project Common European Numeracy Framework with the support of the Erasmus+ programme of the European Union.


Beth Kelly
email: beth.kelly@ucl.ac.uk

What motivates adults to learn mathematics through trade unions in the workplace: social factors and personal feelings

Keywords: Mathematics, motivation, teaching methods, affect, emotions

This 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 experiences. The term an '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.

Introduction

There is much research into learning mathematics that seeks to explain why children and adults have problems with the subject and subsequently struggle to learn mathematics [START_REF] Boaler | Multiple perspectives on mathematics teaching and learning[END_REF][START_REF] Coben | Effective teaching and learning numeracy[END_REF]. Those interviewed for this research reported obstacles to their learning such as; previous unsuccessful learning experiences, demands of shift work and family responsibilities. Yet they had chosen to re-engage with learning mathematics at or below GCSE level in the workplace through opportunities negotiated by trade unions with employers. It was not a requirement of their current job, so this group of adults must have been highly motivated to return to learning mathematics and were chosen to develop a better conceptual understanding of motivation.

The adults in my research have taken up a learning opportunity at work enabled by national trade unions allocating resources to this purpose. Trade Union officers, usually ULRs, are activists whose job it is to encourage workers to develop their skills, give advice and guidance, identify workers needs, and negotiate on-site learning opportunities with employers). Trade unions fund the teachers and equipment needed to support that learning, while employers provide the physical space and sometimes time off work to learn. In 2016, nearly 10,000 ULRs across the UK were involved in training through TUC Education and over 27,000 individuals took up an opportunity to improve their English and maths skills through their union (Unionlearn, 2017, p. 4). This research identifies the importance of how people feel about the learning in relation to their motivation. Adults need to trust people who are encouraging them to join the classes and feel safe in the learning environment in order for them to develop confidence with mathematics. Union Learning Representatives (ULRs), are trusted colleagues that play a key role in encouraging and supporting adults to join classes in the workplace. Trade unions also promote a 'different' learning approach in their courses, influenced by the 'collective ethos' of trade union culture. The learners identify this different pedagogic approach as key to them persisting learning [START_REF] Kelly | Motivating adults to learn mathematics in the workplace: a trade union approach[END_REF].

Research into learning in the workplace has usually focused on mathematical concepts used for work or the more contentious notion of the 'transferability' of mathematical concepts between school and the workplace [START_REF] Lave | Situated learning: legitimate peripheral participation[END_REF][START_REF] Fitzsimons | Review of research into adults learning mathematics[END_REF]. In an endeavour to understand differences in learning approaches research has also usefully included descriptions of pedagogical characteristics [START_REF] Evans | Critical perspectives on adults' education[END_REF] in terms of formality: this research concentrates on adults' motivations to learn mathematics in the workplace, to gain a formal qualification in a less-formal learning setting.

Research into affective issues in mathematics education have been explored by [START_REF] Mcleod | Research on affect and mathematics learning in the JRME: 1970 to the present[END_REF], who conceptualised emotions, attitudes and beliefs in relation to motivation in terms of stability and intensity. Eynde, De Corte and Vershaffel (2006), posited motivational concepts act in different levels of social contexts. [START_REF] Hannula | Exploring new dimensions of maths-related affect: Embodied and social theories[END_REF] further theorised affect in mathematics education suggesting it could be explored in three dimensions, as concepts: cognitive, affective or motivational; as varying in relation to time and in relation to ontological traditions of research. This moves us on from simply identifying 'maths anxiety' [START_REF] Buxton | Do you panic about maths?: coping with maths anxiety[END_REF] as an emotional barrier to learning, to exploring the wider, more complex domain of affect in relation to motivation to learn.

During this research, the concept of an Affective Mathematical Journey (AMJ) was developed to describe changes in learners' feelings in relation to mathematics. An AMJ occurs when positive (or negative) learning experiences inspire adults to modify their feelings towards mathematics. In this research, the change in affect becomes more positive and this change not only contributes to adults' increased confidence and motivation to learn the subject but also influences their potential to use their new skills beyond the classroom. [START_REF] Elliot | Handbook of competence and motivation: theory and application[END_REF] research supports these findings recognising the possibility of 'socialisation practices' to foster different mind-sets in learners which have a 'cascade of effects, altering their meaning systems and their academic outcomes' (p. 135).

Research by [START_REF] Hannula | Affect and mathematical thinking. Exploring developments, trends and future directions[END_REF] highlighted the tradition of research into affect and mathematics education in CERME conferences, but the focus is still on school education. [START_REF] Boaler | Multiple perspectives on mathematics teaching and learning[END_REF] research into motivation and mind-sets has also focused on children. This research points to the contribution that adults who experience a 'stalled education' can add to research into affect and mathematics education.

Methodology and methods

The analysis is based on twenty in-depth semi-structured interviews with members of trade unions, aged between 25 and 65, twelve males and eight females. They come from a range of trade unions, employed in a variety of contexts, in small and large organisations, publicly and privately owned. They studied in a variety of environments; individually following on-line courses with tutor support or in teacher-led groups. However, this is small sample with no claim to be representative of the characteristics of the broader union learning population, especially since learners who were studying basic mathematics up to level 2, the equivalent to a GCSE.

The identification and recruitment process of interviewees relied upon the agreement and support of trade union officers and ULRs. Hence the method of sampling was a combination of purposeful, 'criterion sampling' (Bryman, 2012, p. 419), as I was seeking adults that fulfilled a particular set of criteria and snowball sampling (Atkinson & Flint,p. 275) in that I needed key gatekeepers (ULRs) to agree to help me access potential participants. The research centres on individuals and their 'interpretations' of situations, so naturally relies on qualitative data, which was tested for the trustworthiness and authenticity of the findings (Bryman, 2012, p. 390). The whole interview and storage process followed BERA (2011) guidelines, ensuring interviewees anonymity especially when their words would be used in the research. Nvivo ™ was utilised to help track and analyse the learner's words. Enabling me to interrogate the data and build and group ideas in different ways; as well as identifying commonly used words such as 'confidence'.

The analysis takes a critical approach because the researcher is a teacher trainer who explores how research into adults learning in the workplace might usefully influence mainstream mathematics education practice. The research is also built on the feminist research tradition of exploring personal viewpoints, as Reinhartz and Davidson (1992) suggests when citing Datan (1989, p. 175) 'it is an axiom of feminism that the personal is political' (p. 234). Therefore, when analysing the adults' words there is a subjective encounter with the individual that seeks to understand different perspectives, relating this motivation at the individual level to emotions and cognition, within particular social contexts.

Hence in this research, motivational factors are considered at three levels: the individual, the social and the societal, which all influence each other. At the individual or personal level [START_REF] Mcleod | Research on affect and mathematics learning in the JRME: 1970 to the present[END_REF] focuses on needs and goals. Evans, Wedege and Yasukawa, (2013) broaden the notion of personal motivational factors to include 'the cognitive and affective as part of the whole person ' (p.222). In this research individual learners feelings are key to their descriptions of their motivations to learn mathematics, reflecting [START_REF] Hannula | Exploring new dimensions of maths-related affect: Embodied and social theories[END_REF] dimension of the 'domain of affect' linking cognitive, motivational and emotional factors.

At the social level, local face-to-face groups include people who adults trust and influence their motivation [START_REF] Barbalet | Social emotion: confidence, trust and loyalty[END_REF], such as; fellow learners, families, union members and Union Learning Representatives. Families play an important support role but ULRs and fellow learners also play a key role in motivating trade union members to engage with learning and persist in classes.

The wider socio-economic environment, is a context where individual members are given a 'second chance' to learn mathematics through trade unions who have negotiated resources with both central government and employers to enable this to happen. Therefore, the social context of learning is influenced by the values, history and culture of both the trade union movement and the workplace, acting in wider United Kingdom society.

Findings Motivation to learn mathematics: initial and continuing

Adults in the sample indicated their initial motivation differed from that required to continue or persist in learning. Their initial motivation to re-engage with learning was related to their individual needs and goals, such as: improving job security, filling a perceived personal knowledge or skills gap, gaining public recognition of their knowledge or skills through certification and helping their children. However, their motivation to re-engage with learning was also supported and encouraged by significant social local face-to-face groups such as work colleagues, ULRs, fellow trade union colleagues, other learners and family members. At the wider societal level, the opportunity for learning is reliant upon trade unions negotiating funding with central government and workplace agreements with employers, as stated above.

Motivation to continue, or persist, with learning relied more on the action of the social face-to face groups, both teachers and fellow learners providing what the interviewees called a 'different' learning experience to that previously encountered. The adult's described the characteristics of the learning as: more collaborative than previously experienced, taking place in smaller classes, in a relaxed atmosphere where they felt they could talk openly about mathematics and their problems. They also spoke about the mathematical topics being relevant to their life experience, in the sense that the concepts, where possible, were related to practical applications (such as building a shed) or finances (interest rates) or trade union issues (regarding health and safety).

Emotions and motivation

During the analysis it became clear that emotions played a key role in the adults descriptions of their learning experiences. Eight people, out of twenty interviewed, used such words as hate, phobia, fear and horrendous to describe their previous experiences with mathematics. When talking about the learning organised through the trade unions seven interviewees spoke about feeling more "relaxed", four described the learning experience as "fun" and five as "brilliant". When discussing their motivations to learn and use mathematics eleven interviewees used the word 'confidence' to describe the changes in their feelings and beliefs about their abilities and motivation.

The notion of developing confidence through the achievement of mathematics contributed to the interviewees being more motivated to learn in the classroom but also to act differently when outside. For example, when they were negotiating on behalf of fellow trade union members, dealing with home finances, training to become trade union mathematics teachers or supporting children to learn mathematics at home. Combining personal feelings (of confidence) developed through social experiences (learning mathematics) creating those intentions to act differently, I describe this as adults experiencing a positive Affective Mathematical Journey, relating motivation to emotions, and cognition.

Relevance of the Findings to the Research Field

When considering the notion of increased confidence adults' motivation to learn mathematics relates to affective domains, reflecting [START_REF] Hannula | Exploring new dimensions of maths-related affect: Embodied and social theories[END_REF] two dimensions of motivation acting at an individual emotional and cognitive level but something that can change over time. However in this research the support of various members of local social groups was also key to learning, reflecting [START_REF] Eynde | Accepting emotional complexity: A socioconstructivist perspective on the role of emotions in the mathematics classroom[END_REF] focus on the dynamic interaction between the individual and the social group.

Developing individual's Confidence: an Affective Mathematical Journey

Interviewees often used emotional words when discussing their motivation to learn mathematics, indicating that successful learning experiences, especially after previous 'failures', lead to changes in how they saw themselves and how they act in social situations. Psychological research into motivation reinforces the idea of a positive correlation between self-confidence and achievement in mathematics [START_REF] Mcleod | Research on affect and mathematics learning in the JRME: 1970 to the present[END_REF][START_REF] Zan | Affect in mathematics: An introduction[END_REF]). This research suggests that achievement increases self-confidence while [START_REF] Hannula | Exploring new dimensions of maths-related affect: Embodied and social theories[END_REF] argues self-belief and achievement are reciprocal influences.

Although theoretically confidence can be a problematic concept when used to judge others, the way the interviewees use the term to describe their own beliefs and feelings about themselves and their willingness to act in different social situations, suggests it as an 'emotion of self-projection' as described by [START_REF] Barbalet | Social emotion: confidence, trust and loyalty[END_REF], p. 77). McLeod (1994) argues confidence is mainly cognitive but the emotional words used in this research reinforce [START_REF] Barbalet | Social emotion: confidence, trust and loyalty[END_REF] argument. [START_REF] Mahn | The gift of confidence: A Vygotskian view of emotions[END_REF] further maintain developing confidence has a strong social dimension developed through group activities, which strengthens the findings of this research suggesting the use of the word confidence links the adult's motivation to emotions and social experiences.

When adults report feeling more confident about themselves and more positive about learning mathematics it suggests a change in their identity development at a 'profound' level after a 'transformative' learning experience [START_REF] Illeris | Transformative learning and identity[END_REF]. The conjecture is that feelings towards mathematics can change and effect an individual's motivation and this concept contributes to the theoretical construct an Affective Mathematical Journey (AMJ), where adults' beliefs, motivation and feelings about mathematics change. [START_REF] Debellis | Affect and meta-affect in mathematical problem solving: A representational perspective[END_REF] argue a range of emotions are experienced during problem solving in the mathematics classrooms, describing them as 'pathways', this research builds on that idea but points to changes in adult's emotions happening over a longer period of time, in some cases years, thus it is more of a journey.

Further research could be undertaken, exploring ways that enable adults to experience Affective Mathematical Journey in relation to 'different' learning experiences described earlier. Particularly so for teachers in Further Education in the UK who have to teach 16-19 year olds, who have already experienced failure to achieve GCSE mathematics, at least once.

The role of different social face-to-face groups in motivation [START_REF] Bandura | Social cognitive theory for personal and social change by enabling media[END_REF] identifies social persuasion as key to motivation, this goes beyond verbal persuasion alone, because getting someone to attend a mathematics class would not be enough to develop their self-efficacy. Adults also need to feel supported in order to achieve and to develop their own self-belief. In this research trusted ULRs take on this role persuading members to join mathematics classes; ensuring the classes are accessible and providing a supportive and 'emotionally safe' environment (discussed below), leading to positive learning experiences.

Helping children with their skills development is also an important motivator for adults, particularly for females, findings supported by research by [START_REF] Swain | Beyond the daily application': making numeracy teaching meaningful to adult learners[END_REF] and [START_REF] Coben | Effective teaching and learning numeracy[END_REF]. This research accentuates the importance of supporting children as both an initial motivational force to re-engage with learning as well as a reason to persevere. For those adults who are newly arrived in the UK, with English as their second language, supporting children to learn can also help them to understand and integrate into the wider 'home' society. Hence, learning mathematics can be a positive contributor to a better functioning family in both economic and social terms.

Motivation related to pedagogical approaches used in the learning group

The adults spoke regularly of the importance of having a 'different' learning experience, when reengaging with mathematics. The pedagogical approach promoted by trade unions [START_REF] Rees | New developments in union Learning[END_REF] and described by adults was 'more relaxed', 'more social' and 'supportive'; closer to notions of informal or non-formal [START_REF] Coben | Effective teaching and learning numeracy[END_REF][START_REF] Evans | Critical perspectives on adults' education[END_REF] ways of learning. The learning experiences were described by some as being 'easier' than previous ones linking cognitive to emotional factors in motivation. [START_REF] Bibby | Shame: An emotional response as an adult and a teacher[END_REF] suggests there is a connection between traditional mathematics class and shame, where mathematical tasks are often seen as right or wrong and in the classroom are open to public scrutiny, judgement and, if not 'correct', humiliation and shame. Wojecki calls these remembered encounters as 'wounding learning experiences ' (2007, p. 170) although he posits that adults can and do get over them, something this research supports. [START_REF] Schorr | Student's expression of affect in an inner-city SimCalc classroom[END_REF] argue it is possible to facilitate changes in emotions given the right circumstances, they describe as 'emotionally safe environments ' (p. 131). This is a place where it is safe to expose misconceptions and where learners' will not suffer public humiliation for getting a calculation wrong.

The interviewees also spoke about the importance of relating the mathematical practice to previous and current experiences in life and work, an approach trade unions promote [START_REF] Rees | New developments in union Learning[END_REF]. Linking mathematics to life or work links to the idea of social belonging and relatedness, [START_REF] Ryan | Self-determination theory: basic psychological needs in motivation, development, and wellness[END_REF] where motivation is something that is culturally relevant, such as trade union activities, or meaningful to the individual, for example calculating interest rates on loans.

Conclusion

The findings indicate that adults' initial motivation to re-engage with learning mathematics can be understood differently from their motivation to continue learning, but that both rely on the dynamic interaction between the individual and social groups. Motivation can be seen as a change in behaviour, which is a function of an individual's motivation, emotions and cognition; developing within social face-to-face groups, all acting within a wider enabling social context.

The research has focused on a special group of learners, that have not succeeded in mainstream education but have achieved in an alternative learning environment that gives us a real insight into ways that adults' motivation and confidence to learn mathematics can be developed. It foregrounds the important role that social face-to-face groups such as trade unions play in developing 'different' learning experiences using less traditional or formal methods but that can be successful in developing adults' mathematical skills and confidence. It also highlights the important role personal emotions play in a motivation, and how trust in others plays a key role in re-engagement with learning, overcoming long held negative views on mathematics and developing confidence with mathematics. This change in feelings in relation to mathematics can be described as them experiencing a positive Affective Mathematical Journey (AMJ), where they feel more confident and motivated to both learn and use the subject.