Alina Redmer
email: alina.redmer@uni-hamburg.de

Anke Grotlüschen
email: anke.grotlueschen@uni-hamburg.de

Numeracy practices in older age

Keywords: Numeracy, practices, older adults, PIAAC, large-scale assessment

Demographic changes render basic skills important at an increasingly older age. Among them, skills in financial matters are becoming increasingly relevant. The skills measured in PIAAC and CiLL represent comprehensive cognitive abilities that can be seen as the basis for successful participation in social and economic life. In a secondary analysis, this paper shows that financial practices remain important for older people, although their numeracy skills proficiency appears to be lower than those of younger people. The analysis also indicates a gender difference in dealing with financial matters: men handle bills and bank statements, women manage the household.

Introduction

As a result of demographic changes, it is becoming increasingly important to actively shape one's life in a longer post-employment period [START_REF] Kruse | Weiterbildung in der zweiten Lebenshälfte: Multidisziplinäre Antworten auf Herausforderungen des demografischen Wandels[END_REF]. Financial security and the ability to act have an ever-increasing importance in times of possible restrictions in income among the elderly. People who are low-skilled or whose employment biographies show gaps are more likely to be disadvantaged in this process [START_REF] Wienberg | Aktives Altern" unter Vorbehalt: Ungleiche Teilhabechancen von der Wiege zur Bahre[END_REF].

The question is whether the increased requirements become problematic due to higher financial burdens in older age, if they are accompanied by lower numeracy skills, or whether existing numeracy skills will prove to be sufficient. Instead of merely declaring a decrease of skills in relation to older age, the actual numeracy practices of people of higher age need to be more closely considered.

This article is dedicated to examine the numeracy skills of people of older age (66-80 years at the time of assessment) and their everyday numeracy practices. According to anthropologist Jean Lave (1988), mathematics is used in various situations in everyday life. This includes the comparison of prices, but also the calculation of costs and the household budget. PIAAC (Programme for the International Assessment of Adult Competencies) and CiLL (Competencies in Later Life) define numeracy as the ability to apply and interpret mathematical information and ideas to deal with different everyday situations [START_REF] Zabal | Grundlegende Kompetenzen der erwachsenen Bevölkerung in Deutschland im internationalen Vergleich[END_REF].

In this paper, data from the CiLL survey (Friebe, Schmidt-Hertha, & Tippelt, 2014) is used to compare numeracy practices of individuals aged 66 to 80 years with the practices of individuals aged 16 to 65 years as surveyed by PIAAC. The focus of the following secondary analysis is on practices (also referred to as skills-use variables) that provide information on how often numeracy skills are applied in the lives of older people.

Relevance of numeracy and financial skills in older age

Images of older age have changed in Germany in recent years [START_REF] Vogel | Die Rückkehr der Altersarmut[END_REF]. The boundaries indicating who belongs to the older adults may not simply be determined on the basis of the calendar age. However, since PIAAC looked at people up until the age of 65 and CiLL assessed participants in the ages of 66 to 80, the integral boundary between these two surveys is used as a proxy to distinguish adults and older adults. Older people are thus in this paper the over 65-year-olds.

Medical and technical developments make an ever-increasing lifespan possible, thus creating a longer post-employment phase of life [START_REF] Mahne | Alter im Wandel: Zwei Jahrzehnte Deutscher Alterssurvey[END_REF]. The aspect of financial security plays an important role for a good way of living in this post-employment period. In Germany, income in retirement is mainly determined by the statutory pension as well as by assets and inheritances. Therefore, people whose employment biographies show gaps or who had had low incomes are disadvantaged and in danger of poverty in older age. In addition, this group of people often inherited less frequently and accumulated less assets during their employment period [START_REF] Mahne | Alter im Wandel: Zwei Jahrzehnte Deutscher Alterssurvey[END_REF]. In this respect, the proportion of people who have to work after retirement has increased [START_REF] Franke | Länger zufrieden arbeiten? Qualität und Ausgestaltung von Erwerbstätigkeit in der zweiten Lebenshälfte[END_REF]. This raises the questions about how older people can get along with limited financial resources. Increasing financial skills in older age will not entail an increase of funds or pensions, but can help to manage low income successfully.

Referring to older people, Kruse describes competence as skills and abilities of the human being [START_REF] Kruse | Bildung und Erwachsenenbildung im Alter[END_REF](Kruse, , p. 1193) that contribute to the preservation as well as to the possible restoration of an independent, self-responsible and meaningful life. For older people there maybe a reduced ability with regard to orientation in new problem situations and new learning, but there is the necessary skill to solve familiar problems and to expand existing knowledge systems into old age [START_REF] Kruse | Weiterbildung in der zweiten Lebenshälfte: Multidisziplinäre Antworten auf Herausforderungen des demografischen Wandels[END_REF]. As a result, it can be assumed that basic mathematical skills also contribute to solving financial tasks in everyday life.

According to Geiger, Forgasz, & Goos (2015) the financial competence can be seen as an extension of numeracy that is expressed in numeracy practices. From this perspective, a financially literate person must have both the necessary knowledge to carry out financial transactions and plans and the ability and confidence to make financial decisions (Geiger, Goos, & Forgasz, 2015).

The results of the CiLL study show that the average everyday mathematical competence of the population between 66 and 80 years of age is 240 points [START_REF] Knauber | Alltagsmathematische Kompetenz[END_REF]. This corresponds to the lower third of competence level 2 and indicates that on average older people are capable of dealing with mathematical information embedded in everyday contexts, solving everyday mathematical tasks with little competing information and processing steps [START_REF] Knauber | Alltagsmathematische Kompetenz[END_REF].

Nevertheless, in adult education research in Germany, financial literacy is discussed more frequently than numeracy. [START_REF] Mania | Finanzielle Grundbildung -Ein Komeptenzmodell entsteht[END_REF] describe financial literacy as a sub-area that concerns the existentially basal and practical demands of everyday life and lifestyle in monetary matters [START_REF] Mania | Finanzielle Grundbildung -Ein Komeptenzmodell entsteht[END_REF].They developed a competence model describing the requirements for dealing with money at the level of basic education. The model covers six domains of basic financial education (income, money and payments, spending and buying, households, lending and borrowing, pensions and insurance). Thus they can make an important qualitative contribution to the discourse. Representative results in the field of financial literacy are still lacking. Our study refers to the numeracy aspect of financial literacy in order to explain the financial capacity of older people to act.

Research questions

Starting from a position that skills are complemented by practices, the focus of this contribution is on numeracy practices of older adults (66-80 years). Accordingly, the overarching question is how numeracy practices are used in older age and whether they help people to be financially capable.

In particular, the following questions are addressed: a) How do numeracy skills work with increasing age? b) How do numeracy practices work with increasing age? c) How do numeracy practices change according to gender at the ages of 66 to 80?

Methods

The data from the PIAAC survey and the supplementary survey CiLL can provide a differentiated picture of numeracy skills and practices for the German population. The representative survey PIAAC was initiated by the OECD, and it was conducted for the first time in 2011/2012 and reached a sample of 5,465 persons aged between 16 and 65 years in Germany. In addition to the survey of the domains literacy, numeracy and problem solving in technology-rich environments, questions on the use of skills were asked in an extensive questionnaire. The subsequent supplementary survey CiLL used the same background questionnaire and the skills were measured in the same way as in PIAAC. This enables a comparison of these two surveys (e.g. Schmidt-Hertha 2018). The representative sample of 1,339 persons aged between 66 and 80 years was conducted in Germany in 2012 (Friebe et al., 2014). In both surveys, twelve questions on work-and non-workrelated practices were asked. We are looking at post-employment phases, only the questions that are not work-related are relevant for this research.

Numeracy skills for different birth cohorts

First, the numeracy skills for persons between 16 and 65 years and between 66 and 80 years were evaluated. It should be noted, that ages in PIAAC and CiLL are derived from the indicated birth year. Comparisons between age groups are therefore not statements on an individual decline in competence over the lifespan but statements on differences between groups in the birth cohorts. The mean skill values for numeracy indicate a decrease for both men and women for the older birth cohorts.

When comparing the data of the two birth cohorts 16 to 65 and 66 to 80 year olds, men in the older groups score almost 50 skill points lower and women almost 60. For both groups these are differences equivalent to one whole skill level (men level 3 to level 2; women level 2 to level 1) and indicate that women on average tend to do the simplest calculations. The differences may point to an historical effect, especially for the oldest cohort. Those who are born between 1931 and 1936 often have had interrupted educational careers and different social and cultural expectations to those of the younger cohort.

The average skill value of the entire CiLL cohort (66-80 years) is 240 points, which is below the average skill value of the entire PIAAC cohort (16-65 years) of 272 points. The next step is to check how the skills-use variables (indices) of the two samples differ. In order to assess how the practices are used in general, the analysis took into account the skills-use variables of literacy skills (frequency of reading and writing in everyday life). For all everyday practices, a decreasing frequency of use across age groups can be observed (see Figure 2). However, this development is not comparable to the skills variable described above. For numeracy, there is a small increase in the average frequency of use from the 55-65 age group to the 66-70 age group.

Numeracy practices in older age

Based on practices to be considered in the context of the social and economic system, the variable H_Q03b (Calculation of prices, costs and budgets)1 is divided by gender. There is a highly significant difference between men and women in the application of simple numeracy skills: 42.3% of men said they used calculation of prices, costs and budgets at least once a week or daily, while 58.8% of women gave the same answer (see Figure 3). 20.3% of men said they "never" used the practices. Among women, 16% said they "never" calculate prices, costs and budgets.

It shows overall both groups frequently used the practices, but women use them significantly (level of significance below 0.01) more than men do. The variable that asks for the reading frequency of bills or bank statements (H_Q01g, see Figure 4)2 reveals that men say they use this practice more frequently. 51.2% said they read bills or bank statements at least once a week, while 42.9% of women gave the same answer.

Discussion

We see that the variables on the use of skills recorded in PIAAC and CiLL can provide some information on the frequency in which the skills required for financial decisions are applied.

a) How do numeracy skills work with increasing age?

Despite a lower average skill proficiency in older age groups, numeracy practices are still regularly used. This should enable them to act financially for themselves. After a peak in the age cohorts of 25 to 34 year-old women and 35 to 44 year-old men, the mean skill values are steadily declining. This finding reinforces the results on the course of numeracy skills in PIAAC [START_REF] Maehler | Grundlegende Kompetenzen in verschiedenen Bevölkerungsgruppen[END_REF] and in CiLL [START_REF] Knauber | Alltagsmathematische Kompetenz[END_REF][START_REF] Gebrande | Lesekompetenz[END_REF], which have only been reported separately so far.

b) How do numeracy practices work with increasing age?

The comparison shows that the mean values of the skill-use indices for numeracy between the age groups 55 to 65 and 66 to 70 increase slightly. This could also indicate that the transition to the post-employment phase is accompanied by an increase in the use of numeracy practices. Such as dealing with pensions, paying of mortgages or applying for new state benefits.

Similar to reading and writing practices, a decrease in numeracy practices can be observed with increasing age, with reading overall remaining at a higher level. However, changes in skill-use remain marginal compared to the differences of skills.

c) How do numeracy practices behave according to gender between the ages of 66 and 80?

The index of numeracy practices, consisting of six questions, was further analysed, these included the handling of budgets and prices as well as the handling of decimal and percentage values, which increased. Another variable is used from the index of reading practices, which concerns the reading of bills and bank statements. The results show that 58.8% of women perform simple calculations weekly to daily. Only 42.3% of men do this weekly or daily.

It can be assumed that women manage household budgets weekly and are apparently more often confronted with numeracy practices than men are. The handling of bank statements and bills per week less frequent for men in the 65 to 80 age group. It remains to be examined whether this distribution will change for in the now younger cohorts or whether it is essentially due to a still classical distribution of household chores (c.f. [START_REF] Hobler | Wer leistet unbezahlte Arbeit? Hausarbeit, Kindererziehung und Pflege im Geschlechtervergleich[END_REF].

Overall, it can be seen that numeracy practices in the area of "calculating prices, costs and budgets" are still used at least once a week by about 42% of men and more than half of women (59%) despite their older age. Having lower skills could make these activities more difficult, but at the same time qualitative studies that this older age group also show a considerable range of numeracy practices (card games, mental arithmetic, payroll tax, photography/development, club treasury, cf. [START_REF] Knauber | Alltagsmathematische Kompetenz[END_REF].

Numeracy practices could guarantee self-confidence in everyday life and in socio-political judgements to a certain extent, because "being numerate is being critical" [START_REF] Tout | Some reflections on adult numeracy[END_REF].

Figure 1 :Figure 2 :

 12 Figure 1: Numeracy skills average for the PIAAC Germany (N=5,379) and CiLL (N=1,338) samples compared

Figure 3 :

 3 Figure 3: Frequency of calculating prices, costs and budgets by gender in the CiLL sample (N=1,338)

Figure 4 :

 4 Figure 4: Frequency of reading bills and bank statements by gender in the CiLL sample (N=1,338)

Background questionnaire H_Q01b: "In your everyday life, how often do you normally calculate prices, costs or budgets?"

Background questionnaire H_Q01g: "In your everyday life, how often do you usually read bills, bank statements or the similar?"