Clotilde Coron
email: clotilde.coron@universite-paris-saclay.fr

Questioning 'feminine managerial behavior' -a European study considering gender ideology

Keywords: Gender, perceived managerial behavior, European sample, quantitative methods

The concept of a 'feminine managerial behavior' which implies that women in managerial positions behave differently to men in similar positions by exhibiting a more supportive management style, is both widespread and controversial. To gain new insight into the debate, this study looks at the role national gender ideology plays in structuring the relationship between sex and perceived managerial support. Based on a representative sample of 22,391 employees from 26 European countries, our findings reveal that, on average, perceived managerial support is higher when supervisors are women. However, if we control for the moderating role of national gender ideology, this difference disappears. Therefore, this article contributes to the nature/nurture debate by showing that gender differences in perceived managerial support stem primarily from persistent gender stereotypes.

Introduction

Gender equality is one of the United Nations' Sustainable Development Goals, and gender equality in the workplace and access to managerial positions are an important dimension of this field. The assumption of the existence of 'feminine management,' i.e., of female managers exhibiting different managerial behaviors from male managers, is both widespread and controversial [START_REF] Peterson | Exit the king. Enter the maid : Changing discourses on gendered management ideals in Swedish Higher Education[END_REF][START_REF] Stewart | Does supervisor gender moderate the mediation of job embeddedness between LMX and job satisfaction? Gender in Management[END_REF]. Previous research has shown that women tend to adopt transformational and supportive leadership more often than men [START_REF] Eagly | Transformational, transactional, and laissez-faire leadership styles : A meta-analysis comparing women and men[END_REF][START_REF] Peterson | Exit the king. Enter the maid : Changing discourses on gendered management ideals in Swedish Higher Education[END_REF][START_REF] Ye | Gender and managerial coaching across cultures : Female managers are coaching more[END_REF]. This result supports the argument that, as managers, women bring unique contributions to organizational performance through their supportive managerial behavior, which leads to higher team efficiency [START_REF] Hoobler | The Business Case for Women Leaders : Meta-Analysis, Research Critique, and Path Forward[END_REF].

Gender role theory has been widely used to explain this gender difference: due to gender norms and socialization, women are encouraged to engage in more supportive and caring behaviors, and therefore to adopt a more supportive management style [START_REF] Eagly | Transformational, transactional, and laissez-faire leadership styles : A meta-analysis comparing women and men[END_REF]. However, the idea that women managers are more supportive than men also fuels an essentialist discourse that naturalizes the differences between women and men [START_REF] Pigeyre | Le « management au féminin » : Entre stéréotypes et ambigüités[END_REF].

Most of the studies dealing with managerial support do not measure it concretely, but only assess managerial support as it is perceived by subordinates. Yet, perceived managerial support can differ from concrete managerial support, according to the supervisor's sex1 . Indeed, as gender role theory has shown, perceptions of 'feminine management' could be based on differences in social expectations of women and men [START_REF] Pigeyre | Le « management au féminin » : Entre stéréotypes et ambigüités[END_REF][START_REF] Saint-Michel | Style de leadership, LMX et engagement organisationnel des salariés : Le genre du leader a-t-il un impact ? @[END_REF], with the same behavior eliciting different perceptions based on whether a woman or a man performs it. More precisely, women and men are not similarly evaluated: for example, employees perceived female managers who self-described as communal to be more transformational than male managers [START_REF] Saint-Michel | Leader gender stereotypes and transformational leadership : Does leader sex make the difference? M@n@gement[END_REF]. In addition, the subordinate's sex can also influence perceived managerial support. Particularly, it has been shown that supervisors and subordinates in same-sex dyads report better managerial relationships, with a more common understanding of their roles and expectations [START_REF] Bender | Claims of employment discrimination and worker voice[END_REF][START_REF] Ghaus | Much of a Muchness? The Role of Gender Similarity in a Relationship between LMX and OCB[END_REF]. Hence, sex (dis)similarity between supervisor and subordinate can shape perceived managerial support.

While gender role theory, in contrast to the essentialist discourse, highlights the importance of gender stereotypes in shaping perceived and actual individual behaviors, research on gender differences in managerial support has not paid much attention to national cultures and, in particular, to gender ideologies. Gender ideologies refer to perceptions of the appropriate behaviors for women and men [START_REF] Dolan | Exploring the moderating effect of gender in the relationship between individuals' aspirations and career success among engineers in Peru[END_REF][START_REF] Gaunt | Job insecurity, stress and gender : The moderating role of gender ideology[END_REF]. Gender socialization during childhood shapes these perceptions [START_REF] Grunow | Gender Ideologies in Europe : A Multidimensional Framework[END_REF]. They vary across countries and can influence sex differences in managerial behavior, as well as the expectations that employees can have of their managers. This, in turn, can impact the perceived managerial support depending on the supervisor's sex [START_REF] Beblo | On the nature of nurture. The malleability of gender differences in work preferences[END_REF][START_REF] Ye | Gender and managerial coaching across cultures : Female managers are coaching more[END_REF]. Finally, depending on the subordinate's sex, these perceptions can also shape subordinates' expectations of their supervisor [START_REF] Ghaus | Much of a Muchness? The Role of Gender Similarity in a Relationship between LMX and OCB[END_REF].

Overall, the existing literature on perceived managerial support has not simultaneously taken national gender ideology, supervisor's sex, subordinate's sex, and sex (dis)similarity into account. The current study attempts to fill this gap by investigating the extent to which perceived managerial support is simultaneously shaped by the supervisor's sex, the subordinate's sex, sex (dis)similarity between subordinate and supervisor, and the extent to which national gender ideology plays a role in this relationship. To do so, this article uses a unique combination of two data sources. It relies mainly on the European Working Conditions Survey (EWCS) database, which offers the opportunity to study a representative sample of employees (representative at both the national and the European level and allowing comparisons between countries). The paper also aggregates data from the European Values Study (EVS) database to measure gender stereotypes among employees at the national level.

Overall, the study is based on a representative sample of 22,391 employees from 26 European countries. The results reveal that, on average, perceived managerial support is higher when supervisors are women, regardless of the subordinate's sex and the sex (dis)similarity between subordinate and supervisor. However, this difference between male and female supervisors disappears when the moderating role of national gender ideology is controlled for.

The current study provides several insights and makes various contributions to the debate on gender and perceived managerial support, heeding Ye et al.'s (2016) call to simultaneously take national cultures, the supervisor's sex, and the subordinate's sex into account when studying perceived managerial support. The article's main contribution is to show that gender differences in perceived managerial support stem primarily from persistent gender stereotypes. This insight jeopardizes the essentialist discourse which presumes that women are intrinsically different to men, with different skills and abilities. By doing so, the article contributes to the nature/nurture debate related to sex differences in professional behaviors and choices. This study is also important regarding the managerial issues. The article shows that, instead of assuming that female managers behave differently from their male counterparts, organizations should pay attention to gender stereotypes, which can influence how female and male subordinates perceive female and male supervisors' managerial support.

The article is structured as follows. The first section is devoted to a literature review of the debate on 'feminine management' and managerial support. The second section presents the data used and the methodology, and the third section presents the results obtained. Finally, the fourth section discusses the results.

I. Literature review

I.1. The relationship between sex and perceived managerial support: A rich and inconsistent literature

Previous studies have debated the existence of sex differences in managerial behaviors [START_REF] Stewart | Does supervisor gender moderate the mediation of job embeddedness between LMX and job satisfaction? Gender in Management[END_REF], inviting scholars to take this debate further. Some authors use the gender role theory [START_REF] Eagly | Role congruity theory of prejudice toward female leaders[END_REF] to suggest that gendered socialization, as well as gender stereotypes and social expectations of women and men [START_REF] Essig | Re-thinking gender inequality in the workplacea framework from the male perspective[END_REF], are the reasons for feminine and masculine management styles, with feminine management styles focusing more on managerial support than the men's [START_REF] Saint-Michel | Leader gender stereotypes and transformational leadership : Does leader sex make the difference? M@n@gement[END_REF]. A meta-analysis indicated that female leaders exhibit more supportive and transformational leadership (focused on subordinates' development) than male leaders, who tend to display transactional leadership (focused on subordinates' performance) [START_REF] Eagly | Transformational, transactional, and laissez-faire leadership styles : A meta-analysis comparing women and men[END_REF][START_REF] Maher | Gender-Related Stereotypes of Transformational and Transactional Leadership[END_REF]. The current study focuses on managerial support, which refers to how managers 'provide organizational support through their managerial role by caring for subordinates' well-being and providing work advisory' (Sawang, 2010, p. 248). Previous studies suggest that female managers engage in more supportive managerial behaviors [START_REF] Peterson | Exit the king. Enter the maid : Changing discourses on gendered management ideals in Swedish Higher Education[END_REF][START_REF] Ye | Gender and managerial coaching across cultures : Female managers are coaching more[END_REF].

However, several scholars have criticized this idea of feminine management behavior being distinct from and more supportive than men's (Due [START_REF] Billing | Questioning the Notion of Feminine Leadership : A Critical Perspective on the Gender Labelling of Leadership[END_REF][START_REF] Pigeyre | Le « management au féminin » : Entre stéréotypes et ambigüités[END_REF]. Particularly, this idea is the basis of an essentialist discourse, challenged by critical gender studies, in which the differences between women and men are naturalized in that all women are assumed to have different managerial styles from men, and that these differences are supposedly innate [START_REF] Hoobler | The Business Case for Women Leaders : Meta-Analysis, Research Critique, and Path Forward[END_REF]. Consequently, some academic work has highlighted the limitations of the research that shows differences in managerial support between women and men. First, the magnitude of the sex differentials has been questioned, with some scholars arguing that the similarities in management style between women and men outweigh the differences between them [START_REF] Wajcman | Desperately Seeking Differences : Is Management Style Gendered[END_REF]. Relatedly, previous studies have revealed that non-compliance with management ideals can result in negative evaluations, leading women to strive for compliance with the same management ideals as men [START_REF] Billing | Questioning the Notion of Feminine Leadership : A Critical Perspective on the Gender Labelling of Leadership[END_REF][START_REF] Wajcman | Desperately Seeking Differences : Is Management Style Gendered[END_REF]. Second, studies about female and male managers might be subject to selection bias. In a situation where access to managerial positions is more difficult for women than for men, women who become managers might be chosen because they comply with gender stereotypes or other types of expectations [START_REF] Eagly | Transformational, transactional, and laissez-faire leadership styles : A meta-analysis comparing women and men[END_REF]. Third, and most importantly, most studies dealing with sex and management measure perceived (either selfperceived or perceived by employees) managerial behaviors instead of concrete managerial behaviors. Yet, an employee's perception of their manager's support might be influenced by the manager's sex [START_REF] Saint-Michel | Leader gender stereotypes and transformational leadership : Does leader sex make the difference? M@n@gement[END_REF][START_REF] Saint-Michel | Style de leadership, LMX et engagement organisationnel des salariés : Le genre du leader a-t-il un impact ? @[END_REF].

Particularly, sex (dis)similarity between supervisor and subordinate can influence perceived managerial support [START_REF] Douglas | The moderating role of leader and follower sex in dyads on the leadership behavior-leader effectiveness relationships[END_REF]. For example, supervisors and subordinates in same-sex dyads reportedly have better working relationshipsa result which can be explained by the greater ease of working with similar people [START_REF] Vecchio | Leadership and Sex-Similarity : A Comparison in a Military Setting[END_REF], or by the fact that supervisors and subordinates in same-sex dyads may have a greater shared understanding of role expectations than otherwise [START_REF] Ghaus | Much of a Muchness? The Role of Gender Similarity in a Relationship between LMX and OCB[END_REF]. Incidentally, having a supervisor of the same sex is associated with fewer discrimination claims [START_REF] Bender | Claims of employment discrimination and worker voice[END_REF]; also, sex (dis)similarity between subordinate and supervisor moderates the effect of the managerial relationship quality on voice behavior [START_REF] Wijaya | Proactive Personality, LMX, and Voice Behavior : Employee-Supervisor Sex (Dis)similarity as a Moderator[END_REF].

Overall, both supervisors' and subordinates' sex, and sex (dis)similarity between subordinates and supervisors, must be considered when we study perceived managerial support.

This limitation applies to a study [START_REF] Ye | Gender and managerial coaching across cultures : Female managers are coaching more[END_REF] conducted on cross-national and gender differences in managerial support, which did not take the subordinate's sex into account.

I.2. The moderating role of gender ideology

While gender role theory underlines the importance of gender stereotypes in shaping perceived and concrete individual behaviors, research on gender differences in managerial support has not paid much attention to national gender ideologies. Yet, the relationship between the supervisor's and the subordinate's sex and perceived managerial support can be shaped by beliefs about and expectations of the adequate roles and behaviors of women and men, which will be referred to as gender ideology: 'gender norms influence individuals' beliefs regarding the appropriate behavior for men and women in the context of work and family' (Gaunt & Benjamin, 2007, p. 343). [START_REF] Gaunt | Job insecurity, stress and gender : The moderating role of gender ideology[END_REF], as well as [START_REF] Davis | Gender Ideology : Components, Predictors, and Consequences[END_REF], use the term 'gender ideology.' Other papers use terms like gender roles, gender orientation, gender stereotypes, or gender differences [START_REF] Bagger | How much do you value your family and does it matter? The joint effects of family identity salience, family-interference-with-work, and gender[END_REF][START_REF] Bueno Merino | Contribution of Psychological Entrepreneurial Support to the Strengthening of Female Entrepreneurial Intention in a Women-Only Incubator[END_REF][START_REF] Dolan | Exploring the moderating effect of gender in the relationship between individuals' aspirations and career success among engineers in Peru[END_REF]Zhao et al., 2019). All these terms relate to beliefs and representations about acceptable behaviors for women and men, which are internalized during the socialization process. More precisely, gender ideology is generally operationalized as a unidimensional construct that ranges from traditional to egalitarian [START_REF] Gaunt | Job insecurity, stress and gender : The moderating role of gender ideology[END_REF]. Traditional gender ideology refers to a world vision in which men should be the breadwinners and women are responsible for the household and childcare [START_REF] Bagger | Being important matters : The impact of work and family centralities on the family-to-work conflict-satisfaction relationship[END_REF]. Egalitarian gender ideology corresponds to a universe in which women and men have the same roles and responsibilities in the personal and professional spheres [START_REF] Davis | Gender Ideology : Components, Predictors, and Consequences[END_REF][START_REF] Gaunt | Job insecurity, stress and gender : The moderating role of gender ideology[END_REF]. Overall, gender ideology can influence perceptions of the behaviors observed, depending on whether they are performed by a woman or a man. In addition, gender ideology varies according to sex, with men generally reporting a less egalitarian gender ideology than women do [START_REF] Davis | Gender Ideology : Components, Predictors, and Consequences[END_REF][START_REF] Grunow | Gender Ideologies in Europe : A Multidimensional Framework[END_REF].

Gender ideology is deeply connected to national cultures [START_REF] Alesina | On the Origins of Gender Roles : Women and the Plough[END_REF][START_REF] Fernández | Cultural change as learning : The evolution of female labor force participation over a century[END_REF][START_REF] Fernández | Culture : An Empirical Investigation of Beliefs, Work, and Fertility[END_REF][START_REF] Fogli | Nature or Nurture? Learning and the Geography of Female Labor Force Participation[END_REF]. For example, [START_REF] Beblo | On the nature of nurture. The malleability of gender differences in work preferences[END_REF] show the differences between East and West Germany in the 1980s concerning the expectations of women and men regarding paid and unpaid work. [START_REF] Sweet | Are women less career centric than men? Structure, culture, and identity investments[END_REF] emphasize differences between China and Japan concerning women's and men's career centrality. In a comparative national approach, [START_REF] Fortin | Gender Role Attitudes and the Labour-market Outcomes of Women across OECD Countries[END_REF] studies the gender ideology differences across different countries and the influence national gender ideologies have on women's labor market outcomes. Other studies have highlighted national differences in the prevalence of gender stereotypes [START_REF] Sweet | Are women less career centric than men? Structure, culture, and identity investments[END_REF].

Overall, national cultures vary in gender ideology, ranging on a continuum from egalitarian to traditional. These differences can shape both the sex differences in managerial behaviors and the expectations and perceptions that employees have concerning their managers, depending on the manager's sex. Therefore, gender ideology might play a moderating role in the relationship between the supervisor's sex, the subordinate's sex, and perceived managerial support. Further, if sex differences are due to gender stereotypes, they should disappear if the moderating role of gender ideology is controlled for. This article delves into these relationships. Previous research has examined separately, and without reaching consensus, the roles of the subordinate's sex, the supervisor's sex, sex (dis)similarity between subordinate and supervisor, and gender ideology. This study aims to integrate these different variables in the same analysis, to get a deeper understanding of this complex phenomenon. More precisely, the study investigates the extent to which perceived managerial support is shaped simultaneously by the supervisor's sex, the subordinate's sex, sex (dis)similarity between the different parties, and the extent to which national gender ideology plays a role in this relationship (and moreover, what happens to the relationship between sex and perceived managerial support if the moderating role of gender ideology is controlled for).

A standard hypothetico-deductive approach would require precisely defining the effect of each variable, which in this case, is impossible because previous research has not reached consensus on these effects. That is why the current study uses an exploratory approach rather than a hypothetico-deductive one.

The following section explains the concrete steps and characteristics of the research design, thereby also clarifying this approach.

II. Data and methodology

II.1. Data

The main individual data used in this study is drawn from the 2015 EWCS (European Working Conditions Survey). This survey provides insight into working conditions throughout 35 European countries (28 EU member states, as well as Norway, Switzerland, Albania, North Macedonia 2 , Montenegro, Serbia, and Turkey). The sample is a multistage stratified random sample of the working population in each country. It was restricted to employees who answered the question, 'Is your immediate boss a man or a woman?' The EWCS dataset is a stratified weighted sample, i.e., it is provided with individual weights which allow researchers to produce representative results at the national and international levels. The sample accounts for underrepresented categories by assigning higher weights to correspond to the target population in each country (all individuals aged 15 or more living in private households and employed). In the following analyses, weights (provided with the dataset) were used to produce representative results (both nationally and for national comparison).

To measure gender ideology at the national (and not at the individual) level, data from the 2017 European Values Study (EVS) was used to complement the EWCS data with those national measures. The preceding wave, a World Values Survey, which took place between 2 FYROM (Former Yugoslav Republic of Macedonia) in the survey. The data is from 2017, before the country changed its name to North Macedonia in 2019.

2010 and 2014 covered only a limited number of countries. We assume that national gender ideologies remained stable between 2015 (EWCS) and 2017 (EVS). The EVS survey is a representative, large-scale, and cross-national survey on human values. It contains individual declarative data on gender stereotypes, and weights are provided with the data to extrapolate the results at the country level. Owing to differences in perimeter and country coding, the initial EWCS sample had to be restricted to the 26 countries that appear in both samples, therefore Belgium, Cyprus, Greece, Ireland, Latvia, Luxembourg, Malta, the UK, and Turkey were eliminated. The final sample has data of 22,391 employees (11,889 female and 10,502 male). Table 1 gives the unweighted distribution of respondents across the different countries.

II.2. Measures

Perceived managerial support. Seven items from the EWCS can be used to measure perceived managerial support. Respondents were asked to rate seven statements regarding their applicability to their manager's behavior, e.g., 'Your immediate boss helps and supports you'

(see Table A1 in Appendix for the full list of items). Each item was measured on a Likert scale, in which respondents indicate their degree of agreement ('Strongly disagree'/'Tend to disagree'/'Neither agree nor disagree'/'Tend to agree'/'Strongly agree'). The Cronbach's alpha for the seven items is 0.89. An index corresponding to the aggregation (average) of the seven items was computed. Items were reversed so that a high score would indicate high perceived managerial support, as in previous research that uses the same items [START_REF] Ariza-Montes | Work environment and well-being of different occupational groups in hospitality : Job Demand-Control-Support model[END_REF][START_REF] Artz | How Common Are Bad Bosses? Industrial Relations[END_REF].

Subordinate's sex, supervisor's sex, and sex (dis)similarity. The interviewers reported each subordinate's sex (contrary to the other variables that correspond to the interviewees' answers).

It is coded as 'woman,' 'man,' or 'Don't know.' However, only nine individuals in the whole sample correspond to 'Don't know.' Therefore, only the answers 'woman' and 'man' were kept.

This is a binary variable, which does not consider the possibility of a nonbinary vision of humans. Although this binarity is regrettable, it is necessary, as this study focuses on the possible differences between women and men. The supervisor's sex is determined by the question: 'Is your immediate boss a man or a woman?' Again, only the answers 'man'/'woman' were kept. The subordinate's sex, the supervisor's sex, and their sex (dis)similarity were integrated into one only variable, called 'Dyad,' which corresponds to the combination of the subordinate's and the supervisor's sex. Four categories were created: the woman-woman dyad, the man-man dyad, the woman-man dyad, and the man-woman dyad. Such a strategy was used previously in research on sex differences in managerial behavior [START_REF] Stewart | Does supervisor gender moderate the mediation of job embeddedness between LMX and job satisfaction? Gender in Management[END_REF].

National gender ideology. First, the individual's gender ideology in the EVS sample was measured by averaging responses to seven statements about traditional gender roles and behaviors, e.g., 'When a mother works for pay, the children suffer' (see Table A2 in Appendix for the full list of items). Each item was measured on a Likert scale, with respondents having to indicate their degree of agreement ('Agree strongly'/'Agree'/'Disagree'/'Disagree strongly'). The Cronbach's alpha for the seven questions is 0.88. A high score, therefore, represents an egalitarian ideology, while a low score corresponds to a traditional one. Although the unidimensionality of gender ideology has been debated [START_REF] Grunow | Gender Ideologies in Europe : A Multidimensional Framework[END_REF], those items have been used as a proxy for gender ideology [START_REF] Davis | Gender Ideology : Components, Predictors, and Consequences[END_REF][START_REF] Fortin | Gender Role Attitudes and the Labour-market Outcomes of Women across OECD Countries[END_REF].

Second, national gender ideology was computed as the average of the gender ideology of the country's employees (restricted to employees and based on weighted data for representativeness). This was added to the EWCS database.

Control variables. Several variables supposed to affect perceived managerial support were used in this study. Previous research has shown that temporary employment can influence employees' experience and, specifically, their satisfaction at work [START_REF] De Cuyper | Job insecurity, employability and satisfaction among temporary and permanent employees in post-crisis Europe[END_REF], as well as their psychological contract and behavior [START_REF] Chambel | Different Temporary Work Status, Different Behaviors in Organization[END_REF]). In the EWCS data, the type of contract is an unordered categorical variable classified into four categories:

unlimited duration, limited duration, temporary employment agency contract, and no contract.

Part-time work has also been linked to job expectations and satisfaction [START_REF] Booth | Job Satisfaction and Family Happiness : The Part-Time Work Puzzle[END_REF].

Working time reported in the EWCS data can be either full-time or part-time. Education level, which can influence work expectations and, thus, expectations of management and perceived managerial support [START_REF] Ganzach | Intelligence, Education, and Facets of Job Satisfaction[END_REF][START_REF] Tlaiss | Predicting women's job satisfaction with personal demographics : Evidence from a Middle Eastern country[END_REF], is an unordered categorical variable recorded using the ISCED classification: primary education, lower secondary education, upper secondary education, postsecondary non-tertiary education, short-cycle tertiary education, bachelor or equivalent, master, or doctorate. Age has been shown to influence employees' expectations and experience at work [START_REF] Broadbridge | Experiences, perceptions and expectations of retail employment for Generation Y[END_REF]

II.3. Methodology

The study uses an exploratory approach, in the sense that it is not based on a precise hypothesis drawn from previous literature. Indeed, the lack of academic literature on perceived managerial support which simultaneously studies the supervisor's sex, the subordinate's sex, sex (dis)similarity, and gender ideology, does not allow the formulation of a specific hypothesis, particularly not about the precise moderating effect of gender ideology.

First, this research started with descriptive statistics on the different variables of interest (perceived managerial behavior, sex dyad) in a comparative national logic. National gender ideology was then added to the analysis. Descriptive statistics were used to measure the variations in gender ideology across countries, which prompted us to investigate the moderating role gender ideology plays in the relationship between the supervisor's sex, the subordinate's sex, their sex (dis)similarity, and perceived managerial support more deeply. To this end, hierarchical linear modeling, i.e., multilevel modeling, was used. Multilevel modeling is useful in studying phenomena that are explained both by the individual and at a more general level (organization or, in this case, country) [START_REF] Renkema | Advancing multilevel thinking and methods in HRM research[END_REF][START_REF] Smyth | Field of Study and Gender Segregation in European Labour Markets[END_REF].

Multilevel models give unbiased estimates for nested data and help determine the percentage of variation explained by the different levels [START_REF] Goldstein | Partitioning Variation in Multilevel Models[END_REF]. In this study, two levels are considered: individual and national. SAS software and the mixed procedure were used for the analysis.

III. Results

III.1. National differences concerning perceived managerial support

First, the data was used to produce statistics concerning perceived managerial support across the different countries for the whole sample. A GLM (generalized linear model) showed that the differences between countries were significant (Denmark as the reference group). Table 3 reports the results (the average perceived managerial behavior and, in parentheses, the standard deviation). The stars indicate the significance of the difference in perceived managerial support between countries, with Denmark as the reference group.

Significance levels: *: p<0.1; **: p<0.05; ***: p<0.01

Table 3 shows huge differences between countries related to perceived managerial support.

The countries with the lowest reported average levels of perceived managerial support are, in ascending order: Italy, Montenegro, Serbia, Estonia, Germany, France, and Slovakia (indicated in bold in Table 3). The countries with the highest reported average levels of perceived managerial support are, in descending order: North Macedonia, Norway, Bulgaria, Romania, Portugal, Spain, and Hungary (indicated in italics in Table 3).

III.2. The relationship between sex (dis)similarity and perceived managerial support

Besides the national differences, the study focuses on the effects of the subordinate's and the supervisor's sex in relation to perceived managerial support. To measure these effects, GLM (generalized linear models) models were used to check the significant differences in perceived managerial behavior according to the subordinate's sex (Table A3 in the Appendix) and the supervisor's sex (Table A4 in the Appendix).

Nine countries exhibit significant differences between subordinate women and men concerning their perception of their manager's support (indicated in bold type in Table A3). In all these countries, women consistently report higher managerial support from their managers than men do.

In some countries, there are also differences in perceived managerial support according to the supervisor's sex. In nine countries (indicated in bold in Table A4), employees with a female supervisor report higher perceived managerial support. In the other countries, the difference is not statistically significant. In none of the countries did employees with a female supervisor report lower perceived managerial support than employees with a male supervisor. This finding tends to support the idea that female managers are more supportive than male managers.

In addition, statistics concerning perceived managerial support across the different countries for each of the dyads (subordinate's sex-supervisor's sex) were produced to determine whether perceived managerial support varies according to the sex similarity or dissimilarity between subordinate and supervisor. In each country, GLM models were used to check the significance of the difference according to the dyads (Table A5 in the Appendix).

There are significant differences between the dyads in twelve countries (indicated in bold type in Table A5). The differences vary across the countries. For example, in Austria, Croatia, Germany, the Netherlands, North Macedonia, Serbia, and Spain, female subordinates with a female supervisor report a higher level of perceived managerial support than all the other dyads.

In Albania, female subordinates with male supervisors report the highest level of perceived managerial support from such men, followed by male subordinates with female supervisors. In Switzerland, male subordinates with female supervisors report the highest level of perceived managerial support from such women, followed by female subordinates with male supervisors, female subordinates with female supervisors, and male subordinates with male supervisors.

To summarize, the descriptive statistics provide initial insights. First, there are significant cross-country differences in perceived managerial support. Second, in some countries, there are also differences between female and male subordinates, and in others between subordinates who have a female supervisor and those with a male supervisor. In countries where there are differences, female subordinates consistently report a higher level of perceived managerial support, as do subordinates with a female supervisor. This finding prompted a further analysis that considered the sex (dis)similarity between the subordinate and the supervisor, which showed that the relationship is somewhat more complex than what was expected, with some countries where female subordinates with a female supervisor report higher levels of perceived managerial support than all the other dyads, and others where this is the case for male subordinates with a female supervisor. This finding highlights the importance of considering the dyad (subordinate's sex-supervisor's sex) and not only the supervisor's sex as in previous research.

III.3. The moderating role of gender ideology

It is worth considering whether this sex difference in perceived managerial support can be shaped (at least, in part) by gender ideology, and what happens when the moderating role of gender ideology is controlled for. As underlined in the literature review, gender ideology often differs across countries. Besides, gender ideology can influence both sex differences in managerial support and sex differences in perceived managerial supportsex differences referring to both the subordinate's sex and the supervisor's sex. First, Table 4 gives the average gender ideology for each country: a higher score indicates a more egalitarian gender ideology;

the score is given at the country level, not at the individual level. The countries with the most egalitarian gender ideology are Sweden, Denmark, Spain, France, and Finland (indicated in bold in Table 4). The countries with the most traditional gender ideology are Lithuania, Bulgaria, Slovakia, the Czech Republic, and Montenegro (indicated in italics in Table 4).

A correlation matrix shows no correlation between gender ideology and perceived managerial support (Table 5). However, a quick analysis indicates that countries with significant differences between the dyads have a higher average gender ideology (i.e., a more egalitarian gender ideology on average) than those without significant differences (average gender ideology in countries with significant differences: 3.19; average gender ideology in countries with no significant differences: 2.94).

Thus, we assume the national and sex differences are partly shaped by national gender ideology. Specifically, we propose this ideology leads to differences in perceived managerial support between female and male supervisors, as well as for female and male subordinates. This would explain both the national differences and why, when there are significant differences, they all go in the same direction, indicating that perceived managerial support is higher for female supervisors.

To check this proposition, hierarchical linear models with and without interaction effects are used. The dependent variable for both models is perceived managerial support. Model 1 includes only individual variables and does not include gender ideology or interaction effects.

Model 2 includes individual variables and gender ideology (i.e., a country-level variable).

Model 3 includes individual variables, gender ideology, and interaction effects between gender ideology and the supervisor's/subordinate's sex dyad. The results are reported in Table 6. To simplify the presentation of the results, the effects of control variables are not reported in [START_REF] Woehr | Justifying Aggregation With Consensus-Based Constructs : A Review and Examination of Cutoff Values for Common Aggregation Indices[END_REF]. The ICC(1) of 0.04 indicates 4% of the variation in perceived managerial support can be attributed to the country-level [START_REF] Biemann | agreement : On the use (and misuse) of rWG and rWG(J) in leadership research and some best practice guidelines[END_REF]. It is below the suggested 0.1 cutoff, but this is frequent in cross-national comparisons [START_REF] Autio | Consequences of cultural practices for entrepreneurial behaviors[END_REF]. Further, [START_REF] Bliese | Back to Basics with Mixed-Effects Models : Nine Take-Away Points[END_REF] show that when the data is nested, as is the case here, multilevel modeling is appropriate even when ICC(1) values are small and non-significant. The ICC( 2), which provides an estimate of the reliability of the country-level means [START_REF] Bliese | Group size, ICC values, and group-level correlations : A simulation[END_REF][START_REF] Shieh | Choosing the best index for the average score intraclass correlation coefficient[END_REF][START_REF] Woehr | Justifying Aggregation With Consensus-Based Constructs : A Review and Examination of Cutoff Values for Common Aggregation Indices[END_REF], is around 0.52. This is slightly lower than the suggested 0.7; however, this cutoff has been criticized, and another proposal is that an ICC(2) between 0.4 and 0.7 indicates good reliability [START_REF] Lance | The Sources of Four Commonly Reported Cutoff Criteria : What Did They Really Say?[END_REF][START_REF] Shieh | Choosing the best index for the average score intraclass correlation coefficient[END_REF]. The pseudo-R2 were also computed and are low. However, the current study pays attention to the effect some variables have on perceived managerial support, and does not aim to predict or find all the determinants of perceived managerial support. In this case, a low R2 does not preclude a model from being informative [START_REF] Rights | Quantifying explained variance in multilevel models : An integrative framework for defining R-squared measures[END_REF]. To check the robustness of the results, non-hierarchical models were also estimated and give the same overall results for the effect of the dyad subordinate's sex and supervisor's sex, with and without interaction effects with national gender ideology. Table 6 shows that even when controlling for type of contract, working time, education, age, tenure, skills, sector, and workforce, the effect of sex (dis)similarity on perceived managerial support remains significant. Perceived managerial support is higher in dyads where the manager is a woman, i.e., in Woman-Woman and Man-Woman dyads. The effects are the same when controlling for gender ideology (Model 2). However, Model 3 shows that when controlling for gender ideology as a moderator of the relationship between the dyad and perceived managerial support, the effect of the subordinate's sex and the supervisor's sex becomes insignificant.

IV. Discussion and conclusion

The current study contributes to the debate on sex differences in perceived managerial support by simultaneously considering the subordinate's and the supervisor's sex, their sex national gender ideology plays a role in this relationship. The results show that, on average, perceived managerial support is higher if supervisors are women, regardless of the subordinate's sex and, thus, of sex (dis)similarity between the subordinate and the supervisor.

In addition, the difference between female and male supervisors disappears when controlling for the moderating role of national gender ideology.

IV.1. Theoretical contributions

This study finds that female managers are perceived as being more supportive than their male counterparts. This finding aligns with prior studies that found women exhibit more supportive and more coaching managerial behaviors than men [START_REF] Eagly | Transformational, transactional, and laissez-faire leadership styles : A meta-analysis comparing women and men[END_REF][START_REF] Ye | Gender and managerial coaching across cultures : Female managers are coaching more[END_REF]. In a context where companies increasingly value transformational leadership based on managerial support [START_REF] Hoobler | The Business Case for Women Leaders : Meta-Analysis, Research Critique, and Path Forward[END_REF][START_REF] Saint-Michel | Leader gender stereotypes and transformational leadership : Does leader sex make the difference? M@n@gement[END_REF], this finding argues for greater inclusion of women in managerial positions. Previous research has explained these gender differences by gender role theory, which suggests that managerial behaviors are influenced by gender roles [START_REF] Saint-Michel | Style de leadership, LMX et engagement organisationnel des salariés : Le genre du leader a-t-il un impact ? @[END_REF][START_REF] Ye | Gender and managerial coaching across cultures : Female managers are coaching more[END_REF]: 'to the extent that gender roles exert some influence on leaders, female and male occupants of the same leadership role would behave somewhat differently' (Eagly et al., 2003, p. 572). In particular, there exist a lot of injunctions for women to display more supportive and service-oriented behaviors than men (Saint-Michel, 2018). However, gender role theory not only emphasizes that individual behaviors are constrained by gender norms; it also highlights how perceived individual behaviors are constrained by gender norms. For example, behaviors that correspond to masculine stereotypes (e.g., authority, ambition) are evaluated less favorably in women, because with such behavior, women deviate from the gender norms [START_REF] Eagly | Role congruity theory of prejudice toward female leaders[END_REF].

That is why the current study emphasizes perceived managerial behavior. The data used here relates to how subordinates perceive their supervisors' behaviors and shows that female managers are perceived as more supportive than their male counterparts.

While gender role theory underlines the importance of gender stereotypes in shaping perceived and concrete individual behaviors, research on gender differences in managerial support has not paid much attention to the role national gender ideologies play. Gender ideology refers to beliefs and representations regarding acceptable behaviors for women and men, which are internalized during the socialization process. Gender ideology, as a set of beliefs, can be measured at the individual level, but by aggregating individual beliefs it is also possible to produce a national index, as in other research on values and cultures [START_REF] Hofstede | Dimensionalizing Cultures : The Hofstede Model in Context[END_REF][START_REF] Leung | Values, schemas, and norms in the culture-behavior nexus : A situated dynamics framework[END_REF]. The current study has focused on the role national gender ideology plays in the relationship between sex and perceived managerial support, based on the fact that gender ideology shapes individual behaviors and expectations [START_REF] Gaunt | Job insecurity, stress and gender : The moderating role of gender ideology[END_REF]. In fact, the results show that, although gender ideology does not directly affect perceived managerial support, the effect of the dyad subordinate's sex-supervisor's sex becomes non-significant when gender ideology is introduced as a moderator. The fact that the difference between the dyads disappears when we control for the moderating role of gender ideology indicates that it is mostly national gender ideology that creates differences in perceived managerial support between female and male supervisors and for female and male subordinates. This result brings a new insight to the debate related to existence of a specific feminine management style.

Indeed, it jeopardizes the essentialist discourses which presume that women are essentially different from men, with different skills and abilities [START_REF] Ellingsaeter | Scandinavian welfare states and gender (de) segregation : Recent trends and processes[END_REF][START_REF] Grunow | Gender Ideologies in Europe : A Multidimensional Framework[END_REF].

It shows that female managers are not intrinsically different from male managers, but their behavior can be perceived as different due to beliefs about gender. With this insight, the current study brings an additional argument to the nature/nurture debate related to sex differences in professional behaviors and choices [START_REF] Beblo | On the nature of nurture. The malleability of gender differences in work preferences[END_REF][START_REF] Eagly | Gender Identity : Nature and Nurture Working Together[END_REF]. Overall, the study underlines the importance of considering and measuring gender ideology at the national level in cross-cultural research on gender issues.

Previous research has shown that subordinates and supervisors in same-sex dyads have better working relationships [START_REF] Ghaus | Much of a Muchness? The Role of Gender Similarity in a Relationship between LMX and OCB[END_REF][START_REF] Vecchio | Leadership and Sex-Similarity : A Comparison in a Military Setting[END_REF]. This result has been explained by the 'similarity-attraction paradigm' (or the homophily paradigm), namely by the principle that an increase in demographic similarity leads to an improvement in interpersonal attraction and relations. However, the current study's results suggest that this paradigm does not hold. More precisely, the supervisor's sex plays a more important role than the sex similarity between subordinate and supervisor. For example, Models 1 and 2 show that perceived managerial support is higher among subordinates with female managers, whatever the subordinates' sex. Descriptive statistics are useful to explain this result. Indeed, they show that the effect of sex (dis)similarity is not the same in every country. In most countries where the dyad has an effect, female subordinates with a female supervisor report a higher level of perceived managerial support than all the other dyads, but in other countries (e.g., Switzerland), male subordinates with female supervisors report the highest level of perceived managerial support. These national differences, compared to the fact that female managers are perceived as more supportive than male managers in every country where the effect is significant, suggest that the similarity-attraction paradigm is less adequate than the gender role theory in explaining variations in perceived managerial support according to gender.

IV.2. Methodological contributions

This study is based on the unique combination of two databases (EWCS and EVS). Its measurement of gender ideology, because of the representative and large-scale EWS sample, allows for generalizing the aggregate at the country level, unlike other data sources on national cultures, like researcher-administered surveys [START_REF] Saint-Michel | Leader gender stereotypes and transformational leadership : Does leader sex make the difference? M@n@gement[END_REF][START_REF] Ye | Gender and managerial coaching across cultures : Female managers are coaching more[END_REF]. Notably, it shows that the usual comparative data sources used in cross-cultural research, such as Hofstede's project [START_REF] Hofstede | Dimensionalizing Cultures : The Hofstede Model in Context[END_REF], are not sufficient in investigating gender issues, as they do not contain information about perceptions and norms related to gender. Particularly, gender ideology does not correspond at all with Hofstede's 'masculinity' dimension, which is supposed to measure the differences between men's and women's values. Lithuania, which has a very traditional gender ideology according to the EVS survey, has a low masculinity score, i.e., close to Sweden's, which according to the survey has a very egalitarian gender ideology. This result highlights the absence of data on concrete gender norms in Hofstede's dimensions, a point previously raised by feminist scholars regarding cross-national typologies [START_REF] Orloff | Gender and the Social Rights of Citizenship : The Comparative Analysis of Gender Relations and Welfare States[END_REF] yet not in management cross-cultural research.

The cross-country differences also underline the relevance of a multilevel approach, which incorporates both country-level and individual-level variables, in management research related to gender and perceptions. As [START_REF] Figureau | Experimentally Validated Surveys : Potential for Studying Cognitive and Behavioral Issues in Management[END_REF] underlined, multilevel data collection is required in management research on behavioral and cognitive issues.

Another key methodological issue relates to the approach this study uses, which deviates from the typical quantitative research in management. The study employs an exploratory approach since the complexity of the topic, the relationships between the variables, the conflicting results from previous studies on the existence of a specific feminine managerial style, and the limited research on the impact of national gender ideology made it difficult to formulate specific hypotheses [START_REF] Min | Support exchanges between adult children and their parents across life transitions[END_REF][START_REF] Witt | Our Board, Our Rules : Nonconformity to Global Corporate Governance Norms[END_REF]. Therefore, instead of a precise hypothesis, a general research question was formulated based on the literature review and the theoretical framework of gender ideology or gender norms, asking whether national gender ideology plays a role in the relationship between the subordinate's and the supervisor's sex and perceived managerial support. For this reason, the approach used here has been described as exploratory, rather than hypothetico-deductive. Adopting an exploratory approach in quantitative research helps avoid the pitfalls of hypothetico-deductive research, such as defining hypotheses after the results are known (HARKing) [START_REF] Murphy | HARKing : How Badly Can Cherry-Picking and Question Trolling Produce Bias in Published Results[END_REF] and being locked in a binary position of (in)validating hypotheses without being able to describe concrete situations and formulate new theoretical proposals [START_REF] Locke | The Case for Inductive Theory Building[END_REF][START_REF] Ketokivi | Two Strategies for Inductive Reasoning in Organizational Research[END_REF]. This exploratory approach also heeds some scholars' call for more variety in research design in management research [START_REF] Alvesson | Beyond Formulaic Research : In Praise of Greater Diversity in Organizational Research and Publications[END_REF][START_REF] Greasley | HR analytics : The onto-epistemology and politics of metricised HRM[END_REF][START_REF] Harley | The one best way? 'Scientific' research on HRM and the threat to critical scholarship[END_REF]. Additionally, it is all the more important that new paradigms are seen to emerge in management research due to 'big data,' namely, data-driven science [START_REF] Kitchin | Big Data, new epistemologies and paradigm shifts[END_REF]. Several scholars in management research have underlined the advent of this new paradigm: 'For the most part, OMS (organizational and management scholarship) largely confines its interests to preconceived hypotheses […]. However, Big Data can contain analytical value that exceeds any a priori conception […]. Phenomenon-driven, exploratory approaches may not precisely explain why something is happening, but they can identify (ir)regularities or shed light on boundary conditions, thereby generating novel questions' (Wenzel & Van Quaquebeke, 2018, p. 555). Data-driven science seems closed to exploratory approaches like the one used here, where data is used to enrich existing knowledge without hypothesizing a priori what the results will be.

IV.3. Practical implications

Although gender equality is one of the United Nations' Sustainable Development Goals, women still lag behind men in access to leadership positions. The current study can help companies try to close this gap. Indeed, it challenges the 'think managerthink male' paradigm [START_REF] Peterson | Exit the king. Enter the maid : Changing discourses on gendered management ideals in Swedish Higher Education[END_REF] and the notion of a specific 'feminine managerial behavior.' It shows that when controlling for the moderating role of gender ideology, there is no (subordinate's and supervisor's) sex difference in perceived managerial support. Hence, the article shows that, instead of assuming that female managers behave differently from men, organizations should pay attention to gender stereotypes among their employees and managers, which can influence how female and male subordinates perceive managerial support from female and male supervisors. This is especially true for multinational corporations, which face different national gender ideologies. This paper shows that average perceived managerial support and national gender ideology vary drastically across countries. Companies should try to reduce gender stereotypes among their employees. This could imply training and sensitization workshops focused on unconscious biases and stereotypes, dedicated to both women and men [START_REF] Madsen | Unconscious Gender Bias : Implications for Women's Leadership Development[END_REF]. Besides, the study suggests that, due to gender ideology, subordinates do not evaluate their female and male managers in the same way. Therefore, companies should also limit the effects of gender stereotypes on individual evaluations or be cautious when using individual evaluations.

IV.4. Limitations

Unlike most quantitative studies, the current study does not suffer from common-method bias, because two different data sources were used. In addition, the representative sample allows the results to be extrapolated to the countries included in the survey. However, the study does present some limitations that offer new research areas. The main limitation relates to the scale used to measure gender ideology. It is a unidimensional scale, as in most previous research, but in recent years, some scholars who advocate a multidimensional approach have questioned this unidimensionality [START_REF] Grunow | Gender Ideologies in Europe : A Multidimensional Framework[END_REF]. The second limitation refers to the low value of The stars indicate the significance of the difference between women supervisors' perceived managerial support compared to their male counterparts in each country (GLM models), with men being the reference group.

Significance levels: *: p<0.1; **: p<0.05; ***: p<0.01

(

  dis)similarity, and national gender ideology. Indeed, most previous research has focused on the relationship between the supervisor's sex and perceived managerial support, without looking at the role played by national gender ideology. Besides, only a few studies pay attention to the effect of the subordinate's sex. A research question was formulated based on the literature review and the theoretical framework of gender ideology or gender norms, asking to what extent perceived managerial support is simultaneously shaped by the supervisor's sex, the subordinate's sex, sex (dis)similarity between subordinate and supervisor, and to what extent

Table 1 .

 1 Sample

	Country	Number of respondents
	Albania	452
	Austria	788
	Bulgaria	793
	Croatia	727
	Czech Republic	741
	Denmark	833
	Estonia	803
	Finland	738
	France	1,188
	Germany	1,545
	Hungary	732
	Italy	718
	Lithuania	774
	Montenegro	606
	Netherlands	738
	North Macedonia	647
	Norway	881
	Poland	759
	Portugal	592
	Romania	718
	Serbia	573
	Slovakia	764
	Slovenia	1,232
	Spain	2,378
	Sweden	865
	Switzerland	806

Table 2 .

 2 . The EWCS survey records age in years. Previous research also used tenure (Saint-Michel, 2018), which the EWCS data expresses in years. Phenomena of employees being overskilled or underskilled can also affect their expectations of their work and relationships (McGuinness & Wooden, 2009). The EWCS survey recorded skills in three categories, as an unordered categorical variable: 'I need further training to cope well with my duties,' 'My present skills correspond well with my duties,' and 'I have the skills to cope with more demanding duties.Descriptive statistics

		Whole sample	Women sample	Men sample
		N=22,391	N=11,889	N=10,502
	Employee's sex/
	Supervisor's sex
	Woman-Woman	25.99%	52.12%	0%
	Woman-Man	23.88%	47.88%	0%
	Man-Woman	7.19%	0%	14.35%
	Man-Man	42.94%	0%	85.65%
	Perceived manager's	m (mean)=3.92,	m=3.94, sd=0.85	m=3.89, sd=0.88
	behavior	sd (standard
		deviation)=0.87
	Type of contract
	Unlimited duration	81.41%	81.67%	81.16%
	Limited duration	13.60%	13.89%	13.32%
	Temp. empl. agency	1.26%	1.14%	1.38%
	No contract	3.73%	3.31%	4.15%
	Time
	Full-time	84.51%	76.99%	91.93%
	Part-time	15.49%	23.01%	8.07%
	Education

'

Two additional company-related control variables were also included. The sector is an unordered categorical variable with four modalities: private, public, joint private-public, and other. The workforce is an unordered categorical variable referring to the number of employees within the organization (1, 2-9, 10-249, 250+, and don't know).

Table 2 contains univariate descriptive statistics of the variables used in the study for the whole sample, and for women and men samples.

Table 3 .

 3 Perceived managerial support across the countries

	Country

Table 4 .

 4 Gender ideology across countries

	Country

Table 5

 5

	. Correlation matrix
		n	M	SD	1	2	3	4
	1. Perceived MS	22,391	3.92	0.87	-
	2. Gender ideology 22,391	3.07	0.32	0.01	-
	3. Age	22,391	41.56	11.90	-0.04*** 0.02*** -
	4. Tenure	22,391	9.72	9.65	-0.01	0.02*** 0.57*** -
	Significance levels: *: p<0.1; **: p<0.05; ***: p<0.01

Table 6 (

 6 see TableA6in the Appendix for the full detailed results with the effects of control

variables). ICC(1), ICC(2), and rwg were computed and justify aggregation, as indicated in

Table 6 .

 6 The rwg index, which measures the within-group agreement to justify aggregating individual-level data, is around 0.65 and shows moderate agreement (close to strong agreement, at a 0.71 threshold)

Table 6 .

 6 Results of the models

	Model 1	Model 2	Model 3
	Estimate (SE)	Estimate (SE)	Estimate (SE)

Table A3 .

 A3 ICC(1) and pseudo-R2. Although academic research has shown that multilevel modeling is adequate for nested data even if ICC(1) is low, it means that the country level does not play a very important role in perceived managerial support. Similarly, although the low pseudo-R2does not preclude the model from providing information about the relationships between the variables, it does indicate that the models do not predict perceived managerial support very well. One explanation for this low value could be that several factors can influence perceived Perceived managerial support according to the subordinate's sex

	Zhao, K., Zhang, M., & Foley, S. (2019). Testing two mechanisms linking work-to-family Tables A3, A4, and A5 report the average perceived managerial support and, in parentheses,
	conflict to individual consequences : Do gender and gender role orientation make a the standard deviation.
	difference? The International Journal of Human Resource Management, 30(6),
	988-1009. https://doi.org/10.1080/09585192.2017.1282534
	Country	Women	Men
		m (sd)	m (sd)
	Albania	4.03 (0.64)*	3.91 (0.76)
	Austria	3.96 (0.91)	3.86 (0.90)
	Bulgaria	4.18 (0.73)	4.15 (0.88)
	Croatia	3.88 (0.86)	3.78 (0.92)
	Czech Republic	3.96 (0.79)	3.96 (0.79)
	Denmark	3.89 (0.90)	3.93 (0.87)
	Estonia	3.77 (0.73)*	3.67 (0.86)
	Finland	3.93 (0.90)	3.99 (0.90)
	France	3.71 (0.96)	3.78 (0.96)
	Germany	3.79 (0.81)**	3.70 (0.86)
	Hungary	4.03 (0.87)	3.97 (0.88)
	Italy	3.66 (0.71)	3.68 (0.70)
	Lithuania	3.90 (0.71)	3.87 (0.80)
	Montenegro	3.69 (0.91)	3.68 (0.86)
	Netherlands	3.98 (0.60)**	3.84 (0.89)
	North Macedonia	4.35 (0.74)*	4.23 (0.85)
	Norway	4.22 (0.78)	4.18 (0.81)
	Poland	3.82 (0.84)	3.75 (0.85)
	Portugal	4.18 (0.69)*	4.07 (0.87)
	Romania managerial behavior, such as those related to the work environment. Future research using 4.13 (0.76) 4.15 (0.71) Serbia 3.78 (0.99)** 3.60 (0.94)
	Slovakia	3.77 (0.70)	3.75 (0.81)
	micro-level and qualitative data could provide deeper insight into the organizational and Slovenia 3.99 (0.93) 3.98 (1.02)
	Spain	4.08 (0.92)**	3.98 (0.93)
	managerial environment. Sweden	3.78 (0.81)	3.79 (0.79)
	Switzerland	4.

05 (0.74)*** 3.90 (0.79)

	The stars indicate the significance of the difference between supervisors' perceived managerial
	support among female employees compared to their male counterparts in each country (GLM
	models), with men being the reference group.
	Significance levels: *: p<0.1; **: p<0.05; ***: p<0.01

Table A4 .

 A4 Perceived managerial support according to the supervisor's sex

	Country	Women	Men
		m (sd)	m (sd)
	Albania	4.04 (0.61)	3.96 (0.73)
	Austria	3.98 (0.94)	3.88 (0.89)
	Bulgaria	4.23 (0.68)	4.13 (0.85)
	Croatia	3.91 (0.85)	3.79 (0.90)
	Czech Republic	4.03 (0.77)*	3.93 (0.80)
	Denmark	3.94 (0.89)	3.89 (0.88)
	Estonia	3.73 (0.78)	3.72 (0.79)
	Finland	4.02 (0.89)	3.91 (0.91)
	France	3.79 (0.94)	3.73 (0.97)
	Germany	3.84 (0.76)***	3.71 (0.86)
	Hungary	4.00 (0.97)	4.01 (0.83)
	Italy	3.67 (0.77)	3.67 (0.67)
	Lithuania	3.88 (0.72)	3.90 (0.77)
	Montenegro	3.69 (0.89)	3.68 (0.88)
	Netherlands	4.12 (0.80)***	3.82 (0.92)
	North Macedonia	4.34 (0.79)	4.26 (0.81)
	Norway	4.25 (0.79)	4.17 (0.79)
	Poland	2.13 (0.80)*	2.25 (0.87)
	Portugal	3.87 (0.72)	3.75 (0.79)
	Romania	4.13 (0.76)	4.14 (0.72)
	Serbia	3.86 (0.85)***	3.61 (1.01)
	Slovakia	3.84 (0.69)**	3.71 (0.77)
	Slovenia	4.05 (0.92)	3.96 (1.00)
	Spain	4.10 (0.92)***	3.99 (0.93)
	Sweden	3.87 (0.76)***	3.72 (0.83)
	Switzerland	4.08 (0.70)**	3.93 (0.79)

Table A5 .

 A5 Perceived managerial support according to the dyadsThe stars indicate the significance of the difference between the dyads in each country (GLM models), the dyad Man-Man being the reference group.

	Country	Woman-	Woman-Man	Man-Woman	Man-Man
		Woman	m (sd)	m (sd)	m (sd)
		m (sd)
	Albania	4.01 (0.62)	4.05 (0.67)**	4.20 (0.58)*	3.88 (0.77)
	Austria	4.00 (0.93)*	3.92 (0.90)	3.86 (0.99)	3.86 (0.89)
	Bulgaria	4.22 (0.68)	4.14 (0.79)	4.25 (0.71)	4.13 (0.90)
	Croatia	3.92 (0.86)*	3.83 (0.87)	3.85 (0.85)	3.77 (0.92)
	Czech Republic	4.03 (0.77)	3.88 (0.81)	4.03 (0.74)	3.95 (0.79)
	Denmark	3.91 (0.91)	3.87 (0.89)	4.02 (0.83)	3.90 (088)
	Estonia	3.75 (0.75)	3.80 (0.71)*	3.66 (0.99)	3.67 (0.83)
	Finland	3.95 (0.91)	3.87 (0.89)	4.34 (0.66)***	3.93 (0.92)
	France	3.76 (0.94)	3.66 (0.98)	3.85 (0.96)	3.77 (0.96)
	Germany	3.85 (0.76)***	3.74 (0.85)	3.81 (0.76)	3.69 (0.87)
	Hungary	4.02 (0.94)	4.05 (0.79)	3.87 (1.10)	3.99 (0.85)
	Italy	3.63 (0.77)	3.68 (0.66)	3.76 (0.77)	3.66 (0.68)
	Lithuania	3.87 (0.68)	3.97 (0.75)	3.91 (0.94)	3.87 (0.78)
	Montenegro	3.66 (0.91)	3.71 (0.92)	3.77 (0.85)	3.67 (0.86)
	Netherlands	4.17 (0.80)***	3.82 (0.95)	3.94 (0.79)	3.82 (0.90)
	North Macedonia	4.40 (0.75)*	4.30 (0.74)	4.05 (0.96)	4.25 (0.84)
	Norway	4.29 (0.76)	4.12 (0.80)	4.14 (0.89)	4.19 (0.79)
	Poland	3.85 (0.81)	3.77 (0.87)	3.98 (0.69)	3.73 (0.86)
	Portugal	4.20 (0.67)	4.16 (0.71)	4.01 (0.98)	4.08 (0.85)
	Romania	4.16 (0.74)	4.09 (0.78)	4.01 (0.86)	4.17 (0.69)
	Serbia	3.90 (0.80)***	3.66 (1.13)	3.70 (1.02)	3.58 (0.91)
	Slovakia	3.84 (0.65)	3.68 (0.74)	3.83 (0.96)	3.74 (0.79)
	Slovenia	4.04 (0.91)	3.93 (0.96)	4.05 (0.93)	3.97 (1.03)
	Spain	4.12 (0.90)***	4.04 (0.94)	4.05 (0.99)	3.97 (0.92)
	Sweden	3.85 (0.76)	3.66 (0.88)	3.91 (0.76)*	3.75 (0.81)
	Switzerland	4.03 (0.70)**	4.06 (0.78)***	4.31 (0.63)***	3.86 (0.79)
	Significance levels: *: p<0.1; **: p<0.05; ***: p<0.01

Table A6 .

 A6 Full results of the models

		Model 1	Model 2	Model 3
		Estimate (SE)	Estimate (SE)	Estimate (SE)
	Individual level variable
	Employee's sex/Supervisor's sex (ref. Man-
	Man)
	Woman-Woman	0.06 (0.01)***	0.06 (0.01)***	-0.10 (0.14)
	Woman-Man	-0.00 (0.01)	-0.00 (0.01)	0.05 (0.14)
	Man-Woman	0.06 (0.02)***	0.06 (0.02)***	-0.14 (0.22)
	Country level variable
	Gender ideology	-	0.02 (0.11)	0.01 (0.11)
	Cross-level interactions
	GI x dyad	-
	GI * WW			0.05 (0.04)
	GI * WM			-0.02 (0.05)
	GI * MW			0.07 (0.07)
	Individual control variables
	Type of contract (ref. Unlimited duration)
	Limited duration	-0.08 (0.02)***	-0.08 (0.02)***	-0.08 (0.02)***
	Temp. empl. agency	-0.25 (0.06)***	-0.25 (0.06)***	-0.25 (0.06)***
	No contract	-0.08 (0.04)**	-0.08 (0.04)**	-0.08 (0.04)**
	Time (ref. Full-time)
	Part-time	0.02 (0.02)	0.02 (0.02)	0.02 (0.02)
	Education (ref. Upper secondary)
	Primary
	Lower secondary
	Postsec. non-tertiary
	Short-cycle tertiary
	Bachelor
	Master's or Doctorate

Throughout the article, the word 'sex' will be used to refer to the statistical variable indicating whether someone identifies as a woman or a man. The term 'gender' will be used mostly when referring to 'gender role theory,' which relates to the roles and behaviors expected from a woman and a man.

Acknowledgements

I would like to thank the reviewers and the editor for taking the time and effort to review this manuscript. I sincerely appreciate their valuable comments and suggestions, which helped me to improve the quality of the manuscript.

Appendix

Table A2. Items used to measure national gender ideology Item Type of question 'When a mother works for pay, the children suffer.' Likert scale, with respondents having to indicate their degree of agreement: 1='Agree strongly' 2='Agree' 3='Disagree' 4='Disagree strongly' 'A job is alright, but what most women truly want is a home and children.' 'All in all, family life suffers when the woman has a full-time job.' 'A man's job is to earn money, a woman's job is to look after the home and family.' 'On the whole, men make better political leaders than women do.' 'A university education is more important for a boy than for a girl.' 'On the whole, men make better business executives than women do.'


Katalin Gosztonyi
email: katalin.gosztonyi@umontpellier.fr

Simon Modeste

Towards a model of problem networks for studying problem solving in mathematics education

Keywords: Problem networks, inquiry based mathematics education, ckȼ model

In this paper, we present an ongoing research project developing a modelling tool for conceiving and analysing problem-based teaching and learning trajectories, in form of networks of problems. We explain the origins and the emergence of the model, the underlying theoretical choices. We emphasise how the model allows taking into account argumentation and proving aspects in the context of problem-solving activities.

Introduction

This paper discusses a part of an ongoing project named Problem-based learning trajectories in discrete mathematics education (ProDiME), realized as a collaboration between the two authors and V. Durand-Guerrier. This project seeks to conceive and analyze teaching trajectories and their different implementations in French and Hungarian classrooms, with the help of a modelling tool based on "networks of problems". The first step of the project is to develop this modelling tool by crossing our earlier works. In this paper, we explain how our respective preliminary research motivates the emergence of a model in terms of "networks of problems", focusing particularly on aspects concerning argumentation and proof in the context of problem solving activities.

Our project is closely connected to questions of Inquiry Based Mathematics Education (IBME). Although different interpretations and approaches exist to IBME, with slightly divergent characteristics (Artigue & Blomhoj 2013), they share a number of main principles, one of them being the central role accorded to problems in the learning of mathematics, and the conviction that new mathematical knowledge can only emerge by active participation of the learners in the construction process of mathematics. From this conviction follows the challenge of concerting problem solving activities with curricular learning. Different strategies exist, in different approaches to IBME, to conceive complex problem-based teaching and learning trajectories [START_REF] Artigue | Inquiry based mathematics education and the development of learning trajectories[END_REF], but a general observation is that the conception, analysis and implementation of these trajectories as well as the dissemination of the conception strategies behind them remain difficult.

In several approaches, and more particularly in our respective research, the idea of representing the structure of these trajectories in form of graphs emerged as a powerful tool both for research purposes and for communication with teachers. Our aim is to develop these attempts of representation into a theoretically well based modelling tool, relying on Vergnaud's Theory of Conceptual Fields (2009) and Balacheff's ckȼ model (2013). We thus discuss the question:

 How to establish a theoretical model allowing the description of set of problems, their interrelations, and their relations with mathematical content, for didactical purposes?

In this paper, we emphasize particularly a specific aspect of this work:

 What is the role of argumentation and proof in problem solving activities and problem-based mathematics education, and how can we take into account this role in the frame of our model?

Reflecting argumentation and proof processes appears to be a crucial aspect in this work, for several reasons. First, the process of problem solving implies a series of decisions based on (correct or wrong) convictions and conjectures. A (rational) problem solver has implicit or explicit arguments to choose steps of the problem solving procedure, and solving a problem supposes reasons to be convinced that the solution is found and correct. We suggest interpreting problem solving steps as transformations of the original problem, and Balacheff's ckȼ model allows us to take into account the abovementioned argumentative aspects as controls applied during the choice and execution of these steps.

A second aspect is the connection of problem-solving activities with the construction of mathematical knowledge. Argumentation and proving play a crucial role in assuming the emergence of new elements of mathematical content.

In the following, we first present the emergence and the role of problem networks in our respective preliminary research, then we discuss how a combination of these works leads us to the definition of a modelling tool. We will emphasise questions related to argumentation and proof in this presentation.

"Series of problems" in history of mathematics and in mathematics education

Gosztonyi's research was partly inspired by an interdisciplinary history of sciences project introducing the term "series of problems" as a methodological tool for the analysis of texts from various historical contexts [START_REF] Bernard | Les séries de problèmes, un genre au carrefour des cultures: Une première synthèse[END_REF]. She developed analytical tools and methods by reinvesting them from one context to another.

Series and networks of problems in the Hungarian context of mathematics education

Gosztonyi's work set out from the analysis of Hungarian traditions of teaching by "Guided Discovery". She showed that one of the specificities of Guided Discovery is the conception of long and complex teaching trajectories based on "series of problems", meaning a conscious ordering of problems for reaching one ore several specific teaching purposes [START_REF] Gosztonyi | Séries de problèmes dans une tradition d'enseignement des mathématiques en Hongrie au 20 e siècle[END_REF]. She showed that, although this idea appears quite coherently in different resources related to the approach, the identification of series problems, the reconstruction of the underlying principles of structuration, the related teaching purposes and aspects of implementation is quite a challenging task, both for researchers and for teachers who are novices to the approach. Based on the analysis of some specific texts where the design choices where explicated, she suggested a representation of the underlying structure of series of problems in form of problem networks, making the dense links between problems visible. In a collaborative project involving expert teachers of Guided Discovery, the problem network representation was reinvested as a mediator in discussions between teachers and researchers helping to explicate teachers' design choices [START_REF] Gosztonyi | Teachers' practices and resources in the Hungarian "guided discovery" approach to teaching mathematics. Presenting and representing "series of problems[END_REF]).

An analysis of Clairaut's Elements of Geometry

The idea of problem network representation was also reinvested in the analysis of a text from another historical context, namely Clairaut's Elements of geometry (1741) from 18 th century France, which shows interesting analogies with texts of Hungarian Guided Discovery [START_REF] Gosztonyi | Series of problems in Clairaut's Elements of Geometry: interaction between historical analysis and mathematics education research[END_REF]. Although we don't exactly know its original destination, the preface talks about introducing "beginners" into geometry, and the book was interpreted by many as a didactical text, a kind of teacher guide. Clairaut proposes to write an "Elements of geometry" not in a deductive order but structured by problems.

The intention is to allow beginners to "acquire more easily the spirit of discovery" (p. x). We present here some elements of the analysis of this book, which emphasise the connections between problem solving, proving and the construction of mathematical knowledge.

The text is organised into four books, and structured mainly by numbered "paragraphs"; but different types of content (as definitions, theorems, demonstrations, problems, methods) are not explicitly distinguished. Gosztonyi, in collaboration with Bernard carried out a discursive analysis of the text to identify these types of content and interrelations between them, explicating the hidden structuration strategies of the text. One notable characteristics is the rich fabric of metadiscourse explaining interrelations between elements of the content and arguing transitions from one problem to another. The analysis of the metadiscourse allowed us to represent the structure of the text in form of a graph (Figure 1), and to interpret different types of links (represented on the figure as annotations of arrows).

From a point of view of argumentation and proof, it is interesting to observe Clairaut's ambiguous use of the term "problem". In the few cases the term explicitly appears in the main text, it is used in the classical sense of construction or measurement problems. In the preface however, the term is used in a more general sense, covering also research questions leading to the discovery of theorems, which is closer to modern uses in the context of teaching by problem solving, as Pólya's (1945). In our analysis, we distinguished 'problems' and 'propositionproblems' (PrB on the figure): problems are construction or measurement problems the solution of which is a procedure, while 'proposition-problems' are conjectures or research questions leading to a theorem, and their solution is a demonstration. The discursive structures of these two types of problems and their solutions are quite similar. Clairaut suggests that two types of motivations can lead to the introduction of new problems: need or curiosity. The first book is mainly structured around problems motivated by practical needs of land surveying but we can already observe a combination of the description of procedures and argumentative, demonstrative passages in the description of the solutions. Our analysis pointed out that new 'propositions' emerge repeatedly as constitutive elements of a problem's solution. In later books, proposition-problems and their demonstrations take a more central role, with a shift to motivation by 'curiosity'. Figure 1 illustrates the structure of an extract from book III, underlying rich connections between problems and proposition-problems. The structuring strategies appear to be quite coherent throughout the different books of the text, nevertheless, new types of links (as inductions or generalisation by analogy) emerged together with the proposition-problems.

Analysing problem solving paths in teaching mathematics and computer science

Modeste, the second author of this paper, arrived at the idea of problem networks from another direction. He was interested in the analysis of problem-solving strategies and processes and of proof and validation, at the interface of mathematics and computer science [START_REF] Modeste | Proof, reasoning and logic at the interface between mathematics and Computer Science: toward a framework for analyzing problem solving[END_REF]. [START_REF] Modeste | Prendre en compte l'épistémologie de l'algorithme. Quels apports d'un modèle de conceptions ? Quelle transposition didactique ?[END_REF][START_REF] Modeste | Proof, reasoning and logic at the interface between mathematics and Computer Science: toward a framework for analyzing problem solving[END_REF] as well as Giroud ( 2011), inspired by the notion of problem used in the theory of algorithmic complexity (e.g., see Garey & Johnson, 1979), described problems as follows. A problem P is considered as a couple P=(I,Q) where I is a family of mathematical objects (instances), and Q is a single question that can be asked about any element of I. This definition allows considering various levels of generality of a problem, and various domains of application.

For example, we can consider the problem P1=(I,Q) where I is constituted of all the pairs of positive integers (m, n) (that is, the set N²), and Q is the question "how many different words can be written with m letters 'a' and n letters 'b'?". This description allows to distinguish this general problem from more restricted problemslike the problem P2, asking the same question Q but for a reduced set of instances where m is fixed at 2, or P3 asking the same question for the subset of instances where m=n.

It also allows to consider instantiated problems, where you only take one element of the family of instances. A problem can also be modified by adding constraints to the question Q or by changing it.

Another important choice coming from Modeste's earlier work is the use of Balacheff's ck¢ model. Based on Brousseau's Theory of Didactical Situations (Brousseau & Warfield, 2020) and Vergnaud's Theory of Conceptual Fields (2009), [START_REF] Balacheff | cK¢, a model to reason on learners' conceptions[END_REF] describes a concept as composed of 4 elements: a set P of problems that give meaning to the concept, a representation system L (the signifier), and invariants that are separated into a set R of operators which permits to act on problems (that is, transform a problem into another), and a set Σ of controls that permit to decide whether and how an operator r applies to a given problem, and to evaluate the effects an operator has had on a problem (in particular to determine whether or not a problem is solved).

This last aspect, the distinction between operators and controls is one of the notable developments of Balacheff's model, and this is the aspect on which we can rely for considering the important role of argumentation and proof in problem solving activities. The separation of operators, controls, and representation system is relevant to deal with problem solving activity, to describe the actions one can have on problems, but also to explicate controls as elements justifying the action, thus emphasising proof and proving aspects. The representation system component allows taking into account the semiotic registers in which the problem is represented, and to describe operators and controls available in this register (including those linked with changing register).

Combined with the above-mentioned definition of problems, Modeste at al. ( 2019) implemented the ck¢ model for a priori analyses, describing possible problem solving paths, the way new problems are generated in the process and their relations to the previous problems.

They illustrated how such a model can be used to make an a priori analysis of a given problem, modelling and analysing solving strategies, and explicating the possible operators and controls. This can be represented with a flowchart as in Figure 2, presenting a mathematical solution of Problem 0 (Figure 2).

Arrows between problems (blue) represent the transformations by an operator (green) selected according to a control (red) on operators and on the status of problems.

Towards a modelling tool based on networks of problems

In the ProDiME project, we combine the analysis of "series of problems", that is, longer teaching trajectories conceived by a conscious ordering of problems, based on complex systems of problems, with the analysis of problem solving paths, that is, a set of problems generated by successive transformations of a given problem during problem-solving process. Although these are different types of analyses, there are numerous possible connections between them. For example, a teacher can conceive a series of problems with the intention of facilitating the solving of more complex problems, by posing some reduced versions of the target problem earlier in the series. In this case, the a priori analysis of possible problem solving paths can contribute to the conception of series of problems or to the interpretation of an existing one.

We suggest that modelling in terms of networks of problems can be efficient in both cases, facilitating the connection of these two aspects. We consider several possible uses of networks of problems:

-a priori analysis of problems and their possible solutions; -a priori analysis of series of problems conceived by teachers; -a posteriori analysis of students' problem solving processes and learning trajectories; -design of teaching trajectories (by researchers as well as by teachers). For developing our model, we rely on Balacheff's ckȼ model and, more indirectly, on Brousseau's and Vergnaud's related theories. We consider mathematical activity from a cognitive and epistemological point of view as confrontation with problems and as successive transformation of problems into other ones up to reaching a problem which is solved (i.e. not being a problem anymore). A problem is considered, for an individual, as a question or a task of which the solution method is not directly accessible; or, as a disequilibrium between the individual and the "milieu" (TDS).

Vergnaud introduced the term "conceptual field" as "a set of situations and a set of concepts tied together". As he explains, "a concept's meaning does not come from one situation only but from a variety of situations and that, reciprocally, a situation cannot be analysed with one concept alone, but rather with several concepts, forming systems" (Vergnaud, 2009, p. 86). By modelling in terms of networks of problems, we intend to describe the structure of such conceptual fields, and potential or actual trajectories constructing and exploring them. In the case of series of problems conceived by teachers or researchers, we may identify the "conceptual field" covered by the problems of the series, in case of problem-solving paths, we may consider the conceptual field generated by the initial problem (which might be different according to different paths traced by different individuals).

A crucial question is to describe the nature of the relations between two (or more) problems, the solution processes and the related mathematical knowledge, in order to explicate the ways one can go across various problems making meaning (that is, learning). In the following, we discuss this question by referring back to the examples presented above.

Our respective preliminary works contain elements to interpret these links, but the different approaches have to be conciliated with each other and with some other works, as Giroud (2011). Most notably, we consider that Pólya's (1945) classical list of questions, intended to support problem solving processes, can also be read in this sense: indeed, many of Pólya's questions suggest transforming problems into other problems or linking new problems with earlier solved ones. In this sense, Pólya's suggested problem solving strategies do not deal with individual, isolated problems, they rather situate new problems in a progressively constructed network of earlier solved ones.

The types of problem-solving steps implied by Pólya's questions are quite coherent with the types of links we could identify in series of problems of different authors and teachers related to the Hungarian Guided Discovery approach [START_REF] Gosztonyi | Séries de problèmes dans une tradition d'enseignement des mathématiques en Hongrie au 20 e siècle[END_REF]: as finding a simpler or a more general problem, an analogous problem, reinvesting a method of solution… Many of the types of links identified in Clairaut's book (see Figure 1) seem to be also similar to those described in the Hungarian context. Nevertheless, new types of problems emerged in our successive analyses. We are thus not proposing, at this stage of the research, a full classification of possible links, rather an open inventory of links.

At the same time, the definition of problems as P=(I, Q) explained above allows us to give a coherent description of many possible relationships between problems.

A first example comes from the idea of generality or specificity of problems. A problem P' is a specialization of a problem P if P' can be obtained by a restriction of the set of a problem's instances or by fixing the values of some variables describing the set of instances. A particular (extreme) case is the instantiation of a problem, when all the parameters of the set of instances are fixed, that is when we look at one particular case. Another way of specializing a problem P into a problem P' is when adding constraints to the question (see for ex. Figure 2).

Another type of relation is sufficiency. When you try to solve a problem P, you can transform it into a problem P' (or many problems P'1,…,P'n) such that solving P' (or P'1,…,P'n ) is sufficient to solve P. We can also say that P is necessary to P'. Of course, identifying this relation relates to the controls of the operators used to transform P into P', and can be or not identified by the individuals solving the problem. This type of relation was clearly identified and detailed by Giroud (2011).

Clairaut's 'need-based' transitions (i.e. when a problem is reduced to another one the solution of which is necessary to solve the original problem) can be interpreted in this sense. And indeed, Clairaut often arguments these reductions at length, and these are typically the points where new 'propositions' emerge in his text [START_REF] Gosztonyi | Series of problems in Clairaut's Elements of Geometry: interaction between historical analysis and mathematics education research[END_REF].

The relationships above (generality/specificity, necessity/sufficiency) are mostly of a logical or epistemological nature. We can explore relationships of different kind. For example, if we consider representations as semiotic registers, we can describe certain links between problems as conversions from one register to another. The new problem is not always equivalent to the first one.

Another type of link between two problems can be related to operators. If an operator r has been useful to solve the problem P, and if there are similarities between P and P' (which can be of various nature) it might be relevant to use r, or an adaptation of r, to operate on P'. This could be qualified as operator analogy. We can imagine that there are also analogies that concern controls and could be called control analogy. In fact, when two problems belong to a bigger family of problems, there can be general principles applied on them which can be considered as controls. In this case, the model allows clarification of the relationship between the two problems, by identifying a kind of "ancestor", a problem of which they both are specialisations. In this case, we can consider that using similar operators or control are something deeper than an analogy.

The search for generalization (of problems, operators, or controls) and for ancestor problems can play crucial role in establishing new mathematical knowledge and in increasing abstraction of concepts.

In the context of Guided Discovery for example, mathematical knowledge is often built through progressive generalisation, by exploration of analogies between problems [START_REF] Gosztonyi | Séries de problèmes dans une tradition d'enseignement des mathématiques en Hongrie au 20 e siècle[END_REF].

The interpretation and classification of possible links will be developed through further analysis of examples. At the conference, we will present the analysis of specific problems and series of problems from the domain of discrete mathematics. We will emphasise the emergence and role of controls (i.e., proving) in these series of problems. We will also explain how we plan to use the problem network model in collaboration with teachers.

Figure 1 :

 1 Figure 1: Scheme for the analysis of Clairaut's Elements of Geometry, Book III, paragraphs 9-22 (Gosztonyi, 2022)

Figure

  Figure 2. A mathematical solving of Problem 0 (Modeste & al., 2019)

Acknowledgment: This research was realized in the frame of the Problem-based learning trajectories in discrete mathematics education (ProDiME), founded by the European Union.


Roxane Cattan-Jallet
email: roxane.cattan-jallet@essca.fr

Charlotte Derouet

Camille Doukhan

Lynn Farah

Corinne Hahn

The impact of registers of semiotic representation on problem solving for students entering tertiary education in France: case of the mean

Keywords: Secondary-Tertiary transition, descriptive statistics, registers of semiotic representations, non-specialist students

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

Context

In this poster, we focus on the concept of the mean, studied in grades 7 to 10 in French classrooms. In the 1990s, several researchers studied the concept of the mean (notably [START_REF] Pollatsek | Concept or Computation: Students' Understanding of the Mean[END_REF][START_REF] Batanero | Errors and difficulties in understanding elementary statistical concepts[END_REF][START_REF] Gattuso | La moyenne: un concept inexploité, d'une richesse exceptionnelle [The average: an unexploited concept of exceptional richness[END_REF]. Based on these foundational studies, several researchers explored the statistical knowledge of students at different levels. They widened the scope to study, among other things, the notions of variation and distribution (see for example Garfield & BenZvi, 2005). These studies have shown that, beyond the computational process, the meaning of the mean remains poorly understood by students. We wonder if such conclusions are still relevant in the French context today.

Theoretical framework and research question

We only have access to mathematical objects through their semiotic representations. [START_REF] Duval | Understanding the mathematical way of thinking -The registers of semiotic representations[END_REF] defines a register of semiotic representation as a system of signs allowing three cognitive activities: formation, processing and conversion. The coordination of several registers of semiotic representation is necessary for a good conceptualisation of a mathematical object. In statistics, several registers of semiotic representations exist: the register of natural language, the register of diagrams (bar charts, etc.), the register of histograms, the register of frequency tables, the register of cross-tabulations, the algebraic register (formula), the numerical register… Moreover, institutional expectations for the flexible use of different registers are increasing in tertiary education [START_REF] Gueudet | Secondary-tertiary transition in mathematics education[END_REF]. Hence, we believe that this approach in terms of registers of semiotic representation would be appropriate to the study of statistical knowledge of students entering tertiary education.

Our research questions at this stage are the following: what do non-specialist students know about the concept of the mean as they enter tertiary education? What influence do registers of semiotic representation have on their knowledge?

Method and data

To address our research questions, we designed and administered an online questionnaire (14 items) about descriptive statistics concepts studied in secondary education. Six items tackle the concept of the mean. In this poster, we focus on 4 items centred on the same type of task, comparing the mean of two statistical distributions.

In these items, the registers of representation in which the distributions are given are diverse. In the first item, the two distributions list few ordered values, while in the second only one is represented explicitly in the form of an unordered list of values. In the other two items, the distributions are given in the form of a bar graph. In all four items, the mean of the two distributions is being compared (in the natural language register).

We administered the questionnaire in Fall 2022 to non-specialist entering students before their first statistics course. We obtained 382 complete responses from students who had completed their secondary education in France and were starting their tertiary education in one of the following fields: biology, psychology, economics and social sciences, or management.

Analysis of results

In the poster, we will present the a priori analysis of the four items described above and the student answers to these items. The obtained results show a very limited understanding of the concept of mean, like in the 1990s. However, the analysis of the student answers highlights the impact of the initial register of representation considered on the students' success in similar tasks.