Andualem Tamiru Gebremichael
email: andualemtam@yahoo.com

The relevance of mathematics and students' identities

Keywords: relevance, identity, mathematics curriculum, dialectical relation

This paper reports on a project, which explores the relevance of mathematics to students in an Ethiopian school. The data collection, which involved group interviews, was undertaken in 2010-2011. The analysis process identified themes that captured diverse perceptions of relevance and students' identities. The focus of this paper is on diverse aspects of students' identities and their relationship with students' perceptions of the relevance of mathematics. The analytic framework is cultural historical activity theory. The results indicate that the students' perceptions of relevance and identity are dialectically related.

Introduction

This paper reports on data collected in 2010-2011. It is part of a research project, which was launched in 2009, and had two phases: a pilot study and a main study. The participants were students of a preparatory school (equivalent to an upper secondary school) in Ethiopia. The study examined students' perceptions of the relevance of mathematics to their future goals, their everyday life and society. Previous studies have investigated perceptions of mathematics' relevance focusing on upper secondary school (e.g. [START_REF] Sealy | On the relevance of the mathematics curriculum to young people[END_REF][START_REF] Black | Developing a 'leading identity': The relationship between students' mathematical identities and their career and higher education aspirations[END_REF] and middle school (e.g., [START_REF] Masingila | Examining students' perceptions of their everyday mathematics practice[END_REF]. [START_REF] Sealy | On the relevance of the mathematics curriculum to young people[END_REF] identified three different categories regarding the perceived relevance of mathematics to students' future goals: usefulness, exchange value and transferable skills. They note how students' perceptions are varied across different schools, and argue that such variations are linked to differences in the social and cultural situations of students as well as differences in the social context of the schools. They assert that the mathematics curriculum and its pedagogy should be framed in accordance with these varied perceptions of relevance. [START_REF] Black | Developing a 'leading identity': The relationship between students' mathematical identities and their career and higher education aspirations[END_REF] relate students' identities to the exchange value and use value of mathematics focusing onstudents' aspirations to study mathematically demanding subjects and their future career plans. They argue that the exchange value and use value of mathematics frames students' motives for studying mathematics. Use value relates to how mathematics enables one to pursue a vocation, and exchange value refers to what mathematics offers, e.g. it may enable one to join the university or obtain an academic degree. Black and colleagues suggest that identity develops alongside the motive, students' identities influence their perceptions of the use value and exchange value of mathematics.

Students' perceptions of the relevance of mathematics to their everyday life has also been examined previously [START_REF] Masingila | Examining students' perceptions of their everyday mathematics practice[END_REF]). Masingila's study is about students in Kenya. Her aim is to close the gap between students' experiences of mathematics in the classroom and its use in out-of-school contexts. She identifies six mathematical categories which are consistent with [START_REF] Bishop | Mathematics education in its cultural context[END_REF]: counting; measuring; explaining; playing; locating; and designing. Furthermore, some studies relate mathematical identities and perceptions of relevance to students' cultural backgrounds (e.g., [START_REF] Nasir | Identity, goals, and learning: Mathematics in cultural practice[END_REF][START_REF] Nasir | Ethnic and academic identities: A cultural practice perspective on emerging tensions and their management in the lives of minority students[END_REF]). [START_REF] Nasir | Identity, goals, and learning: Mathematics in cultural practice[END_REF] study highlighted how students establish a sense of identity as learners in relation to their African-American cultural backgrounds. She argued they learn better when tasks relate to their sociocultural backgrounds, such as playing dominoes and basketball. A similar study by [START_REF] Nasir | Ethnic and academic identities: A cultural practice perspective on emerging tensions and their management in the lives of minority students[END_REF] reported that students experience tensions between how they are situated in school and in out-of-school cultural practices [START_REF] Nasir | Ethnic and academic identities: A cultural practice perspective on emerging tensions and their management in the lives of minority students[END_REF]. This body of work highlights how identity relates to race, and its influence on power [START_REF] Nasir | Power, identity, and mathematical practices outside and inside school[END_REF] supporting [START_REF] Gutiérrez | The sociopolitical turn in mathematics education[END_REF] who also calls for the sociopolitical turn in exploring racial identity in mathematics education.

The Ethiopian sociocultural context has peculiar features and its schooling has a peculiar history. Before the introduction of modern education in Ethiopia, there have been traditional education associated with Islam and Orthodox Christianity. Modern education was introduced in Ethiopia around the beginning of the 20 th century [START_REF] Wagaw | Education in Ethiopia: Retrospect and prospect[END_REF]. In 1908, a modern school was opened in Addis Ababa [START_REF] Hoot | Early education in Ethiopia: Progress and prospects[END_REF]. It was named after the Emperor, Minilik II, who ruled the country in that period. Modern education was not in line with the country's needs in some ways, where Many of the subjects were languages such as Arabic, English and French [START_REF] Wagaw | Education in Ethiopia: Retrospect and prospect[END_REF]. Wagaw assert that though local languages such as Amharic and Ge'ez were taught as subjects, the language of instruction was not any of the local vernaculars (the language of instruction for the participants in the current study is also English). Although education was considered as a gateway to getting a job, some parents were sceptical about 'modern education' fearing it would not maintain cultural values [START_REF] Wagaw | Education in Ethiopia: Retrospect and prospect[END_REF]. According to Wagaw, the clerics also expressed concern about maintaining Ethiopian values, a worry shared by some scholars.

Participants of the current study were 16 to 18 years old, yet distinguished from the rest of their age group as they are considered as elites (high ability). Because they joined the preparatory school based on their success in the second of the three national examinations in the Ethiopian school system. They expect to join university after sitting the third and final national examination.

It is important to examine if students perceive that mathematics is relevant to their everyday life, society and future studies, and how students perceive the reason for learning and engaging with mathematics. Preliminary results of this study were reported earlier [START_REF] Gebremichael | Ethiopian preparatory students' perceptions of the relevance of mathematics to learning goals[END_REF]. The purpose of the current paper is to show the relationship between perceptions of mathematics' relevance and students' mathematical identities. This introduction provides background information, which can help the reader to understand the results. In the following sections, I provide the theoretical and methodological issues before presenting the data and its analysis. I finalize the paper with my concluding remarks.

Theoretical issues

Cultural historical activity theory (CHAT) is adopted as the theoretical perspective in this study. I employed Engeström's mediational triangle [START_REF] Engeström | Expansive learning at work: Toward an activity theoretical reconceptualization[END_REF] to understand how the students spend their time at home and at school. In the current study, the school and the family-life are considered as two different activity systems (see Figure 1). In the school, students experience and use artifacts, signs and tools such as textbook, lessons, etc. while undertaking actions towards the goals of learning mathematics and the other school subjects. The students are subjects of the schooling activity, and interact with the object [START_REF] Roth | Re/Making identities in the praxis of urban schooling: A cultural historical perspective[END_REF]. The object of an activity system consists of material objects, the motive, and the goals [START_REF] Roth | Emotion at work: A contribution to third generation cultural-historical activity theory[END_REF]. The object of the school activity system consists of the motive and the goals, including students' future goals. The school's motive could be producing students who can succeed in the national examination to attend university and become capable of solving problems of everyday life and society. The school's goals could be providing adequate knowledge for realizing the motive.

The artifacts, rules, community and division of labor mediate this interaction [START_REF] Roth | Emotion at work: A contribution to third generation cultural-historical activity theory[END_REF]. There are rules, which govern members of the community. According to the rules, students should follow the classroom norms, attend to specific topics and sit examinations. Students are members of the school community, which also interacts with the object. These relationships are also mediated by the division of labor among members of the school community; the students and the teacher have different roles. There could also be differences in the roles of students. There are authorities, who have roles in deciding who will go to preparatory school and university. These authorities are represented by the school administrators, who enforce the rules set by these authorities and form part of the community. The students are divided into social science (SS) and natural science (NS) streams. These are also communities. The mathematics teacher also classifies them in terms of achievement level. The teachers of the various school subjects are members of the school community. According to the school rules, the students are operating in a school, which implements a nationally designed curriculum. Mathematics teachers are required to follow the nationally produced textbook (Federal Democratic Republic of Ethiopia Ministry of Education, 2010). This artifact influences another school artifact, the lesson. It is from this textbook that teachers give classwork and homework.

Figure 1 shows that the components of the activity systems are related to each other, and the double arrows represent the dialectical relationship between them. The figure represents two activity systems: school and family-life. As in the school activity, the student is subject of family-life activity, and interacts with the two objects as an actor. The object of the family-life activity system consists of material objects, home tasks, motive and goals. The motive of the family-life activity could be survival of the family, and one goal of this activity is a student's personal development. As part of this goal, the student engages in school tasks and family members provide support. Participants experience contradictions between the components of activity systems, and contradictions are considered as sources of development [START_REF] Roth | Re/Making identities in the praxis of urban schooling: A cultural historical perspective[END_REF]. Students may experience contradictions between objects of the school and out-of-school activities. For example, as we will see in the data presentation, preparatory mathematics is hardly seen as useful in everyday life and society. These contradictions can be sources of development of particular perceptions of relevance and identity.

As long as there is activity, there is an acting individual, and the individual's identity is integral to the activity [START_REF] Roth | Emotion at work: A contribution to third generation cultural-historical activity theory[END_REF]. [START_REF] Roth | Re/Making identities in the praxis of urban schooling: A cultural historical perspective[END_REF] distinguish between social and personal identity. In my study, the students seem to form aspects of identity as they participate in the school and out-ofschool activities. Personal identity is where the individual student recognizes her/himself as such, and the student can also have a social identity where other members of the community recognize her/him as such [START_REF] Roth | Re/Making identities in the praxis of urban schooling: A cultural historical perspective[END_REF]. For example, a student is recognized by others as a high achiever. [START_REF] Roth | Vygotsky's neglected legacy": Cultural-historical activity theory[END_REF] suggest a third aspect of identity, collective identity, which an individual holds as a member of a group. Students are recognized as members of SS or NS streams.

Collective identity is associated with the structure, which includes the components of activity systems [START_REF] Roth | Vygotsky's neglected legacy": Cultural-historical activity theory[END_REF]. The national curriculum and the instruction language (which is foreign to them) are parts of the structure.

Artifacts, rules, community and division of labor mediate students' identities [START_REF] Roth | Re/Making identities in the praxis of urban schooling: A cultural historical perspective[END_REF]. For example, the teacher recognizes students as such based on their roles in the classroom. The division of labor also mediates the perceptions of relevance as the teacher directs students' attention to what might be of value for future studies in both the SS and NS communities. The school rules, which enforce the significance of success in examinations, can mediate the students' personal as well as social identities. The rules direct the students' attention to the importance of examinations scores for recognizing themselves or others. The teacher's recognition of students as such is also based on examination scores. In addition to the rules, which enforce the adopted curriculum, the school and out-of-school artifacts mediate students' identities as Ethiopians and perceptions of mathematics' usefulness. The contradictions, which they experience between the objects of the school and out-of-school activities, can be sources of development both in relation to their national identities and in addressing doubt about mathematics' usefulness in everyday life and society.

Methodological issues

Group interviews were undertaken where each group had three students of the same sex and same class. Each group had high, medium and low achieving students. The intention of the grouping is to make students feel comfortable and remind each other of their respective experiences. Following [START_REF] Miles | Qualitative data analysis: An expanded sourcebook[END_REF], I conducted data reduction through listening to the video-recorded interviews. This was done to simplify the data and obtain focus by deciding on which pieces of data provided the best idea of what each participant is saying. This data reduction process was influenced by the research questions and aimed to expose the students' perceptions of the relevance of mathematics. I made a table indicating the time and a description of what the interviewer and interviewees said. I transcribed the utterances of students that possibly exposed their perceptions of the relevance of mathematics and related issues. Then, I followed Miles and Huberman, and undertook coding. [START_REF] Miles | Qualitative data analysis: An expanded sourcebook[END_REF] explain that coding of written transcripts can be undertaken based on the research questions and can employ concepts that are often used in certain theoretical perspective. The excerpts reported in the data presentation and analysis are students' responses to the interview questions about the relevance of mathematics. After coding was completed, the next step was to examine the coded text to obtain themes (e.g., [START_REF] Miles | Qualitative data analysis: An expanded sourcebook[END_REF]. I grouped the data segments with the same recurring topic into a particular theme.

Data presentation and analysis

This section presents examples and analysis of the interview data under two subsections as follows.

Students' personal and social identities and the relevance of mathematics to future goals

Students tell stories which relate to their identities and mathematics' relevance to future goals. Identities and perceptions of relevance seem to emerge as students participate in school and out-ofschool activities (e.g., Sofia's story). They emerge in the interactions between the subject, the communities and the objects of the activities (Milkias's story). The rules and the division of labor mediate students' identities as well as their perceptions of relevance (Beliyu and Milkias's stories).

The school rules, which enforce that students should master certain mathematics topics and are ranked in the class based on examination scores, direct their attention to the relevance of mathematics to their future goals and to recognize themselves as such (see Beliyu's story). The division of labor also mediates their identities; for instance, their teachers recognize Beliyu and Sofia as high-achieving students. In the family-life activity, there is interaction between the community and the object, in which members of the family have roles in the student's personal development (e.g., Sofia's story). The rules in family-life activity enforce that students should follow what their elders tell them to do and that elders should care for the younger. Members of the family recognize each other as such. The rules and the division of labor in family-life activity direct students' attention to recognize themselves as such in connection with mathematics. The division of labor also determines their perceptions of relevance as the teacher directs the students' attention to the relevance of mathematics with respect to their current community and their future study (Milkias's story). The students also perceive that mathematics has exchange value (e.g., Beliyu's story). The school rules, which mediate the students' identity, mediate this perception of relevance.

Other members of the community such as the mathematics teacher, recognizes the individual student as such (e.g., Beliyu and Sofia). This could be their social identities (e.g., [START_REF] Roth | Re/Making identities in the praxis of urban schooling: A cultural historical perspective[END_REF].

There is a distinction between the areas of study, which students of SS and NS streams can progress to at university. There is also evidence of a collective identity (e.g., [START_REF] Roth | Vygotsky's neglected legacy": Cultural-historical activity theory[END_REF]. Based on their membership of either SS or NS streams, students recognize themselves or others recognize them as members of these communities, particularly, in connection with the implications this membership has for the use of mathematics in future studies (e.g., Milkias's story).

Students' national identities and the relevance of mathematics to the nation

Students tell stories relating to national identity and mathematics' use in everyday life and society.

Andualem: Is preparatory mathematics useful in everyday life? In the society?

Akalu: I think that is the difference between the others [other countries/ societies] and us. They are more practical, but we grasp the theory and we get confused where to apply it. The students perceive that preparatory mathematics is hardly useful in everyday life and society.

The lessons and the textbooks mediate the perceptions of relevance by offering mathematics topics with hardly any usefulness in everyday life and in Ethiopian society (e.g., Haleluya's story). These artifacts direct students' attention to recognize themselves in connection with their attending of a national mathematics curriculum. The rules and artifacts, which enforce the content of school subjects for SS and NS communities, mediate students' identities by directing the students' attentions to the available gaps (e.g., Haleluya's story). The division of labor, which gives students the role of grasping the theory and does not allow students to have a clear idea of where to apply it, directs students' attention to recognition of themselves as such (e.g., Akalu's story). In addition, participants experience contradiction between the objects of the school and out-of-school activities in Ethiopian society (e.g., Akalu's story). The experiencing of this conflicting situation can be a source of the development of national identity. The students belong to a society, which uses artifacts and has a cultural heritage that are Ethiopian, though diverse. They operate in an Ethiopian school system, which enforces Ethiopian education policy. Students are experiencing a national mathematics curriculum, hardly embraces the situation in Ethiopian society (e.g., Haleluya's story). They learn in a foreign language. These national features form the structure in which national identity emerges.

Concluding remarks

The different aspects of students' identities and their perceptions of relevance are dialectically related (one presupposes the other). They recognize themselves as preparatory students, who have a future goal of joining university. Then, they perceive that preparatory mathematics has an exchange value in giving access to this future goal. Thus, an exposition of students' perceptions of relevance presupposes their identities. In addition, students' expositions of identities presupposes their perceptions of mathematics' relevance as having exchange value in giving access to university study, as well as in being ranked high in examinations. The teacher's recognition of students as SS and NS students presupposes the relevance of mathematics to these students' future study, which influences their perceptions of relevance. The students' perceptions of relevance also presupposes their identities as SS and NS students as well as their identities as future students of certain disciplines. The exposition of perceptions of mathematics' relevance to everyday life and society appears to presuppose who they are as mathematics learners in this Ethiopian situation. On the other hand, their expositions of national identities are based on perceptions of mathematics' relevance to Ethiopian society and everyday life.

The current study exposed national identity, which is different from ethnic or racial identity (e.g., [START_REF] Gutiérrez | The sociopolitical turn in mathematics education[END_REF][START_REF] Nasir | Identity, goals, and learning: Mathematics in cultural practice[END_REF][START_REF] Nasir | Power, identity, and mathematical practices outside and inside school[END_REF][START_REF] Nasir | Ethnic and academic identities: A cultural practice perspective on emerging tensions and their management in the lives of minority students[END_REF]. These works attempt to examine identity of particular minority groups within a larger national culture (e.g., American culture, American school system where school rules enforce the American curriculum). But each national context has peculiar features. The sociocultural situations, including out-of-school activities, and the artifacts students experience in the school and outside of the school are likely to be different in various national contexts. National identity is a collective identity, and consistent with [START_REF] Roth | Vygotsky's neglected legacy": Cultural-historical activity theory[END_REF] the focus is on the structure. It is important to examine this identity in national contexts with peculiar features of the structure including adopted curriculum, use of language foreign to them, etc.

Therefore, mathematics education in Ethiopia must consider the sociocultural situation in which it sits. The situations that separated parents from students, in the introduction of modern education, still appear to prevail in the students' learning of mathematics. Students feel detached from their society with respect to the usefulness of mathematics. Further investigations need to be undertaken in this direction.

Figure 1 :

 1 Figure 1: Network of activity systems adopted from Engeström (2001)

  That is the problem. [11 th grade, NS, male] Haleluya: We learn to change our situations in a scientific way. Therefore, what we learn has to be related to our everyday life ... After consuming two or three pages, you get … numbers. …, what is its importance in my life? …. Or something related to … society. Our country might not have a need for mathematics. However, in economics, many people want to know …, and about business, about alleviating unemployment ... How we can use our resources. [12 th grade, SS, male]


Belgüzar Kara
email: belguezar.kara@uni-due.de

Bärbel Barzel
email: baerbel.barzel@uni-due.de

Social inequalities in mathematics from a socialization theoretical point of view -Analysis of problem-solving processes of students

Keywords: Problem-solving, diversity, inequality, habitus

Theoretical background

In the recent German school curriculum, mathematical problem solving is anchored obligatorily as an important competence for mathematics learning. Since 2008, competence expectations for problem solving have explicitly indicated what the pupils are supposed to have acquired at the end of grade 4 (Ministerium für Schule und Weiterbildung des Landes Nordrhein-Westfalen, 2008). But empirical research results indicate that the demands can generate inequalities with regard to social origin [START_REF] Cooper | Assessing children's mathematical knowledge: Social class, sex and problem-solving[END_REF][START_REF] Lubienski | A clash of social class cultures? Students' experiences in a discussionintensive seventh-grade mathematics classroom[END_REF]. Achievement studies confirm the impact of socio-economic status on mathematic achievement [START_REF] Stubbe | Soziale Disparitäten der Schülerleistungen in Mathematik und Naturwissenschaften[END_REF]. Research in mathematics education on social inequalities in the acquisition of mathematical competences observes different achievement in dealing with problem solving tasks specific to social class [START_REF] Cooper | Assessing children's mathematical knowledge: Social class, sex and problem-solving[END_REF][START_REF] Piel | Social origin and success in answering mathematical word problems. The role of everyday knowledge[END_REF]. The study by [START_REF] Lubienski | A clash of social class cultures? Students' experiences in a discussionintensive seventh-grade mathematics classroom[END_REF] shows differences between students with different socioeconomic status (SES) dealing with problem-centered materials. The analyses show, for example, that, In general, lower-SES students focused more on giving the right answer to a question (...), whereas higher-SES students were more inclined to discuss a method or an idea. (...) lower-SES students were more likely to use language in "common-sense" reasoning that was closely tied to the context of the problems. (...) Higher-SES students were more likely to contribute in relation to abstract, strictly mathematical contexts and to use generalized language and reasoning. (p.

369)

Based on the theories of [START_REF] Bourdieu | Distinction: A social critique of the judgement of taste[END_REF] and the research work of [START_REF] Lareau | Unequal childhoods: Class, race, and family life[END_REF], in the study presented here, the problem-solving process is analyzed from a perspective of socialization. The theory of habitus [START_REF] Bourdieu | Distinction: A social critique of the judgement of taste[END_REF] is used as an analytical tool for the mediation between the individual problem-solving process and the structural conditions of the social background. [START_REF] Lareau | Unequal childhoods: Class, race, and family life[END_REF] identified two milieu-specific child-rearing approaches and characteristic practices associated with the two educational styles: Middle-class families concerted child rearing on cultivation, working-class families concerted child rearing on accomplishment of natural growth. In particular, [START_REF] Lareau | Unequal childhoods: Class, race, and family life[END_REF] describes the two approaches as differently beneficial learning environments and socialization conditions for school practice. For example, while the discussion and argumentation between child and parents is an essential feature in the child-rearing approach of concerted cultivation of the middle classes, the child-rearing approach of accomplishment of natural growth of the working class is characterized by direct instructions. If the characteristics of childrearing approaches are considered in competence expectations and the inherent structure of problem-solving, origin-related differences in competence development can be expected through habitual dispositions. The danger that mathematical competence expectations have undesired impacts for less privileged students, which in turn can have the consequence that accessibility to mathematical contexts and the intended learning processes become more difficult or blocked, is subject of this investigation.

Design and methods of the study

The praxeological approach theory establishes dialectical relationship between objective structures and structured dispositions of the acting actors and serves as the epistemological fundamentals for the investigation. A mathematics performance test and student and parent questionnaires were used to select and classify pupils according to mathematical achievement level and social background. Data was then collected within the framework of individual clinical interviews during the processing of a total of three mathematical problem-solving tasks to get an insight into possible differences in the process.

Results

The preliminary analysis according to the phases model of [START_REF] Schoenfeld | Mathematical problem solving[END_REF] gives an indication of milieu-specific ways of acting through the phenomenon of the abridged analysis and exploration of mathematical problem-solving tasks of students from families of less privileged origin. The focus on a direct approach like one-word-answers without complete comprehension of the task has so far only been observed in students of less privileged origin. The correct answer seems to be more important than the understanding or discussion of the task which is unfavorably supported by the abridged analysis and exploration.


published or not. The documents may come

Cardioprotective effects of omega 3 fatty acids: origin of the variability


Maike Lüssenhop
email: maike.luessenhop@uni-hamburg.de

Gabriele Kaiser
email: gabriele.kaiser@uni-hamburg.de

How do we teach mathematics to refugee students? A qualitative study of the teaching and learning of mathematics in International Preparatory Classes

Keywords: Refugee students, teaching practices, mathematics classroom, diversity, mathematics education and language

Although migration itself is not a new phenomenon, the rising numbers of refugees in Germany and Europe are influencing the population composition leading to greater diversity within our society. This particularly affects school systems, as a great number of refugees are of school age and have to be provided with adequate schooling. In this paper, we discuss how the German education system, in particular in the city of Hamburg, deals with the schooling and integration of refugee students in what are called International Preparatory Classes. We describe a research study, which is examining what aspects influence teaching practices in mathematics lessons in these classes for refugees. The study investigates how teachers in these classes design their lessons from a mathematical and a linguistic perspective, how they deal with the task of providing refugee students with adequate instruction and how students experience mathematics.

Introduction

[H]ow school systems respond to migration has an enormous impact on the economic and social well-being of all members of the communities they serve, whether they have an immigrant background or not. Some systems need to integrate large numbers of school-age migrants and asylum seekers quickly; some need to accommodate students whose mother tongue is different from the language spoken in the host community or whose families are socio-economically disadvantaged; some systems are confronted with all three challenges at once. (OECD, 2015a, p.1) Indeed, it is a major challenge for the German school system to teach refugee students with limited knowledge of the German language and to provide them with adequate support. Refugee children and young adults are perceived as a vulnerable group with a diverse range of special needs. In Hamburg, refugees and other immigrant students attend International Preparatory Classes (IPCs). IPCs prepare them for admission into German mainstream classes. Currently, scarcely any research studies how mathematics teachers work with students in these particular IPCs. Moreover, little is known about how they design lessons while they are confronted with linguistic, cultural and performance-related heterogeneity [START_REF] Benholz | Neu zugewanderte Schülerinnen und Schüler eine Gruppe mit besonderen Potentialen: Beiträge aus Forschung und Schulpraxis[END_REF]. Supporting refugee students' development in general, and specifically in mathematics, is crucial for their participation in society and their ability to contribute to the society as a whole. Our study aims at contributing to close the existing gap in research but will also allow others (e.g., authorities) to use the knowledge gained for educational changes on several levels (Cornely Harboe, Mainzer-Murrenhoff, & Heine, 2016). In this paper, we present two case studies from two different mathematics classrooms. Sharing ideas and best practice examples in order to improve the schooling of refugees in the European Union and worldwide is of great importance for the international community as recognized by the OECD report (OECD, 2015b). In the following, we introduce the current state of research on IPCs from a German perspective and present first results of our ongoing research.

National frameworks for IPCs

Although German federal states implement different school-organizational models of teaching children and adolescents from refugee families, three basic models can be recognized:

• Integrative model: In this model, the students of refugee families are taught in regular classes with additional language support in German. • Parallel model: In this model, children and adolescents from refugee families are taught in separate classes that are set up for them. The greatest amount of teaching is dedicated to the acquisition of the German language. • Semi-integrative models constitute a transition between these two models: Students who are taught in IPCs can already take part in regular classes, depending on the level of subject-specific competences in individual subjects [START_REF] Massumi | Neu zugewanderte Kinder und Jugendliche im deutschen Schulsystem[END_REF].

The three models outlined here are also prevalent internationally (OECD, 2015a). Countries deal very differently with the schooling of refugee students. Some countries such as Iceland and Norway follow the integrative model [START_REF] Ragnarsdóttir | Successful educational contexts in Iceland for immigrant students: Case studies of nine Icelandic schools[END_REF], others such as Spain follow the parallel model [START_REF] Gorgorió | Teaching mathematics in multilingual classrooms[END_REF].

The framework for IPCs of the local education authorities in Hamburg provides insight into the current situation of schooling of refugees in Hamburg and Germany, since each year school-aged refugee children and adolescents move to Hamburg with little to no knowledge of the German language. According to the Geneva Convention, all children are guaranteed the right to education. The Geneva Refugee Conventions (Article 21), the UN Universal Declaration on Human Rights (Article 26) and the UN Convention on the Rights of the Child (Article 28) as well as the German Basic Law (Article 3) explicitly encourage compliance with this right [START_REF] El-Mafaalani | Bildungsteilhabe geflüchteter Kinder und Jugendlicher im regionalen Vergleich: Quantitative Annäherungen an ein neues Forschungsfeld[END_REF]. In IPCs at different types of schools (primary/middle/high school etc.), refugee students are prepared for entry into mainstream classes. 1 The objective of IPCs is to enable children and adolescents to learn the German language and thus to facilitate their integration into the Germanspeaking environment as smoothly as possible. Students that are able to write in the Latin script, start in IPCs that focus on learning the German language and basic knowledge of other subjects. Students who are illiterate 2 or literate in a different scriptsuch as Arabicgain German reading and writing skills in literacy classes. These classes focus on reading and writing in the Latin script and on basic German skills. IPCs aim at providing a quick transition to an age-appropriate class, suiting students' individual education level at general or vocational schools (mainstream classes). Refugee students can be admitted into IPCs at any time of the year and stay in them for a period of twelve months. The number of students in each class is around 15. The language learning process in all different forms of IPCs is divided into two phases in the first year of schooling. In the first phase (basic level), students acquire basic knowledge in German and receive guidance in dealing with, and acting within, the German-speaking world. This first phase should not be longer than 6 months. At the end of this phase, the parents receive information about the performance and learning development of their children in learning development talks organized by the teachers. These meetings advice parents on further schooling and acts as an exchange on how best to support the development of their children.

The second phase (advanced level) focuses on the preparation of the students for classroom communication and the use of subject-specific language in different subjects. After six months, students receive a transition certificate. At the end of the second phase, students transfer into mainstream classes and receive additional language tutoring for another year.

The basis for teaching in IPCs are the frameworks and curricula of German as a Second Language in IPCs at each school level, in which minimum requirements and assessment for performance are set. The level of difficulty of teaching is oriented towards the Common European Framework of Reference for Languages. Subject-specific instruction, such as mathematics, is based on the minimum requirements laid down in the curricula offered in each grade. The framework for IPCs in Hamburg focuses more on the macro level of the regulation of the schooling process than on the micro level of actually teaching the subject. It recommends, however, that teachers orient themselves towards the minimal requirements of the mathematics curricula for mainstream classes.

Current state of research on IPCs

In empirical educational research (in general), there are numerous and comprehensive analyses of the educational opportunities of various groups of refugees. In particular, the research field covers studies on participation in education, educational transitions, acquisition of competencies, and school success. In-depth analyses focus especially on social background, gender and ethnic background and/or migration background. Despite this highly differentiated field of research, the long history and increased presence of refugees, it is surprising that the educational opportunities and conditions of refugee students have not been steadily researched [START_REF] El-Mafaalani | Bildungsteilhabe geflüchteter Kinder und Jugendlicher im regionalen Vergleich: Quantitative Annäherungen an ein neues Forschungsfeld[END_REF]. Unsurprisingly, however, since Germany had a high influx of refugees in 2015, interest in education policy and research into the education of refugees has increased. This interest is also reflected in the literature and more recent research publications [START_REF] Kleist | Flucht-und Flüchtlingsforschung in Deutschland: Bestandsaufnahme und Vorschläge zur zukünftigen Gestaltung[END_REF].

Nevertheless, a paucity of research is still especially obvious when the focus turns to subjectspecific instruction in IPCs, including mathematics education. Some studies use 'wordless methods' of teaching mathematics to circumvent possible language barriers. In his study, [START_REF] Oswald | Beweise ohne Worte mit jugendlichen Geflüchteten[END_REF] used 'wordless methods' with adolescent refugees and found that it had a positive impact on the motivation of the students and the teachers. Furthermore, Mink & Sauerwein (2017) conducted a teaching project in one IPC on figured numbers and proofs without words. They found that only after intensive study of the material the mathematics could be understood, and it remained questionable whether students who see a visualization for the first time can use this to carry out complex proofs, since they are still unfamiliar with elementary mathematical activities. Despite the intention of these 'wordless methods', it is not possible to avoid oral explanations, which were difficult for participating students to understand. Most of the respective research projects implemented so far focus on language and mathematics [START_REF] Prediger | Mathematikunterricht: Interkulturelle Perspektiven auf Mathematikunterricht -Ein Überblick zum Diskussionsstand[END_REF], or mathematics in the context of migration [START_REF] Deseniss | Schulmathematik im Kontext von Migration[END_REF]. In other words, this research does not consider mathematics lessons with refugee students and their teachers. Exceptions are the studies by [START_REF] Oswald | Beweise ohne Worte mit jugendlichen Geflüchteten[END_REF] and [START_REF] Mink | Mathematikunterricht in einer Internationalen Vorbereitungsklasse[END_REF], which focus on new teaching methods for IPCs. However, there currently exists no research on the actual classroom practices of teachers in IPCs.

Although an increasing number of textbooks and teaching materials for refugee students are published by publishers all over Germany, there is clearly a gap in scientific research concerning the needs and challenges of teachers and students in IPCs.

Apart from the German state of research, there exists an extensive discussion on the teaching and learning of mathematics in multilingual classrooms that highlights social, cultural and linguistic aspects of mathematics teaching and learning [START_REF] Gorgorió | Teaching mathematics in multilingual classrooms[END_REF][START_REF] Moschkovich | Language and Communication in Mathematics Education: International Perspectives[END_REF]. As most of these studies focus bilingual or multilingual classrooms and not classrooms with only refugee or immigrant students, we can only draw partially on these studies. For example, [START_REF] Barwell | Mathematics education and language diversity: The 21st ICMI Study[END_REF] stress that situations of language diversity have been extensively examined, such as bilingual and multilingual context. They underline that there is a need in researching trilingual contexts where learners are exposed to home languages, a national and an official language. However, they do not focus on students that are second languages learners (sometimes referred to as minority learners, immigrant students) with different native languages. Furthermore, refugee students might come from countries with multiple national languages, which equals bilingual students, but most of them have no knowledge of the academic language, in this case German, at all, when they enter IPCs. However, Barwell et al. focus partly on second language learners. [START_REF] Gorgorió | Teaching mathematics in multilingual classrooms[END_REF] conducted research on language and socio-cultural aspects of mathematics education in a classroom with a high percentage of immigrant students and they are particularly convinced of the importance of continuing to study the teaching and learning of mathematics in multiethnic and multilingual situations. Moreover, they emphasize the need for continuing studies not only in terms of equity and social justice, but also in terms of the richness of the research ground.

Overall, these studies and the above-mentioned gaps in the research highlight the need for a deeper understanding of what is happening in the IPC mathematics classrooms, what affects the (teaching) practices of teachers and the challenges they face when it comes to teaching materials, previous knowledge of refugee students etc. This research project aims at narrowing this research gap.

Research questions and aims of the study

Overall, it can be stated that the framework for IPCs does not offer advice for teachers on how to teach mathematics to refugee students except to focus on the minimum requirements of the same or similar grade of general schools. Furthermore, the analysis of the current state of research on IPCs in the previous section shows that there is a research gap in the field of mathematics in the context of IPCs relating to best practices, teaching materials and theoretical approaches. Hence, our study highlights findings from research that explores mathematical teaching practices and aims at researching mathematical teaching approaches and mathematical practices within the IPC mathematics classrooms.

Our research project investigates the following research questions:

How do teachers deal with the task of designing and implementing mathematics lessons for refugee students? What kind of practices and which materials do they use, and how do social, cultural and linguistic aspects influence the learning and teaching of mathematics in IPCs?

Overall, the aim of this study is to generate knowledge about how teachers design lessons, which theories from other subjects their lessons are based on, and whether these fit the needs of the students they teach. More insight is necessary on the challenges mathematics teachers are encountering in IPCs, and how they deal with them within the mathematics lessons in order to develop supportive learning environments for refugee students.

Design of the study

The study is embedded in the project "Mathematics and refugees" at the University of Hamburg, a subproject of the state-funded research network "Everyday Mathematics as Part of Basic Adult Education", the so-called "Hamburg Numeracy Project". The research network focuses on (adult) numeracy practices and their use of everyday mathematics, focusing on particularly vulnerable groups, for example refugees and asylum seekers, over-indebted, and disabled persons. The presented study is qualitatively oriented and focuses on the IPC mathematics classroom and the practices of mathematics teacher and their students. The sample of the study consists of teachers and students from three comprehensive schools at lower and higher secondary level in Hamburg.

For the data collection, we use different methods such as classroom observation, guideline-based interviews with the mathematics teachers and narrative interviews with 3-4 students from each class.

The study has been conducted in the last 6 months of 2018 and this paper emerges from a larger research project which we described above. At first, informal talks to gain a first impression of the research field have been carried out with teachers outside of the classroom. After that, participatory observations of four lessons each lasting 45 minutes have been conducted. From these observations and the literature we draw on, we generated questions for the guideline-based interviews. Each IPCteacher has been interviewed for approximately one hour. The interviews focused on the question how teachers design mathematics lessons in IPCs, for example how they deal with the particular heterogeneity of the learning group, how they proceed methodically and didactically, which challenges occur and how they deal with them, and so forth. Moreover, we conducted narrative interviews with refugee students from each class. The interviews focused on the experiences of the students inside the mathematics classroom of their country of origin and in their new school in Germany. The data collection and analysis has been based the coding paradigm of Grounded Theory [START_REF] Corbin | Basics of qualitative research: Techniques and procedures for developing grounded theory[END_REF]. All interviews conducted have been recorded and transcribed. Furthermore, the data collected has been or will be evaluated with MAXQDA.

Preliminary results

In the following, we focus on two IPCs at two different comprehensive schools in Hamburg, which were selected as paradigmatic cases reflecting general characteristics of the whole database. We summarize selected aspects of the observation of four lessons in the mathematics classroom and informal conversations with teachers and students. In this paper, we name these classes IPC A and IPC B to distinguish certain aspects of the observation. Both IPCs are linked to grade 7/8 teaching groups (inter-year groups) situated in parts of Hamburg with a low social index.

Both classes consisted of approximately 12-15 students at the age of 12-16 years. The majority of the students came from Syria, Afghanistan and Iraq. According to both mathematics teachers, the students were very heterogeneous in terms of their mathematical competences and educational biography. Some students from Afghanistan had never gone to school before having fled to Germany, others had finished primary or secondary school in Syria and directly joined the German system, and again others had not gone to school for many years. In addition, both classes had students from Eastern European countries, for example from Bulgaria, or students from countries such as Thailand, Turkey, Ghana and Togo.

Teachers' mathematics and teaching experiences

The mathematics teachers of both IPCs had a very different educational biography but both were class teachers of the observed classes, meaning that they taught most of the subjects and lessons of their IPC. The teacher of IPC A completed her teacher education, worked and researched abroad, had a PhD and worked as a lecturer at university level in educational science, particular in multilingualism, migration and education. It was her first year teaching an IPC and the first time teaching mathematics to students. She emphasized that she did not study mathematics as a subject in her studies, but she had a very good knowledge of the didactics of German and German as a Second Language and carried out extensive research in the field of migration. The teacher of IPC B has completed his teacher education and taught mathematics in IPCs for seven years. He did not study mathematics as a subject at university but completed an additional course in mathematics some years ago. At that time, there was no specific teacher education program for IPC classes.

Since the increase of new immigrant students in recent years, many teachers who have been employed for the newly created school classes have completed a German as a Second Language degree. Often these teachers have to teach mathematics lessons in their IPCs, also due to the existing lack of teachers of mathematics. This means that many teachers lack the didactic and mathematical knowledge of mathematics. In some cases, underqualified teachers were also recruited because the positions in the IPCs could not be filled otherwise. This highlights the fact that apparently many teachers have very different experience with IPCs and teaching mathematics as a subject.

Linguistic aspects

Teacher A paid significant attention to the native languages of the students, which is in line with the accepted research results. A table with important German sentences hang on one of the walls of the classroom, such as "What does that mean in German?", "Good Morning", "Can you help me?". Students added a translation of these sentences in their own mother tongue. Throughout the school year, the teacher studied all sentences of the languages of her students herself. At the beginning of each week, students were allowed to test her language skills. Apart from this, language comparisons between German and the native language of the students were an integral part of the lessons. In addition, two of the weekly lessons in mathematics focused on subject specific vocabulary. With teacher B the picture was contrary. Even if some posters with German grammar were hanging on the walls of the classroom and some dictionaries were available, a set of rules was attached to the teachers table forbidding or avoiding the usage of native languages of students. This seemed to be a general rule for the IPCs of this school. This handling of the native language of the students contradicts findings from language teaching research and general pedagogy, and yet seems to be widespread.

Mathematical Aspects

In IPC A, students were taught in cross-class teaching two hours per week in homogenous groups regarding their performance in mathematics, and the other two-hour classes on subject-based mathematical language in their original classes. Some students sat in groups according to mother tongue, topics or performance. There were chalk-and-talk phases within the lesson and reflections on mathematical vocabulary. Work sheets with topics such as addition, subtraction, multiplication of fractions and subtraction of fractions with different denominators, text problems etc. mainly focused on basic mathematics. Students were working on different worksheets with diverse topics.

After each lesson, the teacher collected the sheets, corrected them and handed them back during the next lesson. Furthermore, the teacher offered some enactive materials such as wooden fraction models. In IPC B, each student worked on his/her own work plan. The teacher talked to each student for 5-10 minutes and gave personal support, no chalk and talk phases occurred. Topics were similar to IPC A and focused on basic mathematics such as subtraction, addition, fractions and so on. In both classes, some students worked with the mathematics textbook that was being used in the mainstream classes they are supposed to transfer to after completing IPC.

Discussion and looking ahead

The two case studies presented here emphasize the great diversity of the design of the lessons in IPCs and the learning opportunities provided by the two teachers based on the heterogeneous group that they were teaching. Socio-cultural aspects played a crucial role, as students seem to have very different backgrounds according to their previous educational development and different linguistic background (and possibly experiences with flight, conflict and/or war). The role of the linguistic aspects in the observed lessons was handled very differently by the two teachers and let to different learning environments. Drawing on the research results of Prediger and Schüler-Meyer (2017) and other scholars, who emphasize that the appreciation of the students' native language has positive effects on students' motivation and confidence, we find it a concern that only one of the two teachers was drawing on these findings. This might indicate that there is a need of further education in terms of the inclusion of the native language of the students.

Furthermore, the initial results of the study point out that the two teachers design their mathematics lessons very differently. Nevertheless, both of them tried to adapt their classes to the need of the students, in a way that they think is best for them. Earlier on, we demonstrated that there is no official curriculum for mathematics in IPCs and few adequate teaching materials that have been developed in the last years. This is also reflected in the initial results, as both teachers' lessons are mostly based on self-developed materials. Overall the two case study presented in this paper provide first insights into how teachers deal with the task of carrying out mathematics lessons to refugee students. The research seems to suggest that the educational background of the teachers influences the way in which they carry out this task, especially concerning lingual aspects and the role of mathematical content and the kind of its inclusion, but also how external factors influence the learning and teaching of mathematics, such as socio-political and socio-cultural aspects.

Immigrant students from EU-countries or other countries are also schooled in IPCs. In our study, however, we focus on refugee students who recently and forcibly fled from their country of origin.

By illiterate, we mean that these students can neither read nor write.


Szymon Gres

Palle Andersen

Rasmus Johan Johansen

Martin Dalgaard Ulriksen

Lars Damkilde

A comparison of damage detection methods applied to civil engineering structures

Keywords: Structural health monitoring, ambient excitation, damage detection, control chart-based algorithm fusion

The early damage detection, among localization and quantification methods, is crucial for the in-time repairs and cost-optimized maintenance plans of civil engineering structures. Preferably, the damage detection is performed by use of the output vibration data, hereby avoiding modal identification of the structure. The most of the work within the field assumes that the unmeasured excitation signal is time-invariant with a constant covariance, which is hardly achieved in practice. In this paper, we present a comparison of a new Mahalanobis distance-based damage detection method with the well-known subspace-based damage detection algorithms robust to the changes in the excitation covariance. The joint features of the methods are concluded in a control chart. The described methodology is implemented in ARTeMIS-a software for health monitoring and operational modal analysis. The performances of the methods are evaluated both on the numerical simulations and a full-scale experimental example---namely data obtained from the Dogna Bridge---using the output ambient vibration signals. The results reveal a positive capability of the damage detection from tested methods, with a less computational complexity of the Mahalanobis distance-based approach. The control chart presents a comprehensive overview of the progressively damaged structure.

INTRODUCTION

In the field of damage diagnosis, the vibration-based methods are effective in the detection of a small deviations solely based on the operational data from the structure [START_REF] Farrar | Vibration-based structural damage identification[END_REF]. The practical aspect of using only the output measurements cause several difficulties due to the variations in the ambient excitation, which can be due to variability in environmental conditions, for example, wind, temperature and precipitation, and operational conditions, [START_REF] Balmès | Handling the temperature effect in vibration-based monitoring of civil structures: a combined sub-space-based and nuisance rejection approach[END_REF]. This issue is addressed in several detection techniques available in the literature [START_REF] Balmès | Handling the temperature effect in vibration-based monitoring of civil structures: a combined sub-space-based and nuisance rejection approach[END_REF][START_REF] Bernal | Kalman filter damage detection in the presence of changing process and measurement noise[END_REF][START_REF] Döhler | Subspace-based damage detection on steel frame structure under changing excitation[END_REF][START_REF] Döhler | Structural health monitoring with statistical methods during progressive damage test of S101 Bridge[END_REF][START_REF] Döhler | Subspace-based fault detection robust to changes in the noise covariances[END_REF][START_REF] Döhler | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF] and effectively implemented for the condition based maintenance and the early damage detection [START_REF] Döhler | Structural health monitoring with statistical methods during progressive damage test of S101 Bridge[END_REF]. A general review of the damage detection strategies is presented in [START_REF] Doebling | A summary review of vibration-based damage identification methods[END_REF][START_REF] Carden | Vibration based condition monitoring: A review[END_REF].

The concept of vibration-based damage detection relates to identification of the deviations in the damage-sensitive features of the collected output data. The deviations are defined by a relative comparison of the reference and operational states of the system. The commonly used features for the detection are the modal parameters (natural frequencies, mode shapes and damping ratios) identified from the data. However, some research questions the use of the modal parameters, arguing that the modal data itself is not sensitive enough to detect the local faults [START_REF] Carden | Vibration based condition monitoring: A review[END_REF], especially when in practice the structure is excited by low-frequency inputs. One workaround for the system identification when estimating the features is to use the statistical properties of the data. The statistical methods use the probability distributions of the deviations, which differ between the damaged and the undamaged states indicating the faults in the system. The focus of this work is twofold. One is to present a simple statistical approach for the damage detection based on the Mahalanobis distance featured with the Hankel matrices. The distance metric is calculated on the output vibration data processed in the framework similar to subspace-based methods. The test is robust towards the changes in the excitation covariance. The damage is detected by the deviations of the distance from the reference test state. The results are compared with the xi2-test build on the residual from the subspace-based methods [START_REF] Döhler | Subspace-based damage detection on steel frame structure under changing excitation[END_REF][START_REF] Döhler | Structural health monitoring with statistical methods during progressive damage test of S101 Bridge[END_REF][START_REF] Döhler | Subspace-based fault detection robust to changes in the noise covariances[END_REF][START_REF] Döhler | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF][START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF]. The second objective is to present a complete, practical overview of the damage indicators combined from the both methods in the Hotelling control chart [START_REF] Lowry | A review of multivariate control charts[END_REF].

The methods are tested both on the numerical simulations and the full-scale bridge. The academic example is a coupled springmass system, excited by white noise of random variance. The comparison is concluded by use of data from the full-scale Dogna Bridge, Italy. The bridge is artificially progressively damaged and excited by the wind.

The structure of the paper is as follows. The basic principles of the robust subspace-based damage detection method and the methodology for the Mahalanobis distance calculated on the Hankel matrices is presented in Section 2. The two test examples are described in Section 3. Both the comparison and joint performance of the methods on the numerical and full scale detection cases are presented in Section 4. The results are concluded in Section 5.

METHODOLOGY

In this section we recall the basic principles of the subspace-based damage detection [START_REF] Döhler | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF] and present the algorithm for the Mahalanobis distance-based damage detection. Both methods are data-driven.

SUBSPACE-BASED DAMAGE DETECION

The damage detection consists of monitoring the deviations of the system from its reference state, characterized by some nominal property repeatable for every healthy conditions. The deviations are defined by a residual that differs between the healthy and the damaged states [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF]. Consider the discrete state space model

𝐱 k+1 = 𝐅𝐱 k + 𝐯 k 𝐲 k = 𝐇𝐱 k + 𝐰 k (1)
with the state transition matrix 𝐅 ∈ 𝑅 𝑛𝑥𝑛 , the observation matrix 𝐇 ∈ 𝑅 𝑟𝑥𝑛 , the states 𝐱 𝐤 ∈ 𝑅 𝑛 and the outputs 𝐲 𝐤 ∈ 𝑅 𝑟 ,where 𝑟 is the number of the sensors and 𝑛 denotes the order of the system, [START_REF] Van Overschee | Subspace identification for linear systems: Theory, implementation, applications[END_REF]. The unmeasured white noise input 𝐯 𝐤 drives the dynamics of the system and 𝐰 𝐤 is the measurement noise. Any perturbation in the structural properties of the system, manifested in the stiffness or mass, leads to deviations in the state matrices and are subsequently reflected in 𝐲 𝐤 . As a consequence, damage changes the eigenstructure of the state space model. Hence, the damage-sensitive system property relates to the 𝐅 and 𝐇 matrices.

Let 𝚺 𝑠 = Ε(𝐱 k+1 𝐱 k+1 𝐓 ) be the state covariance matrix and 𝐆 = Ε(𝐱 k+1 𝐲 k 𝐓 ) = 𝐅𝚺 s 𝐇 𝑇 the cross-covariance between the states and the outputs [START_REF] Van Overschee | Subspace identification for linear systems: Theory, implementation, applications[END_REF]. The output covariance yields 𝚲 i = Ε(𝐲 k+i 𝐲 k 𝐓 ) = 𝐇𝐅 i-1 𝐆 and can be structured in the block-Hankel matrix

𝐇 𝑝+1,𝑞 = [ 𝚲 1 𝚲 2 𝚲 2 𝚲 3 ⋯ … 𝚲 𝑞 𝚲 𝑞+1 ⋮ ⋮ ⋱ ⋮ 𝚲 𝑝+1 𝚲 𝑝+2 … 𝚲 𝑝+𝑞] = Hank(𝚲 i ). (2)
𝐇 𝑝+1,𝑞 ∈ 𝑅 (𝑝+1)𝑟 𝑥 𝑞𝑟 where 𝑝 and 𝑞 are parameters such 𝑞 = 𝑝 + 1. The 𝐇 𝑝+1,𝑞 can be factorized into the observability and controllability matrices such

𝐇 𝑝+1,𝑞 = 𝐎 𝑝+1,𝑞 𝐂 𝑞 (3)
where the observability matrix, 𝐎 𝑝+1,𝑞 ∈ 𝑅 (𝑝+1)𝑟 𝑥 𝑛 , and controllability matrix, 𝐂 𝑞 ∈ 𝑅 𝑛 𝑥 𝑞𝑟 , are given by

𝐎 𝑝+1,𝑞 = [ 𝐇 𝐇𝐅 ⋮ 𝐇𝐅 𝑝 ] , 𝐂 𝑞 = [ 𝐆 𝐅𝐆 … 𝐅 𝑞-1 𝐆 ]. (4)
The subspace-based fault detection test detects if the residual of the characteristic property of the reference (healthy) state of the system based on 𝐎 𝑝+1,𝑞 , or subsequently 𝐇 𝑝+1,𝑞 (Eq. 5), is significantly different from zero. Consider the space, or a property of the reference state, where

𝐒 𝑇 𝐇 ̂𝑝+1,𝑞 ≈ 0. (5)
𝐇 ̂𝑝+1,𝑞 is the empirical output block-Hankel matrix and 𝐒 is the left kernel 𝐔 0 of the reference state 𝐇 ̂𝑝+1,𝑞 , hence

𝐇 ̂𝑝+1,𝑞 𝑟𝑒𝑓 = [𝐔 1 𝐔 0 ] [ ∆ 1 0 0 ∆ 0 ] [ 𝐕 1 𝑇 𝐕 0 𝑇 ], (6)
where ∆ 1 contains non-zero singular values . The reference and the subsequently tested states share the same left null space only when no damage occurs. Bearing that in mind, the residual vector is defined as

𝛏 = √𝑁𝑣𝑒𝑐(𝐒 𝑇 𝐇 ̂𝑝+1,𝑞 ). (7)
Note that the excitation of the system is ambient, which leads to the changes in the cross-covariance matrix, 𝐆, (recall 𝐆 = Ε(𝐱 k+1 𝐲 k 𝐓 ) = 𝐅𝚺 s 𝐇 𝑇 ), and consequently in the output block Hankel matrix 𝐇 ̂𝑝+1,𝑞 . That results in non-zero residuals (Eq. 7)

between the healthy states. To avoid that, [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF] use the fact that 𝐇 ̂𝑝+1,𝑞 shares the same null space with its principal left singular vectors 𝐔 1 , so for each healthy state it holds that 𝐒 𝑇 𝐔 1 ≈ 0. Matrix 𝐔 1 contains 𝑛 independent column vectors that spans the column space of 𝐇 ̂𝑝+1,𝑞 and are principal directions of the data, invariant to the change in excitation covariance [START_REF] Yan | Null subspace-based damage detection of structures using vibration measurements[END_REF]. The robust residual is defined as

𝛏 = √𝑁𝑣𝑒𝑐(𝐒 𝑇 𝐔 1 ), (8)
which is tested for being significantly different from zero by use of the 𝜒 2 -test

𝜒 𝜉 2 = 𝛏 𝑇 𝚺 𝜉 -1 𝛏, (9)
where 𝚺 𝜉 is the empirical covariance of the residual calculated as in [START_REF] Döhler | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF].

MAHALANOBIS DISTANCE-BASED DAMAGE DETECION

The Mahalanobis distance, MD, is a metric used in the multivariate statistics to calculate the distance between a point and a distribution. The formulation of the metric takes into account the correlations between different data dimensions by the covariance matrix of the data. For the multivariate normally distributed variables the squared MD follows the 𝜒 2 -distribution [START_REF] Hardin | The Distribution of Robust Distances[END_REF]. One of its practical uses is detection of outliers [START_REF] Filzmoser | A multivariate outlier detection method[END_REF] and recently detection of structural damages based on the transmissibility functions or AR-coefficients as a feature [START_REF] Zhou | Damage detection using transmissibility compressed by principal component analysis enhanced with distance measure[END_REF][START_REF] Cheung | The application of statistical pattern recognition methods for damage detection to field data[END_REF].

The square Mahalanobis metric of the observations in the data vector 𝐱 𝑖 , from the reference data vector with the sample mean 𝛍 and the covariance matrix 𝚺 is defined as

𝑀𝐷 𝑖 = (𝐱 𝑖 -𝛍) 𝑇 𝚺 -1 (𝐱 𝑖 -𝛍) (10)
In this paper the Mahalanobis distance is calculated on a damage sensitive parameter of the system between a reference and damaged states and used directly as a damage indicator. Proposed metric is robust towards the variations of the excitation covariance hence can be used with the operational measurements. Consider 𝑚 reference data sets and 𝑖 tested states, so that

𝑀𝐷 𝑖 < 𝑇 𝑚 → ℎ𝑒𝑎𝑙𝑡ℎ𝑦 𝑀𝐷 𝑖 > 𝑇 𝑚 → 𝑑𝑎𝑚𝑎𝑔𝑒𝑑
where 𝑇 𝑚 is a threshold calculated for the reference state. The MD increases as the system is subjected to any change that affects its vibration characteristics. Here, we consider the empirical block-Hankel matrices with the output correlations as the input for the damage detection test. The modified version of the squared MD featured with the Hankel matrices is defined as

𝑀𝐷 𝑖 = 𝑣𝑒𝑐(𝐇 ̂𝑐𝑜𝑟𝑟 𝑝+1,𝑞 ) 𝑇 (𝚺 𝐇 ̂𝑝+1,𝑞 𝑐𝑜𝑟𝑟,𝑟𝑒𝑓 ) -1 𝑣𝑒𝑐(𝐇 ̂𝑐𝑜𝑟𝑟 𝑝+1,𝑞 ) (11)
Let 𝑌 ∈ 𝑅 𝑟 𝑥 𝑚𝑁 where 𝑚 is the number of the reference sets and 𝑁 is the length of one data set. The formulation of the empirical block-Hankel matrix yields 𝐂 ̂𝑖 = ; 𝐇 ̂𝑝+1,𝑞 𝑐𝑜𝑟𝑟,𝑟𝑒𝑓 = Hank(𝐂 ̂𝑖).

Consequently, 𝐇 ̂𝑝+1,𝑞 𝑐𝑜𝑟𝑟 is determined using Eq. 12 for each tested data set with 𝑚 = 1. The covariance of the empirical vectorized Hankel matrix is estimated from the covariance of the sample mean with the methodology recalled from [START_REF] Döhler | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. The threshold above which the data set is considered damaged is defined as one standard deviation above the mean value of the reference state, that theoretically should approach zero.

TEST CASES

Methods described in the Section 2 are tested with a both numerical example of a simple system excited by the white noise of random variance and a full scale progressively damaged bridge excited by the wind.

SIMPLE SYSTEM SIMULATION

For the numerical testing we considered a coupled spring-mass system with 15 DOF, Figure 1. Output accelerations are simulated using white noise input of variance taken randomly from the normally distributed vector in between [1 10], acting on the last DOF. The data is recorded in all DOFs along the system. To account for the noise in the measurements 3%, 10% and 30% of white noise is added on top of the response signals. The damage is simulated as a progressive reduction in the stiffness of the 8 th spring by 2%, 5%, 10% and 30%. The sampling frequency is 50Hz. The system is exited for 1000s in blocks of 50 data sets for each simulated case. That results in 250 data records per sensor per noise level with a random variance in between each set.

DOGNA BRIDGE

The Dogna bridge, Figure 2, crosses the River Fella and connects the villages of Crivera and Valdogna (Dogna) in Friuli Venezia Giulia, a region located in the North East of Italy. The bridge is a four-span, single-lane concrete bridge. The span is 16m, with the 4m lane. The deck is made of a 0.18m reinforced concrete (RC) slab, supported by a three longitudinal simply supported RC beams with a rectangular cross-section 0.35x1.20m. The beams are connected to the supports with the rectangular RC diaphragms 0.3x0.7m. The piers are a RC walls approx.1.5m thick, 4.5m deep, and 3.6m high. The pylons are fixed in a piled foundation that consists of a 1m thick RC slab supported by a drilled RC piles of 1m in diameter and 18m in length.

For a traffic safety reasons, the Dogna bridge was demolished on May 2008 and has been replaced by a new bridge built about 200m downstream. A test campaign was carried out from April 02 to April 11 2008, and consisted of a series of tests progressively damaging one of the bridge spans. The tests were carried out under similar environmental conditions so the influence of temperature and humidity on the structure is insignificant. Figure 3 shows the artificially damaged bridge span during the tests. The bridge was equipped with 10 accelerometers mounted on its deck. The samples were taken with the frequency of 400Hz.

The measurement campaign lasted 50 minutes in total while the reference state was measured for 20 minutes. The damaged was introduced in approx. three blocks: the first cut, the second cut and the third cut with the damaged center span.

RESULTS

In this section we compare the results from the classical subspace-based damage detection implemented in ARTeMIS -software for operational modal analysis and health monitoring [START_REF]Structural Vibration Solutions A/S. NOVI Science[END_REF] with the Mahalanobis based damage detection described in the paper.

The results for the each test case are concluded in the control chart.

Note again, that both test cases are excited by a ambient, unmeasured load with changing variance.

SIMPLE SYSTEM SIMULATION

The damage is detected for the three noise levels present in the response of the simple system. The 30 out of 50 healthy data sets are used to calculate the reference state. The thresholds for both of methods are defined as a number of standard deviations above the mean metric of the reference state. One standard deviation for the unsafe zone, and two standard deviations for the damaged zone.

Figure 4 and Figure 5 illustrate the comparison of the damage indicators for the 3% and 30% of white noise it the response data. The percentage of the detected damage sets with respect to the simulation case and degree of the damage is summarized in the Table 1. Both methods detect each degree of the damage present in the response data containing 3% of white noise, disregarding the variance of the excitation, Figure 4. While increasing the noise level to 30%, the Mahalanobis-based approach perform more accurate in detecting the smaller deviations, see Figure 5 and Table 1. The control chart, created for the 30% noise case, gives ability to detect all the 5% damage and half of 2% damaged sets, what was not possible with a single method, see Figure 6.

DOGNA BRIDGE

For the general overview of the measurements, the first six singular values of the cross-spectral densities from the output acceleration data for the reference state are plotted in Figure 7. In the frequency range of 0-40Hz there are three visible modes based on the singular values, Figure 7. Hence the damage detection does not involve the system identification, the modal properties won't be analyzed further.

The total number of 22 data sets are analyzed. The reference measurement model for the full scale example is calculated from 6 out of 8 healthy data records. The comparison of the subspace-based and Mahalanobis-based detection methods and the control chart combining the methods is illustrated in Figure 8. Both the subspace-based and Mahalanobis-based method detect the damages in all the test cases. The time stamp for the particular type of damage in the damaged data (number of cuts in the beams, damage of the centerline of the tested span) is unknow, hence the exact performance of single method is difficult to assess. However, each method clearly distinguish the healthy data from the damaged and detects the last stage of the campaign (the damage of the centerline of the span), see Figure 3.

Figure 1 .

 1 Figure 1. Simple system scheme.

Figure 2 .

 2 Figure 2. Dogna bridge, Italy, after [17].

Figure 3 .

 3 Figure 3. Artificially damaged beams. The cuts introduced progressively during the tests, along with the damage in thecenterline of the span. After[START_REF]Structural Vibration Solutions A/S. NOVI Science[END_REF]

Figure 4 .

 4 Figure 4. Damage detection in simple system. Case-3% of noise in the response. Damage indicators for the Mahalanobisbased (left) and robust subspace-based (right).

Figure 5 .

 5 Figure 5. Damage detection in simple system. Case-30% of noise in the response. Damage indicators for the Mahalanobisbased (left) and robust subspace-based (right).

Figure 6 .

 6 Figure 6. Hotelling T 2 control chart for simple system. Robust subspace and Mahalanobis-based detection. Case: noise 30%.

Figure 7 .

 7 Figure 7. First six singular values of the cross spectral densities of the output acceleration data for the reference state.

Figure 8 .

 8 Figure 8. Damage detection in Dogna bridge. Damage indicators from top: Robust subspace (left). Mahalanobis-based (right). From bottom: Classical subspace (left), Hotelling control chart (right).

Table 1 .

 1 Comparison of the detection methods. Ability to detect damage in the simple system for different noise levels.

	Case/Method

Damaged sets detected 2% damage 5% damage 10% damage 30% damage Noise 3%

	Subspace-based	100%	100%	100%	100%
	Mahalanobis-based 100%	100%	100%	100%
	Subspace-based	10%	26%	100%	100%
	Noise 10%
	Mahalanobis-based 40%	100%	100%	100%
	Subspace-based	2%	8%	100%	100%
	Noise 30%
	Mahalanobis-based 0%	40%	100%	100%