Soryna Sabbah
email: sorynas86@gmail.com

Einat Heyd-Metzuyanim

Agency and identity of female Arab students entering a technological university

Keywords: Agency, identity, gender, Arab students, undergraduate education

This report studies the progression of cultural and mathematical identities of female Arab students during their first semester of engineering studies in a technological university in Israel. The report focuses on the cases of two students, Mira and Lena, who demonstrated remarkably different trajectories of identity development. Data was analyzed by closely examining narratives in students' Space of Authorship related to gender and ethnicity. Findings show that Mira's Space of Authorship included multiple narratives relating to ethnic and religious conflicts, upon which Mira enacted agency by debating and choosing specific courses of action both in social life and in mathematics learning. In contrast, Lena's Space of Authorship drew upon psychological discourse to explain her difficulties in ways that did not afford her agency to act back on the challenges she was facing.

Introduction and theoretical framework

The Arab minority in Israel is an involuntary minority (dependent on their collective identities in relation to the dominant group) [START_REF] Flum | Higher education in a transforming society: The case of Arabs in Israel[END_REF] which consist of 20.5% of the general population. There are long standing tensions between the Jewish majority and the Arab minority in Israel, which have encouraged geographical, political and social separation between them [START_REF] Flum | Higher education in a transforming society: The case of Arabs in Israel[END_REF][START_REF] Kaplan | Jewish-Arab relations in Israel: Perceptions, emotions, and attitudes of university students of education[END_REF]. This separation also exists in the education system, where the K-12 education systems of the two populations are completely separated. Thus, universities are usually the first multicultural encounter between Arab and Jewish students. University courses are taught in Hebrew, while in school Arab students study mostly in Arabic. In addition, Arab students have multiple challenges connected with their minority status and the political and ethnic conflicts it entails. For female students, these are compounded by difficulties relating to changing gender roles in this relatively traditional society [START_REF] Arar | Cultural pluralism increases difficulties in learning experiences yet advances identity formation for Muslim Arab female students at higher education institutions in Israel[END_REF]. As female students enter the predominantly Jewish university, they are exposed to social narratives that question their ability to succeed (similar to Jewish female students). Yet, for Arab females, university studies are often the first occasion to leave their home-village and be exposed to Western, secular culture. This is a transition that often causes "culture shock" [START_REF] Arar | Higher education and the Palestinian Arab minority in Israel[END_REF].

The report is a part of a large study whose goal was to explore the process that Arab female students' identities undergo through their entry into a predominantly Jewish university, both in relation to these students' academic and mathematical learning, and in relation to their sense of ethnic and cultural belonging. Data shows high rates of failure of this particular minority on entrylevel mathematics courses at the Technion. We employ a socio-cultural lens on the activity of mathematics learning to study the relation between cultural transitions and the development of a student's mathematical and learning identity [START_REF] Sfard | Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity[END_REF][START_REF] Wenger | Communities of practice: Learning, meaning and identity[END_REF]. In particular, we rely on a sociocultural view of learning as becoming a participant in a certain community [START_REF] Lave | Situated learning and legitimate peripheral participation[END_REF][START_REF] Sfard | Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity[END_REF]. Communities are characterized by shared values, forms of action and interpretative schemes. These have been defined by Holland, Lachicotte, Skinner, and Cane (1998) as "figured worlds", where a figured world is defined as "a socially and culturally constructed realm of interpretation in which particular characters and actors are recognized, significance is assigned to certain acts, and particular outcomes are valued by others" (p. 52). In the case of Arab students, the home, family and village figured world is extremely different from the predominantly Jewish, technological and western figured world of the university.

In our work, we rely on Holland and colleagues' (1998) concept of "space of authorship", which links between figured worlds and the identity participants author for themselves within these figured worlds. Holland et al. point out that the space of authoring, in which people figure their identity, is always a contested space. We link these ideas with [START_REF] Sfard | Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity[END_REF] definition of identity as a set of reifying, significant and endorsable narratives about a person. Space of authorship, for us, is the realm of narratives from which identity narratives are picked up, contested against and negotiated. Agency, defined by Holland and colleagues (1998) as "the realized capacity of people to act upon their world" (p. 42), is enacted through the orchestration "arranging the identifiable social discourses/practices that are one's resources" and form his/her space of authorship (p. 272). Thus, the question of this study is: How do cultural and mathematical identities of female Arab students develop during the initial phase of their college level mathematics studies and in what ways do these students enact agency in relation to the narratives regarding ethnicity and gender in their space of authorship?

Method

The cases of Mira and Lena were taken from a larger study in which we followed 13 female Israeli Arab undergraduate students during their first semester at the Technion, a technological university in Israel. All of the participants were studying in either the Computer Science faculty or the Electrical Engineering faculty, where 40-50% of the courses taken in the first semester are mathematics courses (usually Calculus I and Algebra I), both of which are among the most coveted faculties in this institute and have very high entrance qualifications. The students' ages ranged between 18 and 19.5 years. They were recruited voluntarily, during the first week of the semester, through an institutional mentoring project for Arab students in which senior students mentor the incoming students. Students were promised complete confidentiality and signed informed consent forms. The study was approved by the Institutional Review Board of the Technion.

All the students were interviewed at the beginning and at the end of their first semester. Interviews were semi-structured and conducted by the first author in Arabic. They lasted between 30 and 90 minutes. Students also completed two midterm diaries, which included answers to questions sent to them by email. The final interview consisted of questions eliciting reflections about the past semester. In addition, students were requested to reflect upon and answer some of the questions from their final calculus exam. Data was analyzed using a combination of inductive and deductive codes [START_REF] Saldana | The coding manual for qualitative researchers[END_REF]. Inductive coding searched for recurring themes in students' interviews. For example, a common conflict described by the students regarded social encounters, whether to "hang out" with certain people and with whom to study. Deductive codes were based on the theoretical orientation of figured worlds. In particular, we searched in the interviews' transcripts for valued actions, valued outcomes, significant actors and roles in the home vs. Technion figured worlds. The main themes arising in all the interviews in relation to conflicts between the home figured world and the Technion figured world related to narratives about ethnicity and gender. In relation to these conflicts, we searched for indications of agency both in the actions descried by participants (e.g., "I try to study hard"), or lack thereof ("I feel I am not focused"with no mention of possible action to become focus). We also searched for agency in the manner in which interviewees spoke about the conflictsas a given situation or as something that can be acted upon.

Findings

Mira and Lena were chosen for this report since they exemplified very different trajectories of acclimation to the figured world of the Technion. While Mira ended the first semester with passing grades and relative success, Lena failed several courses and was quite discontented with herself at the end of the semester. The different trajectories afforded opportunities to view and contrast different processes of change in identity narratives and the ways in which these changes drew upon certain spaces of authorship. We first explore the figured world of Mira and Lena, as seen through the collective narratives told by them. We pay attention here to what they say about gender/ethnicity and their positioning with relation to the conflicts they talk about.

Mira's stories about her family and village drew a figured world that is mono-ethnic and monoreligious. She reported in her first interview that she was having trouble living in a multi-cultural environment. She also identified herself as a "non-social person by nature", perhaps to minimize the potential conflicts that could arise as result of the multi-cultural environment at the Technion. Mira also reported being the first in her family to attend university and that she felt pressure from her family to succeed: I'm the first one [from my family] to ever attend a high academic institution. Everyone at home has expectations (from me) and that's a bit stressful. (Mira, 1st interview) Positioning her academic skills in relation to her family seemed to be important to her: I have four older brothers and they were not strong at school, and a sister (who is) a year older than me. I'm better than her academically. She has not yet begun her academic studies. (Mira, 1st interview) Although Mira authored herself as "strong" academically, she confessed in her first interview that mathematics was not a subject she preferred: "to tell (you) the truth, I do not like mathematics". Her goal was to study computer science and mathematics was seen only as a means towards that cause. Unlike her academic identity, which included narratives of relative strength and did not include many conflicts in its space of authorship, regarding her ethnic identity as an Arab student Mira's space of authorship was much more conflictual. In the initial interview, Mira reported that she had not encountered people from other religions before she started studying at the Technion. Her first close encounter with someone of a different religion occurred when she entered the dorms: I live with a Christian student in the dorms; she is from a different religion. At first, I worried about the idea of living with her, because I am a Muslim and she is Christian. I was worried I would feel restricted and not able to behave freely. (Mira, 1st interview) Mira's justification of her worries as based on the fact that the student was "Christian" while she was "a Muslim" indicated that, in her space of authorship, the mere religious identity of a person could cause problems while living together and "not behaving freely". Notably, the fact that the Christian student was also Arab was not mentioned but taken for granted, and the possibility of living with a Jewish student was not mentioned at all. Yet, like in many other instances, where Mira only reported about conflicts in retrospect, this encounter was already authored as resolved. She ended the story by: Later I discovered that if I give her the space to behave according to her beliefs and religion, she will give me back the same space I need. (Mira, 1st interview) Her agency in relation to this conflict could be seen in the actions she described ("give her space") that resulted in what she needed ("the same space I need"). As the semester progressed, Mira repeatedly recounted encounters with students of other religions and gave them significance. Although Mira expressed an acceptance of students from different religions, her diary entries clarified that she was engaged with conflictual narratives around ethnicity and religion and that these were entering her space of authorship: Every day and all the time, I face differences. First, the atmosphere at the Technion is not like the atmosphere at home at all. Here everyone is independent and each person is responsible for himself and his own decisions. There are people from different places, different religion[s], different thoughts and different values and cultures. For example, I came from a village where everyone is Arab and we all belong to the same religion, whereas at the institute I meet people from all over the world and from different religions. I see people behaving in a way that goes against my values and that is a new atmosphere for me, I am constantly trying to stay on the side and not get involved with anyone. (Mira, 1st diary) Several issues relating to this excerpt are worth mentioning. First, Mira is explicitly talking about "differences" between the "Technion" and "home". In other words, it is clear that she is constructing the two locations as different figured worlds. The differences mainly point to differences in relations between the individual and the social community ("here everyone is independent"), hinting that in her home figured world, people are interdependent and not responsible for their own decisions. The interdependence is linked in Mira's words to the Arab ethnicity and to the religious homogeneity of her home village. Thus, it is clear from Mira's words that she is able to locate the conflict, the values that are at stake, and her positionality with relation to this conflict. Notably, there are signs of agency in this story too. Mira writes about the conflict and immediately states her actions in relation to it ("I try to stay on the side and not get involved"). Later in the semester, this action of "staying on the side" seems to mature into a more nuanced positioning and identification of herself with relation to the "others": I came from a traditional and relatively conservative family, where I grew up with values that I believe are correct. However, at the Technion, I encounter many people who behave against my beliefs, and it makes me feel that I am an extreme person or something like that. Yet I continue to believe in my tradition as they continue to believe in theirs. (Mira,2nd diary) This excerpts hints at several processes that Mira underwent during the semester. First, there are beginnings of separation from "her beliefs". Whereas these beliefs were stated in the first diary as simply "her values" ("I see people behave against my values"), here, these beliefs are located in time and space ("I came from a traditional … family", "I grew up with values"). Moreover, the statement about values "that I believe are correct" shows hints of voices that position these beliefs as "incorrect". These voices are later located in people "who behave against my beliefs" and that, whether directly or indirectly, make Mira feel she is "an extreme person". The taken-for-granted "beliefs" and "values" have thus been questioned and contested for Mira, and this has raised a conflict with regard to identity narratives concerning religion and ethnicity. In the face of these conflicts, Mira enacts agency in choosing to adhere to her home values.

With relation to her mathematical identity, Mira reported finding her mathematics studies to be surprisingly satisfying. In her first diary, she wrote:

The most interesting lecture in my weekly schedule is a Calculus lecture, because I learn mathematics differently… I learn how to learn mathematics, how to think outside the box and how to find ideas or solutions for different mathematics problems. It is a hard course but the lecturer is a wonderful person and he makes it an enjoyable course. (Mira, 1st diary) Her satisfaction with her mathematical studies continued throughout the semester despite hurdles and setbacks. In the last interview Mira reported failing in the first take of the calculus exam because of being "unprepared unlike I should have been", then studying better and passing the second take with a satisfying result.

To summarize, Mira retold her identity at the end of the semester as stronger in mathematics than at the beginning, despite recounting many stories of conflicts with relation to her ethnic and religious identity in a foreign figured world. With relation to almost all the descriptions of conflict, Mira's talk was full of agency, whether it was related to decisions she made to adhere to her religious values, or with regard to changing her methods of study. Along with that, some changes in the space of authorship of her ethnic and religious identity occurred. These could be detected mainly in how she narrated her "values" and "beliefs"from being inherently "hers" to being a result of her "upbringing" and her home figured world. In addition, the space of authorship came to include also third person narratives about her being identified as an "extreme" person by others, something which forced Mira to reexamine her beliefs and values and to make choices regarding them. Unlike Mira, who was hesitant about her connection with mathematics in the first interview, Lena's first person mathematical identity was authored in the interview as highly successful. To signify her success, she narrated the following story: I don't know if you know the Elkasmi competitions? (Interviewer nods.) I participated in this competition from fourth till ninth grade, almost I won the first place, once I won second place and once third place… that means it (mathematics) was for me. (Lena, 1st interview) Elkasmi is a competition for gifted students in mathematics in the Arab community. Lena was signifying her strength in mathematics by drawing on the significance that participating and winning these competitions has in her community, a significance she made sure was shared by her interviewer.

Apart of the Elkasmi story, Lena relied, similarly to Mira, on her family's narratives when authoring her first person academic and mathematical identity. She reported that her family thought of her as a strong student, who could achieve anything she wanted. These expectations were talked about both as a reassuring element and as a stress factor: They (parents and family) always remind you that you have potential. So I somehow try to turn this potential (to reality) ... So that's my parents' expectations in particular, and my sisters think that I'm 'Rambo'. That I could do anything and get along. Most people expect so. (Lena, 1st interview) Unlike Mira, Lena was not a first generation university student in her family. Her parents both had academic degrees. Yet similarly to Mira, she also reported feeling stress from her family's high expectations. Again, it was clear from her talk that she was highly accountable to her family and community with relation to her academic choices and achievements. Yet unlike Mira, Lena's talk about her studies was mostly devoid of agency. Her authoring of herself as successful was based on the diagnosis that "math is for me", justified by her winning math competitions during her childhood. Also in revoicing the voices of parents and significant narrators ("they always remind you"), these voices narrated her as "having potential" and "turning this potential into reality". This diagnostical discourse was most noticeable in the main conflict Lena talked about from her first interviewthat relating to her gender: I will not begin now by saying that boys are smarter than girls, but I believe that boys have more self-confidence in the science field than girls. They (boys) always think that the science field is for them, and girls think that they are less … that is what I read and see around me. Here I see the difference between boys and girls, but that does not mean anything about the final results because sometimes the way the girl thinks help her beat all her competitors. (Lena,1st interview) Notably, already at this initial phase of her studies, Lena was contesting voices that were saying "boys are smarter than girls". She was negating them, yet endorsing the narrative that "boys have more self-confidence than girls", which she justified based on what she "read and saw around her". Again, there were no signs of agency in Lena's talk. Her talk was diagnostic ("here I see the difference between boys and girls"), but there was no mention of her own actions in relation to this conflict.

As the semester progressed, the gender related narratives turned into first person identity narratives. In her first diary, she wrote: Despite all the faith I have in myself, I feel that I'm not confident enough, in comparison with boys, at least with a boy who I study with. I think it comes from old stereotypes about the composition of the boy's brain versus the girl's brain. Sometimes I fear that these stereotypes are true and that my mental ability is not enough. (Lena, 1st diary) Again, Lena's authoring of the narratives about gender and mathematics/technological studies was solely diagnostic and devoid of agentic descriptions. She was examining her "beliefs in herself" and her "self-confidence" against "old stereotypes", indicating at least a partial endorsement ("sometimes I fear these stereotypes are true") of these narratives about "the composition of the boy's brain versus the girl's brain". Importantly, these "stereotypes" were about "brains" of males and females, something that was clearly beyond Lena's agency to change. Lena was examining and identifying herself within this figured world of fixed-ability gender-related narratives, without any indication that she had a choice to accept or reject them. The gender-related narratives evolved in the second diary into a mini-theory of the relation between her "self-confidence" and her actions: When a person is less confident, I don't know how (to explain this) … But sometimes you do not believe in yourself, sometimes you do not want to meet anyone. That (decision), not wanting to meet anyone, is not the best decision in the world. You have to be with people who you see every day and share with them what you solved, and what you did not (solve). So that it helps you keep going. Sometimes it made me stop or slow down. (Lena,2nd diary) This quote exemplifies well the psychological discourse that Lena drew upon to explain her perceived failure during the first semester. Explanations about her failure (which is not mentioned here explicitly, only hinted at) use psychological keywords ("self-confidence", "belief in oneself") to justify her decision "not to meet anyone". The self-diagnostic statements are written about as pregiven and beyond Lena's control. Further, these psychological states have led Lena, according to her narrative, to certain decisions that she regrets ("you have to be with people"), yet there is no indication that Lena had a choice regarding these decisions. They were a direct result of her low "self-confidence". In the final interview at the end of the semester, Lena's discourse was full of such regrets: I started the semester as a little girl, I was not focused … I was very nervous about people around me so I did not study in the library and stayed at the dorms. Everyone asks you 'what's going on?', 'Did you solve the homework?' Why should they ask me? I felt pressure. (Lena,2nd interview) Again, this quote is full of psychological terms relating to maturity, cognitive and emotional functioning ("a little girl", "not focused", "very nervous"). As was common in all her interviews and diary entries, there is no mention of any difficulties relating to her ethnic or cultural status.

The final interview also had indications that Lena was re-authoring her identity as a high-school student (at least to the audience of the interviewer). She said: "I have never been in a place like this, even in high school there were difficulties but it isn't like here." This narrative was different than those authored in the first interview, where Lena talked about herself as successful and being "made for" mathematics. She now talked about her high-school career as containing difficulties too, yet not as extreme as in the Technion. The lack of agency with relation to these difficulties was consistent. Lena talked about wrong decisions that she made during the semester (such as not asking for help) but always in retrospect, and always with relation to how these decisions were unhelpful.

Summary and Discussion

Multiple studies have discussed the relationship between being part of a minority group and success in school or academic studies [START_REF] Arar | Higher education and the Palestinian Arab minority in Israel[END_REF][START_REF] Flum | Higher education in a transforming society: The case of Arabs in Israel[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]. According to these studies, higher education allows minority students to move from the margins of society closer to its centers of power. Yet, at the same time, institutions of higher education pose significant barriers and conflicts for minority students. In the case of the Arab minority in Israel, and in particular, female students, these barriers are only starting to be understood. In this study, we shed light on two contrasting cases of first semester acclimation. These two cases hint that agency may be a crucial aspect of such acclimation, in the face of complex social and political conflicts. Mira's story demonstrates a case of high agency, along with very explicit positioning of herself in relation to ethnic and religious conflicts. This agency seems to have helped her overcome difficulties in the social, as well as the academic and mathematical aspects of the Technion's figured world. In contrast, Lena's story demonstrates a case of very little agency, along with a total neglect of her positionality with regard to ethnic and religious conflicts. Paradoxically, although Lena seemed to start the semester as better acclimated to the figured world of the Technion, her lack of awareness of the constraints imposed on her by her minority status may have led to lack of agency. The significance of our study lies in turning attention to the importance of Arab female students' awareness of the social and ethnic conflicts they face upon entering a foreign figured world. This awareness may cultivate choice, which is the necessary condition for these students' agency to cope with challenges.


Sabrina Bobsin Salazar
email: sabrina.salazar@ufpel.edu.br

Locally disrupting institutional racism by enacting mathematics in a U. S. laboratory classroom

Keywords: Mathematical knowledge, institutional racism, normativity

In this study I investigate whether mathematical knowledge inherently carries fragments of its socio-political-historical development or if mathematics itself is value-free and can only get a political dimension attached to it by how it is used in society. I empirically approached the problem by studying a laboratory classroom collectively solving a problem. This classroom served as a proxy for a community of practicing mathematicians and I particularly investigated how conceptions of mathematics endorsed in this classroom could support (local) disruption of racism.

Theories guiding this study include critical realism and critical race theory and analyzed data includes video records of instruction in this laboratory classroom. Results show that creating opportunities for students to engage in more authentic mathematical practices can support (local) disruption of racism.

The problem I am addressing and how I am doing it

Mathematics often enjoys a distinctive status among the sciences. Such highly regard status comes from the idea accepted by most practicing mathematicians that mathematics is universal, objective, and certain [START_REF] Ernest | Is mathematics discovered or invented[END_REF], or, in other words, mathematics is an a priori knowledge that deals with truths that are true by virtue of necessity, and with objects that are abstract [START_REF] Linnebo | Philosophy of mathematics[END_REF]. These ideas however, are not unanimous, and disagreement about such assumptions can be traced back to the work of Plato and Aristoteles [START_REF] Machado | Matemática e realidade: Das concepções às ações docentes[END_REF]. Whereas refined arguments were developed to support one or another claims, the debate is still open. The work of [START_REF] Kitcher | The nature of mathematical knowledge[END_REF] and [START_REF] Ernest | Social Constructivism as a philosophy of mathematics[END_REF] are relatively recent work supporting an a posteriori view of mathematics that is intrinsically dependent of human activity in society, but introductory texts on philosophy of mathematics such as [START_REF] George | Philosophies of mathematics[END_REF] still presuppose that mathematical knowledge can be philosophically investigated disregarding its social and historical construction.

Mathematics education researchers that are concerned with socio-political dimensions of mathematics education have been interested in these philosophical disputes. Such scholars have a long-term commitment in understanding the relationships between mathematics education and society, particularly relationships involved in social inequality and injustice. To these scholars it is important to understand if mathematics inherently carries fragments of its socio-political-historical development or if mathematics itself is value-free and can only get a political dimension attached to it by how it is used in society. It is clear for socio-political researchers that mathematics and mathematics education are deeply connected with our contemporary social structure that privileges a group of people over others. What is not so clear is whether this connection comes from pure mathematical knowledge itself or not. So far, many philosophers have argued in favor of one option or the other employing a variety of philosophical techniques to support their claims. I am adopting a different strategy. To investigate whether mathematical knowledge is independent of the human activity that produced or discovered it, I study a small community solving a mathematical problem together. This small community is an elementary mathematics laboratory classroom and it serves as a proxy for a broader community of practicing mathematicians. There are striking differences between both communities, and yet many similarities I expect to be helpful in illuminating the problem I am investigating. For example, in this classroom, the teacher functions as a permanent knowledge authority, being responsible for, ultimately, legitimating a mathematical claim in this environment, while in a community of practicing mathematicians, there is an institutionalized peer-reviewing system responsible for legitimating mathematical claims. The discussions the students participate in, however, are very comparable with mathematics seminars run by mathematics departments. There are discussions about whether a mathematical idea is true and arguments are laid out so other participants can be convinced of the validity of mathematical claims.

Moreover, I will look to a specific social dimension, namely institutional racism, so I can investigate it in depth and gather a better understanding of the connection of mathematical knowledge and the (re)production or disruption of racism. A pilot study showed that, in this classroom, there are instructional practices that support local disruption of racism [START_REF] Salazar | Explaining how racism can be locally disrupted in a mathematics classroom[END_REF], the question now is if these practices were the only ones responsible for promoting such disruption or if the mathematics itself was also important in promoting local disruption of racism in this classroom. The research question I want to answer in this study is: What conception of mathematics is endorsed in this laboratory classroom and how can it support (local) disruption of racism?

Conceptual framework

Critical race theory

Critical race theory (CRT) is a theoretical framework for research that foregrounds race, racism, and racialized experiences. To critical race scholars, race is a complex social construct that goes beyond the color of skin and citizenship [START_REF] Ladson-Billings | Just what is critical race theory and what's it doing in a nice field like education[END_REF], and brings real consequences to people once they are identified as member of a particular racial group [START_REF] Bonilla-Silva | Race with racists: Color-blind persistence of racial inequality in the United States[END_REF]. The kind of racism CRT is interested in exposing and challenging is institutionalized racism that permeates daily social interactions as opposed to individual blatant acts of racial prejudice. Finally, CRT posits that people of color experience racial oppression differently based on their individual background and multiple identities and insists in a non-essentialist approach (Delgado & Stefanic, 2001).

Critical realism

Critical realism (CR) is a philosophy of science that starts from a realist conception of the world, and the real things that make the world are then structures and mechanisms, or, in other words, causal laws [START_REF] Bhaskar | A realist theory of science[END_REF]. In this perspective, in spite of being socially constructed, race is real because it has causal powers; race brings real consequences for the lives of people [START_REF] Bonilla-Silva | Race with racists: Color-blind persistence of racial inequality in the United States[END_REF]. In CR phenomena cannot be completely determined by scientific laws, they are only influenced by them. When I say that race causes segregation within schools through tracking, I am describing a tendency of over-representing White and Asian students while under-representing African American and Latinx students in advanced classes, and over-representing African American and Latinx while under-representing Whites and Asian in lower tracks [START_REF] Oakes | Two cities' tracking and within-school segregation[END_REF]. Race influences but does not completely determine what is going to be a student placement.

In CR the real things in the world can be combined in a way that, because of their structure and not only its individual properties put together, a new thing emerges in the world. This new thing, also called "entity" or whole, possesses "properties or capabilities that are not possessed by its parts" (Elder-Vass, 2010, p. 4). The whole is then not just the sum of its parts, but something else, with a new causal power that is, of course, derived from the individual properties of its parts. But not only this, the way the parts interact and relate with each other is also responsible for the emergence of the new thing.

The concept of emergence forms the laminated view of the world. A particular whole is said to be in a higher level than its parts. The same whole, however, can be a part of another emergent structure; in this case the whole is in a lower level than the new emergent structure. Individuals are seen as the lowest level of the social world, the whole society as the highest level, with many intermediate levels in between, such as social institutions. The immediate higher level to an individual is, according to Elder-Vass's (2010) definition, a norm circle. The norm circle is defined by the group of individuals who hold a normative belief of endorsing a social norm or, in other words, each individual in the norm circle acts to reinforce the norm and discourage behavior that does not conform to the norm. Elder-Vass's argument is that the norm circle is actually an emergent structure rather than only a group of people because it has a new causal power: to increase conformity to the norm.

The content within instruction

In this work, I view instruction as defined by [START_REF] Cohen | Resources, instruction, and research[END_REF] as "interactions among teachers and students around content, in environments" (p. 122). Figure 1 shows a representation of this definition. I am looking to social normativity as a particular environment, and to classroom interactions as individual actions in light of norms enforced by norm circles when some or most of these norms are oppressive and perpetuate institutional racism. In this study, I am narrowing my gaze to the role of the content in these relationships.

Methods

The laboratory setting and collection of data

In this study I analyzed episodes of instruction from a mathematics elementary laboratory classroom. This classroom is the main part of a summer program held each summer at a large research university in the United States. In this program, an experienced White teacher teaches lessons to a group of rising fifth grade students (students going to start fifth grade at the subsequent fall) in public--that is, while over 70 other educators observe. Because this summer program is a site of research for student learning and teaching practices, different types of data are collected by the research team. The data set includes video records of instruction, video records of pre-brief and de-brief meetings with learning teachers, copies of students' notebooks, pictures of classroom records such as charts, lesson plans, class materials such as handouts for students, etc. The majority are African American and most come from low-income households. Only a few of them are White. The students have different levels of English proficiency and their mathematical performance in school is homogenously low.

The train problem

At this edition of the summer program, the students worked on collectively solving the train problem (Figure 2) over the span of the entire two weeks of the program. Students worked on this problem every day since the second day of the program at the second half of class that lasted about one hour each day. In this study, I focus on data related with students working on this problem, so video records of classroom, detailed lesson plans for each class, copies of student work (notebooks, homework, and assessments), photos of every collective record produced in classroom (such as charts and white board records). I am focusing on this problem because it is unusual to work in problems of this kind at this level, but some of its characteristics are very common in the field of mathematics, which makes it interesting to investigate practices of doing mathematics in a community. The solution of part 1 is that the SP company can build trains that hold 1 to 15 passengers. Part 2 actually does not have a solution and, in this case, solving the problem means mathematically explaining why it is not possible to find a train that meet all the conditions of the problem.

The Train Problem

The SP Train Company has five different-sized train cars: a 1passenger car, a 2-passenger car, a 3-passenger car, a 4passenger car, and a 5-passenger car. These cars can be connected to form trains that hold different numbers of people.

Analysis

I watched and wrote fieldnotes from all video records. From the fieldnotes, I wrote a description of how the class solved the train problem. Then I selected episodes from interactions of the teacher and the students that showed students working and discussing pieces of the solution of the train problem. Each episode was then re-watched and a more detailed description was written. In each episode I investigated mathematical normative practices that could illuminate the nature of the knowledge produced in this laboratory classroom. I also look for schooling normative practices that could intersect with mathematical practices. Additional data from student work and detailed lesson plans were used to check and refine hypotheses.

Enacting math: Collectively solving the train problem

The class worked to find the solution of the train problem. The process started in class 2 and involved individual and small group work, as well as whole class discussions. To solve part 1, the strategy consisted in finding the smallest train that the SP company could build, then the greatest, then find trains "in between". 1 In this initial discussion, La'Rayne2 (La-rain')3 said wr4 , which holds 3 passengers, is the smallest possible train because w=1 and r=2. Other students loudly say "no", disagreeing. La'Rayne immediately revised her thinking, saying that you could use just the red or the white, unless no train would be made at all. They agreed 1 (or zero) was the smallest train. The class did not reach a consensus about the greatest train as quickly, so they worked independently or in small groups on this question. During this work, the teacher discussed with La'Rayne about the train she was testing. She first thought her train held 19 passengers, but after discussing with the teacher she realized it actually held 15. Then she and the teacher had a very dynamic interaction, with questions and answers being said immediately on top of each other. In this interaction, La'Rayne explained that the biggest train held 15 passengers because it used all the rods, and that it was not possible to build a train for 19 because there would be no rod left to add to the train. The class continued their collective work and concluded that the company could build trains of size 1 to 15 by the end of class 3. 1 I am using the idea of size to talk about the number of passengers a train holds. So, a train of size 7 holds seven passengers, a train of size 7 is smaller than a train of size 11, and the greatest train is a term used to refer to the train who holds the greatest number of passengers.

The solution of the part 2 spanned across the remaining 7 classes. They started doing some exploratory work to understand the problem and make sense of the conditions. During this part of the work they build collectively one train that holds 15 passengers, namely yrpwg, and broke it apart in different ways, trying to understand how they could break it, how they could not, and what were smaller trains they could built from it. They agreed that this train did not work because they could not build 9, 13, or 14. Then they started testing out other trains of size 15 and begun making records of all the trains they were testing that did not work. In this work, the teacher talked with the pair Layla and Olivia who said they were done, meaning they found a train. As a result of this interaction, teacher and girls realized they "were moving" things around, or, in other words, they were violating the condition that the cars had to be next to each other to form the smaller trains. Many students had similar confusion and interactions with the teacher. With practice and interactions with the teacher, the students seemed to overcome such difficulty, and also progressed to make more organized records of their attempts. They continued the exploratory work of testing out trains for about three days.

On class 7 they made some important progress with respect to understanding the mathematical work they were doing: they discussed why it is always possible to make trains for 1 to 5 (only one car is needed to build these trains) and for 15 (all cars are needed), how to decide a train did not work (only one number that does not work is enough), and started to think what are the numbers that are difficult to make and why it that so ("big" numbers, such as 11, 12, 13, and 14).

On day 8, the students received the red and white clue: "If there is a train that can be made for Ms.

McDuff, it will have to have the red car (2) on one end and the white car (1) on the other end." This clue contains a logic implication that was not easy for the students to understand. Moreover, the clue was given to the students with the idea that it could be a "wrong" clue, so before they could use it, they would need to find out whether the clue was valid. Students were confused in the beginning, they thought the clue was wrong because they were testing trains with the red and white cars in the ends of the trains and still could not find a train that works. It was only on class 9 that the students agreed the clue was right, meaning that if they were to find a train that works, they needed to build trains for 13 and 14, and that was only possible with red and white in the ends. Then using the clue, they arrived at the conclusion that they only needed to test six trains: wgpyr, wgypr, wpgyr, wpygr, wygpr, and wypgr. On class 10, the students were divided into six groups, so each group were to test one of the six possible trains. After a final discussion in which each group reported their train did not work, the class seemed convinced there was no train that could fulfill Ms. McDuff's request.

When school norms intersect with mathematical norms

There were often school norms and mathematical norms shaping the interactions observed in this laboratory classroom, and in some instances such norms were conflicting. In other words, in some cases, a school norm assigned a behavior for a particular situation, while a mathematical norm assigned a different behavior for the same situation. For example, this interaction occurred after the brief discussion described above on day 7 about what numbers need to be checked in each train. After talking about not needing to test one to five, the teacher asked if he needed to check 15. He answered:

Kassim:

No.

The teacher asked why not, which seemed to be interpreted as a wrong answer, so Kassim changed it: Kassim:

Oh, yes.

Teacher again immediately asked why. Then Kassim presents a full explanation:

Kassim: No, because all of these is equal to fifteen. [Pointing to the rods.]

In this interaction between the teacher and Kassim, the boy tried to guess what would be the answer the teacher was expecting. He answered, waited for the teacher's response to his question, and quickly changed his first answer after the teacher's reaction was not confirming. This behavior can support school success given the role teachers have in assigning grades often based in students correct responses to questions. In this classroom, however, the teacher tried to promote mathematical reasoning. Thus, frequently her responses to students' answers were follow-up questions asking for more explanation. In the end, to answer in a way that aligned with teacher expectations, Kassim would need to do a mathematical explanation, and so did he.

Another issue generated by the conflict between school norms and mathematical norms arises when students are "solving" a problem that has no solution. It is common in school, even more at this grade level, that mathematical problems always have a solution, often a unique one. Problems with no solutions, or with multiple solutions will most likely only appear in algebra in secondary school mathematics. Moreover, even when students deal with this kind of problems in algebra, it is not always discussed how this kind of problems are "solved" by mathematicians. In the field of mathematics, "solving" a problem with no solutions means to prove the problem is unsolvable. In this laboratory class, the students had to negotiate between a school norm, in which they had to find a solution to the problem, and a mathematical norm, in which they had to prove there was no solution to be found. On day 7, Luiz commented when testing one train:

Luiz: I was so close, I needed two more, 13 and 10.

He expressed the frustration of not finding a solution. Later, on day 10, the class received the visit of Ms. McDuff and they had to report how her request cannot be fulfilled. They had discussed before she arrived how they were going to report it to her. They were all sure there was no solution, and they discussed how they were going to talk about the red and white clue, how they tested all the "possible trains" and none works. But when she arrived, they struggled to say it. Even Olivia who had said before very clearly "there is no train" was speaking more quietly, seemed ashamed of not having a train to report, saying "we couldn't figure out a train". The discussion proceeded and the teacher supported students in reporting what they had actually discovered. Eventually, Miah (Maya) said very clearly that her train was impossible, and Chandler spoke loudly, raising her hand to her front with the palm upwards when Ms. McDuff asked what that meant:

Chandler: Your train is impossible, we can't make it.

And Jerone added:

Jerone: So what we're saying is that your train could never be made.

Ms. McDuff replied that she will have to find another company and students responded loudly at the same time "no no no".

They continued a brief discussion afterwards, and students talked to her about the red and white clue and how they tested all the "possible trains" and none works. They were then able to convince Ms. McDuff her train could not be made and she did not need to keep requesting it from any other company.

Discussion

In this classroom, students engaged in doing mathematics in similar ways mathematicians do. They explained their reasoning mathematically, why some trains always worked, so they did not need to test them; why they did not need to find all small trains they could not build from the whole train to be sure the whole train did not work; why the red and white clue was right and how to use it to solve the problem. They used strategies to solve problems frequently used in mathematics, such as finding the small and greatest possible train for part 1, then look for trains "in between". When engaging in these practices of doing mathematics, the students took an active role in producing mathematical knowledge in their community. The active enactment of mathematics by these students was crucial to support local disruption of racism. When the students described in this report, mostly African Americans, one Latinx, boys and girls, performed mathematics they countered the idea that mathematics intelligence is a function of being White (and male). In this classroom, mathematics is something you do, rather than know, and they could all do it. In spite of the intersection between school normativity and mathematical norms, creating the opportunity for students to engage in more authentic mathematical activity opened up the classroom space for students, specially students in this classroom, to participate in mathematical discussion they usually cannot, and that is disruptive.

Figure 1 :

 1 Figure 1: Representation of instruction from Cohen, Raudenbush, and Ball (2003, p. 124)

  only these five types of cars to build trains, and you can use at most one of each type of car in each train. What are the different numbers of people that the SP Train Company can build trains to hold? Part 2 Ms. McDuff wants to order a special 5-car train that uses exactly one of each of the different-sized cars. Ms. McDuff wants to be able to break apart the 5-car train to form smaller trains that hold exactly each number of people from 1 to 15. The customer wants to be able to build these smaller trains using cars that are next to each other in the 5-car train. Can the SP Train Company build Ms. McDuff's order? Explain how you know.

Figure 2 :

 2 Figure 2: Summer program train problem

The research consents and IRB approval permit us to use the children's real first names. Other identifying information (e.g., last names) are not used.

Names of African Americans in the U. S. often have unusual spellings. Additionally, their names are frequently misspelled or mispronounced. In this work, I am trying to use their first names with correct spelling and pronunciation as a sign of respect. When I cite a name with an unusual spelling and/or pronunciation, I will write the pronunciation in parenthesis with emphasis after the first time I write the name, so La'Rayne is pronounced La-rain'.

The letters used here are a reference to the first letter of the name of the color of the train cars. So the train wr is the train white-red which holds 1+2=3 passengers. Similarly, yrpwg is the train yellow-red-purple-white-green which holds

5+2+4+1+3=15 passengers.


Noureddine Seddari
email: 1n.seddari@univ-skikda.dz

Mohammed Redjimi
email: medredjimi@gmail.com

Mohamed Belaoued
email: belaoued.mohamed@gmail.com

A DEVS/MAS-based Framework for Modeling/Simulation of Complex Systems

Keywords: Complex Systems, Discrete Event Systems Specification (DEVS), Modeling and Simulation, Multi-Agent Systems (MAS), AGR Model (Agent/Group/Role), MAD-KIT

This paper presents an approach for modeling and simulation of complex systems. This approach is based on the decomposition of the considered systems into sub-systems which appear on two levels: On the lower level; the decomposition concerns the division of a global system into atomic and coupled models based on DEVS formalism (Discrete Event Systems Specification). The system components are then represented using the DEVS mathematical equations. This step allows the formal system checking. At the higher lever, the implementation of the obtained DEVS models is realized using Multi-Agents Systems (MAS) based on Agent/Role/Group (AGR). Moreover, a set of procedures and functions permitting to transform DEVS to MAS models are defined and implemented. The main advantages of this approach are its adaptability for various domains, its flexibility (easy to implement), its extensibility (adding new components). A version of this work will be implemented using a functional extension of the Multi Agent Development KIT platform (MAD-KIT).

Introduction

Nowadays, complex systems are made of a large number of interacting entities into interaction, and the overall behavior of these systems emerges from these interactions [START_REF] Simon | The sciences of the artificial[END_REF][START_REF] Bertalanffy | Théorie générale des systèmes[END_REF]. The increasing complexity of the studied systems and the requirements of the designers have led to the introduction of many categories of formalisms into the modelling and simulation process. Indeed, there is no single model, or formalism, to model all aspects or components of a system. Thus, different levels of explanation, different theories, and different formalisms or dedicated languages are necessary to properly describe the different aspects or components of the system. The modeling and simulation [START_REF] Fishwick | Simulation Model Design and Execution[END_REF][START_REF] Sonnessa | Modelling and simulation of complex systems[END_REF][START_REF] Shannon | Simulation modeling and methodology[END_REF][START_REF] Oussalah | Modèles hiérachisés multi-vues pour le support de raisonnement dans les domaines techniques[END_REF][START_REF] Ingalls | Introduction to simulation[END_REF][START_REF] Shannon | Introduction to the art and science of simulation[END_REF][START_REF] Fishwick | Computer simulation: growth through extension[END_REF][START_REF] Vangheluwe | Foundations of modelling and simulation of complex systems[END_REF] of complex systems can be used for several purposes. For instance, in ca be used in order to handle the modeled system without risks, to observe and analyze the phenomena emerging when we change the system inputs, and also to improve the understanding and to prevent eventual damages of the system. Therefore, the challenge of the modeling and simulation process of complex systems is to provide adequate formalism to model these systems on one hand, and to use the appropriate tools for implementation, on the other hand.

This work presents a hybrid method that combines DEVS (Discrete Event System Specification) formalism [START_REF] Zeigler | Theory of Modeling and Simulation[END_REF] and AGR model (Agent/Group/Role) for modeling and simulating complex systems.

First, DEVS formalism [START_REF] Zeigler | Theory of Modeling and Simulation[END_REF] is used to define the structures of entities that make up the system by using both atomic and coupled models. The DEVS representation encapsulates the structure and the behavior of the model. Thus, DEVS models provide mathematical representations that can be formally checked and validated.

Second, multi-agent systems (MAS) present a strong theoretical and practical tool to implement the solution and provide several advantages such as autonomy, proactivity, distribution, self-organization and adaptation. The agents cooperate for collective problem solving, since an isolated agent cannot solve the entire problem because it has not large perception and capabilities. Therefore, the system of agents provides a collective solution that emerges from collective actions within the population of the agents.

The idea developed here is focused on the transformation of models checked and validated in DEVS to agent model representation. The Agent/Role/Group (AGR) concept presents the most convenient paradigm in this scope. Moreover, a set of procedures allowing transition between DEVS model and AGR model has been determined and integrated in MAD-KIT platform. The latter has been developed by the team of Professor J. Ferber in LIRMM [START_REF] Gutknecht | The MadKit agent plateforme architecture[END_REF] and based on the principles of AGR.

This paper is organized as follows: The second and third sections present the Multi-Agent Systems and DEVS formalism. The fourth section describes related works, where we categorize combinations of DEVS and MAS from which our approach was inspired. Both approach and algorithms transformations that we propose are shown in the fifth section. Finally, section 6 concludes the work and underlines our perspectives.

2

The Multi-Agent Systems (MAS) J. Ferber [START_REF] Ferber | Les systèmes multi-agents : vers une intelligence collective[END_REF] [14] [START_REF] Michel | Multi-Agent Systems and Simulation : A survey from the agent's community perspective. Multi-Agent systems : simulation and application[END_REF] represents a multi-agent system by the couple <A, W> where A is an agent and W an environment, and considers an agent in interaction with the world as a system composed of dynamic coupled two subsystems, the coupling taking place through perceptions that the agent has for environment and actions that modify this environment. A = (P a , Percept a , F a , Infl a , S a )

(1)

W = (E, Г, Σ, R) (2)
With -P a represents the function of perception of the agent, -Percept a the set of stimuli and sensations, -F a the function of behavior of the agent that determines the agent's state, -Infl a the function of action of the agent,.

-S a the set of the agent's internal states, -E the space in which the agent evolves, -Γ the space of influences produced by the agent and having like consequences to modify the evolution of the environment, -Σ the set of states of the environment, -R the law of evolution of the environment,

AGR model

The model (AGR) is based on the concepts of agents, groups and roles [12] [16].

Agent. An agent is only specified as an active communicating entity which plays roles within groups.

Group. Groups are defined as atomic sets of agent aggregation representing any usual multi-agent system. Each agent is part of one or more group.

Role. The role is abstract representation of an agent function, service or identification within a group. Each role handled by an agent is local to a group [START_REF] Seddari | Operational approach for modeling and simulation of an industrial process[END_REF][START_REF] Seddari | Multi-agent modeling of a complex system[END_REF][START_REF] Seddari | Agent/Group/Role Organizational Model to Simulate an Industrial Control System[END_REF][START_REF] Seddari | Multi-Agent Modeling of a Complex System[END_REF]. Figure 1 represents the diagram of AGR model. The Discrete Event System Specification (DEVS) [START_REF] Fishwick | Simulation Model Design and Execution[END_REF] [21] is an environment for systems' modeling and simulation. DEVS provides a means of specifying a mathematical object called a system. In addition, DEVS allows gathering, in coherent way, other formalisms of modeling. It is, in fact, a formalism fitted to significant number of scopes of application [START_REF] Barros | Dynamic structure discrete event system specification : a new Formalism for dynamic structure modelling and simulation[END_REF][START_REF] Uhrmarcher | Dynamic Structures in Modeling and Simulation : A Reflective Approach[END_REF][START_REF] Ntaimo | Expressing a forest cell model in parallel DEVS and timed cell-DEVS formalisms[END_REF][START_REF] Troccoli | Implementing parallel cell-DEVS[END_REF].

Formal Specification of a DEVS Atomic Model

A Discrete Event System Specification is a structure ADEVS = (X, Y, S, δ int , δ ext , δ con ,t a , λ)

Formal Specification of a DEVS Coupled Model

A DEVS coupled model is described by the following equation:

CDEVS = (X self , Y self , D,{ Md/dЄD}, EIC, EOC, IC) ( 4
)
Self: is the model itself.

Related works

In this section, we presnt the most relevet tools and platforms for systems modeling and simulation frameworks that combined both DEVS and MAS. The GALATEA platform [START_REF] Davilla | GALATEA: A multi_agent simulation platform[END_REF] [27] provides a high level simulation concepts for specification and design of MAS systems, Thus, one can modeling roles and action plans and establishing communications, negotiations and dialogues between agents to allowing the emergence of efficiency distributed solutions. GALATEA is based on the Zeigler theory of simulation and the logical agents. This platform uses a set of logical programming oriented languages and allows the modeling by using DEVS formalisms. The major disadvantage of this platform is its difficulty of implementation [START_REF] Mattei | Towards Discrete Event Multi Agent Platform Specification[END_REF].

The JAMES platform [START_REF] Uhrmacher | Agents in Discrete Event Simulation[END_REF][START_REF] Himmelspach | Component based modelling and simulation for valid multi-agent system simulations[END_REF][START_REF] Himmelspach | JAMES II: Extending, Using, and Expeiments[END_REF] allows dynamic simulations of MAS systems. An agent is represented by atomic DEVS model, a group of agents is represented by coupled DEVS model and the communication between models is determined by the inputs and outputs events. The agent environment plays an important role and it is perceived as a shared framework for the agents interactions [START_REF] Weyns | Environments for Multi agent Systems Statec of the art and Research Challenges[END_REF][START_REF] Weyns | Environments in Multi-agent Systems[END_REF][START_REF] Wooldridge | An introduction to Multi-agent Systems[END_REF][START_REF] Steiniger | Modeling Agents and Their Environment in Multi-level-DEVS[END_REF].

The VLE platform [START_REF] Ramat | Virtual Laboratory Environment (VLE): a software environment oriented agent and object for modeling and simulation of complex systems[END_REF][START_REF] Quesnel | The Virtual Laboratory Environment -An operational framework for multi-modelling and analysis of complex dynamical systems[END_REF] is a high level simulation framework based on DEVS formalism. This platform allows the simulation of the MAS systems by using DEVS. An agent is represented by a DEVS atomic model, an external stimulus by a message and the environment is characterized by CELL-DEVS [START_REF] Ilachinski | Cellular Automata, a Discrete Universe[END_REF][START_REF] Wainer | Application of the Cell-DEVS Paradigm for Cell Spaces Modelling and Simulation[END_REF].

The DEVSimPy platform [START_REF] Wooldridge | An introduction to Multi-agent Systems[END_REF] considers the advantages and the disadvantages of these latter platforms and provides a framework for agents modeling and simulation by using DEVS formalism. Thus, both the agents and the environments are represented by DEVS atomic models. A group of agents is represented by a DEVS coupled model [START_REF] Seddari | Using of DEVS and MAS Tools for Modeling and Simulation of an Industrial Steam Generator[END_REF].

By analyzing the aforementioned works, one can conclude that in these approaches; the agents, the environments, the interactions between agents and the groups of agents are represented by atomic and coupled DEVS models.

Multi-agent systems (MAS), for their part, present a strong theoretical and practical tool in the field of simulation. MAS include tools for managing the agents. At this level, the implementation of the system can be easily realized. For these reasons, we opted in our work for the combination

Proposed approach

In this work, we propose a new approach for modeling and simulation of complex systems and precisely industrial ones. After studying the most relevant existing approaches we distinguished the following correspondence between DEVS and AGR models : According to the aforementioned points, we provide three major contributions:

 Industrial systems are classified as discrete event dynamic systems (DEDS) and are characterized by a complex and hierarchical structure [START_REF] Ben-Naoum | Methodologies for discrete event dynamic systems: A survey[END_REF]. In our approach, the structure of industrial system's elements is formally defined and represented in DEVS, and according to B.P. Zeigler [START_REF] Zeigler | Theory of Modeling and Simulation[END_REF]we can prove that this formalism is adequate to model of this type of systems.

 Providing a mechanism which can systematically transform the DEVS to AGR model.

 Integrating this mechanism in MAD-KIT platform in order to create a new extension for DEVS formalism. Choosing MAD-KIT platform for implementation can be justified by its advantages such as, its genericity, its independence to a specific field of application, and finally, its simplicity of implementation.

Mechanisms of DEVS to AGR transformation

The proposed Algorithm below transforms atomic and coupled model of DEVS model to agent and groups of AGR model [START_REF] Ben-Naoum | Methodologies for discrete event dynamic systems: A survey[END_REF].

The CDEVS_G (CDEVS) function creates the groups Gi, then, the procedures CDEVS_EIC,CDEVS_IC and CDEVS_EOC define the various interconnections between the agents (AGAj "agent corresponding to atomic model ADEVSj" created by ADEVS_AGA (ADEVSj) function, AGCi_R "Agent representing the input ports of coupled model CDEVSi" and AGCi_E "Agent representing the output ports of coupled model CDEVSi ") and the groups Gi according to coupling type CDEVSi as shown below. The CDEVS_AGC (CDEVSj) function creates two coordinators agents AGC for each coupled model CDEVSi, the first one is a receiver coordinator agent (AGCi_R) and the second one is a transmitter coordinator agent (AGCi_E). The main objective of those both coordinators is well to conserve the encapsulation of DEVS formalism. The model is represented as a black box that acts with environment only via the input and output ports .However, the agents have free interaction with their environment. In this approach; the agents want to communicate with internal agents of a group corresponding to a DEVS coupled model should necessary go through coordinators agents of this group.

The CDEVS_G ( CDEVSi ) function creates groups Gi for each CDEVSi coupled model; then it creates AGAj agents that it will gather them in this group Gi. Then; it creates both coordination agents AGCi_R and AGCi_E through CDEVS_AGC function. The figure 3 below illustrate a graphical representation of elementary transformation of our approach DEVS/AGR.

Fig. 3. DEVS/AGR transformation

Architecture for Simulation System

The figure 4 presents a global system architecture based on the proposed approach. The interaction module (1) allows the system agent to determinate all coupled and atomic models of system to be simulated through a set of pre-defined models in base of models [START_REF] Bertalanffy | Théorie générale des systèmes[END_REF]. This base contains a set of re-used models (The system agent can grow it progressively).

The capture of models is done through a set of files code of system simulation. Thus, the system agent may follow the simulation evolutions on the display system [START_REF] Ingalls | Introduction to simulation[END_REF] by having the possibility to interact with the system thanks to interaction module (1). The parameters are the components of equation of atomic and coupled models. An analysis of data integration is carried out in this level.

The files are saved in XML models (3). Then, the system transforms them thanks to CDEVS/AGR extension (4). The management module of agents (5) allows obtaining a set of agents and groups of agents according to AGR model. Those agents are integrated into MAD-KIT multi-agent platform [START_REF] Shannon | Introduction to the art and science of simulation[END_REF] to obtain the executable code of system simulation.

DEVS AGR

Atomic

Conclusion

In this paper we have proposed a new framework for modeling and simulation of complex systems using multi-agent platform MAD-KIT. The proposed approach is based on the DEVS formalism for system modeling and the AGR of MAD-KIT for their simulation. The main idea behind our approach is to take advantage of the formal power of DEVS formalism for verification and validation and the power of the MAD-KIT platform advanced simulation tools. The advantage of our approach is that it offers an environment for the design and development of a very flexible and extensible modeling and simulation framework. The later will allow adding DEVS formalism to the modeling formalisms already existing in the platform such as Petri Nets, UML, interaction diagrams, etc. As a perspective, we will focus on testing our extension by modeling and simulating real complex systems such as industrial petroleum systems.

Fig. 1

 1 Fig. 1. AGR model

WhereX

  is the set of inputs S is a set of states Y is the set of outputs δ int : S → S: is the internal transition function δ ext : Q × X → S: is the external transition function, where Q = {(s,e) | s ∈ S,0 ≤ e ≤ ta (s)} is the total state set e is the time elapsed since last transition. λ: S → Y is the output function t a (s) is the time advance function

X

  self is the set of inputs of coupled model. Y self is the set of outputs of coupled model. D is the set of names associated with the components of the model, self is not in D. {Md / d Є D} is the set of components of coupled model. EIC, EOC and IC define the coupling structure in coupled model. EIC is the set of external input couplings. EOC is the external output couplings. IC defines internal coupling.

Fig. 2 .

 2 Fig. 2. Graphical representation of an atomic model and a coupled model consisting of two atomic models (A) and (B)



  A DEVS atomic model can be represented by agent model. The inputs and outputs of the atomic models are represented by a set of stimuli and action of the agent. The internal transition function and the external transition function are represented by the function of behaviour and the function of perception of the agent respectively. The output function represented by the function of action of the agent. The sequences of states (S) of the DEVS atomic model are by the set of the agent's internal states.  A coupled model materializes a group. The inputs and outputs of coupled model are represented by transmitter coordinator agents. The external input couplings, the external output couplings and the internal coupling are represented by different interconnections between agents and groups of agents.

  The ADEVS_AGA (ADEVS) function allows the creation of AGAj agents for each atomic model ADEVSj. Thus, both receptors and effectors of AGAj agent represent respectively the input ports (ADEVSj. Inportsi) and output ports (ADEVSj.Outportsi) of atomic model. The function of perception AGAj.Pa corresponds to the external transition function ADEVSj.δext. The behavior function AGAj.Fa represents the internal transition function ADEVSj.δint. The action function of AGAj.Infla agent corresponds to the output function ADEVSj.λ The dynamic evolution of an agent represents time advance function ADEVSj.ta and the internal states of AGAj agent. This corresponds to the set of states ADEVSj.S.

Fig. 4 .

 4 Fig. 4. Overall architecture of the simulation system.

Table 1 .

 1 Correspondence between DEVS and AGR.


Paul Andrews
email: paul.andrews@mnd.su.se

Eva Jablonka
email: eva.jablonka@fu-berlin.de

Constantinos Xenofontos
email: constantinos.xenofontos@stir.ac.uk

Thematic working group 11: Comparative studies in mathematics education

Introduction

As with earlier CERMEs, TWG 11 adopted an eclectic perspective in its interpretation of comparison as referring to any study that documents, analyses, contrasts or juxtaposes cross-cultural or cross-contextual similarities and differences across all aspects and levels of mathematics education. In this way the TWG aimed to encourage critical but supportive discussions around a number of themes, some of which were predetermined by the papers received by the group, while others emerged from them.

A recurrent but very productive aspect of this working group has been the relatively small number of paper presentations, although this has also led to the group being constantly on the verge of extinction. This year, thirteen papers and one poster were received. Unfortunately, for various reasons, two papers were withdrawn, leaving eleven papers and a single poster. This created space not only for colleagues to share their research in detail but for everyone to engage in lengthy and inclusive discussions on the nature of comparative mathematics education research and the means by which it can be meaningfully and rigorously undertaken. In particular, as is discussed in more detail below, the group came to realise that concepts like 'unit of analysis' meant different things to different people according to the research traditions from which they derive. The various contributions fell naturally into five themes that framed the different working group sessions.

Theme 1: Assessment in mathematics education

The first theme drew on two papers with very different foci and methodological conceptualisations focused on assessment in comparative mathematics education research. On the one hand, Cascella and Giberti's paper, drawing on Italian national test data, highlighted significant gender differences in the ways in which students respond to different presentations of the same arithmetical task. Such findings remind us that the debate concerning gender differences in mathematical understanding are by no means resolved. On the other hand, Simsek, Jones, Xenidou-Dervou and Dowens' study, framed by the important assertion that if "the same instrument is used in different cultures, no bias should arise from the particular characteristics of an instrument or its administration, focused on children's understanding of mathematical equivalence. Their use of an extant American scale with English and Turkish students yielded a different factor structure from the original, highlighting the difficulties of assuming the validity of a test developed in one cultural context for use in another.

Theme 2: Mathematics teacher education

The second theme, drawing on three papers, dealt with issues concerning mathematics teacher education. The first of these papers, presented by Clivaz and Miyakawa, reported on how two groups of students, one Swiss and one Japanese, collaborated, both at a distance and face-to-face, on the planning of the same problem-solving lesson. After planning, each group of students taught the lesson repeatedly in their own cultural contexts. Analyses of the two sets of plans and their respective lessons highlighted well the cultural norms that underpin beginning teachers' practices. The second paper, presented by Gorgorió, Albarracín and Laine and written with the aim of evaluating admissions policies, examined the impact of recent changes to the admission requirements for primary teacher education programmes in Barcelona and Helsinki. The study found that such tools are too contextually situated to facilitate genuine comparison and argue for a better theorisation of the expected knowledge of such students and a measure appropriate for crosscultural application. Finally, in this session, the paper by Zhang and Wray examined similarities and differences in the sequences and patterns of teachers' questioning practices in secondary mathematics classrooms in the UK and China. Drawing on observations of a dozen teachers in each country, the authors' discourse analyses identified clear distinctions between the collectivelyfocused questioning of the Chinese teachers and the individually-focused questioning of the English.

Theme 3: The analysis of mathematics textbooks

The third theme, based on two papers and a poster, focused on the analysis of mathematics textbooks. The first paper, presented by Hemmi, Lepik, Madej, Bråting and Smedlund, examined the presentation of material focused on expressions, equations, inequalities, and the meaning of the equal sign in textbooks in grades 1-3 in Estonia, Finland and Sweden. They found considerable variation in the presentation of such material, with concomitant opportunities for the later learning of algebra, both between countries and between different language groups within countries. The second paper, presented by Xenofontos, examined the use of the word 'problem' in the state mandated upper primary and lower secondary textbooks of the Republic of Cyprus. He found, with inevitable consequences for the coherence of children's learning of mathematics, that while the word was used frequently and consistently in the primary books, it was used rarely and less consistently in the secondary. Finally, Palm Kaplan presented her poster comparison of the algebra tasks found in year 8 Swedish textbooks and the TIMSS 2003 assessments. Her discourse analytic approach showed considerable differences in the privileged content of the two sets of data.

Theme 4: Curriculum

The fourth theme, comprising two papers, focused on curriculum. The first paper from a multinational group of scholars, Van Steenbrugge, Krzywacki, Remillard, Koljonen, Machalow and Hemmi, examined recent mathematics curriculum reform efforts in the US, Finland, Sweden, and Flanders. Drawing on a range of data sources they showed that curriculum reform that is both region-wide and supported by teachers is a major challenge, not least in systems where teachers enjoy autonomy and textbooks are largely unregulated. Jablonka's study, the second in this theme and initiated by international tests of achievement, found that the mathematics taught in Germany was intellectually more challenging than that taught in China. However, bringing to the group a concept that provoked much discussion, Chinese teachers experienced greater discretionary space than their German colleagues. That is, they had more adaptive freedom in the planning of their teaching.

Theme 5: Methodology

The final theme, drawing on two papers, differed from the others in its explicit focus on methodology. The first of these, Petersson, Sayers and Andrews, presented an analysis of textbooks based on the use of moving averages to highlight the location and extent of different content emphases. Such an approach, which facilitates comparison between textbooks, offers an important visual support to conventional frequency analyses. The second, Chan and Clarke, alerted group members to the importance, particularly for comparative research, of identifying an appropriate unit of analysis. This particular presentation, despite the issue having been discussed in previous CERMEs, provoked a long discussion, which is revisited below, on how the expression 'unit of analysis' is construed differently by different cultural groups.

Discussion

As with previous CERMEs, the variety of topics and methodologies brought to the group highlighted well the diversity of work undertaken within the broader field. All papers offered insights to remind us that comparative research, which includes much mainstream work undertaken in single cultural contexts, is a messy and complex undertaking reliant on strong theoretical foundations and clearly-defined constructs. In this latter respect, the group's discussion on unit of analysis was particularly interesting as it became increasingly clear that colleagues were not talking about the same thing. On the one hand, colleagues working in a theory-driven tradition tended to construe the unit of analysis as the theoretical framework within which they worked. On the other hand, colleagues working in a problem-driven tradition, typically Anglophone, saw the unit of analysis as the individual piece of data to be analysed. These very different conceptions of the phrase highlighted well the difficulties faced by all researchers and the dangers of assuming a common understanding of what one might typically expect to be unproblematic. Of equal interest, as indicated, was the notion of discretionary space. Here, colleagues were able to see the methodological advantage such a simple yet powerfully evocative concept brings to analyses of curricula and teachers' engagement with them.

As always, the various contributions brought to the group reflected not only cultural diversity but also methodological pluralism. For example, studies included those that were informed by a priori theoretical assertions and those that were not. There were equal numbers of qualitative and quantitative studies focused on a range of aspects of children's and teacher education students' learning of mathematics. All studies confirmed the extent to which mathematics and its teaching and learning are culturally normative.