text:
Sabrina Bobsin Salazar
email: ssabrina@umich.edu

Teaching practices in a mathematics classroom and their connection to race and racism in the United States

Keywords: Racism, teaching practices, critical race theory, critical realism

In this paper I introduce critical realism to investigate the relationship between a mathematics classroom and the broader social context in which it is inserted. To ground the new approach, I focus on race and racism, and use critical race theory to guide the use of critical realism in this investigation. This is a report in an ongoing study about the work of teaching and social change with respect to race and racism in the United States. Data comes from a laboratory mathematics classroom held every summer by a large research university in the United States. Although more analysis is still necessary, initial results reveal how broader social context can influence and be influenced by broader structures of race and racism. Moreover, the framework shows potential to illuminate the relationship between classroom interactions and social systems of inequality.

The problem

This study starts from a perspective that education and schooling play a role in our society that can serve both to sustain and reproduce dominant social structures or to challenge them and promote change [START_REF] Freire | Pedagogia da autonomia: Saberes necessários à prática educativa[END_REF]. It is generally accepted that our social world is unequal and unjust, and under such a view, education is frequently seen as a source of and/or a solution to social inequalities. It is therefore important to educational research, mathematics education research in particular, to better understand the relationships between educational and schooling practices and the broader social context they are situated in.

In this study, I want to zoom in on classroom practices and investigate how mathematics classroom practices can reproduce or challenge macro social systems of inequalities, which I understand as social structures that privilege certain social groups over others. Racism in the United States is for example generally characterized by a set of social privileges rather than by individual acts of hate or prejudice [START_REF] Bonilla-Silva | Race with racists: Color-blind persistence of racial inequality in the United States[END_REF]. Even though individuals being overtly racist still exist, I am referring to a different problem, in which Whites enjoy better social opportunities only because they are White, such as African Americans being incarcerated in disproportionately higher rates than Whites (Barish, DuVernay, & Averick, 2016), and Latinxs having disproportionately fewer opportunities to obtain higher educational levels in comparison to Whites [START_REF] Yosso | Critical race counterstories along the Chicana/Chicano educational pipeline[END_REF].

The main purpose of this paper is to report the application of critical realism in an ongoing study to offer new possibilities to understand the relationship between classroom interactions and the broader social context they are inserted in. I am particularly investigating race and racism in this first exploratory study, because it brings interesting dynamics between local social situations and broader social institutions. Such dynamics come from the fact that racism is deeply ingrained in American society [START_REF] Ladson-Billings | Just what is critical race theory and what's it doing in a nice field like education[END_REF]. I hope that a critical realist lens can help to illuminate how racism occurs within classroom interactions, specifically I want to investigate how racism can be challenged or disrupted in mathematics classrooms through teaching practices. In this study, I am particularly trying to answer the research question: How can the critical realist concept of norm circles help us better understand how teaching practices in a mathematics classroom (can) challenge and/or disrupt structural racism? In this paper, I will present the framework, describing a few core concepts of critical realism and critical theory and discussing how these two theories can be combined to better explain the connection between classroom teaching practices and structural racism, and then I will illustrate the use of the framework with initial findings.

Conceptual framework

Mathematics education research has focused its interest on the social and political dimensions of mathematics and education for quite a while now. To better understand such dimensions, some researchers are now foregrounding power relationships in research and using a variety of critical perspectives and methods in what has been called a sociopolitical turn [START_REF] Gutiérrez | The sociopolitical turn in mathematics education[END_REF]. In her argument, [START_REF] Gutiérrez | The sociopolitical turn in mathematics education[END_REF] describe some of the theories used by critical scholars to address power relationships in society. Critical realism is not directly indicated in her article, but it is a possible sociopolitical theory. Critical realism is a philosophy of science originated out of the need for better theories to understand the social world; in particular, critical realism explores power relationships in the social world. Specifically the focus of this study, critical realism proposes a new approach that can illuminate the relationship between social structures and micro social interactions by elaborating an analytical mechanism that focuses on the interaction between individual agency and social structure [START_REF] Elder-Vass | The causal power of social structures: Emergence, structure and agency[END_REF]. Such a mechanism has its foundation in the critical realist concept of emergence and is operationalized in terms of norm circles. These theoretical constructs however, are not tied to any specific method to conduct research though under a Critical Realist approach. What I expect in this study is that Critical Race Theory can provide theory and methods to investigate the relationship between mathematics instructional practices and racism, and that Critical Realism can complement CRT supporting the explanation of mechanisms of (re)production and disruption of racism.

Critical realism

The basic premise of critical realism is that the world is made by real things that have real causal powers [START_REF] Bhaskar | A realist theory of science[END_REF]. Phenomena are interpreted as outcomes of causal powers of such real things. At a first glance these ideas look very similar to positivist ideas, but they are not. In positivism phenomena can be completely determined by scientific laws, whereas in critical realism, phenomena are only influenced by scientific laws. The main idea is that these laws impose constrains and prevent possibilities otherwise available, describing a tendency rather than a certain outcome. The example cited by [START_REF] Bhaskar | A realist theory of science[END_REF] is that the path of his pen does not violate any law of physics, nevertheless it is also not determined by such laws (p. 95). There is a limitation of what a pen can do that is described by the laws of physics, yet such laws do not determine what is being traced by the pen. What is important in these basic ideas is that the world, which includes the social world, is made by real things; and that scientific laws, even social laws, describe tendencies rather than determination.

One concept that is central for critical realism and that will be very relevant for this study is the concept of emergence. Here, I am particularly adopting the compositional version of emergence as described by [START_REF] Elder-Vass | The causal power of social structures: Emergence, structure and agency[END_REF]. In this version, 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. Elder-Vass (2010) also refers to this new thing as an 'entity' or whole, and it possesses "properties or capabilities that are not possessed by its parts" (p. 4). The idea is that the whole is not just the sum of its parts, but it is 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 is what forms the layered or laminated view of the world under the critical realist perspective. A particular whole is said to be in a higher level or layer 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. In our social world, an individual can be interpreted as the lowest level, and 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, in Elder-Vass' (2010) definition, a norm circle. The norm circle is defined by the group of individuals who hold a normative belief of endorsing a social practice. By endorsing, he means that each individual in the norm circle acts to reinforce the norm and discourage behavior that does not conform to the norm. [START_REF] Elder-Vass | The causal power of social structures: Emergence, structure and agency[END_REF] argues that the shared endorsement of a norm when combined with these sorts of parts, provide a generative mechanism that gives the norm circle an emergent property or causal power: the tendency to increase conformity by its members to the norm. The property is the institution and the causal power is the capability that the group has to affect the behaviour of individuals. That causal power is implemented through the members of the group, although it is a power of the group, and when its members act in support of the norm, it is the group (as well as the member concerned) that acts. (p. 124) With this argument, Elder-Vass ( 2010) is explaining why the norm circle is actually an emergent structure rather than only a group of people. He is explicitly pointing out what is the new causal power by showing the tendency it describes: to increase conformity to the norm.

Particularly relevant for this study is how agency is viewed in this critical realist account of the social world in which the social wholes with causal powers are norm circles. It is important to consider that causal power in critical realism describe tendencies, so the fact that a norm circle enforces compliance with a particular norm indicates that someone in this norm circle will have the tendency to act in conformity to such norm, but this is not determined. With respect to the focus of this study, race and racism, and the guiding provided by critical race theory, a central aspect of this investigation will be, in a context of racialized circumstances, the situation of an individual being in different and conflicting norm circles, i.e. individuals that are part of norm circles that enforce opposite norms, and how the study of this conflicting positions can reveal routes for social change.

Critical race theory

Critical race theory is a theoretical framework for research that foregrounds race, racism, and racialized experiences. Critical race scholars ground their argument in the idea that racism is a social construction and is different than individual prejudice. Gloria Ladson-Billings (1999) reports that race is a complex social construct that goes beyond the color of skin, citizenship, and individual acts of prejudice, forming what can be called a system of racial inequality usually hidden under a colorblind discourse [START_REF] Bonilla-Silva | Race with racists: Color-blind persistence of racial inequality in the United States[END_REF]. To overt the practices of colorblind racism, critical race scholars anchor their work on five tenets: permanence of racism, Whiteness as property, interest convergence, critique of liberalism, and counter-storytelling [START_REF] Decuir | So when it comes out, they aren't that surprised that it is there: Using critical race theory as a tool of analysis of race and racism in education[END_REF]). I will briefly describe three of these tenets because they will be more salient to the preliminary results to be presented later on this paper.

Whiteness as property refers to the idea that that Whiteness can be viewed as a set of social (privileged) possessions, that can operate similarly to property in a capitalist society. In the context of education, Whites have some sort of control of what is valuable knowledge and who gets access to it, which can be interpreted as a kind of intellectual property. Critique of liberalism is a direct critique to liberal economic-based ways of understanding and living in the world, grounded in freemarket ideologies, under which people believe that best outcome for all is achieved when there is no external regulation of the market. Meritocratic and individualist discourses are frequently associated with liberalist discourses [START_REF] Solomona | The discourse of denial: How white teacher candidates construct race, racism and 'white privilege[END_REF]. Counter-storytelling is the main methodological strategy used by critical race scholars to challenge inequality and White privilege. One important aspect of counter-storytelling is the double-consciousness or angled vision attributed to individuals living in the margins of society [START_REF] Anzaldúa | Borderlands/La Frontera: The New Mestiza[END_REF] that is going to be one of connections between critical race theory and critical realism.

Before I discuss the issue of marginality in society in light of critical realism and critical race theory, I will briefly elaborate in another connection between the two theories. Such connection will not be the focus of this study, yet it is necessary to understand why the two theories are compatible and suited to be thought together. [START_REF] Ladson-Billings | Just what is critical race theory and what's it doing in a nice field like education[END_REF] says that racism describes a norm in current American society and "because it is so enmeshed in the fabric of our social order, it appears both normal and natural to people in this culture" (p. 12). This idea is the gist of permanence of racism. So, in a critical realist account of it, there must be a norm circle enforcing and endorsing racist practices. To present one example of a norm in this circle, I will need to unpack the idea of discourse within the critical realist framework.

Dave Elder-Vass (2012) elaborates on the concept of discourse as discussed by Michel Foucault (1969). He emphasizes the idea that Foucault is concerned with the content of what we express using language norms. Moreover, he refers to Foucault asserting that there are normative practices that dictate what we can say and what we should not say, and, in some way, they also dictate how we should act. Elder-Vass (2012) constructs a realist ontology explaining how discourse in this sense can have causal powers. He argues that discourse is shaped by the normative rules enforced by members of a discursive circle through the discourse such members produce. Discourse, therefore, is the means to what the causal power of a discursive circle exerts causal effect.

With this idea of discourse, I will point the emergence of a racist norm circle to the slavery system in the colonial period. As an example of a racist norm originated in the colonial United States, I point to the normative discourse that says "African American are less than Whites". Specifically in educational contexts, such a discourse is reported in the autobiography of the former slave Frederick Douglass (1892) as a way to justify slavery. Once the norm circle reinforcing this discourse had emerged, it started to operate downwards, constraining the individuals in the circle to act accordingly. The way the discourse is reinforced has changed throughout time. For example, in the beginning of 20 th century, IQ tests [START_REF] Karier | Scientists of the mind. Intellectual founders of modern psychology[END_REF] helped to disseminate the idea that Blacks were less intelligent than Whites; and recent research reports such as Robert [START_REF] Berry | Access to upper-level mathematics: The stories of successful African American middle school boys[END_REF] reveals that African American boys are more likely to be placed in lower track courses in mathematics, in comparison to their White peers, as they advance their studies.

Critical race scholars ground their argument in this kind of normative racism in opposition to individual and overt prejudice. Theoretically, they understand this kind of racism as a social construction that brings real consequences to people of color [START_REF] Chapman | You can't erase race! Using CRT to explain the presence of race and racism in majority white suburban schools[END_REF]. This view gains a total new meaning under a critical realist perspective: Social constructions are real and have real causal powers. Now, focusing on norm circles and searching for routes for social change, I will discuss the doubleconsciousness or angled-vision usually explored through counter-storytelling by critical race theory [START_REF] Anzaldúa | Borderlands/La Frontera: The New Mestiza[END_REF]. This is usually a characteristic attributed to individuals that live in the margins of society. This place is viewed as a space of conflict of identity, a space of belonging and not belonging at the same time [START_REF] Anzaldúa | Borderlands/La Frontera: The New Mestiza[END_REF]. In a critical realist perspective, I interpret the angle vision as a product of participating in conflicting norm circles. It is at this conflicting space that critical realism leaves room for human agency to act in a way that might not conform to a social norm. [START_REF] Elder-Vass | The reality of social construction[END_REF] discusses that when an individual participates in two or more conflicting norm circles, the outcome in terms of individual behavior can be very poorly predicted in the sense that the individual can decide for either norm, or can even create an innovative action to escape the ambiguous situation:

In contexts of complex normative intersectionality, skilled social performances depend upon the possession by the individual of a sophisticated practical consciousness of the diversity, applicability and extent of the normative circles in which they are embedded, and indeed of others to which they are exposed, even though they may not be parts of them. Whether or not they are able to articulate this consciousness discursively, members of such societies depend upon it whenever they act. (Elder-Vass, 2010, p. 133) My idea in this study is to explore the norm circles with respect to race and racism that exist in a mathematics classroom environment. Particularly I want to investigate what are the norm circles the teacher participates in, what are norm circles created in the context of the classroom interaction, and how particular teaching moves are interpreted in light of agency within complex normative intersectionality.

Methods

I am conducting my research as a secondary case study on a summer program held by a large research university in the United States, in which an experienced teacher publicly teaches lessons to a group of elementary students. This summer program serves different purposes: one is to be a site of learning for practicing teachers and teacher educators, because of the nature of the public teaching and the professional development sessions that follows it; another is to be a site of research for both student learning and teaching practices.

The student body of the laboratory is composed by a sampling of students from one school district in the Midwest United States. It is made to represent the demographic distribution of this district. Its composition counts with students of different ethnicities, but mostly are African American; all (or most) students come from low-income households; the students have different levels of English proficiency; and their mathematical proficiency is homogenously low.

Because the EML is a site of research for student learning and teaching practices, different types of data are collected by the research team organizing the EML. 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, etc. Since I am analyzing data already collected, I did not engage in a relationship with the participants. I am observing and respecting their voices the best way I can by triangulating different sources of data. The data corpus is composed by video records of classroom interaction (approximately 2.5 hours per day across 10 days), detailed lesson plans for each class, copies of student work (notebooks, homework, and assessments), and photos of every collective record produced in classroom (such as charts and white board records). The high quality documentation of the laboratory classes allows detailed observation of classroom interaction that are usually not possible in regular school settings. Also, the composition of the student body, the qualification and experience of the teacher, and the laboratory setting provide a fruitful environment for the existence of multiple norm circles operating in this same space, which is important for this research.

I am following an analytic induction approach [START_REF] Erickson | Qualitative methods in research on teaching[END_REF] to code and interpret the data. The videos are being watched with a focus on interactions that involve students of color. Written records similar to fieldnotes are produced and used to identify episodes. Each episode is then re-watched; better-detailed descriptions are produced and interpreted according to the conceptual framework. In the next section I will illustrate how the episodes are interpreted in light of the conceptual framework. I will focus particularly on episodes that feature three Black girls.

Illustrating the framework

The initial analysis was focused on the interactions involving three Black girls, and findings show they can engage in mathematics discussions, and are interested in learning mathematics. These findings firstly challenge the idea that girls, specifically Black girls, are not suited for mathematics.

In one example, one Black girl is behaving in a way that could have been interpreted as if she was not listening to what the teacher was discussing about the conditions of a problem they were about to work on. She was not looking to the teacher at the board and she was not quiet in her seat, but she promptly raised her hand when the teacher asked for a "wrong answer". She correctly shared a solution to the problem that violated one of the conditions, making it a wrong answer. This example reveals a Black girl accessing mathematical content, usually a White intellectual property [START_REF] Moses | Radical equations: Civil rights from Mississippi to the Algebra Project[END_REF]. This girl, however, was only accessing such content because the teacher ignored her apparently disruptive behavior and believed in her raised hand. The calling of the student to share her answer was a choice of the teacher; she had other students with raised hands to call at that moment, still she decided to call on that Black girl. She was at that particular moment deciding her action based in different norm circles, one that told her the student was not paying attention to the lesson, and another that told her the student knew something and wanted to share with the class. Initial findings also reveal that the teacher consistently engages in what [START_REF] Boaler | Promoting 'relational equity' and high mathematics achievement through an innovative mixed-ability approach[END_REF] calls assigning competence. This means that the teacher calls a student to publicly share an answer to a problem to raise their mathematical status among their peers. To do that, the teacher specifically points to an aspect of the proposed solution and indicates why that aspect is mathematically relevant. In one example, the teacher called on a Black girl to share how she had recorded an answer to a mathematics problem in her notebook. The teacher focused in how she wrote the answer using complete sentences, and that writing complete sentences foster writing clearly and is an important mathematical practice. This girl was not positioned as a competent mathematics learner or doer, other students in class frequently did not listen to what she was publicly sharing. In this example, the teacher had to intervene so all students were actively listening while she was showing her notebook. The students were also asked to comment what was good in her answer. With this move, the teacher raised her status as a mathematics learner and doer among her peers. The teacher was actively pushing back some kind of liberalist practice in classroom. In a liberalist practice, the teacher might pursue a meritocratic ideal and let the classroom "regulate itself", without intervening in the social relationships being (re)produced there. By raising the mathematical status of some students, in particular two Black girls, the teacher might be changing the power relationships being established among students, which can promote some sort of local socio-mathematical justice. By consistently engaging in assigning competence, this teacher is actively challenging liberalist norm circles.

Concluding comments

The implementation of this new framework still brings methodological challenges. As it stands, the framework shows potential to illuminate the relationship between mathematics classroom practices and the broader context they are inserted in. The theoretical articulation of critical race theory and critical realism can help to refine both theories, providing methodological tools to apply critical realism in empirical research, and providing new analytical tools to understand how racism can be challenged. Particularly, the first example showed how the concept of norm circles was helpful to interpret a teacher action that counter racist discourses, and the second example show how a teaching practice can be used to counter liberalist practices that promote racism. I expect that such initial results can be elaborated with the further development of this study, providing more refined analysis. Finally, I expect that this study, when completed, can provide a more detailed description of classroom interactions in light of the social context they are inserted in.

Proceedings of CERME10


domain: math shs

text:
Elisa Bitterlich
email: elisa.bitterlich@tu-dresden.de

Judith Jung

Marcus Schütte

Subjective theories of teachers in dealing with heterogeneity

Keywords: Diversity, heterogeneous grouping, teacher beliefs, group discussion

This article presents initial results of a research project which investigates subjective theories and typical action strategies of teachers and student teachers in dealing with heterogeneity in school with a focus on the subject mathematics. These results are ultimately intended to contribute to the development of future teaching approaches. To reconstruct subjective theories of teachers and student teachers, group discussions were carried out. The initial results show the aspects of heterogeneity the participants deem important and the possible actions they discern for coping with pupil diversity.

Fundamental theoretical considerations

The fact that children and young people often differ in terms of their needs and preconditions for learning, and that this heterogeneity of learners sometimes presents teachers with significant challenges, are not new phenomena [START_REF] Trautmann | Heterogenität in der Schule. Eine kritische Einführung[END_REF]. The heterogeneity of learners relates to different dimensions, such as cognitive performance, age, gender, linguistic-cultural background, social class and many more (Hinz, 1993). In accordance with current political discourses and social developments, these different dimensions are given unequal attention in pedagogical discussions. The impetus for a renewed focus on heterogeneity in Germany was provided by the results of international comparison studies (in particular PISA, 2000), which highlighted especially the sizeable differentiation in pupil achievement, the alarmingly high number of very-low-achieving pupils, and a close relationship between social background and academic success [START_REF] Trautmann | Heterogenität in der Schule. Eine kritische Einführung[END_REF]. The UN-Convention on the Rights of Persons with Disabilities, which came into force in Germany in 2009, made the inclusive schooling of children with and without disabilities the subject of renewed debate. In addition, the phenomenon of increased linguistic-cultural differences among learners has come into focus in the last two years by the increased number of refugees entering Germany. For most student and practising teachers the heterogeneity of learners represents an important problem area in planning and teaching lessons which seems to be complex and fundamental. [START_REF] Askew | Diversity, inclusion and equity in mathematics classrooms: From individual problems to collective possibility[END_REF] expounds that the teacher's ways of thinking and talking about heterogeneity impact how they react to the differences that learners bring to the mathematics classroom. The following questions arise: "Why have so many, essentially wellfounded pedagogical ideas not been realised? What prevents teachers from seeing heterogeneity as enriching, and dealing with it productively?" (Trautmann & Wischer,p. 9, translated by the authors). In order to answer these questions and create concepts for future seminars, not only a scientific reflection of this topic is supposed to be considered. Especially the perspective of those facing heterogeneity daily in their pedagogical work is to be included by analysing their subjective theories about heterogeneity, too.

In the framework of the Germany-wide "Qualitätsoffensive" for the improvement of teacher training, the project "Synergistic teacher education in an excellent framework" 1 at the TU Dresden includes the sub-study "Heterogenität in der Lehrerbildung von Anfang an" (Heterogeneity in teacher training from the start). Based on qualitative questionnaires, group discussions, and participatory observations of everyday teaching in schools, subjective theories 2 and predominating patterns of action among teachers and student teachers will be surveyed in different kinds of school. On the basis of the survey results, the project intends to develop concepts for teaching events to make student teachers sensitive to the different facets of heterogeneity.

The concept of 'Heterogenität' (heterogeneity) is defined in various ways in the relevant Germanlanguage scientific literature, and indeed is often used without specific definition. A number of terms are used synonymously, ranging from 'Vielfalt' (plurality) to 'Unterschiedlichkeit' (difference), 'Unbestimmbarkeit' (indeterminableness) and 'Beliebigkeit' (arbitrariness), or English words like 'diversity' 3 . In many scientific articles, the focus is placed on only one aspect of heterogeneity (such as language, culture, gender, or disability), and the relevant definitions insinuate a polarisation between 'normal people' and 'the others'. However, authors like [START_REF] Prengel | Pädagogik der Vielfalt. Verschiedenheit und Gleichberechtigung in Interkultureller[END_REF] and [START_REF] Krüger-Potratz | Intersektionalität[END_REF] articulate a different understanding of the concept of heterogeneity. We share this understanding of the concept of heterogeneity, which finds placing the focus on a few 'dominant' characteristics to be a reductive approach [START_REF] Krüger-Potratz | Intersektionalität[END_REF].

Diversity education is based on the 'indeterminability of people'; it is therefore unable to diagnose 'what somebody is' or 'what shall become of somebody'. It [diversity education] opposes all reification in forms of definitions of what a girl is, or a boy, a behavioural deviant, a Turkish woman… If people must be characterised, then this must be based on their dynamic development and the context of their environment. (Prengel, 2006, p. 191, translated

by the authors)

To analyse teachers' subjective theories, and to develop concepts for teaching events that are based on the teachers' views, against our understanding of the concept it nevertheless appears sensible to establish a theoretical categorisation of some of the individual facets of heterogeneity. Some of the existing studies on 'beliefs', as well as on teachers' implicit or subjective theories about heterogeneous contexts in school, show a focus on selected individual aspects of heterogeneity in this way. On the aspect 'heterogeneity', in an interview study on belief systems of primary-school 1 This project is part of the "Qualitätsoffensive Lehrerbildung", a joint initiative of the Federal Government and the Länder which aims to improve the quality of teacher training. The programme is funded by the Federal Ministry of Education and Research. 2 In the literature, concepts such as subjective theories, implicit theories, or naive theories, as well as teachers' ideas and attitudes, or "beliefs", are sometimes used synonymously in accentuating different aspects, and cannot be clearly differentiated (Törner, 2002). In this paper, we will discuss teachers' subjective theories based on Heymann's (1982, p. 146, translated by the authors) understanding of "the totality of knowledge elements and orientations affecting teachers' actions in the conducting of lessons".

3 While in the English speaking area, the Term "diversity" is preferred and well-known, in Germany many publications tend to use only "heterogeneity". But in some more actual publications we can see that "diversity" is more and more a common term for describing the plurality of pupils.

Thematic Working Group 10 teachers working with children with special educational needs concerning cognitive development, Korff (2014) identifies a central challenge for didactic activity in mathematics teaching in the establishing of links between different approaches and varying levels of representation. On the aspects of gender and ethnicity, the ProLEG study4 addresses the question of how ethnic-cultural and gender-related perceptions influence teachers' educational activity [START_REF] Winheller | Professionalisierung von Lehrkräften für einen reflektierten Umgang mit Ethnizität und Geschlecht in der Grundschule[END_REF]. The results show that the respondents overwhelmingly see questions of children's gender as unimportant and considered themselves to be sufficiently competent in this area. In relation to ideas about ethnicity, the respondents attached equally low importance to 'Intercultural Education'.

Topics such as 'Individual Support', 'Social Learning', and 'Inclusivity' are given primary importance, while culturally sensitive approaches have the lowest priority, followed by 'genderaware education'. (Winheller et al., 2012, p. 10, translated by the authors) However, [START_REF] Zobrist | Zum Umgang mit Heterogenität auf der Sekundarstufe i am Beispiel des Mathematikunterrichts[END_REF] investigation attempts to approach heterogeneity from a broad perspective, without restriction to particular aspects of the concept. Using semi-structured interviews and 'simulated recall' in addition to teaching observation, the author attempts to produce a comprehensive view of the ways teachers deal with heterogeneity in mathematics teaching in secondary school. The results show that teachers tend to define heterogeneity especially in terms of different preconditions for learning and different kinds of social behaviour. Furthermore, special educational needs assessment is seen as highly relevant in dealing with diversity, but personal competences in this area are considered inadequate. [START_REF] Schönknecht | Heterogenität aus der Perspektive von Studierenden oder der Wunsch nach Normalität[END_REF] point out that in describing heterogeneity student teachers often seem influenced by an idea of polarisations and dichotomisations, as well as a limited perspective focusing on supposed 'problem children'. Also in relation to perspectives on the dimensions of heterogeneity, little differentiation is evident, with the use of a number of generalising (stereo)types (e.g. 'normal' and migrant children). They summarise: Thus, addressing the construction of normality contributes to dealing with heterogeneity and difference, and can be a significant building block towards realising equality of opportunity in school. (Schönknecht & de Boer, 2008, p. 258, translated by the authors) Regarding to Garmon (2004), both dispositional factors (like openness to diversity as well as receptiveness to others' arguments and ideas, self-awareness and self-reflectiveness, and commitment to social justice) and experiential factors (like intercultural experiences, support group experiences with individuals who encourage another person's growth and educational experiences) influence teachers' attitudes towards diversity.

Method

In order to gain a first impression of student teachers' ideas about the concept of 'heterogeneity' in the school context and to discover how they encounter diversity among children, a qualitative questionnaire with three questions requiring written answers was distributed to around 80 student teachers training to teach mathematics. The students answered the following questions in their own words in running text or bullet points: What does the word 'heterogeneity' make you think about in the school context? To what extent have you addressed this topic (in your studies)? To what extent do you feel prepared to deal with heterogeneity in the school context?

In addition, we held group discussions with student teachers 5 on the topic of 'heterogeneity'. The group discussions were video recorded and transcribed and evaluated using the documentary method [START_REF] Bohnsack | Documentary method and group discussion[END_REF]. Group discussions can help to identify and analyse the implicit or tacit knowledge of the participants while they talk about a specific topic (e.g. heterogeneity). Between three and seven participants talked about a given topic for around 60 minutes. They have some special experiences in common or commonalities in their history of socialisation, thus sharing a "conjunctive space of experience" [START_REF] Mannheim | Structures of thinking[END_REF]. The momentum of the discussion process, uninterrupted by the researcher, is important to discover these conjunctive spaces of experience, which become visible through "focusing metaphors" in which the group adjusts itself to those specific topics that are most relevant in their common experience [START_REF] Bohnsack | Documentary method and group discussion[END_REF].

Concerning group discussions, the immanent meaning comprises that stock of knowledge which is made explicit by the participants themselves. This has to be distinguished from knowledge of experience, which is so much taken for granted by the participants that it must not and often cannot be made explicit by themselves. The participants understand each other because they hold common knowledge without any need to explicate it for each other. (Bohnsack, 2010, p.103)

Results

For reasons of space, this article will highlight a few clear results of the qualitative questionnaires as well as extracts from the overall transcript of the group discussions with teacher students.

Qualitative questionnaire

The students gave highly diverse answers to the question "What does the word 'heterogeneity' make you think about in the school context?" While one respondent (female, 23 years, fourth semester) answers only with a few key words ("diversity, differentiation, boys and girls, high-achieving, lowachieving"), other students give more complex answers, making clear their awareness of the unbounded, indefinable nature of the concept: Every class is different (age, background, etc.). Every child therefore has different preconditions for learning, which one should include in the teaching. Differentiation is important (natural differentiation, internal differentiation, external differentiation). The application of learning environments to enable different approaches (with different difficulties/materials, etc.). (female, 21 years, sixth semester). This student's response also suggests how she would deal with diversity among children and where she thinks particular emphasis should be placed. On the question of how well prepared the students 5 Student teachers in this way includes teachers who have finished their university degree and are now in training for one or two years before receiving their final teaching license and students who want to become a teacher and who still have lessons in university.

Thematic Working Group 10 feel to deal with heterogeneity in the school context, the majority of responses are sobering. Most of the students complain about a lack of practical experience, stating that although the university education in many respects provides a lot of theory, there are few opportunities to reflect upon the ideas and to try them out in practice. Furthermore, it is criticised that not enough attention is paid to the topic of heterogeneity (overall) in the study course; it is often covered quickly as a "marginal topic", but not "dealt with in depth" (female, 22 years, fourth semester). On the basis of their experiences in the course of their studies, several of the students differentiate between the different teaching subjects; for example, one female student (21 years, sixth semester) remarks: "I feel better prepared in maths than in German. For example, by the '(Maths) Learning Under Conditions of Heterogeneity' course".

Extracts from the group discussions

The participants in the group discussions presented below were student teachers in training for primary-school and high school with the subject mathematics. The students are in the middle to last phase of their studies or have finished their university degree and are in training before receiving the final teaching license, meaning that all have already completed placements in schools. During the 60-minute discussion on the topic "What experience do you already have of diversity respectively heterogeneity among children in school and in teaching?" it becomes clear that those facets of heterogeneity that are dominant in the social discourse, such as native language, disability, social status, achievement and migration background also dominate the students' discussions.

After a group of five female teacher students for primary school have talked about topics such as German as a foreign language and the meaning of academic language, one of the participants turns to the topic of inclusivity and the schooling of children with special educational needs in regular teaching. The following extract is a part of the discussion that develops on this point. The following transcript extract (the original version is in German) shows how the students encounter diversity among children, the challenges and opportunities they see in such diversity, and what experiences they have already gained in dealing with it.

Tina:

I also think it's very important how children gain another view of what is actually normal. A person sitting in a wheel chair is just as normal and can also move around. And that simply this acceptance and tolerance can develop amongst each other. That you simply know how to deal with the person and that it becomes natural from an early age on.

Bianca: I think this is also easier for children. I also always like that about children that they very openly go to other children who are a little bit different. I also think that this should be encouraged but it is also a fact that there are also mentally disabled children. I don't know if they are also affected by inclusion? Sarah: Yes.

Bianca:

Well, I think that is difficult. Well, I was at a school for children with special needs and sat in on classes and I thought it was really bad.

Sarah:

Well, otherwise, in front of the same class plus children with special needs that is not possible, I think. Then also structurally things would have to be changed.

Diana:

Well, for all of us it is a challenge to stand in front of a class after finishing our studies. Even if they are top-performing and are all a relatively homogenous group.

Tina: But you will never have this homogeneous group (smiles and shakes her head).

Diana:

(nods) Yes. You also don't have that in society. The whole society is extremely heterogeneous.

Extracts from a discussion between four female student teachers for high school shows a similar view on heterogeneity, however with a greater focus to specific problems of the subject mathematics.

Wiebke:

The heterogeneity of the teachers also effects the lessons and what the children learn in the end. Therefore, also the teacher's competence of explaining.

Nathalie: (Laughs) Yes, especially in mathematics.

Tamara: And I also think what attitudes the teacher has towards heterogeneity. Meaning, is my attitude that I take everyone along or do I only take the top 50 percent along? Or drastically said, what is my opinion about somebody from a migrant background? That also plays a huge role.

Vera: Yes, that's true.

Wiebke: I mean, at the university this is addressed but how I should really deal with it … It is nice to say that you need difficult tasks for those who are good and easy ones for those who are not so good at it. Yes, but in the end, all of them take the same test and are marked according to the same grading system.

A third extract is from a group discussion of four female primary-school student teachers which have already been in training at school for a few months. This brief passage of the discussion shows that student teachers who are already teaching in school seem to be more aware that there are differences between the theory they have learned in university courses and the dealing with heterogeneity in real life. Additionally, they critique some aspects of the education in university.

Linda:

The only option is individualized teaching, if you want to give every single child the chance to take part in the lesson and to have fun.

Beate:

And you have to accomplish this without straining oneself.

Isabel: Exactly! And I would like to know, how that can work (laughs). How can I differentiate without constantly feeling stressed at home? Linda: Well, when I was at university, I often thought, "Bla, bla, heterogeneity, differentiating. How can this work?" They [the university teacher educators] always treated this like a big cloud but they never told us specifically. And then, in school, you think, "Well, how does this work?" And only through experienced teachers you understand "Oh, this is how you can approach this!" And it doesn't always have to be three different worksheets. A more open form can work as well.

But I think that in university it was something which existed somewhere up high in the universe but which cannot be implemented.

Denise:

In my point of view, such opened instruction was seen [in university] as a kind of sanctuary and I always thought, "No. I can't do it. I somehow am not able to do it at all!" Because it is a Utopia to do this. But in the end, it is somehow possible and [at university] one should have used that as a starting point. Even though we heard keywords like "weekly schedule work", we never spoke about this in depth.

It was more like scratching the surface.

Isabel: Then we received lots of academic texts about this which we were supposed to read. Afterwards I knew as much as beforehand because the time to really understand the content was too short. This was easier with conversations. But I did not have a concrete plan either.

Beate: But now, in the courses for trainee teachers, we recognized that we all open our lessons. We do not carry out frontal teaching as we always imagined. That's why it is helpful to have somebody with experience, who has stood in front of children for many years and who was able to teach this to us in a normal way.

It becomes clear during the group discussions that the students are aware of the problem of how to judge children fairly despite potentially enormous diversity. They principally discuss the questions of how fairness can and should look in the school context, how it can be realised, and which obstacles and problems can exist in its realisation. They argue that differences play a secondary role for children and that "it is easier for children" to accept and tolerate each other because they are "a lot more open in that". The participating students seem to agree that it is important to develop a broader perspective of what is 'normal' as early as possible. At the same time, they consider it to be difficult to hold collective lessons for 'normal' children and children with the need for special education, especially when there is only one teacher in class. Furthermore, they are aware that there is a dilemma between the need to differentiate in school and the society being extremely heterogeneous. While in a school context, every child is supposed to receive the best possible support suitable for its own needs, in society, diversity is nearly never considered or discussed.

Perspective

The results collected so far already provide initial insights into student teachers' subjective theories and guiding ideas on heterogeneity in the school context. Further group discussions will be carried out in the near future, and questionnaires distributed. The framework will be expanded to include practising mathematics teachers in different kinds of school in Saxony. Thus, data on teachers' and student teachers' subjective theories on diversity in pupil populations will be available for comparison. Consequently, individual teachers who participated in the group discussion will be selected to receive classroom visits. Through participatory observation we hope to be able to undertake a comparison between the collective opinion emerging in the group discussion and the models of activity that are actually applied by teachers for dealing with diversity. All these data will ultimately serve the development of concepts for events for student teachers with the aim of making them more sensitive to heterogeneity and more prepared to deal with it.

Proceedings of CERME10

"Professionalisierung von Lehrkräften für einen reflektierten Umgang mit Ethnizität und Geschlecht in der Grundschule" (Professionalisation of Teachers for a Considered Relationship with Ethnicity and Gender in Primary School) Thematic Working Group 10 Proceedings of CERME10


text:
Anette Bagger
email: anette.bagger@umu.se

"It is only a test" -social aspects of displaying knowledge in mathematics for second language learners

Keywords: Second language learner, assessment in mathematics, opportunity to display knowledge

This article discusses social dimensions connected to assessment in mathematics for second language learners in Sweden. The data consist of two semi-structured interviews with students in the ninth grade of compulsory school. Foucault's thinking on discourse and positioning was advocated as a frame for analysis. The units for analysis were students' statements about caring and the other in connection to the display of knowledge in mathematics. Results show that caring of and for others are important resources in managing assessment and believing in the future.

Introduction of the problem area

Measures of achievement are often situated as measures of quality in education (Lundahl & Tveit, 2014) that promote striving towards high quality, yet also threaten equity at times [START_REF] Mendick | Does every child count?: Quality, equity and mathematics with/in with neoliberalism[END_REF]. An example of this are recent educational reforms including earlier and extended testing in Sweden [START_REF] Regeringen | Uppdrag till Statens skolverk att föreslå mål att uppnå och nationella prov i årskurs 3. Dnr U2006/8951[END_REF]. These reforms have led to an enhanced focus on the measures of knowledge, while at the same time school agencies generate reports on inequalities in the measured knowledge and grades in mathematics between schools and groups of students. Differences are connected to gender, class and ethnicity (e.g., Skolverket, 2015). Achievement in national tests is central in grading as the results of the national tests in mathematics often are used to indicate students' grades [START_REF] Skolverket | Redovisning av uppdrag om relationen mellan provresultat och betyg i grundskolans årskurs 6 och årskurs 9[END_REF]. Not having an approved grade in mathematics at grade nine in Sweden means that a student does not have access to public programs at the upper secondary school in the following year. The opportunity to display knowledge then becomes a critical point of departure for the individual's possibilities for positive development and positioning in mathematics. This turns the assessment grade in mathematics into a gatekeeper to get access to higher education, and is a gatekeeper that keeps out second language learners (SLL) more often. This portrayal of circumstances alludes to societal, historical, social and political discourses influencing an individual's mathematical development and life-choices by making some positions available and other positions not available. Researchers are currently engaged in issues of for whom education functions and thereby which students can have access to success in education and life [START_REF] Au | Devising inequality: A Bernsteinian analysis of high-stakes testing and social reproduction in education[END_REF][START_REF] Peters | Achieving quality and equity through inclusive education in an era of high-stakes testing[END_REF]. The focus on disadvantaged groups of students then affords new ways of understanding and approaching mathematics education [START_REF] Gutiérrez | The sociopolitical turn in mathematics education[END_REF]. Alternative ways of relating to and understanding assessment are needed and in this, listening to the students as opposed to only labelling them, are of core importance [START_REF] Hodgen | Mathematical 'ability' and identity: A sociocultural perspective in assessment and selection[END_REF]. Research that is paying attention to students' stories also contributes to knowledge about the lived and social dimensions of assessment that needs to be paid attention to according to [START_REF] Black | Inside the black box: Raising standards through classroom assessment[END_REF].

In this article, I strive to recognize and study the production of (in)equity in connection to grading and national assessment in mathematics. I highlight aspects that might be foreseen in the general debate and work in the area of mathematics education. Through this, I contribute to the identification of possible resources that could promote paths for increased equity. In regard to the concept of equity, I draw on [START_REF] Boaler | Promoting respectful learning[END_REF] concept of relational equity. In a school where participants (do not) learn to respect differences and each other in the mathematics classroom, relational equity is (not) lived and taught. This understanding can be applied on both the micro-, meso-and macro-levels of education and opens up studying equity beyond the gap-gazing of diverse achievement levels in mathematics (e.g., [START_REF] Gutiérrez | A "gap-gazing" fetish in mathematics education?: Problematizing research on the achievement gap[END_REF]Rodrigues, 2001) something that actually might counteract equity and hold disadvantaged groups of students behind [START_REF] Gutiérrez | Beyond gap gazing: How can thinking about education comprehensively help us (re)envision mathematics education[END_REF]. Following from this point of departure, resources for producing equity are to be found in lived, diverse, relational and social aspects in the process of assessment and displaying knowledge. I contribute to the identification of some of these resources through the study of social and relational aspects connected to assessment and the opportunity to display knowledge in mathematics for SLL. In this paper, I specifically contribute to the act of listening to and exploring experiences of students within subordinated groups by highlighting experiences of SLL in connection to displaying knowledge in mathematics. The purpose of this paper is to contribute to knowledge about social dimensions of displaying knowledge for SLL in mathematics and how this can be related to future prospects connected to the subject. The investigation examines three research questions: Q1) How do statements about the other and caring appear in talk of displaying of knowledge? Q2) How do statements about the other and caring appear in talk of future prospects connected to the subject? Q3) What discourses are activated and what positions are available in the students' talk? Here, caring refers to both care for oneself and for/from others.

Theoretical framework

A statement works as a mediator of knowledge and truth that exists in a field of power-relations and is embedded in discursive formations with other statements [START_REF] Mcilvenny | New perspectives on discourse and governmentality[END_REF]. Discourses are understood as governing and positioning individuals through power and knowledge [START_REF] Foucault | The essential works of Foucault, 1954-1984[END_REF], and positioning is understood as "the discursive process whereby selves are located in conversations as observably and subjectively coherent participants in jointly produced story lines" (Davies & Harré, 2001, p. 264). Therefore, analysing students' statements will reveal discourses that are activated in connection to the display of knowledge and assessment in mathematics. In order to capture social aspects of displaying knowledge in mathematics, the role of others and caring for/by others are used in this project [START_REF] Black | Mathematics as caring: The role of 'others' in a mathematical identity[END_REF]. [START_REF] Black | Mathematics as caring: The role of 'others' in a mathematical identity[END_REF] have shown that these phenomena may be powerful cultural resources in shaping a positive identity in mathematics. [START_REF] Black | Mathematics as caring: The role of 'others' in a mathematical identity[END_REF] have drawn on Bakhtin. Instead, I use the concepts as signal-words, which reveal the representation of a social and lived aspect of displaying knowledge in mathematics.

Method and selection

This paper presents the analysis of two interviews with SLLs. Their names are fictional. Amina and Ahmed are 15 years old and have both struggled with their learning in mathematics since third grade. They did not pass several of the goals in mathematics in their third and sixth years and have had special support during this time. Amina achieved the lowest passing grade (E) in her ninth grade and was given support in the form of special instruction. In the ninth grade, Ahmed did not receive any special support or adaptions in mathematics and got a D, which is the average grade. The data were collected right before the final choice of program to the upper secondary school was made and the final grade in compulsory school was given. Amina and Ahmed had just finished their last national tests in mathematics. Interviews conducted on this occasion were assumed to contribute to a concentration of their experiences of displaying knowledge and assessment. This selection of students is meant to bring specific questions concerning assessment in mathematics for SLLs into the forefront.

The interviews were semi-structured. This approach was used to promote the student's possibilities to talk freely and to display as much of their understanding and experience as possible (Kvale & Brinkman, 2014). The students could talk about anything they chose in the areas of support, assessment, national tests, grades, mathematics and the future. For the most part, I asked open-ended questions to follow each theme. Questions were, for example: why do you take these tests; what did you think/feel; did you talk about it to anyone and how; was it possible to get help; if so, how? These open-ended questions were followed up with more specific questions connected to what the participant had expressed, in order to get as good an understanding of and rich information about their experiences as possible. The positions and activated discourses were constructed through analysing statements about caring and the other in the context of displaying knowledge in mathematics. Key markers in students' talk were statements involving others as for example peers, friends, family and school-staff and statements regarding caring about or being cared for by others. These statements also had to relate to assessment and/or displaying knowledge in mathematics. An interpretative reading of statements was done back and forth in order to identify the discourses and positions involved. For this purpose, an adaption of [START_REF] Foucault | Vetandets arkeologi[END_REF] description on how to find discursive formations was used: 1) First, statements regarding caring and others were identified. 2) Secondly, the form of these statements was described. 3) Thereafter, the relations between these statements were described and the correlations and contradictions between these statements were explored. 4) Then, the statements were grouped and the correlations and contradictions between these groups were explored. 5) In the final step of the analysis, the discourses were construed.

Ahmed and Amina: Interview data

The interview data connected to the first two research questions are presented here. This presentation derives from the first four steps of the analysis. Overall, these statements concerned family or peers, and notably teachers were not mentioned at all in connection to statements concerning caring and the other.

Statements about the other and caring in talk of displaying of knowledge

The students talked about care of and from others primarily in relation to peers and as peers as a point of reference for the achievement or possibilities to succeed. Ahmed mentions that the girls talked about when the tests were, and, in a way, that could make each other nervous: "The girls are like: it is mathematics (national test) tomorrow, tomorrow! They mentioned it several times" (transcript 005). But he was not nervous himself but rather preferred to take it all in time and put the test into a larger context of living: "If I make it, I make it... There is no point in worrying, life will continue anyway" (transcript 005). Even if he was at ease, he expressed concern about a friend who did not manage Swedish well enough in relation to the support given and the construction of the tests. The friend was very good in mathematics in his homeland but after coming to Sweden he almost did not pass: "He is not so good at Swedish so he thought it was a Swedish word he asked for help with. The teacher could not do anything but read it aloud again. In his homeland, he had like a high grade and here he barely passed" (transcript 005). Ahmed talks about care of himself in relation to effort and outcome on the tests: he is at peace himself with not being able to solve all the tasks, since they are constructed for all levels of difficulty. This circumstance also makes it hard for him to know if he passed the test, "I might think it is hard but I am on an E, the ones being on an A may not think it is so difficult. So, I would say it was ok, even if it was hard for me" (transcript 005). In this way, Ahmed refers to peers as a point of reference. Ahmed says that he made a deliberate choice not to study before the test: "I have myself to blame if it went bad, I accept my choices" (transcript 005). Amina talks about care of herself in relation to her knowledge in the subject, her effort and grades. She thinks mathematics is hard but does not think the grades reflect her experience of the subject as interesting and of herself as someone who is interested, learns and works hard: "I think math is easy, or easy, it is hard but I think it's fun. What comes out shows in the grade… I put a lot of effort in math but I do not get good results. It does not show in the grade. It makes me feel disappointed, but at the same time it challenges me" (transcript 002).

Statements about the other and caring in talk of future prospects

The students' statements regarding the future prospects are often connected to the family's care about them. Both students expressed that parents and relatives had high expectations and beliefs in them and their engagement in mathematics. Amina connects the big expectations she has for herself to her parents' expectations: "I think my expectations come from mum and dad, they expect big things from me" (transcript, 002). The family stressed that they should do what they could to enter upper secondary school. For example, if Ahmed did not get the lowest approved grade, the family would encourage him and not let him give up: "They would be grumpy with me and they would think that I should go back and keep on fighting and not stop" (transcript 005). In particular, Ahmed's brothers had given him advice on how he should choose a program at the gymnasium in relation to mathematics and also had given him a good trust in upper secondary school, the mathematics involved in the program he chose and the teachers: "I have lots of expectations since I have a family from whom I have taken like a lot of advice. All have said very good things about the school and the one (brother) who studied construction has said a lot of good things about a teacher working there" (transcript 005). He also compared and talked about his siblings and how they succeeded and what they had done in their time at the upper secondary school: "You know, my brother, he says that there are three days of practice a week and that you get to learn a lot out in the field. Three weeks before finishing school he was offered an employment… He is 19 and he has a job" (transcript 005). Motivation was in this way connected to talk about parents' and relatives' anticipation of and belief in them and their engagement in mathematics. Although Ahmed could feel that they nagged at him, he understood and appreciated the advice to put effort into the learning in mathematics: "I understand their arguments and so and I really appreciate that they help me there and I understand the point. It seems to be important to get a grade in math" (transcript 005).

Statements about the other and caring in talk of future prospects

Care about themselves in connection to the future were expressed in relation to belief, struggle, worries and seriousness. Ahmed worried about the test a great deal afterwards, if some of the harder tasks would deprive him of his grade, his time in the upper secondary school and stop his journey in life and companionship with friends: "This is life, this is it. I would be very disappointed if I did not pass. Then I will miss a whole year… I do not want to wait a whole life for life to continue" (transcript 005). The students meant that future choices may be limited depending on their knowledge, which made them both choose a program at the upper secondary school with a low level of mathematics. Statements about expressing care for others concerned peers and primarily gatekeeping functions in the assessment in mathematics, but also their own learning in positive anticipation regarding their ability to develop. Amina talks about her peers as participants in discussions about the grades, something that has been intensified over time as it is connected to mathematics as the gatekeeper to the upper secondary school: "We did not talk as much about it (earlier) but more later. That it is the grades that decide if we get into upper secondary school" (transcript 002). Here, Ahmed points towards the gatekeeping function in the national assessment of mathematics, which is worrying: "If you fail in math, then you are done… That is why I have been lying awake at night and thought about the test" (transcript 005). Both students anticipate that math will be hard in the upper secondary school but they are confident that they will learn. Amina says she is eager and ready to take on the challenge: "I am going to study at the upper secondary school and I have to be prepared that mathematics is the hard thing. I am very excited" (transcript 002). Both Amina and Ahmed are very confident that they can learn the mathematics they need when they finally begin the upper secondary school. Ahmed for example states "but I think that when I finally go to the construction program I will learn it, how to count with area and stuff" (transcript 005).

Analysis

The analysis answers the third research question and explores the discourses that are activated and the positions available. A discourse on managing assessment (connected to statements about peers), a discourse on progress (connected to statements about family), a discourse on future challenges in mathematics (connected to care about oneself) and a discourse on fairness (connected to care about oneself), were construed from the analyses of statements connected with caring and the other in connection to displaying knowledge and mathematics. The activated discourses led to some available positions for the students.

The discourse on managing assessment concerns support, comparison and monitoring of support, grades and tests. Ahmed talked more about his peers and talked overall more than Amina. Amina referred to peers as a help in focusing on the grades and Ahmed expressed care for others. He then positioned both himself and his peers as disadvantaged test-takers due to language and the settings and construction of the national test. He also positioned girls as more nervous in their monitoring of test occasions.

The discourse on progress circles about responsibility, advice and expectations stemming from the families. Expectations were then blended with demands on focus and progress. Hard working and you can if you want to were positions connected to the family discourse. These positions had connotations of personal responsibility, achievement and future prospects.

The discourse on future challenges in mathematics held statements in which caring about oneself was connected to a position in which limitations in knowledge blended with striving to learn and the outcome that learning was to be conquered.

The discourse on fairness connects the student's individual responsibility, effort and knowledge to the achievement and assessment in mathematics. If the effort is made, the knowledge should be retrieved and following from that, the grades should be accordingly high or low. In the discourse on future challenges in mathematics both students positioned themselves as capable of trying and working on improving, although within certain limits. A position of struggling while learning was identified, this position is possible to connect to the hard-working and you-can-if-you-want-to positions in various combinations. In the discourse on fairness, a position of choosing your achievement was shown in Ahmed's talk as he chose not to study and accepted the consequences. Amina also spoke out from a discourse of fairness as she questions the grade, and that the hard work should have been seen in the grade. At the same time, she capitulates and says that it is hard for her to learn and remember for example the methods to useso the grade may be fair after all. This could be described as a position of being unable to succeed.

Discussion

The semi-structured interviews contributed to a trustful and open climate for conversation. This made it possible for the two students to display important social dimensions of displaying knowledge and learning mathematics. One example of this was when Ahmed talked about how his brothers and family supported him: "I took their advice… It seems to be important to get a grade in math" (transcript 005). This happened on an occasion when he was actually skipping school for a day and hanging out with his brothers. This occasion proved to be an important moment in his positioning as a mathematics learner. The main concepts in the analysis were statements about the other and caring (also see [START_REF] Black | Mathematics as caring: The role of 'others' in a mathematical identity[END_REF] In many ways, it is possible to assume that there are many differences between Roz, the adult mathematician in Black et. al.'s (2015 study and the two students in this study when it comes to opportunities to learn and display knowledge. What they have in common is that the female mathematician and immigrant students are groups both governed by different types of gatekeeping functions in their access to the subject. Interestingly, the same socio-cultural resources as found in Black et. al.'s (2015 study, seem to work well in illuminating prerequisites for a positive development of identity in mathematics among students that have struggled with their learning. The statements about assessment in mathematics and the future were in many ways a narrative drawing on the community of the family and the peers. This could be a sign that relations outside school, in families or between peers, are important resources in the building of a positive identity in and a relation to the subject mathematics.

Both students in this study were willing to learn and develop their skills in mathematics, although they knew that they were in some way limited because of their lack of knowledge at some times. Although struggling with mathematics, they still had a positive way of approaching the subject, which is not always the case for students in need of support in mathematics after nine years in school. The families' expectations and talk with their children about the future and the role of mathematics in it may have contributed to a discourse about struggling when learning. This discourse could contribute to a positive identity in mathematics rather than devaluing the struggling student as a learner in mathematics. This research contributes to knowledge about the social dimensions of testing for SLL. Social and lived dimensions of assessment may get lost in translation if the measures on achievement are interpreted without taking the social, cultural, political and relational contexts into account. Conclusions from assessment that mainly focuses scores and levels of achievement might reveal differences and tendencies of segregation or lack of knowledge but without affording means to counteract these inequalities. Therefore, more research in the socio-political area of mathematics education is needed. This paper joins the socio-political research in mathematics education as I contribute to knowledge about social dimensions of displaying knowledge in mathematics for disadvantaged groups of students. The aim was to identify valuable resources in providing access to and success in mathematics for all students. I also emphasise an alternative way of understanding assessment beyond measures of knowledge and quality, namely as a means of promoting social and relational aspects of becoming more mathematically able. Since, as the students' statements revealed, it is (not) only a test of knowledge but also an occasion of caring about oneself, caring for others, and being cared about by others.

Proceedings of CERME10


text:
Melissa Andrade-Melina
email: melissa.andrade.mat@gmail.com

Incepted neoliberal dreams in school mathematics and the 'Chilean experience'

Keywords: Governmentality, neoliberalism, historization, rhizome, school mathematics

This work aims at portraying a rhizome of circulating naturalized truths about who citizens should be and how they should act within neoliberal governmentality. It does this by a historization of an incepted belief entangled in diverse social spheres. It unfolds how the ideas of human capital and welfare become a top right in mathematics education. The 'Chilean experience' is used as an example to construct a rhizomatic historization of events, strategies and technics of government that enabled the inception of neoliberal dreams into school mathematics.

Introduction

An idea. Resilient... Highly contagious. Once an idea has taken hold of the brain it's almost impossible to eradicate. An idea that is fully formed-fully understood-that sticks; right in there, somewhere (Dominic Cobb, Inception).

It is intriguing how highly perceived the Chilean models are-economy, education or health systems-to other countries. According to [START_REF] Taylor | The reformulation of social policy in Chile, 1973-2001: Questioning a neoliberal model[END_REF], Chilean systems have been taken as models 'worthy of emulation'. Is Chile doing something marvellous? The country has been seen as an example of organization and 'proper' policies for economic progress and welfare [START_REF] Silva | Capitalist coalitions, the state, and neoliberal economic restructuring: Chile, 1973-88[END_REF]. Its policies are considered as trendsetters among privatized pension systems (see [START_REF] Mesa-Lago | Introduction: Background, structure and method[END_REF], among health care reforms (see [START_REF] Bruce | The Chilean health care reforms: Model or myth[END_REF], and it was one of the first countries implementing neoliberalism as a framework in education [START_REF] Aravena | Neoliberalism and education on an international perspective: Chile as perfect scenario[END_REF]. The results in PISA, particularly in mathematics literacy, have progressively increased over the years-2000 (384); 2006 (411); 2009 (421); 2012 (423). And so, Chile has risen to be seen as one of the most developed countries in Latin America [START_REF] Gregorutti | What if privatising higher education becomes an issue?: The case of Chile and Mexico[END_REF]. Chile is considered, by the World Bank's annual reports on development, the proven example of the benefits embedded in 'conforming' to a neoliberal approach to social policy.

[Chile] is often viewed as a trendsetter in introducing fundamental and far-reaching neoliberal reforms […] the Chilean example as been heralded as proof of the success to be gained from an uncompromising commitment to neoliberal policy prescription (Taylor, 2003, pp. 21-22) But… it is not all sunshine and roses! By building on Foucault's work, this paper aims at portraying how neoliberal discourses about mathematics education have been (re)produced and how they have circulated amongst diverse spheres of human interaction, (re)shaping citizen ways of being and acting in the world. It does this by taking "a critical attitude towards those things that are given to our present experience as if they were timeless, natural, unquestionable" (Rose, 1999, p. 20). This paper deploys a historization of the present of entangled historical events, strategies and techniques that made possible to incept the neoliberalism into school mathematics in Chile. This narration is not a critique about the implementation of educational policies in Chile; rather it is the tracing of naturalized truths in mathematics education as an assemblage of diverse governmentality techniques [START_REF] Foucault | Governmentality[END_REF].

These naturalized truths are traced in five moments. First, regarding the introduction of neoliberalism as a set of political movements. Second, regarding neoliberalism as a system of reason for economic improvement. Third, regarding the specific type of citizen that the new economy requires, a consumer of goods. Fourth, regarding the productive subject of schooling for the market, a competitive subject. And finally, regarding how school mathematics becomes the vehicle to shape the desired subject for economic growth.

The plot of the movie "Inception" inspires the style of writing of this paper. In this movie a series of dreams are unfolded. Each dream should be understood as a new and deeper dream occurring inside the previous one. The dreams do not follow a chronological arrangement. It is not a lineal story; it is a rhizomatic construction. A rhizomatic network allows a non-hierarchical multiplicity of entryways, of dimensions, lines; it has no beginning or end, but always a middle: "The rhizome is altogether different, a map and not a tracing […] The map is open and connectable in all of its dimensions; it is detachable, reversible, susceptible to constant modification" (Deleuze & Guattari, 1987, p. 12). Hence, all dreams are connected not as a sequence of events, but as continuities and discontinuities. All narrations are entangled, even in different times, even in different spaces, and even in different voices. The paper is written in this form in alignment with Foucault's rejection of causality.

We consider the understanding of the way one event succeeds another as a specifically historical issue, and yet we do not consider as an historical issue one which in fact equally so: understanding how two events can be contemporaneous […] History is quite frequently considered as the privileged site of causality […] But we have to rid ourselves of the prejudice that history without causality would no longer be history. (Foucault, 1999, p. 92)

First dream: The Cold War and the neoliberal revolution

It is the late 60s, in a country apparently far from the War, but close enough to be in the spotlight.

There is the danger of it becoming the first socialist nation in South America, and this is threatening for the US. Silent voices were saying: "under no circumstances should Allende be elected!" But, he was… Salvador Allende became the first democratically elected socialist president in the Western hemisphere. What a revolutionary! Fighting for the people! Chile has begun to increase its role in the provision of social services.

By subsidising the reproduction of the labour force through allocating resources to the development of state systems of health, education, housing, staple-food subsidies and social insurance, universalistic social policies tended to reinforce the purchasing power of wages thereby expanding domestic markets for industrial goods. (Taylor, 2003, p. 23) Something is starting to go extremely wrong in Chile. Suddenly there commenced a crisis that led to most of the population clambering for improvement. The, so-called, socialist experiment "united capitalists, landowners, the middle classes, and their political party allies against labor, peasants, and leftist part" (Silva, 1993, p. 535). Apparently the US government, also pressuring the World Bank and the Inter-American Development Bank to do the same, minimized the aid they provided to Chile… And so, Chilean foreign reserves plunged from $400 to $13 million in one year [START_REF] Moreno | Silent revolution: An early export from Pinochet's Chile[END_REF]. Discontent people wanting the president out are growing in number. It is socialism versus capitalism… "$7 million channelled to anti-Allende groups", according to a report of the US senate (Moreno, 2008, p. 93). And he was overthrown on September 11 th 1973, by the military force commanded by General Pinochet. Now, neoliberal ideas are being forced into Chilean minds that are afraid all the time, afraid for their lives, afraid to raise their voices. Meanwhile, those in favour of the new regime are enjoying the pleasures of the new order [START_REF] Salazar | Historia de la acumulación capitalista en Chile[END_REF].

Second dream: The Chilean experiment, Friedman and the School of Chicago

The year is 1950; the place, Chicago. Milton Friedman is developing a new approach to economy theory. This new theory is in opposition to socially conscious economies, which have been prominent in Western governments after 1929. Friedman believes that "economic benefit could best be optimized if the individual has the autonomy to pursue his or her own self-interest" (Moreno, 2008, p. 92). This new theory was the hope for a group of technocrats that moved to Chicago, the "Chicago Boys" [START_REF] García | Chile: A laboratory for failed experiments in capitalist political economy[END_REF]). In the 70s, Pinochet decides to leave the economical management of Chile on the hands and knowledge of the Chicago Boys. This is going to be the first time that a group of Friedman has "an opportunity to influence governmental policy and put their theories into practices […] They already have a complete programme aiming to re-structure the economy and to reverse Allende's social reforms" (Moreno, 2008, p. 94). The military regime and the Chicago boys established neoliberal economic and social policies here [START_REF] Salazar | Historia de la acumulación capitalista en Chile[END_REF]. "[T]he market supplanted state intervention in the economy, except in labor relation" (Silva, 1993, p. 527).

Within the first six years of dictatorship, the 'shock therapy' was the only approach to curb social policy and state expenditure [START_REF] Huber | Options for social policy in Latin America: Neoliberal versus social democratic models[END_REF]. Chilean reform "has been led by both the advocates of monetarism, located principally in US institutions and universities, and by the Chilean reformers themselves" (Taylor, 2003, p. 22). Neoliberal ideas were taken as a sort of 'second independence' and, also, an entrance to the first world of developed countries [START_REF] Salazar | Construcción de estado en Chile[END_REF].

Third dream: Consumerism as the ever-growing economy

Here, in this place of earth, everything could be marketed, everything could be sold, and most people would feel the urge to buy it. Health and education are, by constitution, social rights to every citizen. But here, those basic social rights fade into consumer goods. Public and private enterprises competing with each other, providing services for customers willing to pay for them, after all it is their choice [START_REF] Taylor | The reformulation of social policy in Chile, 1973-2001: Questioning a neoliberal model[END_REF]. Parents have the opportunity to choose freely the type of school-municipal, subsidized private or fee-paying private schools-and the type of education they want for their children [START_REF] Mizala | School performance and choice: The Chilean experience[END_REF]. Free choice… if they can afford it! Public against private institutions… In a place where private institutions have the right to charge in excess to ensure better and better quality. Private schools enjoy, without any guilt, "having greater resources, enabling a stronger quality of education to be taught, and thereby reinforcing the desire of parents with available income to send their children to such schools" (Taylor, 2003, p. 34). After all, the more you pay the better you get; the less you pay the worst you obtained. In a time and place where education policies are transformed into economic policies of education [START_REF] Castiglioni | The politics of retrenchment: The quandaries of social protection under military rule in Chile[END_REF].

Fourth dream: Competitiveness in schools, education and freedom of choice

After the introduction of 'welfare' as a method to increase efficiency, "the element of competition and the response of enterprises to public desires as indicated by market forces were suggested to create an optimal allocation of resources throughout welfare provision" (Taylor, 2003, p. 26). The reform of the 80s, under the military regime, changed Chilean education system. Decentralization was key to encourage private providers to enter the market [START_REF] Mizala | School performance and choice: The Chilean experience[END_REF]. And there was more, so much more than that. This reform involved a reformulation of the interplay between state and schools, a voucher system that indirectly funded schools by assigning the resources to students [START_REF] Parry | Theory meets reality in the education voucher debate: Some evidence from Chile[END_REF]. This measurement left schools receiving financial aid depending "on the number of students that they could attract […]. If schools were unable to compete in this new marketplace environment, they would be allowed to fail and face dissolution" (Taylor, 2003, p. 33). A highly competitive system generated by an educational market and by the policies aiming at improving the quality of education [START_REF] Mizala | School performance and choice: The Chilean experience[END_REF] was shaped. And so, state accountability systems were able to reward and/or punish schools by allocating resources regarding the performance of each school (Elacqua, Martínez, Sontos, & Urbina, 2012). A system in which, schools, teachers, students are constantly competing and being assessed.

Fifth dream: The sky is the limit! Mathematics to the people Welfare and mathematics, always hand by hand. Here, mathematics has been granted with a great importance and status. In the 60s, logic was taken as the foundation of every science, reasoning accurately and rigorously was the core of any argumentation and of critical thinking [START_REF] Diaz | Matemática Básica 101[END_REF]. Mathematics was the one that helped to develop reasoning and logical thinking and reading proficiency was thought as a tool to better understand mathematical instructions (Ministerio de Educación & CPEIP, 1967). In the 80s, the military regime reformed the curriculum and school textbooks to reflect the regime's doctrine: "education was recast to promote studies functional to the new productive structures of Chilean society, whereas traditional arts and humanities studies were discouraged" (Taylor, 2003, p. 32). It was indispensable to embody in individuals certain knowledge skills-mathematical knowledge-, and attributes to facilitate the creation of personal, social and economic well-being (OECD, 2001). Economic growth was about human capital.

National assessment started to be taken as the key to achieve economic progress, on the one side, to test current policy changes, on the other, as a mean to set standards. And so, competitiveness and accountability, within school mathematics testing, led to higher performances, higher incomes, higher social mobility and welfare (OECD, 2014). Nowadays, by knowing students' numeracy proficiency in PISA it is possible to predict, amongst many others, their likelihood of being employed or to calculate how different their hourly earnings would be (OECD, 2015). And so, the promised state of welfare is side by side with mathematics proficiency. Mathematics is now the key for a brighter future, all students have to do is to be good at math and the sky will be their only limit! [End of dream 5]

The standardized test SIMCE has been a key element to promote competitiveness and pressure to the system. Since its results are publicly published, it becomes an objective indicator to assess school performances (Mizala & Romaguera, 2000, p. 393). It also enables parents, as consumers, to demand better services for their children [START_REF] Meckes | Two decades of SIMCE: An overview of the National assessment system in Chile[END_REF], for students to be successful and entrepreneurs.

[End of dream 4]

School mathematics is now an investment! Reforms have shaped education into a capitalist marketplace, by promoting entrepreneurial profit-minded investment and by remodeling education to consolidate the productive structures of economy [START_REF] Taylor | The reformulation of social policy in Chile, 1973-2001: Questioning a neoliberal model[END_REF].

[End of dream 3]

And the so-called 'economic miracle', product of the economic growth in the late 70s, helped raising the prestige of neoliberalism "under the banner of 'the Chilean model'" (Taylor, 2003, p. 25). By now, Chile has become famous for its neoliberal restructuring followed under General Pinochet [START_REF] Silva | Capitalist coalitions, the state, and neoliberal economic restructuring: Chile, 1973-88[END_REF][START_REF] Aravena | Neoliberalism and education on an international perspective: Chile as perfect scenario[END_REF].

[End of dream 2]

This is it! Chile is no longer an underdeveloped country [START_REF] Salazar | Construcción de estado en Chile[END_REF]. Chile is now part of the first world, the "tiger" of Latin America [START_REF] Teichman | The politics of inclusive development. Policy, state capacity, and coalition building[END_REF].

[End of dream 1]

Incepted neoliberalism

You create the world of the dream. You bring the subject into that dream and they fill it with their subconscious (Dominic Cobb, Inception)

From a Foucaultian perspective, conduct is governed through diverse techniques, strategies, and devices [START_REF] Foucault | Governmentality[END_REF], within a space of government that "is always shaped and intersected by other discourses" (Rose, 1999, p. 22). In doing so, each individual conducts him/herself by (re)shaping his/her own modes of being and acting in a space of 'regulated freedom' and under a promised state of welfare. In this sense, "people are governed by and through their own interests" (Cotoi, 2011, p. 113). This is precisely the idea behind the 'inception' of a neoliberal mentality. A set of naturalized truths circulating amongst diverse times and places, knitting a web to govern the self and to regulate habits and desires of cultural and historical subjects through school mathematics.

These discourses help governing productive citizens, in the sense that intend to insert subjects in regulatory practices that (re)shape their conduct "without interdicting their formal freedom to conduct their lives as they see fit" (Rose, 1999, p. 23). Reforms, according to Dussel (2003, p. 94), "have to be understood as part of government technologies that intend to shape the way people are to act, think, and feel about the world, that combine the old and the new in unique ways".

One possible narrative to understand the success of neoliberalism in Chile could be grasped through the articulation of certain discourses about consumerism and competiveness. SIMCE in mathematics, for example, became the first step of knowledge consumerism and of a marketable education/society. SIMCE's results are publically published in national newspapers and widely discussed through other means of public communication, so parents and society could judge schools by their performance in standardized tests. 'Judge' in the sense of deciding which school is the best option for their children's future. This marketing of schools and teachers leads to the most utopian non-sense practices. For example, within the belief that welfare is only achieved by a high quality education, parents, in order for their children to be enrolled in those schools with "higher quality"-with good scores in national tests-are willing to stay all night in line, outside a school, to submit the admission application. Figure 1 shows a Chilean newspaper, Las Últimas Noticias, reporting the news: "Parents slept on the street under -3,6 o Celsius because of enrolment. They are trying to enroll their children in pre-school for next year in Santa María School in Osorno". One of the parents, who waited in line for 12 hours said: "It is demeaning but what else can we do. This school is good and affordable. I have three kids and they all need to study". These discourses are not isolated from the ones of school mathematics. The importance that mathematic literacy has within OECD's indicators help move research to be about how to improve students' performances in mathematics. For example, in order to be successful in SIMCE, students should not be allowed to miss classes. If students are "absent 9 days during the school year (the sample average of absences) reduced performance by at least 23% of the standard deviation of the score on the SIMCE mathematics test" (Paredes & Ugarte, 2011, p. 199). Welfare can also be measured in relation to students' performances in national tests, by correlating SIMCE scores in mathematics to predict a student's future income [START_REF] Bharadwaj | The gender gap in mathematics: Evidence from low and middle-income countries[END_REF].

The Chilean Ministry of Education released the "Learning standards" in school mathematics to help teachers evaluate "what students should know and are able to do for displaying, in national tests, appropriate levels of achievement" (MINEDUC, 2013, p. 4, my translation). These learning standards categorized students in three levels of achievement that, at the same time, predict their future outcome in SIMCE in mathematics. So, if students do not want to be label at the lowest level, they have to engage in regulatory school practices, they have to compete with their classmates and with themselves. In this fashion, SIMCE in mathematics also operates as a technique to generate 'selfentrepreneurs', "individuals that self-regulate, self-direct and are continuously in a process of redefining their competences" (Cotoi, 2011, p. 116).

As portrayed within the dreams, Chilean neoliberalism have been (re)producing discourses that circulate within diverse time and places in order to obtain economic growth, progress and welfare through school mathematics. School mathematics, since it was thought to shape productive citizens, was taken as the key for Chile to become a developed country. Mathematics needed to be a good that people wanted and were willing to consume. With marketable school mathematics, whomever wanting to achieve welfare would have to pay for higher quality. And, therefore, Chilean economy should increase. This would not have occurred without the dictatorship and Friedman thoughts: Economy would be best optimized if people have the freedom to pursue their own self-interest.

Figure 1 :

 1 Figure 1: News about parents staying all night outside a school

Proceedings of CERME10

Acknowledgments

This research is funded by the National Commission for Scientific and Technological Research, CONICYT PAI/INDUSTRIA 79090016, in Chile. I would like to thank TSG10's leader, co-leaders and participants for their comments to previous drafts of this paper.