domain: math shs

text:
Swapna Mukhopadhyay
email: swapna@pdx.edu

Brian Greer

Towards cultural responsiveness in mathematics education

Keywords: Culturally responsive, mathematics education, multiculturalism, educational politics, assessment

We describe ongoing work on the Culturally Responsive Elementary Mathematics Education (CREME) project, in which we work with teachers and students in two schools with contrasting student populations and communities. We present core principles of our emergent theoretical framework as we partner with the teachers and students to realize (in both senses of 'understand' and 'make happen') what culturally responsive elementary mathematics education might be. The backdrop for this activity is the educational/political arena within the United State, in particular in Oregon where we work, and in the schools themselves. We outline future development of CREME, and end with what we see as the implications of our experience with CREME for the ethical and political responsibilities of educational researchers.

Introduction

As an extension of the paper presented at CERME9 (Mukhopadhyay & Greer, 2015) we describe our continuing work on the Culturally Responsive Elementary Mathematics Education (CREME) project, at the core of which is a long-term exercise in teacher development, embedded in the schools and their communities.

We are not doing research in the narrow sense of stating research hypotheses, gathering data, conducting analysis. On the one hand, the paper provides a "thick description" (Geertz, 1973, pp. 5-6) of the evolution of a community of practice [START_REF] Lave | Communities of practice: Learning, meaning, and identity[END_REF], and, on the other, describes an effort to actualize a conception of culturally responsive elementary mathematics education (Greer, Mukhopadhyay, Powell, & Nelson-Barber, 2009) in a particular environment. As such, it can be characterized as raw material contributing to an emerging theory, elements of which we describe. As a development from the CERME9 paper, we devote more attention to educational politics within the United States, in particular the state of Oregon, making the argument that research cannot be considered politically neutral, and we raise meta-theoretical issues about the ethical and political responsibilities of researchers.

Principles and values of CREME

CREME is very much work-in-progress, but many influences are clear. Within mathematics specifically, we acknowledge Critical Mathematics Education (albeit fuzzily defined), and Ethnomathematics as major inspirations. Within critical education more broadly, we have worked to establish the centrality of mathematics (not always recognized) within multicultural and intercultural education (e.g., Greer & Mukhopadhyay, 2015).

Of particular relevance is the concept of "Funds of Knowledge" (Gonzalez, Moll, & Amanti, 2005) which is "based on a simple premise: People are competent, they have knowledge, and their life experiences have given them that knowledge" (Gonzalez et al., 2005, p. ix). As an example, we studied Sandoval-Taylor's (2005) account of her development of a curriculum module for a second grade bilingual class composed mainly of Native American and Hispanic students, on the theme of local building construction. We later had the opportunity to talk about this work with her when she visited our class.

More recently, a related concept termed "Funds of Identity" has been developed [START_REF] Esteban-Guitart | Funds of identity: A new concept based on the funds of knowledge approach[END_REF], introduced by the authors thus (p. 31):

We use the term funds of identity to refer to the historically accumulated, culturally developed, and socially distributed resources that are essential for a person's self-definition, self-expression, and self-understanding. Funds of knowledgebodies of knowledge and skills that are essential for the well-being [sic] of an entire householdbecome funds of identity when people actively use them to define themselves.

This statement provides a concise rationale for the emphasis we put on identity work in CREME and in this paper.

From the above, it will be appreciated that CREME, both in terms of the development of a theoretical framework and as a form of political activism, is very much work-in-progress, guided by beliefs and values, such as a commitment to asset pedagogy and to valorizing diversity in all its forms (Mukhopadhyay & Greer, 2015). The very different natures of the two schools described below is illustrative of the diversity pointed out by [START_REF] Skovsmose | Towards a critical mathematics education research programme[END_REF], relating not only to ethnicity, class, language, forms of life, and so on, but also to: variety of sites for learning mathematics (in contrast to the stereotype of the prototype mathematics classroom (p. 345); variety of forms of mathematics in action; variety of educational possibilities. The two schools represent two examples of the demographic and sociopolitical variety in schools within the US and, mutatis mutandis, anywhere else. Accordingly, the concept of culturally responsive mathematics education (Greer et al., 2009) must be adapted to the particular contexts of schools and communities. In the case of School A, one major focus of our work is support of cultural identity for the children who are recent immigrants, often as refugees. We have become acutely aware of the tensions for immigrants between the pressing issues of adapting to a new society and the practicalities of day-to-day existence, and the maintenance of cultural identity. (Other kinds of tensions for non-dominant groups in relation to dominant groups were discussed in [START_REF] Mukhopadhyay | Cultural responsiveness and its role in humanizing mathematics education[END_REF]. In School B, two important aims are to enable the children who are White to become more aware of cultural diversity, and to model the use of mathematics as a tool for addressing issues of social justice. These contrasting foci are illustrated in activities being carried out at the two schools, as discussed below.

National political background

In [START_REF] Mukhopadhyay | Cultural responsiveness and its role in humanizing mathematics education[END_REF], we sketched the political landscape for mathematics education globally and within the United States. No matter where you are, developments within the US are likely relevant because of the extent to which they directly or indirectly influence what is happening in your country. One point of this paper, accordingly, is to offer an example of how mathematics education and realpolitik interact within a particular context. Readers will recognize parallels with what is going on in their own contexts.

The importance of demographic changes within the United States is made clear by the fact that White children (as officially defined) now constitute less than 50% of the school populationwhile it is still the case that about 80% of the teachers are White. The legislation called No Child Left Behind that was passed in 2001 did address diversity among students in that it required the reporting of performance on standardized tests broken down by ethnicity, a major change welcomed by, for example, African-American organizations as a means to identify and then address inequity. However, the subsequent information on differences in standardized test scores (generally referred to as "achievement gaps", a term that connotes a blatantly deficit model) was instead used to label schools, teachers, and students as "failing" without providing resources to reduce the differences; instead, the data provided fodder for denigration of teachers and the furtherance of privatization (encrypted as "choice"). Now NCLB were has been replaced by the Every Student Succeeds Act (ESSA), a change that can be characterized as rebranding [START_REF] Karp | ESSA: NCLB repackaged[END_REF], with the continuing requirement to administer standardized tests to all students each year from Grade 3 to Grade 8. At the same time, there is a growing national movement against the excessive, inappropriate, and expensive use of standardized testing [START_REF] Hagopian | More than a score: The new uprising against high-stakes testing[END_REF]).

An overwhelming political fact is the increase in wealth and income inequality over recent decades in the United States and many other countries (OXFAM, 2017;[START_REF] Piketty | Capitalism in the 21st century[END_REF]. Education, far from the conception that it should act as a force to reduce such inequality through offering something like equality of opportunity, is now acting as an accelerator, as has been clearly shown in the United States by the statistical analysis carried out by Reordan (2011). In the period 1979-2009, the gap in academic achievement (as conventionally measured) between poor and rich children grew by about 40%.

For many reasons, the election of Donald Trump as president magnifies all of these issues, particularly given his choice as Secretary for Education of a billionaire supporter of the privatization of schools. All indications are that education will become, to an even greater extent, a driver of economic inequality and a generator of corporate wealth.

Local political background

One potentially positive aspect of ESSA is that it grants more power to states to make decisions. It remains to be seen whether this degree of autonomy will be sufficient to counteract federal policy under the new president.

In 2014, the Oregon Department of Education (ODE) issued a call for proposals for the Culturally Responsive Pedagogy and Practices Grant whose stated mission was to work towards formation of a culturally responsive teaching force in the state. CREME was the only project funded within this initiative focused on mathematics.

In June 2015, an extremely progressive and constructive statement on assessment, titled A new path for Oregon was published (Oregon Education Investment Board, Oregon Education Association, & Oregon Department of Education, 2015). This document represents the work of a very broad coalition that worked together over an extended period of time and consulted widely with teachers and others. It clearly shows the influence of [START_REF] Stiggins | Revolutionize assessment[END_REF], who acted as consultant.

From our perspective, A new path for Oregon is particularly notable in the attention it pays to "culturally responsive assessment". It is stated unequivocally (p. 8) that:

A successful system of assessment should not simply highlight problems or generalize about groups; nor should it ignore conditions that influence performance. Instead, a successful system of assessment recognizes the myriad strengths of various learners within their respective communities and within the collaborative nature of the classroom. In addition, such a system is culturally responsive, and implemented by teachers who are assessment literate. (Emphasis added).

As just indicated, the report also highlights the concept of "assessment literacy", not just for teachers and students, but also for families, community members, educational officials and policymakers, andperhaps most importantly of allpoliticians. The Oregon Education Association (teachers' union) is actively leading further efforts to develop quality assessment, and we anticipate working with them and other important actors in exposing and undoing the negative effects of excessive standardized testing, and in developing culturally responsive assessment. Mathematics is of particular importance in these efforts, given the prominence that it is accorded within standardized testing.

At the time of writing, a bill is being promoted in the Oregon Senate that would require educational state agencies to ensure that educators are providing culturally responsive education.

CREME schools, students, and teachers

The project is based in two urban schools in Portland, Oregon. Though less than three miles apart, the schools are contrasting in many respects. One (which will be referred to as School A) serves a very diverse population of children speaking more that twenty languages at home, many of whom are recent immigrants and came to the US as refugees, with indications that many have experienced trauma [START_REF] Sottile | The newest Portlanders: Meet the refugees from the most violent places on the planet[END_REF]. The other school (School B) is a public charter school within the Portland Public Schools system. Unlike many charter schools that are corporate, this school adheres to the original concept of charter schools as test beds of alternative approaches to education. It is based on a progressive set of principles such as constructivism and democratic education and serves mainly White children, not particularly affluent.

School A is required to follow the prescribed textbooks and other instructional materials adopted by the district, and adhere to strict testing requirements. As a result, as CREME teachers have testified, there is excessive emphasis on test preparation. Although the children are smart and creative, as evidenced in many of their open-ended projects, their performances in tests do not reflect that. And although the teachers from School A are kind, caring, and full of creative ideas they are compelled to follow the curriculum and testing regime thrust upon them.

For School B, not only is the demographic different, the curriculum and pedagogy are also in stark contrast. The teachers have intellectual freedom in designing their curricula, with minimal emphasis on performance on standardized tests. The teachers design and develop project-based curriculum spanning weeks at a time. Thus, while the students were learning how to respond to standardized test items in School A, their peers in School B were learning about election and democracy in a national program called Every Kid Votes 2016 (https://www.studiesweekly.com).

The participating teachers contribute to the richness of diversity of the project. They differ ethnically and linguistically, and in years of teaching experience. CREME is a teacher development project that differs in major respects from many endeavors labeled as such. It is not formulated as a group of researchers/academics proscribing, as experts, curricular content or teaching styles for teachers to follow. Rather, over two years and continuing, it is a collaboration founded on mutual respect wherein the mathematics educators propose ideas to the teachers, while the teachers and their students educate the academics about the realities of the circumstances in which they are teaching and learning. In the slow and organic development towards the idea of teachers as intellectuals, the teachers have had the opportunity to interact face-to-face and electronically with the advisors for the project, Marta Civil, Geneva Gay, and Danny Martin, and other scholar activists. And they are beginning to attend and present at conferences and to write papers.

What happens in CREME

We present here some examples of activities that we have collectively carried out (see Ford et al., (in press) for more details).

In School A, a very simple exercise with powerful effect has been to ask children to record, on paper and audio, numbers from 1 to 20 in their home language. Ongoing discussion was related to the large world map that hung in the room: In which part of the world is this language spoken? Where else is this language spoken? The list of languages spoken at home was long: Arabic, Bulgarian, English, Fulani, Kirundi, Korean, Oromo, Portuguese, Russian, Spanish, Sudanese, Swahili. (For a similar activity in Greece, see [START_REF] Chronaki | Number words in 'her' language, dialogism and identity-work: The case of little Mariah[END_REF].

As a more general approach to identity work within an asset pedagogy, the children at both schools record, on strips of paper, lists of what they can do, including but not confined to school achievements. These lists are place in 'talismans', old medicine bottles that are decorated by the children. Recently, all the children and their teachers wrote autobiographical poems based on the prompt "I am from", and drew self-portraits. A compilation has now been published as "We are from", and the book was launched in the library at School A with children reading their poems and the self-portraits on display.

Another aspect of identity that we work on is that of being a potential college student. In School A, in particular, students tend not to have a clear notion of what college is like. Accordingly, we have organized field trips for students to Portland State University, during which children interacted with students and faculty in Architecture, Engineering, and Earth Sciences.

In School B, an example of teaching to make students aware of other cultures is that the teacher, having covered a standard account of the Lewis and Clark Expedition of 1804-1806 that opened up the West to colonizers, then presented an account from the perspective of the Native Americans. We have also connected teachers, through visits to the Portland Art Museum, to Native American artifacts that illustrate the richness of design that serves as a context for the study of geometrical concepts, in particular symmetry.

School B has a tradition of working through Storyline projects, an approach in which students, together with their teacher, explore a theme in depth through creating characters within that theme. Koopman (in press) describes such a project that began by students looking at the labels on their t-shirts to see where they were made. The places were recorded with pins on a world map. Koopman reports that looking at the concentrations of pins in particular areas came as a revelation even to him. From there, the class collectively did extensive research about factories in the early 1900s where t-shirts were originally made. They used extensive arithmetic in calculating, based on historical data such as the prices of groceries and payscales at that time, the cost of living of a family in relation to the pay. They took on roles of workers and factory owner, and acted out situations of conflict. Koopman (in press) related the conflicts enacted by the students to his own experiences during a teacher strike when he was coerced by the school district to substitute for striking teachers. Later the principal of the school joined in the activity. The project culminated with research into contemporary conditions for workers in sweatshops in countries such as Bangladesh.

Another such project involved students building models of food carts (for which Portland is famous). Students created characters for the cart owners and operators, and devised detailed business models. They filled in actual forms relating to hygiene requirements, for example, for these simulated businesses. The teacher posted reviews on Yelp.

In these and similar projects, a great deal of arithmetic was done in context, illustrating how, in elementary school, students may:

(a) Be introduced to the conception of mathematics as a tool for interrogating sociopolitical issues, thus appreciating at an early age a sense of agency that will stand them in good stead as citizens.

(b) Consolidate computational and planning skills in complex contexts that afford relevance, interest, and motivation.

(c) Learn and practice mathematics integrated with other school subjects (science, obviously, but also art, social studies, language arts)something that elementary teachers are in the best position to do.

Future of CREME

At the time of writing, we have submitted proposals for further funding to continue CREME, but it is a measure of the group solidarity that the teachers are keen to keep the project going even without funding. As we continue, we plan to pay more attention to explicit mathematical content, always within the framework of culturally responsive pedagogy (see Ford et al. (in press) for more details). We also plan to explore and expose cultural biases in assessment, including the more subtle ways in which they are manifested. In light of the extremely encouraging political developments within the state, we are developing links with important actors within educational politics in Oregon. And we will continue to foster identity, agency, and autonomy in the CREME teachers, and to help them develop into mentors of new teachers joining the project.

Final comments: Ethical and political responsibilities of researchers

The naive conception of research and researchers as politically neutral, providing systematic objective evidence for the guidance of state educational policy, if ever viable, is certainly not sustainable in the circumstances that pertain in the United States, as a neoliberal agenda is pursuedas, in the face of global corporatization and economization, is happening in many parts of the world [START_REF] Spring | Globalization of education[END_REF]. The experiences described in this paper makes clear that as educators we are confronting issues of ethnic and cultural diversity, inequity, massive social engineering through the mechanisms of mass testing, the characterization of education as a market with huge profits to be made, the use of education as a means to maintain and even exacerbate economic inequality. The role of mathematics in all of these issues is particularly important, and we hold to a vision of mathematics education as providing people with tools for understanding and acting upon issues important in their lives and those of their families and communities. Along with mathematicians and mathematics teachers [START_REF] D'ambrosio | A nonkilling mathematics[END_REF], mathematics education researchers cannot absolve themselves from ethical and political responsibilities.

Proceedings of CERME10


text:
Alex Montecino
email: alexm@plan.aau.dk

The mathematics teacher's quasi-Darwinism: Problematizing mathematics education research

Keywords: The mathematics teacher, effects of power, quasi-Darwinism

This paper discusses the configuration of a quasi-Darwinian view of mathematics teachers, where the survival of the fittest is the cornerstone of a network of practices and discourses. It aims to contribute to the problematization of how mathematics education research and its discourses have effects of power in the fabrication of mathematics teachers' subjectivities, by unpacking naturalized truths of researchtruths regarding a productive and successful mathematics teacher. It deploys a Foucault-inspired discourse analysis, and it argues how the research on the mathematics teacher becomes a practice that governs mathematics teacher's subjectivities through the enunciation of the desire subject, a productive, successful and effective teacher.

Introduction

Providing quality mathematics education has been a concern within the mathematics education community, research on education and international agencies. It is argued that the quality of education and the development of mathematical knowledge is essential for society and social development [START_REF] Gellert | Research methods in mathematics teacher education[END_REF]OECD, 2010b). The idea is circulating that success in school mathematics is a prerequisite for personal and social success. Nowadays, it is considered that mathematics is a powerful mean to understand and control one's social and physical reality (OECD, 2010a), by being a tool and skill that helps people to undertake diverse tasks and problems of everyday life, and of their contexts (OECD, 2014b). However, according to OECD (2014a), modern societies valorize individuals not for what they know, but for what they can do with what they know, in other words, by their mathematical literacy:

Mathematical literacy is an individual's capacity to formulate, employ, and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts, and tools to describe, explain, and predict phenomena. (OECD, 2010b, p. 4) Research in the field of mathematics education is seeking to modify realityin the frame of social changesthrough its findings, proposing rationalities, knowledge, and ways of improving education practicesfor ensuring the quality of teaching and learning of mathematics. Furthermore, OECD (2014a) stated that "[h]igher educational achievement benefits both individuals and society, not only financially, but in the well-being with which it is also associated, such as better health outcomes and more civically engaged societies" (p. 104). In this fashion, the mathematics teachers become relevant, since they are considered as a central element in the establishment of quality education [START_REF] Jong | Assessing attitudes toward mathematics across teacher education contexts[END_REF][START_REF] Luschei | Evolution of policies on teacher deployment to disadvantaged areas[END_REF]OECD, 2005OECD, , 2014b)). Several studies have argued on the relation between the quality of the mathematics teacher and the shaping of successful students (cf. [START_REF] Castro-Rodríguez | Prospective teachers' understanding of the multiplicative part-whole relationship of fraction[END_REF][START_REF] Da Ponte | Preservice mathematics teachers' knowledge and development[END_REF][START_REF] Hemmi | Effective mathematics teaching in Finnish and Swedish teacher education discourses[END_REF]. Also, it is argued that the teacher is open to policy influences, whereas factors regarding students and the classroom context are not open to the same policy influences, at least in the short run (OECD, 2005).

All the aforementioned, the circulating discourses around 'success' in mathematics, are positioning a way of thinking and understanding mathematics education through the configuration of valid methods of doing research and of arguing about the diverse issues involved in the teaching and learning of the mathematics. For example, "[a]ll research is built around a set of assumptions about the world and how it should be understood and studied" (Jablonka, Wagner, & Walshaw, 2013, p. 41). Thus, this paper problematizes mathematics education research and its effects of power on teachers' subjectivities and fabrication. A discourse analysis, inspired by Foucault's ideas, is deployed to unpack the naturalized truths and discursive formations about the effective and competitive mathematics teacher.

Movements to examine the mathematics teacher as a discourse formation

According to [START_REF] Pais | Researching research: mathematics education in the political[END_REF], mathematics education research produces languages and tools that shape what researchers see and say in the world of education and of mathematics education. Mathematics research as a field of inquiry is not an innocent or a neutral activity [START_REF] Halai | Research rationalities and the construction of the deficient multilingual mathematics learner[END_REF]; it has been considered a "social institution which is inseparably linked to power" (Jurdak, Vithal, de Freitas, Gates, & Kollosche, 2016, p. 10). In this fashion, mathematics education, and also its practices, is considered to be political because it operates within governmentality techniques. Hence, by building on these techniques of government, this paper aims to understand how mathematics education research fabricates the mathematics teacher's subjectivity through regulatory practices embedded within naturalized truths. In other words, it addresses how research sees and talks about the mathematics teacher, by establishing regimes of power/knowledge.

According to [START_REF] Foucault | Archaeology of knowledge[END_REF], "[w]e shall call discourse a group of statements in so far as they belong to the same discursive formation" (p. 117). Hence, discourse as a group of statements, provides a particular language and knowledge, assembling regimes of truths. Circulating discourses describe rules and enunciations of a particular body of knowledge from specific spatiotemporal conditions [START_REF] Arribas-Ayllon | Foucauldian discourse analysis[END_REF]. This paper deploys a "research on research" [START_REF] Pais | Researching research: mathematics education in the political[END_REF] strategy built on Foucault's discourse analysis. This analytical strategy helps to unpack naturalized truths within research, that seek to generate a productive and successful mathematics teacher, and, at the same time, to trace the power effects on the fabrication of mathematics teachers' subjectivities. So, by problematizing the discourses, it is possible to understand research as a practice that governs subjectivities through the enunciation of the desired subject.

First, repeated statements about the 'must be' of the mathematics teacher are identified. Second, these statements are analyzed to trace their knowledge/power relationships, and their continuities and discontinuities amongst each other. It does this by analyzing published studies about teaching and learning of mathematics. The empirical materials consist of research about mathematics teachers released within the last four years of three journals: Journal of Mathematics Teacher Education, ZDM Mathematics Education, and Educational Studies in Mathematics.

Finally, it problematizes how research and its discourses have effects of power in the fabrication of mathematics teachers' subjectivities. It does this by portraying how certain rationality is circulating within research in mathematics education. As will be described, such rationality promotes a quasi-Darwinism, in which the survival of the fittest and the idea of evolution are the cornerstone of a network of practices and discourses.

The mathematics teacher research and the survival of the fittest

In navigating through the discourses that are circulating about the mathematics teacher, amongst the materials analyzed, it is possible to identify some enunciations that are continuously repeated. By following a Foucaultian chain of thought, these particular enunciations constitute statements about how mathematics teachers are supposed to act and be within their practices, their 'must be'. Such statements respond to concerns raised by research in the field of mathematics education. For example, who is taken as valid for arguing about mathematics teachers, what does a mathematics teacher ought to do, and how to seek for the improvement of the teaching and learning of school mathematics. From the analysis, some discourses about the 'must be' of the mathematics teacher are configured as truths. These truths are advertised as desired features that teachers should have if they want to perform successfully, namely: a high knowledge [START_REF] Fauskanger | Challenges in measuring teachers' knowledge[END_REF], an updated repertoire of techniques [START_REF] Subramaniam | Prospective secondary mathematics teachers' pedagogical knowledge for teaching the estimation of length measurements[END_REF], and a personality consistent with their practicespersonal aspects such as beliefs or attitudes [START_REF] Jacobson | Knowledge and motivation as mediators in mathematics teaching practice: The case of drawn models for fraction arithmetic[END_REF].

These discourses are naturalized under a competition and comparison system of reason. International standardized testing -PISA and TIMMS -, and its reports are examples of how competition and comparison become part of society, by shaping social discussions, decisions, efforts, and initiatives. At the same time, through those tests' outcomes, diverse countries could monitor themselves to improve the weakest areas, since "[a]ll countries are seeking to improve their schools, and to respond better to higher social and economic expectations" (OECD, 2005). In this fashion, a variety of studies, that seek to improve the teaching of mathematics (see [START_REF] Boston | Connecting changes in secondary mathematics teachers' knowledge to their experiences in a professional development workshop[END_REF][START_REF] Lewis | How does lesson study improve mathematics instruction[END_REF][START_REF] Pang | Improving mathematics instruction and supporting teacher learning in Korea through lesson study using five practices[END_REF], are aimed to identify how mathematics teachers could achieve a successful practice by analyzing their students' achievement on national and international tests. But, as discussed elsewhere, what is taken, by research, as a successful practice leads to a system in which teachers compete against others teachers, against what is considered as a desired teacher, and, also, against themselves [START_REF] Montecino | Mathematics teachers as products and agents: To be and not to be. That's the point![END_REF]. So, research discourse is raising comparison as a mean for knowing the characteristics of competent and effective teacherthe fittest teacher -, effective practices or successful experiences. Within these discourses, it is possible to see statements such as: By comparing and contrasting the practices of LS [Lesson Study] in mathematics in different countries, it will be possible to explicate the local theories of teaching and learning of mathematics, highlight educational values in each culture, and understand why and how these values support certain teacher development processes that are unique to the culture. (Huang & Shimizu, 2016, p. 394) In the unpacking of naturalized truths of the analyzed materials, it is possible to see that some statements highlight mathematics teachers' deficits and flaws. These statements pay attention to what teachers need to improve in their lessons for increasing students' achievement [START_REF] Spitzer | Developing prospective elementary teachers' abilities to identify evidence of student mathematical achievement[END_REF]. On one hand, by emphasizing that teachers need to achieve a higher expertise on school mathematical topics (e.g. [START_REF] Karakok | Secondary teachers' conception of various forms of complex numbers[END_REF][START_REF] Magiera | An exploratory study of preservice middle school teachers' knowledge of algebraic thinking[END_REF]. On the other hand, by focusing on the need for developing more effective teacher's practices (see [START_REF] Lee | Pre-service teachers' conceptions of effective teacher talk: Their critical reflections on a sample teacher-student dialogue[END_REF]. This type of research acknowledges that mathematics teachers have a 'responsibility' for students' performances and, therefore, teachers ought to be highly trained. Alongside the statements about what needs to be improved, other statements exist that pay attention to what teachers lack, in other words, to skills that teachers are required to develop to reach what those studies perceive as 'successful professional development': on the one hand, studies regarding teachers' belief system (e.g. [START_REF] Conner | Impact of a content and methods course sequence on prospective secondary mathematics teachers' beliefs[END_REF][START_REF] Cross Francis | Dispelling the notion of inconsistencies in teachers' mathematics beliefs and practices: A 3-year case study[END_REF]; on the other hand, studies regarding teachers' attitudes (e.g. [START_REF] Hannigan | An investigation of prospective secondary mathematics teachers' conceptual knowledge of and attitudes towards statistics[END_REF][START_REF] Jong | Assessing attitudes toward mathematics across teacher education contexts[END_REF].

According to some research, "[h]ow teachers perceive and adapt their roles will have great impact on overall classroom interactions, such as the teachers' questioning strategies or feedback patterns" (Lee & Kim, 2016, p. 366). This implies that teachers' decisions have an impact on students since it is believed that students' intellectual autonomy could be favored by teachers' practices [START_REF] Goldsmith | Mathematics teachers' learning: A conceptual framework and synthesis of research[END_REF]. And so, the decisions made by the mathematics teacher have a high impact not only on students but also on their learning [START_REF] Stockero | Characterizing pivotal teaching moments in beginning mathematics teachers' practice[END_REF]. This type of research shows that mathematics teachers should be constantly seeking to improve their professional development, practices, knowledge and skills not only for themselves but also for the sake of their students [START_REF] Afamasaga-Fuata'i | Student teachers' mathematics attitudes, authentic investigations and use of metacognitive tools[END_REF]. Since professional development has been understood as a form of lifelong learning in which mathematics teachers are responsible for their own development and achievements, these types of statements, from a Foucaultian approach, are tracing the ways in which the mathematics teacher should become an effective and competitive teacher, through processes of self-regulation.

According to these studies, teachers should aim at improving, by themselves, diverse personal and technical aspects. Such aspects are supposed to encourage the development of a more effective and competent teacher, by recognizing their own deficits and flaws with the goal of overcoming them. This naturalized truth resonates not only within research but also amongst other discourses on education. For example, OECD (2012) states that effective teachers are a key to close achievement gaps between advantaged and disadvantaged students. And, therefore, the aim should be to (re)train and (re)shape teachers to become the desired effective teacher. In this regard, research is tracing a sort of 'evolutionary line' for mathematics teachers, in which at the end of the line rests the desired mathematic teacher. Teachers should evolve when they achieve the desired levels of knowledge and skills established by society, becoming the productive, successful and effective teacher. However, these desired levels are in constant movement, being redefined by new social interests, concerns, desires and demands as well as new mathematical knowledge that the modern citizen should have. This means that mathematics teachers have to govern themselves into a constant process of change, of (re)training and (re)shaping. As Deleuze (1992) asserts, currently nothing is considered to be finished; all is in a constant becoming.

The idea of the 'evolutionary line' helps to tell the narrative of the becoming of the mathematics teacher as the survival of the fittest, since research in the field highlights the features of the 'fittest subject'. This portrays that the survival of the fittestthe desired mathematics teacherinvolves practices of self-regulation, but also of competition against other teachers, practices that could lead to the exclusion of certain teachers labeled as 'inferior subject', unproductive, unsuccessful, and inefficient. For example, [START_REF] Lee | Pre-service teachers' conceptions of effective teacher talk: Their critical reflections on a sample teacher-student dialogue[END_REF] have argued that mathematics teacher training programs "should include more specific investment in the effective use of classroom dialogue for learning" (p. 378), a 'fittest subject' should evolve in an effective classroom communicator whereas the 'inferior subject' will not evolve as a classroom communicator, and, will therefore be taken as ineffective. Consequently, the survival of the fittest governs the self and conducts mathematics teachers' practices towards the desire to evolve, (re)shaping the research about mathematics teachers within a system of reason rooted in a quasi-Darwinism, since it traces the paths for teachers to increase their abilities to survive, compete and evolve.

Quasi-Darwinism of mathematics education research and its effects of power

The analysis deployed has pointed to the existence of statements on the desired mathematics teacher, a self-regulated and evolved subject. These statements have been (re)producing certain truths about who the effective teacher is. For example, mathematics teachers should perceive themselves as responsible for others -i.e., their students' performances -, as promoters of social changei.e., by closing achievement gaps -, and also, as responsible for themselvesi.e., tracing their professional development and learning the best possible way. These statements are building a quasi-Darwinian view of mathematics teachers; an 'evolutionary line' that is embedded within the above discourses and shapes the fabrication of the fittest subject.

The quasi-Darwinism (re)shapes mathematics teachers' ways of being and acting at a particular time and place, through discourses that are produced and reproduced under certain regimes of power/knowledge [START_REF] Foucault | The subject and power[END_REF]. The naturalized truths are constituted, on the one hand, within a particular regime of knowledge, which delineates who is the one to discuss about the mathematics teacher and how, and in what way the knowledge regarding the teacher should be generated. On the other hand, within a regime of power which defines what understanding is meaningful to be studied what discourses are taken valid regarding certain issues or aspect of the mathematics teacherand which practices, knowledge and techniques should be targeted. Therefore, a quasi-Darwinian view (re)produces what the mathematics teacher should bethe becomingtowards the development of the 'human capital' (OECD, 2001). Human capital voices the value that subjects have in correlation with their knowledge, skills, education and preparation for the future, which translates into personal, social and economic well-being. Alongside, a quasi-Darwinian view (re)shapes a discourse aimed at optimizing the becoming of the teacher. Moreover, the research on mathematics teachers seeks to minimize all aspects that could lead to an 'inferior subject'. In order to be the fittest, teachers should engage in practices that turn them into accountable and measurable agents.

In this regard, it is possible to state that research in the field of mathematics education becomes a technology of the self [START_REF] Foucault | Technologies of the self[END_REF] that regulates mathematics teachers' conducts towards the shaping of the desired mathematics teacher. By promoting 'cultural thesis' [START_REF] Popkewitz | Cosmopolitanism and the age of school reform: Science, education, and making society by making the child[END_REF] about the desired mathematics teacher, the analyzed research has effects of power on teachers subjectivities, meaning how mathematics teachers understand themselves and their becoming. Only the 'fittest subject' is the one able to develop the skills and knowledge that society demands and requires, is the only one who can evolve in a 'superior subject'; subjects able to adapt themselves to the new social and professional demands. In other words, teachers are to evolve in subjects that have the tools, skills, and knowledge to survive to all social changes and challenges; becoming a successful, effective, competent and fittest subject. Thus, within the circulating discourses is configured a narrative in which if the mathematics teacher does not adapt or evolve, he/she is excluded or labeled as deficient. The teacher who survives through social changes and challenges is neither the knowledgeable teacher, nor the successful teacher, nor the most intelligent teacher; rather he/she is the most adaptable to change.

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Acknowledgements

I would like to thank Melissa Andrade Melina and Lisa Björklund Boistrup for their comments to previous versions of this paper.


text:
Judith Jung
email: judith.jung1@tu-dresden.de

Marcus Schütte

Content-related and social participation in inclusive mathematics education

Keywords: Inclusion, student participation, interaction

In the 1980s, the WHO presented a differentiated model of disability. The central feature of disability according to this model is not the impairment but the resulting limitation in social participation. The UN Convention on the Rights of Persons with Disabilities (CRPD) goes a step further by demanding that persons with disabilities must have access to an inclusive school education together with persons without disabilities. From our point of view, not only social but also content-related participation is important for an inclusive school education. The following theoretical paper seeks to explore a theoretically grounded concept of participation in mathematics classes for a planned exploration of how these forms of participation are implemented in inclusive mathematics education in Germany.

Introduction

In Germany, most children first attend primary school from the age of five or six to nine or ten and are then separated by a supposedly achievement-based selection process into different school forms. This is a system that has long been criticised in several respects [START_REF] Muñoz | Report of the special rapporteur on education[END_REF]. Furthermore, from the mid-18 th century a parallel branch of schooling for children with special educational needs was established alongside primary school and the four types of secondary school. The special needs school system continued developing into the 2000s, with different schools specialising in particular needs, e.g. social-emotional development, learning, language, vision, hearing, or mental development. In the 2009 school year in Germany, a total of 483,267 pupils with special educational needs went to school. Of these, 387,792 (80.24%) were schooled at special needs schools and 95,475 (19.76%) in regular schools (Sekretariat der Ständigen Konferenz der Kultusminister der Länder, 2016a).

With the EU's ratification of the UN Convention on the Rights of Persons with Disabilities, this praxis of the separated school system came under renewed criticism and also its legality came into question. The UN Convention requires that persons with disabilities should not be excluded from the general educational system because of their disability (CRPD, 2007, article 24). This legal right surpasses the mere freedom of choice to attend a regular school. According to the UN Convention, adequate arrangements have to be made within regular schools in order to ensure the educational success of each individual. Though the ratification of the Convention potentially necessitated little change in other countries, in Germany it opened a political and social discussion on structural changes to the school system. Since 2009, a policy of inclusion  in the sense of integrative schooling of pupils with special educational needs in the regular school system  has been implemented in the German states. Thus, in the 2015/16 school year, 322,518 (62.34%) out of the 517,384 pupils with special educational needs attended a special needs school and 194,866 (37.66%) a regular school. The ratification of the convention therefore appears to have led to a thorough implementation process, as the number of children with special educational needs in regular schools has doubled in only six years. However, looking at the figures on the background of demographic developments, which have seen a reduction in pupil numbers, the proportion of pupils with special educational needs in the total number of pupils in Germany clearly rose between 2009 (6.17% of all pupils between Years 1 and 10) and the 2015/16 school year (7.1%). Thus, the proportion of pupils with special educational needs who are taught at special needs schools has barely fallen. Above all, children who were attending regular schools in any case are now more commonly being given special educational needs status (Sekretariat der Ständigen Konferenz der Kultusminister der Länder, 2016aLänder, , 2016b;;[START_REF] Klemm | Inklusion in Deutschland. Eine bildungsstatistische Analyse[END_REF]. So, it seems there is still a long way to go to achieve a school system that can be described as inclusive. We follow [START_REF] Katzenbach | Widersprüchen denken, und dennoch handlungsfähig bleiben: Inklusion in einem segregierenden Schulsystem[END_REF] in holding that there is little difference between the terms integrative and inclusive in school life. However, there are some clear conceptual differences: the idea of integration depends on a categorisation of people. There are "normal" people and "the others", the non-disabled and the disabled who need to be brought together. The concept of inclusion, however, is based on the premise of diversity. Disability is only one characteristic among many, and school is a place where people with extremely different characteristics intermingle. Following UNESCO's (2005) understanding of the concept, inclusion can be understood as an ongoing process to find better ways of responding to diversity.

Towards providing an empirical base for this process of changing a traditionally strongly separated school system such as that in Germany, in respect to content-related  here mathematical  learning conditions, this paper discusses some theoretical considerations on mathematical learning and learning under inclusive conditions, and synergises these for a planned research project. According to [START_REF] Prediger | Networking strategies and methods for connecting theoretical approaches[END_REF], different networking strategies can be used to connect theories. The first strategy they mention, "having an understanding of the different theories", can be seen as the starting point for all other strategies, allowing them to be compared, combined or integrated in a further step. Therefore, in a first step we will present specific theories for mathematical learning and inclusion theories separately before we coordinate them with each other. This can be done because the theoretical concepts have consistent assumptions. The coordination creates a conceptual framework that helps in identifying the students' participation in inclusive settings and learning in mathematics.

Mathematics learning from an interactionistic perspective

For our understanding of content-related learning in school we refer to [START_REF] Miller | Kollektive Lernprozesse. Studien zur Grundlegung einer soziologischen Lerntheorie[END_REF] antagonistic differentiation between the research traditions of genetic interactionism and genetic individualism [START_REF] Schütte | Das Implizite beim fundamentalen Lernen von Mathematik[END_REF]. However, this is with the goal of bringing the two positions closer together. According to Miller (1986, p. 17), learning can either be anchored in the individual as a process of monologue, in the sense of genetic individualism 1 , or be understood as a process of dialogue between individuals, in the sense of genetic interactionism 2 . Miller considers genetic interactionism, in contrast to genetic individualism, to have a more convincing empirical and theoretical base, at least in relation to learning processes in the early development of individuals, which he considers "fundamental" 3 . Above all, in primary school, where young learners come together, Miller sees learning processes taking place which are primarily collective and based on dialogue. Miller (1986, p. 223) describes individualised learning processes as "autonomous learning" and attributes them to the later development of the individual, locating these processes in moments of reflexive consolidation of things originally learned collectively. From a social-constructivist perspective, learning cannot be seen as a primarily internal cognitive restructuring process. Rather, it is a dualistic process which takes place both within the individual, in the sense of cognitive restructuring, and within interaction processes in which the person participates, which go before these restructurings [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses and mathematizing[END_REF]. This kind of sociological or social-constructivist consideration of learning processes has in recent years gained increasing influence in the theoretical design of content-related learning, and has been taken up and further developed in mathematics education research [START_REF] Lerman | The social turn in mathematics education research[END_REF]. Both nationally and internationally, mathematics has increasingly come to be seen as a cultural tool, constructed and mediated through language (Schütte, 2014). Since the mid-1980s interactionistic approaches of interpretive (classroom) research in mathematics education have engaged with the sociologically based social-constructivist perspective on learning processes (e. g. [START_REF] Bauersfeld | Interaction, construction and knowledge. Alternative perspectives for mathematics education[END_REF][START_REF] Krummheuer | Lernen mit »Format[END_REF] using theories of symbolic interactionism [START_REF] Blumer | Symbolic interactionism. Perspective and method[END_REF]. With this kind of basic theoretical understanding of content-related learning the concept of collective argumentations gains central significance in the analysis of mathematical learning processes. According to [START_REF] Krummheuer | Paraphrase und Traduktion. Partizipationstheoretische Elemente einer Interaktionstheorie des Mathematiklernens in der Grundschule[END_REF], pupils are usually engaged in interaction processes in the classroom conversation, producing an argumentation in the totality of their actions. In this way, participation in a collective argumentation concerning statements about (mathematical) content, terms and/or methods creates the basic conditions for learning opportunities. This interplay of individual and social constituents is difficult to describe. If participation in collective argumentation provides orientation and convergence, then learning success can be seen as the improved coordination between individual attributions of meaning and the results of the interactive negotiation of meaning in the respective group. On an interactional level, this is manifested in an increasing adaption of the (verbal) acts of the learners to argumentations established collectively over the course of several interactional situations. The coordination of an individual's interpretations and actions can be reconstructed empirically as the increasingly autonomous adoption of steps of action within the collective argumentation. The learning of mathematics can thus be described as the "progress" of participation in mathematical collective argumentations. This idea of learning through participation can be linked back to the notion of equal opportunities for participation in educational institutions, according to the Convention on the Rights of Persons with Disabilities. Thus, the following will seek to explain how participation in learning processes in school can be understood from an inclusive-educational perspective.

Increasing participation as a goal of inclusion

Among the goals that are being set by the increasing implementation of the inclusion concept in teaching, ignoring for a moment the thoughts outlined above on the different understandings of the learning of mathematics, it can be noticed that providing all pupils with equal opportunities for participation in learning processes occupies a central position. This reflects UNESCO's principle whereby Inclusion is seen as a process of addressing and responding to the diversity of needs of all learners through increasing participation in learning, cultures and communities, and reducing exclusion within and from education. (UNESCO, 2005, p. 13) This principle, formulated in the framework of the UNESCO "Guidelines for Inclusion", is reflected in the UN Convention on the Rights of Persons with Disabilities (CRPD), ratified by Germany in 2007. The Convention states:

Persons with disabilities are not excluded from the general education system on the basis of disability. Persons with disabilities can access an inclusive, quality and free primary education and secondary education on an equal basis with others in the communities in which they live. (CRPD, 2007, article 24) In addition, the Convention asserts that children/persons with disabilities should be supported within general education, according to their needs, to best progress their education. A "full and equal participation in education" (CRPD, 2007, article 24) should be made possible for them. Some years previously, [START_REF] Booth | Index for inclusion: Developing learning and participation in schools[END_REF] already published an "Index of Inclusion", which is intended as a tool to support an inclusive school development and contains a detailed description of how barriers to learning and participation for all learners can be dismantled. Here, too, the goal of a "greater participation of students in the cultures, curricula and communities of their schools" is cited (Booth & Ainscow, 2002, p. 2). In summary, according to the mentioned literature inclusion can be understood as an unending process of increasing learning and participation for all students [START_REF] Booth | Index for inclusion: Developing learning and participation in schools[END_REF]UNESCO, 2005)  and thus also as an ideal, which will never be fully realised but is already applied with the start of the process of increasing participation. However, it seems necessary to clarify at this point what is understood by full and equal opportunities of participation. Therefore, in the following the concept of participation in learning processes is developed. According to Booth and Ainscow (2002, p.3), participation can be understood as "learning alongside and in collaboration with others in shared learning experiences". They see participation as demanding active involvement in learning processes and the opportunity to express one's own learning experiences.

Adopting a broader definition developed to consider social structures, according to the German sociologist [START_REF] Bartelheimer | Was bedeutet Teilhabe?[END_REF] five requirements can be distinguished for a sufficiently defined term of participation. These five requirements for a general social concept of participation can be transposed onto teaching processes in the classroom as follows. First, Bartelheimer states that participation is only to be understood as historically relative. Transposed onto the school situation, participation in processes that enable learning is to be understood only in relation to the given education system and fundamental features of the educational processes and paths that currently predominate. Secondly, participation is multidimensional and there are always different forms of participation. One approach to consider participation in teaching processes in various dimensions is provides by [START_REF] Roos | Inclusion in mathematics in primary school[END_REF], who distinguishes a spatial, social and didactical/content-related dimension. Thirdly, participation does not describe a simple in or out; rather, there are always gradations of unequal participation. Furthermore, participation appears as a dynamic concept, rather than a condition at a given moment. Lastly, [START_REF] Bartelheimer | Was bedeutet Teilhabe?[END_REF] emphasises that participation is active, that is, it is striven for and realised through action and in social relationships. Summarising the last three points of Bartelheimer's participation concept for teaching processes, an exclusively quantitative description, that is, a definition of participation in the sense of "takes part/does not take part", appears inadequate. As far as possible then, the description of participation must be considered qualitatively over a longer time-frame, and might only be valid for the respective participating individuals within this time-frame, since this is a question of a dynamic concept that is constantly changing.

The underlying research project is concerned with describing scenarios from the process of progressing inclusion in German schools, and on this basis to design steps to further this process. To enable a description of this process the concept of participation in inclusive mathematics learning needs to be rendered more precisely, in order to open it to empirical description and analysis. For this purpose, in the next section the theoretical understanding of mathematics learning and the theoretical conception of participation in inclusive discourse, discussed individually above, are brought together.

A theoretical concept of participation in mathematics education

With the goal of describing the process of inclusion in mathematics learning in German schools using a participation-theoretical model, we take the above theoretical considerations on inclusion as a starting point and link these to our essential theoretical assumptions on mathematical learning. Taking Bartelheimer (2001) into account, it seems important to adopt a longer-term perspective on participation (cf. also Brandt, 2004). Although [START_REF] Krummheuer | Paraphrase und Traduktion. Partizipationstheoretische Elemente einer Interaktionstheorie des Mathematiklernens in der Grundschule[END_REF] go so far as to attempt an interactionistic theory of participation in mathematical learning, focusing also on individual learning using their system of categories, content-related learning and individual cognitive restructuring unfortunately seem to fall out of focus. Yet, our research is also and especially guided by looking at content-related learning. In this context, we are not trying to leave the basic interactionistic orientation. Instead we are trying to connect the ideas of learning and participation within interaction with individual learning. From an inclusion-theoretical perspective, realising inclusive teaching is a question of increasing participation. In order to make this useful for empirical investigations, [START_REF] Bartelheimer | Was bedeutet Teilhabe?[END_REF] more general participation-theoretical model is taken as a basis and combined with the participation-theoretical understanding of mathematics learning (e.g. [START_REF] Krummheuer | Lernen mit »Format[END_REF][START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses and mathematizing[END_REF]. To be able to describe participation in inclusive mathematics learning in Bartelheimer's perspective, one has to view it in relation to its historical context. The image of mathematics learning for all learners is taken to represent present conditions. Whereas at the end of the 1970s, national and international research on mathematics in primary schools was still focusing above all on the learning of skills and isolated concepts of a 'complete mathematics', with an emphasis on arithmetic, a shift took place around the mid-1980s. A new understanding of mathematical learning developed. According to this, children in school not only needed to acquire mathematical skills, but also to discover and understand the mathematical concepts behind them, and to argue and communicate with teachers and classmates using these concepts in order to ultimately be able to autonomously give reasoning for mathematical actions [START_REF] Boyd | Mathematics education and special education: Searching for common ground and the implications for teacher education[END_REF]. However, in adopting an orientation towards this image of mathematics learning we nevertheless acknowledge that it is (still) controversial for learners with special educational needs in special needs schools. The above-described changes in the understanding of mathematics learning can certainly be seen also in the area of special needs education in mathematics [START_REF] Sullivan | The challenge of reporting research to inform the creation of inclusive mathematics learning environments[END_REF]. But these changes are the subject of heated debate and are taking place in the context of a (tried and tested) teaching tradition which is characterised by a reduction of learning content, an isolation of difficulties and a "small-steps" approach with specified solutions to problems [START_REF] Boyd | Mathematics education and special education: Searching for common ground and the implications for teacher education[END_REF]. In addition, Bartelheimer describes participation as an active process. Participation is pursued and achieved through social action and within social relationships. Therefore, it is observable in everyday mathematics teaching as active participation in classroom interactions. Bartelheimer also describes participation as multidimensional, which we have linked to Roos' (2014) spatial, social and didactical/content-related dimensions in the above theoretical section on inclusion. The spatial dimension of participation, according to Roos, relates fundamentally to how much time a student is spending in the same room as his or her classmates. However, in our perspective it also relates to the spatial configuration during time spent learning together in the mathematics classroom, for example in rotating through different tasks or group work. The social dimension focuses on social relationships (with fellow pupils, teachers and pedagogical staff) which emerge in mathematics teaching and which mediate to a great extent an increasingly autonomous participation in collective argumentations. The third dimension addresses participation in didactical/content-related negotiation. Didactical inclusion relates to pupils' participation in subject teaching, focusing on their engagement with the teaching approach and content, as well as any explanations or material supplied by teachers to support the learning process. For the purposes of analysis of didactical/content-related participation, the approaches developed in mathematics education for determining participation in collective argumentations [START_REF] Krummheuer | Paraphrase und Traduktion. Partizipationstheoretische Elemente einer Interaktionstheorie des Mathematiklernens in der Grundschule[END_REF]) can be made use of. With reference to Goffmann (1981), [START_REF] Krummheuer | Paraphrase und Traduktion. Partizipationstheoretische Elemente einer Interaktionstheorie des Mathematiklernens in der Grundschule[END_REF] distinguish two types of involvement in a lesson: the active, verbally productive act, and the passive, receptive non-verbal act. The aim is to identify the type of authenticity, originality and responsibility of speakers, and to identify for recipients the type of nonactive participation. This means that mathematical learning through active participation can be distinguished from learning through non-active participation, which explains "quiet" yet successful pupils. Learning situations become beneficial for learning, according to [START_REF] Krummheuer | Paraphrase und Traduktion. Partizipationstheoretische Elemente einer Interaktionstheorie des Mathematiklernens in der Grundschule[END_REF], when children participate increasingly in ways which permit a shifting from minor responsibility for content and form towards greater responsibility. In this way, participation in a collective argumentation concerning statements about (mathematical) content, terms and/or methods creates the basic conditions for mathematical learning opportunities in inclusive learning settings. Since this model remains more of a formal analysis of the content-related negotiation in the conversation, it will be complemented by the curricular concept of mathematical activities, developed by [START_REF] Bishop | Mathematical enculturation. A cultural perspective in mathematics education[END_REF], in order to approach also the mathematical content of the activity children participate in. [START_REF] Bishop | Mathematical enculturation. A cultural perspective in mathematics education[END_REF] differentiates six activities  counting, locating, measuring, designing, playing, explaining  which are used for the analysis of moments of subject-specific mathematical participation, following Brandt (in press) and [START_REF] Johannson | Perceptions of mathematics in preschool[END_REF]. In addition, [START_REF] Bartelheimer | Was bedeutet Teilhabe?[END_REF] focuses on the principal dynamic, i.e. the changeability of participation over time, and the impossibility of a dichotomous categorisation of inside and outside, participation and nonparticipation. These considerations are in tune with an interactive understanding of mathematics learning and are taken into account within the research project by the theories and methodologies being used. This theoretical conception of participation will be used to address the empirical aspect; the theoretical conception will be further developed through the interrelationship between theory and praxis, with the goal of enabling a description of participation processes in inclusive mathematics. This is while acknowledging that the goal of inclusion, which has the principle of egalitarian difference [START_REF] Prengel | Pädagogik der Vielfalt. Verschiedenheit und Gleichberechtigung in Interkultureller[END_REF] at its base, cannot be for all children to participate actively in class in the same way. The barriers to participation should be reduced for all children, and they should be given the opportunity for participation according to their abilities, so that they move forward in their mathematical learning process.

Genetic individualism is in the tradition of the later Piaget and Kohlberg(Miller, 1986, pp. 15 ff.).

Genetic interactionism draws from the basic assumptions of sociological and psychological studies such as those of Durkheim, Mead, the early Piaget and Vygotski. These studies see social cooperation or interaction as fundamental to individual learning processes(Miller, 1986, pp. 15 ff.).Thematic Working Group 10Proceedings of CERME10

These kinds of learning processes concern the development of "rationality, or rational knowledge structures"(Miller, 1986, p. 15).Thematic Working Group 10Proceedings of CERME10

Proceedings of CERME10