Helena Roos
email: helena.roos@lnu.se

Mathematics classroom setting influencing inclusion

Keywords: Discourse analysis, inclusion, participation, special educational needs in mathematics

This research reports on a study investigating how three students perceived as being in special educational needs in mathematics (SEM), either as students in access to mathematics or as students in struggle to get access, talk about participation in mathematics education. Discourse analysis is used as a theory and a tool to investigate the students' own stories of learning and teaching in an inclusive mathematics classroom. The results show that students' participation in mathematics education is influenced by how the mathematics education is set up. That is, how the organisation of the mathematics classroom and students being in a small group influence students' participation. The results also show that even though the same issues of the organisation influence the three students, there is a diversity within the issues, calling for a critical question, is the inclusive classroom setting really inclusive in terms of participation and access to mathematics?

Introduction

How to set up mathematics lessons to facilitate participation for every student within an inclusive classroom is not an easy task. Nevertheless, it is a particularly important educational task in order to promote an equitable mathematics education and create opportunities for every student to learn [START_REF] Moschkovich | Principles and Guidelines for Equitable Mathematics Teaching Practices and Materials for English Language Learners[END_REF][START_REF] Askew | Diversity, Inclusion and Equity in Mathematics Classrooms: From Individual Problems to Collective Possibility[END_REF]. The setup of the mathematics classroom, including didactical choices of the mathematics teacher(s), is influenced by many different aspects. For example, [START_REF] Askew | Diversity, Inclusion and Equity in Mathematics Classrooms: From Individual Problems to Collective Possibility[END_REF] identifies a Western way of teaching mathematics with a one-size-fits-all approach, influencing the setting of the mathematics classroom. Hence, the set-up of the mathematics education is not in full an actual choice of the teacher, but a choice of the school, or even a choice of the society. This implies there are cultural, social, and institutional factors, such as social patterns, strongly impacting mathematics education settings. For example, patterns of advantage and disadvantage [START_REF] Civil | Participation in the Mathematics Classroom: Does every student have a voice? For the learning of mathematics[END_REF]. Then, the awareness and the ability of the teacher to know how to act and care in the moment in order to promote learning of every student [START_REF] Mason | Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment[END_REF] becomes even more of importance. This implies, the awareness of factors, external of the actual classroom interaction, influencing the mathematics education, the set up, and enactment of the teaching inside the classroom is of importance to promote participation and learning. So, if defining inclusion in mathematics as processes of participation for every student (Roos, 2015), inclusion is strongly connected to the promotion of equitable mathematics education in terms of how the mathematics education is set up [START_REF] Moschkovich | Principles and Guidelines for Equitable Mathematics Teaching Practices and Materials for English Language Learners[END_REF]. Moreover, since inclusive mathematics classrooms are diverse in terms of students' backgrounds (gender, ethnicity, culture, achievement etc.), the mathematics education needs to provide a diverse education [START_REF] Roos | Diversity in an inclusive mathematics classroom: A student perspective[END_REF]. This suggests a diverse setting of the mathematics classroom considering every students' needs. Within the research field of mathematics education, the notion of inclusion is often connected to special educational needs in mathematics (SEM) (Roos, 2019a). Often when discussing SEM, the student in mind is a low achiever, struggling to get access to mathematics education. However, a high achiever can also be in SEM even though she or he has access to the mathematics education, because she or he might need specific solutions in order to have optimal opportunities to learn. Accordingly, the notion of SEM can work in two directions, one direction towards mathematics difficulties, and one direction towards mathematics facility, showing differences in access to mathematics education. Inclusion in mathematics education for both ends of the SEM-continuum can be provided by the promotion of participation focusing on teaching practices and intervention strategies taking off from learning situations enabling meetings among differences [START_REF] Scherer | Assistance of students with mathematical learning difficulties, how can research support practice[END_REF]. Hence, setting up teaching promoting participation becomes an important challenge. To take on this challenge, we need to know the meaning(s) of students regarding participation in the inclusive mathematics classroom. Hence, the aim of this paper is to describe SEM-students meaning(s) of inclusion in their talk about learning and teaching in an inclusive mathematics classroom to have the best opportunities to learn.

Methodology

In this section the site and the participants of the study are described, as well as the theoretical and analytical approach and the data analysis.

The site and participants

A lower secondary school in an urban area in Sweden that has set out to implement inclusive work was chosen for this investigation. The inclusive work at the school aims at including all students in the ordinary teaching in every school subject, and integrate the special education into the ordinary teaching. A grade 7 and a grade 8 classroom at the school were observed during one semester. At least one approximately 50 minutes long mathematics lesson each week for each class was observed. After each observation student interviews were carried out. The selection of students for the interviews was made in cooperation with the students, parents and teachers and took ethical and organisational issues into consideration. The mathematics teachers suggested students they perceived as being in some kind of SEM. Thereafter, if the students and parents gave their consent, they were offered to take part in this study. Six students took part in the study. The interviews were conducted when the organisation and students allowed it. The interviews took place in a small room familiar to the students once a week and were based on the observations made the same week. Hence, open questions about situations and content of the mathematics education were asked in a discussion type of manor. This paper focuses on three students: Veronica in grade 7, Ronaldo and Edward in grade 8. Veronica says that "math is pretty hard" and she states "I don't like math". The mathematics teachers perceive her as a student who struggles to get access to mathematics. Ronaldo describes himself as a student with learning difficulties "I have difficulties within all subjects, and it's like concentration and all that." He also experiences that he forgets stuff "I don't remember, I have to repeat a lot". The mathematics teachers perceive him as a student who struggles to get access to mathematics. Both Veronica and Ronaldo have just about a passing grade. Edward describes himself as a person that thinks mathematics is really easy and does not need much help at all. He does not have to make any effort, mathematics works "automatically" for him and he "already knows" most of what they are doing in math class. The mathematics teachers perceive him as a student with access to mathematics, and he has the highest grade possible in mathematics (grade A1 ).

Theoretical and analytical approach

When investigating students' stories of their own participation in mathematics education, there is a need to identify critical aspects of learning and teaching in the stories. In this research this identification is made by using Discourse Analysis (DA). DA is a helpful tool since the focus of DA is the study beyond text, and because DA has an explanatory power of social contexts. Thus, by analysing the use of language in a certain type of situation we can say something about the social world. When going beyond the text in this particular study, construed discourses of what influences students' participation in mathematics education makes it possible to describe the social world from the students' perspective. In this study, the perspective of Gee (2014a[START_REF] Gee | How to do Discourse Analysis: A toolkit[END_REF]) is used, since Gee's focus on DA is descriptive and this study aims at describing students' view of participation in an inclusive mathematics classroom to have optimal opportunities to learn.

Big and small discourses (henceforth Discourse with capital D and discourse with lowercase d) are used by Gee (2014a[START_REF] Gee | How to do Discourse Analysis: A toolkit[END_REF] as theoretical notions in the use of DA. Here Discourse(s) are describing a social and political context and are always embedded in many various social institutions at the same time. For example, a Discourse can be "school mathematics". Discourses are language plus "other stuff" (Gee, 2014a, p. 52), such as actions, interactions, values, beliefs, symbols, objects, tools and places. Small d discourse has a focus on written and spoken language in use, what stretches of languages are visible in the stories we investigate (Gee, 2014a). Stretches of languages is Gee's notion to describe small conversations within the stories. In this study, big and small discourses will be the theoretical perspective. Gee provides a toolkit for analysing different forms of language, both spoken and written. These tools highlight the communication and pose questions to the text to investigate what is beyond the text in terms of Discourses and discourses. In this paper, the toolkit is used as a methodological tool and is exemplified in the section data analysis. Summarising, DA is used both as a theory and an analytical tool and provides a set of theoretical lenses in this study.

Data analysis

In this paper seventeen interviews, five with Veronica, six with Ronaldo and six with Edward and eight classroom observations have been analysed. The observations were used as contextualisation for the interviews as well as for supporting construction of Discourse(s). The analysis of the data was guided by questions asked to the text. These questions were adapted from Gee's (2014b) toolkit, and with their help both small and big discourses could be constructed. That is, the questions were posed to the text and the answers made stretches of language(s) visible, indicating small discourses. When adding analysis of the data from the observations, such as text on the blackboard and events in the classroom, big Discourses could be construed. The analysis opened up for the construction of three Discourses, the Discourse of assessment (described in [START_REF] Roos | The Influence of Assessment on Students' Experiences of Mathematics[END_REF], the Discourse of accessibility (described in [START_REF] Roos | Challenges at the border of normality: Students in special educational needs in an inclusive mathematics classroom[END_REF] in mathematics education and the Discourse of mathematics classroom setting. In this paper the Discourse of mathematics classroom setting is in focus.

The Discourse of mathematics classroom setting

From the stretches of language identified in the texts from interviews with Ronaldo, Edward and Veronica, three (d)iscourses were construed, indicating a (D)iscourse of mathematics classroom setting influencing the students' participation. The three small discourses are described below.

The discourse of classroom organisation

This discourse is construed by stretches of language where the students highlight how the organisation of the education influences their participation.

A topic visible in the talk of all three of the students was the use of the textbooks. The textbook used in mathematics in both classes has six chapters divided into sections with specific mathematical areas. Each section has four levels, where "level one is easy tasks and level four gives real challenges". Veronica says "I start at level one or two [in the textbook]. Well, it is easier to start, so that you don't start with something difficult right away, that you don't master. Because it is a little easier. So, I don't make mistakes later on. I usually work by myself". She mentions the textbook in relation to preparing for tests "Before a test, we have the mixed tasks that sum up what we have done in the whole chapter." Also how to write in her notebook "well, it's like these small squares in it, so I usually try to write in them, so it doesn't get too big so you have space". Even Ronaldo mentions the notebook, and it seems like he struggles with the writing of the tasks "well it gets, if I write it often gets bloody smudgy, it gets so bloody messy [in the notebook]". Edward mentions the textbook as the one deciding the severity "It is the book that chooses what level you want". He also talks about the levels in the book. "I skip it [skip the first out of four levels in each chapter of the text book], it is too easy (laughing). To have a soft start in the specific area [he starts with level two]". For Edward the textbook also guides him in solving tasks, this is visible when he says that "I look in the key first, or I test first, then I look in the key and then I can see how it should be". Even Edward talks about the notebook in terms of what he doesn't write "when I just calculate ordinary in my book, in the textbook, then it's mostly mental calculations". […] "It is more convenient not to write everything down". In the observation notes it is visible that teaching by the textbook, where the students sit and work separately or in pairs, is the most common way the mathematics education is organised in both classes. Ronaldo talks about the textbook in terms of sections he feels secure or insecure of as well as levels in the textbook: "Well, I start, I think that number one has become easier now [level one in the textbook], because it feels like I have become better at math now, so I sort of starting with number two.

Another topic regarding organisation influencing the students' participation being visible in the talk of all three students was to talk, discuss and work with peers in the classroom. Veronica talks about this as a hinder in her participation "Well, I have always been like afraid of that if I raise the hand, I am wrong and everybody thinks like.. that you are.. like… […] I get unsure of myself, if I am right or wrong, and don't dare…" She also feels insecure when having discussions with peers in the classroom "Well, I have trouble with explaining… I don't know why. I don't know what to say, so they (the peers) get it. Ronaldo sees discussions both as a hinder and as a help for his participation. When he talks about discussion, he says that he can discuss "in groups, but not as much in the whole class. Or I can, but I don't want to." He says that the reason is that he is uncomfortable, but in a group situation he perceives it as a help "I like to cooperate. It is fun to be with others, like Edward or Leo, because they are really good at math and they can explain really well. So you get it more". Though, it depends on whom he cooperates with, "I feel comfortable with everybody in the group sort of … well some I can trust, or I know better than others. So it depends on whom you have in the group". Also for Edward it´s an issue of with whom to discuss. "It's not super easy… because often I have come a lot further, so I have to explain to them … it never happens that I discuss … I mean with somebody else, that we discuss like that.. […] It depends on whom I sit next to". When Edward gets the question if there is somebody in the class that he feels that can challenge him, he says that there are two, but he does not sit next to them, but he would like to "I think I would get more out of it".

Going through2 was yet another topic all of the three students discussed. Veronica reflected on when she thinks she learns the best by saying "I think it is when it is going through and when I work by myself actually". Describing what's good with going through, she says that "I don't really know (laughs), it's just nice when he [the teacher] stands there and talks, shows and explains". Ronaldo talks about going through in relation to him forgetting and implies that others think he should have more going through by saying: "I was supposed to have an extra going through like this, it is just as well, so that I don't forget that, too". He refers to going through in a negative way "it is so bloody much of going through now. It is so boring, you can't stand listening" and he says that "going through does not matter that much I think". Although, he thinks that "sometimes it [going through] can be good, but well, if you are entering a new area, you are supposed to work with, it's good to have a going through. But then, like we have now, a going through each lesson, about the same thing. It is … well I do know it´s just to repeat and all that, but it is so damn hard". This implies going through is a challenge for Ronaldo's participation. Edward says that he listens and learns from going through, but not always. He thinks they have a lot of going through, but that they aren't always good, and when they interfere with his previous way of solving, it gets messy. "I think it gets messy when you have to mix in a lot of stuff to think, because I have my own way of thinking, so it only gets messy to mix in something else". He also thinks the tasks in the going through are basic "it is not really that advanced up on the blackboard, but, well, it´s enough that I get the basics, then I can work from there" implying he would like more challenges, and that he is left alone to work further. Edward mentions that he liked the going through of a substitute teacher "I thought it was good when we had that secondary teacher, then I learned a lot in the going through.

[…] I learned a lot, because it was kind of on another level. It feels like it was a much higher level than the regular teachers".

Yet another topic talked about by all the students was teaching approaches. Veronica talks about the need of hearing stuff, "it is good to listen [to the teacher], like I always learn a little more when we have going through". She likes it when the teachers "talk". She also states that when she is working by herself, she learns the best. Also, she likes problem solving, because "you can choose what you want to use, and, like which way you [can] think. […] It is really fun!" Ronaldo on the other hand talks about problem solving as something he "despises more than anything" as he finds it hard to "connect the text to the task, it is too much". When asked how the mathematics education could be more fun for him, he says that he doesn't know and "I just don't like math, or I think it´s interesting, but difficult". When he elaborates on teaching approaches he likes, he says "[…] you should have more math games or something". In another interview he elaborates on how he learns best and says "not just sit down and work, but like be more active also, you might do some math outdoors, or like do math games or something, not just sit down with the text book all the time, it gets so bloody trite, or like really boring in the end. So, vary things". Here, Ronaldo indicates that he would like variations of ways of working to enhance his participation. This is emphasized in another interview when Ronaldo contrasts variations with working in the textbook. "It is a bloody lot of [work in the textbook] … you could play more math games or stuff like that". Even Edward highlights this contrast when he states "when you are doing more practical stuff, then it is fun, instead of having the nose in the textbook all the time." "Instead of keeping turning pages, drawing lines and writing the number of the task and all that, these things takes such a long time", he would like "those whiteboards in front of you, and sit and sketch and experiment. Because then it's much faster. I want to spend the time on the math". Hence, Edward thinks that the work with the textbook and notebook hinders him from spending time on math, consequently it hinders him from participation.

The discourse of being in a small group outside the classroom

This discourse focuses students' talk about being outside the classroom in a small group. This is mostly mentioned by the students struggling to get access in mathematics, Ronaldo and Veronica. Both Veronica and Ronaldo address feelings when they talk about being in a small group sometimes.

When Veronica was asked if she has been outside the classroom during lessons, she answers "Yes, god yes!" and laughs. Veronica often talks about being in a small group outside the classroom and often says that it "feels good", and that she feels secure in the small group. The reason why she feels good is that "you get help right away and doesn't have to sit and wait (for help) so long" and "it is an extra time, so if you didn't get it when Oliver [the ordinary mathematics teacher] did the going through, you get it [the going through] once more", "It's like you get an extra occasion. If you don't get it the first time, you can get one more time". Also she says that she feels good because "It´s less people, it like just three or four persons". Ronaldo says that "I dare to say stuff to, it feels like I am developing more" and "when you are in smaller groups, you dare to say more", when he talks about being outside the classroom. He contrasts it with the way it was before and says that "it has become a lot better now, we have started to be outside [the classroom] in small group, which we didn't do before, and it is much better now. I concentrate better and it is peaceful and quiet". This implies he needs a place where he can feel secure and a place where it is calm to be able to concentrate. Also, he connects being in a small group to the level of security in relation to the mathematical content "But if I feel in between [of being sure or unsure], or a bit insecure like don't really know, then I go [to the small group]". Even Edward talks about the small group, but in a different way. When he gets the question if he is in the small group sometimes, he laughs and says "No, but she [the special teacher] takes the ones who wants to go through the basics, she usually takes [them] with her and does some going through and stuff like that. […] I don't think I would get something out of it, I don't." Here Edward talks about "them", as the ones who want to go through the basics" and he does not include himself in that group. The observation notes show that on several occasions the special teacher brings a few students with her out into a small room beside the classroom. Often it is Ronaldo and Veronica.

Discussion

The Discourse of mathematics classroom setting pinpoints critical issues for the three students' participation in the mathematics education in terms of the organisation and being in a small group. It seems like the Western way of teaching with a one-size-fits-all mathematics education [START_REF] Askew | Diversity, Inclusion and Equity in Mathematics Classrooms: From Individual Problems to Collective Possibility[END_REF] influences the setting of the mathematics classroom, which is visible in the students' description of the use of the textbook, the mathematical discussions and the going through. This certainly "turns diversity into a problem" (Askew, 2015, p. 129), since there seem to be contradictions in the students' stories, even though they talk about the same issue, it's not in the same way. For instance, Veronica thinks she learns best when listening at going through, but Ronaldo seems mostly frustrated, and Edward thinks they are too basic. Also, when Ronaldo talks about going through in relation to him forgetting, he is placing the struggle getting access to mathematics within himself, creating frustration. Hence, how to realize going through in the classroom is a critical issue. The "problem with diversity" is also seen in the way the students describe discussions. Veronica feels insecure and does not know how to explain her thinking to her peers. Ronaldo on one hand feels the same as Veronica, insecurity when it comes to whole class discussions, but on the other hand he really likes discussions with peers, mostly because they can explain well to him. Hence, it is not him discussing, but using the peers, for instance Edward, as tutors. Edward recognises this, that he has "to explain to them", and he feels that most often he does not get any challenges in the discussions. In this study, textbooks and the work the students do in the notebooks, seem to govern the mathematics education, and is an important factor for participation. This can be seen in the way the students describe how the different levels in the textbook affect their participation, and how they are supposed to write in the notebooks. Thus, it is not just the decisions of the teachers influencing the setting, but also the textbook and the mathematical norms that are established, play a crucial role. This makes the textbook, how it is organized and how it provides access to knowledge, as well as how to write in the notebook important factors to consider, which is also seen internationally [START_REF] Fan | Textbook research in mathematics education: development status and directions[END_REF]. Is being in a small group outside the classroom including or excluding the students? Both students struggling to get access talk about the small group as a secure space and the classroom as an insecure space. Edward on the other hand laughs and does not think he would gain anything from the small group. Veronica answers "Yes, god yes!" and laughs when she is asked if she has gone out of the classroom sometimes. The laughter can be an indication of roles taken by the students as either students in access or in struggle to get access, and the roles seem to influence the students' participation. This is indicated by the fact that the struggling students feel insecure in the classroom situation. This is in line with the finding of [START_REF] Civil | Participation in the Mathematics Classroom: Does every student have a voice? For the learning of mathematics[END_REF], that participation is influenced by organizational structures. Consequently, the result in this study calls for a critical question: Is the inclusive classroom really inclusive for the students in terms of participation and access to mathematics?

In the Swedish grading system A is the highest grade, then B, C, D, E and F. E is the lowest acceptable grade and F is the grade you get when you don't pass.

In Swedish mathematics education, it is common for the lesson to start or end with a "going-through" (in Swedish, genomgång).[START_REF] Andrews | Gjenomgang and Genomgång: Same or Different[END_REF] point out three instances of what can be considered "going-through": when the teachers inform the students of what to work with, when presenting new models, and when demonstrating solutions to problems the students find difficult.


Angeliki Stylianidou
email: a.stylianidou@uea.ac.uk

Elena Nardi
email: e.nardi@uea.ac.uk

Mathematical discourses of a teacher and a visually impaired pupil on number sequences: Divergence, convergence or both?

Keywords: Discourse, inclusion, VI pupils

Research on the inclusion of visually impaired (VI) pupils in mainstream classrooms is underdeveloped in mathematics education. This paper investigates mathematical discourses of a teacher and a VI pupil in a classroom episode on number sequences. We see that, while the teacher considers the pupil's enthusiastically demonstrated mathematical contribution which differs from the institutional one, she draws on it to guide him in responding within the institutional mathematical discourse. We then see the pupil responding within the teacher's expectations but less enthusiastically. We explore the teacher's reaction in terms of whether it evidences valuing of the pupil's initial mathematical contribution and whether it is potentially excluding. We propose an alternative, more explicitly inclusive way forward in which the teacher can play along with nonprevalent mathematical contributions and bring these also to the whole class, to the benefit of all.

Introduction

Inclusive education has been an issue of international consideration especially since the Convention on the Rights of Persons with Disabilities (CRPD) (United Nations, 2006). According to Article 24 of the CRPD, the signatory countries are committed to ensuring the right of persons with disabilities to education within an inclusive education system. Educational benefits from inclusive education are not limited to persons with disabilities though: "[T]here are educational benefits for all children inherent in providing inclusive education" (UNICEF, 2012, p. 11). But is this good intent implemented in the classroom? And if so, howand, how is it experienced by staff and pupils?

In the study we draw from in this paperpart of the first author's doctoral thesiswe focus on the inclusion of VI pupils in mathematics lessons. We see inclusive education as occurring when the VI pupils are invited to participate in a lesson activity on an equal basis with everyone else in class, albeit not necessarily with the same sensory, material and semiotic tools. Apart from equitable participation, we consider equitable value as an equally important element of inclusive education. With the latter, we denote the value attributed to VI pupils, even in those contributions which differ from the prevalent, institutional ones, potentially as a result of the different tools through which VI pupils may construct and convey mathematical meaning. Reference to participation and value as contributors to inclusive education is present in international documents (e.g. United Nations, 2006) as well as in research studies (e.g. [START_REF] Nardi | Feeling' the mathematics of disabled learners: supporting teachers towards attuning and resignifying in inclusive mathematics classrooms[END_REF]. We consider equitable participation and equitable value as interrelated elements of inclusive education: our study's rationale is that one element exclusive of the other does not suffice for implementation of inclusive education. Our discussion of the data we present in this paper evidences the necessity of both elements for what we call "inclusive education".

In this paper, we first discuss key developments in the inclusion of VI pupils in mathematics classrooms. We then zoom in on the study's focus and research questions and present the study's theoretical framework and methodology. Finally, we sample from the study's data and analysis with an episode from a Year 1 (Y1) class.

Literature review and theoretical framework

A limited number of studies have been conducted in the area of inclusion of VI pupils in mathematics classrooms. The existing literature focuses mainly on the following themes: VI pupils' experiences in mainstream mathematics classrooms (e.g. [START_REF] Bayram | An exploratory study of visually impaired students' perceptions of inclusive mathematics education[END_REF]; VI pupils' forms of accessing, expressing mathematics and development of inclusive teaching strategies (e.g. [START_REF] Nardi | Feeling' the mathematics of disabled learners: supporting teachers towards attuning and resignifying in inclusive mathematics classrooms[END_REF]; and, design of inclusive mathematics teaching and learning materials to be used by VI and sighted pupils (e.g. [START_REF] Leuders | Tactile and acoustic teaching material in inclusive mathematics classrooms[END_REF]. While the existing literature has certainly set the foundations towards more inclusive mathematics classrooms, research studies that investigate how VI pupils are included in the classroom are sparse. We argue that an investigation of VI pupils' inclusion in mainstream mathematics classrooms combined with the existing foundations for such an inclusion can lead to even more inclusive mathematics classrooms. As our study explores (e.g. in the episode sampled in this paper), there may be significant gaps between inclusive education policy and practice: while appropriate inclusive education aspirations may appear in international and national policy documents, institutional, curricular and attitudinal constraints may lead to these gaps. Our study aims to investigate these and eventually contribute towards their reduction.

Our study has two phases and investigates: (a) how inclusion and disability are constructed in the discourses of teaching staff and pupils in mainstream mathematics classrooms (both phases); (b) how collaboratively designed mathematics lessons impact upon teaching staff's and pupils' discourses on inclusion and disability (Phase 2). We endorse a sociocultural theoretical framework that draws upon: Vygotskian sociocultural theory of learning [START_REF] Vygotskii | Mind in society: the development of higher psychological processes[END_REF]); Sfard's discursive perspective, known as the theory of commognition [START_REF] Sfard | When the rules of discourse change, but nobody tells you: making sense of mathematics learning from a commognitive standpoint[END_REF]; the social model of disability [START_REF] Oliver | Understanding disability: from theory to practice[END_REF]; and, the theory of embodied cognition [START_REF] Gallese | The brain's concepts: the role of the sensory-motor system in conceptual knowledge[END_REF]. We discuss such use briefly in [START_REF] Stylianidou | Inclusion and disability in the primary mathematics classroom: Examples of teaching staff discourses on the participation of visually impaired pupils[END_REF] and exemplify this use in this paper.

In the episode we present in this paper, we focus on the discursive activity of a teacher and a VI pupil, in interaction concerning number sequences. Our Vygotskian influence is evidenced in our focus on: the semiotic and sensory tools used by the two interlocutors in the setting of the school classroom; and, the mathematical meaning each interlocutor conveys as a result of using the particular tools. We show how particular elements from Sfard's discursive perspectiveword use and visual mediatorsare played out in the mathematical discourse of each interlocutor and we examine a case of commognitive conflict [START_REF] Sfard | When the rules of discourse change, but nobody tells you: making sense of mathematics learning from a commognitive standpoint[END_REF] arising as a result of the teacher's and the pupil's different, incommensurable discourses. The visual mediators in the particular episode constitute the gestures, made by both the teacher and the pupil. Apart from speech, the gestures act too as a vital tool for mathematical communication and the construction, and conveying, of mathematical meaning. This role of gestures makes the theory of embodied cognition [START_REF] Gallese | The brain's concepts: the role of the sensory-motor system in conceptual knowledge[END_REF] pertinent in our analysis of the episode.

In what follows, we present the study's context, participants and methods. We then sample from the data with an episode from a Y1 lesson on number sequences. We conclude with implications that this episode has for our ongoing analyses, particularly those included in the first author's doctoral thesis.

Methodology: the context, participants and methods of the study

Data collection was conducted in four UK mainstream primary mathematics classrooms (Y1, Y3 and two Y5 classes; pupils' ages: 6-10). The VI pupils' presence and the willingness of teaching staff and pupils to participate in the study constituted our criteria for the selection of the classrooms. We collected data after securing ethical approval by our institution's Research Ethics Committee and ensuring participant anonymity, confidentiality and right to withdraw from the study.

We collected data through observations of 29 mathematics lessons (33.5 hours in total); individual interviews with 5 class teachers (6 interviews, 2 hours and 10 minutes in total); individual interviews with 4 teaching assistants (6 interviews, 2 hours and 15 minutes in total); focussed-group interviews with 35 pupils (16 interviews, 2 hours in total); 2 ten-minute individual interviews with one pupil; written transcripts of the teaching staff's contributions in the design of the three Phase 2 lessons; photographs of the pupils' work in the three Phase 2 classes; and, pupils' evaluation forms of the Phase 2 lesson in two classes. During observations, written notes were kept in all lessons. 21 lessons were audio-recorded and 14 lessons were also video-recorded. All interviews were audiorecorded, except four, following interviewee requests. For these, written notes were kept instead.

Data collection for Phase 1 was completed in March 2018 and for Phase 2 in July 2018. Data analysis is ongoing. While a major rationale for Phase 2 is to explore the impact of the co-designed mathematics lessons upon the participating teaching staff and pupils, we note that the Phase 2 episode we have selected to discuss in this paper does not stress the elements of co-design and its impact upon the participants. Instead, our emphasis here is on the mathematical discourses of a teacher and a VI pupil in a classroom incident in which we discern a genuine mathematical contribution by the pupil that diverges from the one expected by the teacher. We discuss the different mathematical discourses of the teacher and the pupil and explore the challenges of implementing inclusion. We first present a factual accountand then a preliminary analysisof the episode. We conclude with a discussion of the episode in the context of the entire study.

A Y1 episode A factual account of the episode

The episode is from a lesson on number sequences in a Y1 class. Ned is the VI pupil of this class and has severe, congenital visual impairment in both his eyes. The class has two general teaching assistants who support pupils that need help at particular instances and their role does not focus on supporting the VI pupil specifically. The class was asked to find the next number in a number sequence, which was on a worksheet that each pupil had in front of them. The number sequence was 8 10 12 14 16. The following dialogue occurs between the teacher and Ned:

1

Teacher: Is it increasing or decreasing? 2 Ned, happily: Creasing. Ned puts his hand straight up, doing a similar gesture to the one the teacher had done when, earlier, she explained "increase". 3

Teacher: Increasing or decreasing? "Increase" means it gets bigger, or is it getting smaller. Whilst saying "bigger", the teacher does a similar gesture to the one Ned had just done. Whilst saying "smaller", she does a gesture, too, pointing down. 4

Ned:

Increasing. He says this with less enthusiasm than before and without doing any gestures this time. 5 Teacher: Yeah. You are right.

A preliminary analytical account of the episode

Ned's reaction to the teacher's question (2)

Ned responds to the teacher's question in (1) by resorting to verbal and gestural discourse. His word "creasing" may suggest that he does not recall the difference in the meanings of the words "increasing" and "decreasing". Ned uses the root word, "crease"1 , of both verbs to show, possibly, that he refers to one of the two verbs, although it is unclear from his speech per se whether he means "increasing" or "decreasing", thus whether he answers the teacher's question correctly.

His gesture, which accompanies his speech, makes it clear that Ned refers to the term "increasing".

Here, the gesture is vital in Ned's conveying the mathematical meaning he has constructed of what an increasing sequence of numbers is. Ned's re-enacting of the teacher's gesture for the term "increasing" resonates with the theory of embodied cognition, according to which concepts are embodied [START_REF] Gallese | The brain's concepts: the role of the sensory-motor system in conceptual knowledge[END_REF]: "the sensory-motor system can characterise a sensory-motor concept, not just an action or a perception, but a concept with all that that requires" (p. 468).

As mentioned above, Ned's speech alone does not suffice for our understanding as to whether Ned refers to "increasing" or "decreasing". Similarly, we can argue that his gesture alone does not suffice either for our understanding of Ned's mathematical expression. His gesture, isolated from his speech, may signify a variety of things, one of which is the term "increasing". The combination of Ned's speech and gesture, though, strengthens our speculation that Ned refers to the term "increasing".

The contribution of both speech and gesture in Ned's mathematical expression, and in our interpretation of it, demonstrates how merging Vygotskian (1978) sociocultural theory of learning and the theory of embodied cognition [START_REF] Gallese | The brain's concepts: the role of the sensory-motor system in conceptual knowledge[END_REF]) is important in our analysis. Take, for example, Ned's response in (2) where we need both his speech and his gesture to grasp how he conveys what he means by "increasing". Both theoriesspeech as the vital tool for mathematical meaning making and expression [START_REF] Vygotskii | Mind in society: the development of higher psychological processes[END_REF]) and the embodied nature of concepts [START_REF] Gallese | The brain's concepts: the role of the sensory-motor system in conceptual knowledge[END_REF] are combined to help us analyse Ned's response in (2). One theory, deployed exclusively, would not suffice for our interpretation of Ned's response.

In (2), we mention the adverb "happily". With our use of an adverb that indicates emotion, we aim to show that we are interested not only in what our participants utter but also in how they feel as they do so. The 'how' part may be expressed with tone of voice, facial expressions and gestures and it provides us with information that frequently adds to, and further illustrates, the 'what' part. Ned's happiness is evidenced in his tone of voice and facial expression -when he says "creasing"as well as in his gesturewe see excitement in the way he gestures. We attribute his happiness to a sense of fulfilment he may derive from the correctness of his answer. Later, we see that this correctness is cast into doubt, if not disapproved, by the teacher. Our exploration of the relation between emotion and cognition resonates with discussions in the literature of the relation among mathematical achievement, enjoyment and self-efficacy in mathematics. For example, we agree with Goldin, Epstein, Schorr, & Warner (2011, p. 553) that "affect, cognition and motivation interact to influence students' mathematical engagement in classroom social environments".

Another reason why we have included a reference to Ned's emotion is to later compare this emotion to that in his second answer, in (4), and associate the fluctuation between the two with the teacher's reactions in the two cases.

Teacher's reaction to Ned's response (3)

The teacher does not explicitly react to Ned's response, either by approving it or by disapproving it. We see in her response, though, implicit dissatisfaction: Ned's answer does not include any of the two terms that the teacher expects to hear. Hence, the teacher repeats her previous question, albeit with a different sentence and tone of voice.

Her omission of the part "Is it", which is in (1), and her direct use of the verbs separated with "or" show her emphasis on these two verbs, which she seems to expect from her interlocutor too. She pronounces the underlined prefixes "in" and "de" in (3) in a different tone of voice. This may indicate some concern about Ned's wording, which does not include any of the two prefixes.

We interpret the teacher's explanation of each of the two terms in (3) as showing her interpretation that Ned may not recall the meaning of the two terms. As with her question, her explanations suggest, too, her emphasis on the two mathematical words, which she expects to hear from Ned.

While the teacher's mathematical discourse is mostly verbal-institutional and the teacher expects such kind of discourse from Ned, too, in this part of the episode, the teacher accompanies her speech with gestures. Thus, she creates a verbal-institutional and gestural mathematical discourse, with each of its componentsspeech and gestureserving the same purpose: that of explaining the meaning of the key terms. We interpret the teacher's use of a similar gesture to Ned's one as a manifestation of "attuning" (in the sense of [START_REF] Nardi | Feeling' the mathematics of disabled learners: supporting teachers towards attuning and resignifying in inclusive mathematics classrooms[END_REF] to the gestural part of her pupil's mathematical contribution and of her approval of gesturing as a form of mathematical expression, albeit not independently from 'approved' speech. Through her gesture, we see that the teacher considers Ned's gesture, too, butalongside speechshe draws on it to guide him in responding within the mathematical discourse which she, and the educational institution she represents too, approves. Therefore, the teacher seems to be concerned about Ned's wording but she seems to partially approve his gesture. Her expectation is of an answer within the verbal discourse aligned with institutional (the National Curriculum's) standards. The teacher's attuning to Ned's gesture resonates with the first camp of participants in (Nardi et al., 2018, p. 157): that of using the VI pupil's contribution to focus on a more conventional contribution.

Drawing upon [START_REF] Sfard | When the rules of discourse change, but nobody tells you: making sense of mathematics learning from a commognitive standpoint[END_REF] theory of commognition, we see evidence of commognitive conflict in the exchange between the teacher and Ned. We define commognitive conflict as the situation occurring when seemingly conflicting narratives originate in different, incommensurable discourses [START_REF] Sfard | When the rules of discourse change, but nobody tells you: making sense of mathematics learning from a commognitive standpoint[END_REF]. We see the different mathematical discourses of two interlocutors, both of whom aim to convey their mathematical meaning of an increasing sequence of numbers. We see the commognitive conflict in the teacher's non-playing by the meta-discursive rules set by Ned: she considers Ned's speech in isolation from his gesture and vice versa. Indeed, as discussed in (1), Ned's speech isolated from his gesture -and Ned's gesture isolated from his speechdo not suffice for conveying the meaning he seems to have constructed of an increasing number sequence. We see that the commognitive conflict arose as a result of the teacher's consideration of Ned's mathematical discourse as consisting of two separate, unrelated elements: speech and gesture. We argue that, had the teacher considered Ned's verbal-gestural discourse as an entity, commognitive conflict might have been avoided: instead, an acknowledgement of two different, but equally acceptable, mathematical discourses might have occurred.

Ned's reaction to the teacher's response on his answer (4)

We then see that Ned changes communicational mode from verbal-non-institutionalwhich was seen as problematic by the teacher (3)and gesturalwhich was partially accepted by the teacher (3)to verbal-institutionalwhich, as he knows from (3), is accepted by the teacher. Here, Ned responds to the teacher's question by playing by the meta-discursive rules set by the teacher, which, as discussed in (3), puts the emphasis on endorsed forms of speech. However, his shift in communicational mode does not seem to be a pleasant experience. His tone of voice and his facial expression evidence this reduced enthusiasm. This raises issues on how included Ned feels his mathematical contribution in (2) is. We see that in (3) the teacher's non-developing of Ned's mathematical contribution in its own right -the camp of teachers' developing of VI pupils' contributions in their own right is the second camp discussed by Nardi et al. (2018, p. 158)is excluding. The teacher restricts Ned to responding within the boundaries of an institutionally endorsed mathematical discourse and this limitation seems to dissatisfy Ned. Drawing upon the relation between affect and cognition, we associate Ned's apparently reduced enthusiasm with the teacher's concern over his initial contribution and her expectation of his responding in one, particular way. Ned's non-gesturing here may be attributed to several reasons. It may be attributed to the non-contribution of a gesture to his conveying of his mathematical construction of an increasing number sequence, since his use of speech suffices as an answer to the teacher's question, unlike in (2). It may also be attributed to Ned's interpretation from (3) that gesturing is not a necessary, or even appropriate, form of mathematical communication.

Teacher's reaction to Ned's response (5)

We then see that the teacher clearly approves Ned's response: his response in (4) resonates with the teacher's expectations, in which speech seems to dominate over gestures.

Discussion of the episode in the context of the entire study

This episode is selected to evidence mathematical discourses of a teacher and a VI pupil in a mainstream classroom and discusses the extent to which mathematical communication is achieved between the two interlocutors.

Zooming out to the entire study, we see that, despite the invitation towards VI pupils to participate in the mathematics lesson on an equal basis with their peers, their mathematical contributionswhich may differ from the ones expected by the teachermay not always be as valued as they could. In other words, we see evidence of equitable participation but not equitable value in the mathematics classroom. We see this discrepancy as a hurdle to implementing inclusion. We are unsure whether this discrepancy arises as a result of undervaluing of the mathematical discourses of VI pupils, as a result of a persistent adherence to the institutional mathematical discourse, as a result of both and/or other influences. In the episode we discuss here, though, we do not consider the teacher's concern with the VI pupil's initial mathematical contribution as ableist 2 and we note that there was hardly any evidence of ableism in the observation and interview data from this teacher. We attribute her concern to this contribution's divergence from the prevalent one and we thus stress that her concern in similar cases is likely to occur with every pupil, not just with a VI one.

In the episode, we see the reported interaction as associatedbut possibly not limitedto visual impairment. We argue that the pupil's contribution may be seen through existing findings on visual impairment: for example, VI pupils use their hands to construct and convey mathematical meaning [START_REF] Nardi | Feeling' the mathematics of disabled learners: supporting teachers towards attuning and resignifying in inclusive mathematics classrooms[END_REF]. We also argue that the teacher's response in (3) was specific to the pupil's genuine mathematical contribution.

Regardless of the reason attributed to the teacher's concern with Ned's mathematical contribution, we set the following questions. Do we, as educators, need to accept only those mathematical contributions that resonate with the institutional ones? Or, do we need to reflect upon the different contributions of our pupils, discern whether they are mathematically valid too and act accordingly to include them in the lesson? Without ignoring the necessity for the pupils' familiarisation with, and endorsement of, established mathematical discourse, we propose that we should heed the reactions of our pupils, even in cases when we, either explicitly or implicitly, have concerns about mathematical contributions which differ from the institutional ones: different pupils construct different meanings of mathematics. This is how we see inclusive education implemented in the classroom: through equitable participation and equitable value. We argue that our valuing, attuning and integrating of pupils' non-prevalent mathematical contributions into lessons may not only encourage participation in the mathematics lesson but may also benefit all pupils in class, who may experience mathematics from a different, and potentially enriching, point of view.

"crease" originates from the latin word "crescere" ["grow", Oxford English Dictionary (http://www.oed.com/)].

Acknowledgements

2 Ableism is defined as "a network of beliefs, processes and practices that produce a particular kind of self and body (the corporeal standard) that is projected as the perfect, species-typical and therefore essential and fully human. Disability, then, is cast as a diminished state of being human" (Campbell, 2001, p. 44).

We thank: UEA's Research in Mathematics Education Group and Professor Anna Sfard for their invaluable feedback on the episode, discussed in a commognition workshop at UEA on 21/06/2018; the participating teaching staff and pupils for their commitment to the study; and, UEA for funding the study through a Social Sciences Faculty Doctoral Studentship.