domain: math shs

text:
Konstantinos Tatsis
email: ktatsis@uoi.gr

Bozena Maj-Tatsis

Authority structures in preservice teachers' talk

Keywords: Authority, preservice teachers, pair work

In the paper we draw on the authority structures that were observed in preservice teachers' talk while working with collaborating pairs of primary school pupils. We have found some interesting structures that emerged, especially when the preservice teachers had to consider the goals of their working sessions with the pupils, as conceived by them. Particularly, the preservice teachers switched to a more authoritative stance, usually when there was time pressure.

Introduction

In any learning environment when there are more than one participants present, various interactions take place. Among these, the verbal ones have been the focus of numerous studies in the last decades. Mostif not allof these studies originate from linguistics; in one the first papers that explicitly refer to the relationship between language and mathematics we read that "the processes of learning and communication are closely inter-related and both present the investigator with problems of bewildering complexity." (Austin & Howson, 1979, p. 162). Today, almost 40 years since that statement was made, researchers in mathematics education are still struggling to interpret the complexities of learning and communication. However, the main interests of research remain on how students learn (at the same time, what constitutes learning may be by itself the topic of investigation) and what (can) teachers do to enhance their students' mathematical thinking. In our view, mathematical thinking relates to effectively participating in the mathematical classroom community [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF]. This in turn requires a particular distribution of authority in the classroom: the teacher is expected to "step back" from the official role and give room to the students to make conjectures, pose problems and generally think like mathematicians [START_REF] Mason | Thinking mathematically[END_REF].

Based on the above assumptions, we designed our study with preservice mathematics teachers. We engaged them in a research-type activity: they were asked to find a mathematical problem and use it as a tool to enhance pupils' creative thinking. Our main interest was in the ways these preservice teachers will manage the interactions and particularly their authority in order to achieve that aim. Our theoretical framework is presented in the next section, followed by our methodology.

Theoretical framework

Language use in the mathematics classroomor, more generally in mathematics learning contextscan convey various aspects of the interactions or the mental processes that occur. There are at least two ways that we can view and analyse language use: at an object level, we may focus on the mathematical register and its use among the students and the teacher. From a teacher education point of view, this can be also related to mathematical knowledge (Rowland & Ruthven, 2011). Then, if we move to a meta-level, we realise that "sense-making in school mathematics is not solely a matter of private interpretation within some absolute, secure reality of 'real' objects ('people' and the like); it is also one of linguistic enculturation, of initiation to discursive practice". (Walkerdine, 1988, p. Thematic Working Group 09 Proceedings of CERME10 128, as cited in Rowland, 2000, p. 194). Mathematics learning is thus seen as a social process, regulated by rules and norms [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF], in which the participants are continuously negotiating meaning by exercising their agency [START_REF] Cobb | An interpretive scheme for analyzing the identities that students develop in mathematics classrooms[END_REF]. This requires a distribution of authority by the teacher. This is because verbal interactions between teachers and students are usually characterised by the teacher's authoritative stance: the teacher represents the mathematicians' community, thus s/he is an authority because of his/her content knowledge; at the same time, s/he is in authority, because of position [START_REF] Wagner | Identifying authority structures in mathematics classroom discourse: A case of a teacher's early experience in a new context[END_REF]. Thus, depending on the didactical approach, the students may only have to follow the teacher's guidance, or they may be given space to actively participate in the learning process. For the purpose of the present paper we adopt the following definition for authority: Authority concerns the degree to which students are given opportunities to be involved in decision making about the interpretation of tasks, the reasonableness of solution methods, and the legitimacy of solutions. Authority is therefore about "who's in charge" in terms of making mathematical contributions. (Cobb et al., 2009, p. 44) The analysis of authority can be done by the use of positioning theory [START_REF] Van Langenhove | Introducing positioning theory[END_REF] as implemented by [START_REF] Wagner | Identifying authority structures in mathematics classroom discourse: A case of a teacher's early experience in a new context[END_REF]. According to this, there are some pervasive "lexical bundles" (i.e. speech patterns) in classroom talk, which are related to participants' positioning and authority. The main categories of these are: personal authority, discourse as authority, discursive inevitability, and personal latitude. Personal authority, which is the most common in the classrooms, relates to teacher's own authority; in other words, the students are expected to follow (the authority of) their teacher. A characteristic case of personal authority is when the teacher says, e.g., "I want you to solve this equation for me". Discourse as authority relates to the rules which must be followed in the interaction, "which come from outside personal relationships, [and] may be attributed to the discipline of mathematics (or perhaps school mathematics)." (Wagner & Herbel-Eisenmann, 2014, p. 873) An example of discourse as authority is the utterance "We need to follow the multiplication rules". Discursive inevitability relates to events or actions that are verbally presented as inevitable; as Wagner & Herbel-Eisenmann (2014) note: "there is no explicit reference to obligation, but rather a sense of predetermination" (p. 873) An example may be the utterance: "We are going to calculate the average of the given values". Finally, personal latitude relates to the students making their own decisions during the interactions, thus exercising their own authority; this is related to conceptual agency [START_REF] Cobb | An interpretive scheme for analyzing the identities that students develop in mathematics classrooms[END_REF] and mainly expressed by students' questions.

The above are usually accompanied by other verbal strategies; speakers sometimes feel the need to convey other messages related to power or solidarity. Politeness strategies [START_REF] Brown | Politeness: Some universals in language usage[END_REF] have been the focus of a number of studies, either solely [START_REF] Tatsis | Vague language in Greek and English mathematical talk: A variation study in face-work[END_REF] or juxtaposed with authority structures' analysis [START_REF] Tatsis | Authority and politeness: Juxtaposed analyses of mathematics teaching episodes[END_REF]; these studies stress the fact that students and teachers continuously interpret each other's actions, in order to perform their own actions.

Summing up, although various verbal patterns coexist in the mathematics classroom, there are situations in which particular patterns are prevalentthese patterns characterise the interactions and influence its outcome to a significant degree. Thus, in our study we were expecting to locate and identify such patterns in preservice teachers' discussions with pupils. Our methods of analysis are described in the next section.

Context of the study and methodology

Our research was based on the work of six preservice teachers, at the third year of their studies in mathematics. They already had three weeks of practice in a primary school, during which they had taught for 18 hours per week. They had attendedamong other courses in pedagogy, psychology, psychology of mathematical thinking, and didactics of mathematicsa course led by the second author of the paper in problem solving, during which they solved problems and analysed pupils' solutions and discussions; they had also watched and analysed a video with a study that involved a pair of young children (see [START_REF] Maj-Tatsis | Investigations in magic squares: A case study with two eightyear-old girls[END_REF].

For the purpose of the present study, the preservice teachers were asked to find an interesting mathematical problem (the only clue given was it should offer the possibility for mathematical explorations) and give it to a pair of pupils from the class that they had already taught. The ages of the pupils varied from 10 to 12 years. There were no restrictions on how to choose the pair; this resulted in pupils from varying backgrounds and of various mathematical attainment. Other instructions provided to the preservice teachers were related to the distribution of their authority: they were asked to let the pupils talk and, generally to try not to impose their own way of thinking. The sessions lasted from 20 to 45 minutes and were videotaped. Then, the preservice teachers were asked to transcribe their sessions and analyse them, according to whether they had achieved their aim (as perceived by them) and the possible reasons or events that were responsible for thatincluding their own decisions and actions.

For the purpose of the present paper, we mainly focused on preservice teachers' transcribed discussions with the pupils; this data was complemented by their interviews given to the second author of the paper, after the completion of the sessions. Our analysis was a meta-level analysis [START_REF] Rowland | The pragmatics of mathematics education: Vagueness in mathematical discourse[END_REF], since we focused on utterances related to the participants' interactions and we searched for manifestations of authority structures. Following Wagner & Herbel-Eisenmann's (2014) framework we located utterances or exchanges that accounted for personal authority, discourse as authority, discursive inevitability, and personal latitude. Particularly, personal authority "was identified by the presence of first-and second person pronouns together" (Wagner & Herbel-Eisenmann, 2014, p. 873). Discourse as authority was identified by the presence of modal verbs such as "need to" and "have to", which explicitly express a strong obligation; it was also identified by the use of "they", which refers "to a non-specified entity or group who have potentially made decisions about the mathematics students encounter" (Wagner & Herbel-Eisenmann, 2014, p. 873). Discursive inevitability was identified by utterances like 'you are going to' and 'it is going to'. Finally, personal latitude was mainly identified by the presence of questions by the pupils. These initial categories were then utilised in order to establish the various authority structures; these were not predetermined, but established during the course of the analysis.

Additionally, we looked into other verbal phenomena, such as the use of personal pronouns, such as "we" and "you" [START_REF] Rowland | The pragmatics of mathematics education: Vagueness in mathematical discourse[END_REF]. During our analysis, there were instances when we had to "make a step back" into an object level analysis, by looking into the mathematical concepts and processes that were discussed and established during the interactions; this was deemed necessary in order to fully comprehend the participants' actions and to study their effect on the authority distribution.

Results

As we mentioned in the previous section, our analysis focused on the verbal interactions of the preservice teachers with the pupils. We will firstly present the analysis of the interactions between Tomek and two grade 6 boys (12 years old). We present this case because it demonstrates in a clear way the authority structure which was the most prevailing among the cases we analysed.

The case of Tomek: Problem posing that leads to generalisation

The situation given to the pupils by Tomek included the following text: John is creating "chairs" by using chips:

The text was followed by these drawings (Figure 1): Below we read the initial discussion1 between Tomek and Pupil 1:

1 Tomek: OK, the task is simple: firstly, read it. Have you ever seen something like this? 2 Pupil 1: Hmm, no, but I know how to do it. 3 Tomek: OK, so now think up of some questions. What questions would you ask? [small break] Usually, there are ready questions, and here…

Then, the pupils started working on making questions for the given situation. In turns 4-35, Tomek talked only four times, and among these only the next was related to the problem posing process:

10 Tomek: Just write your ideas.

In the next turn there is the following exchange:

35 Pupil 2: Do we have to answer every question? [which they have posed] 36 Tomek:

Hmm, later we will choose two questions which we will answer. 37 Pupil 1:

For example, what would be the sum of the second and the third figures? What would be the sum of the chips of the second and the third figures? 38 Pupil 2:

[He writes Pupil 1's question to their worksheet] 39 Tomek:

You can always draw more figures. 40 Pupil 1 or 2: Aha. 41 Tomek:

You are not limited to three [figures] only.

Then, until turn 59 the two pupils are working without any intervention by Tomek; they are making figures and discussing on the pattern. Then they draw the fourth chair shown in Figure 1.

60 Tomek:

Are you sure about that chair? 61 Pupil 1:

Why not? 62 Tomek:

Calculate once again.

The preservice teacher started his research by asking pupils to put questions to the presented situation. By such way of guiding the process he gave them the possibility to engage in problem posing by formulating their own questions. Although his request is an expression of discursive inevitability, it resulted in the pupils being interested in the task and eventually the authority was passed to the pupils (personal latitude). Up to turn 59 Tomek did give space to the pupils, by letting them discuss. By his question in turn 60 he exercised personal authority, he used direct formulation "are you sure?", and then asked them to correct it (62). We may claim that it was a good moment for an intervention, since the pupils had made a mistake and this would result in an improper rule and generalisation. However, that could be done in a less explicit way, e.g. "how did you know how many chips should be in the fourth chair?". After the boys corrected the mistake, Tomek gave them again space to work, in other words, he let them exercise their personal latitude.

In turn 70 Tomek suggested: "So maybe let's do the second question [which was formulated by the pupils and written at their worksheet: "How many chips will the next chair have?"] because I think it is the most interesting". In this utterance we can find two different authority structures: personal latitude (inclusive imperative) and personal authority ("I think"). By that intervention Tomek directed the work of pupils into generalisation. But still the question was formulated by the pupils so it can be considered as an appraisal of their work.

The next intervention of Tomek took place when the pupils put a false hypothesis:

96 Pupil 1: So we can conclude from it [the number of chips in chairs 1, 2, 3 and 4 were calculated by the pupils as follows: 5, 9, 14, 17note that 14 is incorrect] that every figure increases by 4. So, the next figure will have 21. And so on. 97 Tomek:

So, for example, how many will be in the tenth chair? Calculate precisely.

In the above transcript we see that Pupil 1 has formulated the assumption that the number of chips increases proportionally to chair's number, thus the number of chips of the tenth chair is double the number of chips of the fifth chair. This incorrect assumption was formulated despite Pupil 1's correct observation that "every figure increases by 4 [chips]". Tomek decided to resolve that situation by offering a counter-example, which falls into the discursive inevitability category:

111 Tomek: So, in that case, chairs 2 and 4 should… so, the fourth should be twice bigger than the second. 112 Pupil 1:

Oh! That's right. Something went wrong (…)

It is obvious that Tomek's intervention helped the students realise their false assumption. Generally, the previous excerpts demonstrate the prevailing authority structure in Tomek's verbal actions: he mainly let the pupils exercise their personal latitude and he exercised his personal authority and discursive inevitability structures only at crucial moments (according to him, but also according to our interpretation). In other words, if he did not intervene at these moments, the pupils could have spent much time in explorations that would not lead to the desired generalisation. However, this authority structure changed significantly in the 16th minute of the session, when the person who was handling the video camera (who was Tomek's colleague) informed him that it is the 16th minute (turn 171). Tomek switched his behaviour to a more authoritative structure, by asking closed questions, prompting the pupils and generally drastically increasing his interventions. This is eloquently demonstrated in Tomek's own account of his actions:

Generally, making such a research was very tiring. Despite my earlier preparations I did not expect that it will go in such a way. I expected that they [pupils] will calculate chair x just by adding. And they started multiplying from the beginning. I made many mistakes, it could be conducted in a better way. Thankfully, the children liked it. Close to the end, more or less at the 14th minute, I started prompting them too much because I was afraid that I will not manage in time. Because of that [prompts] I limited their mental actions.

Tomek's case demonstrated how we performed our analysis over the six chosen cases. The following authority structures were identified: a) Structure 1: Personal latitude accompanied with mainly personal authority and discursive inevitability; followed by a blunt switch to personal authority, eventually accompanied with discursive inevitability. b) Structure 2: Personal latitude accompanied with mainly personal authority and less frequently with discursive inevitability. c) Structure 3: Personal authority.

The above structures are presented according to their frequency; the first one was found in four cases, while the other two appeared in one case each.

Conclusions

Our study was inspired by studies on the authority structures that can be identified in the mathematics classroom [START_REF] Wagner | Identifying authority structures in mathematics classroom discourse: A case of a teacher's early experience in a new context[END_REF]; we were interested to see whether similar structures would appear in preservice teachers' talk. Our results have shown that the preservice teachers of our study initially had been less authoritative, thus allowing the pupils to work collaboratively and exercise their personal latitude. However, during the interaction, and due to some events, they switched into more authoritative structures. One of the basic reasons for that switch was the time pressure together with the obligation to fulfil the request of their educators. The interesting thing was though that all of them were aware of this switch, as it was expressed during their interviews.

Following [START_REF] Cobb | An interpretive scheme for analyzing the identities that students develop in mathematics classrooms[END_REF], we agree that the distribution of authority is closely linked to the ways that students exercise their agency in the mathematics classroom, including, as our study has shown, collaborative work. Thus, the teacher should be aware of that fact, in other words, s/he needs to know when it is time to exercise authority, and when s/he can deviate from the lesson plan; this is related to the notion of contingency as presented in [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] "knowledge quartet" framework. Particularly, the teachers' responses to students' ideas as well as their ability to deviate from their agenda affect the knowledge construction in a significant way. These considerations should be discussed and analysed in pre-and in-service teachers' training courses, preferably by involving teachers in self-monitoring with regards to the distribution of authority during their interactions with the students.

In our study, the preservice teachers' awareness of the nature and the effect of their own verbal actions was a clear result of their training courses. However, this did not stop them from exercising their personal authority at particular moments. Thus, there is still lots of work that needs to be done in the field of authority distribution by mathematics teachers in order to establish an active learning community in their classrooms.

Figure 1 .

 1 Figure 1. The "chairs" made by chips.

Proceedings of CERME10

All the discussions were translated from Polish by the second author of the paper.Thematic Working Group 09Proceedings of CERME10


text:
Alexander Schüler-Meyer
email: alexander.schueler-meyer@math.tu-dortmund.de

Multilingual learners' opportunities for productive engagement in a bilingual German-Turkish teaching intervention on fractions

Keywords: Mathematics education, multilingualism, positioning theory, agency

A teaching intervention should provide learners with opportunities for productive engagement with mathematics. Teaching learning situations are organized in patterned ways along culturally shared expectations of how these situations unfold. However, in multilingual classrooms, expectations of language learning can compete with mathematical expectations. Drawing on positioning theory, I will reconstruct these expectationsstorylinesin a bilingual Turkish-German teaching intervention, specifically for the case of two learners, Akasya and Ilknur. I will show how the teacher establishes two expectations, one of "deciding who has downloaded more gigabyte" and one of "giving a Turkish explanation". After some effort on forming an explanation, Ilknur's and Akasya's shift their activities towards fulfilling the second expectation, which results in them being less agentic for the mathematics at hand. Instead, Ilknur and Akasya focus on the form of their explanation.

Introduction

As learners construct mathematical knowledge by participating in social practices, they need to engage in interpretations and reflections on the meaning of mathematics, rather than being mere "receivers of predetermined knowledge" (Boaler & Greeno, 2000, p. 179). Productive engagement is an engagement in the classroom where learners are making each other understandable and where the students are responsible for the mathematics at hand, in ways that allow the learners to actively develop their interpretations of knowledge involved in a teaching-learning environment [START_REF] Turner | English learners' participation in mathematical discussion: Shifting positionings and dynamic identities[END_REF].

Multilingual mathematics learners are especially at risk to not engage productively. For example, it has been documented that many multilinguals do not actively engage in classroom discussions because of the demand to speak in the language of instruction [START_REF] Planas | Bilingual students using their languages in the learning of mathematics[END_REF] or that a focus on correctness in the language of instruction excludes multilingual learners and interrupts the engagement with mathematics (Planas & Civil, 2015, p. 47). Such mechanisms of inclusion and exclusion from productive engagement can be understood with positioning theory, where individuals in a conversation dynamically take up specific rolespositioningsand act according to the duties and rights that come with these positionings. To make sense of their positionings and their rights and duties, learners relate to storylines, where storylines stem from culturally shared patterned ways in which conversations are organized [START_REF] Wagner | Re-mythologizing mathematics through attention to classroom positioning[END_REF]. For example, in a storyline where the teacher asks questions and evaluates students' answers, the teacher is positioned to evaluate these answers. The learners are positioned with the duty to give answers, but also with the right to not contribute for a certain time. When the learners perceive this storyline as "I may be negatively evaluated", they may refuse to contribute, and might be less productively engaged with mathematics.

This study addresses two questions: How does the use of two languages in a bilingual teaching intervention influence the learners' positionings, and how does this affect their productive engagement with mathematics? Accordingly, the paper presents the here employed theoretical framework of positioning theory; analyzes data from a bilingual Turkish-German teaching intervention on fractions, specifically the case of Ilknur and Akasya; and shows that by switching the understanding of the storyline of the intervention "giving a Turkish explanation" towards "giving an answer with the form of an explanation", the learners are less agentic for the mathematics at hand.

Positioning theory for investigating productive engagement in multilingual settings

In a teaching-learning situation, teacher and students fluidly take up certain roles, similar to the roles actors take up in a stage play, where these fluid roles are called positionings. "Positioning […] is the discursive process whereby selves are located in conversations as observably and subjectively coherent participants in jointly produced storylines" (Davies & Harré, 1990, p. 48) The individual in a conversation will take up, resist or reframe positionings as it tries to be subjectively coherent in his or her actions. At the same time, the actions of the individuals become intelligible to the others as they assume that the individual's actions are coherent with his or her position (Langenhove & Harré, 1994, p. 362). The theoretical construct of positioning resonates with classical interactionist perspectives on classroom interaction in its assumption that learning, identities and even competence are a result of social interactions (e.g. [START_REF] Cobb | The emergence of mathematical meaning: Interaction in classroom cultures[END_REF]. Positioning theory has been used in mathematics education to investigate classrooms in regard to issues of opportunities to participate.

Storylines are jointly produced, they "stem from culturally shared repertoires" (nurse/patient; coach/athlete) [START_REF] Wagner | Re-mythologizing mathematics through attention to classroom positioning[END_REF] and the culturally shared patterned ways in which conversations are organized. Different storylines provide learners with different means for productive engagement [START_REF] Herbel-Eisenmann | Positioning in mathematics education: Revelations on an imported theory[END_REF]. The use of more than one language reinforces these issues, because it brings more diverse storylines to the classroomfor example of the political value of languages or storylines of language learningthat might affect mathematics learning. Studies on multilingual mathematics learning from the perspective of positioning theory suggest which storylines might lead to productive engagement, and which not: in his research with Latino learners in the US, Domínguez (2011, p. 325) shows that students' home language "figures as a language to discuss, argue, take risks, and learn with others, whereas English tends to be reserved for enacting more traditional schoolwork". Storylines that allow learners to position themselves as mathematically competent while relating to everyday experiences tend to lead multilinguals to productive engagement [START_REF] Moschkovich | A situated and sociocultural perspective on bilingual mathematics learners[END_REF]. [START_REF] Norén | Agency and positioning in a multilingual mathematics classroom[END_REF] gives an example where multilingual students are productively engaged because the teacher establishes a storyline that allows for talking in non-formal ways and making use of out-of-school-experiences. On the other hand, multilingual students may position themselves and others based on storylines from previous classrooms where they have been positioned as less proficient language learners (cf. [START_REF] Civil | Participation in the mathematics classroom: Does every student have a voice? For the Learning of Mathematics[END_REF].

As storylines are a product of the individuals' interpretations in a conversation, more than one storyline can coexist in a given situation. These different storylines mirror themselves in the different positionings that are on the one hand assigned, and on the other taken up by the individuals: while in a teaching learning situation the storyline of teacher/learner is jointly produced, the individuals in the conversation might have different notions of what this storyline is and how they are positioned in it. In this paper, I will analyze the storylines that are jointly produced and the notions the students have of this storyline. I investigate the question: How does a jointly produced storyline in a multilingual teaching intervention influence the students' productive engagement, and hence, their agency?

A bilingual teaching intervention on fractions

The here presented case study is part of the research project MuM-Multi which investigates how to foster multilingual learners in the mathematics classroom. A multilingual Turkish-German teaching intervention on fractions was conceived. The teaching intervention builds on a previous project [START_REF] Prediger | Fostering German-language learners' constructions of meanings for fractions-design and effects of a language-and mathematics-integrated intervention[END_REF] in that it combines a conceptual learning trajectory with a language learning trajectory. Furthermore, it builds on the relating registers approach [START_REF] Prediger | Purposefully relating multilingual registers: Building theory and teaching strategies for bilingual learners based on an integration of three traditions[END_REF]. In this approach, the mathematical-technical-register, the academic-school-register and the everyday register are continually interlinked. For example, the multilingual students are asked to reflect on the Turkish way to express fractions as '5 de 3' ('5 therein 3') (Turkish mathematicaltechnical register) for which they are asked to employ the graphical representation of a fraction bar and the activated everyday contexts (sharing a bar of Baklava-cake). The relating registers approach serves as a heuristic tool for designing activities and learning opportunities in which the multilingual learners can access their multilingual resources for developing conceptual understanding of fractions.

Each session of the teaching intervention starts with an everyday, out of school contexts that connect in culturally sensitive ways to the multilingual experiences of the learners. For example, the first task starts with a short story about a traditional Turkish figure, the "Nasrettin Hoca" and children sharing a Baklava with him in an unfair way. The following tasks in a teaching intervention are then anchored in the culturally sensitive out-of-school context, so that the students can relate to their everyday experiences while working on the mathematical tasks. During all sessions, the fraction bar is implemented as a central graphical representation. All worksheets are given in both Turkish and German, assigned by the teacher. 41 multilingual students in grade 7 with low mathematics achievement and heterogeneous German and Turkish language proficiency participated in eleven small groups in the bilingual teaching intervention lasting five sessions of 90 minutes each. The intervention was conducted by four specifically trained bilingual Turkish-German teachers who were sensitized for specific affordances of teaching in multilingual settings, for example strategies of revoicing. However, the teachers were not aware of the specifics of positioning students in conversations.

The eleven groups of the bilingual intervention were videotaped over the course of the intervention, and specific groups and sessions were selected for transcription based on considerations of key points in the learning trajectory that occur after the students got familiar with speaking Turkish (Sessions 2 -4). The video material was transcribed and, where necessary, translated into German. The data is analyzed qualitatively in a turn-by turn analysis of the learners reflective positionings and the teachers interactive positioning, that is, the positioning of learners themselves within a conversation (reflective) and the positioning of the learners by the teacher in the conversation (interactive) that affects how the learners position themselves (reflective).

I illustrate the case of Akasya and Ilknur, as examples of students who successfully engage with mathematics in Turkish language in the last intervention session. I employ a positioning analysis that focuses on the students' reflective positioning in reaction to the teachers' interactive positioning in the unfolding mathematical conversation. By investigating the positionings the storylines of learners and teacher can be reconstructed. Productive engagement is analyzed with the construct of agency, according to which students are coded as productively engaged in moments when they are

• influencing the direction of the discourse • asking for clarifications or clarifying • taking charge of ideas

• establishing competence This analysis allows investigating in which storyline the students are highly productively engaged, and when less. The analyzed transcript belongs to the first task of the second teaching intervention (Figure 1), in which ordering fractions for different referent wholes is approached by the everyday context of downloading movies.

Downloading movies

Selin and Bleda are downloading two movies in the internet, with different sizes.

-copying monster.mp4 to "movies" --copying horsedream.mp4 to "movies"a) Where has more GB (gigabyte) been downloaded b) Why is Selin's fraction bar longer even though she has downloaded fewer GB?

Analysis: Storyline of Turkish speaking and less productive engagement

Two girls, Akasya and Ilknur, work on the first task in Session 2 of the intervention. Briefly, I will refer to another group of two students who work separately at another table on the same task. I will show here how the teacher changes and refines the storyline that guides his actions over the course of the task. As storylines are jointly produced, the students have to actualize their storylines in line with how they perceive this changed storyline. While Ilknur adapts her notion of the storyline to the newly established storyline of the teacher, Akasya seems to subsume this change under her notion of the storyline as "the teacher guides the discourse and will give the resolution of the task in the end".

Episode 1: The teacher changes the storyline

The teacher introduces the subtask b) to the students, and with it, also changes the previous storyline of "deciding who has downloaded more". The students have signaled, by hand signal, that they have a solution to subtask a). The teacher now establishes a storyline which is not centered on finding a solution anymore, but one where explanations in Turkish are valued.

In the following transcript, the teacher comes to Ilknur and Akasya, after he talked to the other two students at another desk. His utterance marks the first change of the storyline: The teacher metadiscursively addresses the rules of the conversation. In this way, he positions Akasya, and implicitly Ilknur, as competent regardless of the correctness of their solution, as long as she writes down her ideas, and justifies "who has won". The students are explicitly positioned as explainer of their mathematical thinking in line with a storyline of "formulating a written explanation of your decision". This is in contrast to the previous storyline of the teacher, where the students were asked to cooperatively decide whether Bleda or Selin has downloaded more.

In the following utterance, sometime after the previous transcript but still in the beginning of the work on subtask b), the teacher gives a reason why the students need to explain their thinking:

83 Teacher Aber das ist gut, weil dann lernt ihr But that is good, because then you auch. Dann könnt ihr euch auch also learn. Then you also could think überlegen, wie ihr das auf Türkisch about how to write that in Turkish. schreibt. Das ist eine gute Sache.

That is a good thing.

The teacher asks the students to think about how to "write [their explanation] in Turkish". Again, he makes the rules of the mathematical discourse explicit by valuing the Turkish language and adding "That [Turkish] is a good thing", which he repeats in the next turns. Valuing the Turkish language is a recurring theme in this storyline (turns 69, 83, 85) and is tightly interlinked with working on subtask b). In summary, by explicitly positioning the students in these ways, the teacher establishes a storyline where students are required to explain their thinking in (written) Turkish, perhaps best described as "Solving subtask b means giving a Turkish explanation of your solution". This is a refinement of the new storyline.

Episode 2: Ilknur takes over the teacher's storyline

Ilknur seems to adopt the storyline established by the teacher. She makes several attempts to explain why, from her standpoint, Bleda has downloaded more. Her choice of language, her pauses and by "ehm", suggest that she specifically tries to explain her thinking in Turkish: Ilknur attempts an explanation in Turkish, positioning herself in a way that acknowledges the position offered by the teacher (see turn 75). At first, she seems to be uncomfortable with explicitly using only Turkishpreviously she only spoke Turkish in a mixed German-Turkish mode with focus on German which is indicated by the hesitant way in which she forms her explanation. All her following taskrelated utterances after this turn are in Turkish (105, 108, 110), which suggests that she positions herself as Turkish speaker and that the storyline "giving a Turkish explanation" guides her actions.

Episode 3: Students agency and potential for productive engagement

In engaging in Turkish, the students Akasya and Ilknur do not appear as agents for the content of their explanation. Ilknur explicitly positions herself as less competent with the Turkish language: When discussing which language their worksheets should have, Ilknur explicitly wants German worksheets, and she positions herself as being more proficient in French than in Turkish (94). This is reinforced by Ilknur and Akasya's expressing the amount of effort it takes them to form a Turkish explanation (see turns 86-88 and linguistic markers "Üfff" (Boah), "Heh?" in turns 106, 110). Above that, Ilknur and Akasya's struggles also make them less competent in the eyes of the teacher. On the one hand, the teacher addresses those students who have already generated a Turkish explanation and positions them as competent by praising their solutions. Ilknur and Akasya, on the other hand, are addressed with "He şimdi kızlar (So, now, gals)" (turn 107) as a result of them not yet having generated a written explanation, this way positioning the students as being behind in the group.

In the following episode, Ilknur and Akasya seem to explicitly give up on their position as Turkish mathematics learners, and take up a position where they are responsible for the form of their solution.

It seems that they give up their position because of mathematical difficulties, not language difficulties. When Ilknur and Akasya do not make progress in generating a Turkish explanation, they explicitly position themselves as learners who struggle with their task. This goes hand in hand with Ilknur switching back to German (turn 114). Akasya, in line with her storyline where the teacher guides the discourse, assumes that the teacher is likely to correct them in the end, so that only the form of their explanation matters, but not the content. It might be that in accordance with this storyline, Akasya and Ilknur will take up positionings where they exercise less agency for the mathematics at hand.

In summary, over the course of working on subtask b), the students have less capacities to engage with the mathematical content of the task within the storyline "giving a Turkish explanation". They position themselves as insecure and less Turkish proficient, while they receive no help from the teacher or their peers, but are instead positioned as being behind the group. As Ilknur and Akasya

have not yet written down their explanations, and as the teacher was not present, their efforts are not valued. In this sense, turns 114 and 115 can be read as a way to reclaim their agency: Within the teacher's storyline of generating a Turkish explanation, the students change their activities towards writing "something" that conforms with the form of an explanation, but give up improving the content of the explanation. This is consistent with Akasya's take on the storyline -in which the teacher is the guide of the conversation and responsible for the content of the explanations -seems to be taken up.

The potential for productive engagement, that is, the potential for becoming responsible for the mathematical content, diminished in subtask b) as Ilknur and Akasya did not engage with finding a mathematical explanation in the task, but instead reclaim their agency with producing a Turkish sentence that complies with the form of an explanation. The storyline of the teacher allows for this, as the teacher gave a loophole with saying that "you don't have to be afraid if it is wrong" (turn 75).

Discussion

In order to be coherent with the teacher's expectations of giving an explanation in Turkish, the students in the episodes try to form a Turkish explanation. This leads the students to focus solely on forming a correct Turkish sentence with adequate formal words, while the mathematics behind the explanation is not in focus anymore. As the students are not successful in forming a Turkish explanation, the students might take up a position of being less agentic for the content, resulting in less productive engagement with mathematics. It seems that the teacher, by asking for a Turkish explanation and by valuing the Turkish language, and the students, by taking over these positionings, jointly produce two foci: one on the mathematical content in Turkish and one on the form of a Turkish explanation. In shifting to the second focus, the students lose their focus on the content of the explanationwhich interestingly goes hand in hand with a code-switch back to German. It may be that the two foci are a product of valuing Turkish without giving the students the support they seem to need for talking about the content of the task.

Positioning theory and storyline analysis here proved to be a useful tool to reconstruct the participation of multilinguals and the opportunities for productive engagement and mathematics learning, especially in regard to the multilingual teaching-learning situation. However, it is an open question whether the students multilinguality is a resource for the students in regard of what the storylines tell us in this research. Data suggest that difficulties with mathematics led Akasya and Ilknur to move the focus of their activity away from the content of the explanation. Previously, however, Ilknur struggles with the Turkish language.

It is an open question whether Turkish might have added to Ilknur's mathematical difficulties in form of increasing cognitive-load, or if Ilknur is in a natural translanguaging mode where her choice of language is not relevant and where her difficulties would have remained if she had used German instead of Turkish.

Figure 1 :

 1 Figure 1: First task of Session 2 of teaching intervention

Proceedings of CERME10

Acknowledgement

The project "MuM-Multi: Fostering Language in multilingual mathematics classroomseffects and conditions of a content-and language integrated intervention" is funded by the German ministry BMBF (grant 01JM1403A, grant holder Susanne Prediger). I thank Susanne and my project partners T. Kuzu, L. Wessel, A. Redder, J. Rehbein, J. Wagner, and M. Celikkol for the cooperation.