Marei Fetzer
email: fetzer@math.uni-frankfurt.de

The role of writing in the process of learning to speak mathematically

Keywords: Language skills, writing processes, mathematical learning

A lot of research is being done on the interplay of mathematical learning and language skills. However, graphic aspects have been met with little response so far. What role does writing play in the process of improving language proficiency? This paper empirically reconstructs how language use develops within one lesson that provides manifold opportunities for students to negotiate and discuss. Special attention is paid to the role pupil's written notes play within that development.

Introduction

We know that language plays an important role in mathematical learning. Consequently, mathematics education is not only about teaching and learning mathematical ideas and concepts, but also about the development of a language that reaches the demands of describing mathematical activities and expressing fundamental ideas. Mathematical and language learning accompany each other. One approach to improve mathematical and language skills in an interconnected way is to initiate interactive learning situations. Such discursive situations are meant to encourage language use and mathematical learning. Why should that work? First, language serves a communicative function. So it appears to be a platitude that learning language requires using language and interacting with others. Second, language fulfills a cognitive function. The challenge to develop mental mathematical objects is strongly linked to the availability of appropriate means of expression and language skills. Not only theories on language support the assumption that interactive learning situations are beneficial for cross-linked mathematical and language learning. However, the conceptualization of learning as a social process backs that idea. Learning takes place in the process of participating in interactive situations [START_REF] Miller | Kollektive Lernprozesse. Studien zur Grundlegung einer soziologischen Lerntheorie [Collective learning processes[END_REF]Fetzer, 2007a). Referring to social frameworks of learning, all participating students should benefit from such a proceeding. Those who actively contribute to a mathematical discourse improve their language skills by developing an appropriate language that has the potential to convey their idea. Those who rather receptively attend the situation might profit by being part of a mathematical discourse. They might not only catch up mathematical ideas, but also ways to express them. But how does language use develop in classroom interaction? Graphic aspects seem to play an important role in settings that challenge mathematical language development; taking notes on solving processes, recording explanations and accounting for answers in a written way are constituent elements of cooperative classroom settings. Consequently, when studying processes of learning to speak mathematically, the role of writing should be taken into account. What role do graphic aspects play within that development? This paper provides a closer look at the two essential situations concerning these research questions: the phase of writing, and the phase of presenting on the basis of graphic notes.

Development in language use

In order to describe developments in language use, we need to specify. [START_REF] Koch | Sprache der Nähe -Sprache der Distanz [Language of proximity -language of distance[END_REF][START_REF] Oesterreicher | Types of orality in text[END_REF] offer a distinction in two dimensions of language: the medial and the conceptual. The medial dimension is dichotomic: language can be either specified as phonic, when something is said and can be heard, or, it is assigned to graphic forms like writing. In contrast, the conceptual dimension is gradual. Language can be conceptually rather oral or rather written. It stretches between the poles of communicative nearness (oral) and communicative distance (written). This specification applies for both, medially graphic and medially phonic, forms of language. Talking with a good friend for example is closer to the conceptually oral pole than a presentation at a conference. A newspaper article is 'more' written than a twitter post. If we talk to our child when having breakfast, we might use short and incomplete sentences and combine them with gestures. As we are in a face-to-face situation, we can react directly to our counterpart and spontaneously negotiate our roles. Koch and Oesterreicher call this a language of nearness. The strategies of communication belong to the pole of orality. In contrast, conceptually written language refers to the pole of literacy and language of distance. It is used when the interlocutors are not necessarily in direct relation and the processes of speech production and speech reception might be separated from each other. Thus, aspects of the situational and cultural context have to be made explicit. Consequently, sentences are more complex, main clauses and subordinate clauses are formulated. Specific terms come into play to be precise and explicit. Koch and Oesterreicher call this a language of distance. Examples are a text of law (graphic) or a scientific lecture (phonic).

Koch and Oesterreicher help to classify the registers of everyday language and academic language [START_REF] Cummins | BICS and CALP -Empirical and theoretical status of the distinction[END_REF]. Halliday describes a register as "a variety of language, corresponding to a variety of situation" (1985, p. 29). According to him, the term register points to the detection that individuals adapt their use of language to a given situation. Thus, the register of everyday language is rather conceptually oral, no matter which medium is realized. It has to fulfil the function of a fast and unproblematic communication in our everyday life. As such, the oral language may be supported by gestures or by reference to a context. Words do not have to be clearly defined and sentences may be short or even incomplete. In contrast, the register of academic language is conceptually written, again irrespective of its medium. In academic contexts, language should be as explicit and precise as possible and intelligible without any reference to a specific situation. For that reason, words have to be well defined. Sentences might be complex in order to reflect relations.

Eventually, there is the register of mathematical language. It is an academic register related to a certain field; mathematics. Technical terms like probability, multiplication or subtracting, as well as specific sentence structures, are constituent elements of mathematical language. Especially on the medially graphic level, very short and decontextualized sentences or graphic representations are characteristic for that register.

In mathematics classes, we face the challenge to develop individual language use from orality towards literacy, from everyday language in the direction of learning to speak and write mathematically. The learning and teaching situation remains the same: students and teacher know each other and interact face-to-face. Nevertheless, a development towards precise and decontextualized academic, respectively mathematic, language has to be initiated. Fetzer (2007a;b) proves empirically that writing in mathematical classes and mathematical discourse on the basis of students' written works fundamentally support mathematical learning. In this paper, special focus is put on language development, and the way in which the implementation of medially graphic elements helps to provoke a move in the direction of literacy is reconstructed.

Data and methodological aspects

The examples analyzed in this paper are taken from the empirical study MitMaL (Miteinander Mathe Lernen) (Co-operative Mathematical Learning). In this study, cooperative and individual learning settings are compared to each other. Two math classes were observed and video-recorded for a couple of weeks. Both groups dealt with the same mathematical subject, but implemented different methodical approaches. In one class (group i), individual learning situations dominated. In contrast, the other teacher predominantly initiated cooperative group work and provided the opportunity for mathematical whole class discussion (group c). Pre-test, post-test and follow-up-test constitute the quantitative part of the study. Analyses on the transcripts of the videos account for the qualitative part of the study. They are conducted in a reconstructive manner applying analyses of interaction (Cobb & Bauersfeld, 1995). This method refers to the interactional theory of learning and is based on the ethnomethodological conversation analysis. In contrast to conversation analysis, it focuses on the thematic development of a given face-to-face interaction rather than on its structural development.

The videos for this paper are taken from group c, who worked in a cooperative manner. In order to reconstruct the development of language use, the paper concentrates on one single math lesson on probability in grade four. Debora is accompanied throughout the whole lesson. In this way, her individual performance and development can be reconstructed from the beginning until the end of the lesson. Special attention is paid to the role the notes taken in the group work play in the presentation. How do they influence the verbal language skills?

Empirical example

The empirical example is taken from a math lesson on probability. 24 students of grade four try to find out which of the four given rules is the best to win when turning a colored wheel of fortune. The wheel of fortune is equally divided into eight sections, numbered 1 to 8. Two fields are red, two are blue, three are green, and one is white. The rules are: (1) You win with 1, 2 or 3; (2) You win with red; (3) You win with white or blue; (4) You win with 2, 4, 6 or 8. Card four is the most probable to win.

The lesson starts with a thematic introduction in a whole-class situation. The tutor presents the wheel of fortune, and some students go for a first attempt. Afterwards, the children work in groups of four. They are equipped with a wheel of fortune, one set of cards with rules, and with a task sheet: Work together and find out which rule is likely to be the best to win. Why do you think so?

The challenge is to work on the tasks, take notes and give reasons in written form. At the end of the lesson, children and tutor come together. They discuss their findings with their written notes at hand.

Language use in the context of probability:

Talking about probability requires mathematical terms like probable, improbable and equiprobable. In German, 'probable' sounds familiar to children. The expression is used in everyday language and is applied in the sense of 'perhaps' or 'might be'. In contrast, 'improbable' and 'equiprobable' sound rather strange to young students. These expressions belong to the academic, respectively mathematical, register. However, talking and writing mathematically requires more than academic expressions and mathematical terms. Describing, explaining and giving reasons require academic language proficiency on the sentence level. In this particular task, the students first have to describe their selection in written form. This is possible using a main clause. Conceptually spoken language in everyday register meets the demands. In addition, an explanatory statement on their selection is required. This is much more challenging, as German language demands subordinate clauses in order to formulate causal connections. In contrast to main clauses that are structured subjectpredicateobject, subordinate clauses are put together in an inverted structure: subjectobjectpredicate. This inversion does not necessarily belong to the everyday register. The outstanding challenge in the context of probability is the fact that we do not only need a causal conjunction followed by a subordinate clause with inverted sentence structure, but also an infinitive clause as a third component. This last aspect clearly belongs to academic language. To put it in other words: in order to compliment the decision with reasons and justification, not only mathematical terms, but academic registers and complex sentence structures are especially necessary.

Material for the group work:

The rules rely on conceptually spoken language. They address the students directly and refer to the concrete given context of turning the wheel here and now. Mathematically spoken, it rather aims at a situational or context-bound understanding of probability and representativeness of personal attempts rather than at abstraction. The rules provide for a strong link to the students' experience on the one hand, and to their everyday language on the other hand.

The tasks (see above) apply conceptually rather spoken language as well. Again, the students are addressed directly. None of the mathematical terms of probability is used. Instead, the children are asked to choose one of the cards. A second sentence aims at an explanation. Interestingly enough, these colloquial questions can hardly be answered on a conceptually spoken level: if you work on the task in the order of appearance, you first have to determine the selection. That is easy. Afterwards, you need a causal subordinate clause and the infinitive clause, which is grammatically complex. In principle, conditional connections could be a solution: "If you choose…, you are likely to win. That is because…" This structure is much easier in German than the causal connection. Nevertheless, it clearly belongs to the academic register. However, such an answer is only possible if the students have already reached a rather high level of abstraction at this time. Condensed: if students work on the tasks and provide answers, they either have to fulfil a move on the conceptual level from spoken to written language, or they have to develop an abstract understanding that enables them to choose a slightly less complex language use.

Beginning of the lesson (phonic):

Children and tutor are sitting in a circle around a colored wheel of fortune ready to be spun. Debora does not contribute to this opening. The tutor starts the lesson referring to the students' experiences with wheels of fortune: "Perhaps you know something like this. Such a wheel of fortune has something to do with winning or not. This is why it's called wheel of fortune, cause if you win, you are lucky". (Wheel of fortune is 'Glücksrad' in German, and being lucky is 'Glückhaben'). In the first introduction to the theme, the tutor links not only the context of own experiences with the mathematical context of probability, but also different language registers. She plays on the fact that Glück bridges both registers. Pointing at the paper wheel of fortune, she moves her finger around in circles and says: "Today we have a closer look at the wheel of fortune." Repeatedly in this introduction, she moves her finger in this manner as if she spun the wheel. By doing this, she might strengthen an activity-orientated representative understanding of probability: Rule X wins because my own (little) random sample 'proves' it. This understanding does not claim for decontextualized academic language. The tutor's language changes within the introduction. First, she seems to pick the students up and see them through by applying the everyday register. She distinguishes "good rules" you are likely to win with, rules you might "not necessarily" win with and rules that make it "quite impossible" to win. When explaining the task, she even grades further down and speaks of "a good way to win" and "stupid cards". This colloquial language is accompanied by the turning gesture. Acting and speaking like this, the tutor marks the concrete situation as the starting point of the experiment on probability. Not before her last sentences, she uses the mathematical expressions: "For example, with this card, you will most probably win; with these two cards, it is perhaps equiprobable to win; and with this one, it is rather improbable." This is the only time these terms are quoted in the introduction scene.

Group work (phonic and graphic):

For a closer look at the group work, Debora's group is chosen exemplarily. Three phases of language use can be reconstructed within the working process: (1) conceptually spoken language, (2) everyday language with scattered mathematical terms, (3) everyday language with specifications and conjunctions.

(1): Debora is the girl to take the initiative in the group. She suggests cutting the wheel out. While doing this, she lays the foundations for an activity-orientated approach and context-specific experiences. This first phase is dominated by the register of everyday and conceptually spoken language. Sentences remain incomplete; gestures play an important role for understanding. The challenge of writing is no issue of negotiating yet. The following scene emerges as soon as the girls have read out the rules. It illustrates the register of everyday language the girls apply.

Debora:

Points looking at the wheel, fingers still fixed on fields makes no difference-it's equal\ However, they work straightforward on the task and find a first solution that is mathematically correct. The choice is formulated using conceptually spoken, deictic and context-bound language: "That's best." No mathematical term is applied. A first approach to explanation might be the physical detection (by touching) of four (out of eight) 'lucky' fields.

(2): The second phase can be characterized by conceptually spoken language with scattered mathematical terms. Graphic aspects still do not come into play. In this scene, one can reconstruct one example of how mathematical language works its way into the discourse: Debora picks up a comment from a neighbouring group "Red is the most improbable." She spontaneously integrates the term in her active language use and confirms with a look on the wheel: "Yes, red is the most improbable." In the following, this term remains a constituent part of the interactive process. All four girls stick to conceptually spoken language using the register of everyday language. Their sentences remain incomplete with scattered mathematical expressions. For example, Debora sums their findings up as follows: "Red is the most improbable. One, two or three is same, and Dalya's is best."

(3): As soon as it comes to writing, there is no way of supporting words by pointing or showing.

Written language has to carry the whole meaning. Accordingly, writing mathematically arises the necessity to develop language use. Debora's group works sequentially. First, they answer the first part of the task: Work together and find out which rule is likely to be the best to win. They choose the conceptually spoken wording of their verbal formulation from the beginning "That's best" and elaborate on it: "Best is card four which says you win with 2, 4, 6 or 8." The deictic component of the face-to-face interaction is replaced by naming the card "card four". The necessity to write results in specifying the concrete situation. The rule is taken from the card and copied word by word. This appears to be an appropriate strategy to create an answer that is socially accepted as 'good mathematical language'. At the same time, the sentence structure remains conceptually oral. The elements 'specification by concreteness' and 'copying phrases' are put together one after the other. Though the subordinate clause is introduced with the appropriate conjunction, it is not conducted in the right grammatical arrangement.

Later in the group work, the tutor reminds them to provide reasons for their decision. Subsequently, Dalya picks up the pencil. Well-we've found a because-uhm I've written reads out best is card four which says you win with 2 4 6 or 8 because they are well spread on the wheel of fortune.

Debora: spread thumb up

In this short scene, the justification is graphically recorded. The aspect of the winning fields being evenly or well spread on the wheel of fortune was mentioned earlier only once by Julia. At that time, her idea would not come through in the interactional process. Why is she successful now? The tutor directly addresses the group and asks them to find a justification. The girls are running out of time. In this situation, Julia starts her suggestion with "because". Her first utterance is rather vague. Perhaps feeling supported by Dalya's "yes", she then elaborates "the turntable". As a third move towards improved language, she implements a mathematical term "wheel of fortune." Summing up, her suggestion is put forward at the right time within the solving process. Julia is working with the elements of specification and integrating mathematical terms. The conjunction 'because' indicates a justification. That was the task. The sentence structure itself still reminds of spoken language. This form of speaking mathematically appears to be convincing to the group members. Debora seems to try that wording on her own by repeating it. Her thumb up can be understood as acceptance. Again, she picks up an expression and integrates it spontaneously in her active vocabulary.

End of the lesson (phonic and graphic): Gathering for the end of the lesson, the students have their sheets with their written works at hand. Debora is the first child to contribute in the presentation. As she hesitates and starts over twice, it seems as if she has to orientate herself first. In this situation, her group's notes appear to be helpful. After a quick glance at the paper, she performs much more fluently: "Best we found out is card four." In German, this is an example of conceptually spoken language: her word order is irritating. Precisely, 'best' is used as an adverb referring to 'to find out'. Taking into account the given context, it appears to be more reasonable that she uses 'best' to characterize card four. Thus, her sentence structure is grammatically incorrect. Instead, it might reveal her order of remembering: we had to select ("best"), we had to work ("we found out"), we made a decision ("card four"). Here, the notes appear to support the memory: what did we do? In the following, her language develops. "Cause best is card four". This is, in fact, only half a sentence. But, in contrast to the first attempt, it is in the right German word order. Then, she reads from the notes, probably in order to make sure to get the correct wording: "which says you win with 2 4 6 or 8". Turning her eyes away from the notes and looking to the tutor, she proceeds: "Cause they are quite well spread on the wheel of fortune." On the notes, the subordinate clause is in the wrong word order. Speaking aloud, she corrects the sentence structure. This can be interpreted as a step towards abstraction and language of distance. Her language develops towards academic register. Summing up, the notes warrant three aspects: orientation, reminder and thematic reassurance that helps to develop language skills.

Analysing the other contributions in this presentation confirms these findings. Every single contribution is thematically on point. Obviously, graphic notes help to orientate. Moreover, the students supply their comments with phrases that structure the whole presentation process: "We found out the same" or "Not as the others". That provides for orientation and structure not only within individual understanding processes, but also within the interactional discourse. The aspect of reminding of the own working process appears to take affect strongly. Every single student who contributes to the whole class presentation thematically closely links his or her comment to the notes. The quick glance at the paper is to be reconstructed in each sequence. Especially the observation that medially verbal language use improves compared to the graphic version in the notes confirms as an empirical result. For example, in one group, the notes say: "Weil Gewinnkarte vier das halbe Glücksrad einnimmt." (Cause winning card four takes half the wheel of fortune.). Contributing in the class, the presenting boy turns 'half' into a noun and combines it with a genitive construction ("nimmt die Hälfte des Rades ein"). Using substantives and genitive constructions is a German speciality, which is especially characteristic for academic register. Accordingly, this is understood as development in language skills.

Conclusions

How does language use develop in cooperative classroom settings? What role do graphic aspects play? This paper provides a closer look at the two essential situations concerning these research questions: the phase of wiring, and the phase of presenting on the basis of graphic notes.

The necessity to write mathematically activates a development in language move at the conceptual level. Mathematical terms are embedded in rather spoken sentence structures; a first move towards mathematical language. Specification on the concrete context serves for more precision in descriptions. Even at the level of sentence structure, first developments can be reconstructed. On the one hand, students implement phrases that were socially accepted earlier as mathematically elaborated. On the other hand, they implement conjunctions. If-clauses and causal-clauses are characteristic elements of academic and especially mathematical register.

Presenting is eventually a medially verbal action. But, in the given case, graphic aspects work their way into interaction. They contribute fundamentally to orientation, structuring and reminding both individual learning processes, and the interactional process. Particularly the graphic notes thematically reassure students and thus provide capacity to develop language skills.

  to the paper Well-because they are well spread on the


Rachel-Ann Friesen
email: rachel-ann.friesen@tu-dresden.de

Marcus Schütte
email: marcus.schütte@tu-dresden.de

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

Solving problems collaboratively in multi-age classes a possibility for learning?

Keywords: Multi-age education, collaborative learning, interactional analysis

, this is seen as learning. In multi-age classes, students have even more diverse framings then in single-age classes and when they are given the opportunity to learn collaboratively, an increased need to coordinate their framings might lead to an optimised possibility for learning. This research project has the objective of identifying how students participate in processes of collective negotiation of meaning and therefore how possibilities for learning mathematics emerge. First analyses show that the possibilities for learning emerging in the interaction vary depending on whether differences in framings can be coordinated and a takenas-shared meaning is brought forth.

Multi-age education

No matter the school, students in a classroom are never homogeneous, also in relation to their learning preconditions. Each individual is unique and therefore has differing abilities and experiences. Using age as the distinguishing factor for placing students in specific classes is the norm, because it is seen as a possibility to make classes as homogeneous as possible. But even in learning groups which are homogeneous in age, the differences in students' preconditions for learning can be up to five years at the time they enter into school (Lorenz, 2000, p. 22). One concept with the aim to purposefully value and meet this existing diversity is merging students of different ages and grades in one class deliberately (Prengel, 2007, p. 69), which can be referred to as multi-age classes. The benefit in making learning groups heterogeneous in age is e.g. seen in that, because the diversity of learning preconditions increases, the necessity for differentiation increases also. However, mathematics is often presumed to be a subject built up rectilinear whose content can only be learned following a specific order (Lorenz, 2000, p. 19). [START_REF] Nührenbörger | Mit Unterschieden rechnen. Lernumgebungen und Materialien für einen differenzierten Anfangsunterricht Mathematik[END_REF] found that because of this common assumption students in multi-aged classes are rarely given the opportunity to learn from and with others in mathematics classrooms. Instead, this often causes extremely individualised and separated learning to take place as teachers either feel they need to separate the learners back into the different age groups or they need to let each student learn individually at their own pace in order to meet the increased diversity of the students. However, both of these waste the opportunity seen in collaborative learning (Nührenbörger & Pust, 2006, p. 17), especially from an interactionist perspective on mathematics learning.

Besides overarching concepts, which have been developed for collaborative learning within multiage mathematics teaching (e.g., Nührenbörger & Pust,3006), interactions between learners who are heterogeneous in age have been examined within multi-age mathematics education research. These studies predominantly focus on dialogues between two students from different grade levels and concentrate either on how knowledge is constructed in the interaction [START_REF] Nührenbörger | Students' mathematical interactions and teachers' reflections on their own interventions. Part 1: Epistemological analysis of students' mathematical communication[END_REF] or on when interactions are beneficial for learning in these settings [START_REF] Gysin | Lerndialoge von Kindern in einem jahrgangsgemischten Anfangsunterricht Mathematik. Chancen für eine mathematische Grundbildung [Children's learning dialogues in a multi-age mathematics education[END_REF]. Our focus is also on possible benefits for students' learning conditions when children with diverse preconditions interact with each other while solving problems collaboratively. However, we base our study on the interaction theory of learning, which will be described below, and include analyses of group settings of three students working together.

Interaction theory of (mathematical) learning

From an interactionist perspective, the individual process of learning is constituted in the social process of the negotiation of meaning. The social interaction, therefore, holds the potential to generate meanings new to the individual. Especially for young learners, interaction is seen as a foundational constituent of learning. In this context, [START_REF] Miller | Kollektive Lernprozesse: Studien zur Grundlage einer soziologischen Lerntheorie [Collective learning processes: studies on the basis of a sociological learning theory[END_REF] creates a sociological theory of learning of at least two individualsthe theory of collective learning processesin contrast to psychological theories of learning which predominantly focus on isolated individuals. Hereby, he does not, on principle, question that learning is also an individual process, but rather that, when learning fundamentally in the early stages of a child's development, an interactive collective process precedes or determines this process of learning something new [START_REF] Jung | An interactionist perspective on mathematics learning: Conditions of learning opportunities in mixed-ability groups within linguistic negotiation processes[END_REF]. In this sociological theory of learning, the concept of argumentation is essential and Miller (1986, p. 37) differentiates it from "communicative action." Argumentation, and hereinafter more specifically collective argumentation, is mainly characterized by the aspect of rationality, whereas communicative action is based on something uncontroversial. Mathematical learning, therefore, takes place when two or more people negotiate what is viewed from a mathematical perspective as rational. [START_REF] Krummheuer | Paraphrase und Traduktion. Partizipationstheoretische Elemente einer Interaktionstheorie des Mathematiklernens in der Grundschule [Paraphasing and traducing: Participation theory elements of an interactional theory of learning mathematics in primary school[END_REF] draw on this aspect of learning in collective argumentation and create an interaction theory of mathematical learning. For them, it is of great importance that students participate in collective argumentation within classroom interaction in which meaning is collectively negotiated (Krummheuer & Brandt, 2001, p. 20). This, at best, will lead to learning to occur which is seen as the construction and/or modulation of meanings which outlast the situationalso called framings (see below) [START_REF] Krummheuer | Lernen mit "Format". Elemente einer interaktionistischen Lerntheorie [Learning with "format[END_REF][START_REF] Jung | An interactionist perspective on mathematics learning: Conditions of learning opportunities in mixed-ability groups within linguistic negotiation processes[END_REF], p. 1091) and can be reconstructed in the increasingly autonomous participation in these collective argumentations [START_REF] Krummheuer | Paraphrase und Traduktion. Partizipationstheoretische Elemente einer Interaktionstheorie des Mathematiklernens in der Grundschule [Paraphasing and traducing: Participation theory elements of an interactional theory of learning mathematics in primary school[END_REF]. Based on this understanding of learning and the significance of interaction in mathematics learning, diverse theories with rich empirical content have been developed in recent years from a foundation of "sensitizing concepts" (Blumer, 1954, p. 7)a kind of theoretical skeleton for the interaction theory of learning, which will be described below.

At the beginning of an interaction, the participants develop preliminary interpretations of the situation on the basis of their individual experiences and knowledgecalled definitions of the situation [START_REF] Krummheuer | Lernen mit "Format". Elemente einer interaktionistischen Lerntheorie [Learning with "format[END_REF]. Furthermore, participants in the interaction attempt to attune their definitions of the situation to each other, ideally leading to the production of taken-as-shared meaning (Voigt, 1995, p. 172), a working consensus. The participants' interpretations of a situation can never be exactly the same, however, by creating an understanding between them concerning the objects, ideas and rules of the interaction, the participants are enabled to work together. The result of the participants' negotiation only is something "temporarily" which serves as a basis for further processes of negotiation but might also be rejected or transformed as the interaction unfolds [START_REF] Bauersfeld | Interactional theory of learning and teaching mathematics and related microethnographical studies[END_REF][START_REF] Voigt | Thematic patterns of interaction and sociomathematical norms[END_REF]. The taken-as-shared meaningthe working consensus (Goffmann, 1959, p. 9) is on the one hand socially constituted and, on the other, potentially novel for the individual if it pushes systematically beyond his or her interpretive capacities. The working consensus represents the 'stimulation potential' of individual cognitive restructuring processes. When taken-as-shared meanings are brought forth repeatedly in the interaction, the participants' individual definitions of situations can become standardized and routinized and can therefore be produced by the individual in similar situations. Drawing on [START_REF] Goffmann | Frame analysis. An essay on the organization of experience[END_REF], [START_REF] Bauersfeld | Interactional theory of learning and teaching mathematics and related microethnographical studies[END_REF] call these definitions framings. Thus, ideally, mathematical ascriptions of meaning which sustain beyond the situation (framings) are constructed, or the working consensuses, which are repeatedly collectively negotiated, 'converge' in individual mathematical framings.

However, it often happens that the framings produced by young learners either among themselves or in interaction with individuals with advanced skills are not in alignment with each other [START_REF] Krummheuer | Lernen mit "Format". Elemente einer interaktionistischen Lerntheorie [Learning with "format[END_REF]). Students, for example, each interpret situations based on framings from the environments they have experienced outside of school and in their previous school career. In order to maintain the progressive mutual negotiation of content relating to a particular theme, the differences in framing between the participants need to be coordinated. This coordination should be done by an individual who is advanced in the subject-specific interaction. While these differences in framing can make it more difficult for the participating individuals to adjust their definitions of the situations to fit each other, they also provide the "'motor' of learning" (Schütte, 2014, p. 927) since, on the interactional level, they generate a certain necessity for negotiation.

Analysing students' participation in processes of negotiation of meaning

One of the unique characteristics of multi-age education is that learners most likely have even more diverse framings then in single-age classes. We believe that this can provide and optimized opportunity for learners when working collaboratively, as there is an increased need to coordinate their differences in framing by negotiating meaning. In order to coordinate the differences in framings, the students have to bring forth a working consensus collectively. This can then lead to the construction and/or modulation of framings which is seen as learning. Based on this interaction theory of learning, the overall question this research seeks to address is: How do students participate in processes of collective negotiation of meaning within multi-age mathematics education and do possibilities for learning mathematics emerge.

Interactionist approaches of classroom research

Methodologically, this work can be located within qualitative methods of social research which follow a reconstructive-interpretative qualitative approach and have the aim to 'understand' the actions of the individuals participating in class and develop local theories (Schütte, 2011, p. 776).

More specifically, this work is located within interactionist approaches of mathematics educational classroom research (e.g., [START_REF] Krummheuer | Representation of the notion 'learning-as-participation' in everyday situations of mathematics classes[END_REF]. For this study, the classes are filmed several times over a period of one to two years in order to reconstruct possible age-specific changes concerning the participation in collective argumentation of the children. The interactions of the students are then transcribed and analysed using the interactional analysis, which was first deployed in the area of interpretive classroom research in mathematics education in studies by [START_REF] Bauersfeld | Interactional theory of learning and teaching mathematics and related microethnographical studies[END_REF]. The interactional analysis allows to reconstruct the ways in which negotiations of mathematical meaning are interactively constituted by individuals. Furthermore, it can help to reconstruct patterns and structures of verbal actions of the teacher and the students. Overall, this research seeks to describe how mathematical learning takes place collaboratively in multi-age education and will use the sensitizing concepts in order to describe these processes.

Collective argumentation of students in multi-age education

The group works analysed in this paper are from a mathematics lesson in a multi-age class including 24 children from first, second and third grade. At the beginning of the lesson, the students are divided into groups of three which mostly contain one student of every grade level. The students are given this task: "Tina and her family want to decorate their Christmas tree with hand painted Christmas ball ornaments1 . They want every ornament to look different. To paint the ornaments, they have the colours red, green and blue. With these colours, they can paint dots or stripes. For each ornament they want to use two colours and one pattern at the most. Which possibilities do they have when painting their ornaments?" To solve this task, the students have material on their tables which includes circles, stripes and dots of each colour to try out different combinations without painting them. The interactions of two groups of three students, including a student from every grade level, working on this task were analysed and the summaries of these analyses will be presented below. Afterwards, the analyses will be compared to each other and a conclusion will be drawn.

Analysis of the first group work

The first group includes Isabella (1 st grade), Hans (2 nd grade) and Elias (3 rd grade). During the first part of this group work, the students start by creating two green circles, one with red stripes and one with blue dots. Then the following interaction emerges. In this excerpt, Isabella argues that they should not only use green circles but also the two other colours because they should not only use the same colour <1>. Elias then picks up this idea and starts suggesting to put blue dots on a red circle <2>. Hans also follows up on Isabella's idea and suggests to put green dots on a red circle <3>. Elias seems to misunderstand Hans' utterance and reiterates Isabella's idea <4>. Then Hans repeats his idea to clarify it <6> and Hans and Elias agree on creating a red circle with green dots <7>. Hans and Elias then briefly disagree on the number of dots they should use but both seem to implicitly agree that for the differentiation of the possibilities the number of dots is not relevant. After settling for three, they also disagree on which background colour they should continue with. Elias now suggests to use a blue circle but Hans disagrees and suggests to create a second red one using the other colour left for the pattern (blue). They then agree on using a red circle and add two blue stripes. Isabella says that she has found another possibility and points to a blue circle with one green stripe. Contrary to Elias who then says he wants to create another red circle with a pattern, Hans explains that they have created two green and two red ones so far and suggests to now create blue circles. All three quickly agree that they will first use a blue circle with red dots.

In this part and in the continuation of this group work, one can see that all three children participate in the creation of the possibilities. All of them contribute suggestions but also disagree with each other directly. However, before they continue working on the next possibility, they always come to an agreement concerning a possibility. In doing so, they collectively negotiate their suggestions. In the arguments, which are brought forth in the processes of reaching agreements on the procedure and correctness of the specific possibilities, various strategies of the children can be reconstructed (even usage of all colours; maximum difference in the materials used; alternating colours and patterns at the same time; aesthetic reasons such as using the same or deliberately different number of dots and stripes; conservative usage of materials to avoid wasting materials). The two strategies which prevaileven usage of all colours, as well as the variation of the number of dots and stripes for aesthetic reasonsare both introduced by Isabella first and then picked up by Elias and Hans. It is noticeable that Isabella contributes a lot of ideas but does not implement them directly but rather formulates them in the subjunctive. Because she mainly relies on verbal suggestions without using the material, whereas Hans and Elias often directly use the material while bringing forth a suggestion (e.g. <3, 4>), her suggestions and arguments are often very explicit. Elias, in particular, takes up Isabella's suggestions over and over again and implements these as well as his own suggestions. Hans primarily takes up Elias' suggestions but often also contradicts them and starts acting directly on his own. However, in these contradictions -at the beginning especially between Elias and Hansno competition develops for who prevails, instead the contradictions are used for finding the best possible strategies and methods for solving the given task. Elias, in particular, involves everyone in the process of solving the task by asking for their approval. Therefore, a working consensus seems to establish that all three participants should agree on a suggestion (or at least no longer disagree on a suggestion) before moving to the next possibility. This leads to the group bringing forth collective argumentations in order to solve the emerging differences in framing. Throughout this entire task, they all continue to work collaboratively because, even though they briefly start working alongside each other once in a while, they always agree collectively on a possibility before moving forward.

In this group work, the differences in framings seem to be coordinated as the children bring forth a working consensus through collective argumentations. Therefore the possibility for learning can be reconstructed as this working consensus can lead to the students constructing new or modulating existing framings. In further analyses, different roles of the individual students within these negotiations could be reconstructed more specifically. Elias, for example, seems to act as one who coordinates the differences in framings and therefore seems to help them come to a working consensus as a group.

Analysis of the second group work

The second group includes Kim (1 st grade), Erich (2 nd grade) and Hannes (3 rd grade). In this group, we can also see that all three children participate in the creation of the possibilities by making suggestions, responding to the suggestions of others, but also by acting immediately on their own suggestions. As a group, however, they only agree on a possibility briefly before revising it again. At the beginning, they start by working on one possibility collectively, however Erich's proposal is rejected by Kim and Hannes without giving clear reasons. Kim only says "no I believe I don't know if that will work with red dots I don't believe [takes the red dots off the blue circle]" <78> and after Erich moves the blue circle Hannes says "no Erich red needs to go [takes the blue circle which Erich moved and puts it in front of himself] <82>. From then on Erich withdraws from working collaboratively. The disagreements between the children are often solved by exercising authority and rarely by giving arguments and if an argument is given, this is not taken up by the others. This is also reflected in the fact that from the start Hannes repeatedly contradicts suggestions by referring to the taskonly two colours may be usedbut then does not comply with it himself. Therefore, a pattern is established in which all three children work side by side. Kim and Hannes seem to interpret this situation as aimlessly trying out various possibilities, as Kim, for example, repeatedly rejects or reverses possibilities and never leaves a possibility permanently. Erich, on the other hand, seems to be looking for possibilities in a more structured way. His structured approach can also be reconstructed by him letting his suggestions remain visible and not taking them apart again like the other two. At different times during the group work, Kim looks at other tables and then introduces new suggestions for strategies for solving the task (using the sheet of paper; gluing the patterns onto the circles; using a more structured approach) but her suggestions are not taken up or even rejected by Hannes and Erich. All in all, it can be seen that all three act very autonomously right from the start but do not find a working consensus about how to solve the task. They then start working side by side, presenting their own suggestions to each other but when contradictions arise they neither bring forth arguments (other than aesthetics), nor agree on possibilities and secure their results as a group.

In this second group work, the few attempts of the participating children to coordinate the differences in framing, which emerge, fail and therefore no working consensus seems to emerge. This may be due to the fact that they bring forth separate arguments but no collective argumentation is brought forth. From an interactionist perspective, the possibility for learning in this group work is therefore limited as without bringing forth a taken-as-shared meaning the possibility for constructing new or modifying existing framings is impeded.

Conclusion

When comparing the interactions of the two groups, even though all children participate in solving the task, from an interactionist perspective, only in the first group an optimized possibility for learning can be reconstructed. Therefore, solving problems collaboratively in multi-age classes can vary in the way it enables possibilities for learning to emerge. In further analyses, collective argumentations seems to be initiated e.g. when learners explicitly verbalize disagreements or ask questions about the correctness of a solution as this seems to put the other learner(s)often the more advanced ones -into an interactive obligation to bring forth arguments. In the process of the students then bringing forth the argumentation collectively, both the advanced and the less advanced student seem to have the possibility for learning even though these may be different (cf.

Friesen, forthcoming). However, as seen in the two group works presented in this paper, this interactive obligation does not always lead to a collective argumentation in which a taken-as shared meaning is produced. In both groups, there are arguments that are brought forth but only in the first group these are brought forth collectively as can be seen in the excerpt. In the second group, the students do sometimes give an argument when a disagreement is verbalized, however, these are not taken up by the others and therefore no argumentation is brought forth collectively. In the future, these analyses will be compared with analyses of further group work in order to more clearly describe the patterns that enable children to bring forth and participate in collective argumentation.

Transcript 1: Excerpt of the first group work

			but green dots [places his two green dots on the red circle in front of Elias]
	7	Elias	yes okay . that that yes
	1	Isabella I wouldn't just use green now but (if we) also the other two colors [points to the blue and red circle]
	2	Elias	on red wait [takes a big red circle from the pile] on red we could [picks up a small blue dot]
	3	Hans	no now now [touches red dots] now green dots [picks up two small green dots]
	4	Elias	no we already have green dots . twice [first points to a big green circle and then with two fingers to
			the two green circles with patterns on them]
	5 < Elias	now we have to

In Germany, Christmas trees are often decorated with different kinds of ornaments (e.g. ball ornaments) which can be in one or more colors. Therefore, children in Germany are usually familiar with ball ornaments.

Acknowledgment

The project "Collaborative Learning in Multigrade Mathematics Education at Primary Level" is part of a greater project called Sylber (Synergistic teacher education in an excellent framework) at the Technical University of Dresden, which is funded by the German Federal Ministry of Education and Research in the framework of the joint state and federal "Teacher Education Quality Offensive."


Daniela Götze
email: daniela.goetze@tu-dortmund.de

The importance of a meaning-related language for understanding multiplication

Keywords: Multiplication, mathematical concepts, verbal learning, low achievers

Multiplicative thinking is much more than recalling number facts. Furthermore, it is much more than repeated addition. It is about links and relationships and knowing how multiplicative ideas are connected and why multiplicative grouping processes work as they do. Supporting this conceptual understanding of multiplication among underachieving primary school children is the goal of a single-case intervention study presented in this paper. These children were fostered in a language sensitive way placing emphasis on the relationship of different multiplicative representations and a meaning-related vocabulary associated with this operation. The in-depth analysis of an individual learning trajectory of a low-achieving child shows that the progress may depend on whether and to what extend the meaning-related vocabulary becomes the vocabulary of thinking for these children.

Theoretical background: Fostering a conceptual understanding of multiplication [START_REF] Kilpatrick | Adding it up: Helping children learn mathematics[END_REF] define the conceptual understanding as an understanding of mathematical concepts, operations and relationships. A mathematical concept is said to be understood when it constitutes a part of the mental knowledge and is said to reach a higher level if it is interconnected with existing knowledge and can be used to solve diverse mathematical tasks [START_REF] Kilpatrick | Adding it up: Helping children learn mathematics[END_REF]. What this means for the multiplication specifically is that children need to understand that cardinal quantities are combined with each other, wherein, formally speaking, the multiplier represents the frequency of the cardinal quantities and the multiplicand represents the size of the cardinal quantity (Figure 1). Both factors are therefore not equal [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF]. However, multiplication is often introduced in the form of repeated addition, which is the preferred strategy used to solve the multiplication tasks particularly in low-achieving children [START_REF] Moser Opitz | Rechenschwäche / Dyskalkulie. Theoretische Klärungen und empirische Studien an betroffenen Schülerinnen und Schülern[END_REF]. But this equating of multiplication with repeated addition leads to an incorrect understanding of multiplication [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF], in the case of addition, both numbers represent a respective quantity, which can be cardinally translated and merged. Multiplication, however, means to coordinate interrelated units. If children tend to equate the multiplication with repeated addition or with count-by sequences ("four, eight, twelve")as another typical strategy of low-achieving childrenthis differentiation between multiplier and multiplicand does not become clear. Furthermore, the knowledge in relation to associative and distributive contexts is not accessible to them. Additionally, those solution strategies are very limited and cannot, for example, be used when it comes to fractions or decimal numbers [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF].

In this respect, many researches have shown that a conceptual understanding of multiplication instead of automaticity needs to be fostered in the first place. Many of these studies take a strategy-training approach (e.g. [START_REF] Gaidoschik | Do lower-achieving children profit from derived facts-based teaching of basic multiplication: Findings from a design research study[END_REF][START_REF] Woodward | Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills[END_REF]. Particularly remarkable is that predominantly lowachieving childreneven after a strategy-trainingseldom use derived-fact strategies compared with normal-achieving children [START_REF] Zhang | Improving multiplication strategic development in children with math difficulties[END_REF] or have difficulties in deriving [START_REF] Gaidoschik | Do lower-achieving children profit from derived facts-based teaching of basic multiplication: Findings from a design research study[END_REF]. On the one hand, this can be traced back to the fact that the use of decomposition strategies is very working-memory demanding and low-achieving children are characterised by their limited working memory [START_REF] Zhang | Improving multiplication strategic development in children with math difficulties[END_REF]. On the other hand, the cause can be seen in an insufficient conceptual understanding of multiplication [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF]. These children do not seem to have fully understood what phrases such as "three times four" or "four multiplied with three" actually mean. This means that perhaps the differentiation between multiplier and multiplicand needs to become more obvious for these children as a central basis for developing derived-fact strategies. However, current research projects have repeatedly shown that the transition from the repeated addition or count-by strategies to the understanding of multiplication is aif not thecritical obstacle to the conceptual understanding of multiplication especially in the case of low-achieving children (e.g. [START_REF] Zhang | Improving multiplication strategic development in children with math difficulties[END_REF]. Therefore, in future research, this should be the starting point to identify which steps could help children overcome this critical obstacle. But if phrases like "three times four" or "four multiplied by three" are initially of no meaning to these children other phrases such as "three groups of four", "the four three times", "three rows of four columns each" may support the development of a conceptual understanding. It has still not been researched sufficiently whether and to what extent this "basic meaning-related vocabulary" [START_REF] Prediger | Fostering German language learners' constructions of meanings for fractions -Design and effects of a language-and mathematics-integrated intervention[END_REF]Pöhler & Prediger, 2015) supports a conceptual understanding of multiplication in low-achieving students. In terms of increasing a conceptual understanding of division among low-achieving children, it has been shown that the understanding of division can be fostered with the help of meaningful verbalisations such as, for example, "12 divided by 4 means how many fours fit into 12" [START_REF] Götze | Fostering a conceptual understanding of division: Results of a language and mathematics integrated project in primary school[END_REF]. Whether or not students benefit from such support seems to primarily depend on whether this meaning-related vocabulary will become the students' language of thinking (Pöhler & Prediger, 2015). This means that in the course of such fostering, it might primarily depend on if and to what extent the students are able to internalise these verbalisations and how they can contribute to the forming of mental models. In this respect, the single-case research project presented in this paper focuses on the following research question:

How does a language sensitive support of multiplicative grouping promote a conceptual understanding of multiplication in low-achieving children?

Methodology of the study

The study presented in this paper pursues the goal of conducting a detailed in-depth analysis of the learning processes for the development of a conceptual understanding of multiplication among lowachieving children. Therefore, the study is structured as a single-case didactical design research (Pöhler & Prediger, 2015). The data presented are from the first circle.

A total of ten pairs, consisting of one normal-and one low-achieving child, were assisted by one trained preservice elementary teacher. The cooperative setting was selected because the discursive negotiation of meaning among heterogeneous children has proven itself to be very beneficial for learning in other studies as well [START_REF] Götze | Fostering a conceptual understanding of division: Results of a language and mathematics integrated project in primary school[END_REF][START_REF] Zhang | Improving multiplication strategic development in children with math difficulties[END_REF]. The pairs of children were provided with fostering help on a total of four remedial lessons. Each of these remedial lessons lasted approximately 25 to 30 minutes. They were provided by three pre-service teachers who had been intensively trained beforehand.

At the time of the study, the participating children were in the third grade in a German primary school and had already learnt multiplication in their second year in school. As a side note, in Germany the rows of times tables are determined by the second factor. So, 1 x 2, 2 x 2, 3 x 2, 4 x 2... are second-row tasks, 1 x 5, 2 x 5, 3 x 5, 4 x 5... are tasks from the fifth row. For these children, the tasks 3 x 4, 3 x 6, 3 x 9 belong to different rows.

Based on the review of the current research, these children were fostered in a language sensitive way placing emphasis in the relationship of different multiplicative representations (concrete, iconic and symbolic representations) and a meaning-related multiplicative (not additive) vocabulary associated with this operation. For doing this, the following central learning and diagnosis steps were applied in order to foster a conceptual understanding (based on Pöhler & Prediger, 2015):

-Step 1: Informal thinking starting from students' resources.

-Step 2: Focusing meaning-related vocabulary for understanding multiplicative grouping and connecting this vocabulary with concrete, iconic and symbolic representations. -Step 3: Independent use of the meaning-related vocabulary.

-Step 4: Use of meaning-related vocabulary in order to initiate decomposition strategies.

Especially in the second support unit (step 2) the children had to find reasons for the assignment of iconic rectangular arrays and symbolic terms. At the same time, the multiplicative understanding was addressed directly: "three times four" means that there are three fours or there are three groups, each of which has four in one group. The children should always explain this relationship: Where do you see the number of the groups? Where do you see the group size? Subsequently, during the third support unit (step 3), this understanding was deepened further through more game-based assignments (for example pairs card games). The children were furthermore required to name the matching terms for orally described rectangular arrays. In the case of difficulties, they were allowed to lay the rectangle. Then, the children should lay rectangular arrays selected by themselves with manipulatives and describe these in a similar fashion. The children were repeatedly asked to interpret the group number (the multiplier) and the group size (multiplicand) in terms of content. The constant alteration between language reception and language production was intended to form mental models. During the transition to the decomposition strategies in the fourth support unit, the children needed to think about which arrays could be laid next to each other and which overall term would match. They were given some rectangular arrays, which they could distributively combine in sometimes different ways. What was important here once again was the fact that the children had to explain why, for example, 3 x 4 and 2 x 4 together make 5 x 4, but also why 3 x 4 and 3 x 1 together make 3 x 5. This required them to pay attention to the differentiation between multiplier and multiplicand, as well as to verbally describe their meaning and importance. This makes it absolutely necessary to have mentally understood the previous actions carried out on the rectangular arrays or to be able to fall back on these once again.

Therefore, the following section gives an insight into the learning pathway of the low-achieving child Ilay. The goal of the analysis is to ascertain under which conditions the use of meaning-related vocabulary fosters learning for the development of a conceptual understanding of multiplication and as a basis for developing derived-fact strategies. For a deeper insight and a comparison between different children's development see Götze (under review).

The individual learning trajectory of Ilay

Ilay (I) worked together with his normal-achieving classmate Nelli (N) during the fostering course. Ilay is a very communicative child who likes to contribute to class discussions and really likes to explain how he arrived at his solution. This is less the case with Nelli.

During the first support unit Ilay knew many multiplication tasks from the times tables off by heart. Otherwise he used the strategy of repeated addition or count-by sequences. While doing so he usually started to recite the times tables from one, something which was very time consuming. For instance, he needed a total of 45 seconds until he could provide the correct solution to the problem 6 x 7. While reciting, he raised individual fingers one after another sometimes even two at once, which would indicate additive calculation. He explained how he arrived at the solution as follows.

56

Ilay:

Because when you now, when you now... aahmmm... then that's 7 (points to his thumb) plus 7 (points to his index finger) are then 14. And then you work out these 2 (points to his middle finger and his ring finger) are also 14. There are 28 in total. And then you only need to add this 14 (holds up the next two fingers). In total around 42. 57 Teacher:

And what comes out at the end of this problem? (points to the task 7 x 7) 58 Ilay: Around... 48 (he starts to count his fingers forward once again, whispers) 14, 28, 42. So... (after 25 seconds) 49.

Ilay worked out the result by availing of additive references (turn 56 and 58). For him it appeared that being asked to multiply was more a request to count up. In doing this, Ilay displayed typical one-sided additive ideas with respect to multiplication and little strategic knowledge [START_REF] Gaidoschik | Do lower-achieving children profit from derived facts-based teaching of basic multiplication: Findings from a design research study[END_REF]. Furthermore, he was not able to derive the solution for the new task 7 x 7 from the previous task 6 x 7.

In order to emphasise the differentiation of multipliers and multiplicands for Ilay, during the second support unit the meaning-related vocabulary for the same groupings was introduced (step 2). In the following scene, approximately ten different rectangular arrays were laid out in front of the children, which also included the correct array for the task 3 x 5 (for Ilay and Nelli a task from the fifth row).

7

Teacher: I will now give you the problem. Do you know the game, "I spy with my little eye"? We will now play using the cards: I spy with my little eye lots of 3 groups of 5 each (places the note "3 groups of 5 each" in front of the children). 8

Nelli: 3 groups of 5 each? 9

Teacher: Do you have any idea which dot image I mean? 10 Ilay:

(points uncertainly at the card containing the dot image relating to 3 x 5) 11 Teacher:

Why could that match? 12 Ilay:

Yes, because here are 5 (points to the uppermost row) 13 Nelli:

... because there are 5 in each row. 14 Ilay:

... because there are 5 in each row, so... 15 Nelli: ... 15. 16 Ilay:

So when you now calculate (takes the card with the rectangular array 3 x 5), 1, 2, 3 (tips each of the respective first dots of the first column), there are 3 below and here, calculate 1, 2, 3, 4, 5 (tips each of the respective dots of the first line), then these are 15 and here are 3 groups of 5 each (points to the note with the same sentence). 17 Teacher:

Okay, so where are the 3 groups of 5 each? 18 Ilay: 3? 19 Teacher:

You have already said that these are 3 and 5. But where are the groups of 5? 20 Nelli:

Here at the top (moves along the top line). 21 Ilay:

Exactly, and we then have a group of 5 here (moves along the top row), two groups of 5 together, three groups of 5 together (moves along the other rows).

It can be seen that the children initially react in an uncertain manner to this unknown vocabulary (turn 8, turn 10). Even when Ilay directly found the requested dot image (turn 10), he was still not able to immediately explain why the description "three groups of five" matched this dot image. He pointed to the first column and the first row (turn 12, turn 16). This focus can be seen in many lowachieving children and indicates a limited conceptual understanding [START_REF] Moser Opitz | Rechenschwäche / Dyskalkulie. Theoretische Klärungen und empirische Studien an betroffenen Schülerinnen und Schülern[END_REF]. Nelli, on the other hand, seemed to understand the content immediately, which can often be seen in the case of normal-achieving children [START_REF] Götze | Fostering a conceptual understanding of division: Results of a language and mathematics integrated project in primary school[END_REF]. She emphasised that in each row there must be five dots (turn 13), which Ilay then repeated word-for-word (turn 14). Thus, Ilay had not yet understood the content of this formulation because he could not answer the follow-up question of the teacher (turn 17, turn 18). The teacher correctly recognised that although Ilay pointed to the numbers three and five of the term in the rectangular array in his customary manner, he had not yet started to think of cardinal groups of always five dots. It was only when Nelli pointed out the group of five in the dot image to him that Ilay rethought (turn 21) and recognised this new perspective.

The overall evaluation of the data from the support units of all ten low-achieving fostered children demonstrated that this scene appeared to be a key scene in the learning process for low-achieving children such as Ilay. If they were to understand the grouping concept, they had to be given a new way of looking at the rectangular arrays by means of the new language being offered. In order to ensure that the children did in fact understand the content of the subject matter, substantial repetition was needed in order for the children to make this new way of looking at rectangular arrays a part of their own mental models.

In this respect, in the third support unit Ilay showed that he began to integrate the meaning-related vocabulary into his own mental thought patterns (step 3). It was astounding that already in the third support unit he seemed to give up on his additive viewpoint that he had so strongly been focussed upon up to that point. He explained why the term 6 x 6 matched to the corresponding rectangular array without having to be asked to do so by the teacher.

123 Ilay: So, I took this one here because that is 6, so 6 is here (moves along the left column with his finger) and there are 6 groups (moves his finger from left to right several times) ... And here 1, 2, 3, 4, 5, 6 (taps on each of the respective points in the first row), there are 6 in a group, and when you count down (moves the finger from top to bottom), then 1, 2, 3, 4, 5, 6 (taps on each of the respective points in the first points), that's actually really easy.

First, he addressed the groups number, in other words the multiplier, by pointing to the six groups in the right corner ("there are 6 groups"). Subsequently, he described the meaning of the second six in the term ("there are 6 in a group"). Drawing on the meaning-related vocabulary he had learnt, he completely independently explained the differentiation of both sixes in the term 6 x 6: one indicates the number of groups, the other the group size.

In order to obtain first indications about a possible further development of derived-fact strategies, in the fourth support unit, the children were supposed to try to make deductions between the tasks. In the following scene, Ilay was asked to think about why 7 x 5 is always seven more than 7 x 6.

47 Ilay: Because, that (points to the multiplier) stays the same and that (points to the multiplicand), that should, if you jump over one, then that should now also be 7s. 48 Teacher: I don't quite understand that. What do you mean, if you jump over one? 49 Ilay:

If you count one to here (points to the multiplicands 5 and 6), then that will, then it is, if you, a 7, if you (points to both multiplicands) count up to 7. 50 Teacher:

So, if we have 7 times 5 and 7 times 6. Why is it 7 more? 51 Ilay:

Nelli, you count it. 52 Teacher:

That's right. But I am not convinced yet. 5s become 6s.

53 Ilay:

Oh! Because here, here, you have 6. And then one always comes in here, (points to both multiplicands), it only comes, if we do that with the dots, it only comes, so one, so one, so one row, so one dot always becomes more.

At the beginning of the scene, Ilay initially tried to explain the correlations with his own linguistical means and using lots of pointing gestures (turn 47). Indeed, he also occasionally used meaningrelated vocabulary such as "sevens" but could not yet use it to explain the content-related correlation of 7 x 5 and 7 x 6. Only the direct reference of the teacher in turn 52 ("fives turn into sixes") seemed to activate the meaning-related vocabulary for him. He was then able to use it to explain the multiplicative (not additive!) correlations very independently (turn 53). In contrast to the first support unit, he did not use his finger to count the rows but thought purely in terms of content. This content-related thinking in multiplicative structures was shown several times in the fourth support unit.

It can be seen in the above analysis that Ilay completed the intended four learning steps. After the introduction of the basic meaning related vocabulary, he first needed some time for practice and networking with concrete representations (step 2), before this vocabulary slowly became Ilay's own vocabulary of multiplicative thinking (Pöhler & Prediger, 2015; step 3). Perhaps this newly formed content related multiplicative thinking provided the basis for using derived-fact strategies. Repeated addition was a dead end for him in multiplicative understanding. With this respect, the meaning related vocabulary fulfilled a cognitive and epistemic function for him. Expressions such as "3 times 5" were no longer a "secret language", and the multiplication in rectangular arrays was really perceived in a multiplicative way via these expressions.

Discussion and limitation

Ilay's learning trajectory stands representative for other fruitful learning trajectories of other children. All these learning trajectories have in common that the children start to think in groups of the same size. The fostered vocabulary seems to support the development of a mental model in all these children. The meaning-related verbalizations become the language of thinking and therefore, the children can understand how to decompose or derive multiplication tasks. This multiplicative point of view has apparently not been obvious to the children in their previous learning biography through formal expressions such as "3 times 4" or "3 multiplied with 4" or "4 multiplied by 3". However, basic meaning-related phrases such as "3 groups always with 4 (dots) in a group" or "3 groups of 4" illustrate thinking in multiplied groups of equal sizes. This vocabulary then fulfils epistemic functions in the learning process and contributes to conceptual understanding. The problem with this is repeatedly having to detach low-achieving children from using their old, supposedly safer strategy of repeated addition. Throughout, the children use both additive and multiplicative ideas at the same time. The in-depth analysis of children such as Ilay therefore shows how central it is to continue to promote this new multiplicative point of view. Furthermore, the indepth analysis demonstrates that the multiplicative thinking seems to be a good basis for the development of decomposition strategies. If children can think in a content-related way they can also derive tasks from each other and recognise correlations between tasks.

Despite that, not all supported children show these learning developments (for details see Götze, under review). The most striking factor however is that these children do not start to think multiplicatively. The meaning-related vocabulary does not become the vocabulary of thinking for these children. This is mostly caused by the teacher who starts to retrieve facts and vocabulary (for example "Where are the groups in this term? Where are the number of groups in the rectangular array?") instead of supporting multiplicative thinking. Especially in this paper only one single-case study can be presented. For a comparative analysis it would be helpful to contrast Ilay's individual learning trajectory with those of other children (see Götze, submitted). But that would go far beyond the scope of this paper.

This study is laid out as qualitative Didactical Design Research (Pöhler & Prediger, 2015). First of all, the aim was to derive hypotheses about the effectiveness of the support and not to operationalize them. Therefore, the duration of the fostering was very short, with just four support units. Yet, the study provides first indications that the conceptual understanding of multiplication can be promoted in a language sensitive manner when we can achieve that the children use the meaning-related vocabulary as their vocabulary of thinking. How long the fostering should actually last for the supported children to be able to independently think in a content-related way, remains unanswered. Future research must therefore check the hypotheses made in this paper.

Figure 1 :

 1 Figure 1: Visualization of multiplier and multiplicand in the rectangular array


Jenni Ingram
email: jenni.ingram@education.ox.ac.uk

Nick Andrews

Claims and demonstrations of understanding in whole class interactions

Keywords: Conversation Analysis, classroom interaction, understanding

One aim of teaching mathematics is to develop students' understanding of mathematics. In this paper we examine how teachers and students do understanding in interaction, and how this understanding is handled interactively during whole class interactions. Taking a Conversation Analytic approach, we look at interactions between teachers and students where there is claim of understanding and where there is a demonstration of understanding. We show how these two situations are handled differently, by the teacher and the students, with only the latter situation being concerned with students' understanding.

Introduction

One aim of mathematics teaching is to both develop students' understanding of key ideas, concepts and methods, and to respond to difficulties students encounter in understanding mathematics. Studying understanding in mathematics education is not original, but the existing research largely treats understanding as something real, as something that students do or do not have. However, understanding is also used by teachers and students in interaction for practical purposes, and this use or display of understanding is observable and analysable. The aim of the analysis in this paper is to investigate what the uses of words like 'understand', 'sense' or 'mean' do in interaction, focusing on how teachers and students are using these words to achieve different things. We use Sack's notions of claiming and demonstrating understanding as is observable in the interaction, rather than making any claims as to whether students actually understand or not, which is not directly observable in interaction.

Pedagogic interactions, and in particular teacher-student classroom interactions, are dominated by the IRE interactional structure. The IRE (or IRF) sequence, teacher initiation-student responseteacher evaluation, has been written about extensively (e.g. [START_REF] Hellermann | The interactive work of prosody in the IRF exchange: Teacher repetition in feedback moves[END_REF][START_REF] Mehan | Learning lessons: Social organization in the classroom[END_REF][START_REF] Sinclair | Towards an analysis of discourse: The English used by teachers and pupils[END_REF][START_REF] Wells | Reevaluating the IRF sequence: A proposal for the articulation of theories of activity and discourse for the analysis of teaching and learning in the classroom[END_REF]. This characterisation, however, only focuses on the broad structure of the interaction, rather than the multiple activities being done within it. The third turn, the evaluation turn, in particular does more than evaluate a student's answer as correct or not, but also comments on its adequacy as a response to the initiation, and its appropriateness for the broader interactional and pedagogical purposes of the interaction [START_REF] Macbeth | Hugh Mehan's learning lessons reconsidered: On the differences between the naturalistic and critical analysis of classroom discourse[END_REF], and influences the interaction that follows. What is being assessed or evaluated by the teacher in this third turn is not necessarily focused on the content of the student's response, but could also, or alternatively, focus on how the response enables the interaction to progress, they are task-oriented rather than content-oriented [START_REF] Antaki | High-grade assessment sequences in the completion of interactional units[END_REF]. [START_REF] Koole | Teacher evaluations: Assessing "knowing", "understanding", and "doing[END_REF] also showed that teacher assessments in this third turn can focus on students' knowing, students' doing or students' understanding. However, positive assessments largely focused on students' knowing, treating their response as shared knowledge on which to build on, but negative assessments largely focused on students' doing or understanding.

In this paper we focus on IRE sequences where the content focus includes understanding, or not understanding. In particular, we examine instances where evidence of understanding (or not understanding) is displayed by teachers or students in whole class interactions. These instances can include situations where the teacher-initiation requests students to claim or demonstrate understanding, which might include teachers themselves claiming or demonstrating understanding or not understanding, or situations where it is the students who make claims or demonstrations of understanding relevant in their response turn, or finally situations where the teacher's evaluation turn treats the student's turn as a claim or demonstration of understanding. Our analysis of these sequences takes a Conversation Analytic approach drawing upon Sacks' discussions of 'how understanding is shown' (1992:II:140). We make use of Sacks' distinction between claiming and demonstrating understanding and show that the teachers in these interactions treat claims and demonstrations differently, being content-oriented when there are demonstrations of understanding but task-oriented when there are solely claims of understanding.

Methodology

This analysis was carried out on a video corpus collected in fifteen mathematics teachers' classrooms in ten schools in England of secondary educations (students aged 11-18). All names used are pseudonyms. These videos were transcribed using Jefferson transcription [START_REF] Jefferson | Transcript Notation[END_REF] and the analysis focused on these transcripts, but they are presented here in a simplified form for ease of reading. The transcripts of all whole class interactions in a total of 39 lessons were analysed using Conversation Analysis (CA) [START_REF] Sidnell | The handbook of conversation analysis[END_REF]. The CA approach was used to develop a collection of cases [START_REF] Sidnell | Conversation analysis: An introduction[END_REF] examining how the topics of understanding and sensemaking are treated by the teachers and students as they interact. These cases include 299 interactions where the teacher explicitly talks about understanding and 58 where a student explicitly talks about understanding (including interactions about making sense, getting it, and finding meaning). In this paper we focus on those interactions that broadly follow the IRE structure in that they involve both the teacher and the students, rather than those that occur during a teacher explanation for example.

From a CA perspective understanding does not refer to a cognitive state but to an interactional object where teachers and students do understanding as they interact. For example a student saying 'I don't know' is often used to request help from the teacher, or to initiate a complaint about an explanation [START_REF] Lindwall | Uses of "understand" in science education[END_REF]. As such, we are not making any claims about what teachers or students understand, instead focusing on how they use understanding in interaction to achieve particular things. Within CA research a distinction is made between demonstrating and claiming understanding [START_REF] Sacks | Lectures on conversation[END_REF]. For example, a teacher can demonstrate understanding of what a student has said by responding with an alternative formulation as in the following extract: Alternatively, the teacher or student could just respond 'yes' or 'good' which would merely claim understanding.

In classrooms teachers generally ask questions that they already know the answer to, and as such student responses are demonstrations that they also know the answer. However, students can respond in a way that shows that they have understood, or in a way that shows what they have understood [START_REF] Koole | Displays of epistemic access: Student responses to teacher explanations[END_REF]. Whilst showing that they have understood can involve either a claim or a demonstration of understanding, showing what they have understood requires a demonstration of understanding. Koole (2010) also showed that teacher questions of the form "do you understand" are usually followed by a claim of understanding and not a demonstration of understanding.

Results

In the data we present here we will demonstrate that claims of understanding support the smooth progression of interaction [START_REF] Stivers | A preference for progressivity in interaction[END_REF]) that is underpinned by shared understanding [START_REF] Weatherall | When claims of understanding are less than affiliative[END_REF]. That is, claims of understanding are affiliative joint accomplishments that enable the interaction, and the lesson, to continue, but tell us nothing about what students understand. On the other hand, demonstrations of understanding tell us something about what students do or do not understand, but also lead to more extended interactions that focus on the content of these demonstrations, rather than moving the interaction on.

Claims of Understanding

Claims of understanding usually follow an understanding check by the teacher, such as "do you understand?" or "does that make sense". Here the students have made a claim of understanding in their answer yeah in turn 71, but there is no evidence of what they understand or the nature of their understanding within the interaction. The teacher acknowledges this claim in turn 72 before moving on to a different student and asking them to explain their method.

Another situation where there is a claim of understanding but without a demonstration of understanding is given in extract 3.

173 Teacher: numbers pick one of the numbers as a value for n. work out the missing numbers, using the same shape, apply that rule to somewhere else, see if you can add them up to be um using this rule like this one if I draw that C anywhere else I should find that I get er five n add two will tell me, if I draw a C in here, er draw this c in one two three four five. five times thirty-two is a hundred and sixty, add two makes a hundred and sixty-two. SNAME 174 Student: I don't understand okay I'll come and see you. it will be seven minutes, to have a go, starting now.

Extract 3: Ryan's lesson on a hundred square investigation

In turn 173 Ryan is describing how to draw a letter C on a hundred square and then sum the numbers in the letter C by treating one of the squares as n. In turn 174 a student makes a claim of not understanding but does not add anything that would indicate what they do not understand, so no demonstration of understanding or not understanding. This claim is requesting help from the teacher. Ryan acknowledges this request but defers dealing with the issue before continuing to talk to the rest of the class.

These teacher turns that include understanding checks are not always followed by student responses as in extract 4:

178 Teacher: …but if I drew another rectangle and had that as two and four, that isn't similar to that cus they're not in the same proportions, yeah? does that kind of make sense? It's a bit of a weird mathematical word. right? but we have to get used to it in a mathematical thing, so it doesn't quite but congruent certainly is going to come up. okay. so we're going to start with reflections, but…

Extract 4: Imogen's lesson on transformations

Here Imogen asks "does that kind of make sense?" but without pausing and without any students responding continues here turn before moving on to a new topic at the end of the extract.

Where there is a claim of understanding (or not understanding) but no demonstration of understanding, the interaction continues without any reference to what has, or has not, been understood. Where the teacher has invited a claim of understanding but no student response is given, again the interaction continues without any reference to what has, or has not, been understood. Since 'do you understand' or 'does that make sense?' questions prefer a claim of understanding rather than a demonstration of understanding [START_REF] Koole | Displays of epistemic access: Student responses to teacher explanations[END_REF], when no response is given it is a claim of understanding that is 'noticeably absent' [START_REF] Bilmes | The concept of preference in conversation analysis[END_REF] and in every case in this data where there is no response, the interaction continues in the same way as if there had been a positive claim of understanding. In contrast, as we will show in the next section, when there is a demonstration of understanding, or not understanding, the following interaction focuses on the content of the understanding and extends the IRE sequence [START_REF] Schegloff | Sequence organization in interaction: A primer in conversation analysis[END_REF].

Demonstrations of Understanding

Teachers can explicitly ask students a question that requires them to demonstrate their understanding. In extract 5, Fiona asks the student to explain so that she and the other students can understand, and the student gives an explanation in turn 188. In the turn that follows Fiona focuses on one part of the student's explanation and asks a follow up question.

187 Teacher: right explain it to me so that I understand but also so that people that haven't done this one understand 'cause I know that there's quite a few people who haven't done it is it because Rio is forty degrees and I've worked out that Khartoum is thirty-five degrees you just add five 189 Teacher: right where did you get thirty-five from for Khartoum, was that one of the ones that we've worked out Extract 5: Fiona's lesson on negative numbers

In extract 6, Tom is asking why it makes sense to take a moving average over four points when dealing with quarterly sales of ice cream. The student, Sayed, gives an explanation which Tom then follows with a prompt to be more specific in turn 70, to which the student responds with more detail.

68 Teacher: yeah it is because of that so why, why does that make sense, can you say so why does that makes sense to use four in the numbers. it is something to do with the fact that it's quarterly. Sayed? 69 Sayed: because there's four quarters 70 Teacher: where 71 Sayed: in the end, in a year 72 Teacher: yeah there's four quarters in a year but, so, you know so …

Extract 6: Tom's lesson on moving averages One of the most common ways that teachers ask about students' understanding is in relation to the meaning of mathematical vocabulary.

11 Teacher: ah that's a good word isn't it isosceles. what do you understand by isosceles 12 Student: erm two lines that are perpendicular they're straight lines and then they have two of the same angles 13 Teacher:

we're getting lots of different mathematical words aren't we. um two lines are perpendicular what do you mean perpendicular.

Extract 7: Dave's revision lesson

In turn 10, before the transcript begins, a student has given a justification for an angle having a particular value as being because the triangle is isosceles. Dave follows this up in turn 11 by asking this student what he understands by isosceles, which he explains in turn 12, (inappropriately) introducing the word perpendicular to describe the two sides with equal length. Dave then follows up on this and returns the turn to the student to explain what he means by perpendicular.

In these three extracts the teacher's initiation includes an explicit request for a demonstration of understanding, which the student gives in their response, and which is followed by the teacher asking a follow-up question focused on the content of the students' demonstration which returns the turn to the student. Demonstrations of understanding can also occur when a student is indicating that they do not understand something. In extract 8, a student has given the equation of a straight line using the gradient and the y-intercept which Dave repeats in turn 71. Dave then opens up the floor to invite questions from the students at which point a student asks how to find the equation and states that he does not understand. This is both a claim of not understanding and a

demonstration of what it is that is not understood1 . The teacher follows this with an explanation for how the class found the equations of straight lines in previous lessons as well as giving a rule.

71 Teacher: …yes three x plus one. um okay that's fine. who wants to ask anything about the first side, you do. go on 72 Student: um how do you find out the um equation I don't understand it 73 Teacher: no okay. this three I remember we went to the computer room and drew some things didn't we this three turned out to be the gradient and this one turned out to be where it goes through the y axis. so once you've found out the gradient is three you can just stick it in there um that requires quite a bit of thought doesn't it um at this stage with the exams coming up just stick it in there in front of the x and you put plus one on the end because of that and it all fits in with the pattern we saw when we drew lots and lots and lots of these in the computer room …

Extract 8: Dave's revision lesson Furthermore, whilst the student in turn 72 treats the issue as one of understanding, the teacher's following turn treats it as an issue of remembering [START_REF] Koole | Teacher evaluations: Assessing "knowing", "understanding", and "doing[END_REF].

In each of these cases where there is a demonstration of understanding, or not understanding, the teacher's follow up turn focuses on the content of the students' responses. This is in contrast to the situations where there was only a claim of understanding where the teacher's follow up turn moved the interaction on. This arises both in situations where the teacher has asked students about their understanding as well as where students indicate that they do not understand. However, this second situation is rare in whole class interactions. Also, where the teacher has requested the demonstration of understanding in their initiation, the teacher's third turn also returns the turn to the same student. The preference for moving the interaction on following claims of understanding is reinforced by the numerous occasions where teachers as "do you understand" but does not wait for an answer before moving on to the next topic (as illustrated in extract 4), which occurred in just over 40% of cases of understanding checks in this data (in contrast to [START_REF] Koole | Displays of epistemic access: Student responses to teacher explanations[END_REF] where the students always responded).

Conclusion

In this paper we have focused on how claims and demonstrations of understanding are dealt with by teachers and students in their interactions. Within the IRE structure, teachers can invite either claims of understanding or demonstrations of understanding in their initiation, and this distinction is further supported by the students who in their responses give either a claim or a demonstration as required by the teacher's initiation. However, teachers and students are doing different things when they invite or give claims to when they invite or give demonstrations. Claims of understanding are treated as ways of supporting the progressivity of the interaction [START_REF] Stivers | A preference for progressivity in interaction[END_REF] and do not deal with the content or issues of what is or is not understood. In contrast, demonstrations of understanding lead to the interaction focusing on the content of these demonstrations and an expansion of the IRE structure.

Transcript Notation [START_REF] Jefferson | Transcript Notation[END_REF] [ text ] Brackets Indicates the start and end points of overlapping speech.

(.) Micropause A brief pause, usually less than 0.2 seconds.

. Period Indicates falling pitch or intonation.

? Question Mark Indicates rising pitch or intonation.

, Comma Indicates a temporary rise or fall in intonation.

  25 Teacher: … what is the difference then, what makes a histogram a histogram. Jin? 26 Student: because the bars can be wider 27 Teacher: exactly in a histogram, you can have different width um bars… Extract 1: Tom's lesson on statistical diagrams

Extract 2: Becca's lesson on multiplying fractions

	70 Teacher:	is (.) does that make sense?
	71 Students:	yeah
	72 Teacher:

okay Simone did you do it exactly the same. (.) no slightly different. thank you Steve. er Simone could you (.) explain what you did. was it-

Whilst the student utterance of "how do you find out the um equation" could be interpreted as an issue of knowledge or knowing, the student treats it as an issue of understanding by following it with "I don't understand it" rather than "I don't know".