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

Nick Andrews

Andrea Pitt

Revisiting the roles of interactional patterns in mathematics classroom interaction

Keywords: Interactional patterns, funneling, questioning

The ways in which teachers and students interact about mathematics in lessons can be more powerful in influencing learning than the materials and resources that teachers use. Interactional patterns structure all interactions and there are many such patterns that occur frequently in mathematics lessons. This paper focuses on one such pattern, the funneling pattern, which is widely discussed in the literature. Three distinct examples described in the literature as a funneling pattern are examined in order to examine the different roles sequences of closed questions can have and the opportunities these patterns can provide or constrains to students in the learning of mathematics.

Introduction

The aim of this paper is to contribute to the discussions around the roles different interactional patterns have in the teaching and learning of mathematics. The simplest, and most prevalent, pattern that is discussed widely in the mathematics education literature is the IRE pattern of teacher initiation, student response, and teacher evaluation [START_REF] Mehan | Learning lessons[END_REF]. Many authors describe this pattern as doing little to encourage students to reason, give explanations or articulate their thinking (e.g. [START_REF] Cazden | Classroom discourse: The language of teaching and learning[END_REF][START_REF] Nystrand | Dialogic instruction: When recitation becomes conversation[END_REF]). Yet the discussion has now moved on, with authors pointing out that it is not the IRE pattern itself that is the issue, but rather how it is used. This IRE pattern continues to dominate classroom interaction because it enables students to know when to speak, how to speak and about what to speak [START_REF] Ingram | Turn taking and 'wait time' in classroom interactions[END_REF][START_REF] Wood | Alternative patterns of communication in mathematics class: Funneling or focusing?[END_REF]. This pattern can be used by mathematics teachers to convey and establish different norms. It creates opportunities for students to communicate in classroom interaction, but it is largely teachers who can both constrain or enhance their students' opportunities to communicate mathematics or to communicate mathematically. This paper focuses specifically on another interactional pattern called the 'funneling' pattern [START_REF] Bauersfeld | Hidden dimensions in the so-called reality of a mathematics classroom[END_REF][START_REF] Wood | Alternative patterns of communication in mathematics class: Funneling or focusing?[END_REF] that comprises of a series of IRE sequences. Four extracts that have many of the features of the 'funneling' pattern are discussed with a view to illustrating here that again it is not the pattern itself that focuses student thinking on "trying to figure out the response the teacher wants instead of thinking mathematically himself" (Wood, 1998, p. 172), but rather it is the way that it is used by the teacher that can affect student thinking.

The funneling pattern

The funneling pattern was initially described by [START_REF] Bauersfeld | Hidden dimensions in the so-called reality of a mathematics classroom[END_REF][START_REF] Bauersfeld | Interaction, construction, and knowledge: Alternative perspectives for mathematics education[END_REF] and consists of a series of teacher questions and student responses that has particular features. The sequence follows an incorrect answer from the student, or some other form of difficulty with the mathematics. The teacher uses "more precise, that is, narrower, questions" (Bauersfeld, 1988, p. 36) to lead the student to the correct answer. This narrowing effect of questions towards a particular correct answer (hence the term funneling) contrasts with sequences of questions that leads students step-by-step through a process (e.g. [START_REF] Herbel-Eisenmann | How discourse structures norms: A tale of two middle school mathematics classrooms[END_REF], however both invite students to do little more than complete the teacher's sentences (e.g. [START_REF] Franke | Teacher questioning to elicit students' mathematical thinking in elementary school classrooms[END_REF]. These different examples have led to further terms, such as leading questions [START_REF] Franke | Teacher questioning to elicit students' mathematical thinking in elementary school classrooms[END_REF], guiding questions [START_REF] Moyer | Learning to question: Categories of questioning used by preservice teachers during diagnostic mathematics interviews[END_REF] and scaffolding [START_REF] Wood | The role of tutoring in problem solving[END_REF], becoming associated with funneling. The distinction between these terms and the precise relationship with funneling is often not made but we believe it is an important one as we outline below. As a result, funneling has become used more broadly in the literature to describe any sequences of IRE patterns that lead students through a series of specific narrow questions, often only requiring short factual response from the students.

Particular concerns have been raised about the implications of such funneling interactions. For example, [START_REF] Brousseau | The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics[END_REF] refers to the Topaze effect in which the sequence of 'funneling' questions disguise the mathematical knowledge that is being targeted by the interaction as a whole. Indeed such interactions in which students do not need think about mathematical relationships, patterns or structures in order to answer the teachers' questions [START_REF] Wood | Alternative patterns of communication in mathematics class: Funneling or focusing?[END_REF] the most frequently cited instantiations of 'funneling' patterns. [START_REF] Wood | Alternative patterns of communication in mathematics class: Funneling or focusing?[END_REF] argues that in funneling patterns of interaction students are only responding to the surface linguistic patterns in order to respond appropriately to the teacher's initiations. However, [START_REF] Temple | Developing fluency in the mathematical register through conversation in a tenth-grade classroom[END_REF] have shown how this aspect of the funneling pattern can be used by mathematics teachers to activate prior knowledge and offer them opportunities to talk about newly learned concepts. This indicates the possibility that the funneling pattern can have a variety of roles within the classroom, some of which support students' learning and communication of mathematics. Wood also connects funneling to "certain beliefs about the nature of mathematics and the relationship between teacher and students" (p. 175) but we would suggest that it is not the pattern itself that indicates these beliefs, but how it is used by teachers.

Data and methods

The data used in this paper to illustrate the different functions of a funneling pattern of interaction comes from two sources. The first is transcripts from two videos of mathematics lessons collected as part of a larger project looking at the role and use of language in mathematics teaching and learning. Both lessons were from the same school, a small inner-city comprehensive secondary school with high levels of students in receipt of free school means and over 50% of the students with English as an additional language. The lessons are taught by two different teachers and the students are aged 11-12 years old. The second source is transcripts from a published article [START_REF] Drageset | Student and teacher interventions: A framework for analyzing mathematical discourse in the classroom[END_REF] focusing on categorizing language use in mathematics classrooms that also uses conversation analysis as its methodology. A conversation analytic approach is taken in the analysis of the transcripts, which is an approach that focuses on the identification of patterns of interaction. Conversation analysis (CA) looks specifically at what participants are doing in their turns at talk through a careful analysis of how the turn is designed, both in terms of its content but also in terms of how it is spoken, i.e. quickly, hesitantly, emphasizing particular words. A key feature of any analysis based on CA is the reflexivity of turns at talk. Each turn is designed in response to the turns that it follows and affects the turns that follow. This makes it a particularly useful approach for examining the relationship between teacher questions and student responses.

The roles of funneling

In this paper we will outline three distinct patterns of interactions described in the literature as funneling. The first is used by the teacher to make assumed knowledge publicly available. The second offers students the opportunity to use recently introduced vocabulary. The third involves two extracts that are used in combination to draw attention to structures within a sequence of mathematical interactions.

Making assumed knowledge publicly available

The funneling pattern of interaction does not occur very often in the lessons collected as part of the larger project, which contrasts with other studies looking at mathematics classrooms (e.g. [START_REF] Temple | Developing fluency in the mathematical register through conversation in a tenth-grade classroom[END_REF][START_REF] Franke | Teacher questioning to elicit students' mathematical thinking in elementary school classrooms[END_REF]). Yet using a conversation analytic approach in the analysis of this pattern reveals that each instance of the pattern is doing different things. For example, in the first lesson the students have discussed the meaning of some key words on the whiteboard associated with probability. The extract in Figure 1 follows this discussion and then is followed by an activity where students are tossing a coin twenty times and then combining the results. No connection is made between this interaction and the tasks that came before it or after it. The teacher asks a series of questions requiring short factual answers, which are given by the students. These questions lead the students through a step-by-step process for calculating a probability. The fact that these responses are given hesitantly, as indicated by the pauses, ums and phrasing the response as a question, is ignored by the teacher. The sequence of questions focuses on the identification of the numerator and the denominator when identifying the probability and this is emphasized through the teacher's choice of accepting the answer three sixths rather than the half, which is acknowledged but not treated as the answer to the probability of rolling an even number.

The interaction ends with the teacher checking that the students are happy with this process and treating them as such by moving on to the next task. Yet there is little in the interaction to indicate that the students as a whole could calculate the probability themselves. This is a feature of the funneling pattern that Wood (1988) draws attention to: that it can give the impression of learning even though it is the teacher that has done the cognitive work. However, what the teacher has done through this interaction is explicitly to make the process public and has involved the students in this process (as opposed to just telling them how to calculate the probability). The ability to calculate the probability of an event is taken as assumed knowledge in the following task where the students have to calculate the relative frequency of getting a head when tossing a coin. So, whilst there is no evidence that the students are doing more than responding to the immediate initiations, the funneling pattern of questions and responses does make public knowledge that is needed later in the lesson. So the teacher's questions are doing something other than just assessing whether students have the required knowledge. This is demonstrated further in other examples where incorrect responses are ignored such as the second student's suggestion of larger in the second extract (Figure 2), which comes from a lesson focused on solving linear equations.

Opportunities to use terminology

The majority of the lesson on linear equations is spent with students working independently through a set of differentiated exercises. At the start of the lesson a student asks a question about the difference between an expression and an equation and the extract follows this question. Again, the teacher leads students through a series of closed questions requiring short factual responses from them. This example shows the teacher using questions that offer students opportunities to talk about newly learned concepts and new terminology in a similar way to the example offered by [START_REF] Temple | Developing fluency in the mathematical register through conversation in a tenth-grade classroom[END_REF]. The questions serve to support the students in recalling processes and words introduced in previous lessons such as simplifying and collecting like terms. Each use of a technical word is connected to the specific example, 3x + x becoming 4x, and 3x + x = 4x being an expression is contrasted with 4x = 12 being an equation. Throughout the interaction student responses that do not fit with the use of the language the teacher is focusing on are ignored or built on by the teacher who turns them into a form that does fit. This sequence of questions again is doing other than assessing students' knowledge. The questions are providing students with the opportunity to use mathematical terminology and hear it used in a mathematical way by the teacher. This sequence could be considered a form of scaffolding [START_REF] Wood | The role of tutoring in problem solving[END_REF] if the support the teacher is giving, through his questioning and phrasing of his responses, is withdrawn over time until the students are using the language in their own descriptions of their work on mathematical tasks.

Drawing attention to regularities

The last two transcripts are taken from [START_REF] Drageset | Student and teacher interventions: A framework for analyzing mathematical discourse in the classroom[END_REF] and have been coded as "closed progress details" which is one of the "main elements of funneling" [START_REF] Drageset | Redirecting, progressing, and focusing actionsa framework for describing how teachers use students' comments to work with mathematics[END_REF]. In each extract the teacher takes a step-by-step approach in posing questions and students are only required to give short factual responses to the question asked immediately before:

Teacher:

How much is one of… one-fifth then of … of twenty-five?

Student: Five.

Teacher: It is five, yes. How much is two-fifths?

Student: …ten.

Teacher:

Then it becomes ten. How much is three-fifths Student: Fifteen.

Teacher: How much is four fifths?

Students: Twenty.

Teacher: And how much is five fifths:

Students: Twenty-five.

Student: One whole.

Teacher

One whole, yes. Yes, good. Great. Extracts 1 and 2 are not just narrowing sequences of questions, but are also specific, structured, and lead to a mathematical regularity within the sequence of questions itself. It is also the repetition of the pattern of interaction itself that offers students an opportunity to see the relationship between the fractions and the quantities. This is pointed to by the teacher in their penultimate turn with the phrase "and then, this time". So, whilst the teacher does not explicitly talk about the meaning of 'one whole' the sequence of questions identifying each of the fractional parts goes in order, and stops when one whole is reached. The teacher does not ask what seven sixths is, and also does not stop at four sixths for example. In both extracts the total number, twenty-five and then thirty, is said alongside one whole. The sequence of closed questions is leading students through a process in a similar way to the example offered by Herbel-Eisenmann (2000, p. 182). However, it is also the repetition of the sequence that makes this process more explicit and affords students' attention to be drawn to it.

Conclusion

In this paper we have explored three different interactional patterns referred to in the literature as a type of funneling pattern: one example of a narrowing pattern, one example of step-by-step pattern, and one example of connected step-by-step patterns. Each pattern includes a sequence of closed questions requiring short factual responses from the students. Each sequence is leading the students to a particular answer. However, we question whether the mathematical knowledge is always being disguised (c.f. [START_REF] Brousseau | The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics[END_REF]. Each pattern is doing something different to the other patterns and, in the final example the repetition of the pattern itself can be used to support the students' thinking.

We have shown the possibilities for how teachers can use these sequences of questions to make assumed knowledge publicly available for subsequent work, offer opportunities to use technical vocabulary, and perceive regularity in mathematical processes. Each of these functions is an important part of the teaching and learning of mathematics. Whilst in each of the examples offered it is the teacher who is controlling the content, in is possible to imagine situations where the teacher is using a similar sequence of closed questions about a student's idea. The funneling pattern can and does have a role in the teaching and learning of mathematics but it is how it is used, rather than the structure of the pattern itself, that can offer or constrain opportunities for students to engage in mathematical thinking and communicating.

Figure 1 :

 1 Figure 1: Calculating the probability of getting an even number See Jefferson (2004) for details of the transcript conventions used here

Figure 2 :

 2 Figure 2: The difference between an expression and an equation

  if I have thirty chips here and then divide them into six equal piles, then how many are there in each pile then?Student1:There are five (hold up five fingers).Teacher: Five. But how much is two-sixths of thirty, then?

Proceedings of CERME10

Acknowledgment

The work discussed in this paper was partly supported by the John Fell Fund.


domain: math shs

text:
Uta Häsel-Weide
email: uta.haesel.weide@math.uni-paderborn.de

Occasions for productive interactions in inclusive mathematics classrooms that arise following mistakes

Keywords: Cooperative learning, discourse, interaction, inclusion, mistakes

When learning mathematics, mistakes can serve as productive occasions if a mistake is followed by a process of negotiation and insight. This paper addresses the question of the extent to which the informal occasion of a "mistake" leads to productive interactions in inclusive mathematics classrooms. Within the context of the project LUIS-M video-based qualitative analysis of cooperative learning situations in inclusive classrooms are made with focus on formal and informal occasions for productive interaction. In the paper the negotiation processes which follow mistakes are exemplified and discussed with respect to opportunity for learning processes.

Introduction

Learning mathematics requires interaction; cooperative learning leads to interaction. This is the hope which is associated with cooperative learning in teaching mathematics. Therein, learning is understood as a co-construction process in the context of social processes and/or interactions. As such, not only social learning objectives but also cognitive learning objectives are of importance in cooperative learning. From a mathematical-didactical point of view the epistemological learning processes based on the exchange of different interpretations are of interest [START_REF] Steinbring | The construction of new mathematical knowledge in classroom interaction -An epistemological perspective[END_REF].

Cooperative forms of learning also gain in importance in the context of fostering teaching, both for children with and without (mathematical) learning weaknesses. Meta analyses in the elementary school area prove that cooperative forms of learning and peer-supported learning show stronger effects with respect to subject matter performance than traditional forms of teaching [START_REF] Rohrbeck | Peer-assisted learning interventions with elementary school students: A meta-analytic review[END_REF]. However, the effects are dependent on both the composition of the pairs as well as the degree of structure of the cooperative learning environment.

For the inclusive teaching of mathematics, the expectation being associated with cooperative learning environments is that children support and correct each other. Further it can be assumed that children with a lower learning performance develop their own interpretations due to the more elaborated interpretation of other children.

This report focuses on the question of in which way children's mistakes in cooperative learning settings of inclusive mathematics teaching can serve as an occasion for productive interactions. For this, transcribed interaction flows are analysed in the sense of a qualitative research paradigm, and theoretical insights are derived in an abductive process. However, before the research design and the results are presented, the essential theoretical foundations and empirical insights will be outlined.

Theoretical and empirical starting points

Cooperation and interaction in mathematics education

When students learn mathematics, they can benefit from sharing their ideas with their peers, especially when their peers have a different point of view. Several studies mention that differences between students' explanations and procedures can increase mathematical understanding and create new insights [START_REF] Pijls | Reconstruction of a collaborative mathematical learning process[END_REF]. However, not every interaction leads to new insight, the interaction must take place in a productive discourse. [START_REF] Dekker | A process model for interaction and mathematical level raising[END_REF] work out that verbalising, explaining, defending, asking and arguing are key activities for productive learning processes. Cooperative learning situations seem to be suitable to make such key activities come up. It is expected that children associate ideas, discuss solution possibilities and discover mistakes when they work in small groups.

Especially, it seems to be productive to develop ideas collaboratively and to find a common way for problem solving. In these phases of cooperative learning a high frequency of communication can be observed and the interacting partners change within the group [START_REF] Brandt | Using Goffman´s concepts to explore collaborative interaction processes in elementary school mathematics[END_REF]. It can be assumed that the children show a high rate of key activities like verbalising, asking or arguing by sharing ideas. Therefore, the cooperative learning situation should be based on tasks and problems which allow a range of solution procedures, so that children need to verbalise and argue with their partner. This could be achieved when the problem is complex and the students do not have a routine which allows them to solve the problem easily [START_REF] Dekker | A process model for interaction and mathematical level raising[END_REF]. Open problems, which allow solutions on different levels, can lead to many key activities as well.

In an epistemological view [START_REF] Nührenbörger | Processes of mathematical reasoning of equations in primary mathematics lessons[END_REF] mention that it is helpful to use tasks which create a "productive irritation" concerning the experiences of the children. While working on the tasks the children discover something surprising which then becomes an amazing phenomenon. As a consequence, the children may provide phenomena that they do not expect, so that "they have to reflect on the given structures and see the need to re-interpret the experienced mathematics behind the problem" [START_REF] Nührenbörger | Processes of mathematical reasoning of equations in primary mathematics lessons[END_REF]. Activities like »comparing« and »sorting« can help initiate such processes e.g. when children reason about the criteria for sorting [START_REF] Häsel-Weide | Replacing persistent counting strategies with cooperative learning[END_REF]. Only if children feel the need or the pleasure to share ideas, mathematical communication takes place.

In addition to the cooperation processes initiated through the cooperative learning technique, the tasks and the work assignment, mistakes can lead to interactions and discourse between children [START_REF] Götze | Ko-konstruktive Lerngespräche unter Grundschulkindern. Ergebnisse einer empirischen Studie zur sozialen Interaktion im Mathematikunterricht[END_REF][START_REF] Häsel-Weide | Vom Zählen zum Rechnen: Struktur-fokussierende Deutungen in kooperativen Lernumgebungen[END_REF]. Götze analysed interaction processes of children in so-called "math conferences", where the children share and discuss their solutions. She points out, that a suggestion of a wrong solution can induce an exchange of arguments and a verbalisation of prior statements -as such key activities take place, that are initiated by the incorrect solution. In my own study regarding the replacement of persistent counting strategies, the children verbalised different interpretations of a structure initiated by mistakes. This interaction had a productive effect and enabled new insights for some children [START_REF] Häsel-Weide | Vom Zählen zum Rechnen: Struktur-fokussierende Deutungen in kooperativen Lernumgebungen[END_REF]. The expectation, furthermore, is that non-comprehension or nonknowledge leads to questions and then, as a result, to explanations, and therefore moments of »gaps« and/or »helping« become productive.

Cooperative mathematics learning in inclusive beginners' classes

Inclusive teaching always is in between the poles of common ground and individualisation. It is necessary to take the individuality of each child into account, his/her competences and difficulties, and to foster it individually. At the same time, inclusive teaching has the objective of joint learning, i.e., an objective of participating in content, of the joint learning of mathematics that goes beyond [just] being a part of a class.

In order for children to be able to work together on content, the learning environment must allow working on different levels. At the same time, a joint focusing on a core idea that can be worked on by all children and that is at the centre of the exchange is necessary. Concepts of cooperative learning furthermore recommend a significant methodological structuring as well as the creation of a positive dependency between the cooperating children, e.g. through role distribution, limiting of the material, or time allowances [START_REF] Johnson | The new circles of learning: Cooperation in the classroom and school[END_REF]. As such, a field of tension exists between a limitation and a common focus for a successful outcome of cooperative processes and the enabling of working on an individual level. In addition, the mathematical competences and verbalising and presenting interpretations, as well as interpersonal competences such as a constructive dealing with conflicts that are necessary for the interaction and cooperation must first be learned.

Objective and design of the study

In project "Learning environments for inclusive mathematics teaching -LUIS-M", learning environments were developed for the entry phase of elementary school and tested in six classes of the first and second school year (children age 6 to 8). The focus in each class is on a cooperative learning environment which is processed in partner work and accompanied by other offers for individual and cooperative learning. Thematically, the learning environments pick up the basic mathematical topics, which means in the first and second school year the presentation of numbers and operations. Children with different competences in mathematics are supposed to interact and cooperate with each other. Two objectives are being pursued by this: in the spirit of mathematical didactics as a design science [START_REF] Wittmann | Mathematics education as a design science[END_REF], learning environments are developed for and with teachers and scientific insights are gained via the analysis of the learning processes of the children. We are interested to explore if and how children work together, which different levels of understanding can be reconstructed, and which occasions during the cooperative learning lead to a productive interaction. In this paper the following questions are discussed:

How can aspects and moments causing a productive discourse be characterised? In what way do mistakes induce productive interactions?

The teachers were asked to choose pairs with different competences in order to work together. Two pairs were video recorded in each of the six participating classes. Chosen for the video were pairs of children where one child shows low mathematical competences. The corresponding transcripts were interpreted by a group of researchers. The analysis was compared in an interactive way with empirical findings of other studies and theoretical approaches. As a result, insights about the communication of children in heterogeneous groups could be constructed. This procedure allows the development of new theoretical elements analysing individual cases.

Analysis of episodes

In the following two examples are presented and analysed where it was possible to reconstruct mistakes. Based on the interpretation of the examples, the theoretical conclusions are being worked out and subsequently presented, going past the examples. In this, analyses of additional scenes are included in the development of a theory.

Case 1: Jana's incorrect interpretation of the dot-strip image

The episode is from the second school year and was recorded during the introduction to the multiplication tables. The children work in a learning environment that aims at comprehension of multiplication as repetition of equal groups, and the connection between addition and multiplication. The children lay out dot images with strips of two, four, and eight dots and interpret them additively and multiplicatively. For each dot image only one type of strip may be used. The children work focused on each other, i.e., one child lays out a dot-strip image and the other child states the matching addition and/or multiplication task. These are jotted down and then sorted in a second step. In this process the children can already recognise first relationships between the tasks.

In this scene, Kadirin accordance with the assignmentlays out two strips of eight in the following dot-strip image. Jana is now required to state the task. In her first statement, Jana mentioned the total number of dots, "sixteen", and the multiplication, "times two." Since she determined "sixteen" dots by counting, she appears to link the number of dots with the operation in her phrasing. In this, she is phrasing a typical sequence of »number operation number«. Kadirprobably as a reaction to Jana's statementlays his head on the table. Jana seems to interpret this as a sign that he considers her statement to be incorrect. She then corrects herself and states "sixty times two". She does not change the structure of her statement, but rather the number.

Jana is able to determine the number of dots correctly and seems to know the sentence structure of a multiplicative phrase. Besides, she shows a behaviour that is typical for children with mathematical learning difficulties: She determines the quantity by counting in ones, but she seems to have difficulties to find a fitting multiplicative expression and to distinguish between the number words sixteen ("sechszehn") and sixty ("sechzig") [START_REF] Anghileri | An investigation of young children´s understanding of multiplication[END_REF][START_REF] Gaidoschik | Einige Fragen zur Didaktik der Erarbeitung des "Hunderterraums[END_REF].

Kadir's statement (5) makes it clear that he considers Jana's interpretation to be incorrect and furthermore shows that he is applying a different way of reading ("You can't read that") the strips than Jana. As such, a disagreement exists between Jana's interpretation and his own. Kadir now interprets the dot-strip image himself and in the process phrases a matching multiplicative interpretation. At first glance, this approach seems to be of little help. Kadir does not provide Jana with any indications how she can correct her interpretation, but rather solves the task himself. However, taking a closer look at his remark, it becomes apparent that he formulates two aspects that can be a learning opportunity for Jana in the spirit of co-construction.

First, he explicates a structured way of grasping the number of dots on a strip. Jana has determined the total number by counting them in ones. Kadir points out to Jana the power of five in the strips, because he separates the dots in the five dots on the left hand side of the mark and the three dots on the right. So he explicates the part/whole structure 8=5+3. He demonstrates and verbalises an option of the structured grasping of the quantity. Through his gesture he additionally makes it clear that only the number of dots on one strip needs to be grasped since these are strips of equal magnitude. In the second part of his statement, Kadir formulates the multiplicative structure and phrases "two times the eight". With the nominalisation, he emphasises the two elements "number of strips" (multiplier) and "number in each strip" (multiplicand).

Therefore, at second glance, it becomes apparent that Kadir is not only providing the correct result, but that there are opportunities of insight for Jana located in his verbalisation and his gesture. The key activities of verbalisation, explaining, and pointing out, described as productive by Plijs et al. (2007), can be reconstructed in the case of Kadir. He shows an understanding of multiplication and an elaborated strategy in the grasping of the quantity. His statement is caused by Jana's solution, which he recognises as being incorrect, and which he introduces in the interaction. To what extent Kadir's interpretations are picked up by Jana cannot be identified in this scene since Jana notes down the solution without it becoming apparent to what extent she is exclusively translating the verbalisation into a term or to what extent she can comprehend it with respect to the dot-strips.

Case 2: Marie suspects a mistake in Milene's solution

In the first school year, the children were given tasks to create "simple" subtraction problems with the subtrahends 1, 5 and 10. In this, the children freely select the minuend, by laying down corresponding dots in the field of twenty, draw an action card, »minus 1«, »minus 5«, or »minus 10«, solve the problem by removing the number of dots or mentally, determine the result and note down the problem on a card. After a while, the cards are sorted in accordance with self-selected criteria, which could be e.g. the minuend or the subtrahend or the size of the numbers. The relationships between the problems become apparent by sorting them, e.g. according to the minuend, the subtrahend, the difference or the size of numbers. So, on the one hand, the focus is on the basic topic of »simple subtraction problems«, on the other hand the structure between problems can be discovered and described. Both the activity of sorting, as well as the potential discoveries are suited to stimulate productive interactions. The step to find the problems can be worked on either individually by each child or the children work with distributed roles. In this case, one child specifies the minuend with dots while the other does the subtraction by taking away one, five or ten dots and notes down the term. Working while focused on each other allows for more cooperation between the children and for a direct reaction in case of erroneous solutions or questions, but also leads to a limitation of the individual level of processing.

Marie and Milene are working on the assignment with distributed roles based on the division of labour, i.e. one pupil lays out the minuend on the field of dots and draws the card with the subtrahend (c.f. Figure 1); the other pupil finds the difference and notes down the task. Maria has already placed a strip of 10 and 3 chips on the field of twenty in the following way: (lifts away the third dot from the bottom row of the field of twenty) 2

Marie: Now you have to guess how much that is 3 Milene: One, two. Twelve! (looks at Marie, puts the chip back onto its original spot on the field of twenty) So, and now I write down the assignment (takes a piece of paper and a pen) 4Marie: (4sec) (shakes her head) You got this wrong, you misunderstood. 5

Milene:

(notes down the task 13-1=12 on the piece of paper) Thirteen minus one equals twelve. (places the piece of paper on the stack with the other tasks) 6Marie: (5sec) (points with the finger to the dots) That equals thirteen (grabs Milene by the arm and points to the 3 dots in the bottom row of the field of twenty). That equals thirteen. 7

Milene:

(lifts away the third dot from the bottom row slowly once more, directly looks at Marie, and nods) 8 Marie Oh (turns away from the field of twenty). 9

Milene: Understood?

Milene, in accordance with the action card, »minus 1«, takes one dot away from the field of twenty, holds it in her hand and states the correct difference (1 & 3). She then places the dot back onto the field of twenty so that thirteen dots lie on the field once again. Marie voices the suspicion that Milene has made a mistake (4) and phrases "That equals thirteen" (6) which probably traces back to the thirteen dots lying on the field. In this situation, there is disagreement between the children.

Both pupils defend their interpretation in the further process: Marie grabs Milene's arm, maybe in order to be heard, repeats her interpretation, and points to the dots in the field of twenty. By taking away one dot, Milene seems to illustrate the operation »minus 1«, looks at Marie and nods and thus demonstrates that the difference of the task is twelve. The sequential carrying out of the actions corresponds to the process of subtraction so that Marie potentially recognises that the 13 dots represent the minuend, one dot was taken away correctly, and the difference of 12 dots was determined. However, due to the replacement of the dot (3) it was not the difference that was visible, but rather the minuend. Marie seems to be ablethrough the gesture of Milene ( 7)to comprehend that the difference was determined correctly, nevertheless. Her statement "Oh" and her turning away from the task could be a sign of agreement and an acceptance of the result.

The suspected mistake with respect to the solution of the subtraction task leads to the connection between quantity, change, and result being demonstrated on the material. The trigger of this elucidation was the dissent between the interpretation of the 13 dots that Milene considered to be the minuend whereas Marie saw the difference in the field of dots. Here, the supposed mistake leads to negotiation and defence of different interpretations. Both children show key activities, pointing out and verbalising their interpretations of the material. In this way, they appear to defend their respective point of view and to negotiate the correct interpretation.

Interpretation and conclusion

Both scenes exemplarily show that mistakes and suspected mistakes can cause verbalisations, demonstrations, and defensive actions. If (supposed) mistakes are recognised in the interaction, they appear to not only be corrected but key activities also take place in the negotiations (Table 1). The children utilise, for example, the material to illustrate something, or (re)phrase it so that other aspects become clear. In this way, (supposed) mistakes function as a trigger for a productive interaction process. Therefore, it does not seem to matter whether a mistake was actually made or if this was only suspected. The children not agreeing in their interpretationmeaning that there is dissentis of central importance. In this, two cases could be distinguished. (1) One solution is incorrect and this is recognised by the partner child who in the context of the cooperation has therefore solved the task himself (correctly). ( 2) A correct solution is interpreted as incorrect by the partner child. Here too, the partner child must solve the task herself. Prerequisite is, in this case, that both partners solve the task and arrive at different results. The activities in the negotiation are gestures and verbalisations which are utilised as justifications and defences. Properly formulated chains of reasoning are not observed in the inclusive beginners' classes. This could be because the children are not yet equipped with respective competencies or also because this does not correspond to the interaction of children among each other and requires a focusing and moderation by a teacher [START_REF] Gellert | Dispute in mathematical classroom discourse -"No go" or chance for fundamental learning?[END_REF].

Regarding inclusive mathematics teaching, it can also be assumed that the cooperation of children is productively becoming an informal learning occasion through mutual helping and correcting. The analyses show that children in the first and second school year not only take corrective action but also display key activities in their correction which can lead to learning processes for themselves and also potentially for the partner children. This, however, requires that mistakes are recognised during the learning process. Yet, this prerequisite does in no way always exist, not even when pairs of children with different competences work together. On the one hand, the children are in part simultaneously busy with their own tasks, while on the other hand even in case of working focused on one another not all mistakes are being recognised and explored. As such, it is not sufficient to trust in informal occasions. Productive interaction should rather be stimulated through suitable activities and through phenomena yet to be discovered.

Figure

  Figure 1: Depiction of the assignment situation 1Milene:(lifts away the third dot from the bottom row of the field of twenty) 2Marie: Now you have to guess how much that is 3 Milene: One, two. Twelve! (looks at Marie, puts the chip back onto its original spot on the field of twenty) So, and now I write down the assignment (takes a piece of paper and a pen)4Marie: (4sec) (shakes her head) You got this wrong, you misunderstood. 5Milene:(notes down the task 13-1=12 on the piece of paper) Thirteen minus one equals twelve. (places the piece of paper on the stack with the other tasks)6Marie: (5sec) (points with the finger to the dots) That equals thirteen (grabs Milene by the arm and points to the 3 dots in the bottom row of the field of twenty). That equals thirteen. 7Milene:(lifts away the third dot from the bottom row slowly once more, directly looks at Marie, and nods) 8 Marie Oh (turns away from the field of twenty). 9Milene: Understood?

Table 1 : Productive interaction triggered by a mistake

 1

Proceedings of CERME10

Thematic Working Group 09Proceedings of CERME10


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

Kerstin Tiedemann
email: kerstin.tiedemann@uni-bielefeld.de

Talking with objects

Keywords: Language, objects, recontextualization, domains of subjective experience (DSE)

Both language and objects seem to play an important role in mathematics learning. In our research, we focus on their interplay: How do language and objects support students' development of mathematical ideas? In order to develop a framework of 'talking with objects', we draw on three approaches. First, we adopt the idea from Bauersfeld that learning is a domain specific process. It is always bound to a very specific situation and context. Second, Aukerman's approach of recontextualization supports our insight in the link between language and context. Third, Latour's Actor-Network-Theory helps us to better understand, how concrete objects take part in the process of constructing social reality in mathematics lessons.

Introduction

Children are supposed to learn what a 'number' is or what we mean by 'addition'. The challenge of mathematics learning is to construct abstract mental objects that can neither be touched nor seen. Even if we cannot see the mathematical objects themselves, there is a lot acting and handling of and with concrete objects to be observed in everyday mathematics classes: Children write, read, and work with different concrete objects like bead frames, hundred boards or Dienes blocks. And they speak: They ask questions, explain their ideas and discuss about different mathematical interpretations. Obviously, both objects and language play an important role, when students are learning mathematics. Children and teachers use physical representations of mathematical objects in order to clarify what they are talking about and what they are referring to. These physical objects help to coordinate children's mathematical communication and their learning processes (Sfard, 2008, p. 147).

As mathematics educators, we see already quite clearly that language is an important aspect of mathematics learning. But how do objects come into play? And how do language and objects interact as means of representation? These questions lead our main research interest. In our research project, we intend to reconstruct mathematical learning processes with a special focus on the interplay of language and objects (Fetzer & Tiedemann, 2015). To begin with, we concentrate on primary school children who learn arithmetic in different German primary schools. We collect data in several schools so that we cover different social and cultural backgrounds and get an impression of our research topic that is as broad as actually possible.

Theoretically, our study is based on two main assumptions. First, we assume, together with many other researchers, that mathematics learning is a social process [START_REF] Bauersfeld | Interaction, construction, and knowledge: Alternative perspectives for mathematics education[END_REF][START_REF] Jungwirth | Banal sozial? Zur Soziologisierung des mathematischen Lehrens und Lernens durch die interpretative Unterrichtsforschung[END_REF][START_REF] Miller | Kollektive Lernprozesse. Studien zur Grundlegung einer soziologischen Lerntheorie[END_REF]. Children do not construct abstract mathematical objects without any suggestions from their environment, but rather in permanent exchange with it. In processes of social interaction and collective argumentation, mathematical objects are constructed, negotiated, and clarified. In this sense, children create abstract mathematical objects on the basis of social processes. But who are the players in these social processes? Usually, mathematics educators think of students and teachers as actors. However, concrete objects influence the ongoing interaction, too. And, in our opinion, it will mean missing important opportunities to support mathematics learning if we neglect them. Especially in primary classes, objects play an important role in the process of abstraction.

Second, we conceptualize mathematical abstraction as a process of becoming aware of similarities in different experiences [START_REF] Skemp | The psychology of learning mathematics[END_REF]. According to that assumption, children have to grasp the similarities in different representations, which they encounter in the context of arithmetic. To make this clear, we can consider children playing with little game figures. Four are sitting in a train and two more are getting on. What do these little figures have to do with the drawing of a number line, with six fingers of our hands, with the arithmetical task "4+2" written on a sheet of paper or with specific arrangements of didactical material in mathematics classrooms? (Compare Fig. 1) In order to express what is similar in all those representations and to come to a social agreement on those similarities, children and teachers need language. It is a tool, which allows individuals to share their interpretations of reality with each other. They can express what they 'see' in a certain representation and can, in this way, develop a shared interpretation. Within that interplay of language and objects, children construct their concepts of addition or number. It is for that reason that children have to develop appropriate language skills in mathematics classes, i.e. that their language has to become suitable for describing similarities in different representations.

We present the theoretical framework that we have developed so far for our research project. It consists of three parts. First, we refer to [START_REF] Bauersfeld | Interaction, construction, and knowledge: Alternative perspectives for mathematics education[END_REF] framework of domains of subjective experience (DSE). He points out that learning is a domain-specific process, i.e. that children's mathematical constructions are always bound to the situation in which they were developed. Second, we focus on the aspect of language and our fundamental assumption that every linguistic utterance, how concrete or abstract its content may be, always refers to a context [START_REF] Aukerman | A culpable CALP: Rethinking the conversational/academic language proficiency distinction in early literacy instruction[END_REF]. Third, there is the question of objects and the role that they may play in the process of mathematics learning. In this regard, we refer to Latour's Actor Network theory (ANT) (2005) which offers a new perspective on objects and their contribution to mathematical communication.

On domain-specific learning: Bauersfeld (1988) [START_REF] Bauersfeld | Interaction, construction, and knowledge: Alternative perspectives for mathematics education[END_REF] approach of Domains of Subjective Experience (DSE) elaborates how individuals organize their construction of mathematical knowledge. He assumes that children do not organize their remembrance of experiences in a hierarchical way, but rather accumulatively in separate domains. Each experience is stored with reference to the very specific and complex situation in which it was made and, accordingly, in its own domain. These different domains of knowledge are called "domains of subjective experience (DSE)". They include their own meaning, language, actions and objects. To illustrate this approach, Bauersfeld (1983, p. 3) reports from Ginsburg's (1977) work about eight-year-old Alexandria. She is not able to solve the task "8:4=" which is written on a piece of paper, She only suggests 0 or 1 as possible solutions. But, surprisingly, she can solve another task without any apparent effort: "Imagine you have 5 dollars and there are four children. How many dollars will each child get?" In fact, this second task is more difficult from a mathematical point of view. So we might ask why Alexandria did not transfer the initial task "8:2=" to the money-world herself. Why did she not solve it with reference to the domain that is obviously much more familiar to her? Bauersfeld's answer points to an important characteristic of DSEs: They are not linked automatically. Thus, from Alexandria's point of view, two different DSEs are affected which are unconnected up to now. In the paper-world, you have to cope with mathematical signs that are written on a piece of paper. In the money-world, you have to cope with banknotes and coins and think about buying attractive goods. Language, actions, objects, but also interests, motivations and feelings are fundamentally different in both DSEs. For that reason, Bauersfeld (1983, p. 6) doubts fundamentally, whether Alexandria regards the number word 'eight' which appears in both domains as the same at all. According to Bauersfeld, mathematics learning can be understood as a process of constructing, deepening and connecting DSEs. However, how can those separate domains be linked? How does mathematics learning proceed? Bauersfeld (1983, p. 31) describes that individuals cross the borders of a DSE by trying, creating and negotiating. In order to link two different DSEs, they have to build a third DSE that exclusively aims at comparing the two already existing ones. Solely in such comparative DSEs, it becomes reasonable to develop a comparative language. In fact, it is this comparative language that allows students to talk about similarities, which they 'see' in different representations. This means that all parts of a DSE, including language and objects, can help children to link DSEs and to get access to abstract mathematical objects. In the following paragraph, we focus on the language at first.

On language and context: Aukerman (2007) [START_REF] Aukerman | A culpable CALP: Rethinking the conversational/academic language proficiency distinction in early literacy instruction[END_REF] points out that it is quite misleading to talk about a 'decontextualized' language because no "text, and no spoken word, ever exists without a context" (p. 630). This approach puts the main emphasis on the content level of a linguistic utterance: Every utterance refers to a context, no matter whether this context is concrete or abstract, close or far, accessible to observation or only hypothetical. It is important to notice that Aukerman does not make any statement about the setting in which language is used or how language is used in it, but rather about the point of reference. Utterances in mathematics classes may be produced in many different ways, e.g. with gestures or not, with a parallel action or not, with pointing at something in the closer environment or not, etc., but they are all produced with the intention of talking about something. Subsequently, we always talk and listen to others with regard to a specific context. We think about a specific context and produce an utterance. We hear an utterance and interpret it against a background that we deem appropriate. Thus, no matter whether we are the ones who speak or the ones who listen, we relate every utterance to a context that we regard as adequate at that very moment. [START_REF] Aukerman | A culpable CALP: Rethinking the conversational/academic language proficiency distinction in early literacy instruction[END_REF] refers to the process of connecting utterances with contexts as recontextualization. In the process of recontextualizing, speaker and listener have to agree to a certain extent on the context of their conversation: What are we talking about? Thus, when students are expected to talk about mathematical objects, they have to re-contextualize their language and match it with rather abstract contexts. Seen from that perspective, the question is no longer, whether a student is able to decontextualize his or her language, but the question is whether students and teachers succeed in finding a shared context: Do their recontextualizations fit together sufficiently?

On objects as actors: [START_REF] Latour | Reassembling the social. An introduction to Actor-Network-Theory[END_REF] When students and teachers are negotiating a shared context for their constructions of DSEs, they can get help from concrete objects, which have a lot to offer. Objects as actors? This conceptualization appears to be unfamiliar at first sight. Nevertheless, we think that it can be very useful to adopt [START_REF] Latour | Reassembling the social. An introduction to Actor-Network-Theory[END_REF] sociological proposal for accepting objects as actors in the course of action. According to him, they participate in the emergence of social reality. [START_REF] Latour | Reassembling the social. An introduction to Actor-Network-Theory[END_REF] goes beyond the traditional understanding of the social, widens the perspective and redefines the notion of 'the social'. He takes a closer look on who and what assembles under "the umbrella of society" (p. 2). As a consequence, he defines sociology as "the tracing of associations" and thus "reassembles" the social (p. 5). In his view, the social refers to any kind of networking: humans with humans, but also humans with any kind of things. Heterogeneous elements that are not necessarily social themselves associate in different ways. According to Latour, all these different associations create social reality. Thus, in his Actor Network Theory (ANT), he extends the list of potential actors in the course of action fundamentally and accepts all sorts of actors: "Any thing that does modify a state of affairs by making a difference is an actor" (p. 71). Consequently, objects participate in the emergence of social reality, too. In this sense, Latour asks for a broader understanding of agency. "Objects too have agency" (p. 63). They are associable with one another, but only momentarily. To say it with Latour's words, they "assemble" (p. 12) as actor entities in one moment and combine in new associations in the next one. Following Latour, there are no longer stable and pre-defined associations and actor entities.

Again, following [START_REF] Latour | Reassembling the social. An introduction to Actor-Network-Theory[END_REF], objects participate in the emergence of classroom reality. In fact, this is true for all sorts of objects: Paper and pencil, as well as manipulatives or even the bottle of water on the table. Should we as researchers in mathematics education not focus on a certain kind of object, on didactical material? From a theoretical as well as from a methodological point of view, we clearly deny that restriction. Just imagine that the bottle was open, and would drop. Not only the table, but also the paper would get wet, the pencil might fall on the floor. This would surely influence any process of social interaction. "Any thing" (p. 71), a human or non-human actor, might become associated with other actors in the course of action, but only momentarily. The association might be dissolved the next minute. However, in that very moment these actors, no matter who and what they are, contribute to the ongoing process of social interaction.

Looking through Latour's sociological glasses, we can see clearly that concrete objects do play a role in the emergence of social reality. This appears to be especially true for manipulatives and other didactical material. They participate in the negotiation of a shared context and, in this way, offer help in the social process of constructing and connecting DSEs. However, how do they contribute? Earlier research revealed different modes of participation that objects might take or have in ongoing classroom interactions [START_REF] Fetzer | Counting on objects in mathematical learning processes. Network theory and networking theories[END_REF]. Our current research on the interplay of language and objects goes one-step further. Now, we try to get hold of objects' contributions on the content level. In our opinion, their most important contribution is to offer various contexts for re-contextualisation from which students and teachers may choose. A short example that we could observe in a second grade class might illustrate this variety of possible offers. The students and the teacher talk about the question what the diagonal might be on their hundred board. On that special hundred board (compare figure 2), the numbers from 1 to 100 are covered with red and blue pieces of paper. Here are some of the offers that the students accept and express in linguistic utterances -always in association with the hundred board in front of the classroom: 1) "The diagonal runs from 10 to 91."

2) "The diagonal runs from one corner of the hundred board to the opposite one."

3) "The diagonal runs from one corner of a square to the opposite corner."

We see that the hundred board suggests a wide variety of contexts, which might be suitable for recontextualization. Most important to us is the fact that those offers range from "concrete" to "abstract". Thus, on the one hand, objects support the opportunity to construct new DSEs because they make a very concrete offer. They are concrete in nature so that students can associate with them and refer to their rather concrete offer: The 10 is at the top right, the 91 is at the bottom left. On the other hand, they make offers that appear to be suitable for comparing. The hundred board has properties that other geometrical forms have as well. It is a square and in every square, "the diagonal runs from one corner […] to the opposite corner." However, in order to grasp that similarity in different representations, you have to construct a DSE of a specific square at first and then a comparing DSE, which aims at comparing different geometrical forms. Those offers of objects may support the construction and connection of DSEs and may support the language development which goes along with it, too.

Integrating the concepts: Talking with objects

According to Bauersfeld (1988, p. 178), the subjective realization of a mathematical object remains always bound to the context of experience, i.e. to the objects and language used in the situation of construction. This approach gives us a clue that we will understand mathematics learning better, if we concentrate not only on language or on objects but on the interplay of both. How do language and objects interrelate in the process of mathematics learning? How can we talk with (the help of) objects? [START_REF] Aukerman | A culpable CALP: Rethinking the conversational/academic language proficiency distinction in early literacy instruction[END_REF] approach of re-contextualization and [START_REF] Latour | Reassembling the social. An introduction to Actor-Network-Theory[END_REF] Actor Network Theory seem to be useful background theories to tackle these questions.

According to Aukerman, every spoken word in mathematics classrooms refers to a context and is recontextualized by the recipient. In order to achieve a shared understanding, the interlocutors have to agree on the 'right' context. What can serve as a solid ground, which a linguistic utterance can be related to? At this point, objects come into play. According to Latour, humans and non-humans associate with one another and create social reality. In terms of mathematical learning processes, children and objects interact in the social process of learning. Objects make offers that students and teachers can accept and refer to in order to coordinate their mathematical communication. The "steely quality" (Latour, 2005, p. 67) is a solid ground that allows individuals to experience reality. Objects are not a mere tool in students' hands that can easily be manipulated. Objects are participants in their own social right and contribute to the ongoing classroom interaction: Objects make offers and students 'listen' to those offers. Students talk to objects, and become associated with one another. At this point students 'talk' together with objects in a combined action.

But how does that work? Objects are concrete in nature. Nevertheless, in their concreteness they prove to be not a limitation, but a chance for development of (mental) mathematical ideas. Indeed, objects offer a variety of possible contexts for re-contextualization ranging from concrete to abstract. Sometimes, objects may provide the context for very specific experiences. In these cases, objects can help to construct a new DSE or to deepen already existing DSEs. At this stage, students try to find words with which they can express the particularity of this specific context and to negotiate it with others. Language is probably the most important tool for such a negotiation: What do I 'see' in that object? What do you 'see'? In a second step, students have to become aware of similarities in different experiences. They are in permanent exchange with their social environment. They listen to as well as talk to and with participating actors. In doing so, they construct new DSEs that aim at comparing already existing DSEs. Again, objects profoundly contribute. They offer a context for comparisons: What do I 'see' as the same in different objects (or in different actions with objects)? What do you 'see' as the same? Where are differences? Do we agree? In this sense, objects help not only to coordinate mathematical communication, but also to develop language more and more. Students are challenged to match their language with a concrete experience at first and with a comparison afterwards.

On closer inspection, we see that objects are actors that students can talk to and talk with. In fact, objects contribute to the process of negotiating mathematical meaning. In most interactional situations, it is not only the child who is responsible for a linguistic utterance. Words are not the only means to negotiate mathematical meaning. Instead, students and objects often associate and convey a mathematical idea together. In these cases, the object actor takes over part of the act of recontextualization [START_REF] Fetzer | The interplay of language and objects in the mathematics classroom[END_REF]. Students talk together with objects. The boundaries between language and objects almost seem to merge.

Discussion

The theoretical framework that we have sketched in this paper raises awareness of some aspects that are not new in mathematics classes, but that are new in our thinking about content-related language use. When students want to express their mathematical interpretation of reality, they are not restricted to the words they have at their disposal. Thus, they can accept one of many offers that the objects in their close environment make. In this process of assembling, the objects achieve two things. They offer their help, but at the same time, they challenge the children to move further in their mathematical development and in their improvement of content-related language use.

For that reason, the framework does not only make us sensitive to the importance of objects in the process of language development, but it points in a direction that might be productive for our further research. We have not only to analyze objects that we use in mathematics classes, but we have to analyze children's associations with them, too. Which offers do the children accept? Moreover, how do these offers support their language development in mathematics classes? These are the questions, which will lead our further steps in that project about the interplay of language and objects.

Figure 1 :

 1 Figure 1: Different representations of 4+2

Figure 2 :

 2 Figure 2: Hundred board covered with red and blue pieces of paper

Proceedings of CERME10


text:
Kirstin Erath
email: kirstin.erath@math.uni-dortmund.de

Implicit and explicit processes of establishing explaining practices -Ambivalent learning opportunities in classroom discourse

Keywords: Explaining practices, student participation, discourse, interaction, implicitness. Thematic Working Group 09

Participation in mathematical practices is widely accepted as important for students' meaningful learning of mathematics. But how do students learn to adequately participate in these practices? This paper addresses the question for the specific case of oral explaining practices in whole class discussions. The study is theoretically based on merging an interactionist and an epistemological perspective to describe explaining practices as interactive processes in a classroom microculture while simultaneously keeping in mind the development of the broached mathematical content. The identified implicit and explicit processes of establishing explaining practices are exemplified and discussed with respect to ambivalences in the differing learning opportunities they offer.

Introduction

This study is based on the assumption that students' learning is inseparably linked to participation in classroom interaction, which is mainly based on verbal communication. Therefore, learning of mathematics is conceptualized as "a process of enculturation into mathematical practices, including discursive practices (e.g., ways of explaining, proving, or defining mathematical concepts)" (Barwell, 2014, p. 332). In this study, the discursive practice of explaining in whole class discussions is further investigated. Erath, Prediger, Heller, and Quasthoff (submitted) show that explaining is the most frequent discursive practice in German grade 5 mathematics classrooms. Furthermore, explaining has an important role for the meaningful learning of mathematics since it serves to communicate about more isolated pieces of knowledge, including talking about meanings and connections [START_REF] Prediger | Content, or interaction, or both? Synthesizing two German traditions in a video study on learning to explain in mathematics classroom microcultures[END_REF]. But how do students learn to participate in explaining practices and the corresponding epistemic processes? This question is investigated in this paper on the level of the interactive processes of establishing explaining practices in four microcultures.

Theoretical background: Explaining as practices of navigating through different epistemic fields

Following Interactional Discourse Analysis, explaining is understood as multi-turn units which are interactively co-constructed, contextualized and serve to convey or construct knowledge (Erath et al., submitted). This definition is extended and intertwined with an interactionist and an epistemological perspective from mathematics education: From an interactionist perspective, explaining can be conceptualized as a mathematical practice [START_REF] Cobb | Participating in classroom mathematical practices[END_REF]) that is interactively established in a classroom microculture and that allows talking about collective mathematical development. Here, this concept is used descriptively to talk about identified ways of collective explaining processes in whole class discussions that are interactively established by students and the teacher. But not every explanation constitutes an own practice. To clarify this point, the notion of mathematical practices is enriched by the following definition from general educational science: "Practices are […] understood as rule-governed, typecasted, and routinely recurring activities" (Kolbe, Reh, Fritzsche, Idel, & Rabenstein, 2008, p. 131; translated from German by the author). Therefore, mathematical explaining practices are conceptualized as recurrent ways of explaining that are treated as matching the classroom microculture from the participants' perspective.

The epistemic matrix was derived from research in mathematics education from an epistemological perspective [START_REF] Prediger | Content, or interaction, or both? Synthesizing two German traditions in a video study on learning to explain in mathematics classroom microcultures[END_REF] and is used to further specify the notion of 'recurrent ways of explaining'. Different possible objects of explanations in mathematics are systematized in the lines of this matrix (not readable in Figure 1 that focusses on the depicted pathways), called logical levels (from top to bottom): Concepts, propositions, representations, and models (conceptual logical levels) and conventional rules, procedures, and concrete solutions (procedural logical levels). Each of these mathematical aspects can be explained by different means that are distinguished in the columns of the matrix [START_REF] Prediger | Content, or interaction, or both? Synthesizing two German traditions in a video study on learning to explain in mathematics classroom microcultures[END_REF] On the one hand, the epistemic matrix is used to characterize the utterances of students and teachers in explanations by analyzing which cells (called epistemic fields) of the matrix they address. On the other hand, the matrix is used to depict the explaining pathways that are interactively established by mapping these characterizations of utterances in the matrix (see Figure 1): Students' utterances are depicted as rectangles (including turn number and name), the teacher's utterances as circles with turn numbers. These pathways give access to the underlying mathematical structure of explaining sequences, which are especially characterized by the teacher's navigations through different epistemic fields (indicated by arrows in the pathways). Therefore, the matrix is used as tool of analysis and at the same time the associated language of 'explaining pathways' serves to specify the definition of explaining practices: Altogether, explaining in mathematics classrooms is conceptualized as practices of navigating through different epistemic fields, which are identified by building categories of pathways with similar structures [START_REF] Erath | Rekonstruktion von Unterrichtsgesprächen in unterschiedlichen Mikrokulturen[END_REF]. Hence, different practices are constituted by different defining patterns of their pathways describing the 'recurrent ways of explaining'. For example, all three pathways in Figure 1 have entries in the lines of "representations", "procedures" and "concrete solutions" in common and after working in the column of "purpose & evaluation" (more on the right) the teacher navigates towards the column of "explicit formulation" (more on the left). This recurring pattern defines the practice of explaining 'good' representations in one classroom: "navigating from the evaluation of a student's concrete solution and suggestions for improvement to deriving more general hints for drawing and characteristics of good representations".

But how do students learn to adequately participate in these interactively established explaining practices? Studies on the establishment of related norms in mathematics education (e.g. [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF] and discourse analysis [START_REF] Heller | Academic discourse practices in action. Invoking discursive norms in mathematics and language lessons[END_REF] identify implicit and explicit processes as oppor-tunities for students learning how to explain in their classroom. This work is extended to the processes of establishing explaining practices in this paper.

Methodology of the study

Larger data corpus. In the larger project INTERPASS, video data was gathered in 10 x 12 mathematics and language lessons (each 45-60 min.) in five different grade 5 classes (age 10-11 years) in an urban area of Germany. Eight lessons were observed in the beginning of the school year directly after the transition from primary school since it could be expected that processes of establishing practices are more explicit in this time of getting to know each other. Further four lessons each were gathered in the middle of the school year in order to get a long-term impression.

Sampling for the case study of this paper. The presented study builds on data of four mathematics classrooms chosen due to the following different characteristics in order to observe a broad range of interactions [START_REF] Erath | Rekonstruktion von Unterrichtsgesprächen in unterschiedlichen Mikrokulturen[END_REF]: Two higher tracked secondary schools (German: "Gymnasium") and two normal secondary schools (German: "Gesamtschule") and within each of these subgroups one classroom with students from a privileged and one with students from an underprivileged quarter.

Data analysis. This paper is based on analyses done for the PhD thesis "Mathematical discursive practices of explaining in different classroom microcultures" [START_REF] Erath | Rekonstruktion von Unterrichtsgesprächen in unterschiedlichen Mikrokulturen[END_REF] enrooted in the larger project INTERPASS. In this context, all explaining sequences in whole class discussions were transcribed and analyzed by means of the epistemic matrix resulting in explaining pathways for each sequence. In a second step, explaining practices were identified in each classroom by developing categories of pathways with similar structures. In this way, three to five different explaining practices were explored in each of the four classrooms. In order to answer the question "How do students learn to adequately participate in the explaining practices of their classroom?" the interactive processes of establishing practices were further investigated. More precisely, it was explored if these processes were explicit or implicit and which turn teachers use in order to express their expectations and if this makes any difference for students learning opportunities.

All presented transcripts were translated from German and simplified (capital letters indicate stressed words, round brackets indicate phrases difficult to understand in the video data).

Empirical results: Processes of establishing explaining practices

The investigation of processes of establishing explaining practices in four German grade 5 classrooms [START_REF] Erath | Rekonstruktion von Unterrichtsgesprächen in unterschiedlichen Mikrokulturen[END_REF] shows that there are some explicit but primarily implicit processes that contribute to the establishment of explaining practices. Furthermore, it comes to the fore that teachers (implicitly or explicitly) explicate their expectations in the turn of demanding an explanation as well as in the turn of responding to a student's utterance.

Explicit processes

Out of 16 identified explaining practices, only the practice of "explaining a concrete solution by means of a conventional rule" in Mr. Maler's classroom is recurrently established in an explicit way. The following transcript from the sequence "rounding on tens" (see [START_REF] Prediger | Content, or interaction, or both? Synthesizing two German traditions in a video study on learning to explain in mathematics classroom microcultures[END_REF] for a longer extract of the sequence) exemplifies how the teacher explicates his expectations for a 'good' explanation in his response to Kosta's explanation why 63 can be rounded on 60: Kostas explains his solution by referring to the meaning related model of distance and closeness on the number line. This is implicitly rejected as a matching explanation by the teacher in #20, immediately followed by questioning the class about a rule that could be applied to explain the solution. In this way, Mr. Maler explicates his idea of a 'good' explanation of a concrete solution, which is underlined by his reaction to Katja's formulation of the corresponding rule. This sequence is an example of explicating expectations in responding to a student's explanation by navigating to the epistemic field (explicit formulation of a conventional rule) that would match for an explanation from the teacher's perspective and directly demanding an answer in this epistemic field. Another conceivable possibility would be that the teacher explicitly talks about which mathematical aspect of an explanation he values or which part did not match from his perspective.

These kinds of explicit processes can also be observed in Mr. Maler's demands for explaining a solution. The following extract from "rounding on thousands" in the context of a homework on rounding the length of rivers illustrates the case of explicating expectations in the turn of asking for an explanation by pointing to the expected epistemic field (explicit formulation of a conventional rule):

6 Tabea: SIX thousand 7

Teacher: GOOD; but now my QUESTION is, HOW did you arrive at this six thousand? Since we also want the RULE 8 Tabea:

Because from e:r, 9

Teacher: to be CLEAR 10 Tabea:

Well up to five, well up to four, you have to round DOWN, and from five six seven eight nine you have to round UP. 11 Teacher:

EXACTLY.

[…]

After naming the right number (#6), Tabea is asked by the teacher in #7 to explain her solution. In this turn of demanding an explanation he directly states that she should refer to the conventional rule (shortly interrupted by Tabea): "HOW did you arrive at this six thousand? Since we also want the RULE […] to be CLEAR" and in this way explicitly points to the expected epistemic field. Tabea follows this navigation (#10), which is explicitly evaluated positive by Mr. Maler in #11.

In both ways, the teacher explicates his expectations and reveals the recurring, typical structure of explaining a concrete solution by means of stating the related conventional rule. That is, this structure is made accessible to all learners and not only to those who can interpret the implicit processes of establishment (see below). Hence, the teacher's explication of expectations in the turns of demanding for and responding to an explanation are major learning opportunities for explaining a concrete solution adequately in this classroom (this must not hold for other classrooms since every microculture establishes different practices). Especially the way of explicating in the demand for an explanation seems to be important: This allows all children to contribute in the subsequent explanation even though they might not yet recognize the recurrent pattern of the underlying practice.

Implicit processes

Explicit establishments (see above) have only been found in rare cases in the data corpus. Instead, processes take course implicitly. Three different ways of implicit processes contributing to establishing explaining practices were identified and are exemplified in the following: (1) marking match or mismatch in responding to a student's explanation without giving reasons for the evaluation, (2) picking up only particular aspects of a student's explanation without explication, and (3) navigating recurrently to specific epistemic fields without revealing the underlying (intended) pathway.

An example for evaluating a student's explanation without further comments is the sequence "distinguish lists" from Mrs. Bosch's classroom. During revision at the beginning of the lesson, students are asked to distinguish the concepts of tally sheets and frequency tables.

12 Teacher:

[…] Now, WHAT was tally sheet, WHAT was frequency Mrs. Bosch marks Maria's explanation explicitly as matching (#16) but does not reveal the underlying pathway in her response: The analysis of several sequences on explaining concepts in this classroom unfolds that in this microculture a concept is adequately explained by means of addressing an epistemic field on the level of procedures, which means formulating an instruction for generating a representation of the concept. Therefore, the sequence is an example of an implicit process that contributes to the establishment of an explaining practice by marking an explanation as matching or mismatching without commenting on the reasons for the evaluation.

The second kind of implicit processes (picking up only particular aspects of a student's explanation in a response without explication) is concretized by the sequence "function of diagrams" from Mr. Schroedinger's classroom in the context of talking about different ways of presenting data.

1 Teacher:

[…] WHY they're doing quite frequently in printed media but also um on TV in the news, um why they're not giving a LIST like that Marcus:

[How] many SIBLINGS they have, because then in parts they would maybe have to always go THROUGH our classroom that small. … 9

Teacher: THIS exactly meets the point, these two utterances. THEREFORE you normally do it in the form of such diagrams, because of the clarity actually […] In his evaluation in #9, the teacher explicitly marks that the students' answers match but in his subsequent summary, only specific aspects are picked up: The teacher takes on the aspects related to functionality but he does not refer to the further issues of meaning (#2, "it's MORE because it's BIGGER") or examples (#4/6, number of siblings) addressed by the students. This selection is in line with the practice of explaining that is established during several sequences: In Mr. Schroedinger's classroom, concepts are adequately explained by referring to purposes and functionality.

The third way of implicit processes contributing to establishing a practice is the teacher's repeated steering to specific epistemic fields without revealing the underlying (intended) pathway. This directly refers to the definition of explaining practices that forms the basis of this study. Figure 1 shows three pathways of sequences that show the same regularities. These kind of pathways are identified five times in Mr. Schroedinger's classroom in the context of explaining how 'good' representations are designed. But, as with the two ways illustrated before, the underlying pathways are not revealed. The following extract from the sequence "lists of pets" exemplifies the navigation from evaluating a concrete solution to formulating hints for generating 'good' representations. Mr. Schroedinger initiates (#1) the evaluation of the representations and after some students stated pros and cons for the different representation, he navigates to formulating suggestions for improving the lists (#37), which helps clarifying how good lists should be designed. By repeatedly starting from a student's concrete solution and navigating from its evaluation and suggestions for improvement to deducing more general hints for drawing and characteristics of good representations this explaining practice is established across several sequences.

Although the three presented ways of establishing explaining practices must be characterized as implicit processes, they serve as opportunities to learn how to participate adequately in whole class explanations, at least for some learners. But the examples also show how challenging it is for other students to interpret these implicit processes (see e.g. Gellert, 2009, for further discussion of divergent learning opportunities related to implicitness). In the cases of marking matches and mismatches or picking up particular aspects without further comments, it might be challenging for students to follow since they must relate the teacher's evaluation to a classmate's utterance that is not present any more. The third way (recurrent teacher's navigations) allows all students to contribute in the explanations as long as the teacher explicitly demands for the shifts of epistemic fields. However, it probably takes students several sequences of one practice to recognize that there is a pattern and that knowing "how explaining works" is important for adequately taking part in whole class discussions.

Conclusion and discussion

The distinct dominance of implicit processes found in the qualitative analysis of the video data corpus is in line with research that identifies criteria of "successful participation" and "expected student contributions" (Gellert, 2009, p. 131, translated from German by the author) as often staying implicit. The presented study deepens these findings for explaining practices. Furthermore, the epistemic matrix offers a possibility to talk about mathematical aspects of 'good' explanations and to make the hidden regularities visible and discussable by means of the pathways. The dominance of implicit processes also suits the observation that explaining (as well as oral communication in general) is not treated as an explicit learning goal by the teacher [START_REF] Erath | Rekonstruktion von Unterrichtsgesprächen in unterschiedlichen Mikrokulturen[END_REF]. Instead, explaining serves as learning medium that is used without talking about adequate participation beyond general social behavior, i.e. the mathematical aspects of 'good' explanations. Moreover, it becomes apparent that learning how to adequately participate in whole class explanations is a learning process (across several sequences and lessons) and hence especially not a feature that students bring to the classroom but a competence that can be acquired in the interaction of collective explanations guided by the teacher.

The explicit and implicit processes of establishing explaining practices relate to different learning opportunities for students as explicated above: More explicit processes are eligible since they reveal the underlying patterns of the pathways and provide more students access to this mathematical aspects of participation, not only those who are able to also interpret the implicit processes. However, this does not imply a call for direct instruction: Talking about language on meta-level while simultaneously talking about mathematical content may ask too much especially from weaker students. Instead, it is about making the criteria for matching and mismatching utterances in relation to a practice accessible in responses to explanations or in the turn of demanding for explanations. Thereto, teachers need to be sensitized for the role of oral explaining as a learning goal in order to be able to deliberately initiate particular practices. But as a first step, research in mathematics education should further specify which explaining practices are reasonable (also related to general discourse acquisition) or rather necessary in which grade in order to help teachers to enable even more students to actively participate in oral explanations and the corresponding epistemic processes.

  , called epistemic modes (from left to right): Labeling & naming, explicit formulation, exemplification, meaning & connection, and purpose & evaluation.

Figure 1 :

 1 Figure 1: Three explaining pathways contributing to the practice of explaining 'good' representations

  And then there, if you take AWAY the three and shift the ZERO to it. So, you could DO that, but actually it's WRONG. You just have to round down and nea.. nearest number with a ZERO you have to write there. … 20 Teacher: […] and you already implied WHY; but does any of you know a RULE, HOW one has to proceed here, and when one here, when the ten stays the SAME? In this case, and the place BEHIND, which is rounded, goes to ZERO? Ha; [4.5 sec. break] Katja. 21 Katja: With zero one two three FOUR you are rounding down and with five six seven eight NINE you are rounding (up). [3.5 sec. break]

	8	Kostas:	°hhh [articulated clearing his throat] Well, if you are rounding DOWN the
			sixty-three on TENS; then it comes, it gets, there must be ALWAYS a zero
			at the end, it MUST be,
	9	Teacher:	[hm_hm	]
	10 Kostas:	[when you are rounding.]
	11 Teacher:	On TENS yes.
	12 Kostas:
	22 Teacher:	Did EVERYBODY understand that?
	23 Class:	YES [affirms in choir]

table ,

 ,

		this PART,
		Barbara,
	13 Barbara:	Tally sheet is where you did strikes; and frequency table is er [4.0 sec.
		break] er-
	14 Teacher:	Can you HELP Maria?
	15 Maria:	YES, when you did it all count up and then wrote it DOWN with numbers
	16 Teacher:	EXACTLY. Well CAUGHT. […]

Proceedings of CERME10

Acknowledgment

The research project INTERPASS ("Interactive procedures of establishing matches and divergences for linguistic and microcultural practices") is funded by the German ministry BMBF (grant 01JC1112 to S. Prediger & U. Quasthoff). I thank Susanne Prediger, Anna-Marietha Vogler, Uta Quasthoff, and Vivien Heller for the collaboration.