META-LEVEL MATHEMATICS DISCUSSIONS IN PRACTICE TEACHING: AN INVESTIGATIVE APPROACH

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1 META-LEVEL MATHEMATICS DISCUSSIONS IN PRACTICE TEACHING: AN INVESTIGATIVE APPROACH Marit Johnsen Høines & Beate Lode Bergen University College This paper refers to a project where the preconditions for a subject based and reflective approach in the context of practice teaching in teacher education. The project is linked to a compulsory 30 ect credits study of mathematics education. Five second-year student teachers and three tutors were invited to participate in the investigation. This paper focuses on how the preliminary analysis provides a basis to inquire further the didactical conditions for including a subject based discussion within the conversation in practice teaching. More specifically it discusses contradictions implied by the findings that practice teaching communication is imprinted by an evaluative approach that can restrain the development of a subject based, reflective approach. BACKGROUND In this paper the notion of practice refers to the study within teacher training where students 1 are teaching in school. Under supervision, the students are allowed to teach and organize children s learning activities. The students are intended to consider, structure, try out and discuss the mathematical and didactical 2 content they have been studying at the university college. The experience gained in practice teaching is intended to give direction to their study of mathematics education and other subjects. The importance of practice teaching can be verified from different angels: Practice teaching has a strong tradition and can be seen as a subculture within teacher education. The corps of tutors (special trained primary and/or secondary school teachers) has been recruited, which has a high degree of autonomy in respect of tutoring the students. This tradition has well-articulated and clearly defined position. The student evaluations indicate that they consider their practice to be one of the most important aspects of their education (Hove, 2005). The teacher training curriculum focuses on practice teaching as an important part of the teacher education. The development of a stronger bond between practice teaching and other parts of the study programme is emphasised. More 1 In this paper student refers to student teachers and pupil to primary school pupils. 2 The term didactical refers to didactics as it often is used in a Norwegian tradition. In this paper, the notion more explicitly refers to an area constituted by how (mathematical) knowledge is developed, used and communicated. Didactics implies (theoretical) considerations relevant for educational practises (inside and outside schools) and deals with conditions for learning, using and communicating knowledge. 1

2 than in the past, the curriculum now focuses on the responsibilities of the mathematics educators (as well as those in other subjects). 3 THE FIELD OF INVESTIGATION Most often, one tutor (T) is responsible for a group of 2-4 students (S) for a set period of time (for instance 4 weeks). The students and their tutors work close during this period. The learning environment is affected by both the tight-knit natured group and the professionalism of the tutors. This is evident in the way the tutors knowledge of their pupils, the curricula and the teaching tradition is handled in interaction with the students. In considering the relationship between the students and their tutor(s) we found it useful to employ the notions of communities of practice and master-learning, as elaborated by Wenger (1998), Lave (1999) and Kvale and Nielsen (1999). The students are engaging in a practice that is already established; their interactions promote their own learning, but also a renewal of the practice. We also find our investigations related to an ongoing project where Nilssen studies the interaction between teacher students and their tutor and discusses the role of the tutor as a mentor and a model in context of the student teachers learning. In Nilssen (1999) she elaborates on how a concept of imitation may challenge the discussions on the processes of student development in the context of established practice. By taking part in the didactical discussions in practice, teacher educators are able to study and join in the interaction with the students and their tutors. As such we wanted to develop our role as practicing teacher educators and as researchers. The participants were invited to cooperate analytically. In this process, insight from the LCM-project (Learning Communities in Mathematics) at Agder University College (Fuglestad & Jaworski 2005) is relevant for our work. The Agder-project introduces mathematical tasks that teachers and researchers explore together. This provides common basis for the implementation of projects in classrooms and for further investigative cooperation. Our ongoing inquiry approach has links to the LCMproject, although our initial focus is on the students teaching practice (Jaworski, 2004; 2005). The field that we are entering bares the imprint of the tutors two foci: the children s learning and the students learning. The students share this dual focus which is a challenge. The tutors and the students observe each other when they are teaching. They discuss the class sessions both beforehand and afterwards. The tutors are responsible for establishing the conversation, and the post-teaching conversations are seen as important parts of the didactical 4 conversations that take place regularly. The Didactical as referred to in footnote 2 2

3 discourse generally is initiated without the intervention of the mathematics educators (M). When present, the Ms are visitors, observing the teaching/learning sessions and taking part in the post-teaching conversations. As Ms we often feel as if we are entering a group where the participants know each other fairly well. We are entering or are perhaps confronted by a discourse that is already established when we join in the didactical post-teaching conversation (SMT-conversation). The discussion has started earlier. 5 PRELIMINARY APPROACH RESEARCH FIELD In order to establish a stronger emphasis on cooperation in the field of practice teaching, we have chosen to focus on the SMT-conversation as a forum for discussions between S, T and M. In this project we want to explore the didactical potentials of SMT-conversations. We invited two groups of students and their tutors to cooperate in an investigation. 6 We all observed teaching-learning situations where students were actively engaged with the pupils. These observations served as a common reference point for SMT - conversations. As Ms, we brought a meta-perspective into these discussions. In order to gain insight into the nature of such didactical discussions and to explore how suitable preconditions for such conversations can be established, the position of mathematics 7 in practice teaching were investigated and also the role of teachers from the college, especially the mathematics educators. The students and their tutors were invited to join in a discussion which would foster our joint understanding of the qualities and the conditions for interaction. METHODOLOGICAL APPROACH The project is initiated and implemented by the mathematics educators (M1 and M2) 8. It is based on data collected during observations and (SMT-) conversations in the field of practice. During the research period, the two groups were teaching in different schools; one group in first grade and the other in second. On an initial visit, the tutors showed to be open and cooperative. It was this positive attitude that constituted the only, but nonetheless important, criteria for the groups to be invited to join the project. The eight persons that were invited went all eager to participate in the project. The project was organized in these phases: 5 The fact that we enter a discussion that we interpret (also by the participants telling us) that started earlier, makes an analytic perspective based on M.M. Bakhtin relevant. In a perspective of continuity we do not see a discussion as started and ended. A discussion is to be seen as a complexity where a variety of discussions are brought to the fore (Johnsen Høines, 2002; 2004). These perspectives are important in the project, but are not developed further in this paper. 6 Five students, three tutors and two mathematics educators (researchers) participated in the project. 7 Mathematics and mathematics education. 8 M1 and M2 are the researchers and authors of this paper. 3

4 1. Observation of sessions in which S,M and T watched students teaching. These observations generated references for 2. During the session notes were taken, some of which were detailed and well-developed text. 2. SMT conversations based on these observations. The discussions were intended to addressing the issues mentioned above: How a subject based discussion can develop and can be understood as part of the didactical conversations. In addition, it was an issue to investigate how the role of the mathematics educator can be viewed. Notes from phase 1 provided the basis and notes from phase 2 were written up. 3. Individual interviews with the students and the tutors based on the theme in 2 was recorded. The emphasise was made on the development of subject-based approaches as part of the SMT-discussion, and the development of a conversation about the SMTdiscussion itself (2). In order to gain insight into the positioning of subject-based perspectives within the didactical discussions in practice, we tried to frame the interviews as investigative dialogues (3). By implication therefore it became important not to pose questions for them to answer - or at best to minimize this. We invited the participants to join us developing and sharing insights; to expound our perspectives; to turn and twist issues and search for possibilities. We build upon a dialogical approach developed by Alrø & Skovsmose (2002) where they underline dialogue as a conversation of inquiry: Entering an inquiry means to take control of the activity in terms of ownership. The inquiry participants own their activity and they are responsible for the way it develops and what they can learn form it. The elements of shared ownership distinguish a dialogue as an inquiry from many other forms of inquiry where, for instance, an authority sets the agenda for the investigation and the conversation. (p.119) This chosen methodological approach had implications for the analysis. The SMTconversations and the interviews were analytical processes in the sense that we discussed how we understand and develop relevant concepts. All the participants made serious contributions to the analytical processes. Consequently analysis on some levels has been included in the empirical data. There appeared to be a complex interaction between the processes of gathering, interpreting and analysing the data. The analytical processes may be characterised as inquiry so far as they are investigative and questioning processes, and they developed on different levels: L1: The didactical post-teaching discussion was to some extent analytical by nature. The discussion was intended to reflect on practice. 9 The teaching/learning situations basically serve as a basis for the discussions, but previous experience and issues from other phases of the study are made relevant. Participants: SMT 9 Even though we have focused mostly on post-teaching discussions, didactical discussions before and during practice teaching are included. 4

5 L2: We initiated a discussion about the discussion itself in order to investigate how it can be described and how it can be developed. We developed criteria for identifying and describing different aspects or types of didactical discussions. The empirical material was related mostly to L2, where our discussions of L1 were discussed. Participants: SMT L3: Analysis of L2 (and L1) discussions was undertaken. Participants: M. The levels described here are not to be interpreted as being sequentially organised. For instance the analyses in L3 would have consequences for further communication with S and T, either in later interviews or in informal conversations. PRELIMINARY FINDINGS AND RESEARCH FOCUS Already in the preliminary discussions effort was made to create an awareness of the kind of discussions we were participating in. We identified the most common approach as the evaluative. All the participants expressed a familiarity to the SMTconversations as evaluative conversations where the focus was on the learning/teaching session interpreted in terms of what had or had not worked well, what could or could not have been done, and why specific choices had been made. This kind of conversation represented a what-happened or retrospective perspective. Quite often the discussion focused on organisational problems, such as good, you asked them to take their caps off, you should have waited until they were quiet before you.. A conversation in this evaluative mode might, as was explicitly stated on several occasions, enlighten students regarding the consequences of their later teaching and learning sessions. However, this approach relies to a dominant subordinate relationship and is to a high degree directed towards the past - an evaluation of what has been done. At an early stage it was made clear that, as M, we intended to challenge the discourse developed within the SMT-conversation by having beyond evaluation to initiate or stimulate an educative 10 discussion about the mathematical content, teaching and learning. We felt that discussions should be based on reflections about what happened in the learning sessions without the common what-i-did-well-or-wrongfocus. We applied the term subject-based discussions to the kind of didactical communication we sought to develop. We recognized the importance of such communication even before we were able to explain what a subject-based discussion should be. Our efforts to foster an alternative kind of discussion implied a challenge to the established discourse. We interpreted this as indicating that our interference made the characteristics of the established discourse more visible to all the participants. We wanted to learn about the established discourse and the discursive possibilities and offered a common focus: How to describe a subject-based discussion? And how 10 We use educative as the English word that we find closest to the Norwegian dannende (danning) 5

6 do we envisage the didactical conditions for including a subject based discussion within the didactical conversation in practice teaching? EXCERPTS FROM SMT-CONVERSATIONS Excerpt 1: The back table did not follow As mathematics educators we sought to establish our background role in the beginning of the discussion by telling the others to: Just carry on as you normally would. T1 11 was in charge and asked S1 to describe what had happened in the lessons in relation to the preparations and previous supervision. The other students were asked to add their comments, as were the mathematics educators. T1 made the point that the group had emphasised on a supportive communication, in which alternatives and possibilities were sought for. She wanted to foster a climate where it was possible to articulate constructive criticism. Phase 1 S1 had been working with 28 six-years-olds on Lurvelegg 12 and number-concepts. The children had made models, they had drawn and they had song and danced about Lurvelegg. We had enjoyed ourselves as we observed how S1 managed to handle various aspects of number concept in a flexible and creative classroom dialogue. The children helped to chose the symbols to be used (drawings); they negotiated numbers, letters and positions and they discussed and made changes on the blackboard. Phase 2 S1 started the SMT-conversation by stating that it went pretty well as planned : Most of the pupils followed what I said and grasped it, I think. He seemed satisfied. We noticed that the back table did not follow, T2 commented. 13 The conversation turned to issues of behaviour, norms and limits. After a while, M2 shifted the focus to number concepts and how communication could be used to foster linguisticmathematical creativity. S1 seemed to resist entering the discussion. Several times he responded with utterances like I muddled the order... I should have done No, I did not I tried to. M2 tried to reassure him that she did not think this was a case of importance. It was obvious, however, that S1 did not change his focus to join the discussion. He returned to evaluate what he might not have done well enough. M1 intervened, emphasising that S1 had taught a very good lesson. Her description was detailed and had a strong evaluative component. She described his muddling with numbers, signs and order as a brilliant didactical example. She gave grounds for why the lesson had been very good: You invited the children to play and argue, and they joined in. You handled a variety of number-concept features brilliantly, she said. M1 stressed that we were interested in going beyond a discussion of what was 11 There are two tutors in this group 12 Lurvelegg is a cultural fantasy-figure. He has one eye, two noses, three ears and four legs. 13 The pupils were organized in groups. The back table refers to one of the groups seated back in the classroom. 6

7 good and was not; that we wanted to establish a basis for a subject-based discussion independent of wether he had succeeded or not. This appeared to help; S1 seemed more relaxed and satisfied. Nevertheless he soon returned to comments like: Yes, I should have. Phase 3 During the phase 3 interview we told S1 that we interpreted the way he interacted as resistance and he commented: Yes, that is how I handled it, because that is what we are used to doing. In practice teaching last year we always focused on what we did well enough and what we did badly. The only interest we were supposed to have was about what we did badly. The post-teaching conversations focused on what should have been done differently. That was the point. As if well, the evaluation was introduced by: yes but... you should have done so and so. The discussions we have had this year has been fantastic in comparison. He was referring positively to the sessions led by his tutors this year. 14 S1 proceeded to tell us about how the comment we recognised that the back table did not follow had influenced him. T2 had referred to we implying himself and M2: I understood that the two of you had been talking about this, and I took this as an indication that you considered it as important. S1 went on to say that he had felt relaxed initially since he felt that the children had been active and well focused. The comment about the back table had changed his attitude. T2 s words had been impossible to erase from his mind and he could not concentrate on the discussion. Even at the time of the interview the back table was what he remembered most clearly from the lesson and from the post-teaching discussion. He referred to it as something that disturbed him: they did not follow, and he had not recognised this. He claimed that even when we discussed the function of such utterances and tried to see how it might restrain or stimulate the didactical discussions, the bad feeling in his stomach constantly repeated: the back table did not follow. Excerpt 2: Is it strictly speaking concrete? Phase 1 S4 had started the class by singing, talking about which day it was and what date was. It was the tenth. S4 took out a small rack with five wooden sticks in it. One child puts nine white balls on the sticks to indicate 9 - yesterday. Then ten white balls were shown. Communicating with the children, S1changed the ten balls into a red ball, put it on the next stick in order to symbolize 10 - today. Phase 2 14 The students have two periods of practice teaching each year. The present period is their fourth. 7

8 In the SMT-conversation S4 received supportive comments from everybody for the interactive way she had started the day. In this context T3 emphasized how well S4 had used the balls on the sticks in order to concretize the numbers M1 followed up with exploratory questions: Is it, strictly speaking, concrete? To use a red ball in order to symbolize ten; to put it on a stick you have decided for the tenths? Is it abstraction, or concretizing? What is symbolizing? The conversation shifted. A new thoughtfulness prevailed. The conversation was characterised by slowness, hesitation and pauses; unfinished sentences were followed by new unfinished sentences, the tentativeness was underlined by the questioning tone of voice. M1: This is not meant to imply that you should have done anything differently. I just consider it interesting. This remark from M1 passed without comments. We could not observe that this comment had any disturbing effects. S4: I have never thought about that. T3: This might be the case most often when we talk about concretizing. Especially when it is concerning the positions the number-system? S4: What about when we are playing to buy in a shop? I have seen that lots of children mixes 10-kroner, 5-kroner and 20 kroner? S5: Is it abstraction or concretization when we use money as teaching material in the context of play shopping? S4: When we say fifty-two and make it concrete by using piles of kroner and changing them into tens and into a 50-kroner-note? These two excerpts were chosen to illustrate the nature of the kind of subject-based discussion that we were trying to implement, as well as the struggle to introduce and position a subject-based discussion within the established discourse. Excerpts like these were used as examples and tools in the SMT discussions and interviews. CONFLICTING APPROACHES In the beginning of the SMT-conversation in excerpt 1, M2 encourage the participants to start in the way they were used to. T1 took the lead in this phase. M1 and M2 participated as visitors, and were invited to join in. Drawing upon Foucault, we view this discourse as being generated institutionally (Mellin-Olsen, 1991; Popkewitz & Brennan, 1998). The participants knew which questions were relevant (and which were not). They knew which way questions should be posed and responded to. When we visit different groups, we find that although the communication pattern differs from group to group, there are marked similarities. The post-teaching conversations have established a discursive identity. The discourse is based in the culture of tutors established through their practice, training and seminars. The tutors are given the responsibility; each is in charge of establishing a constructive basis for the discussions. However, they do their tutoring in a context 8

9 that sets discursive limitations and possibilities. When analysing the data, the power of this discourse became evident. S1 initiated the discussion by commenting that it developed all right and that most of the pupils followed what I said and grasped it, I think. This comment can be viewed in connection with a comment from another part of the interview where he was defending a fellow-student (S2) and explained the following: Of course the focus of S2 is on wether the pupils really did what he intended them to do. What he had planned that they do. That was his focus! S1 felt satisfied, therefore, the pupils had done what he wanted them to do; they joined in. Therefore he had concluded that they had grasped it. His interpretation was questioned by T2 in the comment about the back table that did not follow. This led to a change in his attitude. His body-language, his silence, his tone of voice when he was talking and the brevity of his utterances all indicated to us that he felt confused and insecure. This was highlighted by the evaluative approach: I muddled the order... I should have done No, I did not I tried to. We tried to challenge him by leaving the evaluative mode and starting a subjectbased discussion. However, even when M1 starts by praising S1 s teaching and then explicitly explained the intended discussion concerning children s learning of number concept without discussing what was done well or badly, he did not fully participate. S1 never regained the relaxed and well confident attitude that he had had in the beginning. In the interview, S1 later commented on what had happened by referring to what they were used to. Even though this interview took place at the end of second year, he still referred to the first year in his statement that the only interest we should have ( ) was what we did badly ( ) the evaluation afterwards was introduced by: Yes but..you should have done so and so. S1 described the changes in approach from previous year. By so doing, he also strengthens that the foundation was laid early in his studies. Later in the interview, he commented on the discussion concerning a fellow-student: That is how it is - we easily retreat to the perspective of evaluation ( ) it is difficult to leave ones focus and look for other aspects ( ) The focus is on getting the pupils to do what we have planned. S1 s comments strengthen our view that the dominant approach is evaluative. He moves actually the focus still further. Rather than evaluating what was good, or could have been better in a broader sense, he simply questioned weather he had succeeded in getting things done as planned. S1 s inability to shift focus from the back table that did not follow was also unexpected to us. According to S1, T2 s comment (which included M2) played a critical role for him. He returned to it over and over again. He did so even after M2 said that she had considered saying that she did not quite agree with T2. She had felt a bit uncomfortable. However she had not interpreted the impact the comment had on S2 s impression of his teaching. Later in the interview S1 exclaimed: You should 9

10 have said so! You should just have said explicitly that you did not see it quite as strongly and that it should not have such great importance. M1: Do you think that M2 had the power to do so? S1: Yes well I see I see that T1 and T2 are in charge and that you want to include all the participants in the discussion, but yes, I think you should have. Because I could not let it go. But I see it is not easy. M1 had praised his work. She had said how highly she evaluated his communication of number-concepts. However, neither she nor M2 had commented on the back table. We interpret this as an indication that S1 is describing characteristics of the discourse of practice teaching by describing what is considered relevant to talk about (and in which ways) and what is not. He describes a discourse imprinted by evaluative connotations. He also describes how the discourse is implied as a frame for interpretation. The comment about the back table is interpreted by the student in the light of how such conversations have been practiced. Analyses of the transcripts 15 from the SMT-conversations and interviews highlights the aspects referred to above. It is possible to identify strong evaluative tendencies characteristic of the discourse as it is articulated by all the participants (S, T and M). By trying to foster a didactical discussion without a strong evaluative focus, we challenged an established discourse. The established discourse became more clearly visible when viewed in the light of an approach that the participants described as unfamiliar. The fact that we tried to develop an alternative discussion and to question what kind of discussion we were trying to develop influenced the way we questioned the established discourse. We found the students inquisitive and cooperative when it came to analysing the interactions. The paragraph in which we discuss the back table did not follow sequence shows traces of how a cooperative inquiry has developed (Jaworski, 2005). The following three statements by students illustrate this development: As S2 was on his way out of the interview, he turned back and looked at us: If I were to define a subject-based discussion, well I think we are moving forward from the teaching-learning situations it is about winding up a discussion for later use or for relevant situations later on. Earlier in the interview, S2 commented: I am ready for this. We went through the What? How? Why? Did it work as planned? Last year I need to move on not just to circle around in a what-did-happen perspective and we have learnt to cope with the what you did wrong -comments. Just forget about it. Go ahead. Expel the whole situation. So the comments are for no use. No I am ready to move on. When interviewing S3, we used excerpt 2 - Is it strictly speaking concrete? as one of the examples. She commented: That is the benefit of having you there (at the SMT-conversations). We really discussed things in another way than we usually did. We have an excellent tutor, caring and supportive. But these questions we pose when you are present are different on nature. I would never have thought in that direction. It s 15 The limitation of the paper restricts us from presenting broader analyses here. 10

11 interesting. She explained that she did recognise that she managed to changing from an evaluative approach, but that she thought that her fellow student S5 would not manage. He is showing resistance; I see that. It is not easy for him and it is not easy to push him; he simply does not want to. I think I can understand his resistance. The students talked about leaving an approach, about resisting, about moving on, about later use. They talked about being ready for a change, and they understand the resistance they observe. In studying the kind of expressions that were used, we get evidence for describing the discourses established and the interactions between them. Additionally we have to be aware that these expressions have been given relevance within the discourse we have established in our group. 16 We have identified two different approaches: An evaluative approach that aims to focus on a discussion of the learning/teaching session in terms of what was considered to work well and what did not, what could have been done (and was not), why choices had been made. It represents a what-happened perspective, a retrospective perspective. A conversation within this mode might discuss the consequences one sees for later teaching and learning sessions. However, the perspective is to a large degree directed towards the past through the evaluation of what has been done. A subject based- reflective discussion would be educative approach that aims to explore how the situation might generate discussions for further development; a future-oriented perspective. It is generated in the practice teaching situation, released from the evaluative aspects and developed as a subjectbased interest; as a foundation for subject-based reflectiveness. This perspective implies an investigative and dialogical approach. (Alrø & Skovsmose, 2002, Johnsen Høines, 2002) These two approaches are very different and may appear to be conflicting approaches. They do not occur as equal. The evaluative discussion has a strong tradition in the didactical conversations in practice teaching and in the SMTconversations. It is constructed in exchanges between the tutors and the students 17 ; it is embedded in the discourse shared with other tutors. The dominant position of the evaluative discussion can be attributed due to the discourse that has developed in context of practice teaching (Mellin-Olsen, 1991). Early findings in our project indicate that a strongly evaluative approach might restrain the development of a subject-based, reflective discussion approach. The two approaches can be represented by individuals in the sense that each participant brings their personal approach into interaction with others. It can also be 16 We have actively questioned these issues. As mathematics educators we have stimulated and influenced on how they understand the didactical conversation in practice and how they express their understanding. As this is the case for all the participants; we have influenced each other within a group where the power is not shared equally. The questions about symmetry - asymmetry are important for this project, but it is not elaborated explicitly in this paper. 17 and to a less degree by the teachers from the university college 11

12 identified as intrapersonal, in the sense that each person moves between different approaches in the process of gaining understanding. S1 described how he retreated to an evaluative argumentation. S4 claims that she understands the resistance of S5, and the difficulty he has in changing approaches. Our analyses reveal that this is the case for all of us; both the perspectives were evident in the argumentation of each participant. We move between these partly conflicting perspectives, intrapersonally and interpersonally (Johnsen Høines, 2002: 77; 2004: 65). FURTHER QUESTIONS FOR DISCUSSION When entering into these didactical conversations, it was our intention to initiate a subject-based discussion. Experience had shown that mathematics as a theme was little evident in the post-teaching conversations. We wanted a change and we wanted to identify the preconditions necessary for change. In this respect we see evidence of normative aspects in our conversations: We could be interpreted as saying that we should not rely upon an evaluative approach; that we should develop alternative and more educative discussions. As we see it, this is true, to some extent. However, it is challenging to find a balance on this issue. We do not consider the evaluative approach to be irrelevant, nor would we want to eliminate it completely. We wanted to explore the conditions for implying an educative discourse and to develop a common understanding as to what a subjectbased discussion could be. In fact we learnt about the tension between the two approaches. We learnt about how an educative approach can be blocked by a preexisting evaluative approach. We discussed the preconditions. The participants told us about the restrictions and, about the conflicts they experienced and confirmed that they felt ready for a change. We discussed the difficulties, the risks and the unpredictable changes this could imply. We discussed the social circumstances, referred to the interpersonal relations in the group and the institutional conditions as well as what they personally interpreted as condition for taking part in a subject-based discussion. The analysis and the discussions call attention to the participants readiness as a crucial point. The present analysis provides a foundation upon which to extend the nature of readiness as a conceptual tool for further analyses. We discussed readiness in terms of implying, to make room for, to be motivated for, to adopt an expectant attitude; to be on one s way to realization. This implies to be clear for, to be actively present or actively engaged. This paper does not discuss this issue further, but simply indicates our basis for developing these kinds of discussions. It is intended to provide information about the angle of approach and the early discussions between mathematics educators, students and tutors in the context of practice teaching and the potential for joint investigations. 12

13 References Alrø, H. & Skovsmose, O. (2002). Dialogue and learning in mathematics education. Intention, reflection and critique. Dortrecht: Kluwer Academic Publishers. Fuglestad, A. B. & Jaworski, B. (2005). Læringsfellesskap i matematikk utvikling og forskning i samarbeid. Tagenten 16(3), Hove, S. (2004). Allmennlærerstudenten på spranget til yrket. Hvordan tenker studentene om kvalitet i utdanningen. En kvalitetsundersøkelse. Skriftserien nr. 7/2004. Bergen: Høgskolen i Bergen. Jaworski, B. (2004). Grappling with complexity: Colearning in inquiry communities in mathematics teaching environment. In M. Johnsen Høines & A. B. Fuglestad (Eds.) Proc. 28 th Conf. of the Int. Group for the Psychology of Mathematics Education. (Vol. 1, pp ) Bergen: PME. Jaworski, B. (2005). Johnsen Høines, M. (2002). Fleksible språkrom. Matematikklæring som tekstutvikling. Doctoral Thesis. Bergen: Universitetet i Bergen. ISBN Johnsen Høines, M. (2004). Understanding constituted by understandings. In NOMAD 9 (2) Kvale, S. & Nielsen, K. (1999). Landskap for læring. I K. Nielsen & S. Kvale (Red.) Mesterlære. Læring som sosial praksis. (pp ). Oslo: Ad Notam Gyldendal. Lave, J. (1999). Læring, mesterlære, sosial praksis. I K. Nielsen & S. Kvale (Red.) Mesterlære. Læring som sosial praksis. (pp ). Oslo: Ad Notam Gyldendal. Mellin-Olsen, S. (1991). Hvordan tenker lærere om matematikkundervisning? Bergen: Bergen lærerhøgskole. Nilssen, V. (2002). Mentoring teaching of mathematics in teacher education. In N. A. Pateman, B. J. Doherty, & J. Zilliox (Eds.) Proc. 27 th Conf. of the Int. Group for the Psychology of Mathematics Education (Vol. 3, pp ). Honolulu: PME. Popkewitz T.S. & Brennan, M. (1998). Restructuring of social and political theory in education: Foucault and an epistemology of school practices. In T.S. Popkewitz & M. Brennan (Eds.) Foucault s challenge. Discourse, knowledge and power in education. New York: Teacher College Press. Wenger, E. (1998). Communities of Practice: Learning, Meaning and Identity. Cambridge: Cambridge University Press. 13

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