Towards a collaborative, interactionist model of teacher change

Teaching and Teacher Education 26 (2010) 854e862

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Towards a collaborative, interactionist model of teacher change Raimo Kaasila*, Anneli Lauriala University of Lapland, Finland

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 March 2009 Received in revised form 14 September 2009 Accepted 26 October 2009

The aim of this article is to extend the scope of the models of teacher change to an interactionist view which co-ordinates sociocultural and constructivist perspectives. Accordingly our focus will be on the cultural and situational factors and processes of social interaction, as well as on the development on an individual level. First we study how roles, statuses, expectations as well as social and socio-mathematical norms were formed in the group of four student teachers, when they were co-planning, teaching and assessing mathematics lessons during their teaching practice. Finally, we attempt to construct a collaborative, interactionist model of teacher change integrating sociocultural and constructivist perspectives. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.

Keywords: Teacher education Interactionism Collaboration Roles Norms Identity

1. Introduction There is at the moment an increasing emphasis on the social dimension in the education of teachers. One indication of this is that collaboration is regarded as a key factor in teachers' professional development (see e.g. Gellert, 2008). According to earlier studies, collegiality adds teacher efficacy, which is also shown to predict pupil achievement (Lee & Smith,1991). So it is important that future teachers learn collaborative skills and dispositions during teacher education. In Finland, teaching profession is highly valued and teacher education attracts many more students than can be taken in. In general, students are of high quality, and very committed to teaching profession. Students in primary teacher education take a Master's Degree in education. Finnish teacher education is based on a teacheras-a-researcher eapproach, which is reflected also in student teaching practices where inquiry-orientation is cultivated. The goal is to educate reflective teachers who are able to combine knowledge of educational science with pedagogical content knowledge, e.g. pedagogy of mathematics. Further, training aims at developing future teachers' collaborative skills. Courses in pedagogy in the university and practicums are tightly connected to each other, and form a continuum. Teaching practice takes place mainly in teacher training schools, involving one practicum in field schools. * Corresponding author at: University of Lapland, Faculty of Education, P. Box 122, 96 101 Rovaniemi, Finland. Tel.: þ358 16 3412 407; fax: þ358 16 3412 401. E-mail addresses: Raimo.Kaasila@ulapland.fi (R. Kaasila), Anneli.Lauriala@ ulapland.fi (A. Lauriala).

We understand teacher change as a growth: Teachers are assumed to change (and learn) through acting and interacting in professionally relevant learning communities. Teaching practices are thus seen as fertile sites of teachers to learn both to teach and innovate teaching in authentic, everyday situations. (cf. Clarke & Hollingsworth, 2002, 948.) In earlier studies considering teacher change we can see the dominance of theories of cognitive psychology: Their focus has often been on how an individual's beliefs and practices are changing (see e.g. Kaasila, Hannula, Laine & Pehkonen, 2008; Senger, 1999; Shaw, Davis, & McCarty, 1991; Smith, Williams, & Smith, 2005). We see the models emphasizing the psychological view of teacher change as useful, but inadequate by themselves to explain teachers' change processes. Hence, we see that it is appropriate to extend the view by taking into account cultural and situational factors and processes of social interaction according to the interactionist view adopted in this study. We emphasize the co-ordination of sociocultural and constructivist perspectives: in accordance with Cobb (1994, 13) we state that “mathematical learning should be viewed as both a process of active individual construction and a process of enculturation (socialization) into the mathematical practices of wider group or society”. Focusing on collaboration in mathematics learning, pre-service teachers' collaboration has been quite little studied. As an example of this is Manouchehri's study, which focuses on two pre-service secondary mathematics teachers' peer collaboration. According to him, peer discourse, peer observations and peer feedback can facilitate the development of professional knowledge of teacher students. The peer student teachers helped problematize instructional issues for one another. (Manouchehri, 2002.)

0742-051X/$ e see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.tate.2009.10.023

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It was especially difficult to find studies dealing with pre-service elementary teachers' cooperative or collaborative planning in mathematics. The reason for this can be that in many countries teaching practice is organized in field schools, in which the research is not so easy or customary to carry out. In Finland teaching practice mainly takes place in teacher training school, i.e., Professional Development Schools, which are meant to be experimental developmental schools. However, even in Finland there are few studies focusing on pre-service elementary teachers' cooperative or collaborative planning in mathematics or in any other subject. In the study of Kazemi and Franke (2004) in-service teachers used a common mathematical problem that they adapted for their own students. For each collaborative meeting teachers selected some students' work to share with the teacher group. Teachers learned through collective work to focus on the details of their students' mathematical thinking and to develop instructional trajectories. (Kazemi & Franke, 2004.) As a further example of previous attempts to enhance collaboration is the Lesson study1 idea, which has been developed alongside efforts to facilitate Japanese elementary teachers' professional development since the late 1990. In a US study teachers jointly produced a detailed lesson plan, and then one of them taught a “research lesson” while the other teachers observed and made notes. The results indicate that this kind of collaboration has an impact on teacher efficacy. Teachers themselves emphasized the socio-emotional and cognitive impact of their collaborative process. (Puchner & Taylor, 2006.) As to the research on student teachers' beliefs of mathematics, these have often been studied without taking into account their connections to teaching practices and specific situations. Yet we know that a change in a student's view of mathematics does not necessarily mean a change in his or her teaching practices (see, e.g. Vacc & Bright, 1999): Although pre-service teachers talk about change, their actions in a classroom may be in conflict with their verbally expressed beliefs. So in our study we also pay attention to student teachers' teaching practices. Contextual and situational factors of change can manifest themselves, for example, in the following way: The social and socio-mathematical norms and the ways to act in different practicum classrooms often differ from each other and create different possibilities and obstacles for student teachers' learning and teaching (Lauriala, 2004). The first author's (see Kaasila, 2000) research findings indicate that in some practicum classrooms student teachers developed a rich array of beliefs, whereas in others the change and variety in beliefs was meagre in comparison (Kaasila, 2000). In our earlier article (see Kaasila & Lauriala, 2008) we focused on how four student teachers' former beliefs, experiences and goals influence, and are in dialogue with the situational demands of the classroom, which involved a new approach to teaching and learning mathematics, i.e., the problembased approach. The focus of this article is to describe student teachers' collaboration and its relevance to the change of their beliefs and practices and finally, to attempt to construct a collaborative, interactionist model of teacher change. 2. Theoretical framework The chosen approach of our study, interactionism, has guided our choice of the concepts. However we have aimed at leaving space also for new phenomena arising from the data. The main

1

In Finnish teacher education the Lesson study e principle (joint reflective post analysis) has a long tradition, although we in Finland usually don't use this term. For example, the first author of this article has applied lesson study type approaches in teaching practice and in the mathematics methods course since 1995.

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concepts used are roles, norms, statuses, expectations as well as identity. 2.1. Roles, statuses, expectations and norms According to Woods (1992) student teachers' social interactions, social observations and perceptions and the way they give and receive feedback seem most significant for the construction of their identity and for the determination of their roles. In addition, student teachers also enter into a dialogue between their past, present and future identity (Kaasila, 2000). When familiarizing themselves with a new classroom situation and with a new pedagogical culture student teachers' earlier beliefs about themselves emerge and may end in a dialogue, even conflict, with the beliefs and practices present in the new context (Lauriala, 1997). In the project presented here we tried to facilitate student teachers' collaboration which we conceptualize in terms of the phenomena of group dynamics: Group structure refers to the stable patterns of relationships among group members, involving roles, norms and interpersonal relations. Group cohesiveness is understood to reflect the strength of the bonds and feelings of attraction between group members and the group. (Forsyth, 1999.) In social situations a person must adopt a social role, which refers to a set of expectations of how a member of a special group or community is expected to act in his/her position. In the teaching practice situations there are very clear, built in expectations of how student teachers should act in their roles as teachers. So learning to be a teacher is guided by a special set of norms. ‘Expectation state’ etheory (see e.g., Berger, Cohen, & Zelditch, 1972) suggests that if the members of a recently formed group begin collectively to solve a task, each of the members evaluates the joint work. Each person develops expectations on the basis of his/her own status characteristic, for example, on the basis of his/her skills, experience or gender. A person's status characteristic is associated with his/her performance expectations, i.e., with a belief about how a member having a given characteristic is expected to perform. A member's status defines the weight given to him/her by the other members when decisions are made inside the group. Status and role are defined on the basis of competence: The higher the status and role a member of a group has, the bigger contribution other members of the group expect he/she to have in solving the task. So the members who have a higher status are expected to be more active than the members having a lower status. While studying the norms, we divide them into social and sociomathematical norms. Social norms are mutually constructed expectations for the appropriate behaviour of members in a particular cultural group (Yackel & Cobb, 1996). So they are (unwritten) rules of how people must act in a specific group or community. While social norms can be enacted in any subject, socio-mathematical norms refer to the normative aspects of interaction which are specific in mathematics (Yackel & Cobb, 1996). These are interpretations that become taken-as-shared by a community, for example, a school class or a student teachers' group. One example of socio-mathematical norms is the norm about what constitutes an elegant solution in mathematics. When student teachers are talking about and doing mathematics, norms are implicitly negotiated. When they are talking about talking about mathematics, norms are explicitly negotiated. (McNeal & Simon, 2000, 478.) In our study we focus on partly implicitly, partly explicitly negotiated norms. 2.2. Identity and mathematical identity Social interactions and collaboration have an impact not only on professional thinking and action, but on a person's identity as well.

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In our study identity is understood to mean a person's conception of self at a certain point, not totally or universally, and involving a reference to “we” or a group a person identifies him/herself with (Hall, 1999; cf. Lauriala & Kukkonen, 2003, 2). Identity can be regarded simultaneously as both stable and changing (e.g., Lauriala & Kukkonen, 2001; Demo, 1992). According to ideas of symbolic interactionism, people construct their identity through social interaction, and different sides of identity arise in different situations. Further, social context and a membership in a group especially influence on how a person describes him/herself and what kinds of choices s/he makes. A membership can enhance a positive self-confidence and alongside a need to belong to a group. We can talk about significant other as a kind of wholeness of collective attitudes, roles and rules, which are guiding a person's actions and thinking as a member of a group. Identity is the who-we-are that develops in our own minds and in the minds of others as we interact. As to our method, according to Ricoeur (1992), narrative identity means that people often develop their sense of identity by seeing themselves as protagonists in different stories: What creates the identity of the character is the identity of the story and not the other way around. (Ricoeur, 1992) Student teachers' narratives have an important meaning in our study as a way of revealing their evolving identities. One of the key concepts of our study is mathematical identity: it is a construction which describes a person's relationship towards mathematics (Bikner-Ahsbahs, 2003). According to Op't Eynde (2004), students' mathematical identity is developed in different mathematics education communities (e.g., in a school class) through their interactions with the teacher, the books, and their peers. While these interactions are determined by the social context they are situated in, student teachers have also a lot of experiences about mathematics learning and teaching from their school years. A student teacher's mathematical biography includes a retrospective explanation of how his/her earlier experiences have influenced his or her past and present mathematical identity (Kaasila, 2007). Students' stories reveal stereotypical views and a predominance of traditional methods in our schools, even today. These former views of mathematics learning and teaching are often actualised during first practice teaching phases (Kaasila, 2000). All in all, students' view of mathematics is an important part of their mathematical identity. We distinguish three components in students' view of mathematics: 1) their view of themselves as learners and teachers of mathematics, 2) their view of mathematics and its teaching and learning, and 3) their view of the social context of learning and teaching mathematics, i.e., the classroom context including the socio-mathematical norms of a school class (Kaasila et al., 2008; Op't Eynde, De Corte, & Verschaffel, 2002). 3. Earlier models of teacher change We have earlier constructed a model of the phases of teacher change by combining certain central elements of Smith et al. (2005) and Senger's (1999) models: 1) problematising current beliefs and practices, in which students accept that their views of mathematics are not the best possible for teaching pupils effectively; 2) becoming aware of a new approach, in which students create new personal visions of what mathematics learning and teaching should look like; and 3) exploring and testing alternative beliefs and practices during the mathematics education course, in teaching or in verbalising new beliefs; 4) reflectively analysing benefits of the new approach in which students become more convinced of new beliefs they adopt; and 5) changing one's views of mathematics and one's teaching practices (Kaasila et al., 2008). This model describes well how student teachers' beliefs and practices change, but its main focus is on an individual level. Thus

it is not sufficient or adequate to describe the change processes on a community level. Lewin (1947) has observed instances of successful organizational change. His starting point was that the forces driving change must be increased and the forces resisting change must be decreased. Organizations often tend to resist change and return to the previous state after a disturbance. Our earlier studies indicate that new and ‘deviant’ contexts are needed for pre-service teachers to become aware of the influences of their former experiences and to be able to break the chain of influences of cumulative socialization (cf., Lauriala, 1992, 1997, 128). Here, changes are studied in relation to classroom contexts, interaction and cultures, as well as to student teachers' co-learning and collaboration. One of the most central goals of our study is to familiarize student teachers with a new kind of way to teach and learn mathematics and to facilitate their collaboration. When applying Lewin's (1947) theory of organizational change we distinguish three phases in the process of teacher change: 1) unfreezing, 2) moving, 3) refreezing. Edgar Schein, building on Lewin's work, has refined the model and extended it to the changes in psychological processes that are necessary for an organization to successfully change its culture. Schein (1992, 301) equates a group with an organization: They react similarly in change phases. For example Reio (2005) has applied Lewin's theory to educational reform. According to him, emotions are a key component in school reform and teacher change. Teachers' emotional experiences of reform influence their risk taking. Change can affect teacher development by creating an environment of uncertainty (Reio, 2005, 986). Teaching practice situations can by themselves be anxiety-provoking, and mean a threat to students' identities. 4. The method 4.1. The research questions Our research problems are as follows: 1) How are the student teachers' roles, norms and statuses formed in the student teacher group? 2) How do their mathematical identities develop during the practicum? 3) What kind of negotiations of norms are manifested within the group? 4) How can we describe and explain teacher change in the framework of interactionist theory; can we outline a tentative model to describe teacher change? 4.2. The approach of the study Narratives have considerable significance in teacher education. Here we assumed that it is important to listen to the voices of preservice teachers talking and writing about themselves as mathematics learners and teachers. (Kaasila, 2007.) When applying the narrative inquiry, we are interested not only in the pre-service teachers' views of mathematics as such, but also in the ways they tell about these and the ways they talk about their collaboration. 4.3. The study context The study was carried out as connected to Subject Didactic Practicum 2 (SD 2) at the 3rd class in the Training School of the University of Lapland in February and March 2007. The goal of the four-week SD 2 practice was to familiarize students with planning and teaching lessons in mathematics and two other subjects, as well as with evaluating pupils' development in these subjects. As to the pedagogical approach, in this practice the emphasis was on collaboration and problem-centred teaching and learning. Students gave about 12 lessons each, including 3 to 5 lessons in mathematics. During SD 2, they received guidance from university lecturers

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specialized in education of subjects, and from a cooperative teacher2 in the training school. The teacher of the classroom in question (the cooperative teacher) has worked for some years in the training school, and she has actively developed her teaching and supervision practices during that time. She is regarded as a competent and empathetic supervisor. There are about 20 pupils in the classroom, and they are accustomed to active, collaborative studying and learning. The problem-based learning was introduced to the second year students in the mathematics education course, which was given by the first author of this article. At the end of the course, he provided the students with some advance guidance in making their plans for the mathematics lessons for SD 2. During SD 2, he provided feedback on one mathematics lesson by each student. In the mathematics education course, students were introduced to the basics of problem-based teaching. In problem-based or inquiry-oriented teaching of mathematics, pupils learn through solving problems: a pupil acquires new mathematical knowledge through problem solving and at the same time gets insight into how new content is related to his already existing mathematical knowledge (Nunokawa, 2005). The idea of new learning content is carried out through pupils engaging in solving one or more research tasks related to the contents to be learned. In addition, pupils use manipulative tools or figures as an aid when solving the problem. 4.4. The participants The choice of the participants represents purposive sampling to achieve a many-sided and rich picture of the phenomenon under study. There are 18 classes in the Teacher Training School in question, and so the students were divided into 18 groups, from which we chose one group, consisting of four student teachers e Jari, Kirsi, Risto and Meri, the participants of this study. From the different types of purposive sampling presented by Patton (1990) our sampling represents most closely the case of Intensity, i.e. a rich case that manifest the phenomenon intensely, but not extremely. Accordingly, the choice was based on the following factors: 1) Their biographies varied as to the amount of mathematics teaching experience, and as to their success in learning mathematics at school: Meri had 7 years' experience of acting as a substitute elementary teacher, and Jari had 3 years' experience as a substitute special teacher, and he also acted as a substitute elementary teacher during his teacher education. Kirsi had been nearly half a year as a school assistant in a lower secondary school: She has also been for some time a substitute teacher. Risto's first teaching experiences were gained during teacher education in Subject Didactics 1 (SD 1) practice teaching, preceding the SD 2, under study here. Jari, Risto and Kirsi took advanced courses in mathematics in upper secondary school. Jari and Risto had succeeded quite well, but Kirsi's success had been poor in the mathematics component of Matriculation Examination. Meri took only general courses in mathematics in upper secondary school with poor success in the mathematical section of Matriculation Examination. 2) When starting to plan their mathematics lessons, at the end of the mathematics education course, before the SD 2, the participants' collaboration seemed to begin well. Majority (n ¼ 3) of the participants were above average as to their

2 The teacher in whose classroom the practicum took place. Cooperative teachers guide students' practice teaching daily, while University Lecturers in different subjects supervise and observe them approximately two lessons per week.

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disposition to collaboration, but there was also one deviant case among the students. 3) Jari functioned as a tutor in mathematics for the 18 students' practice group from autumn 2006. The aim of a tutor was to guide the other group members while preparing themselves for examinations in mathematics education course. 4.5. Data gathering The research material consists of: a) the interviews of the four students and one cooperative teacher, b) the observation notes of the university lecturer in mathematics, and c) the student teachers' mathematics portfolios. The portfolios comprise the individual lesson plans and related self-assessments, an assessment of the progress of one pupil in the class, chosen by the student teacher, as well as the students' reflections on two self-chosen articles forming a part of the €nen, Kupari, Ahonen, & Malinen, 2004). required course reading (R€ asa The use of different data gathering methods is an indication of triangulation; i.e. method triangulation, as well triangulation of data sources, and which may be used to confirm the results. When interviewing the student teachers, we also utilised a narrative inquiry (see e.g., Kaasila, 2007). The goal of the narrative interview was to get the interviewee to tell stories about things that were important to him or her. Especially at the beginning of the interview we used narrative questions, for example: “Tell me about the event or thing you best remember from SD 2.” We also asked them to tell about their mathematical autobiographies. After that we asked them to tell about central themes e.g. how their views of mathematics had changed (or not) and how they felt that their cooperation with other student teachers had worked out. The duration of each interview was between 40 and 85 min. 4.6. Data analysis In data analysis we applied narrative inquiry in the following ways. We divided the analysis of our data into two parts. When considering how student teachers' mathematical identity had changed on an individual level, we read their mathematical autobiographies which were included in their teaching portfolios. In a mathematical autobiography a student teacher told about his/her own development in learning and teaching mathematics. (Kaasila, 2007.) Then we constructed student teachers' mathematical biographies: Our task was to elicit how a student teacher's earlier experiences had influenced his or her past and present mathematical identity. Here we used ‘emplotment’: A story line or plot that serves to configure or compose the disparate data elements into a meaningful explanation of the protagonist's responses and actions' (Polkinghorne, 1995). Within each mathematical biography we compared the teacher student's view of mathematics at the beginning and at the end of the mathematics education course. We also looked for principal facilitators of change manifested in the trainees' talk. So each mathematical biography contained a retrospective explanation (Polkinghorne, 1995) linking central events in the student teacher's past to account for how his or her mathematical identity had developed. When considering student teachers' collaboration, we applied narrative analysis, which includes influences from the methodology  & Secord, 1972). The four student of the account of change (see Harre teachers told how they felt the collaboration with each others. We were also interested in the different ways in which a student teacher talked about his/her collaboration with other students. From this we see that student teachers' identity talk is always directed to some audience or listeners. We compared student teachers' narratives (accounts) with each other and with the ways the cooperative teacher and the supervising lecturer (first Author) assessed their

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actions. This may be seen to represent participant triangulation. We constructed a ‘grand’ narrative, which combines the accounts told by the different actors. To confirm our results, we let the cooperative teacher read the ‘grand’ narrative and assess how she felt it corresponded to her own experiences of the classroom events and interactions. Although our analysis was data driven at the beginning, in the later phase we looked for the connections between the results and relevant theories of teacher change. 5. Results 5.1. Student teachers' roles and statuses in the group The roles and statuses within the group were beginning to be formed already at the end of the mathematics method course when the student teachers made their section plan in mathematics, but the events and situations during teaching practice significantly influenced the role and status formation. Jari's role: All the students of the group named Jari as an authority on the grounds of his good mathematical proficiency and wide-ranging teaching experience. Jari was already, at the beginning of mathematics methods course in September, given the central role, because he was chosen to be a mathematics tutor for the whole mathematics exercises group, consisting of 20 student teachers. When student teachers planned teaching periods and lessons collaboratively, Jari's goal was not to tell directly his ideas to others.: “Often I had to think how I could support others without telling the right answer: I think I succeeded very well”. This further indicates that Jari had internalized a socio-constructivist view of a tutor's role as a promoter of insights. He was a very diplomatic negotiator. All in all, Jari noticeably promoted the cohesion of the group. Kirsi acted as an analytic reviewer in the group: She brought many issues that deepened the collaboration between the student teachers. Kirsi mentioned that in some co-planning situations students had to vote on two different alternatives: “Often three of four students had a common view. After voting all acted in accordance with the opinion the majority of the group had presented.” All in all, Kirsi was very eager to collaborate and she would have wished the collaboration to yield even more. On the grounds of her minor teaching experience Kirsi seemed to have a some what lower status in the group than the more experienced group members Meri and Jari. Meri's role was twofold: On one hand, Meri was an authority figure on the basis of her widest amount of teaching experience. On the other hand, she was a novice on the grounds of poor mathematical proficiency. Meri's view of herself as a learner and teacher of mathematics was clearly weaker than that of the other three students who had chosen advanced courses in upper secondary school mathematics. According to the cooperative teacher Meri was quite nervous during her first mathematics lesson: “Later she began to trust herself more.” Meri described her role in the group by the following statements: “I didn't want to show the others who's the boss.”and “I tried to be on the same level with the others”. These statements crystallize Meri's goal to be a conciliatory facilitator and equal member of the group, who also wants to listen to others. On the other hand, when it seemed necessary, Meri could bring forth her wide teaching experience: “If I saw, on the basis of my experience, that something isn't going to work then I said it. I had an opportunity to do it.” This kind of rhetoric, grounded by experience seemed to get support in the group. When applying Halai's (2006) classification, Jari's role can be seen to be a subject specialist, while Meri's one is an ‘expert-coach’. In addition, they can be seen also as critical friends to the other members of the group. Risto was partially unattached to the group. As an example of this Kirsi told about Risto's role in the collaboration: “It is interesting to note

how Risto, although he had less experience than the others, sought others' opinions and guidance least of all. He seemed to take a view that he knew what teaching is about. Certainly, it is a good thing.” It seems that Risto could not seek help from the others in all situations. On the other hand, Risto had positive experiences about collaboration with Jari in science teaching. Risto seemed to have the lowest status in the group. In many respects, he compared his way to plan the lessons to that exercised by the more experienced students Jari and Meri: “I feel as if they were already ready teachers, they had their own, clear thoughts in beforehand. I was may be such a person who needed more time to think about”. All students named Risto as a novice on the grounds of his lack of teaching experience. For example, Jari described Risto's contribution in co-planning in the diplomatic way: “Risto could mention a very good idea. Then Meri noted on the grounds of her experience that ‘it is a good thing but when implemented it results in this or that ’. For example, ‘it is better to use (a learning game) in the lessons in which the class is divided into two smaller groups’.” On the other hand, Risto was an authority figure on the grounds of his good mathematical proficiency. All in all, a reflective thinking was characteristic of Risto, according to both his interview data and teaching portfolio.

5.2. Changes in the students teachers' mathematical identity Meri's view of herself as a learner and teacher of mathematics changed clearly towards a positive direction: “The word ‘math’ makes me no more feel like fainting! I thing I succeeded well in the lessons I taught during SD 2. Now I feel that I am very eager to teach mathematics.” Also Meri's view of learning and teaching mathematics changed into a more versatile direction: “Now I see the meaning of mathematics much more clearly and I think it is a creative subject, and doesn't mean only learning of the rules.” Jari's positive view of himself as a mathematics learner and teacher was confirmed during mathematics method studies and SD 2. The positive development was enhanced by Jari's functioning as a mathematics tutor for his own group: “My view of myself as a mathematics learner was confirmed by being a tutor. The members of my group often asked me for advice in their task, as well during the exercises as before the examination. Tutoring also increased my confidence as a mathematics teacher. I felt myself to be a good teacher.” Jari's view of mathematics teaching and learning also changed towards a more action-oriented direction: “In the mathematics' course I finally comprehended how important it is to use manipulative tools in mathematics. Actually, while planning mathematics lessons I decided to emphasize the use of manipulative tools as much as possible.” The cooperative teacher evaluated Jari's development in the following way: “Jari's own professionalism was well developed. I thought he was already a competent teacher. He presented many ideas and was able to take different roles, thus he succeeded in getting the children to become inspired in learning.” Kirsi's view of herself as a teacher and learner of mathematics ‘didn't change much’ after SD 2: “The SD 2 was the first time that I was acting as a real teacher. At least, based on the experience of the SD 2 I got a view that I am a pretty good mathematics teacher, and I've also a positive view of mathematics as a whole.” The biggest change took place in Kirsi's view of learning and teaching mathematics. Her view changed from a teacher-centred

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one, involving emphasis on 'drill and practice', towards more pupilcentred and problem-based teaching: “In the lectures and the practice of the mathematics education course the emphasis was on using manipulative tools, problembased teaching and discovery learning. They were the skeleton of the whole course. The course has dramatically changed my view of mathematics and teaching of it.” Risto's initially quite positive view of himself as a learner of mathematics didn't change during practice teaching. Although he had no previous experience in teaching mathematics, Risto was rather satisfied with his mathematics lessons. He was also able to present some suggestions for how to develop them. On the whole, Risto's view of teaching mathematics seemed to be eclectic: On one hand, during the interview he said that: “imitating could be the best way to learn”. On the other hand, Risto's portfolio indicates changes in his views of teaching and learning: “Learning is much more than just silent cramming, rote learning and copying the teacher”. It seems that Risto was striving towards interactive and pupil-centred teaching, but the teacher-centred model, dating back to his own school years, was deeply rooted in his mind. To sum up, it seems that a student teacher's earlier teaching experience had a greater influence on his/her status in making the period plan than his/her mathematical proficiency. This could be seen to continue during SD 2 teaching practice. A rise of status was apparent in Meri's case, because she was able to invoke and take into use her wide-ranging teaching experience. In addition, the commitment to collaboration influenced how much a teacher student experienced she or he had benefitted from the group. The higher the status and the bigger the degree of commitment a student had, the better was the possibility that his/her view of mathematics developed. 5.3. Negotiation of norms We divided the negotiation of norms guiding the actions in the student group into two subcategories which are linked to each other: 1) social norms, and 2) socio-mathematical norms of the student group. 5.3.1. Social norms in the student group a) The norm of security manifested itself in all teacher students' narration: “Nobody is left alone”. The students learned to give and receive help from each other. Kirsi also said that “it was always possible beforehand to double-check with others if an idea included in the lesson plan was really working in practice.” In all, the student teachers used each others as reflective mirrors (Markham, 1999) when building their identity. In a wider perspective this refers to the norms of reciprocity (Putnam, 1995): A student teacher can think that helping the other members of the group will increase the possibility that he or she will receive help from the others in the future. This norm was enhanced also by the trust grown within the group. On the other hand, Kirsi told, that when students gave feedback from each others' lessons they avoided giving negative feedback: “I felt that it's very difficult to give constructive criticism”. b) The norm of commitment manifested itself when student teachers made their lesson plans and also during SD 2. Jari crystallized this in the following way: “We worked together as a group”. It is also a matter of the norms of collegiality (Little, 1982). On the other hand, Risto was not present when the other students had their second planning session and made their mathematics period plan together. Also the cooperative teacher had paid attention to Risto's lack of commitment:

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“Sometimes Risto was late”. Kirsi mentioned that Risto acted during SD 2 more independently than the others. Often, in teachers' talk collegiality is connected to the norm of loyalty: it refers to the way to show loyalty so that people don't interfere with the others' different way to act (cf. Hargreaves, 1980). c) The norms emphasizing negotiations were characteristic for the behaviour of this student group. The members of the group underlined that when making the period plan in mathematics ‘everybody's views were taken into account’. Certainly, Jari and Kirsi mentioned that the negotiations did not lead to consensus in every case. In such cases the decision was done by voting. The social norms negotiated in the student teacher group were in many respects parallel to the norms negotiated in the teaching practice class. The cooperative teacher crystallized this in the following way: “To help and pay attention to each other, everyone is accepted such as s/he is, pupils encourage the weaker pupils in the class. Hard work is appreciated (in the class).” When the first author of this article was observing the classroom during SD 2, he noted the following social norms: The environment of the classroom is respectful. Most of the pupils are ready to talk about their ideas. In the whole class and in the smaller group discussion pupils share their ideas with each other. Solutions to the problems come from the pupils, and the role of the teacher is to provide encouragement. These social norms of the practising class are very near the ones Szydlik, Szydlik, and Benson (2003) have noted in their study. 5.3.2. Socio-mathematical norms Socio-mathematical norms reflect the goals of the mathematics methods course and SD 2 practice teaching, that are based on socioconstructivist learning theory: Accordingly ‘effective mathematics teaching’ is characteristic of pupil- and problem-centredness involving the use of manipulative models. Jari crystallized his view of mathematics teaching in the following way: “By using problemcentered approach and manipulative models pupils have to develop themselves solution processes and discover the solutions. When I was planning my mathematics lessons I decided to engage in using the manipulatives as much as possible.” Because Jari acted as a tutor and facilitator of the pedagogical ideas in the group, his crystallization of these ideas can also be seen to reflect the more widely negotiated and accepted sociomathematical norms of the whole group. 5.4. Collaborative, interactionist model of teacher change When trying to form a collaborative, interactionist model of teacher change our aim has been to unite sociocultural and constructivist perspectives. The model is partially data driven, partially theory driven, and in constructing it we have applied following ideas: 1) Kurt Lewin's (1947) theory of the organizational change; 2) Edgar Schein's (1992) theories refining Lewin's model; 3) some ideas including the cognitive models of teacher change (see e.g., Kaasila et al., 2008; Lauriala, 1997; Shaw et al., 1991; Smith et al., 2005), and 4) expectation state theory (see e.g., Berger et al., 1972). The main goal of the mathematics methods course was to develop student teachers' views of mathematics. To aid student teachers to change their view of mathematics, one of our central tools was to support or encourage student groups' negotiations of social and socio-mathematical norms and norms supporting professionalism. These negotiations begun already during mathematics methods course and continued during SD 2. So we related changes on an individual level to social interaction and cultural context. In the emerging model of teacher change we focus on mutual interaction

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and transitions from the group (community) level to the individual level. In the following we will start by describing first the change at the group level and proceed then to the individual level. Here we apply Lewin's theory which involves different phases or aspects. According to Lewin (1947, 228) a successful organizational change includes three aspects (or phases): 1) unfreezing the present level; 2) moving to the new level; 3) (re) freezing group life on the new level. Schein (1992, 303) emphasizes that Lewin's model describes any change process in the individual, group or organizational level. We applied these phases in the process of teacher change in the following way: 5.4.1. Unfreezing 5.4.1.1. Group level. Lewin called unfreezing the creation of a motivation to change. For unfreezing to occur, the forces facilitating change must overcome the forces resisting change. It is a matter of shaking the equilibrium dominated in the group and convincing the need of the change. The first subgoal of the mathematics methods course was to facilitate this by getting students to question or make questionable the socio-mathematical norms and the views of mathematics they had adopted during their years at school. According to Schein (1992) the unfreezing phase can be composed of three different processes: a) enough disconfirming data to cause disequilibrium (in our case one subgoal of the mathematics methods course); disconfirming information is not enough, however, because we can ignore or deny the information. So it is important b) to connect the disconfirming data to central goals (in our case to goals of the course) causing anxiety and guilt; c) in addition enough psychological safety must be created to allow the organizational members to admit the disconfirming data without defensively denying it; for instance in the sense of seeing that it is possible to change one's views without losing one's identity or integrity (Schein, 1992, 299). We see that the ability to tolerate uncertainty plays a central role in the changes both on individual and group level. Especially Jari e as the mentor of the group e was ready to take risks when planning the whole section. Schein (1992, 301) emphasizes the importance of visionary leadership. It seems that Jari's beliefs and actions were reflected in the whole group leading to growing of a climate of uncertainty (cf. Reio, 2005). To conclude, we can say that peer mentoring was in a key position in all phases of change and accordingly relevant aspect in our model. 5.4.1.2. Individual level. According to earlier cognitive models of teacher change we assumed that for a change to occur we need to invoke a disequilibrium on an individual level. This means that a student teacher experiences a cognitive conflict or perturbance in his or her thinking and teaching practice. It is important that he or she will become committed; i.e., he or she will make a personal decision to realize the change as a result of perturbances (see e.g., Shaw et al., 1991). For example, it was regarded as pivotal that all four student teachers started to question and reflect on their current beliefs and practices (cf. Lauriala, 1997). 5.4.2. Moving (changing) The second subgoal of the mathematics methods course was to facilitate new socio-mathematical norms that support socioconstructivistic views of mathematics learning and teaching. These norms were negotiaded within the student group when students planned collaboratively their mathematics section and also later, during SD 2 practice. 5.4.2.1. Group level. Once an organization has taken the first steps to unfreeze the dominant culture to achieve a complete organizational culture change, then the aim of the moving phase is to persuade teachers to accept the new procedures linked with the

change and get them to believe that by changing their work habits their performance develops (Reio, 2005, 990). 5.4.2.2. Individual level. On the individual level moving means that the students of the group become aware of a new approach: They create new personal visions of what mathematics learning and teaching should look like, they explore and test alternative beliefs and practices during the mathematics education course and SD 2 teaching practice, in teaching or in verbalising new beliefs (see Kaasila et al., 2008; Smith et al., 2005) and possible identities. 5.4.3. Refreezing 5.4.3.1. Group level. In the last phase the group must ensure that the changes of the moving stage are institutionalized, new norms are crystallized and stabilized within the group. In the refreezing phase (student) teachers execute the new practices and actions and integrate these into their beliefs (Reio, 2005, 990). Without this phase, there is a danger that the group (or organization) reverts back to its original culture. Once the unfreezing has taken place, initiatives can be put into place that actually serve to change the existing teaching culture and move it into a new direction. Cultural changes must be accepted at the individual, group and organizational levels if they are to be long lasting. (The concept of homeostasis Schein, 1992.) 5.4.3.2. Individual level. On an individual level student teachers reflectively analyse benefits of the new approach, and they become more convinced of the worth of the new beliefs; and finally they are changing their views of mathematics and their teaching practices (Kaasila et al., 2008; Smith et al. 2005). In addition to the above, we tried to conceptualize the process of teacher change by applying expectation state theory (see e.g., Berger et al., 1972) in the following way: Each of the student teachers in the group evaluated both his/her own as well as the others' potential contribution to the group work. So each person developed expectations on the basis of his/her own status characteristic. The more significant the other members of the group are for the individual, the more their views and actions influence the student's views and behaviour in the group. Both a status and role seem to be defined on the grounds of competence: The higher the status and role a member of a group has, the bigger contribution other members of the group expect s/he to have to the co-planning process. Jari had the highest status in the group, so he had the most dominant influence on the other students' view of mathematics and teaching practices. It is to be noted, that a student teacher's earlier teaching experience influenced his/her status in the group more than his/her mathematical proficiency. The rise of the status manifested itself especially in Meri's case, because she could utilise her wide-ranging teaching experience which became actualised and taken into collective use. Further, the higher the student teacher's status and the stronger her commitment was, the better opportunity there was for his/her view of mathematics to develop. We can also say that the bigger the cohesion in the group is, the bigger can the changes be.

6. Discussion The aim of the study presented here was to extend the scope offered by the previous models of teacher change; and from an interactionist view our goal was to connect sociocultural and constructivist perspectives. Teacher change (alike teacher's work) is so complex and many-sided a phenomenon, that it's difficult to describe it by a simplistic or one-dimensional model; also both cognitive models and sociocultural models alone are too narrow. When constructing the model our greatest challenge was how to

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co-ordinate different perspectives and combine them sensitively with each other, which may be seen to lead to too eclectic a model in building a working, appropriate model of teacher change. Here we have here aimed to capture the dialogical relation between the changes on an individual level and on community level, and how they mutually influence each other. It's difficult to imagine community level changes without preceding changes on an individual level. On the other hand, becoming part of a new community, culture may induce changes on a newcoming individual's views and action, which may also happen through a cognitive conflict. We see that roles, statuses, expectations and social and sociomathematical norms are useful tools to conceptualize the process of change. Student teachers' collaboration began when they started to make their section plans in the exercises of the mathematics methods course. At the same time there could be seen role shaping to start in the group. More experienced students Jari and Meri took the role of the mentor when helping Risto and Kirsi to act on a more developed level than they could have done alone. We can conceptualize this by Zone of Proximal Development (Vygotsky, 1978): A mentor and a novice generated shared understandings, through which the novice internalizes new ways to think. Mentoring emerged as a key issue for changes to take place. Also more generally teacher learning associated with mentoring is seen as beneficial for both the mentor and mentee (Garet, Porter, Desimone, Birman, & Yoon, 2001). This phenomenon manifested itself also in our study: Jari was from the beginning as a role of a leader, because he was chosen the tutor of mathematics among the whole exercise group. Jari could have succeeded well during SD 2 practice also alone. On the other hand, his role as the leader of the group seemed to lead to a social pressure, which influenced his actions and reflections. Also Meri got a high status inside the group on the grounds of her wide teaching experience. Kirsi and Risto had an authority on the grounds of their good mathematics proficiency but in the group they were in a novice role, due to their meagre teaching experience as compared to Meri and Jari. Kirsi, Meri and Jari were more strongly engaged with the collaboration than Risto. So they also benefitted more from the other students than Risto. Besides mentoring among the peers, the role of a leader seems to be an important factor for changes to take place. It's important that s/he is convinced of the necessity of the change, and that he internalizes the change goals, although other members' beliefs and dispositions do also matter, and it's important that all become aware of the need of the change. On the basis of the results it seems necessary to take into account the following issues: Firstly, it is possible that student teachers may give in the interviews a more positive view of collaboration than what their collaboration really was (cf. Kaasila, 2007). The video-taping of the student teachers' co-planning sessions could have enlightened the character of the collaboration and the role of each student more clearly. On the other hand, we were able to see the group dynamics to some extent; e.g., that there were some tensions within the group. We see that these tensions give a more authentic view of student teachers' collaboration than the interviews alone did. We also used different data gathering methods and contexts to confirm our results. In addition, the cooperative teacher read the manuscript and confirmed that the things described here were congruent with her own views about the collaboration of this student group. Secondly, belonging to a community can provide much support, but after the collaboration there is the danger to turn back to traditional way of teaching (Gellert, 2008). Also routine can often be seen as a barrier against change (Gellert, 2008). Reforms can be realised, if teachers become convinced that by innovative efforts they can collectively address problems they meet at school (cf. Lauriala, 1997, 130).

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The collaborative, interactionistic model presented here well described the change process on the group level, although students' starting points influenced on which phase they entered in their professional development. For instance, Risto with his small teaching experience was able to reach only the unfreezing phase, and new learning caused a cognitive conflict in him. Jari and Meri reached the phases were they became convinced of the worth of the new approach, and they also changed their beliefs and practices in accordance with the phases of moving and refreezing. As to the different developmental profiles between the participants, Kirsi's case may raise some questions. Although she had a good view of herself as a mathematics learner, which did not change during the teaching practice, she remained in the side role in the group. However, although she was as inexperienced as Risto, she had an important role in strengthening the group cohesiveness. She also wanted more collaboration. As to Risto's case, he did not comply to the norms of collegiality which seemed to bind the others. How to make this kind of students to develop collaborative disposition and skills? Maybe we must pay more attention to the individual starting points in teacher education; some students need more guidance and time to melt themselves into the community and to learn to trust and engage in genuine collaboration. Our data does not directly support the view according to which differences in student learning and action are gender erelated. Earlier teaching experience and mathematical proficiency seemed to be more important in defining the individual's role and status than his/her gender. From our results we can make some recommendations for teacher education: 1) The relationship between theory and practice is in a central role in teacher education: In this study the contents and timing of mathematics education course had a direct relationship to SD 2 teaching practice. Student teachers had an opportunity to try in teaching practice the principles and methods they had learned during mathematics education course in the university. 2) Improving collaborative practices in teacher education may be enhanced through implementing the following ideas: Teacher educators should convince students of the advantages of collaborative practices by collaborating themselves with each other and giving thus students an example of good collaborating practices and culture. They should also emphasize the decisive role of all members, in building and maintaining a working group, despite their differences; e.g., lack of experience. It is also important to understand barriers to collaboration, that may be previous experiences, expectations of others and self, lack of motivation or fear of failure, which may lead to defensive reactions, as is shown in Risto's case. 3) When grouping student teachers into different classes during teaching practice the advantage of heterogenous groups is that students with different backgrounds and capacities compensate each other's strengths and weaknesses, and provide a basis for social comparison and learning. For instance, if a student teacher has a negative view of him/herself as a (mathematics) learner and/or teacher, he/she can in the group at least partially compensate for his/her poor proficiency in the subject matter by a wider teaching experience and in this way get a higher status within the group Later it could be useful to test our interactionist model of teacher change with several student groups teaching in different classes. It would also be interesting to see if the changes we observed in this study will transfer to the next teaching practices the pre-service teachers’ own work, i.e., how situation dependent collaboration is

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