Frida Harvey
email: frida.harvey@oru.se

Anna Teledahl
email: anna.teledahl@oru.se

Teacher Professional Development and Collegial Learning: A literature review through the lens of Activity System

Keywords: Activity Systems, Collegial learning, Communities of Inquiry, Teacher Professional Development

, is used as a theoretical lens which allows for the visualization of TPD as a complex system. The result indicates that, apart from differences in the level of detail in the description of various features, there are differences in the demands the two models place on teachers. Establishing norms that promote collegial learning, in which critical inquiry is expected, emerged as a critical issue. This highlights the importance of viewing any variant of TPD as a process, in which the functions of features shift. Awareness of this process may prove important in designing and implementing future TPD initiatives.

Introduction

In recent years, there have been calls for raised achievement in mathematics in a number of European countries, of which Sweden is one. In an effort to respond to these calls, the Swedish National Agency for Education (SNAE) launched several teacher professional development (TPD) initiatives, which build on the idea of collegial learning. SNAE describes collegial learning as activities in which teachers collectively and systematically analyse their teaching, in an effort to develop their practice and improve student achievement. Several studies have identified a shift in TPD where a traditional culture of viewing TPD as single occasions of "new knowledge being handed to teachers", is replaced by a new culture (e.g. [START_REF] Villegas-Reimers | Teacher professional development: An international review of the literature[END_REF]. Evaluations of the largest of these Swedish initiatives, Matematiklyftet [the Mathematics Lift], show that teachers, who have participated, have developed their ability to plan, carry out, and reflect on their teaching. What is left to investigate is in what way the various features of the initiative have contributed to this development [START_REF] Österholm | Utvärdering av Matematiklyftets resultat[END_REF]. The question of what it is that makes TPD, and in particular, the kind of TPD which is based on ideas of collegial learning, effective, acts as a starting point for our study. We aim to explore this question through an inquiry into previous research on what makes TPD, in general, effective, but also by turning to a specific theory for collegial learning, Communities of Inquiry (CoI). In order to tackle the complexity of TPD as an endeavour we have turned to Cultural-Historical Activity Theory (CHAT). By using Engeström's model of activity systems (1999) we believe we can create a model of TPD and collegial learning that accounts for multiple dimensions, thus offering an opportunity to understand possible tensions between the two models of which the latter is part of the former.

Purpose and guiding research questions

By viewing TPD as an activity system, and categorize findings from previous research as parts of this system, we believe we can gain a comprehensive understanding of different aspects of collegial professional development. We hope that such insights will contain a problematization of prerequisites for TPD, which may prove helpful in future TPD initiatives. First, we investigate the question: What characterizes effective professional development of teachers? When we have mapped aspects of TPD, we look for differences between these and the corresponding aspects of the particular form of TPD known as Communities of Inquiry. The latter is expressed in the question: What are the possible similarities and differences between general models for effective teacher professional development and Communities of Inquiry?

Theory

Cultural-Historical Activity Theory (CHAT) is a theory that offers both an analytical and conceptual framework to understand human practices. The theory takes a social perspective on human actions and learning, meaning that human activities are understood in relation to their history, culture and context [START_REF] Engeström | Learning by Expanding[END_REF]. Within CHAT, [START_REF] Engeström | Learning by Expanding[END_REF] has developed a model for analysis, which he calls activity system. An activity system is "the smallest and most simple unit that still preserves the essential unity and integral quality behind any human action" (p. 81). Activity systems have been used increasingly as a research framework in educational research in recent years [START_REF] Gedera | Activity theory in education. research and practice[END_REF]. The activity system consists of six nodes as shown above. The object is the underlying "true" motive for the activity. Subject(s) are the participants who share the object of activity, for example teachers in a TPD-community. Mediating artefacts are the tools, with which the object is achieved. A tool can be a material object but also a procedure. Rules describe the rules and norms, both explicit and implicit, which are shared by the participants in the activity. Community refers to the context to which the subjects belong. This context may also include others, for example those who, in different ways, contribute to a shared object. Division of labour describes both the expressed roles of the subjects and the implicit hierarchical structures. To every activity system there is also an outcome, which can be defined as the change and/or result of an activity. A central aspect of CHAT is the analysis of tension within and between the nodes of the activity system. It is imperative to development and learning, to identify such tensions. (Definitions are a synthesis of [START_REF] Engeström | Learning by Expanding[END_REF][START_REF] Engeström | Activity theory and individual and social transformation[END_REF]. [START_REF] Engeström | Learning by Expanding[END_REF] stresses that an activity system can have any size, it may consist of one single person or an entire society. All those who share the object of an activity are part of the activity system. In educational research, the activity system has been commonly used as an analytical tool to interpret data, foremost for its usefulness in making the context of educational processes visible (e.g. [START_REF] Jaworski | The practice of (university) mathematics teaching: Mediational inquiry in a community of practice or an activity system[END_REF][START_REF] Walshaw | The teacher's role in classroom discourse: A review of recent research into mathematics classrooms[END_REF]. In this article, TPD is seen as one activity system.

Method

In our aim to investigate what research says about effective collegial TPD, we turned to literature in two different ways. 1) In order to identify key features TPD in general, we turned to studies, which implemented a meta-analysis of previous research. Meta-studies offer opportunities, not only for quick and easy access to numerous studies in a specific field, but also to take advantage of a synthesis aimed at highlighting, comparing or critiquing, the most prominent aspects of a particular phenomenon. It is, however, important to consider that meta-studies often lack the nuances and depth, which systematic reviews of existing literature, provide. To partly counter such drawbacks we chose meta-studies conducted by researchers, which we through extensive reading, perceived as prominent researchers in the field of TPD: [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF], [START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conzeptualization and measures[END_REF] and [START_REF] Darling-Hammond | Effective teacher professional development[END_REF]. 2) The concept of TPD encompasses a number of different models for professional development.

Collegial learning is a specific model for professional learning that has gained popularity in recent years. In Sweden, the model is the preferred model in national development initiatives. In order to compare the key features of TPD with research on collegial learning as a specific model for TPD, we chose to focus on the framework of Communities of Inquiry (CoI), as described by [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF], since this can be understood as a particular framework for collegial learning. To further anchor, the general framework of Garrison, in the context of mathematics education we also chose to include Jaworski's article on CoI in mathematics teaching development [START_REF] Jaworski | Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration[END_REF]. Teacher learning and professional development are complex endeavours, which are difficult to grasp. In order to deal with this complexity we turned to Engeström's description of an activity system, consisting of six nodes. Viewing TPD as an activity system, allowed for the mapping of, not only the various concepts involved, but also the relations between these concepts. The activity triangle thus offered a visual, as well as analytical tool, with which we could manage the inherent complexity.

Viewing collegial TPD as one single activity system, which then constituted our unit of analysis, allowed us to explore the literature through the questions; what does research say about the object, the subjects, the division of labor, the mediating artefacts, etc. Salient features were noted in table 1. After having created an image of collegial TPD as a complete activity system, we explored each node looking for similarities, differences and possible gaps in our collective knowledge. Our aim is to provide insights into PD that will prove helpful in future initiatives.

Key features of an activity system for effective teacher professional development

In this section, key features of effective TPD as well as aspects of CoI are organized in relation to the nodes of the activity system. Similarities and differences found in each node are compiled in Table 1, and further presented in more detail under each of the headlines.

Node

Effective TPD

Communities of Inquiry

Subject

Teachers, facilitator Teachers, facilitator

Object and outcome

Teacher

Subject

In all the three meta-studies, TPD initiatives are described as including more than one participant. This can be seen as a reflection of a focus on collective participation. [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF] mentions teachers and facilitators as subjects, while [START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conzeptualization and measures[END_REF] and [START_REF] Darling-Hammond | Effective teacher professional development[END_REF] mention only teachers as subjects in TPD. The teachers, together with a facilitator, are all participants in the framework of CoI [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF]. Facilitators seem to have a more pronounced and central role in CoI than in research on effective TPD.

Object and outcome

Borko ( 2004) has investigated the impact of TPD on teacher learning whereas [START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conzeptualization and measures[END_REF] makes is clear that TPD should increase teacher learning and change practice to be seen as effective. It may be seen as unproblematic to view teacher learning as the object of the TPD activity system. Desimone's reference to a change in practice, however, highlights an uncertainty regarding what teacher learning entails. When it comes to CoI, [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF] argues that learning is manifested in teachers' different ways of thinking, reflecting and acting, which is the result of activities where the focus is on connecting different experiences and ideas, rather than just sharing them. [START_REF] Jaworski | Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration[END_REF] pinpoints this difference by arguing that the teachers, not only change their thinking regarding certain phenomena, but that they develop inquiry as a way of being, they are "taking the role of an inquirer; becoming a person who questions, explores, investigates and researches within everyday, normal practice" (Jaworski, 2008, p. 312). This indicates that the process of inquiry instead of being a means, turns into the object. Teachers who collectively act as inquirers, into their own teaching, have opportunities to tackle didactical and pedagogical challenges in a systematic and effective way.

Mediating artefacts

The mediating artefacts described in research of effective TPD can be categorized into three overall categories; 1) long term activities built on active learning [START_REF] Darling-Hammond | Effective teacher professional development[END_REF][START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conzeptualization and measures[END_REF], 2) content focus related to subject matter knowledge [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF][START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conzeptualization and measures[END_REF] and 3) tasks with relevance for classroom practice [START_REF] Darling-Hammond | Effective teacher professional development[END_REF]. The content focus limits the scope of the TPD, and acts as a guard against diversions and irrelevant questions, and it can therefore be seen as a mediating artefact. [START_REF] Darling-Hammond | Effective teacher professional development[END_REF] argue for the importance of feedback and opportunities for reflection, while [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF] is more explicit regarding the implementations when she proposes that classroom-material such as video recordings, student work and lesson plans, could be the starting point of professional development for teachers. In CoI there are more specific examples of mediating artefacts. For example [START_REF] Jaworski | Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration[END_REF] proposes that the long term activity should be designed as inquiry cycles where teachers plan lessons, act & observe, reflect & analyse and finally get or give feedback on the lessons. This cyclic process, where the questioning of practice is a starting point, can carry on for longer or shorter periods. Such a model offers a concrete tool to use in the TPD. In relation to the second category, [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF] proposes that the content should be focused on sharing, comparing and developing experiences and new ideas through critical discussions. To the third category, Garrison adds that the tasks should be exploratory in character. Jaworski uses the expression "inquiry as a tool" (p. 310) with which she argues that teachers need to use, for example, questions and exploratory methods to practice and develop inquiry as a way of being. What [START_REF] Jaworski | Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration[END_REF] and [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF] propose can be understood as a refinement of the more general categories found in research of effective TPD.

Rules

Norms concerning collegiality seem to be important in effective TPD [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF][START_REF] Darling-Hammond | Effective teacher professional development[END_REF]. [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF] stresses the importance of trust, which enables challenging discussions on teaching, and she argues that this is one of the most important features of TPD. [START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conzeptualization and measures[END_REF] and [START_REF] Darling-Hammond | Effective teacher professional development[END_REF], stress that teachers are expected to be active learners. In CoI, the importance of a critical review of experiences, ideas and results of the teaching, is stressed. This is something that goes beyond the mere exchange of experiences [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF][START_REF] Jaworski | Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration[END_REF]. [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF] argues that norms of inquiry require open and risk free communication, where all participants can share ideas and experiences, without the risk of being ridiculed or attacked. Before such norms, of active learning, critical reviews and inquiry, are established however, they can be considered artefacts mediating learning, rather than part of a set of ingrained expectations. The rules and norms are quite similar between effective TPD and CoI. There is, however, in the case of TPD as well as CoI, a lack of description of how, and by whom, these rules and norms should be implemented and established.

Community

The participants in a TPD are also participants in other contexts where teacher learning and improvement of practice is the motive. Consequently, for TPD to be effective it must be in consonance with teachers' contexts, for example prior beliefs and knowledge [START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conzeptualization and measures[END_REF]. Desimone also mentions the importance of coherence between the content of TPD and for example school reforms. In the same vein [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF] argues that individuals are inseparable from their context. Examples of elements of the social context, mentioned by Garrison, are educational content and communication medium. Aspects concerning community seem quite peripheral in research of both effective TPD and CoI, which indicates that the concept may not be well defined. (2009) and [START_REF] Darling-Hammond | Effective teacher professional development[END_REF], which only mention teachers as subjects, argue that teachers need to be active learners, which could be understood as taking on a role. [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF] describes the role of the facilitator, whose most important task is to foster trust and norms, which enable discussions. All three studies describe several key features such as time, active learning, norms of collegiality etc., but it is not clear, who is responsible for their implementation, or for upholding them, once they are established. According to CoI, the facilitator has a leading role. [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF] argues that the facilitator's task is to design and organize the activities, as well as to keep the conversations focused and exploratory in order to create rich and meaningful discussions. This could entail posing challenging questions, giving feedback or guiding teachers. The teachers are described as inquirers who participate actively in discussions, as well as explore and analyse their teaching practice. Garrison stresses that the teachers should be increasingly independent in the learning process. Although division of labor is described in more detail in CoI, concerning the role of the facilitator, little is said about how the facilitators should work to achieve this trusting, but also challenging, work environment. This points to an area where more research is needed.

Division of labor

Desimone

Discussion

By using [START_REF] Engeström | Learning by Expanding[END_REF] Aspects of effective TPD and CoI, which have proved complex, are mainly those that can be sorted under the nodes: object, rules and, mediating artefacts. Some key features were sorted under more than one node in our categorization. Inquiry is positioned under each of these three nodes. This demonstrates the complexity of teacher learning, but also the flexibility of this framework. As the activity evolves the content of the nodes shifts, and creates a dynamic system. The activity system as an analytical framework has potential to identify different phases of the learning process. The inquiry process is an example of a process, which initially can be seen as a tool to foster changes in the way teachers approach a certain issue. When the tool however has become a part of teachers' natural approach, its function shifts and it can be seen as part of the rules and norms that govern the teachers' interaction. [START_REF] Jaworski | Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration[END_REF] argues that inquiry may be seen as a way of being and therefor can be seen as the object of TPD. CHAT as an analytic lens allowed us to view inquiry as a process, which can be found under the three nodes, object, rules or mediating artefacts, depending on what stage of a TPD initiative we are analysing. Theoretically, there is a considerable difference between viewing a process as a means or as a goal, which could generate what [START_REF] Engeström | Learning by Expanding[END_REF] calls tensions in the activity. Empirically, however, it is natural that you can adopt a process by using it, by living through its different steps.

If the process of inquiry is viewed as a mediating artefact, something that can be used to achieve the goal of learning, then this is the aspect in which we find the greatest difference in detail, between previous research on TPD in general, and CoI. Inquiry, exploration and challenging of experiences are examples of concepts from CoI that refines some of the key features of effective TPD. [START_REF] Jaworski | Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration[END_REF] and [START_REF] Garrison | Thinking collaboratively: Learning in a community of inquiry[END_REF] stress that simply talking or sharing experiences, is not enough for a collegial activity to result in learning. Collective, critical and systematic reviews and analysis of experiences and practice are crucial to building new collective knowledge and collegial learning. We believe this constitutes a tension, between a new collegial culture and a previous, more individually oriented culture. A culture, which no longer equates, passively acknowledging other's accounts of their teaching, with learning that informs and changes one's own practice, requires engagement from the teacher community. It places demands on the analytical and social skills of the participating teachers, on the pedagogical skills of the facilitator, and on the organization, that supports collegial work. In order for the teacher community to meet such demands, teachers as well as school administration need an awareness of the prerequisites for collegial learning that are highlighted in this article. In order for collegial learning to be successful

Figure 1 :

 1 Figure 1: Activity system, as illustrated by Engeström (1999)

  activity system, we have tried to visually represent what research says about effective TPD and CoI.

Figure 2 :

 2 Figure 2: Activity system for effective collegial teacher professional development

Table 1 : Similarities and differences between effective TPD and CoI

 1

		learning, changes in	Better understanding of what is being
		practice	questioned, developed thinking/reflection,
			inquiry is a way of being, connecting
			ideas/experiences and applying new ideas in
			practice. All the above lead to changes in
			practice
	Mediating	Long term activities built on	Questioning of practice or problems in
	artefacts	active learning, content focus	practice as a starting point for critical
		related to subject matter	discussions, sharing, comparing and
		knowledge, task with	developing experiences and ideas through
		relevance for classroom	critical discussions, exploratory working
		practice	methods, inquiry as a tool, inquiry cycle
	Rules	Collective support, trust,	Critical review of experiences, ideas and
		challenging/critical	results of teaching, exchange of experiences is
		discussions, active learning	not enough, inquiry as a way of being, open
			communication
	Community Teachers' beliefs and prior	Social context; for example, educational
		knowledge, school reforms	content and communication medium
	Division of	Facilitator foster norms of	Facilitator designs and organizes the meetings
	labor	critical discussions, teachers	(structure and content), proposes tasks, keeps
		act as active learners	discussions focused, problematizes
			discussions, gives feedback and guides the
			teachers, teachers are inquirers who explore
			and analyse their teaching practice

  , several important norms, such as trust and risk-free, open communication, must be implemented. Collectively asking questions about, investigate, or critically review, one's own practice requires a social environment, in which trust is considerable. Few examples are given on how the teachers and/or a facilitator should foster such norms. This is characteristic for many of the key features presented in figure2. Research offers pointers on what and who, but very little is said about how the processes can be achieved. The processes leading to collegial learning is an area in need of more research.


Einat Heyd-Metzuyanim

Talli Nachlieli
email: tallin@levinsky.ac.il

Opportunities for Adopting a Discourse of Explorations in a Professional Development Setting

Keywords: explorative mathematics instruction, productive pedagogical discussions, professional development, lexical cohesion

We examine the productiveness of discussions in a professional development (PD) towards explorative mathematics instruction (EMI). The artifacts used in the PD session under examination were a video-clip of a classroom discussion, together with the Quadrants coding scheme, a scheme originally developed for research purposes. We define productive discussions by relying on the linguistic tool of lexical chains that point to lexical cohesion. The analysis shows that the use of the coding scheme as an artifact, together with a video clip, offered increased opportunities for surfacing misalignments between the teachers' framing of learning and instruction and the PD leader's framing of it. These misaligned frames, which draw on the Acquisition Pedagogical Discourse, vs. Explorations Pedagogical Discourse, are usually difficult to surface.

Introduction and theoretical background

For the past three decades, calls for more reform, student-centered, instruction, are voiced in mathematics education [START_REF] Gregg | The tensions and contradictions of the school mathematics tradition[END_REF]. We adopt the Commognitive framework, and define such instruction as explorative mathematics instruction (EMI), teaching that provides students opportunities for explorative participation [START_REF] Heyd-Metzuyanim | Opportunities for learning given to prospective mathematics teachers: between ritual and explorative instruction[END_REF]. Efforts to help teachers adopt explorative teaching practices are widespread, mainly through professional development (PD) [START_REF] Sztajn | Research in Mathematics Professional Development[END_REF]), yet the processes of learning these practices have not yet been sufficiently explicated and theorized.

Whereas teachers who teach in a teacher-centered approach rely on an Acquisition Discourse, those who promote student-centered instruction rely on an Explorations Discourse. Our use of Discourse (with a capital D) here comes to signify the historical, social and institutionalized nature of these pedagogical Discourses, which are not just a matter of how teachers and teacher-educators talk, they exist in documents, regulations and curricula. One of the best examples of the Explorations Discourse can be found in "reform" documents such as the NCTM (2000). The Acquisition Discourse does not have such institutional documents as exemplary instantiations, yet it is the common Discourse that is often seen in classrooms worldwide [START_REF] Nachlieli | Ritual-enabling opportunities-to-learn in mathematics classrooms[END_REF]. In documents, it can often be found as the anti-thesis of actions valued by the Explorations Discourse (for example, by stating that teachers should not be using their time in the classroom for 'drill and practice' or that students' ideas are those that need to be heard, not teachers') [START_REF] Heyd-Metzuyanim | Narratives of 'good' instruction: teachers' identities as drawing on exploration vs. acquisition pedagogical discourses[END_REF]. Pedagogical discourse (with a small d)that discourse which includes teachers' talk (inter or intra-personal) about how, what and whom to teach, draws upon the pedagogical Discourses, often in a multi-voiced combination of Explorations and Acquisition Discourses. There is never a teacher (or teacher educator) that talks "purely" in one of these Discourses, but PD leaders or teacher educators that try to move teachers towards more explorative instruction often draw more on the Explorations Discourse, whereas teachers who are initial participants in such PDs often draw more on the Acquisition Discourse.

Acquisition Discourse rests on the assumptions that learning mathematics entails acquiring a certain set of facts and skills, where the role of the teacher is to "dispense" this knowledge, and the role of the student is to "acquire" it. Accordingly, teachers relying on an Acquisition Discourse value the demonstration (by the teacher) and the execution (by the student) of precise procedures. Different realizations of mathematical objects [START_REF] Sfard | Thinking as communicating[END_REF], as well as different procedures for solving similar problems are less valued since they are not necessarily efficient and since they can be a source of "confusion" for the students. In contrast, Exploration Discourse frames learning as a form of participation in a community, where mathematical objects are explored and narratives about them are authored by students, facilitated by the more experienced participant, namely the teacher. Teachers relying on the Explorations Discourse value the presentation of different realizations of mathematical objects, and the authoring of narratives by students, even if those are not yet articulated precisely according to the rules of the mathematical community [START_REF] Heyd-Metzuyanim | Narratives of 'good' instruction: teachers' identities as drawing on exploration vs. acquisition pedagogical discourses[END_REF].

Previous studies have shown that shifting from Acquisition towards Explorations pedagogical Discourse is a difficult and complex process (Heyd-Metzuyanim, [START_REF] Heyd-Metzuyanim | From ritual to explorative participation in discourse-rich instructional practices: a case study of teacher learning through professional development[END_REF]. It has also been shown that this process involves misaligned frames (Heyd-Metzuyanim, Munter, & Greeno, 2018a). Frames are those parts of the pedagogical discourse that relate to interpretations. They are usually tacit, and take form in what people notice, accentuate and foreground. In relations to these frames of classroom instruction, Heyd-Metzuyanim et al. (2018a) differentiated between the subjectifying aspects of frames, namely, those interpretations relating to who speaks in the classroom, and the mathematizing aspects of frames: what mathematical objects and routines are talked about. In their study of a US teacher participating in a PD intended to promote EMI, Heyd-Metzuyanim et al. (2018a) noticed that the teacher was well aligned with her PD coach in relation to the subjectifying aspects of framing the envisioned lesson that they were planning together: they both valued students' discussion, group work and agency. Yet with relation to the mathematizing aspects of the frame, they differed. While the PD coach aimed to expose students to different realizations of the mathematical object at the focus of the lesson, the teacher was focused only on the execution of specific procedures. This misalignment went unnoticed during the coaching session, yet led to the failure of the lesson planned together by the two parties. This case highlighted the need to study what factors can contribute to more productive discussions in a PD setting, which can surface misalignments and provide opportunities for teachers to engage with more implicit aspects of the Exploration pedagogical discourse.

Productive discussions in PD settings have been the focus of increasing numbers of studies (e.g. [START_REF] Borko | Video as a tool for fostering productive discussions in mathematics professional development[END_REF]. Many have pointed to the usefulness of video to increase the productiveness of teachers' conversations as well as the utility of observation-assisting artifacts such as protocols and scoring rubrics to fix teachers' attention on important aspects of instruction [START_REF] Borko | Video as a tool for fostering productive discussions in mathematics professional development[END_REF][START_REF] Schoenfeld | Uses of video in understanding and improving mathematical thinking and teaching[END_REF]. Yet studies examining the usefulness of video and observational artifacts rarely explicate what "productive discussions" actually means. In our specific context of promoting EMI, our assumption is that teachers in the PD setting start out by mostly drawing on Acquisition Discourse whereas the PD leader draws mostly on the Explorations Discourse. This may lead to tacit differing interpretations (frames) that go unnoticed since the participants (teacher and leader) assume they are talking about the same thing while in fact they are framing teaching-learning interactions differently, in accordance with the differing Discourses on which these frames draw. Thus, a productive discussion, in the context of these sessions, would be a discussion that surfaces misalignment between Acquisition frames of teachers vs. Exploration frames of the PD leader.

We turn to the notions of lexical cohesion [START_REF] Halliday | Cohesion in English[END_REF] and lexical chains [START_REF] Morris | Lexical cohesion computed by thesaural relations as an indicator of the structure of text[END_REF] to conceptualize productiveness of discussion for our purpose. Cohesion is a semantic term that refers to the meanings that exist within the text [START_REF] Halliday | Cohesion in English[END_REF]. Cohesion occurs where the interpretation of some element in the text is dependent on that of another. As argued by [START_REF] Morris | Lexical cohesion computed by thesaural relations as an indicator of the structure of text[END_REF], lexical chains help determine coherence and discourse structure, and hence the larger meaning of the text. We adopt the idea here to study the cohesion of the text that was produced by the teachers and instructors during the discussions in the PD and thus to learn about the productivity of the pedagogical discourse by visually showing the topics discussed and followed and the specific words chosen by different participants. Our research goal is thus to characterize productive discourse processes that promote pedagogical discourse of explorations by examining discourse cohesion and its relation to the surfacing of misaligned frames.

Method

The study took place in the context of the TEAMS PD program (Teaching Exploratively for All Mathematics Students), a program that included 16 4-hours sessions over two years (2016)(2017). The PD introduces teachers to the 5 Practices for Orchestrating Mathematics Discussions [START_REF] Smith | 5 Practices for Orchestrating Productive Mathematics Discussions[END_REF] as well as to Accountable Talk™ tools [START_REF] Heyd-Metzuyanim | From ritual to explorative participation in discourse-rich instructional practices: a case study of teacher learning through professional development[END_REF]. Teachers participating in this program were relatively experienced (ranging from 15-31 years of teaching). 14 of the teachers taught in middle-school and 2 taught in elementary school. 9 teachers continued from the 1 st to the 2 nd year. At the end of the first year, we analyzed the first year data using the Quadrants coding scheme (QCS) [START_REF] Stein | Using Theory and Measurement to Sharpen Conceptualizations of Mathematics Teaching in the Common Core Era[END_REF]. The analyzed data included 3-4 videorecorded lessons of each of the participating teachers. Our analysis revealed that teachers were adopting explorative instructional practices to varying extents. In particular, our findings showed that teachers had trouble to maintain Explicit Attention to Concepts (EAC) whenever they raised Students Opportunities to Struggle (SOS). Through the process of analyzing teachers' videos according to the QCS, we came to the realization that such a process may be beneficial for the teachers as well. Therefore, we chose to open the second year of the PD with introducing the QCS and the findings of our analysis.

For the current study, we watched and partially transcribed all 16 PD sessions. Partial transcriptions were intended to capture the activities of the session, as well as important conversations and themes. Based on these partial transcriptions, we focused on the first session of the 2 nd year since it was a pivotal session in several aspects. First, it was where we made our first use of the QCS [START_REF] Stein | Using Theory and Measurement to Sharpen Conceptualizations of Mathematics Teaching in the Common Core Era[END_REF] together with video, in attempt to assist teachers to better grasp our vision of EMI. We thus chose to present two rubrics (EAC and SOS) to the teachers, together with a de-identified video of one lesson which we thought was particularly rich in opportunities for EMI. The rubrics each include 4 or 5 levels with 1 being lowest. Presentation of the full rubrics is beyond the scope of this report but the relevant parts for the reported discussion will be detailed in the findings section.

Viewing this session at the end of the year, after all sessions were observed and partially transcribed, revealed that this indeed was a session where intense pedagogical reasoning occurred, yet only in specific episodes. To better understand these occurrences and the discursive processes underlying them, we fully transcribed the session, after which we chose two clips from the discussion. These clips were similar in that they both referred to the QCS (in particular, to the EAC rubric) as well as included some "example" of instruction to talk about, yet different in that one of them was before the video was shown and the other after it. This offered us an opportunity to examine more closely the role of the video, together with the QCS, in the rising level of productiveness of the talk.

Findings

The session that we chose to focus on began with Talli (2nd author and leader of the PD) presenting to the teachers the results of the 1 st year video analysis, in order to launch the main focus of the 2nd year PD plan: to work with teachers on how to preserve conceptual depth while letting students author mathematical narratives and struggle with problems on their own. Talli then presented the teachers two rubrics from the QCS: the EAC and SOS rubrics. Teachers were asked to read the rubrics in groups, discuss them, and were then called for a whole-group discussion. She opened by stating that there are no simple criteria for determining if the EAC level is 4, 3, 2 or 1 and that there could be "significant arguments" around them, but that those arguments are important because "they help us notice nuances".

The discussion started in rather unproductive ways. One teacher, Rony, was asked, based on her former experience with coding videos according to the Quadrants scheme, to give an example of a "level 4" of EAC. She tried doing so, yet other teachers related more to the specificities of her example and her choice of concept words, than to levels of the EAC rubric. The lexical chain analysis performed on this episode revealed that the EAC lexical chain died off right after Rony's description of her envisioned lesson and was not picked up by other teachers. Talli cut off this discussion quite promptly, as she was sensing the teachers were not relating to the subject that she considered to be the goal of the discussion. Einat (1 st author) tried giving examples of lessons that were coded as '4' vs. lessons that were coded as '3', yet these were not very productive in eliciting lexical chains focused on elements from the EAC rubric either.

Following that, Talli announced that they will be seeing a 16-minute video-clip of an 8 th grade classroom discussion around a ratio problem. She first gave out the task so that teachers got an opportunity to solve and discuss it shortly before viewing the video. The task read as follows:

Dafna prepared a necklace of blue and red beads. Two fifths of Dafna's necklace were blue and the rest were red. Draw four different possible necklaces.

Following the viewing of the clip, Talli opened the discussion:

Talli

Let's start with the attention to concepts 2.2 Valerie I did not see here much attention to concepts, by the way… attention to concepts… there wasn't much. I hardly wrote any concepts here (on her "scoring" worksheet).

Talli

Which concepts? 2.4 Lena But attention to concepts needs to be at the beginning of the lesson? Already? They just started… 2.5 Veronica But we didn't watch the end of the lesson, when the teacher summarizes all the things that appear on the board. …She called to the board all sorts of student groups so that they show what they-…. And she wanted actually, on the board, to show all sorts of ways, all sorts of different answers, so that other kids from the different groups see what others were thinking. I think that what we are not seeing is the end (made by) the teacher, that should be at the end of the lesson, after all the answers are shown on the board. We see the discourse in this excerpt as productive, since all speakers repeatedly relate to the same lexical chain, that of EAC (underlined). We also see all teachers relating repeatedly to the videoclip (in wavy underline), referring to actions observed in it. This shows us the productiveness of the video as a tool for supporting the cohesion of the talk, as well as for focusing the teachers on EAC. The lexical chain of EAC remained active for 17 turns. Within it, some teachers objected to Valerie's assertion that there was not much attention to concepts. For example:

3.1 Yafit I think that actually there is attention to concepts because, as someone says the answer, she repeats things, and then she asks them "do you agree? Disagree?" It's a type of, it's also a type of explication of a concept. Or clarification. Attention to concept does not have to be writing the concept on the board. Also, along the way they said, someone there said the word ratio, (quoting the teacher) "oh, you talked about ratio, what did you mean?" And then he, like, so I think it's also a type of attention to concepts. As the discussion continued, some ideas for concepts that would have justified a higher scoring of the EAC rubric started being voiced by the teachers. These included "ratio", "expansion", "part whole relationship" and "correspondence". Talli revoiced and summarized these ideas, and then she tried eliciting other ideas. She asked 3.2 Talli What else do you have to say about the… what we saw here? What other concepts she-, what did the teacher do with these concepts? Talli's intention here was to stir the conversation to several mathematical ideas that she thought were prominent in the clip, yet none of the teachers seemed to notice them. In particular, she was frustrated by the teachers' lack of appreciation of the students' ideas, and most importantly one idea which related to the routine of dividing the blue beads to two, finding out the basic unit, then multiplying it by 3. The teachers only noticed the routine (employed by the first two students in the clip) of expansion: multiplying both sides of the ratio by a constant number. Despite her questions, the teachers only repeated the concept names (expansion, part-whole) raised before. As Talli was ready to give up and stir the conversation to the issue of scoring the clip, Valerie said:

3.3 Valerie The fifth, also 3.4 Talli

The fifth, what? 3.5 Valerie I don't know, the fifth disturbs me here

Talli

In what way does it disturb you?

Valerie

Because a fifth of what? Of the blues? The reds? Which fifth (is it) here? 3.8 Naomi

The fifth of the group (Yafit: The fifth as a part, a part) 3.9 Valerie But it's a split group. The group is split. Which fifth are (they) talking (about)? The 1 out of 5 blues? Of the reds? Now, a lively discussion revolved with multiple lexical chain entries, focused on the "fifth". Veronica stated it was "once two fifth, once three fifths", adding that "it's dividing by two or dividing by three. It depends what they want the fifth (for)". One teacher, Lena, got confused by all this discussion of finding "fifths". She said 4.1 Lena But we already know how many reds and how many blue there are. For Talli, this was a good opportunity to highlight one of the student routines she saw as most significant in the video clip:

4.2 Talli

No, but, one student came and said another idea. She said, listen, "you want to create more necklaces? If you tell me how many blue (beads) there are, I will already tell you how many reds there are … I will take the blues, I'll divide them by two because … all the blues are two fifth, I will find this one part. The other part I will multiply by three, to find the red (ones)". That's the fifth. That's the idea of the fifth 4.3 Lena Oh, OK. So that's a different division.

Lena's short exclamation is an important indicator of the productiveness of this exchange, where the lexical chain around the "fifth" was prominent, along with the lexical chain reporting on students' speech in the video. Although there are good reasons to believe most of the teachers did not see the importance of the students' routines of division in search of a "unit", Lena's exclamations showed it most clearly. Her first request for clarification [4.1] made use of the "how many" quantification word. It is therefore highly probable that she was only seeing the routine of multiplying the two sides of the ratio (as this routine relates to natural numbers, 2 and 3, multiplied by a coefficient of expansion). In addition, her use of "we" in 4.1 for objecting on what "we already know", indicates she was not making any differentiation between her own mathematical reasoning and the students' mathematical reasoning. In contrast, when she used the word "division" after Talli's explanation [4.3], Lena was probably already seeing the routine of dividing into fifth and using the fifth as a unit (since only with relation to this routine, the word "division" makes sense). Also, Lena's use of "that" indicates that she was now differentiating between her own mathematical reasoning and that of the revoiced student. This change may have resulted from Talli's repeated reference to the voice of the student (seen in the I pronouns of the video actions lexical chain).

Misalignment of frames

The productiveness of talk, which rose considerably once the video and coding scheme were introduced to the discussion, made certain misalignments between Talli and the teachers more apparent. Interestingly, Talli was able to pick up in real time only on the mathematical aspects. The subjectifying aspects remained hidden until the analysis was performed.

The misalignment of the mathematical aspects of frames was most noticeable in the discussion of the "fifth" and in what the teachers saw as important mathematical narratives versus what Talli saw as important. Talli interpreted the video clip as containing rich stems for discussion of the ratio object, including visual realizations of 2 different "necklaces", the fractions 2/5 and 3/5, the signifiers 2:3 and some verbal statements describing routines for calculating the ratio. She therefore thought the clip should be scored as a 3, where, according to the EAC rubric One or more concepts are discussed and/or defined in some detail. This entails explicit noting of the concept in a whole-class setting. Explanation and/or elaboration of critical features of the concept may be incomplete, and connections to the larger web of mathematical ideas … are notably present, but weak (EAC rubric level 3, the Quadrants coding scheme).

In contrast, the teachers hardly saw any "concepts" in the clip. It seems that for them, "concepts" meant keywords on the board, together with explicit explanations made by the teacher. They were not searching for stems of different realizations of the ratio object. Talli therefore insisted on eliciting more ideas about "concepts", until finally the "fifth" was raised by Veronica [3.3].

The misalignment of frames in the subjectifying aspect was more nuanced. It appears only when one looks closely at the pronouns relating to the performer of actions (bolded) in the lexical chains referring to the video (wavy-underlined). While Talli talked about routines that surfaced in the discussion, focusing mainly on those narratives authored by the students, the teachers almost solely focused on the teacher. Veronica [2.5] as well as Yafit [3.1], even while "defending" the lesson as containing more attention to concepts than Valerie claimed, still based their justifications on the actions of the teacher ("She wanted to show", "she repeats (what the students has said)", "she asked"). This focus on the teachers' actions, both for evaluating what did happen and what did not happen in the lesson in terms of EAC, was prevalent in most of the teachers' talk.

Discussion and Conclusions

This paper's goal was to characterize productive discussions in PD settings that promote EMI, by surfacing misalignments between frames of the PD leader, who was drawing on the Explorations Discourse and frames of the teachers, who were relying on the Acquisition Discourse. Our analysis of the PD pedagogical discussions included identifying lexical chains, both those relating to the rubric of EAC and those relating to the video-clip. This analysis showed that by using a video-clip of a classroom discussion and the Quadrants rubrics, misalignment between the teachers and the PD leader surfaced, and could therefore be referred to and further discussed. The misalignment in mathematizing aspects of the explorative frame, which are often difficult to notice (Heyd-Metzuyanim et al., 2018a), surfaced in this discussion well. However, subjectifying aspects of misalignment that related to whose contribution is valued (students or teacher), remained hidden during the discussion, perhaps because they were not attended to in advance by the PD leader. Our findings strengthen earlier studies regarding the usefulness of using video together with rubric scores in a PD setting (e.g. [START_REF] Schoenfeld | Uses of video in understanding and improving mathematical thinking and teaching[END_REF]. The contribution of this study is also methodological, in introducing and exemplifying the use of linguistic tools to analyze the productiveness of PD discussions.

Acknowledgments

This study and the TEAMS project are supported by the Trump foundation, grant no. 220.


Lara Huethorst
email: lara.huethorst@math.tu-dortmund.de

Algebraic solutions of German out-of-field elementary school teachers

Keywords: Professional development, teacher training, out-of-field teaching, algebraic thinking

In Germany out-of-field elementary school teachers are a frequent phenomenon but not regarded as unproblematic. In order to foster the pupils' algebraic thinking, teachers themselves need to be aware of the structures. Additionally, tasks which foster process-related competences too can only be set when the teacher knows which discoveries can be made by using a specific task. In this article theoretical background is given to process-related competences, teacher training and out-offield teaching at German primary schools. A teacher training course is presented and first insights into the algebraic solutions of the participating teachers are given.

Introduction

In Germany, out-of-field teaching, which is to be understood as trained primary school teachers who have been trained in different subjects than mathematics, exists fairly frequently in primary schools due to terms and structures of the first and second phases of teacher education. As a result, a national study points out that nearly one third of all interviewed primary school teachers, according to their own statements, teach mathematics outside their area of expertise [START_REF] Richter | Aspekte der Aus-und Fortbildung von Lehrkräften in der Primarstufe[END_REF]. Hence, teacher training courses are needed to support teachers in teaching their daily mathematics courses. Therefore, this paper presents a teacher training course for out-of-field primary school teachers with the focus on usage of task formats to foster process-related competences.

Firstly, process-related competences are described and why out-of-field teaching cannot be regarded as problem-free. In a second step, the design of the study is described. Thirdly, the teacher training design is presented to lead to the fourth part of giving insights into how out-of-field mathematics teachers themselves solve arithmetic tasks.

Theoretical background

There are several widespread theoretical topics relevant to fully acknowledge the key findings of the study which are briefly described in what follows below.

Process-related competences

In the German framework, agreements for elementary schools, math competences are divided into two different but intertwining kinds. On the one hand, there are content-focused competences, including numbers and operations; space and shape; data; frequency and probability; and sizes and measuring. On the other hand, there are so called process-related competences. They include problem-solving, arguing, communicating, presenting and modelling (KMK, 2004). By now, the latter are found in almost every curriculum in all German federal states. They are considered to be indispensable for a successful usage, acquisition and implementation of mathematics [START_REF] Kmk | Bildungsstandards im Fach Mathematik für den Primarbereich (Jahrgangsstufe 4)[END_REF].

The following example explains that in more depth: numbers, which can be chosen freely from all natural numbers including 0 (they can also be identical or different). This task is one of those which "have the potential to address algebraic thinking" (Steinweg, Akinwunmi, & Lenz, 2018, p. 283). Not only (early) algebra can be fostered with number-chains but processrelated processes as well. The children need to get adequate assignments and support. An assignment which fosters the competence of arguing is, for example, one which asks the students to ( 1) guess what happens when the first start number is increased by one, (2) calculate four examples and test their guess, (3) derive a rule from that and (4) give justification for the rule. It is important to describe and generalize the finding so that the children acquire knowledge about connections and relationships between numbers. [START_REF] Eichholz | Mathematik fachfremd unterrichten[END_REF] showed in her study on out-of-field teachers in German elementary school that some of the participating teachers show little knowledge about these process-related competences. This indicates that there is a necessity to have a closer look at how teachers can be supported in learning and teaching general mathematics competences.

Out-of-field teaching in German primary schools

Due to the federal education system in Germany, study regulations vary widely in the 16 federal states. Although non-obligatory framework agreements state that studies at university to become a teacher comprise content knowledge and pedagogical content knowledge in both German and mathematics and an additional third subject (KMK, 2013), this is not the case in all states. For an overview on whether the education in the first and second phase of teacher education brings forth more generalists or specialist see [START_REF] Porsch | Spezialisten oder Generalisten? Eine Betrachtung der Fachausbildung von Grundschullehrerinnen und -lehrern in Deutschland[END_REF]. The principle of class teacher [START_REF] Porsch | Fachfremd unterrichten in Deutschland. Definition -Verbreitung -Auswirkungen[END_REF] is dominant in German elementary school so that out-of-field teaching occurs frequently. However, there are enormous variations in percentages of teachers who teach mathematics without being fully educated from about 1.5% to almost 45 % in the different federal states [START_REF] Richter | Aspekte der Aus-und Fortbildung von Lehrkräften in der Primarstufe[END_REF].

Nevertheless, it is widely known that the competences of a teacher have considerable impact on the competences of their pupils. COACTIV, a German wide additional study to PISA, for example, showed that both content knowledge and pedagogical content knowledge of a teacher have an impact on students' performances [START_REF] Baumert | Das mathematikspezifische Wissen von Lehrkräften, kognitive Aktivierung im Unterricht und Lernfortschritte von Schülerinnen und Schülern[END_REF]. There is not only a high correlation between content knowledge and pedagogical content knowledge, content knowledge also proved to be mandatory for pedagogical content knowledge [START_REF] Krauss | Die Untersuchung des professionellen Wissens deutscher Mathematik-Lehrerinnen und -Lehrer im Rahmen der COACTIV-Studie[END_REF]. Moreover, "[m]ore traditional beliefs were associated with more traditional practices" (Stipek, Givvin, Salmon, & Mac Gyvers, 2001, p. 221), such as emphasis on performance or speed, which contradicts the contemporary views on how a good mathematics class should look like. As [START_REF] Porsch | Unterscheiden sich Mathematiklehrkräfte an der Grundschule mit und ohne Fach-Lehrerbefähigung hinsichtlich ihrer berufsbezogenen Überzeugungen? Ergebnisse aus TIMSS[END_REF] pointed out, out-of-field teachers agree less often with constructivist views and therefore, according to selfdisclosure, realize them less often. These results imply that there is a need to focus research on out-of-field teaching in German primary schools and its consequences. However, there are quite contradictory results in German research. While a nationwide investigation showed that elementary school students have poorer scores when being taught by an out-of-field teacher, which is especially true for the weakest 5% [START_REF] Richter | Aspekte der Aus-und Fortbildung von Lehrkräften in der Primarstufe[END_REF], reanalysis of the same data indicates opposing results. [START_REF] Ziegler | Der Einfluss fachfremden Unterrichts auf die Schülerleistung: Können Unterschiede in der Klassenzusammensetzung zur Erklärung beitragen[END_REF] state that other reasons are responsible for the comparably weak performances of these pupils. The fact that children with weaker cognitive skills and migrant background experience more out-of-field teaching could be an explanation for the first result. Nevertheless, it needs to be investigated why there are such differences in out-of-field teaching concerning that group of elementary school pupils.

(Pre-)Algebra in German primary schools "Generalizations are the life-blood of mathematics" (Mason, Burton, & Stacey, 2010, p. 8). However, "German primary curricula and standards mention studies on algebra in a very limited way, if at all" (Steinweg, Akinwunmi, & Lenz, 2018, p. 284). Nonetheless, problems like little conceptual knowledge of variables are as present in Germany as they are in other countries (ibid.). This does not mean that there are no opportunities in German elementary school to foster algebraic thinkingbut it stays implicit: "The objective is to encourage teachers to integrate algebraic thinking into their classroom. Moreover, the aim is to enable teachers to become aware of their already addressing algebraic thinking" (ibid., p. 286). The presented part of the study focuses on the algebraic thinking of teachers based on the assumption that teachers cannot set adequate tasks to foster algebraic thinking if they have not completely understood the mathematical structures themselves. This knowledge provides them with the opportunity to concentrate the pupils' focus on interesting patterns and structures.

Teacher training

"Professional teachers require professional development" (Wilson, & Berne, 1999, p. 173). That there is a need for teacher training is commonly known, so various research results can be used to develop and legitimate the presented teacher training course. There are five core features which can be identified in most research literature: (1) duration, (2) content-focus, (3) active learning, (4) collective participation and ( 5) coherence (i.e., [START_REF] Desimone | Improving impact studies of teachers' professional development: Towards better conceptualizations and measures[END_REF]Garet, Porter, Desimone, Biram, & Yoon, 2001). Derived from these core features of teacher training, the German National Center for Mathematics Teacher Training (Deutsches Zentrum für Lehrerbildung Mathematik, DZLM) has developed six design principles: participant orientation, competence orientation, diversity of teaching and learning, case study, stimulating cooperation and stimulating reflection [START_REF] Barzel | Die DZLM-Gestaltungsprinzipien für Fortbildungen[END_REF]. The following project is affiliated to the DZLM, however, elaborating on the six DZLM design principles is not possible in the scope of this paper (see [START_REF] Barzel | Die DZLM-Gestaltungsprinzipien für Fortbildungen[END_REF].

Design of the study

"Practice, at least in education, requires a cyclic alteration of research and development" (Freudenthal, 1991, p. 159). That is why this study was performed cyclically and was constantly revised. Prediger's working group has developed a Design Research Cycle that is specifically adapted "for teachers with a focus on content-specific professionalization processes" (Prediger, Schnell, & Rösike, 2016, p. 97). The cycle is divided into four working areas which are "(a) specifying and structuring PD goals and contents in hypothetical intended professionalization trajectories, (b) developing the specific PD design, (c) conducting and analyzing design experiments in PD settings, and (d) developing contributions to local theories on professionalization processes" (ibid., p. 98). The examined course was performed three times, with results from the first two courses being presented.

To gain insights into the level of knowledge and the beliefs of the participating teachersabout 25 in the first two circlesbefore and after the course, several research tools were used. At the beginning of the first lesson and in the last there were two different questionnaires. The first examines beliefs of how mathematics is learned, how it must be taught, the teachers' self-concept regarding the process-related competences and other related features. These items are all from tested questionnaires which fulfil common quality criteria. The second questionnaire, which was designed by the author, is about how the participants themselves solve two different elementary school tasks. An operative change must be described in a first step. In a second step, the changes must be explained at elementary school level and, thirdly, the participants are asked to give an algebraic explanation. Each participant generates his or her personal code, so that the questionnaires stay anonymous but can be matched. Data evaluation is made mainly qualitatively by comparing pre-and posttest results and (possible) changes in beliefs. However, categories have been created, to allow statements about trends. Additionally, semi-standardized interviews with volunteers were conducted. The focus is on the questionnaire about the elementary school tasks to gain answers to the research question: How far do the algebraic solutions of out-of-field teachers primary school teachers, who participate in a training course with a focus on tasks (formats) to foster the processrelated competences, change?

Design of the teacher training course

Based on the core features of teacher training and the DZLM design principles, the constructed course is presented briefly, before one exemplary period is described in more detail.

There are five appointments which last three hours. For each meeting one typical elementary school task format is combined with one process-related competence (except modelling), so that the participants gain knowledge in two different ways. On the one hand they are supposed to completely understand the task on a deeper mathematical level. On the other hand, they should be able to use their (newly gained) knowledge to set assignments which foster the process-related competences as well as the content-related ones (with help from Primakom, a platform for out-offield teachers on which they can undertake trainingregardless of point or period of time or place).

Topic of the appointment

Primakom pages 1 Fostering process-related competencesdocumentation with the example of discovery calculation packs Between two appointments there are always approximately four school weeks, distance phase, so that the participating teachers have the opportunity to try out what they have acquired in the presence phase beforehand. Reflecting on one's own lessons can be enabled only when following this concept of duration, which is essential, according to Hippel (2001), when wanting to take the step from professional knowledge to professional action. In order to provide support during the distance phases, each topic is supplemented with one or two pages of an internet site (https://primakom.dzlm.de). The content focus is realized through the concentration on task formats and process-related competences. This covers the point of coherence, since these tasks can be found in almost every elementary school mathematics textbook allocated to different grades and are adaptable for all grades so that every participating teacher can use what is learned during the presence phases in their own class in the distance phases. To satisfy active learning, the participating teachers solve assignments with the focus on algebraic structures of the task. Additionally, they are asked to illustrate their findings using elementary school appropriate material, for example sachets, and variables. In the questionnaire about the teachers' solutions of elementary school tasks, in this example number-chains, the teachers are asked to describe operative changes in number-chains, to give an explanation where these changes derive from what they could use in their elementary school classes and to give an algebraic explanation. It becomes clear that the participating teachers differ widely in their explaining and algebraic competences at the beginning of the course. Two examples, in each case of the same participants, of the preand post-tests for the following tasks are presented in what follows below: "Give an algebraic explanation (with variables) for the changes of the last number. Also give a reason for your algebraic explanation". As shown in table 1, numberchains have been discussed during the course. In contrast, the other task in the questionnaire has not been talked about during the course. The first example of the pre-test illustrates that some conventions are known, for example, that variables are represented by small letters and often start with a. The explanation is, "When I add to a two numbers b + c, I get the final number d. If I increase both numbers by 1, the final number increases by two". But the formation rules do not become clearer by the variables because they do not show the connections between the two starting numbers and the final number. That is why it cannot be counted as an explanation for the increase of the final number. But it has been recognized that the second starting number has increased by one, the third number as well and the final number by two. Nonetheless, the causal relation to the formation rules is not established. In the post-test, it can be seen that some other mathematical conventions have not been noted, such as that the notation of two variables without an operation sign stands for multiplication instead of addition. In the teacher's explanation, however, it becomes clear that she knows it is addition: "In the addition of two variables b is taken twice, so that the result is increased by two times b". This could be because, during the course, the elementary-school-appropriate representation to support the pupils' generalizing competences are little sachets and no operation sign is needed in this case.

First results

Teacher B tries to illustrate the formation rules similar to teacher A. She describes: "The first number-chain consists of (in the little chart) a: starting number, b: 2nd number, c: 3rd number, d: final number. The other numbers need to change". This, again, is no illustration of the formation rules of the number-chains. But the changes are marked by the numbers in the circles, although this is not highlighting why these changes occur. In contrast to the pre-test, the post-test shows a different result for teacher B as well. She highlights the 2b by circling and underlining and explains "When b is increased by one, the final number is increased by two because b is two times in the final number". The formation rule gets evident through the used variables, but she does not use numbers to illustrate the changes. Nonetheless, she explains that because of two b in the final number, by whichever number b is increased, it is doubled in the final number. This is, in contrast to the pre-test, a very good algebraic explanation.

Both examples show that there is a growth in algebraic competences for these two teachers. While in the beginning the formation rules cannot be expressed algebraically, the post-test shows that they can illustrate the changes and the reason for this case after the course. Their explanations change from being descriptive to being explanatory. This tendency emerges in a first analysis for the elementary school appropriate explanation as well as for the algebraic explanation. Before the course, the changes were described, but the answers of the participating teachers were not fully coherent and explanatory. This is only true for some teachers. A few participants were able to express the reasons for the operative changes in the pre-test as well. However, there is a very rare, in this case only one, questionnaire which is categorized as being fully explanatory in both questions (elementary school appropriate and algebraic). Whether the teachers were not able to see where the changes derive from or whether they could not express it properly, either with words or variables, cannot be answered.

Conclusion and outlook

This shows that there is a need for teacher training courses that foster the teachers' algebraic competences. They are often able to name the changes and to describe them, but they are not able so explain why these changes occur and where these changes derive from. In order to foster students' algebraic thinking and use the (full) potential of a task to foster content-focused and process-related competences at the same time, teachers need to be aware of the mathematical patterns and structures of such a task. Teachers need to be able to decide whether a student's explanation is sufficient; teachers themselves need to know what is important to explain and not only to describe a process. In a next step, the other task of the questionnaire, which has not been discussed during the course, will be analyzed further. This may give insights into how far the participating teachers are able to transfer what is learned to other tasks. The solutions for both task formats will be categorized so that more than trends can be named. Additionally, the interviews will be considered to underline detailed analysis of some participants. Further investigations on design principles for teacher training courses and possibilities to support teachers need to be done.

Figure 1 :

 1 Figure 1: Number-chains Number-chains follow the rules of the Fibonacci sequences. The two first numbers are startnumbers, which can be chosen freely from all natural numbers including 0 (they can also be identical or different). This task is one of those which "have the potential to address algebraic thinking"(Steinweg, Akinwunmi, & Lenz, 2018, p. 283). Not only (early) algebra can be fostered with number-chains but processrelated processes as well. The children need to get adequate assignments and support. An assignment which fosters the competence of arguing is, for example, one which asks the students to (1) guess what happens when the first start number is increased by one, (2) calculate four examples and test their guess, (3) derive a rule from that and (4) give justification for the rule. It is important to describe and generalize the finding so that the children acquire knowledge about connections and relationships between numbers.

Figure

  Figure 2: Operative changes in number-chains

Figure 4 :

 4 Figure 4: Post-test teacher A

Table 1 : Topics of the Appointment

 1

	 Mathematics -more than
	calculation


Ronnie Karsenty
email: ronnie.karsenty@weizmann.ac.il

Yochai Peretz

Einat Heyd-Metzuyanim

From judgmental evaluations to productive conversations: Mathematics teachers' shifts in communication within a video club

Keywords: Video club, school-based professional development, peer-discussions

The study reported here explored how secondary mathematics teachers, participating in a schoolbased video club, communicated with each other and with the facilitator along the different sessions of the club. Specifically, their evaluative comments were analyzed, with respect to the nonjudgmental norms that this club aimed to nurture. Three types of evaluative comments were identified, reflecting varying degrees of teachers' capability to interpret and discuss observed teaching moves while attributing possible rationalizations to the filmed teacher's decisions. It was found that the amount of highly judgmental comments decreased as the club proceeded, allowing for a more productive communication.

Introduction

Video club is a form of professional development that has become known since the early 1990's [START_REF] Sherin | New perspectives on the role of video in teacher education[END_REF][START_REF] Sherin | Effects of video participation on teachers' professional vision[END_REF]. In a video club, a group of teachers meet on a regular basis, usually under the guidance of a facilitator, to watch and discuss classroom video selected according to a certain aim. Different video clubs vary in their goals, agendas and the frameworks employed to direct teachers' conversations. For example, a video club can be established around the aim of enhancing teachers' proficiency to notice and analyze students' mathematical thinking (Sherin, Jacobs, & Philipp, 2011;[START_REF] Van Es | Mathematics teachers' "learning to notice" in the context of a video club[END_REF]. Another case is a club focusing on a certain teaching proficiency, such as running class discussions [START_REF] Coles | Using video for professional development: A case study of effective practice in one secondary mathematics department in the UK[END_REF]. Video clubs can be schoolbased, or alternatively region or university-based, but in general groups are meant to be small enough (usually 5-15 participants) to allow for genuine peer-conversations and sharing of ideas and insights.

One of the core issues associated with conducting video clubs is establishing norms of discussion. Apart from the basic intent to ensure a climate of trust, where teachers feel safe to raise ideas, norms also "orient individuals to one another, as well as to how information is communicated, what constitutes an idea worthy of investigation, and how to make sense of one's practice" (van Es, 2009, p. 104). Different video clubs may embrace different norms, according to their focus, goals and agenda. However, there is an apparent consensus in the literature that teachers' comments slipping into evaluations, judgments or negative reactions, lead to discussions with little value [START_REF] Coles | Using video for professional development: A case study of effective practice in one secondary mathematics department in the UK[END_REF][START_REF] Jaworski | Video as a tool for teachers' professional development[END_REF][START_REF] Van Es | Mathematics teachers' "learning to notice" in the context of a video club[END_REF].

In this paper we report on a study conducted within one video club, in which we explored the types of evaluative comments made by teachers and followed their frequency along a sequence of sessions. In the following, we describe the context of this club and the emergent research question, present a brief account of the data collection and analysis means, and demonstrate central findings.

Context, rationale and research question

The video club studied was conducted as part of a professional development project named VIDEO-LM (Viewing, Investigating and Discussing Environments of Learning Mathematics), developed at the Weizmann Institute of Science in Israel. The project is aimed at promoting secondary mathematics teachers' reflective skills, as well as their mathematical knowledge for teaching. A large collection of videotaped lessons serve as learning objects and sources for discussions with groups of teachers within video clubs. The videos are taken, in a sense, as "vicarious experiences" which allow for indirect exploration of one's own perceptions on the practice of mathematics teaching, through the observation of unknown peers in action [START_REF] Arcavi | Using the unfamiliar to problematize the familiar: The case of mathematics teacher in-service education[END_REF][START_REF] Karsenty | Professional development of mathematics teachers: Through the lens of the camera[END_REF]. Peer-conversations are guided by the use of an analytic framework, comprised of six viewing lenses through which the lesson may be discussed: (1) mathematical and metamathematical ideas around the lesson's topic; (2) explicit and implicit goals that may be ascribed to the teacher; (3) the tasks selected by the teacher and their enactment in class; (4) the nature of the teacher-student interactions; (5) teacher dilemmas and decision-making processes; and (6) beliefs about mathematics, its learning and its teaching as inferable from the teacher's actions (for further details on the framework and its utilization, see [START_REF] Karsenty | Mathematics, lenses and videotapes: A framework and a language for developing reflective practices of teaching[END_REF][START_REF] Karsenty | Professional development of mathematics teachers: Through the lens of the camera[END_REF].

VIDEO-LM clubs maintain an agenda based on the working assumption that the filmed teacher is acting in the best interest of his/her students. Thus, observers are required to "step into the shoes" of the filmed teacher in an attempt to understand his/her goals, decisions and beliefs. This 'exercise' of attributing reason to another teacher's moves is meant to encourage reflective thinking about the span of possible actions and decisions available within teaching situations. VIDEO-LM facilitators attempt to establish nonjudgmental norms of discussion, through the redirection of highly evaluative comments into "issues to think about"; instead of judgmental comments about the filmed teacher's decisions, participants are asked to consider alternative paths and their ensuing tradeoffs.

To date, more than 70 VIDEO-LM clubs were conducted across Israel. The specific club at the focus of this study was a school-based club, held during 2017 in an urban high school in the center of Israel. The school's population is characterized by a high proportion of low achievers, and most students study mathematics towards the basic level of Matriculation1 . The mathematics department includes 7 teachers, 5 of them with a teaching experience of over 30 years, and 2 with under 10 years of experience. Only 2 teachers in this team teach the higher Matriculation mathematics tracks exclusively, whereas the other 5 teach mainly in the basic track. The facilitator of this VIDEO-LM club (the second author of this paper) is a member of this department. Although being the youngest teacher in the team, he is the only one who has taken a VIDEO-LM facilitator course and had become a qualified facilitator, and his colleagues all agreed to participate in the club under his lead.

Setting

The club met 8 times during the year (about once a month), with each session lasting around 2 hours. In a preliminary discussion with the head of department, it was decided that the club will focus on videotaped lessons filmed in low-track Matriculation classrooms, due to the characteristics of the school mentioned above. In the first 6 sessions, the facilitator (henceforth: YP) selected videos of teachers unknown to the team, from the VIDEO-LM collection. In the 7 th session, YP showed a video of his own teaching, and in the last session the group watched a lesson taught by one of the other teachers, who volunteered to be videotaped for this purpose.

Rationale and research question

This video club enabled investigation of a situation that is less explored in the literature on video clubs for mathematics teachers; unlike most cases, where the facilitator is external to the group, here the facilitator was a fellow-teacher from the same team. This situation potentially allows for intimacy among participants, who are familiar and comfortable with one another, with no outsiders present. This was an interesting opportunity for us to study what teachers talk about, and how they communicate. The emerging research question was thus the following: What characterizes the peerconversation taking place in a school-based VIDEO-LM club, facilitated by a member of the group? 1. Preforming a coarse transcription of all the recorded data from sessions, and identifying recurring themes/issues of interest. 2. Selecting sessions for in-depth analysis. Of the six sessions in which videos of unknown teachers were discussed, we decided to select the first two and the last two (# 1,2,5,6) to enable a perspective on shifts over time. In addition, the two sessions in which videos of team members were discussed (# 7,8), were also selected for analysis. These six sessions were then fully transcribed. 3. Segmenting transcripts into units of analysis. The basic unit was defined to be an idea unit, i.e., a transcript section in which one idea appears, whether in a single turn or along a number of turns. 4. Classifying each idea unit into one of the themes detected in step 1, using the Atlas.ti software. 5. Identifying categories within each theme (e.g., within the theme of evaluative comments, levels of evaluations were identified). 6. Systematically categorizing all turns of talk according to the identified categories. This classification was validated by two researchers.

Findings

One of the interesting themes identified in step 1 of the analysis, was teachers' evaluative comments. About 18% of the idea units were classified within this theme. As described earlier, VIDEO-LM clubs attempt to establish nonjudgmental norms of discussion. Thus, it was interesting for us to categorize types of evaluative comments and examine whether there were any shifts in the teachers' adherence to the expected norms. In this report, we focus solely on this issue.

We identified three categories of evaluative comments. These categories are stated and illustrated below. Due to space limitations we limit the illustrations to examples from 2 sessions.

Category (a): Decisive negative judgment towards teaching moves observed in the video.

This category was characterized by keywords such as 'you cannot do this'; 'unacceptable'; 'it's disturbing'; 'I would never do that'; 'it hurts to see this', etc.

Example (a)1.

In Session 1, teachers watched a lesson on the topic of exponential growth and decay, given in a low-track 11 th grade. When asked to relate to the teacher's decisions that they have noticed, the following conversation evolved between the facilitator and one of the teachers: In this communication, Sapir not only harshly criticizes the teacher's decision not to use a textbook, but also rejects any attempt to find a possible reason for it. Her use of words such as "extremely disturbing", "very disturbing" and her refusal to respond to YP's requests to take the teacher's viewpoint, are all signals of a high level of negative judgmental talk.

Example (a)2.

In Session 2, teachers watched a lesson in analytic geometry in a low-track 12 th grade, introducing the concept of a canonical circle. At a certain stage the teacher, together with the students, arrived at the equation x 2 +y 2 =13 2 , by applying the Pythagorean theorem to a triangle drawn in the first quadrant (with one side on the x-axis, another side parallel to the y-axis, the point (x,y) as the edge of the hypotenuse, and R=13). Then, he asked what if y=0, wrote on the board x 2 =13 2 and under this, following a student's answer, wrote x=13. He then showed the point (13,0) on the graph. Following the video, Olga criticized the fact that the teacher did not state the negative solution, -13: No, when I begin, let's say, with these students, I would start differently.

In the above episode, we see Olga solely focusing on what she sees as undesirable teaching moves [602,632,634], despite the recurring request of the facilitator's to "detach herself" and to state the teachers' beliefs [605,631,633]. Although, in line 606 she feebly cites Iris's reasoning in response to this request, she immediately goes on to state that it is "unacceptable".

Category (b):

Disagreement with moves observed in the video, and proposing an improvement.

This type of disagreement often came in the form of suggestions, sometimes as if the filmed teacher was present in the room, and avoided decisive negative judgments. Keywords are 'you need to'; 'you should'; 'it's worthwhile'; 'I'm used to'; "I'm not sure about this', etc.

Example (b)1.

In session 1, Uri relates to the teacher's choice to introduce exponential growth and decay as a case, or continuation, of geometric series:

142 Uri: [the teacher says that] exponential growth and decay is like a geometric series and that's it. It's a little different, you need to explain, from daily life. I even would have given an example of materials that decay, even just an example of a person loosing 3 percent of his weight, like, what is decay?

Example (b)2.

In the video screened in session 2, the students phrase a definition of a circle which is incorrect: "a circle is a central point with all points at the same distance from it". The teacher writes it on the board, and gradually refines it by erasing and replacing several words, until the written definition is "a circle is all the points that are at the same distance from a certain point". Olga comments:

380 Olga: Also it's worthwhile to explain what was not good. 381 YP:

Ah, you think this is something that's missing? 382 Olga: I think it's missing.

YP:

But could it be that it would harm the students had he explained what was not good? 384 Olga:

I think it's important to explain, to make an emphasis on it.

Clearly, Olga's words "it's important to explain", as well as Uri's words "you need to explain", signify a more tolerant and less negative critique than the ones raised by Sapir and Olga, as demonstrated above. Such moderate disagreements are more likely to open up inquiries shared by the group, rather than a direct clash as seen earlier in Example (a)2.

Category (c):

Inspecting possible reasons for observed teaching moves, even if the speaker disagrees with them or testifies that s/he would not have performed such moves in his/her class.

This category is less characterized by specific keywords, and more by whole sentences that convey a complex view on the subject at hand, as demonstrated in the next example.

Example (c):

This example goes back to the issue of writing "x 2 =13 2 ; x=13". In contrast to Olga, Iris expresses a much more elaborated stance towards the choice not to write x=-13 on the board:

490 Iris: Perhaps it's a flaw… I would not leave on the board x 2 =13 Category C, as exemplified in Iris' stance, demonstrates the ability to view a certain practice from the perspective of another person.

In addition to these three categories of evaluative comments, we identified a fourth category associated with evaluation, which we termed "adhering to non-evaluative norms". This category included all turns in which an explanation to the observed teaching actions was offered, without personal reservations, or turns in which the speaker objected to judgmental comments of peers.

The frequency of all four categories across the six analyzed sessions appear in Table 1. As can be seen, the amount of decisive negative judgmental comments decreased over the sessions, with the exception of session 6. We note that YP was displeased with his facilitation of session 6, and that this session merits a closer analysis, not carried out yet, in order to better understand its occurrences. We also note that the lack of judgmental comments in the last two sessions may be partly attributed to the fact that the videos were of lessons taught by team members, present in the discussions. We conjecture that gradual adaptation to the VIDEO-LM norms, combined with sensitivity to colleagues, has brought up this absence of decisive negative judgments.

Regarding the other types of evaluative comments, that are more productive in nature, no patterns were discerned in the analyzed data. Inspecting the frequency of idea units classified to the theme of evaluation (see Table 2), clearly shows that teachers allocated less and less of their attention to the act of evaluation (again, with the exception of session 6), hence more attention was given to other foci of discussion (e.g., pedagogy oriented towards low achievers). We see this shift as a positive one; if the video is used more as an artifact around which to raise issues of practice, and less as a source for criticism, it is more likely that reflective thinking will take place.

Discussion and Conclusions

Our goal in this study was to examine the characteristics of peer-conversations taking place in a school-based VIDEO-LM club, facilitated by a fellow teacher who was an integral part of the group. We focused on one of the most important norms of VIDEO-LM sessions: avoiding judgmental talk. We constructed a categorization system that helps to distinguish between different types of evaluative comments. This is, to the best of our knowledge, a novel endeavor. The types of evaluative comments found in our analysis differ in their degree of potential productiveness: while type A is clearly unproductive (and this coheres with conclusions of previous studies), in type B teachers are able to move slightly beyond criticism and offer suggestions for improvements of teaching moves. However, they still can only see the situation from their own perspective, thus their "span of possibilities", or teaching repertoire, does not seem to expand. In type C evaluations, teachers move beyond their own perspective and provide a rationale for a teaching decision that they themselves would probably not have taken. This is a form that seems to be at least partially productive as it offers revisiting one's habitual practices. Finally, type D comments can be seen as productive as they may result in reflecting upon and perhaps modifying one's own practice.

Our findings show a decrease in evaluative discourse over the course of the year, at least in type A comments. This is an encouraging finding, as it points to shifting towards more productive forms of discussions. For facilitators, the awareness of the process by which such a decrease can occur is of practical importance. For researchers, this methodology can provide a fruitful path towards a better understanding of how and under which circumstances evaluative talk can be decreased.

  94 Sapir:Another decision of the teacher that's disturbing, to me personally it's extremely disturbing, the children are passive, there's no textbook to read from, he writes the question and answer on the board for them… which is pretty organized, okay, but […] I wouldn't have written the question, only the answer. But the children see the question in front of their eyes in the textbook. That is, to me it was very disturbing that the children sit like that [crosses her arms] like… 95 YP: So let's ask the same question again. The teacher arrived in class, apparently in the beginning of the year, and said "you have no textbook" […] He arrives and decides to write the exercises on the board, one by one. We'll see it also later, each exercise he writes on the board, the question and sub

		questions…
	96 Sapir:	What's the advantage in this? I don't…
	97 YP:	I'm asking you.
	98 Sapir:	I don't see an advantage.

  This communication between Olga and Iris is an example of an unproductive argument. It starts with a back and forth "you can, you can't" exchanges, devoid of any justifications. Then, after three such exchanges, Iris brings in a justification based on the circumstances of this being 'a weak class', to which Olga simply answers 'no', without relating to the content of Iris' claim. Later in the session, when the facilitator brings forward the issue of the teacher's beliefs, Olga returns to her objection:

	513 Olga:	No.
	601 YP:	What, in your view, are the teacher's beliefs about the students' role in the
		lesson, about the teacher's role in the lesson?
	602 Olga:	Okay, I relate this to what we talked about, let's say… If I would have heard
		such a thing like x 2 =13 2 and the answer… I would have said to the students,
		[I would have] stopped and said to the students, do you agree with what I
		wrote? I give them the floor and I want to hear their response.
	603 YP:	Okay, and what did he [the teacher] do?
	604 Olga:	He didn't do such a thing, at all.
	605 YP:	So what is his belief, derived from that?
	606 Olga:	So maybe like Iris says, that he didn't want to cut the flow and he continues,
		but I think it's unacceptable.
	[…]
	631 YP:	Did you perceive any belief that he might have regarding low achievers? Let
		go of yourself for a moment, detach yourself. […] did you see anything that
		you can say "I think that his credo about low achievers is XYZ", did you see
		something like that?
	632 Olga:	There are several things here that, let's say, bother me
	633 YP:	Not what bothers you…
	634 Olga:
	507 Olga:	You can't write that and not say a word.
	508 Iris:	You can.
	509 Olga:	You can't.
	510 Iris:	You can.
	511 Olga:	No.
	512 Iris:	In a weak class you can.

Table 1 : Frequency of turns classified to categories a-d, out of all turns classified as evaluative

 1

	Category	Session 1 Session 2 Session 5 Session 6 Session 7 Session 8
	(a) Decisive negative judgment	26.67%	21.74%	20.75%	67.74%	0.00%	0.00%
	(b) Disagreement, proposing improvement	34.67%	33.70%	0.00%	22.58%	20.00%	47.06%
	(c) Justifying a different perspective	0088%	32.16%	18.87%	9.68%	50.00%	17.65%
	(d) adhering to non-evaluative norms	08.16%	16.61%	60.38%	0.00%	30.00%	35.29%

Table 2 : Frequency of idea units classified to the theme of evaluation, out of all classified idea units

 2

In Israel, the mathematics Matriculation exam is obligatory for receiving a Matriculation certificate, and can be taken in one of three levels: basic, intermediate or high.

Data collection and analysis

Data was collected mainly through video recordings of 7 out of the 8 sessions (one session was audiotaped). The recordings were supplemented by a researcher journal where YP documented his insights. The content analysis performed on the collected data included the following steps: