Roles, Identities and Interactions of Various Participants in Mathematics Teacher Collaboration

Abstract

Mathematics teacher collaboration often involves additional participants and stakeholders besides the collaborating teachers themselves. These may include facilitators, mathematicians, researchers, administrators, policy makers or other professionals, who assume a variety of roles in regards to the endeavored collaboration. Such actors may have a significant impact on the collaboration’s productiveness, either from within the collaboration, for example when diverse perspectives are shared amongst group members with different expertise, or from the outside, for example when external actors encourage (or hinder) the creation of suitable environments for collaboration. The nature of roles taken up by participants in teacher collaboration can vary in different countries and contexts. Moreover, within teacher collaborative groups, the roles of participants may shift over time. In this chapter, we aim to explore the roles and identities of various actors involved in mathematics teacher collaboration, as well as the nature of interactions between them. We center in particular on the role of facilitators of teacher collaborations. The chapter relates to the following four foci: (1) existing theoretical framings and methodologies for researching the roles of different participants in teacher collaboration; (2) the role of facilitators in promoting mathematics teacher collaboration; (3) the professional trajectory of facilitators; and (4) the role of the environment in which the teacher collaboration is situated. We drew upon previous reviews and studies, as well as on the 16 papers presented in the Theme C sessions at the ICMI Study 25 conference.

Contributors (in alphabetical order):

Hannah Jütte, Zehavit Kohen, Miriam Lüken, Janka Medová, Hannah Nieman, Marisa Quaresma, Bettina Roesken-Winter, Gil Schwarts, Svein Sikko and Wanty Widjaja

1 Introduction

Mathematics teacher collaboration often involves additional participants and stakeholders besides the collaborating teachers themselves. These may include facilitators, mathematicians, researchers, administrators, policy makers or other professionals who assume a variety of roles in regard to the endeavoured collaboration. Such actors may have a significant impact on the collaboration’s productiveness, either from within the collaboration, for example when diverse perspectives are shared amongst group members with different expertise (Robutti et al., 2016), or from the outside, for example when external actors encourage (or hinder) the creation of suitable environments for collaboration (Vangrieken et al., 2015).

The nature of roles taken up by participants in teacher collaboration can vary in different countries and contexts. For instance, in the Lesson Study model, the role of the ‘knowledgeable other’ alters across cultural environments in which the model is implemented (e.g. Adler & Alshwaikh, 2019; Gu & Gu, 2016; Takahashi, 2014). Moreover, within teacher collaborative groups, the roles of participants may shift over time (Jaworski, 2006; Krainer, 2008).

In this chapter, we aim to explore the roles and identities of various actors involved in mathematics teacher collaboration, as well as the nature of interactions between them. This topic was at the centre of discussions within the Theme C group in the ICMI Study 25 conference. In particular, the group was interested in the role of facilitators of collaboration, as we shall detail below. The papers submitted to the Theme C group aimed to address the following questions, appearing in the Discussion Document published as part of the call for papers towards the conference:

• What is the role of lead teachers, facilitators, mentors and teacher educators in supporting teacher collaboration?

• How are different roles and identities shaped and developed among various actors (teachers, leaders, mathematicians, researchers, etc.) within a collaborative group? How do lead teachers negotiate their dual roles and identities as both teachers and facilitators of peer-collaboration?

• What are characteristics of a good facilitator of teacher collaboration? How can these facilitators be prepared and supported?

• How can different stakeholders impact teacher collaboration?

• What types of learning environments enhance or hinder mutual learning of teachers and other participants in collaborative interactions?

A total of 16 papers were accepted to this theme (henceforth, we refer to these as Theme C papers) and the group included 19 participants, representing 4 continents and 13 countries:¹ thus, we enjoyed a diversity of cultural perspectives that enriched our conversations. Unfortunately, we did not receive any papers from Africa, and we note this as a shortcoming that requires the community’s attention.

This chapter draws on the Theme C papers as one source of reviewing the state of the art in researching various roles within mathematics teacher collaboration. Other sources are the plenary talk given by Konrad Krainer at the conference, and current research as reflected in Special Issues, conferences, books and articles dedicated to the topic of professionals involved in collaborative learning of mathematics teachers.

We begin this chapter by reviewing various methodologies for researching roles in collaboration, as revealed in the Theme C papers, as well as from other sources. This review provides general insights on the types of research that are prevalent in the field, pointing also to certain gaps that need to be addressed in the future. Then, we devote the lion’s share of the chapter to a central actor in mathematics teacher collaboration: the professional who leads, maintains and supports teachers’ collaborative activities. There was a consensus among the Theme C group participants that of all actors who may be involved in a collaboration – be it a professional development (PD) course, a community of practice, a school-based initiative or any other form of collaboration – the role of the person who facilitates the work of the collaborative group is most critical.

This stance is in line with the growing interest of the mathematics education community in the work of facilitators, which began in the late 1990s and early 2000s with sporadic pioneering studies (e.g. Even, 2005; Schifter & Lester, 2002; Zaslavsky & Leikin, 1999, 2004) and has, since then, been accumulating, and more so in recent years (e.g. Borko et al., 2014, 2017; Coles, 2014, 2019; Jaworski & Huang, 2014; Karsenty, 2016; Karsenty et al., 2023; Kuzle & Biehler, 2015; Lesseig et al., 2017; Prediger & Pöhler, 2019; Prediger et al., 2019; van Es et al., 2014). The interest in facilitators is also linked to the need to understand their crucial role in processes of scaling-up PD models (Cobb & Jackson, 2011; Jackson et al., 2015; Roesken-Winter et al., 2015; Zehetmeier, 2015).

At this point, a note on terminology and definitions is required. Various names are used in the literature in reference to professionals who support the learning of mathematics teachers. As Even and Krainer (2014) have noted, the term ‘Mathematics Teacher Educator’ (MTE) commonly refers both to those who educate prospective teachers (e.g. Appova & Taylor, 2019) and to those who educate practicing teachers (e.g. Zaslavsky & Leikin, 2004). However, in the literature specifically referring to professionals who work with practicing teachers, we find the terms facilitators (e.g. Karsenty et al., 2023; Prediger & Pöhler, 2019; Tekkumru-Kisa & Stein, 2017a); leaders (e.g. Borko et al., 2014; Elliott et al., 2009; Lesseig et al., 2017); mentor-teachers (e.g. Kuzle & Biehler, 2015); didacticians (e.g. Coles, 2014; Jaworski & Huang, 2014), PD providers (e.g. Even, 2005); multipliers of a PD program (e.g. Maaß & Doorman, 2013; Roesken-Winter et al., 2015); pedagogical instructors (e.g. Yow & Lotter, 2016) and brokers (Eriksen & Solomon, 2022). Sometimes, various terms are used within the same study, for example Lewis (2016), who explored how novice practitioners learn to conduct Lesson Study in the United States, used the terms teacher leaders, facilitators and teacher developers interchangeably.

As we shall show later in this chapter, this multiplicity of terms may result in methodological challenges, particularly when researchers from diverse contexts attempt to communicate about their methods of study. Thus, for the sake of clarity, from now on we use the term ‘facilitator’, which we define as follows:

A facilitator of a group of mathematics teachers involved in collaborative work is the person who manages the activities of the group, by supporting teachers’ exchange of experiences and new insights, monitoring the discussion, ensuring adherence to the norms set for the collaboration and assisting teachers in forming a trajectory toward the goals set for the group’s work.

A facilitator may or may not be the initiator of the collaboration, and s/he also may or may not be the provider of resources (e.g. contents, methods, artifacts) for the teachers’ collaborative work. This is appreciably dependent on the specific context of the collaboration at hand. Also, facilitators may come from diverse backgrounds; they could be experienced mathematics teachers, university mathematics educators, mathematicians, researchers, etc., who take on the role described above within a certain group. Hence, being a facilitator is always attached to a particular context: the same person can be a facilitator in one group, a participant in another group and an external observer in yet a third group. Moreover, this role can change within the same group over time, as shown, for example, by Ribeiro (2020).

We explore the role of a facilitator in two consecutive parts of this chapter. Firstly, we focus on ways to conceptualise this role. Within this focus, we provide a brief account of constructs, frameworks and models developed to describe what being a facilitator may entail, and elaborate upon research findings concerning the knowledge, practices and skills needed for productive facilitation of teacher collaboration. We also refer to challenges faced by facilitators while attempting to lead groups of teachers, and touch upon the model of co-facilitation. Secondly, we review the research concerning professional trajectories of facilitators, including their preparation to take on this role, development and changes that may occur over time in different aspects of their work, and existing means and models for supporting them along their work period.

Facilitators of mathematics teacher collaboration do not act in a void. Even the most capable facilitator needs to negotiate and manage the environment in which the collaboration takes place. The complexity of factors associated with the environment of collaboration is the focus of the penultimate section of this chapter, where we consider the interrelations between the environment and the various actors within it. Included in this account is a review of models that specifically relate to the environment of collaboration; how environments are created and shaped by different participants; and the cultural and contextual aspects of collaborative environments in which various participants act. We also refer to institutionally-imposed environmental factors that impact teacher collaboration. Finally, we challenge the facilitator–environment relationships by looking at collaborative environments that operate without a facilitator. The chapter concludes with a short summary, looking ahead at future challenges in this field.

2 Methodologies for Researching Roles in Collaboration

In general, research on teacher collaboration reflects diversity of theoretical framing and choice of methodology (Robutti et al., 2016). Methodological issues are particularly important when analysing the knowledge, practices, identities and interactions of people in their different roles within a collaborative group, an analysis that, as we show in this chapter, is complicated and multi-variabled. The literature on mathematics teacher collaboration includes a spectrum of foci, ranging from investigating knowledge facets and affective motivational variables at the teacher level (e.g. Stahnke et al., 2016) to the impact of the collaboration on achievement at the student level (e.g. Lomos et al., 2011).

Within this extensive and diversified body of research, investigations concerning the roles of different actors in teacher collaboration also reveal a span of methods and research perspectives. Specific consideration is warranted to extend and consolidate the growing body of knowledge around methodologies for researching such roles, if we are to understand better this new field of research. This is the focus with which we begin this chapter.

In Sect. 4.2.1, we provide a summary of our analysis of methods, approaches and theoretical perspectives described in the set of the Theme C papers. We further explore the variety of methods employed in these studies by applying the RATE tool (Relevant Actors, Targets and Assignments) presented at the ICMI Study 25 conference (Krainer & Spreitzer, 2020) and elaborated in Krainer, Roesken-Winter & Spreitzer (Chap. 8, this volume), as a means for depicting and identifying essential methodological features. We then compare the results of our analysis with previous surveys of research into teacher collaboration. In Sect. 4.2.2, we address methodological challenges, including lack of commonly shared constructs resulting from the infancy of this field, and highlight the need for studies that go beyond self-studies, in order to develop theory and more operationalisable constructs. In Sect. 4.2.3, we underline tensions observed within several aspects of possible research methods. We conclude in Sect. 4.2.4 by reflecting on the methodological issues identified.

2.1 Various Methods Used to Research Different Roles Within Mathematics Teacher Collaboration

Research into the roles of various participants within mathematics teacher collaboration employs a variety of methodologies, grounded in diverse theoretical perspectives. While a full survey of these methodologies and perspectives is beyond the scope of this chapter, we provide here an analysis of methods and perspectives represented in the Theme C papers, which can be seen as an international sample of state-of-the-art research in this domain.

The 16 papers provided descriptions of methods used to explore the different roles that participants undertook within mathematics teacher collaboration. Of this set of papers, 87% used qualitative approaches. The majority of the Theme C papers were written by researchers working in the field of teacher education and, in more than half of the cases, the researcher was part of the studied process. In 67% of the papers, the study included groups of fewer than ten members, or even fewer than three in 40% of the papers. In 73% of the papers, the context was professional development for practicing teachers, whereas initial teacher education or day-to-day work were present in only one or two papers each. Regarding the types of collected data, video or audio recordings of meetings and interviews were the most prevalent means (53% and 40%, respectively) and they were generally transcribed, at least partially. Other types of data included field notes, questionnaires, lesson observations, online forums and protocols. The theoretical underpinning reported by the authors of the Theme C papers varied; however, 93% indicated that their study was situated within a sociocultural perspective, one author took a cognitive approach and one indicated application of a metacognitive approach. In 40% of the papers, the identified themes and categories inductively emerged from the data. In the other 60% of papers, themes and categories were derived deductively from the literature. Two papers employed both inductive and deductive approaches (Nieman et al., 2020; Quaresma, 2020).

In previous surveys that align the theme of mathematics teacher collaboration (Adler et al., 2005; Robutti et al., 2016), the majority of studies were authored by researchers working in the field of mathematics teacher education, most studies were self-studies, qualitative methodologies were applied and the majority of studies were small-scale studies. We see similar trends in the Theme C papers. In order more readily to depict the range of methods used in the Theme C papers we now present selected research papers mapped via the RATE model, which was introduced in the ICMI Study 25 conference (Krainer & Spreitzer, 2020); see also Krainer, Rösken-Winter & Spreitzer (Chap. 8, this volume). RATE assists in identifying the essential features of an interaction, by focusing on the Relevant Actors, Targets and Environments (RATE) of an initiative. The use of arrows serves to indicate the relationship between actors. As applied here, RATE enabled us to make observations related to methodological issues and to summarise key features regarding the different roles involved in the collaboration.

Four studies from Theme C papers were selected for RATE mapping, based on selection criteria aiming to cover a variety of methods, actors, foci and scale: since the majority of papers reported on small-scale qualitative studies, 3 of the 4 selected papers (Clivaz & Daina, 2020; Nieman et al., 2020; Schwarts, 2020) represent this choice: however, these three exemplify different methods of data collection and analysis (e.g. stimulated-recall interviews; field notes; fine-grained analysis of PD interactions), as well as different involved actors (e.g. facilitators; a facilitator’s mentor; school principal). In addition, they represent different kinds of relationships between the researcher(s) and the research subjects (the researcher as an external investigator versus a double-role of being a researcher and a facilitator of the collaboration). The fourth selected paper (Cao et al., 2020) represents a larger scale study, employing a combination of quantitative and qualitative approaches. Finally, the four papers represent very different cultures, as they come from USA, Israel, Switzerland and China. To broaden the picture further, we also used RATE to map a study outside the set of the Theme C papers (Roesken-Winter et al., 2015). Below we detail the RATE analysis for each of these five studies.

The small-scale study by Schwarts (2020) investigated how one novice facilitator perceived her role within a collaborative, video-based, PD project. The focus of the paper was on one person: however, the project as a whole was broader. Mapped using RATE (see Fig. 4.1), we readily see the relationship between the facilitators, the team that supports them, the researchers and the participating teachers. The arrows indicate that the novice facilitator and the mentor (a senior facilitator) who supported her worked together in a coaching model. The target of the larger collaboration was improvement of mathematics teaching via peer-analysis of video recordings of lessons taught by unfamiliar teachers. The target of the specific study was to explore the facilitator’s development of expertise, thus enabling “insight into how novice facilitators’ practices and identities evolve, as well as what is required to support this process” (Schwarts, 2020, p. 540). The data collection means, including questionnaires, stimulated-recall interviews and reports by the facilitator’s mentor are also depicted.

Fig. 4.1

RATE representation for Schwarts (2020)

The study by Nieman et al. (2020) investigated the roles that one school administrator and two facilitators played in establishing an inquiry-oriented mathematics teacher community. Interviews with the three leaders, as well as transcripts of the conversations during planning meetings, were used and analysed, resulting in categories that emerged from the data. The results revealed the influence of the relevant environment and its important role. The influence of the broader community on how leaders see their roles was analysed at many levels: the teachers, the school and the normative expectations set for teachers and leaders in the USA. Mapped onto RATE (Fig. 4.2), we see the three authors following a school leadership team, comprising the principal and two facilitators of professional learning, collaborating with the target of developing a strong professional community of middle school teachers.

Fig. 4.2

RATE representation for Nieman et al. (2020)

Clivaz and Daina (2020) reported their micro-analysis of teachers’ interactions while participating in a mathematics lesson study group. Through the RATE mapping (Fig. 4.3), we see eight teachers in the study collaborating with the four researchers and two facilitators; one of the researchers is a facilitator and one of the teachers is also a facilitator. The role of the school was to enable the collaboration through allocating time to the teachers; similarly, the role of the university was to allocate time for work to facilitators and researchers. The focus of the collaboration was teaching oriented towards problem solving. In this study, video-recorded interactions between teachers and facilitators were analysed to interrogate the influence of those interactions on the development of teachers’ mathematical knowledge for teaching. An existing dialogic framework was adapted to support the development of categories, in order to identify the “specific dialogic role of each facilitator and of each teacher during the phases of the lesson study process” (Clivaz & Daina, 2020, p. 465), which was inferred from the analysis of the interactions. The small scale of the study enabled the development of a very fine-grained analysis of the roles of the participants in this Lesson Study research.

Fig. 4.3

RATE representation for Clivaz and Daina (2020)

The three studies presented above are all small-scale studies, with a focus on 12 or fewer participants. In contrast, the study reported by Cao et al. (2020), with a relatively high number of teachers (n = 72) from three cities in different regions of China, would be regarded as a large-scale study. To distinguish it further from any other Theme C paper, its focus resided in mathematics teachers’ daily interactions and collaborations in middle schools, in the absence of any designated facilitators. Data collection included questionnaires which were statistically analysed, but also semi-structured interviews (not presented in the paper). The survey methodology employed in this study is arguably of a typical ‘classic’ nature, but at the same time uncommon in the context of researching roles and identities within collaboration. The RATE mapping of this study (Fig. 4.4) shows authors surveying 72 teachers from different middle schools in China regarding informal teacher collaboration, and captures the context in which Chinese teachers interact and collaborate, and the influence of groups such as the Teaching Research Groups and the Lesson Planning Groups, as well as the influence of the specific Chinese school collective culture. Despite the rich results reported, the authors concluded that, “results from the semi-structured interviews are needed to understand how teachers seek learning and growth through interactions with colleagues” (p. 451).

Fig. 4.4

RATE representation for Cao et al. (2020)

Moving beyond the Theme C papers, we now consider another unique study to exemplify further variety in methodologies used to research various roles in teacher collaboration. The study by Roesken-Winter et al. (2015), in which 12 facilitators participated, can be distinguished from more prevalent, small-scale studies, due to its comprehensive methodology. The time scale and the complementarity of the descriptive statistical analysis of the two questionnaires (at intervals of 4 months) on the one hand, and the qualitative treatment of the semi-structured interviews on the other, allow examination of many elements about the Continuous Professional Development (CPD) program in which the participants collaborated. However, we acknowledge that details of roles, identities and interactions between the various participants in this study remain challenging to extract. Mapping this study onto RATE (Fig. 4.5) shows both the purpose of the study, i.e. to conduct an in-depth investigation on a one-year CPD course for facilitators, and the purpose of the whole project, i.e. to support facilitators in conducting their own CPD courses for mathematics teachers via the German Centre for Mathematics Teacher Education (DZLM). The topic at the focus of the CPD was stochastics, a relatively new topic within the middle-school mathematics curriculum.

Fig. 4.5

RATE representation for Roesken-Winter et al. (2015)

The five studies pictorially represented above by RATE reveal a range of participants taking on various roles within mathematics teacher collaboration and the application of a range of methodologies used to investigate these roles. To round out our discussion of the various methods used to research different roles and their underlying theoretical perspectives, we consider the methods of meta-studies.

First, we refer to Krainer et al. (Chap. 8, this volume) who presented and used the RATE model as a tool to assist in identifying the relevant actors, targets and environments in teacher collaboration across seven selected studies. To apply their model, they selected research studies for analysis according to strict criteria: a clear focus on the topic of mathematics teacher collaboration (via keywords); recency of publication (2018–2019); journal quality; and geographical spread (one or two papers from each continent). As a result of mapping these studies onto RATE, the authors formulated six observations, four of them related to methodological issues: that small-scale qualitative research predominates; that most research is conducted by teacher educators studying the teachers with whom they work; that only a few initiatives describe the context and relevant environments having a potential impact on the initiative; that most initiatives describe extensively their particular approach. The analysis of the five studies we have conducted, presented above, concurs to a large extent with the observations resulting from Krainer et al.’s meta-study.

Second, we refer to the survey by Robutti et al. (2016). The choice of papers for inclusion in that meta-study was made according to publication date (2005–2015), sources (mathematics education journals, conference proceedings, books and handbooks) and keywords. This led to a set of 316 sources that then were analysed, resulting in the identification of three main themes. We note that one of the themes was entitled, “Theories and methodologies framing the studies”, indicating the importance of considering this aspect of this field and pointing to possible gaps.

In this sub-section, we have highlighted the variability in choice of methodology employed in research into mathematics teacher collaboration through a brief analysis of selected studies. We found the RATE mapping to be useful for exemplifying key methods and as a means for visually depicting the essential elements of studies on teacher collaboration while drawing attention to the various participants. Our analysis concurs with methodological issues identified in previous studies (e.g. Adler et al., 2005; Robutti et al., 2016; Krainer & Spreitzer, 2020), mainly that small-scale studies dominate and that research designs are predominantly qualitative. This analysis has also highlighted the centrality of the facilitator in teacher collaboration, as shown in the RATE diagrams, in most of the analysed papers facilitators played a critical role. We now turn our attention more specifically to methodological considerations regarding the role of the facilitator, broadening our view to include mathematics teacher educators (MTEs) in general.

2.2 Methodological Challenges, Issues and Considerations Framed Within Research on Mathematics Teacher Educators

As with any other research domain, research in mathematics education can be viewed as accumulating through trends, resulting from the recognition of important issues for which the community has yet to develop deep understanding (Karsenty, 2020). Nowadays, we witness the rise of a research trend that can be referred to as studying the profession of the mathematics teacher educator (MTE). Our analysis in the previous sub-section indicated the centrality of facilitators in initiating, promoting and supporting teacher collaboration. Combining this with the fact that facilitators are a specific and important sub-group of MTEs,² our interest in facilitators is well aligned with the current trend of researching MTEs, which is the focus of our discussion here.

As mentioned above, although early work in researching MTEs began around the turn of the millennium (e.g. Even, 2005, 2008; Goos, 2009; Jaworski & Wood, 2008; Zaslavsky & Leikin, 2004), it is only in recent years that the literature is considerably growing (e.g. Borko et al., 2014; Beswick & Chapman, 2020; Lesseig et al., 2017; Roesken-Winter et al., 2015). The recent initiatives of conference groups dedicated to this topic – for instance, ETE (Educating the Educators) conferences beginning in 2014, the new Thematic Working Group in CERME named “the professional practices, preparation and support of mathematics teacher educators” formed in 2022 and several Special Issues in leading mathematics education journals (e.g. Jaworski & Huang, 2014; Krainer et al., 2021; Tekkumru-Kisa & Stein, 2017b) – indicate that this trend’s apogee is still ahead of us. In this sub-section, we focus on MTEs, and explore methodological challenges, issues and considerations that arise in the accumulated research around this role.

One of the characteristics of a research domain in its early period is that theory-building is in the making, and this is often reflected in the following phenomena: (a) a lack of commonly shared constructs; (b) a search for conceptual frameworks, research methods and strategies that would be appropriate for the developing field; and (c) the predominance of self-studies. In the following, we provide a brief account of how these phenomena are manifested in the research domain, focusing on MTEs as supporters of teacher learning and collaboration.

2.2.1 A Lack of Commonly Shared Constructs

To begin with, research on MTEs is far from having a shared language with which to speak. As shown in the introduction section above, apart from the term mathematics teacher educator, many other different terms are used to describe the role of a person who promotes and manages teacher learning in a collaborative group, such as: facilitator; mentor; teacher leader; PD provider; didactician; pedagogical instructor. This abundance of terms not only reflects cultural differences between research projects across the world, but may also sometimes imply, for individual researchers, a choice of a particular theoretical lens. This leads to a challenge when researchers attempt to communicate their work to peers. To illustrate this, we refer to two anecdotes that we have recently witnessed.

• 1. At the ICMI Study 25 conference in Lisbon, it was apparent that some terms had very different meanings to participants from various countries. For instance, for some researchers, the label ‘facilitator’ related to a role which is hierarchically lower, in terms of prestige and status, than the role of a teacher educator, whereas, for other researchers, a ‘facilitator’ could be a senior university researcher who is head of the project. These different interpretations produced some awkward moments in the peer conversation, resulting in the consensus that a glossary is needed for the international community of researchers, in order to reach a shared understanding of objects under investigation within this domain. To the best of our knowledge, such glossary does not yet exist, and creating one can be a challenge for future researchers aiming to review the field.

• 2. Two of the co-authors of this chapter, who submitted a manuscript about the professionalisation process of PD facilitators, received a comment from a reviewer asserting that the term ‘facilitator’ “belongs to the constructivist paradigm”, which was not the explicit theoretical stance taken in the research. This seemingly minor detail suggests that the choice of terms within this emerging field carries a potential for miscommunication about theoretical lenses underlying studies, which is a point we find worth thinking about.

Yet, a lack of shared terms is not the only methodological challenge we witness in the field of studying the role of MTEs. It seems that the constructs used in various studies in regard to this role are ambiguous and may have different interpretations in diverse contexts. Thus, we hear about MTEs’ knowledge, skills, practices, strategies and moves without a clear shared definition of these constructs and what differentiates them from one another. For instance, Jaworski (2008) described the connection between MTEs knowledge and mathematics teachers’ knowledge as a partial overlap (Fig. 4.6), which implies that MTEs’ are not necessarily former (or present) teachers, whereas others (e.g. Perks & Prestage, 2008; Appova & Taylor, 2019) portray MTEs’ knowledge as including teachers’ knowledge (for further examples of models pertaining to MTEs’ knowledge, see Karsenty, 2020).

Another example is the use of the construct of practice: in van Es et al. (2014), the construct of facilitators’ practice is characterised as a set of moves (e.g. the practice of ‘sustaining an inquiry stance’ comprises the moves of highlighting, lifting up, pressing, offering an explanation, countering and clarifying; see also Sect. 4.3.2). Similarly, Lesseig et al. (2017) define facilitation practices as useful skills for leaders to enact in PD sessions (e.g. monitoring, selecting and sequencing teachers’ sharing of their solutions for mathematical tasks). In contrast, Appova and Taylor (2019) related to MTEs’ practices not as moves or skills, but rather in terms of knowledge resources that expert MTEs draw upon (e.g. the MTE uses student work; refers to professional literature; connects to the curriculum). Thus, it seems that the research of the MTE profession is still in need of clarifying and developing further understanding of core constructs.

Fig. 4.6

Interconnections between teacher educators’ knowledge and teachers’ knowledge (Jaworski, 2008, p. 336)

2.2.2 The Pursuit of Conceptual Frameworks, Research Methods and Strategies

A methodological challenge typical of an emerging research domain is how to develop useful frameworks (for examples of such frameworks, developed for researching the profession of MTEs, see Sect. 4.3.1 below). One of the very few works that address this challenge of creating frameworks is that of Konuk (2018). Konuk described four main approaches by which frameworks for conceptualising the knowledge and/or practices of MTEs may be generated: the standards-based approach; the practice-oriented approach; the inquiry-based approach; and finally, the method of extending or revising existing frameworks. Due to space limitations, we do not detail here the characteristics of each of these approaches (which can be found in Konuk, 2018; see also Karsenty, 2020, for a short summary), but focus briefly on the last approach of extending or revising existing frameworks, which is becoming more and more prevalent.

Prediger et al. (2019) presented three strategies for setting research agendas (i.e. developing design principles, research practices and methods) when moving from the classroom level to the PD level. These strategies are lifting, nesting and unpacking. Karsenty (2020) suggests that the lifting and nesting strategies, and sometimes their combination, are also useful in the process of forming new frameworks. According to Prediger et al. (2019), lifting a research practice means that certain types of research questions and/or methods from the classroom level are implicitly or explicitly transferred, and adapted to the PD level and applied in an analogous way. Similarly, Karsenty (2020) refers to lifting a framework as the idea of creating an analogy between an existing framework at the teacher level and a new framework at the MTE level. An example of this idea is how the Mathematical Knowledge for Professional Development (MKPD) framework (Borko et al., 2014; see Sect. 4.3.1) lifts the Mathematical Knowledge for Teaching (MKT) framework (Ball et al., 2008). The nesting strategy is defined by Prediger et al. (2019) as using a similar structure in different levels, so that the content, design principles or practices of one level are included as a component in the next level. Borrowing this strategy to the forming of frameworks, Karsenty (2020) asserts that nesting a framework (or a model) that was originally created for the teacher level within the MTE level results in a complex structure, where the elements that comprise the knowledge and practices of MTEs usually include, as a subset, the elements that comprise the knowledge and practices of teachers. An example of this strategy can be found in the work of Zaslavsky and Leikin (2004), who nested Jaworski’s (1992) Teaching Triad of mathematics teachers within the Teaching Triad of MTEs. It is reasonable to anticipate that in the next few years, we will witness plentiful research practices (including frameworks, methods, strategies, etc.) to study the profession of the MTE, that extend, adapt or revise existing research practices used in the already well-developed study domain focusing on the profession of the mathematics teacher.

2.2.3 The Predominance of Self-Studies

Even and Krainer (2014, p. 203) have noted that, “most research on the professional education and development of mathematics teacher educators includes reflections of teacher educators on their own personal development”. In this type of research, known as self-studies (i.e. “the study of oneself, in particular, one’s thinking and actions” – Chapman, 2008, p. 1), MTEs are researching themselves or are part of the research team. Self-studies are not only an effective way to develop expertise among MTEs (e.g. Baker et al., 2019), but are also the field’s way to accumulate first-hand knowledge about what is involved in the study of MTEs’ knowledge and practices (e.g. Bragg & Lang, 2018; Coles, 2014; van Es et al., 2014). Self-studies enable researchers to pursue the unpacking and the operationalisation of core constructs in the field, which is, as described above, necessary for its advancement. The tendency to engage in self-study is therefore typical of the early stage in which the MTE literature currently stands, and was also reflected in the ICMI Study 25 conference, where many of the papers presented about the role of MTEs (see Sect. 4.2.1) were self-studies.

However, in the ICMI Study 25 conference there were also several researchers who presented work outside the realm of self-study, that is who explored the role of MTEs from ‘an external’ perspective (see, for example, Pöhler, 2020; Schwarts, 2020). Such research is gradually accumulating in the past decade, as exemplified in the work of Borko et al. (2014), Jackson et al. (2015), Lesseig et al. (2017), Karsenty et al. (2023), Prediger and Pöhler (2019), and others. This might be an important step towards the maturity of this research field and, moreover, necessary for the design of large-scale implementations of programs that centralise teacher collaboration, facilitated and supported by professional and skillful MTEs. In the words of Tekkumru-Kisa and Stein (2017b, p. 2), we need, “to learn how designers’ foci, tools, and resources evolve as they transition from a single program to one that is scalable beyond the initial developers’ vision”.

To sum up, we illustrated how three phenomena typical of a developing research field (i.e. the lack of commonly shared constructs; the search for appropriate frameworks and methods; the high proportion of self-studies) are manifested in the study of MTEs’ role. As mentioned earlier, the roles of other participants in teacher collaboration are still under-researched (we briefly touch upon this in Sect. 4.5.3 below). However, it can be expected that similar methodological challenges would be encountered and reported as this research domain progresses.

2.3 Tensions Within Research Methods

We now turn to the tensions that researchers negotiate as they employ research methods to investigate various roles within teacher collaboration. We draw in part on aspects of the keynote presentation by Prediger (2020) at the ICMI Study 25 conference (see also Prediger, Chap. 6, this volume) for this purpose. First, we consider tensions in investigating generic versus content-specific aspects of roles within teacher collaboration. We then reflect on specification of roles versus the complexity of roles. Lastly, we mention tensions that may arise between taking a situated perspective, focusing on analysing the practices of an individual and attempting to draw more general conclusions pertaining to groups of teachers.

2.3.1 Tension in Investigating Content-Specific Versus Generic Aspects of Practice

Prediger (2020) described a study of teacher collaboration, where a community of inquiry was established for mathematics teachers and special needs teachers, in order to develop inclusive teaching practices of mathematics in heterogeneous classrooms. The episode upon which we draw is associated with helping students with learning difficulties in the area of basic arithmetic operations. The following citation, by one of the teachers in this community, represents one outcome of the collaboration, after 9 months of intense work:

Suleika can calculate the subtraction well, only the carries pose problems for her. But we can handle this successfully by differentiated tasks: I only give her subtractions without carries. (Prediger, 2020, p. 4)

Superficially, we see the work within this collaborative group resulting in a teacher believing that the approach for differentiation and teaching inclusively in a heterogeneous classroom is to provide students with individualised exercises that guarantee their success. From a methodological perspective, this incident may be evaluated in different ways, depending on the evaluator’s viewpoint. If analysed from a generic viewpoint, for example by using the TALIS (Teaching and Learning International Survey) distinction of different types of teacher collaboration (OECD, 2014) or the Gräsel et al.’s (2006) framework, one could have assigned a highly-valued category to this teacher collaboration, since the teachers have been working collectively in an intensive manner over a long period of time and took an active part in collaborative professional learning.

However, from a content-specific viewpoint, the conclusion drawn with respect to Suleika’s learning is that the collaboration could benefit from an expert opinion, helping teachers to find ways to foster the student’s learning, going beyond just task completion. This small episode reveals a possible tension, depending on whether the theoretical and methodological approaches are grounded in a content-specific perspective or not. One might go as far as questioning the merit of reporting such data from a mathematics teacher collaboration, if it is being investigated solely from a generic viewpoint. This perspective prompted thought as we discussed methodological considerations in our Theme C meetings. However, whilst other tensions were enunciated in some of the Theme C papers (e.g. Zhao et al., 2020; see Sect. 4.5), the tension of content-specific versus generic aspects of practice was not explicitly addressed in the Theme C collection of papers.

2.3.2 Tensions Between Specifying Roles and Attending to the Complexity of Roles

Methodological issues also arise when considering a potential tension between specifying roles in the collaboration and attending to the complexity of roles. Returning to the collaborating teachers in Prediger’s (2020) study, we see the benefits of the extended collaboration involving an experienced facilitator, who brought in her perspective and well-designed classroom material to help progress Suleika’s learning and the teachers’ mathematical knowledge for teaching. This hints to the tension stated above. In relation to the specification of roles, the teachers are involved as learners – the facilitator is learning as well, and both bring in their expertise. The complexity of these roles calls for attentive scrutiny as to what may be the focus of the investigation in such situations. One could imagine investigating the outcome of the teachers’ collaboration with the facilitator; the learning pathways of the teachers; the learning pathway of the facilitator; the degree to which the facilitator’s expert viewpoint is explicitly brought in; the positioning of the teachers as experts of their classrooms; and more. The question then is: how can roles be specified and, at the same time, how can the complexity of roles be captured? We further ask: what are adequate research methods for resolving this tension? While the answer is likely to depend on how the different aspects of the study are operationalised, what seems apparent is that resolving the tension of specification versus complexity of roles may require qualitative methods such as observations, interviews and the use of narratives that allow for a fine-grained and sensitive analysis, rather than employing quantitative methods.

2.3.3 Tension in Pursuing a Situated Research Approach While Investigating Groups of Teachers

A further methodological tension arises in the pursuit of investigating groups of teachers through a situated research approach. The more a situated approach is pursued, the more the practices of individual teachers, facilitators and other actors are at the focus of investigation, usually involving qualitative research methods (Depaepe et al., 2013). However, within a cognitive methodological perspective, constructs such as knowledge and affective-motivational variables would likely be investigated on a larger scale, involving groups of teachers, often by applying quantitative methods with the use of standardised tests or scales. It is acknowledged, however, that a dichotomy of qualitative versus quantitative methods is not a clear demarcation. In their systematic review on empirical mathematics education research pursued from a situated perspective, Stahnke et al. (2016) distinguished between studies elaborating on perception, interpretation and decision-making, and pointed out how a variety of qualitative and quantitative approaches can be realised.

In the preceding paragraphs, we identified three sources of methodological tensions that have arisen in researching roles in teacher collaboration. It appears that a systematic review, with a focus on research methodologies, differentiating between confirmatory studies, intervention studies and case studies would help progress research into the roles of various participants within teacher collaboration. In the next sub-section, we present concluding comments to summarise our discussion on the methodological aspects of such research, while pointing at needed future directions.

2.4 Methodological Issues in Researching Roles Within Teacher Collaboration: Looking Ahead

From our analysis of methodologies for researching roles in collaboration, we can point to a need for more research from ‘an external’ perspective (i.e. going beyond self-studies) and for more large-scale studies. There is also, however, room for further research using observations or interviews, as these allow fine-grained analysis of the roles of various participants in teacher collaboration. As such, there appears to be a need for the development of explicit research agendas that incorporate the individual and move ‘upstream’ to consider the wider range of participants in the collaboration, but also for a systematic review of research methodologies directed ‘downstream’, in order to favour the development of commonly-shared constructs.

Methodological issues arise as a result of the choice and the development of theoretical frameworks. Various theoretical perspectives on roles within teacher collaboration impact the methodological choices and approaches. There is a variety of theoretical frameworks that can be applied with respect to investigating roles in teacher collaboration, and one might ask whether we even need more theoretical frameworks, or perhaps more specified methodological approaches that allow for fine-tuning of existing theories. Maybe this is not an ‘either/or’ question, but more about where we stand and where we want to go. We hope that our discussion above provides a contemporary basis for thinking about these questions. We now conclude our inspection of methodological issues in researching roles of various participants in mathematics teacher collaboration, and proceed with looking specifically at the important role of one significant participant, namely, the facilitator of collaboration.

3 Conceptualising the Role of Facilitators in Promoting Mathematics Teacher Collaboration

As already noted, it is universally agreed that facilitators take a central role in teachers’ professional development and collaboration (e.g. Borko et al., 2015; Even, 2008; Jackson et al., 2015; Jaworski, 2008), both from a systemic perspective, as they are key to scaling up innovations to a larger number of classrooms (as addressed for example by Banilower et al., 2006, Borko et al., 2011; Zehetmeier, 2015), and from a local perspective, as they catalyse the individual group’s expertise and actions into sustainable progress (Jackson et al., 2015; Tekkumru-Kisa & Stein, 2017a).

In this part of the chapter, we review contemporary research, as well as suggested future directions, regarding the conceptualisation of the facilitator’s role in designing, maintaining, and supporting collaborative activities for mathematics teachers. We begin in Sect. 4.3.1 with a brief account of constructs, frameworks, and models developed since the turn of the millennium to describe what this role might entail. Then, in Sect. 4.3.2, we elaborate on what research so far tells us about the knowledge needed for productive facilitation of teacher collaboration, and which practices and skills for facilitation were examined in various contexts. Challenges faced by facilitators, as found in current studies, are described in Sect. 4.3.3. Finally, a less-studied type of facilitation, where co-facilitators work together to support teacher collaboration, is presented in Sect. 4.3.4.

3.1 A Brief Account of Constructs, Frameworks and Models to Conceptualise the Facilitator’s Role

In the past two decades, several researchers have suggested various conceptualisations of the role of facilitators (specifically, their knowledge and practices), introducing new constructs, frameworks and models. These conceptualisations vary in grainsize and detail. Even (2008) introduced the term knowtice, a combination of ‘knowledge’ and ‘practice’, as a construct that describes the knowledge that PD facilitators need to learn and develop. She suggested that this construct is comprised of four elements: knowledge of mathematics; knowledge of current views of mathematics teaching; knowledge of current views of teaching teachers and of teacher learning; ways of educating practicing mathematics teachers. Borko et al. (2014) have introduced the framework of Mathematical Knowledge for Professional Development (MKPD), which builds on the well-known framework of MKT (Mathematical Knowledge for Teaching, developed by Ball et al., 2008, for characterising the knowledge required by mathematics teachers). MKPD includes specialised content knowledge, pedagogical content knowledge and learning community knowledge. Another framework that builds on MKT is Mathematical Knowledge for Teaching Teachers (MKTT), proposed by Zopf (2010), which, although introduced for the pre-service level, may also be useful for conceptualising the work of PD facilitators. MKTT includes: knowledge on how to unpack MKT for teachers; knowledge on how to develop a precise mathematical language; a connected mathematical knowledge; knowledge of the epistemology of mathematics. A more recent framework for facilitators’ knowledge, published by Lesseig et al. (2017), lays emphasis on three elements: identifying the learning terrain for teachers in the PD and articulating consequent learning goals; orchestrating discussions; and cultivating norms that support the attainment of goals. Karsenty (2020) suggests the meta-lenses framework (MLF) for unpacking knowledge and practices needed for facilitators in a video-based PD. MLF includes six components: the PD’s agenda, ideas and norms; explicit and implicit facilitator goals; PD tasks and activities; facilitator–teacher interactions; facilitator dilemmas and decision-making; facilitator beliefs about mathematics teaching, about how teachers learn and about the facilitator’s role. Another framework for exploring facilitators’ decision-making, and their professionalisation over time, is the ROGI framework (Karsenty et al., 2023), which adapts Schoenfeld’s (2010) ROG framework, i.e. the triad of teachers’ resources, orientations and goals, into a quartet of facilitators’ resources, orientations, goals and identity.

Other researchers have proposed visual models to describe the interrelation between various aspects concerning the role of facilitators (or MTEs in general). For example, the tetrahedron structure suggested by Perks and Prestage (2008) relates to vertexes, such as practical wisdom, professional traditions, and teacher knowledge. The Teaching Triad of MTEs (Zaslavsky & Leikin, 2004, based on Jaworski, 1992) includes challenging content for mathematics teaching, management of mathematics teachers’ learning and sensitivity to mathematics teachers.

As can be seen, these frameworks and models differ from one another regarding their foci, yet, in general most if not all of them refer both to academic knowledge and to social and interpersonal skills needed by facilitators. In the next sub-section, we review empirical research about these aspects in various contexts.

3.2 The Facilitator of Collaborative Teacher Learning as a Professional Expert

Following our definition of a facilitator, we take the position that facilitators are not necessarily the ones who determine the goals for teachers’ collaborative work, but rather the role of facilitation is focused around organising and supporting the activities of the group towards agreed goals. Such goals may be set by the teachers themselves or by some external source, which can be either the PD project represented by the facilitator or educational authorities at the local, regional or national level. For example, facilitators may be expected to promote innovations, such as pedagogy oriented towards problem-solving or language sensitivity (e.g. Prediger & Pöhler, 2019), introduce new teaching content such as statistics (e.g. Kuzle & Biehler, 2015) or present and support the use of technology for distant learning, a challenge that became especially pressing since the global spread of COVID-19 (Bakker & Wagner, 2020).

The role of facilitation emphasises skills such as guiding the exchange of ideas, monitoring the discussions and elevating reflections within the group of teachers. In filling that role, facilitators need to maintain a careful balance between communicating at eye-level and leading the group towards the pre-determined goals. In sum, although the collaborative work under facilitation is likely to have clear goals, the process for reaching these goals is often under-defined, and it is the facilitator’s responsibility to orchestrate it.

These expectations from facilitators mean high demands on their professional knowledge and expertise, which contrasts with the fact that, as we discuss in Sect. 4.4 of this chapter, there are hardly any institutionalised paths to become a facilitator. The frameworks mentioned in Sect. 4.3.1 imply that facilitators, like teachers, must master academic knowledge aspects as well as pedagogical and social skills. For facilitators, these knowledge and skills need to be framed in terms of educating adults (Knowles, 1990), specifically, mathematics teachers in various contexts. This undertaking involves, for example, unpacking the learning goals and their rationale, since teachers are more likely to participate in PD activities if they see their relevance to their teaching (Pinto & Cooper, 2017). Successful facilitation includes promoting knowledge domains for teachers (Park Rogers et al., 2007), so in PD contexts, these domains are to be viewed from the perspective of leading groups of teachers rather than that of teaching students. However, facilitators’ knowledge of the PD content comprises within it also knowledge domains at the teacher level (e.g. as described in the MKT framework), which can be seen as nested in the knowledge at the PD level (Zaslavsky & Leikin, 2004; Luft & Hewson, 2014; Perks & Prestage, 2008; Prediger et al., 2019). In addition, Borko et al. (2014) stress that facilitators, “must hold a deeper and more sophisticated knowledge of mathematics than their colleagues, just as teachers must hold a deeper and more sophisticated knowledge than their students” (p. 165). For instance, facilitators ideally have an extensive specialised content knowledge, in reference to different solution strategies for tasks in a certain content area under analysis, as well as knowledge about the tasks’ conditions and interconnections, and common difficulties which solvers may encounter. Or, if the PD content involves various mathematical representations, the facilitator is expected to have robust knowledge about their advantages, constraints and generalisations. Indeed, in a perfect world, teachers should possess such knowledge as well, but the requirements are stricter for facilitators who are often regarded as experts. According to the MKTT framework (Zopf, 2010), the mathematical knowledge of those who work with teachers needs to be “unpacked, connected, language focused, and discipline oriented” (p. 185). This means that facilitators should have a solid knowledge of mathematical structures and procedures, and be experts in choosing appropriate interpretations, representations and examples, and in highlighting interconnections between them, while at the same time supporting and developing a language that does not compromise mathematical integrity, but is still practical in the classroom. Moreover, facilitators should know how teachers learn to teach the content, which requires them to be familiar with contemporary models of teacher learning with its conventions and restrictions. At best, facilitators have passed through a special training, focusing on how teachers’ professional development can be promoted in general and in regard to the PD content (as described for example in Borko et al., 2015; Maaß & Doorman, 2013; Kuzle & Biehler, 2015; see Sect. 4.4 below).

It is worthwhile to explore not only the knowledge of successful facilitators, but also their practices and orientations, i.e. what they do in collaborative settings in order to promote teacher professional growth, and which attitudes they adopt when they “support learning toward carefully defined goals without undermining learners’ sense of agency” (Tekkumru-Kisa & Stein, 2017a, p. 3). A central issue is promoting fruitful discussions on mathematics and its teaching (Karsenty et al., 2023). For example, this may be done by lifting Smith and Stein’s (2011) five practices from the classroom level to the PD level, as was done by van Es et al. (2014). By utilising facilitation moves (such as highlighting, lifting up, pressing, etc.), van Es and her colleagues suggest that facilitators can direct participants’ attention to interesting and important mathematical ideas, invite them to elaborate their own ideas and encourage them to explore various explanations and interpretations.

Similar facilitation moves were considered in some of the studies presented in Theme C (e.g. Griese et al., 2020; Schwarts, 2020). In addition, van Es et al. (2014) refer to facilitator practices that explicitly address orienting the group towards the task, maintaining the focus on the mathematics and supporting group collaboration. They conclude that coherence of facilitator practices is essential for productive professional development. This means that it is not enough for facilitators to be able verbally to express an attitude of debate and exchange in general, for instance by utilising conversational prompts that spark the discussion, but they need also to keep the intended goal in mind and be aware of the different paths that lead to it. Re-focusing the group may also mean to call a halt to certain threads of the discussion, and to follow up others. This needed coherence is mirrored also in the Theme C papers. For example, Pöhler (2020) found that, “identifying facilitation moves can be insufficient” (p. 522) and that identifying content-related principles that were or were not addressed by the facilitators proved more fruitful in understanding facilitation.

An interesting and somewhat different case of looking into facilitation practices is the case when the facilitators are mathematics researchers (e.g. Goos, 2014; Sztajn et al., 2014), and the PD is framed in terms of “boundary crossing” (Cooper, 2019). The notion of ‘boundary’ (Robutti et al., 2020; Wenger, 1998) implies that mathematics researchers and mathematics teachers belong to different, yet connected, communities of practice. While the two communities have different values and different knowledge, a PD environment provides a space for researchers and teachers “to exchange knowledge from their communities impacting both researchers’ and teachers’ practices without reducing the importance of either” (Wenger, 1998, p. 201). Various artefacts are used as boundary objects to negotiate and establish shared meanings among researchers and teachers. For example, Goor (2022) used videotaped lessons filmed in high-school mathematics classrooms as boundary objects in a collaborative group of secondary mathematics teachers and mathematicians. In this case, facilitation was conducted by mathematics education researchers, who served as brokers between the two communities. Sztajn et al. (2014) reported a shift in their program from using artifacts from the researchers’ community to those from the teachers’ community (e.g. curriculum materials and lesson plans) to support and sustain meaningful collaboration. In this PD, they drew attention to the researchers’ practices, as they took the role of facilitators in their work with elementary school teachers, designed around research-based knowledge on students’ mathematics. These practices included: (1) drawing on teachers’ expertise and understanding of school mathematics; (2) using research-informed evidence (e.g. clinical interviews with students, written diagnostic assessment) to develop teachers’ learning trajectories; (3) bringing into play teachers’ contextualised knowledge of students’ learning to foster exchange of knowledge.

Generally speaking, the exploration of facilitation practices and moves within mathematics teachers’ collaborative settings in different countries has provided important insights into the expertise of facilitators and the outcomes of such expertise. These insights relate both to the PD content (e.g. learning goals and learning pathways for teachers) and to the PD arrangements (e.g. design principles, process quality, facilitation moves), thus defining the specificities of what is to be learned in the PD and of how this is to be orchestrated (Prediger, 2019). For example, in the Unites States, Carlson et al. (2007) found that facilitator behaviour that, “attempts to understand the mathematical thinking and/or perspective of someone else” (p. 841) had a positive influence on the quality of the mathematical discourse of the learning community. The Theme C papers add to the variety of perspectives. In Germany, Griese et al.’s (2020) explorations of a collaborative PD on teaching stochastics resulted in proposing general heuristics for successful facilitation, such as moving from assessing a concrete phenomenon to understanding a structure, or mediating between aspiration and reality by elaborating on students’ learning trajectories. Research from Portugal (Quaresma, 2020) has shown that specific critical incidents may be crucial for the development of collaborative experiences, thus facilitators need to have specific moves for such incidents available.

Another aspect that can impact the success of a PD concerns the facilitator’s orientations, which can exert their influence via the prioritising of goals, for example giving precedence to ‘atmosphere’ goals (i.e. maintaining a stress-free environment) over content goals (Karsenty et al., 2023) exemplify such a case. One risk in such prioritising is that, “politeness is valued over professional debate and controversy” (Jackson et al., 2015, p. 95). Some researchers have found that such phenomena can be overcome if addressed accordingly in facilitator training (e.g. Pöhler, 2020, in Theme C). Jackson et al. suggested that it is important for PD leaders to regard teacher learning as a progression, rather than rectifying deficits in understanding which could be dealt with simply by informing the group how to understand the logic of a teaching approach, as many leaders in this research initially tended to believe. In order to shift facilitators’ orientations, guided reflections on the underlying motives for choosing a certain activity or a specific move can be useful (Masingila et al., 2018). Just as at classroom level, “it is principles (or beliefs) rather than methods or material that underlie practice at a level that makes a difference for students” (Beswick, 2007, p. 116), at the PD level this rationale retains its significance. Since we already know that changing teacher orientations is a multi-step process that requires time, as “it is not the professional development per se, but the experience of successful implementation that changes teachers’ attitudes and beliefs” (Guskey, 2002, p. 383), it follows that the same holds when lifted and applied to facilitator orientations. This stresses the necessity of research-inspired facilitator qualification programs which include facilitation and reflection phases, so that facilitator expertise is supported and monitored over a period of time.

3.3 Situational Challenges of Promoting Collaboration

The challenges for facilitators are located on different levels. We are interested in the specific situational challenges that appear in regard to the interactions of the collaborative group. Researchers agree that promoting a fruitful communication and discussion is a crucial part of successful collaborative facilitation (e.g. Borko et al., 2014; Coles, 2013; Krainer, 2015; Zehetmeier, 2015). Thus, we present below three situational challenges, associated with promoting successful communication: starting and managing a discussion; establishing and maintaining norms; and observing, redirecting and sharing responsibility.

3.3.1 Starting and Managing the Discussion

To spark-off an exchange, often stimuli like video vignettes or problem-solving tasks are utilised (Borko et al., 2014; Coles, 2019). Depending on the framing of the collaborative work, the facilitators might be able to choose from a range of stimuli specifically designed for certain goals, or they might face the challenge of having to find an appropriate stimulus themselves. When there is no material available, the challenge is increased by the necessity to find a discussion starter that is in line with the group’s agenda, and to create activities for the teachers that involve and activate the whole group. One option raised in the Theme C papers was to create relevant professional learning tasks (PLTs), as demonstrated by Ribeiro (2020). In order to be able to decide which tasks are deemed, suitable, facilitators, particularly those who have not been involved in the design phase of the PD program, are in need of clarification and explication of how to select or create tasks, so that they can use them with integrity to the goals of the PD. Ideally, this should be achieved within their qualification as facilitators of the specific program. Even if there is a range of stimuli available from the PD developers, the question remains as to which aspects of which specific stimulus is to be addressed by the facilitator. Karsenty and Arcavi (2017) suggest that records of practice that serve as a base for teachers’ discussion can be inspected by using six different lenses, representing six aspects of lesson analysis (mathematical and meta-mathematical ideas; the teacher’s goals; tasks that students engage in; interactions in the lesson; teacher dilemmas; teacher beliefs). The facilitator can choose one aspect or several different aspects, in order to frame the discussion and enhance various viewpoints.

3.3.2 Establishing and Maintaining Norms

Although it can be assumed that all collaborations build on norms of trust and respect among the group members, still different collaborative settings may embrace further different norms that commonly stem from the approach underlying the collaborative work. For example, one possible approach to a fruitful discussion around video clips, developed in the Open University in the UK (e.g. Coles, 2013, 2014; Jaworski, 1990), advocates that teachers’ contributions should focus at first on describing in detail what they have observed, and only at a second stage they are encouraged to elicit different interpretations and address a wide range of possible alternative reactions. Other approaches encourage teachers to watch videotaped lessons (e.g. Santagata, 2009) or live lessons (as in the Japanese Lesson Study approach and its adaptations, e.g. Fernandez & Yoshida, 2004; Lewis & Lee, 2017; Lim et al., 2005) and explicitly discuss strengths and weaknesses of the lesson (see also the plan–teach–evaluate cycles in Wright, 2020). There are also approaches that deliberately wish to avoid evaluation, and set a norm of restricting the discussion to non-judgmental exchanges (e.g. Karsenty & Arcavi, 2017; Schwarts et al., 2022). This norm has been identified as a considerable challenge for facilitators in certain collaborative settings, as shown in the Theme C paper by Schwarts (2020), since teachers, like other human beings, tend to criticise actions that differ from their own, and so it takes time and effort to uphold a conversation that is not evaluative in nature (Karsenty et al., 2019). The other side of this coin is that teachers may fear being exposed to criticism by peers, and thus they refrain from taking part, for example, in video documentation of practices. In such cases the facilitator’s challenge resides in overcoming such fears, as described in another Theme C paper (Quaresma, 2020). In situations where norms are violated, or when criticism becomes harsh, facilitators are expected to remind the group of the rules or assumptions agreed before, for example the “basic assumption that the [filmed] teacher is acting in the best interest of his/her students” (Karsenty & Arcavi, 2017, p. 438), or the agreement of “starting discussion with accounts of and only later moving to accounts for” (Coles, 2014, p. 269; italics added). This also means that the facilitator must consciously fill the role of a leader, whereas at other times facilitators might stress their alternative role as members of the group (Knapp, 2017). With the objective of supporting collaboration among the group of teachers, facilitators are advised that it is worthwhile sometimes to stand back and simply appreciate the ideas coming up in the exchange (van Es et al., 2014), but this must be utilised in balance with intervening moves of re-establishing norms.

3.3.3 Observing, Redirecting and Sharing Responsibility

Once the discussion has gained momentum, it may divert to themes unconnected to the professional goals. The facilitator should carefully observe the conversation and, if necessary, intervene to redirect the focus of the discussion, while being aware that the responsibility for reaching the PD goals is to be shared between the facilitator and the participating teachers. For the purpose of keeping the conversation on the right track, facilitators can utilise practices designated by van Es et al. (2014), such as “sustaining an inquiry stance” or “maintaining a focus on [the video and] the mathematics” (p. 347), which include several possible moves that guide and encourage reflection and promote discussion around the desired goals. The purposes of these moves are many and varied, and the facilitator is challenged to notice nuances in the discourse, to analyse the situation ad hoc and to decide on a move, following certain principles such as communal investigation, co-operative teaching and open-mindedness towards the teachers (see the Theme C paper by Nieman et al., 2020). Since these principles may be deployed either by the facilitator or by the participating teachers, the facilitator may choose to use them only when there is no teacher doing so. For example, re-introducing an important idea mentioned before for further discussion, or connecting different ideas, is not exclusive to the facilitator role. Some researchers (e.g. Felton & Koestler, 2015) suggest allowing the group a maximum of agency, which they perceive as crucial when aiming to support changes in teaching. By actively sharing the responsibility for the professional development, the facilitator can promote the group’s agency.

In sum, it appears that the work of a facilitator involves specific challenges that portray it as a dual-objective endeavour: creating an environment of support and trust on the one hand, and assuring that the discussions will meet demanded standards and depth on the other, to allow participating teachers to gain new knowledge (Sherin & Han, 2004). In light of this complexity, one can understand the rationale for appointing more than one facilitator per group. The next sub-section deals with the advantages and pitfalls of co-facilitation.

3.4 Co-facilitation

Here, we focus on a special form of facilitation, namely co-facilitation, which brings into consideration not only how this unique kind of facilitation supports collaboration within a group, but also how the collaboration of the co-facilitators themselves is formed and characterised. Co-facilitation is a case when a shared responsibility exists between two (or rarely, more than two) facilitators in regard to leading a certain collaborative group of teachers. Even though several authors (e.g. Gitterman & Shulman, 1994; Rothman, 1981) consider solo-facilitation as the most effective way of leading a group, more recent sources (e.g. Cohen & DeLois, 2002; Nachlieli, 2011; Reid & Demissie-Sanders, 2014) describe cases when co-facilitation has brought a new dimension to this role.

Edwards (2009) understands the knowledge of ‘who can do what in the best way’, as the core of co-working, since collaborative tasks require resourceful use of the expertise of others and mutual alignment of the professional practices of collaborating persons. The ability to offer support and to ask for support from others is closely connected to the sharing of agency, as in general a collaboration includes “negotiating roles and relationships” (Quaresma, 2020, p. 530).

In several reported cases of successful co-facilitation (e.g. Cohen & DeLois, 2002; Nachlieli, 2011; Novotná et al., 2013), at least one of the collaborating facilitators was based in university, and at least one of the two had experience as a teacher. Considering the high expectations from facilitators in collaborative groups, their varied backgrounds may be a positive factor in the successful sharing of responsibilities while leading groups. For example, in the case reported in Theme C by Pöhler (2020), one facilitator was a teacher educator and the other was an experienced teacher. The former provided the theoretical introductive part of the session, and the latter orchestrated the whole-group discussions that followed the small-group work. Nachlieli (2011) presented another example of a researcher and an experienced teacher co-facilitating. Their distinct backgrounds and diverse areas of knowledge enabled them to accommodate different roles during the PD: the experienced teacher encouraged the discussion about specific classes, while the researcher attended to generalising, theorising or hypothesising by moving the conversation to the general classroom. Interestingly, both used the same type of communication move (in this case, confrontation), but with different objectives and effects.

In another Theme C paper, Medová et al. (2020) reported on a process of developing a relationship of trust and of sharing responsibilities between two facilitators from their very first co-facilitated session. In this case, when the less experienced facilitator encountered challenges while leading the PD, the more experienced facilitator first stood back and provided her with an opportunity to gain more experience. He switched to the role of observer of the session, which enabled him to reflect-in-action, which in turn led several minutes later to his decision to intervene. This scenario provided information used in consecutive, post-session reflections shared by the collaborating facilitators. Medová et al. concluded that the time invested in co-facilitation contributed not only to the facilitation itself, but also to the professional growth of the facilitators. This may be connected to the construct of relational knowing (Hollingsworth et al., 1993), constructed in open conversations during which participants spend their time learning, enriching and trying to understand each other’s ideas, rather than just completing the forming of an idea (Hollingsworth & Dybdahl, 2007). In school contexts, such sharing of knowledge can be particularly valuable, especially in view of the long-term relationships between colleagues, whose identities and roles may develop.

Cohen and DeLois (2002) stressed the importance of collaborators feeling pleased with the collaboration from the very first session, despite possible objective discords. Successful co-facilitation depends on finding a balance between co-facilitators filling their individual roles and performing as collaborating practitioners. In order to allow for such successful co-facilitation, it is imperative that the collaborating facilitators spend time together between sessions. A considerable amount of interaction and exchange is necessary for developing productive collaboration, including open communication, mutual regard and trust (Reid & Demissie-Sanders, 2014). In some cases (e.g. Cohen & DeLois, 2002; Medová et al., 2020), a previous friendship or acquaintance may be an asset, but it might also cause an impediment, as sometimes uneasy conversations between co-facilitators may be necessary, particularly when long-held routines are challenged while working with practicing teachers. Meetings between sessions should not be limited to planning future activities and dividing primary responsibility for each part of the content. The practice of shared reflection of co-facilitators on previous sessions appears to be fruitful, not only for the evaluation of the session (Eriksen & Solomon, 2022; Medová et al., 2020; Zaslavsky & Leikin, 2004). The presence of two persons may offer an adequate reflective distance and allow deeper understanding of the situation. The shared reflection can also be the means of enhancing the relational knowing and professional knowledge of the co-facilitators involved, but it requires an openness to honest feedback.

The question of how a collaboration between facilitators may influence the building of community and collaborative relationships within the group of teachers is under-researched. There is a need to explore further co-facilitation settings and identify characteristics that have impact on teacher collaborative work.

To sum up this section of the chapter, the facilitator role was examined from various perspectives, including: a theoretical perspective providing different categories for facilitators’ knowledge; a profession-oriented perspective on facilitators’ practices and moves, elaborating on facilitators; implementation of their knowledge and skills; a pragmatic perspective concentrating on the challenges involved in facilitation, illustrating the complexity of this endeavor; lastly, the unique perspective of co-facilitation. In the next section of this chapter, we explore the question of how one becomes a facilitator of teacher collaboration, and what may support the development of professionals assuming this role.

4 Exploring Professional Trajectories of Facilitators of Mathematics Teacher Collaboration

This section focuses on facilitators’ own professional development. First, we unpack what becoming a facilitator might mean, taking into account that often facilitators hold other positions as well, and may have different starting points such as teachers or researchers (Sect. 4.4.1). Then, we review and compare different types of existing preparation programs for facilitators and the design principles upon which they are based (Sect. 4.4.2). Next, we refer to the development and changes in different aspects of facilitators’ work that may occur over time (Sect. 4.4.3), and review some findings regarding identity and agency shifts (Sect. 4.4.4). Finally, means and models for supporting facilitators along their work period are discussed (Sect. 4.4.5).

4.1 Becoming a Facilitator

The process of becoming a facilitator (or an MTE in general) has only started to gain research attention at the beginning of the 2000s (e.g. Even, 2008; Llinares & Krainer, 2006; Zaslavsky & Leikin, 2004). There is no designated route for becoming a facilitator of mathematics teacher collaboration, especially since the backgrounds of facilitators may vary considerably. For instance, facilitators may hold the following positions: mathematics teachers; researchers; teacher coaches; mathematicians; leaders at different school levels; independent consultants and more. These backgrounds may also overlap and, moreover, they may take different forms within different cultural contexts or traditions. For example, in the Chinese Lesson Design Study (Li et al., 2011; Yang, 2014), facilitators are expert teachers, who also conduct research and publish papers. These expert teachers have gone through a process of shifting from being ‘ordinary’ teachers into exemplary role models that serve as leaders for other teachers.

Even (2014), in her Commentary on the ZDM Special Issue on the practices and professional development of didacticians (Jaworski & Huang, 2014), noted that professional development of those who lead PDs for mathematics teachers is commonly not an organised activity, but rather occurs as a by-product of the project. She pointed to the need to understand better what facilitators should learn, as well as how and when they can learn it. The spectrum of backgrounds that facilitators may have makes this a complex endeavour. Yet, the case of facilitators who are former (or still practicing) mathematics teachers is relatively common (as in the Chinese case above) and merits special attention. Dinkelman et al. (2006) claimed that most practicing facilitators were teachers at some point. As several of the Theme C papers reveal, being a classroom teacher before or simultaneously to being a PD facilitator is a prevalent situation, at least within the cultures represented in these papers (this might not be the case in some Latin American countries, for example).

Perry and Boylan (2018) drew on the concepts of first- and second-order roles of facilitators (Murray & Male, 2005). A first-order role is that of teacher in the classroom, whereas a second-order role is a step removed from the classroom (e.g. researchers or independent consultants). A teacher who is also a facilitator constantly moves between the first-order and the second-order roles. Considering such transition can draw upon other relevant transitions. For example, Labaree (2003) pointed out that, “In many important ways, the transition from teacher to educational researcher is a natural and easy one. As prospective researchers, teachers bring many traits that are ideal for this new role, including maturity, professional experience, and dedication” (p. 15). Similar arguments can be used regarding those who make the teacher–facilitator transition. In other words, an experienced teacher could be considered well equipped to become a facilitator, based on maturity, professional experience and motivation (assuming that facilitation is commonly chosen rather than being imposed).

However, just as Labaree (2005) also noted aspects that may make the teacher–researcher transition complicated, the same could be said about the shift from teacher to facilitator. Indeed, Even (2005) asserts that, “being a good teacher does not necessarily imply the ability to help others develop their teaching” (p. 344), and papers from different countries presented within Theme C demonstrate that switching from a teacher position to a facilitator position is far from being trivial (e.g. Schwarts, 2020; Widjaja & Vale, 2020). The question of whether and how the ability to be a good facilitator can be learned is still an open question in contemporary research, which is being explored in recent years in various contexts.

Maaß and Doorman (2013), based on Müller (2003), have suggested that a longitudinal model of the learning and development of PD facilitators (they use the term ‘multipliers’) should include three phases: Learning-off-job, which consists of gaining fundamental knowledge, preferably within organised seminars or preparation courses; Learning-by-job, which consists of using the knowledge acquired in the first phase for planning and implementing a PD, with a close support and counselling by experts; finally, Learning-on-job, which consists of further growth enhanced by experience, reflection and peer support (alternatively, Zaslavsky & Leikin, 2004, integrated the second and third phases into one complex model of growth-through-practice). This three-phase model was utilised in the EU PRIMAS (Promoting Inquiry-based learning in Mathematics and Science), in which 12 European countries participated (Maaß & Doorman, 2013; reported also by Sikko & Ding, 2020 within Theme C). In some of these countries, the prospective facilitators were schoolteachers who were qualified as facilitators by going through a training period at the university. Schwarts (2020) described a trajectory along which teachers became facilitators in a video-based PD program in Israel: first, they experienced the PD as teachers, then they developed into prospective facilitators by taking a facilitator training course, after which they became novice facilitators supported by more experienced peers. Finally, over time, they became experienced facilitators themselves and could mentor new peers.

In the following, we focus on research exploring the process of becoming a facilitator and developing as a professional within that role. We begin with studies examining the preparation stage, i.e. facilitators’ pre-service period, and continue with studies looking into changes occurring within facilitators’ in-service periods.

4.2 Preparation of Facilitators of Mathematics Teacher Collaboration

Literature on the preparation of facilitators towards leading mathematics teacher collaboration is still relatively limited (Jackson et al., 2015; Roesken-Winter et al., 2015). In particular, little is known about how facilitators may be prepared to become skilled in putting different knowledge aspects into practice. However, the body of research that unpacks what the role of a facilitator may entail, reviewed in Sect. 4.3, can be drawn upon when considering the preparation of facilitators.

Some research suggests, in accordance with the needed requirements of academic knowledge as well as social and interpersonal skills (see Sect. 4.3.1), that key to successful preparation of facilitators is the enhancement of solid acquaintance with mathematical content, skills in establishing norms and using prompts that support productive discussions, and an inquiry stance towards practice (Borko et al., 2008; Elliott et al., 2009). Attention to creating a safe and supportive learning environment and establishing trust, and explicit articulation of key facilitation practices, were also found crucial to the process of preparing facilitators (Borko, 2004; Borko et al., 2014). The importance of having facilitators first experience the PD as learners, before shifting to their role as facilitators, was highlighted as well (Koellner et al., 2008; Kuzle & Biehler, 2015).

Regarding learning artefacts that can serve in the process of preparing facilitators, analogies can be created between preparing teachers and preparing facilitators. At the teacher level, video vignettes and students’ work samples have been found to be useful to capture the complexity and dynamic interaction of classroom practice, and to support teachers in developing their ability to notice, critically reflect and direct productive discussions among students (Borko et al., 2008; Brophy, 2004; Rosaen et al., 2008; van Es & Sherin, 2008). By analogy, repeated viewing of video vignettes with different foci can potentially support the development of facilitation skills and foster productive professional discussion among teachers (Borko et al., 2017). Several researchers have argued that it is critical to select video vignettes (either from mathematics classrooms, as proposed for example by Borko et al., 2017, or from PD sessions, as reported by Lesseig et al., 2017) with clear goals, and embed these in activities that are planned thoroughly to assist prospective and beginning facilitators to learn how to support teachers’ pedagogical reasoning.

4.2.1 Design Principles for the Preparation of Facilitators

Although still relatively scarce, some research has been published on preparation programs for facilitators within various programs across the world, and these allow discussing curricula and design principles developed for this purpose.

In Germany, Roesken-Winter et al. (2015) reported on a 1-year PD course they conducted for 12 PD facilitators. Participants were mathematics teachers (at the secondary school level) who were to become responsible for PD in their federal state. The course was designed by utilising design principles derived from literature on effective PD for teachers, and one of the research aims was to inspect whether such principles could be effectively adapted to the level of facilitators’ PD. The study focused on two design principles, named participant-orientation and competence-orientation, that were rated by the facilitators as the most relevant principles, in surveys that were conducted 6 and 10 months after the course. The principle of participant-orientation means that several modes of participants’ active involvement in their own learning were employed (e.g. self-study, e-learning, practical try-outs, portfolio writing). The principle of competence-orientation involves the operationalisation of several types of facilitators’ expected competencies (e.g. in the cognitive, affective-motivational and technical domains) into a detailed framework used explicitly in the course for discussing how these competencies may be strengthened.

In the Unites States, Elliott et al. (2009) reported on a training model consisting of a series of seminars aiming at facilitators’ learning of how to cultivate mathematically rich PD environments, within a project named Researching Mathematics Leader Learning (RMLL). The design principles of the seminars were based on two frameworks adapted from classroom-based research to the PD level: socio-mathematical norms (Yackel & Cobb, 1996) and the five practices for orchestrating productive mathematical discussions (Stein et al., 2008). These principles include: (1) mathematical coherence of the tasks; (2) socio-mathematical norms as a perspective to explore what counts as adequate mathematical explanations and justifications within teachers’ discussions; (3) purpose-based learning of facilitators; (4) opportunities for facilitators to connect the work in the seminars with their own work of facilitating teacher learning; and (5) cultivating a stance of inquiry, through which facilitators consider the affordances and limitations of specific pedagogical moves and recognise that there is no single best ‘protocol’ for facilitating mathematics PD.

In a second phase of the RMLL project (Lesseig et al., 2017), the researchers have added the MKT framework (Ball et al., 2008) to the other two previously considered, and introduced a revised set of design principles for PD of facilitators, that emphasise mathematical goals. These include the following: (1) facilitators’ mathematical work should be based on clear learning goals for teachers, in order to differentiate distinctly between teachers’ and students’ learning needs; (2) tasks designed for facilitators are more effective if they explicitly relate to Specialised Content Knowledge (SCK); and (3) using video-cases exemplifying instances when mathematical goals for teachers are pursued is helpful in supporting facilitators’ awareness to pedagogical and mathematical aspects of facilitating teacher learning.

In Israel, within a video-based teacher PD project named VIDEO-LM (Viewing, Investigating and Discussing Environments of Learning Mathematics; Karsenty & Arcavi, 2017), a preparation course for prospective facilitators was implemented. Reported design principles for the preparation model (Karsenty, 2016) included the following: (1) Relevance – course activities are directly linked to realistic issues that the facilitators are expected to deal with, according to the experience accumulated in the PDs already conducted; (2) Feasibility of learning goals – prospective facilitators need to learn from the experiences of other facilitators, thus vivid cases of facilitation should be presented in the course, along with a set of tools to analyse these cases (this set of tools was named ‘meta lenses’, see Karsenty, 2020); (3) Commitment to the project’s agenda and norms – in analysing cases, explicit and recurring references need to be made to ideas at the core of the project (e.g. the six-lens framework used with teachers desired norms such as maintaining nonjudgmental discussions); and (4) Modeling – leaders of the facilitator course should maintain facilitating that is aligned with what participants are expected to do as course leaders in the future (e.g. maintain a supportive atmosphere, use diversified and engaging activities).

Looking at these cases from different countries, some general features of what preparation courses for facilitators focus upon can be distilled. The most salient features are the attempted efforts to link the course experiences to actual expected facilitators’ practice; the centrality of carefully-designed course tasks (e.g. analysing video-cases) as springboards to discuss what teachers’ learning within PDs may entail; the use of specific analytic tools provided to participants for making sense of the cases they inspect (e.g. socio-mathematical norms and the five practices in the case of RMLL, meta-lenses in the case of VIDEO-LM); finally, the emphasis put on active involvement of facilitators in their learning. Broadening this last feature, several researchers have advocated that opportunities given to facilitators, actively to try out facilitation skills in their own schools, are critical (Borko et al., 2021; Elliott et al., 2009; Roesken-Winter et al., 2015).

4.3 Development and Change in Different Aspects of Facilitators’ Work

Findings regarding changes and development in facilitators’ proficiency (e.g. their knowledge, practices, competencies and identities) are beginning slowly, yet steadily, to accrue. In this sub-section, we analyse findings from six projects, reported in the following papers (ordered chronologically): Zaslavsky and Leikin (2004); Shagrir (2010); Jackson et al. (2015); Roesken-Winter et al. (2015); Perry and Boylan (2018); Schwarts (2020). The six studies are briefly overviewed in Table 4.1. We draw on collective salient findings from these projects to discuss development and change in facilitators’ identity, knowledge, beliefs and attitudes; development and change in facilitators’ practices; the influence of collaboration and interaction on facilitators’ changes and development.

Table 4.1 Overview of the six studies

4.3.1 Development and Change in Facilitators’ Identity, Knowledge, Beliefs and Attitudes

Different authors have reported an increased involvement of facilitators in the program they represent, over time (Schwarts, 2020; Zaslavsky & Leikin, 2004). With regard to identity aspects, several studies found that facilitators gradually consolidated their identity pertaining to this role, and saw themselves as more proficient and confident as time passed (Schwarts, 2020; Shagrir, 2010; Zaslavsky & Leikin, 2004). Schwarts (2020) characterised the professionalisation process of a novice facilitator, who facilitated a collaborative PD within a video-based project for the first time, over 1 year. Findings showed that, while in her first session the facilitator did not take most of the relevant components of the project’s goals and agenda into account, at the end of her first year she structured the PD sessions according to these components. By that time, the facilitator finally saw herself as a leader whose role in the VIDEO-LM project was to involve teachers in meaningful peer-discussions and to foster collective reflection. Similarly, Zaslavsky and Leikin (2004) pointed out that novice facilitators are often not very confident in their qualification to undertake this role when they begin working, and expressed a need for guidance from more experienced peers or from the project’s team. They noted that, in many cases, the development of facilitators during the project’s duration included growth and transition from the role of mathematics teacher to that of a facilitator. According to Zaslavsky and Leikin, this development is accompanied by facilitators’ enhancement of knowledge.

Roesken-Winter et al. (2015) maintained that the improvement and deepening of facilitators’ competencies over time, for example in terms of pedagogical content or content knowledge, is crucial to the success of the project. Some of the studies have shown facilitators’ changes and development in this area (Perry & Boylan, 2018; Roesken-Winter et al., 2015; Shagrir, 2010; Zaslavsky & Leikin, 2004), with different foci on aspects and dimensions of knowledge and skills. Shagrir (2010), for example, stated that most of the participants in her study (94%) testified that they improved their performance as facilitators during their participation in the program, referring to their professional knowledge and skills, as well as the professional language they use. The facilitators emphasised that the project kept them well-informed of global research, theories and existing approaches in the context of teacher education.

The findings of Roesken-Winter et al. (2015) showed a significant positive development of self-estimated competences of facilitators, in each of the seven considered knowledge and skills dimensions (e.g. mathematical content knowledge; pedagogical content knowledge; PD management), over the course of the continuous PD program. Interestingly, the facilitators acknowledged those dimensions in which they themselves had strong self-efficacy, as important for their participating teachers.

Perry and Boylan (2018) investigated facilitators’ professional learning, practice and change, and found that the most salient outcomes of their ‘Developing the Developers’ program were related to facilitation knowledge and skills, and to knowledge about professional development. Facilitators who participated in this study felt less need for learning in the area of content knowledge (e.g. subject-specific content knowledge or pedagogical content knowledge) or in regard to skills for teaching, which reflects the ‘second order’ nature of their role (see Sect. 4.4.1). Another type of change documented in this study, exemplified by one of the facilitators, was a newly constituted “belief that sharing theories of professional learning with the participants leads to greater impact and engagement” (p. 263). This stemmed from the facilitator’s positive experience with an activity in his own PD session with teachers, in which he explicitly used Clarke and Hollingsworth’s (2002) interconnected model to stimulate a discussion on how teacher learning can occur through professional development.

4.3.2 Development and Change in Facilitators’ Practices

The authors of the six papers referred to facilitation strategies (Perry & Boylan, 2018), facilitators’ managing of teachers’ contributions or ideas (Jackson et al., 2015; Schwarts, 2020), facilitators’ sensitivity to, and view of, teachers’ learning (Jackson et al., 2015; Zaslavsky & Leikin, 2004), facilitators’ design of activities for teachers (Zaslavsky & Leikin, 2004) and facilitators’ pursuit of goals for teachers’ learning (Jackson et al., 2015).

With regard to facilitation strategies, Perry and Boylan (2018) noted that changes in the domain of practice could occur by trialing new strategies by the facilitators. As an example, the authors presented a case of a facilitator, whose PD session openings were too long at the beginning. In later PD sessions, she therefore tried a new opening technique and perceived it as an improvement, since teachers were evidently more quickly engaged with their professional learning.

Schwarts (2020) reported on a development concerning one facilitator’s dealing with teachers’ contributions: whereas the facilitator initially demonstrated a practice of mainly listening, later she perceived her role as facilitator in a more active manner and was able to be more adaptive towards teachers’ contributions. She also drew on her teaching resources and no longer saw her teaching and facilitating practices as mutually exclusive. This case study shows that keeping the identity of an experienced teacher alongside a facilitator identity “can enrich practice and keep the facilitator practice-oriented while working with her colleagues” (p. 547). In contrast, Jackson et al. (2015) could not identify a linear progression in terms of facilitators’ capacity to capitalise on teachers’ ideas in PD sessions. Indeed, they reported that their facilitators initially elicited teachers’ ideas, but tended to give teachers little direction and did not press for elaboration or build on teachers’ contributions in meaningful ways. The quality of the facilitators’ ability to elicit and build on ideas did not increase successively over time, but varied across activities, apparently depending on their nature and on the associated goals for teachers’ learning.

Other research findings concern an evolvement of facilitators’ sensitivity to teachers’ ways of thinking, facilitators’ awareness to what may be expected of teachers within a PD (Zaslavsky & Leikin, 2004) and facilitators’ views of teachers’ learning (Jackson et al., 2015). As mentioned in Sect. 4.3.2, the facilitators in the study of Jackson et al. (2015) initially tended to approach teacher learning as rectifying deficits in their understanding and practice, which could be realised by using an isolated activity or by showing good practice. However, with time, they began to approach teacher learning as a developmental progression and to conceptualise the PD sessions in terms of a sequence of linked activities.

In addition to the reported improvements in facilitators’ ability to design PD activities as a sequence, the focused studies showed an expansion of facilitators’ repertoire of PD activities (Zaslavsky & Leikin, 2004) or a shift from designing PD activities that focused on peripheral aspects to more central aspects of instruction (Jackson et al., 2015). However, despite the finding that the designed PD activities were increasingly focused on the core issues, the facilitators “frequently evidenced a ‘show-and-tell’ approach when they facilitated those activities” (Jackson et al., 2015, p. 102). The authors therefore concluded that, sometimes, the PD activities did not seem to support teachers’ learning in the goal-oriented way intended.

In conclusion, it must be noted that changes and development in facilitators’ practices would be necessary, but not sufficient, for instructional improvement at scale (Jackson et al., 2015.). This is related to the reality that the impact of the PD is also “mediated by other aspects of the contexts of teachers’ work” (p. 94), for example the instructional expectations that school leaders communicate to teachers.

4.3.3 Influence of Collaboration and Interaction on Facilitators’ Changes and Development

Across the six studies, a great importance was ascribed to facilitators’ collaboration with each other as a resource for their professional learning (Perry & Boylan, 2018; Shagrir, 2010; Zaslavsky & Leikin, 2004). Facilitators themselves also perceived such collaboration as positive, seeing their colleagues as a support group that helped them cope with the difficulties and challenges of their new job (Shagrir, 2010).

Correspondingly, facilitators’ changes are often related to input they have received from peers, in terms of peer observations, feedback or suggestions such as new facilitation techniques (Perry & Boylan, 2018). In this context, Zaslavsky and Leikin (2004) argued that facilitators’ learning by collaboration occurs via awareness of differences between their own and their colleagues’ positions, which then leads to discussions of their different stances in an attempt to find a shared position. Furthermore, the exchange of ideas with more experienced persons (for example, project leaders), or the observation of their facilitation moves or strategies, could have a positive effect on facilitators’ change and development as well (Zaslavsky & Leikin, 2004). In addition, they report facilitators could also learn from teachers’ contributions and from a careful reflection on how teachers react to the offered PD activities. As a result of such reflection, facilitators often vary, modify, reduce or expand the PD activities over time.

To conclude, the different studies reviewed here show that facilitators “can undergo a professionalisation process and become proficient facilitators” (Schwarts, 2020, p. 546) and that participation in a longer-term program could motivate a novice facilitator “to act like a professional one” (Shagrir, 2010, p. 52). In this context, Roesken-Winter et al. (2015) stated that, from their data, “a clear picture emerges: convincing multipliers that they can trust what they were taught in their CPD [continuous PD] courses is a direct link to implementation into their own CPD practice” (p. 22).

The six papers discussed here are promising steps towards understanding facilitators’ change and development over time; however, this aspect is still under-researched and more studies are needed to consolidate the results reported and to explore further directions. In particular, the place of mathematics within facilitators’ development of knowledge and skills is still strikingly absent from most existing research on facilitators’ professionalisation processes, and remains a challenge for future research, as was noted in the Theme C group discussion.

4.4 Shifts in Agency of Facilitators of Mathematics Teacher Collaboration

Much has been written about mathematics teachers’ agency and the importance of maintaining this agency in PD initiatives, but ‘lifting’ this construct to the facilitator level has not yet been established. One possible conceptualisation of facilitators’ agency could be made by considering the widely-used definition by Holland et al. (1998), which refers to teachers’ agency as their “realized capacity […] to act upon their world […] purposively and reflectively” (p. 42), and which may be appropriate for facilitators as well. It appears that the notion of agency is highly connected to the notion of identity. As Goos and Bennison (2019) stress in regard to mathematics teachers, “there are many factors influencing identity development, and […] individuals can exercise agency by authoring their own identities” (p. 416). Thus, we may conclude that, both for teachers and for facilitators, such increasing capacity for agency is connected to the possibility of informed identity building. Some questions that are of interest in this emerging field are:

• How is facilitators’ agency manifested?

• To what extent can studies on facilitators’ agency draw on the research regarding teachers’ agency?

• What design principles may inform programs that wish to ensure facilitators’ agency?

• What are the relationships between fidelity, integrity and agency?

These questions are complex to investigate and lie heavily on how one defines each term. For example, Brodie (2019) argued that, “teachers always enact agency, even when they choose not to act, or might seem to ‘passively’ accept policies or practices from others” (p. 562). This implies that understanding the different ways facilitators enact agency requires investigations of their underlying motives. Since there are almost no studies published in this field, we can only refer to few preliminary results available so far regarding facilitators’ agency.

From the analysis and comparison made in the Theme C paper by Sikko and Ding (2020), it can be concluded that agency may take different forms in different cultural contexts, that is different places in the world with different educational systems and cultural assumptions. In addition, Sikko and Ding highlighted that agency may help bridge the gap between research and practice, and serve as the missing link for the successful execution of reforms. Still, developers of PD programs do not always consider teachers’ and facilitators’ agency within their design.

In Müller’s (2003) model, used in the PRIMAS program (see Sect. 4.4.1), the third developmental stage (Learning-on-job) suggests that:

it is the multiplier’s self-education that leads them to develop their competences. […] Once a need is discovered, the teacher educator must search for related information to improve their knowledge and carry out ‘experiments’ by asking questions about their way of teaching or running professional development courses. (Maaß & Doorman, 2013, p. 892)

From these comments, it appears reasonable to suggest that facilitators’ capacity to detect their needs and actively to find ways to respond to them (that is, their agency) is required for facilitators’ learning and development.

The transfer from teacher to facilitator also involves a shift in the focus of agency. A teacher’s prime focus is the development of students at school, while a facilitator’s prime focus is the development of teachers, with student development taken as a secondary goal or as a sign of teacher development. As Perry and Boylan (2018) argued, the impact of PD facilitators on student outcomes is mediated by the ‘second-order’ nature of their role (Murray & Male, 2005; see Sect. 4.4.1) which separates them from direct classroom impact. Thus, for facilitators “to act upon their world” (Holland et al., 1998, p. 42) means different things from what it means for teachers.

Dinkelman et al. (2006) investigated two teachers’ transitions from classroom teachers to teacher educators. They found that the shift in agency was not an easy one, and that both subjects retained elements of their identity as classroom teachers, while at the same time acquiring the new identity as facilitators. Kohen et al.’s (2020) Theme C paper provided insights regarding a PD in the form of a problem-solving forum, that enabled teachers to experience both the role of problem solvers and the role of mentors. A case study of one experienced teacher showed the development from a pure problem solver, with similar behaviour to those of students wanting to demonstrate how clever and quick they are in their solutions, into a mentor who supported the thinking processes of others. This case demonstrates the importance of first-hand experience as a student for development as a good facilitator. It also demonstrated the shifts in agency involved in this development.

Schwarts (2020) provided insight into how facilitators’ practices and identities evolve, via an in-depth case study of a novice facilitator working within a video-based PD program (see detailed account in Sect. 4.4.3). The facilitator’s conflicting identities, that of a teacher and colleague, and that of a facilitator, yielded challenges that were prominent both in the planning of the PD sessions and in managing teachers’ discussions and reflections. Schwarts suggested several means by which the facilitator’s identity as a leader may be strengthened, which, as Goos and Bennison (2019) have argued, may have positive consequences for the shifting of agency. Such means are elaborated in the next sub-section.

4.5 Supporting Facilitators’ Professional Development

The support and guidance provided for facilitators during their professionalisation processes (also framed as part of continuous professional development – CPD – for facilitators; see Roesken-Winter et al., 2015) is another important element that influences the success of facilitators in supporting mathematics teachers’ collaboration. This is especially crucial for novices in the field of facilitation.

As already discussed in previous sub-sections, reviews such as that conducted by Even (2014) have shown that facilitators’ qualification is often informal and spontaneous, with few projects providing formal preparation (see Sect. 4.4.2). Accordingly, institutionalised on-job support for facilitators is also not common and rarely investigated and reported, even within the accumulated literature on facilitators in recent years. The professional support that facilitators may receive is almost always reviewed as background or context for studies on facilitators, and not as the subject under investigation. Thus, within PD research in mathematics education, it appears that minimal attention has been given so far to the question of how to support facilitators in designing and leading high-quality PD (Elliott et al., 2009; Weißenrieder et al., 2015). As Elliott et al. have stated more than a decade ago:

The research community has lagged behind in providing insights on how to best support these new leaders as they facilitate teacher learning. Filling the knowledge gap in the research on leading PD is an urgent issue if teacher learning is to be improved and adequately addressed. (p. 365)

Although some progress has been achieved since then, the gap still exists. In this sub-section, we relate to several core questions regarding this issue, and provide an overview of some suggested answers, gathered from the limited amount of works dedicated to facilitators’ professional support.

Herein, we define professional support for facilitators as the in-service education offered to those taking up the role of facilitating mathematics teachers’ collaboration. Support may include a CPD course, mentoring or guidance sessions, or any other space that allows for learning and reflection about the practice of facilitation. The core questions we address are the following:

• What are the goals set for supporting facilitators and what challenges may be involved in this endeavour?

• Which models and means exist for this purpose?

• What is an effective support and how can it be studied?

4.5.1 The Need for Support and Challenges Associated with It

In making the case for the need to support facilitators in developing their skill and expertise, we reflect upon the literature on novice teachers and their transition into the profession. It has long been established that various challenges associated with entrance to the profession of teaching may be addressed using professional support, such as sessions with a personal, content-specific mentor at the school, practicum sessions for first-year teachers, induction programs that include PD designed at the school or even the district level, etc. The model of apprenticeship, a key term in situated learning (Collins et al., 1988), is common in many professions such as law and medicine, and is based on the assumption that novices may learn the fundamentals of the profession by observing, imitating and holding continuous conversations with experts in the field. Similar to other novices, beginning facilitators experience various challenges, some of which are discussed in Sect. 4.3.3 above. For example, they might find it challenging to manage productive discussions (Borko et al., 2014), to implement suitable activities (Jackson et al., 2015) and to navigate between their professional identities (Knapp, 2017; Schwarts, 2020). Hence, there is an essential need for professional support to assist them with these encounters. However, such support requires human and financial resources that are not always available in PD programs. It appears that provision of such support is dependent on the educational system in which the program is embedded. In many countries, PDs are not institutionalised, and may, for example, be initiated by universities as part of grant-funded projects. As such, the design of scaling up, and henceforth the education of facilitators, will vary according to circumstances and could impact the sustainability of the project. It follows that the fundamental challenge in providing support to build facilitators’ capacity is finding a feasible model that may be implemented within the system’s resources and organisational configuration. The subsequent challenge for PD designers is to incorporate appropriate and relevant content to offer facilitators within the chosen model.

4.5.2 Models of Support

Although scant, the literature provides some examples of models of support for facilitators, and these can be categorised as falling into two groups: the first group comprises models where support is blended with facilitators’ initial training, such as in the iPSC (Implementing the Problem-Solving Cycle) program reported in Borko et al. (2014) and in the CMP2 (Connected Mathematics Project 2) described in Jackson et al. (2015); in the second group, training and support occur in different distinct times. More common to the first group are embedded approaches where the support team and the facilitators meet before and after each PD session, to reflect on the previous session and build the next session together. This approach is typical for programs with a unified curriculum, where all the novice facilitators are about to provide the same content. An example of a model that would fall into our second category is the model offered by Müller (2003), employed in the EU PRIMAS program (Maaß & Doorman, 2013; see Sect. 4.4.1). The support provided in the Language-Responsive Mathematics Teaching Project in Germany (e.g. Prediger & Pöhler, 2019; Pöhler, 2020) and in the VIDEO-LM program in Israel (Karsenty, 2018) are examples of this category as well.

Another differentiation between models is related to their target audience and to the supporters’ other professional roles. Roesken-Winter et al. (2015) described a CPD for facilitators where the participants were facilitators of different PD programs, thus the content provided evolved around general design principles of mathematics teachers’ PD. However, in the majority of the reported models, the target audience comprised the facilitators of a specific program, and the supporters were part of the program’s team, usually researchers, or, much less frequently, district directors or instructors outside the team. This state of affairs does not necessarily reflect the situation in the field; perhaps only support designed and provided within an academic context is published in research journals. Therefore, there is need for more reports of support with an emphasis on sustainable models where other actors in the educational field provide their guidance.

The support itself may be offered in the more formal setting of a PD course, by personal mentoring or coaching, or in a more ‘natural environment’, such as co-facilitation or the program’s team meetings. In the latter setting, novice facilitators who are also members of the team can learn, in a situated manner, from expert facilitators. Zaslavsky and Leikin (2004) frame this kind of support as the learning of newcomers into the facilitators’ community of practice, and emphasise that these interactions are beneficial both for novices and for experts.

4.5.3 Means of Support

The literature describes several means of support that can be divided into reflection-related and content-related means. Reflection-related means are different methods or scenarios that provide facilitators with opportunities to discuss their practice. For example, in the support system developed within the VIDEO-LM program (Karsenty, 2018; Schwarts, 2020), facilitators have opportunities to reflect upon their practice in three different types of communication: with themselves, via writing journals and filling reflective questionnaires; with their peers, in facilitators’ bi-monthly group meetings; with experts, in personal mentoring sessions with senior team members who are experienced facilitators. Another arena for reflection is provided by participation in research. In fact, in many cases means of support overlap with research data collection, when sometimes the first leads to the second or vice versa (Zaslavsky & Leikin, 2004). For instance, Karsenty et al. (2023) described a research design where stimulated-recall interviews served as a research tool; however, a preliminary finding in this research suggested that facilitators considered these interviews as milestones in their professional development. Content-related means for support are the materials provided for facilitators such as suggested activities, manuals, video-clips to use, and so on. Facilitators’ meetings often integrate both kinds of means: they allow time for peer-reflection on practice (for example, while watching and analysing videos of facilitation) and time for engaging with content, for example working collaboratively on planning sessions, including rehearsals or simulations of ‘approximations of practice’ (Borko et al., 2017; Jackson et al., 2015) or discussing theories of professional learning (Perry & Boylan, 2018).

4.5.4 Features of Effective Support for Facilitators’ Professionalisation

Effective support for facilitators may be defined by its results, that is the extent to which facilitators are competent in executing the goals set for the relevant teacher collaboration. The following features appear to contribute to an effective support for novice facilitators’ professionalisation, as found in studies looking at such results:

• observation of other facilitators in conjunction with post-session reflection (Psycharis & Kalogeria, 2018);

• help of a personal supporter who is an expert facilitator for the specific PD program (Schwarts, 2020);

• the opportunity to investigate prior PD sessions (via analysis of videos) and to plan jointly for upcoming PD sessions with accomplished others (Jackson et al., 2015).

However, systematic understanding of how support contributes to facilitators’ development is still lacking. More research is needed in order to unpack how exactly support affects novice facilitators, and what kinds of support can be sustainable in projects that aim for large-scale dissemination.

In sum, it is clear that the professional development of facilitators is complex. Research shows that facilitators may have different backgrounds and starting points, and that they move through diversified trajectories of becoming facilitators and professionalising in their role. Several models for facilitators’ preparation and for their in-service support have been outlined here, and changes in different features of facilitators’ work such as knowledge, practices, beliefs, identities, attitudes and agency have been addressed. Knowledge about the professional trajectories of facilitators of mathematics teacher collaboration is accumulating, but more research is needed to extend our knowledge on becoming and developing as a facilitator. A further important consideration in the development of facilitators is how a facilitator negotiates and manages the environment in which the facilitation occurs. Even if a facilitator might be identified as reaching a high level of expertise, environmental factors over which the facilitator will have little or no control may impact the success of the collaboration. The complexity of environmental factors at play that may impact teacher collaboration is the focus of the next section.

5 The Role of the Environment in Mathematics Teacher Collaboration

At this point in the chapter, we move away from exploring the role of facilitators in mathematics teacher collaboration to look expansively at the environment of collaboration. Apart from facilitators, various participants make up such environments, and their actions and interactions will impact the outcomes of a collaboration. For the purposes of this section, we define environment as the setting in which teacher collaboration takes place.

The complexity of considering the impact of the environment on the roles, identities and interactions of various participants will become evident throughout this section. First, we overview models for analysing teacher collaboration, selected due to their specific reference to the environment of collaboration (Sect. 4.5.1). We then look internally to the environment created by the actors within a collaboration (Sect. 4.5.2). Next, we turn our attention outward to consider cultural and contextual aspects of environments in which various participants collaborate (Sect. 4.5.3). This is followed by an analysis of an often overlooked but frequently mentioned institutionally-imposed environmental factor that impacts teacher collaboration, that is time allocated for teacher collaboration (Sect. 4.5.4). The facilitator–environment relationship is then challenged through reviewing teacher collaboration without facilitators (Sect. 4.5.5).

At several points within this section, we refer specifically to two particular research reports presented at the ICMI Study 25 conference, because they describe teacher collaboration without a facilitator (Cao et al., 2020; Jütte & Lüken, 2020). As such, they provide valuable anchor points for analysis both of the environment and of the role of facilitators within it. As we return repeatedly to these studies, the complexity of analysing the environment in relation to teacher collaboration is highlighted.

5.1 Environments of Collaboration in Research: Review of Existing Models

In their plenary paper to the ICMI Study 25, Krainer and Spreitzer (2020) considered collaborative groups in mathematics teacher education to interrogate the diversity of roles, identities and interactions. They framed their analysis in accordance with three elements in teacher collaboration, one of which was the environment, thus emphasising the importance of this component. The other two elements were identified as actors and targets. To highlight the extensive and expansive nature of the environments in which teacher collaboration occurs, they listed “departments, schools, school boards, districts, committees, ministries or enterprises” (p. 23) as examples of environments that impact teacher collaboration. To analyse teacher collaboration on an international scale, Krainer and Spreitzer identified reports of mathematics teacher collaboration projects recently published (2018–2019) undertaken in Africa, Asia, Australia, North America, South America and Europe. Environments were only minimally addressed in these reports, with Krainer and Spreitzer concluding that there was little discussion about the environments in which the collaborations occurred, or even environments being mentioned as a factor impacting the success or otherwise of the collaborations. As they stated, “it would be interesting to read more about context” (p. 34), as “only a few initiatives describe the context and relevant environments having a potential impact on the initiatives” (p. 35).

Krainer and Spreitzer’s analysis revealed that it is not common for the role of the environment to be the focus of interrogation when large studies of teacher collaboration are reported. Yet, it is an important aspect of consideration in relation to sustainability and scalability if mathematics teacher collaboration is regarded as teacher learning. Underscoring the complexity of learning through collaboration, Krainer (2008) stated that, “it is important to take into account that teachers’ learning is a complex process and is to a large extent influenced by person, social, organizational, cultural and political factors” (p. 2). This statement reminds us of the need to pause and think about the impact of the environment on teacher collaboration, and to give due consideration to the fact that successful collaboration requires more than artful and expert facilitators.

In exploring teacher learning, Llinares and Krainer (2006) discussed the work of Peter (1995), who proposed a model that incorporates four analytic domains (personal, practice, inference, external) and four associated mediating processes (teacher knowledge and beliefs, classroom experimentation, valued outcomes, stimulus or support). This model emphasises the expansive nature of factors that impact teacher learning and, by extension for our chapter, teacher collaboration. It is the external domain and the mediating processes of support that are of interest here. As stated by Llinares and Krainer, “The nature of changes undergone by teachers suggests a link between individual change processes and external conditions determined by school culture” (p. 446). They also referred to Krainer’s (1994) model that includes four dimensions of action, reflection, autonomy and networking, and that can be used both to “design in-service courses and explain how teachers’ learning is generated” (p. 447). In Krainer and Zehetmeier (2013), this model is framed as a learning system, “where action, reflection, autonomy and networking are regarded as important educational dimensions to consider when analysing educational practice” (p. 875). The following statement elaborates the operation of these four dimensions in a teacher professional development project: “the focus is on discussing in groups, negotiating meanings and norms, sharing knowledge, collaborative learning, designing didactical contracts, institutional constraints and organizational and systemic aspects that foster or hinder teachers’ learning” (Llinares & Krainer, 2006, p. 450). In this summary statement, we can see the interconnected nature of the individual and the environment, with individual teachers engaging in group discussions which have their own meanings and norms, of applying their learning and sharing with others the outcomes of their experimentation, together with operating within the constraints and affordances of their working environment. While this model does not explicitly capture the role of the facilitator, it does consider the role of the environment on teacher collaboration. However, the role of the facilitator can be extrapolated here as taking responsibility for managing discussions among teachers, or between teachers and teacher educators, establishing the meanings and norms, facilitating the sharing of knowledge and negotiating the environment of collaboration.

Another model that serves to encapsulate the potentially problematic or supportive role of the environment in teacher collaboration was proposed by Goos (2013). In this model, Goos adapted Valsiner’s (1997) zone theory of child development to propose a zone theory model of teacher development and, hence, teacher learning. Valsiner’s zone theory incorporates Vygotsky’s well-known zone of proximal development (ZPD) and two other zones: the zone of free movement (ZFM) and the zone of promoted action (ZPA). According to Goos, the ZPA can be “interpreted as activities offered via teacher education programs, formal PD, or informal interaction with colleagues that promote certain teaching actions” (p. 523; italics in original). Further, “The ZFM structures the teacher’s environment, or professional context” (p. 523), and elements of the ZFM could include teachers’ knowledge and beliefs, curriculum and assessment requirements, but also organisational structures (school timetable, class allocations, student groupings) and organisational cultures. A succinct summary of these two zones is offered by Bennison and Goos (2013), who stated that, “The ZFM is related to the individual’s environment and includes the set of actions that the individual is allowed to perform, while the ZPA is the set of actions that are promoted by the individuals” (p. 4; italics in original).

From this perspective, the ZPA can be initiated by external influences and suggestions of facilitators of professional learning. The ZPA can also be initiated as a result of informal collaborations with others, and therefore may take place in the absence of a designated facilitator. ZFM is about teaching actions that are permitted; that is, the freedom with which teachers may be allowed to move in relation to developing and applying their learning. This zone captures both restrictions and affordances internal to the individual and external in the environment. As stated by Goos (2013), “Zone theory offers a useful framework for research that aims to understand the complexities of teachers’ learning […] But another important line of inquiry […] is concerned with promoting change in classroom practice, and […] zone theory can also provide guidelines for designing such projects and interpreting the outcomes” (p. 528; italics in original). It is an interesting exercise to consider zone theory in relation to the role of the facilitator and the role of the environment in teacher collaboration. The role of the facilitator is to promote teacher action, and therefore the facilitator operates and is located in the ZPA. The ZFM relates to the environment in which teachers are permitted to apply their new learning, but the environment can also be seen as a factor in promoting teacher action. In fact, as these two zones work together, Valsiner suggested that they be considered as a ‘ZFM/ZPA complex’ (Bennison & Goos, 2013, p. 4).

A further approach for interrogating teacher learning is through the concept of teacher agency. In Sect. 4.4.4, shifts in agency of facilitators were discussed. Here, we outline Imants and Van der Wal’s (2020) model of professional development, teacher agency and school reform, noting that it does not specifically address the role of the facilitator, but is included here as another lens for analysing the role of the environment in teacher collaboration. The basis of this model lies in theorising the extent to which teachers enact new learning, which, according to them, must be considered in relation to the environment in which teachers work. Both agency theory and organisational theory underlie this model. If we consider teacher collaboration as a process for seeking improvement of the current state, then the outcomes of such collaboration are both related to the actions of the teacher and in accordance with the environment in which s/he works. The complexity is that the role of the environment and the role of the facilitator in teacher collaboration are mediated by the activity of the teacher. As stated by Imants and Van der Wal:

Teachers’ enactments of their work environment in the context of professional development and school reform primarily are located in the teachers’ own sphere of influence, this is in the interactions where they practice their agency. Besides, general school conditions will affect teachers’ action in their own spheres mediated by teachers’ interpretations. (p. 5)

Agency theory, then, is potentially useful, by extension, for considering the role of the facilitator in teacher collaboration and how the environment may enhance or hinder mutual learning of teachers and other participants in collaborative teacher interaction.

The models presented in this sub-section are important as they bring our consideration of the role of the environment within teacher collaboration to the fore. They provoke us to look more deeply into various issues associated with the environment in relation to teacher collaboration. In the next sub-section, we explore different actors and the interactions between them, as we look internally to the environment created by various stakeholders in the collaboration.

5.2 The Environment Created by Different Actors Within a Collaboration

Different actors within collaborative communities, and the interactions between them, create the human environment in which teachers’ communities can develop (Cestari et al., 2006; Jaworski et al., 2017). Collaboration between teachers and other members of the school community not only contributes to the professional development of teachers, but also leads to better student achievements (Darling-Hammond et al., 2017; Robutti et al., 2016). In order to realise the foundation for collaboration that supports interaction between various actors, we turn to a fundamental theoretical perspective for collaboration within communities; that is, communities of practice (Wenger, 1998).

‘Community of practice’ is a well-known term used in reference to collectives of people committed to learning and developing through sharing their knowledge and practice (Wenger, 1998, 2011; Wenger & Snyder, 2000). Although regarded as a contemporary term, the premise of groups of people forming together to learn from and with each other is an enduring social practice that is age-old (Wenger, 2011). Of particular note is Wenger’s discussion of spontaneously formed groups in educational settings, where teachers meet together during break times, sit together in the staff room, sharing practice, commenting on student progress and discussing various approaches they have tried and found successful.

A community of practice is a joint enterprise in which members mutually engage to produce a shared practice that is reflective of their collective processes of learning. In communities of practice, different members can interact in different ways to support the community, to maximise its benefits. In some cases, communities of practice develop to solve a particular problem or to achieve a certain outcome, but, fundamentally, they are a group of people who mutually engage “in a process of collective learning in a shared domain of human endeavor” (Wenger, 2011, p. 1). Extending from this perspective, Jaworski (2006) proposed a shift from learning within a community of practice to forming a community of inquiry. This involves the process of critical alignment in which members of the community critically question and reflect upon their roles and on the purposes of the collaboration as they progress in their collaborative learning toward a shared product.

It is important to note that Wenger (2011) defined a community of practice as a group where members join through self-selection. He distinguished this from other organisational working groups in which members may have been forced to join. This is a critical distinction when considering the environment created by the actors in the collaboration. One of the Theme C studies describes how teachers work collaboratively in a self-selected manner (Cao et al., 2020). From this study, we see Chinese teachers’ informal, daily interactions with different people, including friends, leaders of local school-based groups, colleagues in the same subject group, the ‘neighbour who is sitting next to you in the office’ and mentors in apprenticeship programs. Cao et al.’s study suggests that, when seeking help with regard to teaching, Chinese teachers tend to pursue interactions with colleagues in teaching research groups or in lesson planning groups. Whilst not presented as a collaboration that reflects a community of practice or a community of inquiry, this study is interesting to consider in relation to the environment of collaboration and the voluntary roles of the actors.

Research also indicates that, through the experiences and interactions between different actors in a collaboration, actors might shift or modify their roles over time (Krainer, 2008; Kramarski & Kohen, 2017; see also Ribeiro’s, 2020 Theme C paper). According to Hunter (2010), the possible change of roles might be affected by several key aspects, including the nature of the communication and the collaborative partnership between the various actors, the setting for collaboration and the social and cultural backgrounds of the members in the community.

The interest in how teachers learn through interaction with other stakeholders is widely reflected in research on the Japanese Lesson Study model. This form of collaboration focuses on teachers’ interactions and involvement with colleagues and ‘knowledgeable others’ (Takahashi, 2014), through a practice-based approach within a professional community. Japanese Lesson Study was elaborated by Isoda (2020) at the ICMI Study 25 Plenary Lecture for Theme B. A Lesson Study cycle includes the development of a lesson plan, the implementation of the lesson plan in a classroom by one of the group members, while other teachers observe, and a post-lesson discussion to analyse its impact on students’ learning, in which the group may decide to plan an improved version of the lesson, and the cycle begins again (Chen & Yang, 2013; Fernandez & Yoshida, 2004; Lewis, 2002; Lewis et al., 2009). In some Lesson Study scenarios, and particularly so in Japan, actors operate in an environment where roles are clearly defined (as discussed in the Theme C paper by Clivaz & Daina, 2020; see also Clivaz & Takahashi, 2018, and Sect. 4.5.3 below). However, this might not be the case for other scenarios where Lesson Study is implemented. For example, as we mentioned earlier, the role of the ‘knowledgeable other’ varies across and within cultural contexts (e.g. Adler & Alshwaikh, 2019; Gu & Gu, 2016; Lewis, 2016; Takahashi, 2014).

Several Theme C papers described the roles of various participants in the collaboration. While not explicitly stated, the environment created by different actors within a collaboration can be extrapolated as a community is formed and we see actors shifting roles. It is reasonable to assume that the environment of the collaboration enabled forward progress of the group’s work. Pereira et al. (2020) describe a collaboration between three groups of actors: supervising teachers (counselors) who were university lecturers in mathematics, and who formed a link between the university and school administrators; the teachers in the project schools who acted as facilitators and helped pre-service teachers to adapt to the life of the school community; pre-service teachers who were studying for a mathematics degree course. Ribeiro (2020) describes how teachers and teacher educators, working collaboratively, created opportunities to reflect on their knowledge and to share their practical classroom experiences. A dynamic process of changing of roles was reported, with participants sometimes acting as learners and, at other times, acting as educators. Kohen et al. (2020) report on a collaboration amongst mathematics teachers who solved challenging problems in an online environment termed Problem-Solving Forums (PSF). The study’s focus was on the importance of collaboration for the development of teachers’ dual roles – from students in the PSF to mentors of others in this environment. These studies, and others as outlined previously (see Sect. 4.4.4), show a range of actors taking on various roles, as well as shifting roles in teacher collaboration.

At this point in this sub-section, and as we grapple with considering the environment created by the various actors in a collaboration, it is worth considering the seminal study of Horn and Kane (2015), in relation to opportunities for professional learning in mathematics teacher workgroup conversations. From their large corpus of data collected over a six-year professional development initiative, and working with groups of teachers located at their various schools in one district, Horn and Kane noted the progress of some groups over others. In their detailed analysis of collected survey, interview and observation data (during workgroups and in classroom practice), they concluded that, “a teacher community cannot be the only lever for change” (p. 415). They determined that teachers with greater knowledge for mathematics teaching brought much richer conversations to the workgroup, which in turn provided greater opportunities to learn for the whole group. It was noted that the presence of the facilitator promoted greater opportunities in all workgroups, but, in their absence, these opportunities varied considerably. The district contributed to the environment through provision of time for workgroups to meet. Internally, actors impacted that environment.

In this sub-section, we briefly overviewed Wenger’s (2011) concept of communities of practice and Jaworski’s (2006) concept of a community of inquiry. The former concept is useful for reflecting upon collaboration that occurs without a facilitator. We then provided examples of studies that included a range of participants who, in some cases, took on various roles. All participants contribute to the environment of the collaboration and serve to impact it. In the next sub-section, we turn our attention to considering cultural aspects of environments in which participants collaborate, and consider more deeply the facilitator–environment relationships.

5.3 Cultural Aspects of Environments in which Various Actors Collaborate

Facilitators operate to promote collaboration, but they do not operate in a vacuum. Facilitators move and act in varying environments. Negotiating the environment is an important part of the role of the facilitator in teacher collaboration. It is also interesting to consider the role of the environment when teacher collaboration occurs without a facilitator. The impact of the environment on teacher collaboration is considered in this sub-section by interrogating environments and how they served to assist, or worked to impede, the collaboration. From this analysis, we see how facilitators manage environments, but also view how environments without facilitators may support or otherwise hinder teacher collaboration.

Few Theme C papers specifically addressed the role that the environment plays in teacher collaboration, yet some papers referred to environments as contexts in which the studies took place, while showing the influence of context and/or culture. Several studies referred to the solitary, or private, nature of teachers’ work in schools. For example, Nieman et al. (2020) mentioned the “deeply established norms of privatization in US schools” (p. 500), with reference to Little (1990) and Lortie (1975). They further stated that only recently time has been built into the teachers’ workday to enable collaboration. This was echoed in another Theme C paper by Griese et al. (2020), who stated that the “last 40 years have been characterized by a shift from teachers working in isolation to collegial collaboration” (p. 468), and in line with the work of Krainer (2003) pointing to the gradual shift towards teachers’ work in more collaborative ways. Pereira et al. (2020) also mentioned the tradition in their country (Brazil) of teachers “working in a solitary way” (p. 510). Widjaja and Vale (2020) contextualised their study through reference to the work of Robinson and Timperley (2007), who discussed the difficult role of facilitators when teacher autonomy and privacy of classroom practice is the norm. They further referenced the work of Vangrieken et al. (2015), who described the impact of school culture on teacher collaboration where there is a “longstanding culture of teacher isolation and individualism, together with teachers’ preference to preserve their individual autonomy” (p. 35, as cited in Widjaja & Vale, 2020, p. 556). Through analysis of just this small collection of studies, it appears that, although gradually changing, there is a long-standing tradition of teachers working in isolation, where their practice is very private and individualised, in many countries across the world.

In contrast, in another Theme C paper, Cao et al. (2020) stated that their study was located in schools that have an established environment of teachers working collaboratively. They emphasised that, “Chinese teachers have traditionally worked in groups” (p. 444). This feature of Chinese teachers’ practice has been previously noted by Hargreaves and O’Connor (2018). Zhao et al. (2020) elaborate the specific roles of participants in Lesson Study in China, showing that clearly defined roles contribute to the environment for Lesson Study to take place. As stated by Zhao et al., the university educators/researchers bring theory on teaching and learning for teachers to draw upon as they develop lessons that are then viewed and critiqued by all. Zhao et al. reported on the initially passive role taken by the teachers in their Lesson Study research, due to cultural norms of deference to university educators/researchers as knowledgeable others. However, through the course of the project, group members began to engage in collaborative dialogue, facilitated by a teacher educator. Zhao et al. applied Activity Theory to analyse their data, concluding that effective Lesson Study requires “active boundary brokers”, “shared tools” and “more equal rules and division of labor” to “promote interaction between teachers and researchers” (p. 570).

Three specific Theme C studies are outlined here because of their contribution to considering the impact of the environment on teacher collaboration. Jütte and Lüken (2020) investigated teacher collaboration without a facilitator (elaborated in Sect. 4.5.4 below). They described two teachers who were required to teach together due to an organisational shift in policy that resulted in a more inclusive approach for teaching children with special needs in regular classrooms. The researchers reported that this collaboration, which, as was already said, occurred without a facilitator, was not overly successful for a number of reasons. When interrogating the environment, it can be seen that the collaboration was forced upon the participants, that there was little time provided by the organisation for the participants to plan together and get to know each other’s teaching style and strengths, and that, as identified by one of the participants, there was no “structural framework and guidance by the school administration which accompanied the collaboration” (p. 480). Yet, both teachers joined the teaching team voluntarily and shared similar views about the importance of inclusive education. In this case study, we see that the participants pointed to environmental factors as impacting their collaboration. The researchers described how the team was “not yet situated in a positive school culture” (p. 482), and posited the potential value of a facilitator to progress this collaboration in a productive way. In contrast, Cao et al. (2020), who investigated Chinese mathematics teachers’ informal interactions and collaborations without a facilitator (also elaborated in Sect. 4.5.4 below), reported positive outcomes. Their research discussed the environment in which teachers work in Chinese schools, and specifically emphasised that the “Chinese school collective culture provides teachers with opportunities to seek help ‘within their reach’” (p. 450). In this study, we see that the environment played a significant role in supporting teacher collaboration: as collaboration is a typical and unexceptional process in schools, teachers readily collaborated with colleagues, even without the presence of formal facilitation.

From another viewpoint, Choutou (2020) described her research into assisting teachers collaboratively to plan and teach interdisciplinary units in mathematics and arts classes. The researcher was the facilitator and Choutou’s study provides a personal account of the difficulties she encountered in her role. Of the environmental factors that impacted the collaboration, she mentions the difficulty in finding a suitable and quiet space to meet with participants, and the importance of scheduling meetings at a time when all participants could meet. She makes reference to wishing to avoid a ‘top-down’ model (as per Cestari et al., 2006, p. 1357) as the facilitator, but that potentially “a stakeholder from ‘above’ would make things easier” (p. 458), indicating how the environment could have provided greater support in progressing the collaboration in this study.

Of these three case studies presented in Theme C, the study by Cao et al. (2020) clearly emphasises the potential of the environment to impact teacher collaboration positively. In the other two cases presented, we see one report on teacher collaboration without a facilitator and one with a facilitator, and in both studies the environment seemed to have a non-positive impact on the collaboration. Other Theme C papers also touched upon the role of the environment in different collaborations. Clivaz and Daina (2020) discussed Lesson Study where all participants have well-defined roles, and the participation is led by a ‘knowledgeable other’. In this case, it can be seen that the environment provided clear demarcation of roles. Further, Nieman et al. (2020) discussed a whole-school vision of high-quality mathematics instruction emanating from the school principal. As emphasised by Nieman et al., it was the work of the school leader, who valued teachers’ collaboration and who created conditions necessary for it to occur, that ensured that the collaboration was effective. Interestingly, Nieman et al. mentioned the fragile nature of the environment at this school, because the work of the principal and senior leadership team did not readily align with the established tradition of practice in schools in the district, where privatisation of practice was the norm and whole-school visions for teaching and learning mathematics were typically unapparent.

Whilst scarcely emphasised or addressed in studies that discuss the role of facilitators within teacher collaboration, it is clear that the environment does play a role in collaboration. However, the management of the environment to foster collaboration is not clear cut. As stated by Nieman et al., “creating a structure in which teachers are expected to collaborate, on its own, does not necessarily lead to strong community nor does it necessarily benefit teachers or students” (p. 501). To explore this issue more deeply, we consider institutional systems that impact the environment of collaboration.

5.4 Institutional Systems and the Provision of Time for Participants to Collaborate

One of the repeated themes emanating from studies that address the role of facilitators in teacher collaboration is time. While not explicitly addressed as an issue in many reports, it is a theme that is repeated in many studies on teacher collaboration. In the Theme C meetings, it was mentioned in various studies; hence, it warrants elaboration. In this sub-section, a brief summary of how time, as an institutional factor that impacts teacher collaboration, is overviewed. In her study of facilitating collaboration between mathematics and arts teachers, Choutou (2020) frequently makes reference to time, although it was not overly emphasised as a key finding associated with the collaboration. Early in her report, however, she refers to the issue of strict timelines that limit teachers’ options to meet, resulting in situations where teachers may not have sufficient time for collaboration. Throughout her study, we see teachers arriving “tired and under time pressure” (p. 456) to the collaborative meetings. We read of teachers making demands on the facilitator as a result: “Give me already designed tasks or even ideas to use them” or “I would like that what is brought to the group meetings would be something that concerns everyone and that everyone can learn from it […] Why should I waste my time on something that I already know?” (p. 456). These words highlight that teachers need to feel that the time they invest in a collaboration is worth spending, especially if they are under considerable time constraints. This concurs with Cramer and Stivers’ (2007) claim that, ideally, “Collaborative relationships can be a rich source of professional and personal growth, well worth the investment of time and effort that may be necessary to nurture them” (p. 10). Choutou also mentions time pressures in facilitating the collaboration, in terms of managing participants’ contributions while also achieving the goals of the meetings. Specifically, she noted the difficulty of scheduling enough time for the meetings to allow for productive discussions, and the dilemma as a facilitator, not interrupting and redirecting teacher discussion that is not overly focused on the goal of the PD. It appears that giving due consideration of time is an important aspect of the environment for collaboration. In another Theme C paper, Ribeiro (2020) described the importance of taking time to develop the collaborative learning environment, noting that it took several meetings to establish an environment of trust and mutual respect among participants. Widjaja and Vale (2020) also discussed the important benefits that school-based facilitators noted, when they gave teachers sufficient time to perform activities that were at the core of the project.

From a different perspective, Jütte and Lüken (2020) discussed how the lack of time provided by the organisation impacted the capacity for teachers to collaborate and learn together through a ‘forced’ collaboration situation. As previously mentioned, in this study two teachers collaborated to co-teach an inclusive classroom where children with special needs were integrated. Initially, the teachers were keen to work together and learn from each other, but, as time progressed, the teachers developed their collaboration in a such a way that they neither prepared lessons together nor taught together. Surprisingly, and much in contrast to what both teachers reported that they wished for, the children with special needs were taught separately from the whole class in mathematics, with the special education teacher responsible for the preparation and teaching of mathematics for the whole class, while the primary teacher separately teaches the children with special needs. As Jütte and Lüken note, this distribution of roles is rather unusual (Friend et al., 2010; Scruggs et al., 2007). One recurring explanation was the amount of time that would be needed in a collaborative preparation of all lessons. In order to save time, the teachers allocated subjects to each other with respect to their professional competence. Neither of the teachers felt competent in mathematics, but as they wanted to divide the labour, and as the special education teacher had recently taken a course in teaching mathematics, he took over the mathematics instruction. The primary teacher felt uncomfortable with the way mathematics lessons were being executed, but, because of her lack of mathematical knowledge, she did not feel in the position to make alternative suggestions. The special education teacher, on the other hand, resented the double requirement of preparing and teaching the whole-class instruction and additionally preparing the tasks for his colleague who was lacking mathematical knowledge as well as special needs expertise. Both teachers were unhappy about their daily teaching and their collaboration, which was reflected in an increasing tension on an interpersonal level. Professional learning from each other was not happening.

This case study shows how the teachers approached the lack of provision of time for collaboration offered by the organisational environment. It also highlights the lack of growth of mathematical knowledge for teaching that possibly could have been promoted, if the special education teacher could have taken a more facilitative role. Yet this did not occur due to lack of time. This case provides an interesting position from which to consider simultaneously the role of the environment with respect to time and the role of facilitation (or the lack of it) in promoting or hindering the development of mathematical knowledge for teaching within teacher collaboration.

5.5 Collaboration Without Support of a Facilitator

The role of the facilitator can be placed under the microscope through analysis of teacher collaborations in the absence of a facilitator. Analysis of studies on teacher collaboration without a facilitator can identify situations during collaboration in which external advice is needed, hence pointing to the potential value of a facilitator. Analysis may also indicate the kind of support that is missing with regard to professional learning, and thus further define the required role of a facilitator. Two studies of teacher collaboration without a facilitator are detailed here, to draw implications for the role of the facilitator in teacher collaboration.

It is commonly accepted that collaboration between mathematics teachers can occur without the support of a facilitator, through teachers’ informal interactions with colleagues. Significant outcomes of teacher learning can result due to daily conversations, where teachers exchange their ideas and opinions about teaching. These informal talks promote the development of teachers’ understanding of knowledge, their creativity in teaching and any changes that may be occurring in the classroom (Cross et al., 2002). Penuel et al. (2009) studied the role that informal teacher interactions play in helping teachers enact instructional changes in practical teaching. They argued that teachers’ interactions draw not only on the social context of the school, but also on the expertise and resources that can be exchanged through these interactions. Similarly, Chen and Yang (2013) outlined Chinese teachers’ construction of a reform-based teaching strategy, and concluded that teachers within the school context had a shared interpretation system as revealed by their ‘native discourse’ (their daily language use, concepts and interactions). The significance of teachers’ informal collaborations, without facilitators, informed the study by Cao et al. (2020) as further elaborated below.

Cao et al. studied Chinese mathematics teachers’ informal interactions within schools. Their study occurred in two phases. In the first phase, 72 middle-school mathematics teachers from three cities in China were surveyed on collaboration with colleagues on teaching-related issues. Responses indicated that teachers collaborated most frequently with colleagues who taught the same subjects with the focus of collaboration on lesson planning and analysing student learning of mathematical knowledge, but less so on sharing and reflecting on student performances during lessons. The results raised questions on how teachers view informal interactions in their daily practice, which led to the second phase of the study that aimed to examine these questions. Individual interviews revealed that informal interactions helped teachers to prepare and revise their lesson plans. Although no facilitators were involved in these informal interactions, teachers were willing to initiate discussions with colleagues about teaching-related issues, through which they also solved problems or learned from each other. The study emphasised the importance of informal interactions in daily teaching, but suggested further research is needed to investigate how informal interactions without a facilitator might contribute to sustaining mathematics teachers’ professional learning in teaching practice.

As also outlined earlier, Jütte and Lüken (2020) investigated the first 2 years of a German teaching team collaboratively teaching mathematics lessons in an inclusive classroom. The two teachers had different professional backgrounds, with one being trained to teach whole classes and the other to teach children with individual needs. It was expected that, in this inclusive classroom setting, both teachers potentially would profit from their different expertise through formally interacting with each other on a daily basis. As stated by Hargreaves (2001):

Teaching together is reputed to be better than teaching apart. Cooperation and collaboration among teachers give access to the new ideas, creative energy and moral support that helps them to be more effective with their student. (p. 503)

In Jütte and Lüken’s study, the mathematics lessons were observed and further data was collected by informal talks and semi-structured interviews with both teachers. Results showed similar requisites regarding the teachers’ orientation towards inclusion, their didactical–methodical approaches and their idea of working as a team. Both teachers formulated the joint teaching of children with and without special needs in one classroom as their concept of collaborative teaching in an inclusive setting. Both considered frequent exchange about the collaborative work as important. However, it became clear that both teachers had not been trained in mathematics education, resulting in the special education teacher taking responsibility for planning and teaching all mathematics lessons, as detailed earlier. This became a major point of contention in this collaboration. Hargreaves found that collaborating teachers tend to avoid disagreement and conflict. While teachers who teach collaboratively value appreciation and acknowledgement, as well as personal support and acceptance (Hargreaves, 2001), differences in instructional approaches and conflict over strategies need to be actively addressed (Little, 2002), which was not the case here.

This study, where the collaboration occurred without a facilitator, points to the need for a facilitator on three levels. First, a facilitator is needed to initiate professional learning. Second, the facilitator’s role is crucial in mediating conflicts if the collaboration struggles. In this case, regular conversations about the collaboration, which would be coached by an external person, might have helped to clear misunderstandings and turn the swelling conflict into a professional exchange about goals for collaboratively teaching an inclusive classroom. Third, this specific team also needed professional support in teaching mathematics, as both teachers felt incompetent to prepare lessons satisfactorily. Thus, the role of a facilitator as an expert in mathematics teaching appears to be essential.

Upon reflection of both studies presented here on teacher collaboration without facilitators, we see contrasting outcomes. We do not aim to resolve this discrepancy. Rather, we view these studies as revealing the complexity of the facilitator–environment relationships, a complexity that calls for further consideration within future research.

In sum, the studies reviewed in this section engage various types of learning environments that enhance or hinder mutual learning of teachers and other participants in collaborative interactions. Although the studies presented are predominantly of a small scale, they provide some useful insights for considering the role of the environment in teacher collaboration. Llinares and Krainer (2006) stated that there is a need for large-scale studies “understanding from a broad perspective how different contexts influence teachers’ learning”, as it “is essential to investigate their involvement in different relevant environments in which they work, which environments influence them and which are influenced by them through their learning” (p. 451). While not providing definitive answers to these questions, this section of the chapter has highlighted the impact and importance of the role of the environment in teacher collaboration.

6 Concluding Comments

This chapter has explored the roles, identities and interactions of various participants in mathematics teacher collaboration at four levels: (1) research methodologies; (2) the role of the facilitator; (3) the professional trajectory of the facilitator; and (4) the role of the environment. We drew upon previous reviews and studies, and also presented an analysis of all 16 papers submitted specifically to Theme C at the ICMI Study 25 conference. It should be noted that the topic of discussion is incipient, as evidenced also in the way in which these papers are distributed by regions of the world. Seventy-five percent of the papers in Theme C are from countries or regions of the world that generally have significant research advances (eight from European countries, two from Israel, one from Australia and one from USA). The other 25% are evenly distributed between Brazil and China. This distribution can also be compared with the general distribution of papers for ICMI Study 25 by region. We therefore see a gap in research aligned to this topic in other countries that are not represented here. We wonder about the research in other countries published in languages other than English, and acknowledge that there is further work to be done to advance this field of inquiry.

In the second section of our chapter, we focused on methodologies and theoretical perspectives of research into different actors within teacher collaboration, to provide an overview of this growing field of research. We found that application of the RATE tool (Krainer et al., Chap. 8, this volume) assisted us to undertake our analysis. Our results aligned with findings of previous reviews, in that research approaches in this field are predominantly qualitative, there is a dominance of self-studies and the majority are small-scale studies. We noted that there is a need for larger studies that utilise quantitative approaches, yet that there is still room for further small studies to continue to build knowledge and theory in this domain. Many studies in the set of the Theme C papers centralised the role of the facilitator. In order to add to our knowledge of various participants in mathematics teacher collaboration, there is a need to extend the focus of reported research to interrogate the role of other stakeholders in the collaboration.

Because of the key role that the facilitator plays in teacher collaboration, the third and fourth sections of our chapter were dedicated to this focus. In the third section of the chapter, we looked specifically at literature around conceptualisation of the role of the facilitator in promoting mathematics teacher collaboration. From the current frameworks and models, we identified the academic knowledge and social and interpersonal skills needed by facilitators, as key to this profession. This section also discussed facilitators as professional experts, and overviewed research around how successful facilitators promote knowledge of teachers. The delicate balance of managing teacher conversations in a PD program, together with propelling teachers forward in building their own knowledge for mathematics teaching, is an important research finding associated with developing facilitator expertise. Our review pointed to the need for further research into how facilitators encourage the collaborative relationships and build a sense of community amongst participating teachers.

In the fourth section of our chapter, we interrogated the developmental trajectory of becoming a facilitator and professionalising in this role, and considered the types of support required for facilitators to develop their expertise. It is not common for research on teacher professional development programs to report on the role of facilitators, their preparation and the support provided to them to orchestrate such programs. Our review here pointed to issues faced by facilitators, particularly how they often must negotiate shifting roles between being a classroom teacher and becoming a facilitator. The importance of ensuring that facilitators are prepared and supported is evident when considering how to scale-up programs, while warranting that program outcomes can be sustained. We pointed to the need for research-based professional development programs for facilitators, aiming that the preparation and support of facilitators will be an integral component of the teacher professional development programs.

The fifth section of our chapter focused on the environment as an integral component of teacher collaboration. In this section, we overviewed selected models that might be useful in analysing the success or otherwise of teacher collaboration. These models emphasised that environmental factors are at play in any collaboration. In this section, we briefly overviewed the roles that participants other than facilitators may play in teacher collaboration. We also gave due consideration to analysing teacher collaboration with and without facilitators. Such research findings remind us that, while facilitators play a pivotal role in achieving the intended outcomes of a collaboration, facilitators may not be the sole drivers. Despite the expertise and high-level knowledge of a facilitator, the environment may serve to impact the collaboration outcomes. Conversely, there are examples of successful teacher collaboration in the absence of facilitators.

Taken together, these four sections (Sects. 4.2, 4.3, 4.4, and 4.5) in this chapter have drawn upon current research that contributes to our understanding of how various participants take part in, and influence, collaborative endeavours of mathematics teachers. There is further research to be done to build theory around this topic. We need more research to broaden our knowledge of the professional trajectories of Mathematics Teacher Educators (MTEs) and facilitators of mathematics teacher collaboration. We also need more research on the impact of different actors in the environment on teacher collaboration, as well as on facilitators, and how facilitators negotiate the environment of a collaboration.

Another future direction to pursue further concerns addressing the problem of scaling up collaboration programs, going beyond the training of facilitators and the contextual conditions that are required for this to be effectively carried out, and looking into other questions. For instance, how do institutional factors impact the sustainability of mathematics teacher collaboration? What models of scaling-up programs of teacher collaboration exist, and what are the roles of different actors in these models? We also need to know how the topic of actors involved in teacher collaboration is treated in other publications that are not written in English. While there is still much work to be done, we recognise the progress made in recent years in studying different roles in mathematics teacher collaboration, reflected in the considerable body of research that we have drawn upon here to address this important issue. We look forward to seeing this interesting field of research further developing in the future.