1 Introduction

After receiving the invitation from the editors of this volume to write this chapter, we decided to organise our discussion in four perspectives that, in our view, systematise relevant issues highlighted by the papers presented at ICMI Study 25. Such perspectives are:

  1. 1.

    different forms and meanings for collaboration;

  2. 2.

    what do we investigate, how do we investigate, and who investigates collaboration?:

  3. 3.

    the complex relations between collaborative groups and classroom practice;

  4. 4.

    possibilities for scaling up collaborative professional development initiatives.

Through the discussion of such perspectives, on the one hand, we seek to reflect on the advances and possibilities of collaboration considered as a field of investigation and as a process that promotes professional development (PD). On the other hand, we aim to identify some of the challenges that collaboration confronts nowadays.

For this purpose, we build our arguments by drawing on two references. Firstly, the chapters included in this book, the papers presented at the ICMI Study 25 conference mentioned in those chapters, and other complementary literature resources. Secondly, we draw on our experiences of collaboration with teachers in Brazil. To do so, we refer to our participation in the Grupo de Sábado [Saturday Group] (GdS), a Brazilian collaborative group operating for more than 20 years. The GdS gathers schoolteachers, teacher educators, graduate students and pre-service teachers interested in learning about and researching on mathematics teaching practice. As we will show in this chapter, the GdS differs from many groups and collaborative initiatives presented in this book in several ways. Thus, it provides a counterweight that reveals and interrogates some of the ideas assumed by diverse research endeavours focused on mathematics teacher collaboration.

2 Different Forms and Meanings of Collaboration

In England and the U.S., the movement of collaboration among teachers gained visibility and recognition in the 1970s, when Stenhouse (1975) systematised a type of action research or design research aimed at improving learning and teaching through inquiry. Such a framework involved a cycle of four steps: examining current practice, making decisions, planning optimal learning environments; implementing the findings in the classrooms with reflection. This movement gained strength in the U.S. in the 1980s, mainly in the context of in-service teacher education. At that time, it was adopted stressing its potential contributions to teachers’ PD. Considering that not all PD initiatives are collaborative, the term ‘collaborative PD’ has come to be used by the pertinent literature to differentiate those that are.

By the end of the twentieth century, according to Hargreaves (1994), collaboration had already become a “meta-paradigm of educational and organizational change in the postmodern age” especially because it made possible the articulation and integration of “action, planning, culture, development, organization, and research” (p. 244). Collaboration was recognised as a creative and a “productive response to a world in which problems are unpredictable, solutions are unclear, and demands and expectations are intensifying. In this kind of context, the promise of collaboration is extensive and diverse” (p. 244). Since then, collaborative work and collaborative research among professionals from different institutions and levels of education have emerged worldwide as a response to the social, political, cultural, and technological changes that are taking place on a global scale and that jeopardise the traditional ways of organising PD initiatives (Fiorentini, 2004).

This movement gave rise to several models and conceptualisations of collaboration and collaborative research. Besides, models were recovered and adapted according to different sociocultural realities: that was the case of the Japanese Lesson Study (LS). It emerged at the end of the nineteenth century, but only came to the attention of educators outside Japan in the late 1990s, due to the systematisation conducted by Yoshida (1999). Since 2000, educational researchers have tried to use the LS model, adapting it to different cultural realities (Isoda, 2020). We suggest that the study of these processes of modification and adaptation of models of collaboration is a great opportunity for our research field. On the one hand, it would enable us to systematise and discuss diverse theoretical perspectives of collaboration, revealing different meanings assigned to collaboration. On the other hand, it would allow us to identify and problematise each model's educational possibilities, contributions, and limitations.

da Ponte et al. (Chap. 2, this volume), drawing on Robutti et al. (2016), conceptualise collaboration as “a group of participants, who work together pursuing a common aim, by establishing some joint working processes in which active involvement, balanced roles and caring relationships are central features” (p. 21). In our view, such conceptualisation is relevant, since it includes PD initiatives that promote horizontal or dialogical relations among teachers and teachers’ educators. However, we consider that this conceptualisation does not enable us to distinguish between superficial and deeper or sustainable forms of collaboration (Azorín & Fullan, 2022). In addition, this broad conceptualisation makes it difficult to analyse diverse collaborative initiatives in relation to their impact and appropriateness in different circumstances and sociocultural contexts (Hargreaves, 2019).

In what follows, we discuss briefly these different conceptualisations of collaboration, especially since it seems to have been an issue little discussed during the ICMI Study 25 conference. To “work together pursuing a common aim” may happen, for instance, in situations of contrived collegiality (Hargreaves, 1994, 2019), especially when subjects are requested to participate in groups without having the possibility of negotiating the group’s goals. Besides, it could happen in a group that lacks a common culture of sharing and negotiating practices and meanings. In a similar direction, both Azorín and Fullan (2022) and Hargreaves (2019) stress that a group becomes effectively collaborative and sustainable when it develops collaborative professionalism.

Such a notion refers to a learning community whose members, even having different knowledge, develop joint practices, negotiate their goals and carry out collaborative research. Therefore, the development of collaborative professionalism demands time: a time that cannot be pre-established by administrators or teacher educators, since it depends upon the disposition and upon the previous experiences of the members. Similarly, our years of participation in the GdS and other collaborative groups, as well as the research we conducted (Fiorentini, 2004, 2013; Losano et al., 2021), point to the fact that groups are not born collaboratives. Instead, they became collaboratives over time. For example, when new members join the GdS, their participation during the first 6 months involves experiencing and reflecting on practices historically produced by the group over the years—practices oriented at planning, implementing and analysing investigative classroom tasks.

To understand better what we mean, let us consider, for example, the study by Cooper (2019) that was analysed by Krainer, Roesken-Winter and Spreitzer (Chap. 8, this volume), using the RATE tool to highlight the relationships among Actors, Goals and Relevant Environments. It is an Israeli PD initiative that gathered twenty primary schoolteachers, two mathematicians and a facilitator with a Ph.D. in mathematics education, who was the study’s author. The goal of the initiative was to bring together two groups that could share their perspectives on teaching integers, crossing boundaries between the world of primary school and the world of academic mathematics.

Such a goal emerged from the requirement, external to both groups, of the Israeli Ministry of Education, establishing that, “primary-school teachers need ‘to enroll in mathematics PD initiatives to specialize in mathematics’ (p. 71)” (cited by Krainer et al., Chap. 8, this volume, p. 8). Such collaborative arrangement gathered representatives from two fields of study, enabling sharing perspectives about how to teach integers. The group facilitator also gave her feedback, valuing the contributions of the two communities. The author concluded that the initiative allowed teachers to benefit from the mathematicians’ perspective. In addition, it enabled mathematicians to achieve sensitive understanding. According to Cooper, collaboration contributed to mathematicians paying more attention to their students’ ideas, opening spaces for discussion in their university math classes. Primary teachers, on the other hand, were able to build more mathematical confidence.

There is no doubt that there was learning involved in this PD initiative. Each subgroup learned something from the dialogue with the other. Each participant mobilised a surplus of vision and knowledge (Bakhtin, 2003) in relation to the other. Notwithstanding, is it possible to state that such learnings are sustainable and have the potential of transforming the participants’ teaching practices? We suggest that this short PD initiative opened up the possibility of negotiating and developing a joint project around a common goal in the future. From this experience, mathematicians and primary teachers could engage in a sustainable project focused on teaching and learning mathematics at school (Azorín & Fullan, 2022). Thus, they could establish a “genuine collaborative work” according to the expression used by Esteley et al. (Chap. 3, this volume). Our point is to reinforce the etymological meaning of the verb ‘to collaborate’, which means to work together (collaborare from Latin) around a common objective, defined or negotiated jointly by all the participants.

Voluntariness is another relevant condition for participation in a collaborative group (Esteley et al., this volume). Although participation can be initiated on a mandatory basis, a PD initiative may become effectively collaborative if the participants have opportunities to jointly define their goals and actions. Several articles presented at the ICMI Study 25 Conference show that this scenario is feasible within a school (Collura & Di Paola, 2020) or an educational system (Canavarro & Serrazina, 2020; Soto et al., 2020). However, those authors stressed that such success depends on the way in which the leaders build, in collaboration with the participants, the group design and dynamics.

Hollingsworth et al. (Chap. 10, this volume) shed light on this process. They report on a teacher who participated in a collaborative group that was part of the network of Research Institutes in Mathematics Teaching (IREM—France). IREM usually promotes collaboration between the university and the school, improving mathematics teaching and teachers’ PD. Thus, the collaborative IREM group sought to address a problem that had emerged during the annual conference of this network. Teachers lacked didactical resources to teach logic effectively in secondary education, as recommended by the new curriculum.

The group members were two university teachers, three high-school mathematics teachers, and a French language teacher. During the first meeting, they collectively defined the group goals and work schedule. Thus, they agreed the group would produce classroom tasks to develop the students’ logical reasoning. The tasks were tested in their classrooms before being analysed and disseminated to other teachers. The two university teachers initially assumed a traditional role, providing mainly theoretical content and perspectives. Over time, they also assumed a more collaborative role, analysing and discussing the teachers’ proposals. One of them actually ended up leading the group.

To sum up, the analysis of the papers presented at the ICMI Study 25 conference reveals diverse meanings for collaboration, and different and rich ways of promoting it among mathematics teachers. Such analysis also brings to the forefront two relevant features of our conceptualisation of this notion: collaboration requires time and demands shared negotiation of goals and actions. We identified such features by drawing on the literature focused on collaboration as well as on our years of experience collaborating with teachers.

Considering such diversity of meanings for collaboration, we suggest some future directions for research. How do diverse ways of promoting collaboration contribute to transforming teaching practice? What are their contributions to teachers’ learning? How do mathematicians, mathematics educators and teachers negotiate mathematical meanings and reconstruct their professional knowledge? How does that knowledge differ from the school and/or academic tradition?

3 What Do We Investigate, How Do We Investigate and Who Investigates Collaboration?

The works presented at the ICMI Study 25 Conference indicate the presence of two privileged research perspectives. The first concerns the study of collaboration, its resources and its theoretical–methodological bases. This perspective was the focus of three themes covered in this event, highlighting the following descriptive aspects: the theoretical, epistemological, and methodological bases to promote and investigate the collaboration of mathematics teachers (Chap. 2, Theme A, this volume); the design and dynamics of collaboration, with an emphasis on its goals, its environments and the different collaborative actors and their roles, interactions, and identities (Chap. 4, Theme C, this volume); the tools and resources mobilised and produced to support and organise the collaboration (Chap. 5, Theme D, this volume). The second research perspective focused on the effects of collaboration. Such perspective had a dual emphasis: (1) on participants’ PD and learning as well as on the growth of the collaborative community; (2) on the improvement of curriculum and teaching linked to the collaborative process (Chap. 3, Theme B, this volume).

The discussion of the theoretical bases of collaboration is relevant, since it is an emerging field of study. Furthermore, the discussion of this issue during the ICMI Study 25 addressed a limitation noted by Jaworski et al. (2017): only one-third of the papers analysed by Robutti et al. (2016) explicitly stated their theoretical bases. Although the theoretical aspects in collaboration were the main focus of Theme A, they were also discussed by all themes and were present in all parallel plenary sessions.

Regarding the studies focused on the resources and tools produced in/for collaboration, the authors who contributed to Theme A argued that theory can be seen as an important tool for designing and developing relevant and sustainable collaborative projects. Concretely, such theories may be useful for analysing the contributions of collaborative groups to teachers’ learning, curriculum development, or teaching improvement. This point was also made by authors who contributed to Theme D, albeit placing greater emphasis on the resources of collaboration. Brodie and Jackson (Chap. 9, this volume), for instance, considered knowledge and representations of professional practices as resources. Differently, Robutti et al. (Chap. 5, Theme D, this volume) highlighted the dialectical nature of resources. Thus, they argue that, on the one hand, resources are essential for challenging teachers’ thinking and practices, producing the desired results of collaboration. On the other, such results become resources that can lead to new cycles of collaborative learning.

We value the emphasis given to resources in/for collaboration during the ICMI 25 Study Conference. However, we would like to call attention to one resource that, in our view, has great generative power and was not discussed in this volume: the narratives written by participants of collaborative groups. According to our experiences in the GdS (Fiorentini & Carvalho, 2015; Fiorentini et al., 2018; Losano et al., 2021), narratives written by teachers are relevant resources to represent their histories of participation and learning processes. Therefore, narratives increase collaborative work and provide rich material for analysing teachers’ learning. In addition, teachers’ narratives are loaded with affections, meanings and perceptions of the support (or lack of) provided by their schools to introduce innovations in teaching practices. Hence, they also reveal forms of teacher resistance and the strategies they develop to mobilise their agency, implementing in their classrooms aspects of what they learn in the collaborative group.

Narratives written by teachers who participate in collaborative groups are “means and products” (Robutti et al., Chap. 5, this volume, p. 3) of collaboration. From their authors’ point of view, these narratives are not only the means (semiotic mediation), but also a way in which they develop their collaborative professionalism and identity in dialogue with other professionals and members of the group. Thus, by becoming the authors of published narratives disseminated to a wide audience, they also become agents of change in the school culture and productive members of a broader educational community (Hargreaves, 2019; Fiorentini, 2013). From the point of view of teachers’ educators, once published, narratives become relevant resources to support pre-service and in-service PD initiatives (Fiorentini & Carvalho, 2015).

In collaboration, all voices have value and need to be heard, as each member has a surplus of vision and knowledge (Bakhtin, 2003) about the practices of teaching and learning mathematics at school. In this sense, the organisers’ decision to give teachers a platform to share their experiences while participating in collaborative projects was quite pertinent to the purposes of ICMI Study 25 (Hollingsworth et al., Chap. 10, this volume). We suggest that encouraging participating teachers to write narratives about their learning in this context is another way to give them voice and authorship and to value their perspectives.

The discussions presented in this volume, as well as our experiences of collaboration, reveal that the design and the resources have strong impacts on the effects of collaboration, which lead us to the second research perspective on collaboration mentioned previously. Such effects may be analysed in terms of teachers’ learning, community development and/or institutional improvement. These benefits highlight the multifaceted and complex nature of collaboration, given its different purposes and modes of organisation, as shown in Borko and Potari (2020).

The attempt to understand and theorise the learning and the development of teacher’ knowledge for-in-of-practice (Cochran-Smith & Lytle, 1999) from participation in collaborative groups makes us return to the heading of this section. Considering who investigates collaboration, when we examined the papers presented at the ICMI Study 25 Conference, we verified that its authors are mainly mathematics educators, especially teacher educators or graduate students who participated in collaborative groups. Many of them also assumed the role of facilitators.

The focuses of such investigations include the two perspectives described above—that is, the study of collaboration and its effects. There is a clear trend toward developing studies about teachers. Such trend explores teachers’ learning, PD, professional knowledge or their roles in collaboration. These results indicate that there are still few collaborative investigations, that is research carried out collaboratively by university academics with schoolteachers. Of the eighty papers reported in ICMI Study 25, only two are of this nature.

In this sense, we stress that collaboration is also a good opportunity for both parties to investigate together. University academics and schoolteachers engaged in collaborative groups can negotiate the focus of the research and develop joint interpretations about the participants’ knowledge, actions and discourses, revealing knowledge situated in the collaborative practice (Lave & Wenger, 1991). Faced with the challenges of the current school context and the unfavourable conditions for schoolteachers to carry out research, they are left with the possibility of participating in collaborative inquiry groups, as underlined by some studies presented in ICMI Study 25 (Castro Superfine & Pitvorec, 2020; Uzuriaga et al., 2020).

Many of the Brazilian studies that assumed this perspective adopted the Relational Narrative Investigation (RNI) as a research methodology (Clandinin & Connelly, 2000). Cristovão and Fiorentini (2021) consider the RNI suitable for academics to develop investigations with schoolteachers, focusing mainly on teachers’ professional learning and the PD. In this investigative process, “teachers are also encouraged to investigate their practices, narratively, with the collaboration of teacher educators, especially when both are committed to discuss and understand what and how they learn in this context” (p. 35).

Thus, schoolteachers generally explore their professional work. They may, for instance, analyse their students’ or their own learning during a cycle of planning–implementation–reflection–evaluation of lessons. By sharing these investigations and findings in the collaborative group, teachers may problematize their practice, developing an inquiring and critical attitude towards their work and public policies in the educational field (Jaworski, 2008; Fiorentini, 2013).

Collaboration between schoolteachers and university academics is a powerful context for PD and for producing knowledge about school practice from a non-colonising perspective. In addition, we suggest that it is also a rich field of research for both, namely academics at the university and schoolteachers. Thus, we argue in favour of carrying out collaborative research with teachers, instead of conducting exclusively research about them. In recent years, this research perspective has flourished in Brazil and Latin America, fuelled by the expansion of the Lesson Study process. To map these investigative experiences and analyse their findings and contributions seems to be a relevant topic for research and discussion in an upcoming ICMI Study.

4 The Complex Relations Between Collaborative Groups and Classroom Practice

The chapters included in this volume show the wide variety of ways in which mathematics teachers can work and learn in collaborative groups. Chapter 3 (Theme B, this volume) reveals that each one of these ways of collaborating establishes different relations with regular classroom practice. For example, the initiative studied by Kooloos et al. (2020) connects the collaborative setting with teaching practice through the analysis of classroom videos to develop teachers’ noticing of students’ thinking. Soto et al. (2020) employed problems as a linking resource: teachers engaged in a community of practice were invited to solve problems, implement them in their classrooms and discuss such experiences in the community. Also, there are social contexts that developed powerful forms of school-based collaborative professional development. This is the case of Lesson Study in Japan or China, where the cycles of planning, implementing and analysis are job-embedded tasks with a long tradition. On the other extreme, the work of Heck et al. (2020) analysed a PD program based on the mathematics immersion of secondary teachers. The authors admit having trouble attending some of the program’s goals since, “discussions about the connections between what they experienced in mathematical immersion and teaching were infrequent or lacked depth” (Esteley et al., Chap. 3, this volume, p. 33).

These examples highlight that PD in collaborative groups and regular teaching practice are different social, cultural and historical situations. Even in the cases in which the PD is strongly connected to classroom practice, the participants, the activities, the positionings, the times and the spaces specifics of collaboratives groups are not the same as the ones of regular classroom practice. Such understanding is evident in Brodie and Jackson’s words, in this volume, when they state that, “although collaborating with colleagues about teaching has become more common in recent years, by and large, classroom instruction remains a private endeavour” (Chap. 9, this volume, p. 13). In our view, further work analysing the complex relationships among collaborative groups and regular classroom practice would be highly beneficial to the field. To pursue such a research interest encompasses theoretical and methodological challenges.

Considering theory as a way of understanding—i.e. theory as a means “to understand the educational phenomena related to teacher collaboration, by providing conceptual and/or methodological tools to analyse and understand phenomena from different perspectives”, in da Ponte et al.’s words (Chap. 2, this volume, p. 15)—the challenge concerns how to conceptualise the relations between PD in collaborative groups and regular classroom practice. One possibility is to frame the problem in terms of ‘impacts’: we need to study how participation in collaborative groups impacts teachers’ classroom practice. This is a perspective frequently adopted and mentioned several times in this volume. Although we agree with the point being made, we would like to problematize the cause–effect metaphor underneath the notion of ‘impact’. Theoretically, this perspective assumes that teachers learn within the collaborative group and then apply such learning in their classroom.

Such an assumption is strongly questioned by socio-cultural perspectives—extensively employed in our research field—that stress the mutual relations among people, activity, and the social world. According to Lave and Wenger (1991), what teachers learn while participating in collaborative groups is situated in the practices and social arrangements developed by the group. Therefore, we cannot assume such knowledge will be directly transferred into the classroom setting without consideration of the different activities, goals, circumstances, and social positions (Lave, 1996).

The perspective of Kazemi and Hubbard (2008), squarely brought in Brodie and Jackson’s chapter, brings another important point in this regard: the relations between PD and classroom settings are not unidirectional, but multidirectional. Teachers’ participation in PD and classroom practice co-evolves, since they are engaged in knowing in both contexts and bring their knowledge across contexts.

The two premises presented previously—the one that states that there is no direct learning transference between different contexts, and the one that assumes that the relations between collaborative groups and regular teaching practice are multidirectional—bring to the forefront the challenges involved in theorising about the complex relationships between PD in collaborative groups and classroom practice.

In terms of methodology, we identify two main issues. The first one is related to the temporal dimension. It is possible to adopt a short-term perspective, considering only the period in which the teacher participates in the group. Otherwise, it is possible to employ a long-term perspective, addressing the question of the sustainability of outcomes, that is, to analyse if changes made in the context of collaboration sustained long-term changes in the classroom setting (Esteley et al., Chap. 3, this volume). The second issue concerns the analytical procedures mobilised. In our view, it is necessary to develop methodological strategies to establish relations between data coming from the PD setting and data coming from the classroom and school settings. This would require a careful and creative endeavour. Further exploration of this issue might usefully inform directions for advancing research in this area.

Considering the theoretical and methodological challenges involved in researching the relations between collaborative groups and regular teaching practice, we believe that the chapters of this volume, as well as the papers presented at the ICMI Study 25, point to two promising directions to further our understanding of the topic. The first one is the study of the resources and how they are transformed while travelling between the collaborative group and the classroom setting. Considering Brodie and Jackson’s framework in this volume, we refer specifically to the representational resources created for and from collaboration. Frequently, participation in collaborative groups involves creating and/or adapting tasks, lesson plans, curricular material, websites, etc., as well as analysing students’ errors, assessment items or classroom videos. As stated in Chap. 9 (Brodie & Jackson) and Chap. 3 (Esterley et al., Theme B), resources are important products of collaboration and, in our view, can support important links between classroom practice and collaborative groups.

We believe that the situated perspective proposed by Brodie and Jackson, in this volume, is a fruitful approach to the problem of analysing how resources are transformed as they travel back and forth between the collaborative group and the classroom. Such a perspective assumes that, “the use of tools and artifacts to mediate learning relies on people assigning meaning to them […] and these meanings are shaped by the broader contexts in which tools and artifacts are used” (Brodie & Jackson, Chap. 9, p. 4). This assumption is evident in our experiences in the GdS when a task or a lesson plan, carefully planned during several meetings, is subtlety—or sometimes substantially—transformed when implemented by the teacher in her classroom.

For instance, while interacting with her students, the teacher modifies the duration of the task or emphasises one aspect of the task over others. How and why do these transformations take place? What can we, and the teachers, learn from them? In addition, when we consider a long-term perspective, the evolving nature of resources comes to the forefront. Prior resources developed inside collaborative groups are often retrieved and adapted by teachers to be employed in their teaching at present. How are these resources shaped by the users over time? When, why and how are they recovered and adapted? Such questions highlight the complexity of the transformation operated over the resources when they travel from the collaborative group to the classroom, and vice versa. We believe that research focused on such issues deserves our attention and study.

The second promising direction for studying the relations between collaborative groups and regular teaching practice is to focus on the teachers. They regularly cross the boundaries between the collaborative group and the school context, introducing elements of practice and ways of knowing and of being from one context into the other. In this process, they become boundary brokers (Wenger, 1998). To co-ordinate their affiliation to both communities is a delicate endeavour, since it often requires reconciling, implicitly or explicitly, competing expectations. How do teachers manage to develop a sense of themselves among such conflicting practices and discourses? What conflicts do they experience in this process? How do teachers solve them? What do they learn in the process?

Another possibility is to adopt the notions of professional identity and agency. While participating in the GdS, teachers come to know other ways of understanding mathematics teaching and learning. In addition, they engage in reflexive processes that often problematise the implicit rules and identities fostered by the school world. Teachers also implement classroom activities inspired by these new perspectives and, later, they share and analyse such experiences in the group. Thus, participation in the GdS enables teachers to take a stance in front of the demands and expectations of the school world, gradually expanding their room for manoeuvre to make decisions and choices concerning their work (Vähäsantanen, 2015).

The collaborative group allows its members to experience new ways of being teachers, engaging themselves in an evolving process of identity development. Over time, teachers also expand their agency. They become actively involved in conceiving and directing their teaching practice according to their purposes, principles, and interests, as well as to the requirements and possibilities set by the school context (Losano & Fiorentini, 2021). How do teachers develop new positions and roles in the school and the collaborative group? How do they recover practices and discourses coming from one context to develop senses of themselves as mathematics teachers in the other? In our field, research on teachers’ identity and agency has flourished over the last decades. We suggest that further exploration of the process of identity and agency development of teachers who participate in collaborative groups could provide opportunities to develop our understanding of teachers’ professional growth across contexts.

5 Possibilities for Scaling-Up Collaborative Professional Development

In our view, collaboration is a way of transforming the colonising relationship commonly established between the university and the school. In this way, collaborative groups, such as the GdS, are an opportunity for university teachers and schoolteachers to engage in joint learning processes and imagine together ways of facing the current challenges involved in teaching and learning mathematics.

The great potential of these groups is underlined throughout this volume. Thus, the question posed by Hollingsworth et al. (Chap. 10, this volume), is particularly relevant: “How might effective collaborative group activities and outcomes ‘reach’ more mathematics teachers?” (p. 23). To respond to such a question is a complex endeavour, because, in our perspective, collaboration cannot be imposed in small-scale projects and much less in large-scale initiatives.

A partial answer to this issue can be found in our experience with collaborative groups in Brazil. The sustained work of some of those groups allowed many schoolteachers and university teachers to have relevant experiences of participation. Many of these members moved to other regions of Brazil due to personal or career opportunities—a common thing in a country with continental dimensions such as ours—and decided to promote and cultivate collaborative groups in the new institutions in which they began working. In addition, collaborative groups also developed practices oriented at disseminating their work. Thus, they created and organised diverse events—congresses, seminars, etc.—and journals devoted to presenting and discussing teachers’ reflexive work developed from their participation in collaborative groups.

In addition, books containing narratives written by teachers who participate in collaborative groups began to be published. In such narratives, teachers problematised and analysed classroom situations, producing knowledge-of-practice (Cochran-Smith & Lytle, 1999). These dissemination processes inspired many school and university teachers to constitute collaborative groups over the country. In a snowball effect, collaborative groups expanded and gained legitimacy inside the mathematics education community in Brazil (Carvalho, 2014).

The expansion of collaborative groups in different regions of the country happened informally and spontaneously. In this scenario, the participation of schoolteachers, pre-service teachers, mathematics educators and researchers in such groups is still little or not recognised by public policies (Gonçalves et al., 2014). The members of many collaborative groups devote their own time to participating in the group. In this way, they prioritise their PD over other responsibilities. Moreover, several of the members that abandon the GdS make that decision based on their difficulties to balance group participation with work or familiar commitments—to take care of their children or attend to the demands of a new job.

As stated by Brodie and Jackson in this volume, the issue of the resources made available for collaboration by public policies, “draws attention to the deeply cultural, political and historical contexts in which teachers’ collaboration occurs across different contexts, and the many inequalities that still pervade our school systems” (Chap. 9, this volume, p. 8). To expand opportunities for collaborative PD public policies that explicitly support collaboration are needed. Such policies should provide time and spaces for professional collaboration, as well as value and acknowledge teacher collaborative work.

Some of the papers presented at ICMI 25 suggest a promising possibility of scaling up collaborative PD: the development of blended professional networks that gather schoolteachers, pre-service teachers, mathematics educators and researchers. Two examples of these networks are described in this volume. The first one is mentioned in Chap. 3 (Theme B, this volume) and refers to the research conducted by Heck et al. (2020). In this project, groups of teachers from different cities in the United States were engaged in PD involving both synchronous and asynchronous activities. In such a network, teachers were immersed in mathematical activities connected to their teaching practice. The second example is described in Chap. 10 (Hollingsworth et al.) and concerns Shelly, a teacher who participated in an online professional learning network directed at promoting “quality mathematics instruction, mentorships for new teachers, and curriculum development” (p. 3) via social media.

Both examples highlight the potentialities of online or blended collaboration. They enable access to “many participants from different geographically distant regions and from a variety of contexts […] bringing together a myriad of perspectives” (Esteley et al., Chap. 3, this volume, p. 48). In this way, such networks have the potential of connecting teachers in distant or isolated contexts. They also make possible to gather professionals who work with different student populations, enriching the personal learning experience. In addition, social media enable teachers to receive real-time feedback for lesson development, as Hollingsworth et al. stated in this volume. Social media open up new possibilities of collaboration, reaching, in seconds, a vast public and allowing extended discussions in which people from all around the world can contribute. Finally, online communities are often flexible, embracing new educational tends more rapidly than mainstream educational circles.

Inspired by such experiences, as well as by our last research projects, we suggest that blended collaborative networks would be a powerful way for scaling up collaboration. Considering the demands and interests coming from classroom teaching practice, the members of such networks would organise themselves in small groups. Such small groups would congregate teachers, pre-service teachers, facilitators and researchers interested in discussing one topic related to teaching practice considered particularly problematic. Each group would negotiate its goal and the activities it would develop—for example, to plan and implement classroom tasks oriented at teaching a specific mathematical topic or analysing textbooks or classroom material. The small groups would gather periodically face-to-face or online. The network would act as a support space. Thus, all their members would meet more sporadically to share and discuss the work of the small groups.

In addition, the members of the network would be able to help and support each other through interactions via digital technologies. Once the small groups achieve their goals and complete their activities, the members of the network would reorganise themselves, forming new small teams. This kind of collaborative network would be organised according to a bottom-up model, since it would have the autonomy for establishing its agenda through the negotiation of aims and topics explored by the small groups.

Several works discussed in this volume showed that the Covid-19 pandemic challenged many aspects of teaching and teacher education. In the present context, it is unlikely that teacher education would return to its previous traditions. Thus, developing effective online or blended opportunities in which teachers could work and learn through collaboration, such as the ones we are suggesting, is an urgent endeavour.

Despite their potential, we cannot be naïve about the constitution of this kind of network. Thus, we anticipate some challenges to be faced. As Esteley et al. stated in Chap. 3, “collaboration is essentially about relationships, about finding a common ground to have support for the possible changes” (p. 4). Gathering together people who work in different places, with diverse publics and resources could produce rich exchanges. But it also demands establishing shared understandings among the members. In addition, collaboration frequently requires adopting an open attitude and sharing uncertainties, problems or ambiguous situations of teaching practice. Therefore, each member should feel safe and embraced, trusting that the interactions inside the network would be oriented toward seeking alternatives in a non-judgmental way.

The division of responsibilities and roles inside each small group and in the network also requires fine-tuned negotiation processes. Also, it is necessary to set up carefully the processes through which each group delineates its topic and the resources—technological or not—that are best suited for each one of them. If collaboration is not established instantaneously in groups interacting face-to-face, this would neither happen in blended networks.

Considering that many agents would participate in the network, careful scheduling for interactions among the members can also be challenging. Brodie and Jackson (Chap. 9, this volume) point out another challenge located in a different dimension—nowadays, there are still “inequalities between rich and poor in relation to access to the internet” (p. 22). We had already co-ordinated online teacher education initiatives, in which teachers should take a one-hour boat trip on a Saturday to reach their school, since only there do they access a stable internet connection. Thus, when establishing blended collaborative networks, we cannot assume that all the teachers in the region or the country would have the same online accessibility.

How to promote trustful relationships among members of blended networks? What practices and strategies support transparent negotiation processes inside the network? How can technological tools contribute to these processes? How to cope with the inequalities regarding access to technological devices? We suggest that the development of studies centred on these issues would provide a more accurate vision on the advantages and disadvantages of blended collaboration.

6 Final Remarks

In this chapter, we have made an effort to highlight the advances resulting from the ICMI Study 25, drawing mainly from the systematisation carried out in the chapters of this book. The contributions of this systematisation allowed us to understand better the possibilities and potentialities of collaboration for the PD of teachers who participate in collaborative projects.

We believe that this progress would help us—and the rest of the members of the mathematics education community— to design better opportunities of collaboration, as well as to gain understanding about the development of collaborative communities. On the other hand, the studies in this volume also showed that collaboration is a complex and multifaceted undertaking (Theme B), since it depends on the conditions and dispositions of the participants and on the support of the institutions of which they are part.

In this direction, we argued that the possibility of collaboration does not entirely depend on the institutions’ desire to promote it, nor on the willingness of participants who want to work together. Collaboration is a cultural practice that needs time to be developed. In our view, teachers’ communities are not born collaboratively, even if that is the initial intention of their members. Therefore, no ideal model designed to foster collaboration can be applied without adapting it to local conditions and cultures, as Isoda (2020) has shown about the international diffusion of Lesson Study. The chapters of this book also acknowledge the importance of material and theoretical resources to support the design of collaborative PD and to conduct research in this context (Themes A and D).

In this regard, we argued in favour of using teachers’ written narratives (Fiorentini, 2013; Fiorentini & Carvalho, 2015). We believe they provide rich opportunities for teachers to investigate their own practice and reveal their learning processes. We also highlighted the methodological potential of Relational Narrative Investigation as a framework that enables developing joint research with teachers. In this direction, our analysis of the papers presented at ICMI Study 25 suggests that a great challenge in our research field is to describe and characterise the knowledge-of-professional-practice produced inside collaborative groups and how it is co-produced by its members.

In addition, the chapters of this volume also delineate issues concerning the way collaborative groups relate to other settings. In particular, the study of the relations between collaboration and regular classroom practice emerges as a relevant and exciting topic for further research. We suggest that the notions of resources, professional identity and agency may be key concepts to develop nuanced and rich analysis in this direction (Themes C and D). Such theoretical constructs have the potential of moving forward simplistic perspectives based on cause–effect relations to measure the ‘impact’ of collaboration.

Finally, another challenge faced by teachers and researchers interested in fostering and investigating collaboration is how to scale up opportunities of collaborative professional development. Such challenge becomes more urgent since the work of the ICMI Study 25—starting with the literature review presented at ICME 13 (Robutti et al., 2016) and finishing with the synthesis of the papers presented at the conference in this volume—had shown the immense potentialities of collaborative groups. Considering the uncertainties of the global context, the possibility of cultivating blended collaborative networks seems to be highly promising.

The chapters of this volume stress that teacher collaboration, especially among teachers with different knowledge and views on practice, is multi-faceted and takes diverse forms in different parts of the world. However, in each of these forms of collaboration, what is learned and how it is learned has its own singularities and nuances that are different from other traditional learning and PD processes. This is one of our challenges as researchers in the field of collaboration: namely, to systematise and theorise these epistemological processes of co-learning and co-production of knowledge from practice. Collaboration, therefore, is a fertile and still little explored field that demands continuity of studies and socialisation, discussion and systematisation in events, as was the case of ICMI Study 25.