Theoretical Perspectives on Studying Mathematics Teacher Collaboration

Abstract

The aims of this chapter are: (i) to showcase the state-of-the-art for theoretical perspectives for studying mathematics teacher collaboration; (ii) to identify promising theoretical and methodological perspectives for future studies. The ideas that are synthesised in this chapter are based on the presentations, papers, and discussions that occurred as part of the ICMI Study 25 Conference. In this introductory section, we briefly review the results from the previous ICME-13 research survey on mathematics teacher collaboration (Robutti et al., 2016; Jaworski et al., 2017). These results, which functioned as a starting place for the Theme A Working Group, not only informed the organisation of our group but were also reified in the ICMI Study 25 Conference discussion document (IPC, 2019).

1 Introduction

The aims of this chapter are: (i) to showcase the state-of-the-art for theoretical perspectives for studying mathematics teacher collaboration; (ii) to identify promising theoretical and methodological perspectives for future studies. The ideas that are synthesised in this chapter are based on the presentations, papers, and discussions that occurred as part of the ICMI Study 25 Conference. In this introductory section, we briefly review the results from the previous ICME-13 research survey on mathematics teacher collaboration (Robutti et al., 2016; Jaworski et al., 2017). These results, which functioned as a starting place for the Theme A Working Group, not only informed the organisation of our group but were also reified in the ICMI Study 25 Conference discussion document (IPC, 2019).

In this section, we connect results from the previous survey with the research reported by ICMI Study 25 Theme A participants. We note points of convergence among the studies and theoretical additions from ICMI Study 25, which helps document the current theoretical and methodological landscape of research on mathematics teacher collaboration. Next, we outline the research questions posed in the discussion document for Theme A and present the methodology of our work. We conclude this section with an overview of the structure of the chapter.

1.1 Background

The research on mathematics education often requires a theory or a theoretical framework for different purposes (Assude et al., 2008; Niss, 2007a). Theories, and the inherent concepts that come along with them, provide guideposts for analysing phenomena related to the object of study and for making important distinctions, connections, and relationships. Therefore, the identification of theories that may be used to study mathematics teachers’ collaboration is an essential condition for a deeper understanding of the potential and limitations of the collaboration.

1.1.1 ICME-13 Survey and Four Theoretical Perspectives

ICME-13 Survey refers to the published results of an international research survey commissioned for ICME-13 held in 2016 focusing on mathematics teachers working and learning through collaboration (Robutti et al., 2016). Importantly, the ICME-13 Survey found that, while many papers did not declare explicitly the theoretical perspectives, four perspectives were evident for studying collaboration involving mathematics teachers: Community (of Practice or of Inquiry), Activity Theory, Valsiner’s Zone Theory, and Meta-Didactical Transposition.

Theories involving community were used most frequently, and in most cases referred to the theory of Communities of Practice (Lave & Wenger, 1991; Wenger, 1998) or the derived theory of Communities of Inquiry (Jaworski, 2006). Communities of Practice is a social theory of learning, which is understood as a process of social participation. This theory emphasises the negotiation of meaning through participation within the community, the formation of common goals and a repertoire of social practices, and the building of a professional (teaching) identity relative to the shared social norms of the community. Relatedly, the theory of Communities of Inquiry puts special emphasis on how alignment of the participants takes place in order to achieve the aims of the group, stressing the role of critical alignment, in which participants align with the norms of the community while asking questions and critically reflecting through inquiry.

The second most frequently used theoretical perspective in research on collaboration involving mathematics teachers was Activity Theory (Engeström, 2001), in which the activity and its object are achieved collaboratively through the mediation of tools and framed by the communities’ rules and division of labor. Special attention has been paid to the version presented by Engeström that stresses the role of expansive learning, taking place when there are contradictions within an activity system or between different activity systems.

Another theoretical perspective identified in the survey results was Valsiner’s (1997) Theory of Zones, which expands Vygotsky’s Zone of Proximal Development (ZPD) learning theory to distinguish among the free movement of learners in their Zone of Free Movement (ZFM) and the restrictions placed on them by an “other” who seeks particular outcomes for them in their Zone of Promoted Action (ZPA). These ideas were applied to investigate the teacher’s practice as a learner, in which ZFM suggests the teacher’s possible actions and ZPA are the activities offered by the teacher education program.

The fourth identified perspective was that of Meta-Didactical Transposition (MDT; Aldon et al., 2013), derived from Anthropological Theory of the Didactic (ATD; Chevallard, 1985, 2019). ATD characterises mathematical knowledge and its teaching and learning in terms of ‘praxeologies’ and ‘didactic transposition’. MDT considers the transposition (i.e. teaching and learning) of didactic knowledge and practice on teaching, which is conducted in the professional development.

Interestingly, whereas Communities of Practice, Activity Theory and Valsiner’s Zones were developed outside of the field and later applied to mathematics education, MDT is a framework that was developed within the field of mathematics education.

1.1.2 Connections with ICMI Study 25: Discussion Document in Review

A summary of the most frequently used theoretical and methodological perspectives to study teacher collaboration (Robutti et al., 2016) is present in the ICMI Study 25 discussion document (IPC, 2019), including aspects of the dynamics of teachers’ collaborative work and the communities in which they work. This document also references theoretical perspectives developed outside mathematics education that have focused on the nature of the communities in which teachers collaborate. Given their prevalence, special attention was given to Communities of Practice (Wenger, 1998), Communities of Inquiry (Jaworski, 2006) and Activity Theory (Engeström, 2001). The ICMI Study 25 discussion document also attends to Valsiner’s (1997) Theory of Zones and its use in mathematics teacher education (Goos, 2005), in which the emphasis is on the professional learning experienced by participants.

In addition to these three broad perspectives, the ICMI Study 25 discussion document also references theoretical perspectives outside mathematics education that have focused on conceptions of teacher learning. For example, Situative Theories of Learning assume that knowing and learning are situated in particular physical and social contexts, are social in nature, and distributed across the individual, other persons, and the tools used (e.g. Greeno, 1997). Another perspective presented in this document is the Practice-Based Theory of Professional Education (Ball & Cohen, 1999). This theory considers the mechanisms underlying teacher learning, suggesting that professional development programs should situate teacher learning in the types of practice they wish to encourage. The discussion document notes that the themes and questions addressing theories that focus on teacher learning overlap with those identified by the theoretical perspectives that focus on the nature of communities.

Furthermore, theoretical frameworks developed within mathematics education research that allow us to investigate different aspects of teacher collaboration are also reported in the discussion document. For example, the Documentational Approach to Didactics is mentioned, which describes a theoretical approach that focuses on studies of teacher collaboration on the role of participants as designers and users of resources (Pepin et al. 2013). Two frameworks based on ATD (Chevallard, 1985, 2019) are also included: (1) MDT (Aldon et al., 2013; Arzarello et al., 2014; Robutti, 2020), which was referred to in ICME-13 Survey; and (2) Paradidactic Infrastructure (Miyakawa & Winsløw, 2013), which characterises the different settings for teacher collaboration inside and outside school.

1.1.3 ICME-13 Survey and Methodological Approaches

The ICME-13 Survey (Robutti et al., 2016) specifies two methodological approches: (1) research methodologies used to study the work of collaborative groups; and (2) developmental methodologies, which provide directions for the constitution, and development of collaborative groups. The research methodologies used to study teacher collaboration are mostly qualitative and include participant observation, case studies, action research, and design-based research. Data sources include participant journals, interviews, questionnaires, narratives, and audio and video recordings of the collaborative activities or of the activities that the members of the collaborative groups carry out with other participants.

Concerning developmental methodologies, the ICME-13 Survey highlighted Lesson Study, learning study, action-research, design-based research and developmental research in their findings. The discussion document mentioned all these methodological approaches and data collection techniques and added that, concerning methodology, “the important issue is that data is thorough, systematic, reliable and authentic regarding the perspectives and practices of participants” (IPC, 2019, pp. 4–5).

1.2 Questions of the Discussion Document

The theoretical and methodological landscapes mentioned above suggest the following questions to be explored within the context of ICMI Study 25. These were framed in the discussion document (IPC, 2019) in the following terms:

• How do the different theoretical perspectives or networks of theories enhance understanding of the processes of teacher collaboration?

• How do they enhance understanding of the outcomes of teacher collaboration?

• What is illuminated by the different perspectives and methodologies and what needs further investigation?

• What are promising research designs and data collection and analysis methods to study teacher collaboration?

1.3 Work Methodology and Structure of the Chapter

This chapter is based on the work carried out by participants at the ICMI Study 25 Conference whose collective work was organised around Theme A: The Theoretical Perspectives on Studying Mathematics Teacher Collaboration. This body of work included the presentation of 18 papers and associated discussions, as well as overarching discussions on cross-cutting issues and a plenary talk by Susanne Prediger with a reaction from Boris Koichu, which were dedicated to issues of theory and theorising (all included in the references of this chapter).

We began the writing process by creating an overarching structure for the chapter that would address the theories that were discussed at the study conference in the Theme A working group. The strength of our group was both the diversity of theories and the diversity of contexts and aspects of teacher collaboration used by participants in their research. This strength simultaneously presented us with the challenge of writing a clear, coherent narrative capturing the important takeaways from our collective work while avoiding oversimplification or misrepresentation of any of the theories.

We organise this chapter around two complementary perspectives, which allow us to investigate the nature of theories used and developed in the research on mathematics teacher collaboration. The first perspective is the theory itself. We classify the theories we faced in the presentations of the ICMI Study 25 Conference according to their roles, origins, and research issues linked to the theories, and investigate the characteristics and specificities of these theories in our research area. The second perspective is the teacher collaboration. We classify the theories we faced according to the kinds of teacher collaboration, which conceptualise in different settings of teachers’ work, and investigate what kinds of theories may help to address different issues of teacher collaboration, and what kinds of theories are still needed to be developed to productively address these issues.

This chapter is arranged under five sections. After this first introductory section, in the second section, we present a general discussion on theories in mathematics education research. In the third section, we discuss the diversity of theories that we identified in the studies related to teacher collaboration, and subsequently, in the fourth section, consider issues specific to teacher collaboration. The fifth section completes the chapter with conclusions and perspectives for future work. All along the chapter, we only make occasional reference to the methodologies used, because these were scarcely discussed in the study conference and in the 18 presented papers.

2 Generalities of Theories

In this section, we address the broader questions on the notion of theory and its roles within the context of research on mathematics education. We conclude with the assumption that all research is theoretical in some way, even if researchers are unaware of the underlying epistemology that influences their work. To this end, we argue that making these implicit perspectives explicit through continued dialogue within the broader mathematics education research community is generative for the field.

2.1 What Is a Theory?

When answering this question and others closely related to it, researchers in mathematics education often begin with dictionary definitions of the word or its etymology (e.g. Eisenhart, 1991; Jablonka et al. 2013; Mewborn, 2005). For example, the online Oxford Advanced Learner’s Dictionary lists the following:

theory noun

• 1. a formal set of ideas that is intended to explain why something happens or exists

  According to the theory of relativity, nothing can travel faster than light.

• 2. the principles on which a particular subject is based

  This is your chance to put theory into practice.

• 3. an opinion or idea that somebody believes is true but that is not proved

  He has a theory about why dogs walk in circles before going to sleep.

Word Origin: late 16th cent. (denoting a mental scheme of something to be done): via late Latin from Greek theōria ‘contemplation, speculation’, from theōros ‘spectator’.

Idioms: in theory. Used to say that a particular statement is supposed to be true but may in fact be wrong:

In theory, all children get an equal chance at school. (OED, n.d.)

The variability within the everyday usage of theory suggests that the “notion of theory is not exactly a monolithic one” (Niss, 2007a, p. 97) within the relatively young field of mathematics education (Schoenfeld, 2000). Thus, one purpose of this linguistic exercise in scholarly publications is to situate contemporary usage among historical origins, and as a byproduct, bring an established problem in mathematics education into focus: “theory is a value-laden term with a long and convoluted history” (Mason & Waywood, 1996, p. 1055).

This history is reflected in some of the critical questions about theory currently under discussion within the broader mathematics education community. As framed by Niss (2007a): Where do the entities referred to as theories invoked in mathematics education come from? How are they developed? What foundations do they have? What roles do they play in the field? Is it problematic that there is “no such thing as a well-established unified or unifying ‘theory of mathematics education’ that is supported by the mathematics education research community” (Niss, 2007b, p. 1308; see also diSessa, 1991)? Put another way, what this means within our community is that the quintessential scientific and scholarly practice of theorising is often rendered invisible by inconsistent and overlap** usage of theory. Such use, in turn, is reified as a taken-as-shared repertoire of practices for the broader mathematics education research community (Mewborn, 2005; Silver & Herbst, 2007; Schoenfeld, 2007).

With that being said, there is also a notion that “implicit theories” (Sternberg, 1985) can guide research practices without being explicitly articulated. In the ICMI Study 25 Theme A group, which was devoted to theory, the authors did explicate their theoretical stances. However, the papers manifest various implicit theories of theory, as the theory notion is sometimes used at its face value. The role of this section is to make implicit theories of theory explicit as a background for further analysis and synthesis of the contributed papers.

While a shared understanding of theory remains unsettled in the international mathematics education community (e.g. Robutti et al., 2016), some scholars have offered definitions in an effort to address this critical issue in our field. For example, Niss (2007b) proposed the following definition:

A theory is a system of concepts and claims with certain properties.

• A theory consists of an organised network of concepts (including ideas, notions, distinctions, terms, etc.) and claims about some extensive domain, or a class of domains, consisting of objects, processes, situations, and phenomena.

• In a theory, the concepts are linked in a connected hierarchy (oftentimes, but not necessarily, of a logical or proto-logical nature), in which a certain set of concepts, taken to be basic, are used as building blocks in the formation of the other concepts.

• In a theory, the claims are either basic hypotheses, assumptions, or axioms, taken as fundamental (i.e. not subject to discussion within the boundaries of the theory itself), or statements obtained from the fundamental claims by means of formal (including deductive) or material (i.e. experiential or experimental with regard to the domain(s) of the theory) derivation. (p. 1308; italic in original)

Niss’s definition is clearly inspired by how mathematics is (often) conceived, while the second example that follows seeks to encompass the fundamental products of an empirical research process. Radford (2008) specified a theory in mathematics education as a triplet formed by the following elements.

• A system, P, of basic principles, which includes implicit views and explicit statements that delineate the frontier of what will be the universe of discourse and the adopted research perspective.

• A methodology, M, which includes techniques of data collection and data-interpretation as supported by P.

• A set, Q, of paradigmatic research questions (templates or schemas that generate specific questions as new interpretations arise or as the principles are deepened, expanded or modified). (Radford, 2008, p. 320; italic in original)

While a wide range of different theories or theoretical perspectives have been taken up by mathematics education researchers over the last decades, there is also heterogeneity in what is called a theory by different researchers and different scholarly traditions (Niss, 2007b; Prediger et al., 2008b). These conceptualisations may or may not match with the ordinary usage of theory or application of theoretical perspectives in the research community. One consequence of defining the notion of theory in such rigid terms is that it leaves room for the interpretation of elements that are necessarily ambiguous or absent depending on the theories being used. While some researchers understand this as a problem to be corrected, we instead argue that productive discussions around ambiguous or missing elements of purposefully used theories in research can be generative for the field.

2.2 The Roles of Theory in Mathematics Education

Another way of conceptualising the notion of theory is through the clarification of its roles in mathematics education. For example, Niss (2007b) suggested six purposes of theory: as an explanation of observed phenomena, as a predictor of future occurrence, as a set of guideposts for research design, as a structured set of lenses through which research is conducted, as a safeguard against unscientific approaches to myriad facets of research, and as a shield against attacks from skeptics and hostile colleagues from other disciplines. As these purposes are inextricably connected to the design decisions of the researcher, we follow the purposes of the participating authors of the Theme A papers, taking them into account at face value.

However, theories do not always specify all of these details, are not necessarily structured in a hierarchical way, and have the potential to emerge from research contexts (Bikner-Ahsbahs & Prediger, 2010; Prediger et al., 2008a; see also Assude et al., 2008). Even well-established or well-known theories such as the Theory of Didactical Situations (Brousseau, 1997) or Anthropological Theory of the Didactic (ATD; Chevallard, 1992, 2019) are still in a process of evolution and advancement. From this perspective, Bikner-Ahsbahs and Prediger (2010) argue for a more dynamic consideration of theories in mathematics education research in their complex relation to mathematics education reality (see also Prediger et al., 2008b; Prediger, Chap. 6, this volume).

A more dynamic view is also assumed by Silver and Herbst (2007) when they characterise the roles of the theory as a mediator of relationships among practices, problems, and research. In proposing these relationships, which highlight the diversity in approaches to understanding theory, the authors discussed different types of theories: “grand theories” of mathematics education, “middle-range theories” that concern subfields of study, and “local theories”, thereby providing one way to conceptualise the diversity of theories used in mathematics education. The objective of the papers presented for Theme A is not only to better understand the teacher collaboration and to get insights for the practices, but also to study, examine, compare or develop the theories. This suggests that the theory itself can be an object of study, which is not fully captured in the aforementioned examples.

Accordingly, theory can be understood as both a tool and an object of research (Assude et al., 2008). When considering theory as a tool, its functions are to:

• (a) conceive of ways to improve the teaching/learning environment including the curriculum;

• (b) develop methodology;

• (c) describe, interpret, explain, and justify classroom observations of student and teacher activity;

• (d) transform practical problems into research problems;

• (e) define different steps in the study of a research problem;

• (f) generate knowledge. (English & Sriraman, 2009, p. 1622)

When considering theory as an object, the development of theory is also one of its primary functions (English & Sriraman, 2009). For example, in the way that theory drives methodology, such as “what is taken to be data and what data are selected for interpretation” (Kilbourn, 2006, p. 545), the methodology for data collection and analysis may similarly drive theory development. Taken together, in Theme A we argue that making “implicit theories of theory” explicit is a non-linear essential practice that affords the mathematics education research community with important opportunities to engage with and apply a broader range of scholarly results in their own work.

2.3 A Coda on Epistemological Awareness

It has been argued that theories are often “taken to be unproblematically applied to a research study” (Lerman, 2006, p. 12). Simon (2009, p. 486) attributed this phenomenon to the confounding of “what one looks at” and “what one looks with” when using theory in mathematics education research. This leaves room for the choices in a research study to be based entirely on comfort with what the researcher is looking at, rather than deliberate consideration for theoretical perspectives that may be most useful to look with. As a consequence, the theories that are used for and developed from research may be ambiguous or absent to the researcher and/or the broader scholarly community. This can fuel dangerous self-reinforcing research cycles in which assumptions of atheoretical research in mathematics education becomes a self-fulfilling prophecy.

Silver and Herbst (2007) observed this phenomenon as journal editors, noting that manuscripts are often rejected for being atheoretical. Some would argue that it is not possible for research to be atheoretical, rather epistemologically unaware (Koro-Ljungberg et al., 2009). However, the larger point we make here encourages avoidance of the “sort of rigid, blind adherence to a theory that characterises much theory-based research” (Lester, 1991, p. 198; see also Eisenhart, 1991). Instead, we hope to provoke discomforting yet somehow comforting conversations by continually applying pressure to explore theoretical frontiers in and for mathematics education research.

To this end, as we concluded in our ICMI Study 25 Theme A discussions, the practices of interrogating and communicating the theories used for and developed from our research not only support a better understanding of our own studies, but also provide accessible entry points for researchers who use a diversity of theories—and a diversity of approaches to theories—to better understand and learn from the scholarship of others.

3 Diversity of Theories Related to Teacher Collaboration

In this section, we introduce the variety of theories used to analyse teacher collaboration, discuss the roles of theories as well as the origins of theories and their foci, and conclude by addressing research issues related to develo**, enhancing, networking, and analysing theories.

3.1 Diversity as a Result

The 18 papers presented in the study conference referred to a variety of theories and theoretical perspectives, which range from grand theories to local theories in terms of Silver and Herbst (2007). Looking across the 18 papers, we found it difficult to judge what counts as a theory. Some papers referred to well-established theories such as the Anthropological Theory of the Didactic or the Cultural–Historical Activity Theory. Others referred to models or perspectives that may not be called theory, but they nevertheless play important roles for studying teacher collaboration. In Table 2.1, we list the theoretical references explicitly mentioned in the 18 papers as theoretical underpinnings that were used or proposed to study teacher collaboration. Evidently, the references are of different nature.

Table 2.1 List of theoretical references explicitly mentioned in the 18 papers

This table shows a greater diversity of theoretical perspectives as compared to the results of ICME-13 Survey (Robutti et al., 2016; Jaworski et al., 2017), where apparently only four perspectives were evident (see also earlier):

Many papers did not declare explicitly the theoretical perspectives behind a project. Of those that did, four perspectives were evident: Community (of Practice or of Inquiry) (69%), Activity Theory (20%), Metadidactical Transposition (6%) and Valsiner’s Zone Theory (5%). (Jaworski et al., 2017, p. 267)

The percentages provided in the quote refer to papers that explicitly mentioned theoretical perspectives.

The greater theoretical diversity in our Theme A working group can partly be explained by the fact that the nature of papers examined in that survey was very different from ours—our group explicitly focused on theories, and the papers in the survey did not. Even so, the theoretical diversity in our working group reflects a wider use of theoretical perspectives in the research field of teacher collaboration. While the percentage of the papers referring to a specific perspective may not be significant, as there are only 18 papers in our working group, some theoretical references have been mentioned in more than one paper. Those are ATD, DAD, Community of Practice/Inquiry, Activity Theory and Lesson Study.

Considering the theoretical references in Table 2.1, they can be classified according to different aspects such as purpose, origin, focus, and use. Selected theoretical perspectives are specific to teacher education (e.g. IMPG, Zones of enactment). Some theories were developed in mathematics education research, while others were developed in other disciplines (e.g. sociology, psychology) and later used in mathematics education research studies. Some theories are primarily used to understand teacher collaboration, while others are instrumental in designing the professional development (PD) program or in conceptualising teachers’ ordinary work including teacher collaboration. Some studies combine or network the theories to analyse teacher collaborative work. This diversity of theories was one of the results we obtained in the study conference. It reflected the complexity of researching teacher collaboration, its multi-faceted nature, and different forms in different contextual/cultural parts of the world.

Of note is that the theoretical diversity is not specific to research on teacher collaboration. In mathematics education research, a diversity of theories has often been observed and discussed as an issue to be addressed. In fact, several attempts have been carried out to identify the different roles of theories, as well as to explore the relationships between them, in order to develop a coherent view on the theories used and developed in mathematics education research (Niss, 2007a; Assude et al., 2008; Radford, 2008; Bikner-Ahsbahs et al., 2014). In line with the Networking Theories Group (Bikner-Ahsbahs et al., 2014), we consider that a diversity of theories is an indicator for the dynamic character of the field. We also agree with this group in that, “the diversity of theoretical approaches can only become fruitful if connections between them are actively established” (p. 8).

In order to clarify and understand better the roles and functions of different theoretical perspectives and to make connections between them, we investigate in this section the theoretical diversity related to the research on mathematics teacher collaboration, through the 18 papers presented in the study conference. In particular, we identify the special features of theories and perspectives used or developed, for studying and designing mathematics teacher collaboration. For this, we examine the 18 papers, looking for the research issues addressed as well as for roles, origins, and foci of theories and perspectives related to teacher collaboration.

We choose the papers as examples so as to introduce different theoretical perspectives and not to overlap with the ones that will be presented in the next section. In this way, we expect to be able to develop insights that will contribute to the enhancement of theories suitable for studying teacher collaboration. Hence, in addition to providing an overall landscape of different theoretical perspectives used for studies on teacher collaboration, the aim of Sect. 2.3 is to characterise the diversity and suitability of theoretical perspectives as well as to outline some perspectives for future research.

3.2 Roles of Theories

As discussed in Sect. 2.2, theories in mathematics education research play different roles. We consider that the two main roles of the theory in the research on teacher collaboration, when it is used as a tool, are: “a way of producing understanding and ways of action” (Radford, 2008, p. 320). Regarding the former role, theories are used to understand the educational phenomena related to teacher collaboration, by providing conceptual and/or methodological tools to analyse and understand phenomena from different perspectives. There are several aspects in teacher collaboration that can be the object of study. This is one of the reasons for the diversity of theories in research on teacher collaboration. A variety of theories can be used depending on the aspects to be studied, and different theories enable different understandings of the multifaceted nature of teacher collaboration.

Regarding the second role, the theory is used to design the work of a collaborative group, of a PD program or of other settings (including teacher collaboration inside/outside schools). A diversity of contexts and scales of designing a situation or a setting for teacher collaboration also produces a diversity of theories. The object to be designed could be the pre-service or in-service PD program or the community of teachers and/or researchers, at different levels (national, regional, local/district, school, etc.). The different theories would inform how to design a situation according to the contexts. The frameworks used for such studies are often called ‘model’, which implies how to organise PD programs, for instance. The term ‘model’ in the research on teacher collaboration may be used for those that “embody a theory of objects and relations among them” (Schoenfeld, 1998, p. 9) like in scientific fields (e.g. a model of the solar system), and also for those that describe or conceptualise the teachers’ collaborative practices without explicit theoretical underpinning.

Among the theoretical references mentioned in the papers presented in the study conference, one can easily identify these two main roles. Many papers use or develop theoretical constructs to better understand teacher collaboration. For examples, Capone et al. (2020) utilise multiple theoretical perspectives (Semiosphere, Semiotic mediation, Boundary objects) to advance a multifaceted understanding on Lesson Study (LS) in the Italian context. Pepin and Gueudet (2020) discuss how the Documentational Approach to Didactics (DAD), a theoretical perspective which focuses on resources used and/or developed in teachers’ work, allows us to understand the professional learning of teachers (in teacher collaboration) in terms of schemes.

In contrast, some papers referred to theoretical perspectives in terms of designing a PD program or other forms of teacher collaboration. For example, White (2020) designed a collaborative PD program in Ireland by leaning on the idea of Professional Learning Community (PLC) and the insights obtained around this idea in previous studies (see the detail below). Horn and Bannister (2020) developed insights for the intervention design of a form of teacher collaboration that was said to support transformative professional learning, based on the Interactionist perspectives.

The distinction between the theory for understanding and the theory for designing is not clear-cut. The former could be also used for designing a PD program or a teacher community. This depended on how the researchers/educators used the theory and in which ways they were engaged in the teachers’ practices. In fact, the Interactionist perspective (Horn & Bannister, 2020) mentioned above was used, in addition to designing PD programs, to define teacher learning and allow the researchers to analyse the data and get insights for intervention design.

The Fractals perspective (Suurtamm, 2020—see more details below) was used to understand and design the professional learning communities and their relationships, as well as to design research with teachers. The intervention design was also a central concern in the project presented in the plenary lecture by Prediger (2020, and Chap. 6, this volume), who introduced several content-specific theoretical elements for designing and explaining classroom practices as well as teacher PD practices and teacher educators’ practices.

We provide here examples of studies presented in the study conference, which show how a specific theory is used for understanding and for designing PD programs.

3.2.1 Example 1: Theoretical Constructs for Understanding Lesson Study Within ATD

The paper presented by Otaki et al. (2020) proposed theoretical constructs within the Anthropological Theory of the Didactic (ATD; Chevallard, 2019), in order to better understand two aspects of Lesson Study (LS), which are critical to describe the mechanism of this professional development process.

The first aspect is the paradidactic aspect of LS that considers the nature of teachers’ activities outside classroom such as designing, discussing, and analysing mathematics lessons. Otaki et al. (2020) proposed a theoretical construct that describes the factors the teachers consider in such activities, in terms of the dialectics, which is a notion used in ATD to characterise the two opposed types of constraints that influence different activities (Chevallard with Bosch, 2020). There are six dialectics which may happen during the process of LS: dialectics of stakes and gestures, of period and study program, of milieu and infrastructure, of the predidactic and the postdidactic, of school and noosphere, and of the designer and the analyser. We do not go into the detail of all these dialectics. To explain just one, the dialectic of stakes and gestures identifies teachers’ back-and-forth reflection between mathematical knowledge to be taught and way of teaching it, by employing the notions of didactic stake and didactic gesture used in ATD (Chevallard, 2019). These theoretical constructs allowed the research team to characterise teachers’ activities and reflections during the LS.

The second aspect is the sociocultural aspect of LS that questions the viability of LS in a given place. Within the ATD, the so called ‘ecological analysis’ studies the ‘living’ of a given phenomenon (e.g. a specific mathematics teaching, teacher collaboration) by identifying the conditions that made it viable in a given place (called institution) and the constraints that might hinder it. It has been one of the main research issues addressed within the ATD (Chevallard, 2019). Otaki et al. (2020) proposed the scale of levels of paradidactic determinacy as a theoretical tool to investigate and classify the conditions and constraints that support or hinder the existence of LS in a given institution (e.g. another country). This tool is an evolution of the scale of levels of didactic co-determinacy, which has been originally developed in ATD, and that highlight the multilayered nature of conditions on a given phenomenon, going from Humankind to Civilizations, Societies, Schools, Pedagogies and Didactic systems (Otaki et al., 2020).

The theoretical constructs developed in this study provided a terminology and allowed the researchers to arrange “a set of specific observations and interpretations of singular but related phenomena into a coherent whole” (Niss, 2007a, p. 105; italics in original). It was noticeable that the ATD as a principal theoretical framework provided the basic concepts and ideas, as well as “a coherent whole” to characterise the educational phenomena, in this case the existence of LS.

3.2.2 Example 2: PLC as a Perspective for Designing a PD Program

We present here another study wherein a theoretical perspective is used for designing teacher collaboration. White’s (2020) paper already mentioned above presents an attempt to design and implement an effective teacher PD for mathematics teachers through an examination of the different models proposed and developed in previous studies in Ireland. In this study, the author examined first the impact and limitation of LS implemented in the Irish context in the last decade, and then proposed a PD based on the notion of Professional Learning Community (PLC) as an alternative approach for Irish teachers’ collaboration.

PLC is a term widely used in the context of teacher education. Following DuFour (2004), White considered it as “a group of teachers who recognize the need to work collaboratively with a common purpose of improving student learning and achievement” (p. 216). In the literature, several studies have been carried out to characterise PLC, and they identified different elements of effective PLCs. For example, PLCs operate effectively when members have shared values and vision, focus on student learning, take an inquiry stance, make teaching more public, share experience and expertise, and so forth (Scott et al., 2011).

From what we can see in the studies of our ICMI Study group, PLC seems to be a general notion denoting the situation or setting where the professional learning happens, and it is not sufficiently conceptualised. It is more a model that provides ideas to design and structure the PD program, rather than an elaborated theory that is a systemic entity characterising and informing the mechanism of teacher collaboration. This is why several studies have investigated the characteristics of effective PLCs. Similar to LS, PLCs could be also an object of study, and at the same time used for designing PD activities.

3.2.3 Example 3: Fractals as a Perspective for Designing Teacher Collaboration and Research

Another example concerning the roles of theories is the case where the theoretical perspective functions as a tool or model for designing the teacher collaboration as well as for engaging in the research with teachers. Suurtamm (2020) proposed the mathematical idea of Fractals as a model for networked teacher collaborative communities, and at the same time as a model for the research with such communities. In several of her projects, she identified, as a key mechanism of teacher collaboration, the iterative and self-similar nature of the communities of researchers and teachers, which could be described by fractals. According to her, this nature fits well the ways of thinking about how they work and create the teacher collaborative communities, and design educational research with teachers. For example, she noticed self-similarity between the collaborative community of teachers and the collaborative communities developed in their classrooms, with a reciprocal process of feedback between these two kinds of communities.

In another case, the self-similarity can be found in large-scale projects between the multiple networked PLCs, and the single PLC. Further, in terms of research, the iterative dynamic of research design can be found in the data collection (each collection builds on the previous one), and in the ways some participants were engaged in collaborative research with teachers (e.g. mathematics co-ordinators became research participants as the data collection progressed).

The idea of Fractals here is a model that leads us to focus on specific aspects of teacher professional learning communities and understand their characteristics: iterative, and self-similar. This model is likely to have implications when designing and facilitating the nested communities of practice and when setting up a research project. However, this model alone would not inform the design of other aspects of teacher collaboration and research projects. Hence, she relies on several theoretical perspectives of communities and learning, such as Communities of Practice (Lave & Wenger, 1991; Wenger, 1998), theories of constructive learning (Cobb et al. 1993), sociocultural theory (Vygotsky, 1978, 1986), to name but a few. In other words, she synthesises different perspectives into one overarching one: Fractals.

3.3 Origins of Theories and Their Foci

We identified a variety of theories, which could be distinguished in terms of their different origins and different foci. Some have been developed in the area of mathematics education research, and others in for example, general education, sociology, or psychology. Again other theoretical frameworks were constructed specifically for analysing teachers’ practice and later adapted to study teachers’ collaborative work (e.g. DAD). It can be claimed that most of the theoretical perspectives discussed were not created for studying teacher collaboration; instead, they have been enhanced to deal with the issues of teacher collaboration.

Mathematics teacher collaboration includes a wide range of aspects, it is social by nature and takes place in a specific contextual or cultural setting. It is often discussed in the context of PD or teacher learning, which may be connected to the cognitive perspective. It includes the use and development of curriculum materials and resources, which may be considered in terms of theories of tool use and mediation.

Different theories enable us to understand different aspects of teacher collaboration (e.g. Trouche et al., 2019). The aspects the different theories consider are different according to their origins. The theories that originated from mathematics education research were developed especially to better understand the phenomena specific to the teaching and learning of mathematics, as the object of study. The theories developed in other areas do not focus on the aspects specific to mathematics. They typically provide insights into other important aspects of teacher collaboration, such as interactions between different participants, teacher learning, and community development and operation.

In general, the origin of theoretical frameworks and perspectives is not always clear. A perspective may be developed based on the different perspectives by combining and adapting them, so that it may fit well to the analysis of the object of study. For example, DAD is a theoretical perspective that originates in French research of mathematics education and is today used in different fields of educational research. It had been developed based on ideas of French didactics of mathematics, the field of technology use (cognitive ergonomics), and socio-cultural theory, and later further developed including the field of curriculum design (e.g. Trouche et al. 2020).

The 18 papers in the study conference included a wide range of theories, and they originated from different fields. It is not easy to determine a single origin to one theory. Selected theories, such as for example, ATD, Commognition and DAD, have been developed in mathematics education research. Some theories, such as Cultural–Historical Activity Theory (CHAT) and Enactivism have their origins in general education or educational psychology. Theoretical perspectives, such as Community of Practice and Culturally Figured Worlds, have been developed within anthropology.

The theories used or developed in the papers presented in the study conference focus on one or several of the following aspects of teacher collaboration:

• social;

• cognition and learning;

• identity;

• resources;

• teacher mathematical knowledge.

We also found that several theoretical perspectives considered the social aspect in different ways. The theoretical construct within ATD presented above, levels of paradidactic determinacy (Otaki et al., 2020), account for the cultural factors, beyond the teacher’s local setting, that shape the teacher collaborative work. The theoretical framework, Culturally Figured Worlds (presented below), focuses on teachers’ formation of professional identity, and characterises how teacher behaviour shapes and is shaped by their cultural and social contexts and the mutual relations of power. This implies that the diversity of theories is due to the multifaceted nature of mathematics teacher collaboration, as well as to the approach adopted by a theoretical perspective to study a specific aspect. This would be also the case for the aspect of cognition or learning, which is characterised in different ways according to the theoretical perspectives.

In what follows, we provide three examples of how theories from different origins contribute to a better understanding of mathematics teacher collaboration.

3.3.1 Example 1: A Theory Originated in Mathematics Education Research

The first example is a study by Kondratieva (2020). She used the ATD developed in mathematics education research to understand teacher learning in a collaborative setting. The ATD includes several theoretical constructs to investigate the phenomena of mathematics teaching and learning, such as ‘didactic transposition’, ‘praxeology’, inquiry with ‘media’ and ‘milieu’, for example. In her study, Kondratieva used two kinds of ideas as analytical tools to better understand teachers’ collaborative learning in mixed groups of elementary and secondary school teachers, when solving mathematical problems within a teacher education graduate online course.

First, she used the notion of praxeology. This is a model of human activity. A praxeology consists of two blocks: the first that describes praxis or practices including a type of tasks and a technique to solve such a type of tasks, and the second, that describes logos or discourse including technology and theory underpinning the praxis. Kondratieva used this notion to describe the development of mathematical knowledge and practice at stake in teacher collaboration, and revealed the overlap of elementary and secondary school praxeologies consisting of generic exemplification, which belongs to logos in the former and to praxis in the latter.

Second, she used the following three dialectics (from ATD), to analyse teachers’ collaborative learning:

• dialectics of idionomy and synnomy (the individual and the group);

• dialectics of conjecture and proof (also called dialectics of media and milieu);

• dialectics of black boxes and clear boxes.

These are dialectics which can be found in the mathematical inquiry as well as in the inquiry of other domains. The use of dialectics allows Kondratieva to identify various tensions and discrepancies among teachers’ viewpoints, and to consider teacher learning as a result of negotiation and resolution of those viewpoints during the process of mathematical problem solving. Of note is that the notion of dialectics is common in ATD and was also employed in the study by Otaki et al. (2020) presented above.

Kondratieva’s study focuses on the mathematical inquiry and the collaborative development of mathematical knowledge in teacher collaboration. In such a study, the theory developed in mathematics education research like ATD allows us to capture the specificities of mathematics teacher collaboration.

We have also seen that the same ATD notions, dialectics and praxeology, can be used in different ways to analyse teacher collaboration. We will see later that the notion of praxeology is also used to characterise teachers’ and researchers’ practices in Meta-didactical transposition (Aldon, 2020; Shinno & Yanagimoto, 2020). It implies that a single theoretical construct may be employed and developed to analyse different aspects of the phenomena under investigation. The particular object of study we set up for studying teacher collaboration therefore enhances the theories.

3.3.2 Example 2: A Theoretical Perspective Originated in General Education

The next example is taken from the study by Calleja (2020) who investigated teacher learning in the collaborative setting in a continuing PD program. He showed how the theoretical perspective of Zones of Enactment could be used to better understand teacher change, that is teacher learning in his case, through their engagement in a collaborative PD program (learning to teach mathematics through inquiry). It showed in particular that the interplay between teachers’ personal resources in enacting inquiry (e.g. knowledge, beliefs, and practices) and external factors (e.g. pupils, policy, public, private, and professional sectors) could be investigated through Zones of Enactment.

Zones of Enactment is a theoretical perspective that has been developed in general education and curriculum studies, based on Vygotsky’s work regarding the Zone of Proximal Development (ZPD). Spillane (1999) proposed it to understand teachers’ reconstruction of their practice in the implementation of educational reform from the perspective of educational policy. Zones of Enactment are defined as the space in which teachers “make sense of, and operationalise for their own practice, the ideas advanced by reformers” (p. 159). It characterises the teacher’s change or reform of practices by focusing on the role of community and the context of teacher learning within educational settings. Spillane suggested a model to account for the ways teachers responded to and enacted mathematics reform.

Vygotsky’s work is very influential in educational research and has been developed in different directions according to the object of study and to the aspects the researchers are interested in. Valsiner’s zone theory that we already discussed at the Introduction was also based on Vygotsky’s work to characterise child development and then adapted to study teacher learning. In the case of Zones of Enactment, this perspective has initially been a development for the curriculum studies or educational policy. Calleja (2020) further adopted it to understand teacher learning in the collaborative setting, with a special attention not only to the personal or cognitive factors (knowledge, beliefs, and practices), which are often discussed in the context of teacher learning, but also to the external or sociocultural factors. As the theory has been developed in general educational research, the disciplinary specificities of mathematics teacher learning were not taken into consideration in the theory itself. This is a point, which is different from the previous example. However, this is not necessarily a shortcoming. Such specificities would be addressed by combining other theoretical perspectives.

3.3.3 Example 3: A Theoretical Approach Originated in a Social Practice Perspective

The third example is taken from Skott’s (2020) paper, which adopts a social practice theoretical perspective to better understand how contextual and power related aspects influence teacher collaboration. Specifically, her study concerns the dialectical relationships between teachers’ social interactions and the social, cultural and power-related aspects of their local setting and beyond that, when they adapt LS in a Danish educational context. With an example on how an individual teacher participates in teacher collaboration in a LS context, she shows how social practice theory, especially the concepts of Figured Worlds and Cultural models, enable us to both conceptualise adaptations of LS in countries outside East Asia in a new theoretical way and to study teacher collaboration and individual teacher learning in such adaptations from a contextual perspective. The teacher’s learning was characterised by shifts in ways of participating in the collaborative interactions with respect to the Figured Worlds s/he drew on.

The concept of Figured Worlds has been developed in the area of anthropology, in order to investigate how people form their shifting identity in relation to their social worlds and privileges of power. A Figured World is defined as, “a socially and culturally constructed realm of interpretation in which particular characters and actors are recognized, significance is assigned to certain acts, and particular outcomes are valued over others” (Holland et al., 1998, p. 52). This perspective enables researchers to study how people’s behaviour shapes and is shaped by their cultural and social contexts and their privileges of power. The original aim was not to study educational phenomena, but the approach is increasingly employed in educational research.

This theoretical perspective provides a terminology and concepts to understand and explain teacher collaboration and individual teacher’s learning in terms of cultural, social, and power-related aspects both in and beyond their local setting. As we have seen in other examples, the contextual aspect is critical in teacher collaboration and addressed in different ways according to the adopted theoretical perspective.

3.4 Research Issues on Theory: Develo**, Enhancing, Networking, Analysing

In addition to using theory as a tool for understanding or designing teacher collaboration, the researchers may examine and develop it as an object of study. The rationale for carrying out explorations on theories should go towards a better theorising of teacher collaboration. As we have shown, most existing theories were not specific to teacher collaboration and addressed only certain aspects of teacher collaboration.

Furthermore, very often they did not attend to the characterisation of teacher collaboration as a whole, including associated teacher activities, student learning, curriculum resources. Prediger’s plenary lecture for Theme A presented her attempt to theorise teachers’ PD in a collaborative group in more content-specific ways. She based her work on the theory of professional growth (Clarke & Hollingsworth, 2002), and the identification of content-specific theory elements capturing the PD learning content (Prediger, Chap. 6, this volume).

Viewed from this perspective (on theory), one may find the different ways of exploring theories: the theory to be constructed; the theory to be used, adapted, or enhanced; and the theory to be analysed. This corresponds to another diversity of theories, as an object of scientific research from a structural point of view.

In the papers presented in the study conference, we identified two issues on theory:

• 1. develo** theoretical constructs or theorising some aspects of teacher collaboration;

• 2. analysing theoretical perspectives or models.

Regarding the first, some papers proposed a new theoretical construct or model that allowed the researcher to better understand teacher collaboration. Other papers reported on other proposed theoretical constructs that complement existing theories which are not specific to teacher collaboration. For example, Otaki et al. (2020) developed concepts within ATD, which theorise the sociocultural aspect of teacher collaboration and the constraints the teachers encounter. Suurtamm (2020) proposed the idea of Fractals to theorise the self-similar and iterative mechanism within teachers’ and researchers’ work and nested communities. In addition, some papers provided a new methodological perspective on how the researcher can analyse or design the teacher collaboration, by adapting, enhancing, or combining the existing theoretical frameworks.

Regarding the second issue, some papers investigated the theories or models themselves, in order to better understand their affordances and limitations. It is especially in these studies that two or more perspectives were compared or contrasted.

In the study conference many studies considered more than one theoretical perspective. However, the ways to relate two or more perspectives were different. In some cases, there was a dominant theoretical perspective, with the integration of some small ideas or concepts into the dominant one. In other cases, theoretical perspectives were equally combined. In a study by the Networking Theories Group (Prediger et al., 2008b), different networking strategies have been identified (Fig. 2.1). We can also find some of these strategies in the papers of the study conference. For example:

• synthesising, in which a new theory emerges: Fractals model (Suurtamm, 2020) is a result of synthesising the ideas of different theoretical perspectives, mathematical idea of fractals, communities of practice (Lave & Wenger, 1991; Wenger, 1998), theories of constructive learning (Cobb et al. 1993), sociocultural theory (Vygotsky, 1978, 1986), and so forth;

• combining, in which the used theories keep their identity: Hoyos and Garza’s (2020) study combines DAD with Ernest’s idea of personal philosophy or image of mathematics, and also with Clarke and Hollingsworth’s (2002) IMPG. Capone et al.’s (2020) paper also combines three theoretical frameworks: Semiosphere, Semiotic Mediation and Boundary Objects;

• comparing, in which the theories keep themselves apart: Ding and Jones (2020) carried out a comparative analysis of different models of mathematics teacher collaboration: Action-Education model; Learning Study model; Community-Centered model.

In addition to carrying out the theorising work by combining existing theories, we note that it is also possible to undertake such theorising by a methodology involving the construction of hypotheses and theories through the collection and analysis of empirical data, similar to the grounded theory approach. However, we have no examples of such studies in Theme A papers at ICMI Study conference.

Fig. 2.1

A landscape of networking strategies. (Prediger et al., 2008b)

We now present some specific studies that exemplify different ways of exploring theories.

3.4.1 Example 1: Adapting and Theorising

The first example is Brown and Coles’s (2020) study which has attempted to theorise a way of working with teachers of mathematics in collaborative work. They investigated and theorised how teacher learning might take place, based on: (1) their experiences of working in a PD course; (2) what they perceived as enactivist ideas. They proposed a model that conceptualises the process of mathematics teacher learning in collaborative groups. Through the analyses of the specific cases, they showed how this model allowed them to identify mathematics teacher learning in the process of collaborative work.

Brown and Coles’ basic idea of theorising is that ‘learning to teach mathematics’ can be viewed in terms of the development of awareness. They characterised this development of awareness, relying on enactivist ideas, especially the three levels of categorisation of how humans perceive the world, namely at a detailed or subordinate level, a basic level, and an abstract or superordinate level (Varela et al. 1991, p. 177). According to their view, teacher learning takes place through a descent into the detail of experience, from the basic level to the subordinate level. A cyclical process of mathematics teacher learning in five phases was proposed: describing the detail of events; identifying new distinctions; develo** new labels; trying out new actions; develo** new basic-level categories.

Enactivism is a perspective of cognitive science that claims to model mechanisms of human cognition. It is a general perspective, which is not specific to education nor teacher collaboration but offers ideas which enables the researcher to theorise the process of mathematics teacher learning in collaborative work. Brown and Coles’ study is a case of adapting the enactivist perspective to theorise mathematics teacher learning.

3.4.2 Example 2: Combining Theories as a Networking Practice

In terms of combining multiple theories, Capone et al. (2020) explored the possibilities of combining theoretical frameworks in order to understand mathematics teachers’ PD, in particular LS in the Italian context. This paper is reminiscent of Skott’s (2020) paper, which provides a theoretical perspective to analyse the implementation or adaptation of LS in a specific cultural context outside East Asia. While Skott’s paper adapts a single social practice perspective (i.e. the framework of Figured Worlds), Capone et al. propose a study showing how dissimilar theoretical frameworks can highlight different aspects related to the adaptation of LS, and how LS improves teachers’ practices in terms of PD. The following three perspectives were employed: Semiosphere and Semiotic of Cultures; Semiotic Mediation; and Boundary Objects.

They characterised the abovementioned three theoretical frameworks in terms of the three elements proposed by Radford (2008), which describe the theory: principles, methodology, and research questions. They then showed how each theoretical framework enabled the researcher to understand the different aspects of implementation of LS. Semiosphere, which is a concept developed in semiotics, permitted to see the deconstruction of practices and beliefs, and the production of a new awareness on the part of teachers. Semiotic mediation, which is originally Vygotsky’s idea and has been further developed by an Italian group of mathematics education research (Bartolini Bussi & Mariotti, 2008), facilitated a better understanding (and importance) of the teacher’s role for the appropriate choice of artefact.

The concept of Boundary Object, which has been developed within CHAT (Akkerman & Bakker, 2011) gains a significant role here. A boundary object is an object that lies at the intersection of several social worlds facilitating communication between them (Star, 2010). Capone et al. (2020) show how LS helped the boundary crossing of the prospective teachers towards the practicing teaching community. A specificity of their study is that it was carried out explicitly from the perspective of networking theories. The object of study was the LS, and at the same time the theories themselves. Capone et al. combined the theories, in order to answer the common research questions on LS. This was carried out through contrasting the specificities of each theory, in particular the basic principles that constitutes the system of theory and the methodology supported by this system.

3.4.3 Example 3: Comparing and Analysing Different Models for Teacher Collaboration

The last example presents a case of analysing multiple models of teachers’ collaborative work. Ding and Jones (2020) comparatively examined three models of mathematics teacher collaboration and learning, and identified the affordances and limitations of each model. The models analysed were Action–Education model (Gu & Gu, 2016), which is practiced in China; Learning Study (Lo & Marton, 2012), which is a combination of LS and design study; and the Community-Centered model (Borko et al., 2005), which is developed for university-based PD program in US. The authors analysed these models by using Boylan et al.’s (2018) framework that theorises the nature and processes of teachers’ professional learning, in terms of components and relationships, scope, theory of learning and location of agency.

The results of analysis showed the differences in particular aspects of PD: goals, learning processes, learning outcomes and contexts. For example, the learning outcome of the Action–Education model was, “Core elements of practical knowledge & its relationship such as task design and lesson implementation”, while that of Learning Study was, “Lesson design and implementation of necessary conditions for learning according to VT [Variation Theory]” and that of the Community-Centered model was, “Content knowledge, mathematics-specific pedagogical knowledge, and recognition of the importance of learning community” (p. 116).

This paper is similar to Capone et al.’s (2020) study, in the sense that both studies aimed to compare/contrast the multiple theoretical perspectives or models related to teacher collaboration in order to better understand them. In contrast, the natures of perspectives or models seemed very different in the two studies. The three theoretical perspectives investigated in their study were constructed mainly as tools to better understand the educational phenomena, which was not necessarily specific to teacher collaboration, while the three models in Ding and Jones’ study were mainly to design teacher activities or PD, which presupposed teacher collaboration. The roles of perspectives and models and the position of teacher collaboration were also different.

Interestingly, in Ding and Jones’ study, another theoretical tool was employed to analyse the three models of teacher collaboration: the analytical framework developed by Boylan et al. (2018) to study teachers’ professional learning. This framework directed the focus of study to specific elements that constituted professional learning: components, scope, theory of learning, location of agency, and philosophical paradigms. Compared with Capone et al.’s study which adopted Radford’s (2008) framework of theory, Boylan et al.’s framework is specific to teacher learning and provided insights into the models of teacher collaboration for designing and practicing PD. In contrast, Radford’s framework provided insights into the perspectives for designing research on mathematics education.

3.5 To Conclude

In this section, we reported on our observations from investigating the diversity and specificities of theories related to the research on mathematics teacher collaboration, in terms of the nature of theory. Our investigation strategy was to enter and clarify this diversity regarding the nature of theory in terms of three aspects: roles of theories, origins and foci of theories, and research issues related to theories.

We illustrated the diversity in each of these aspects with the papers presented in the study conference. This investigation allowed us to better understand the characteristics of theories used or developed in research of mathematics teacher collaboration, as well as the research issues related to the individual theories and to the set of theories.

Regarding the specificities of theories in research on teacher collaboration, we have observed that there were two main lines (while some frameworks deal with both): frameworks that conceptualise teacher learning in collaborative groups (e.g. teacher knowledge development) and teacher collaborative activities (e.g. PD program, teacher communities); and those that theorise the socio-cultural aspects of teacher collaboration (e.g. under what conditions this is likely to happen).

4 Conceptualising Teacher Collaboration

4.1 Introduction

In Sect. 2.3, we discussed the nature of theories how theories were used, developed, and adapted to study or design teacher collaboration. In this section, we discuss theories taking into account the nature of teacher collaboration. According to the kinds of teacher collaboration, we look for, through the papers presented at ICMI Study 25 Conference Theme A, how different theoretical perspectives enhance our understandings of the processes and outcomes of teacher collaboration situations, and what needs further investigation and theorising. To this end, in this section we first conceptualise how we understand teacher collaboration.

While teacher collaboration may take different forms in different contexts, some aspects are essential. We assume that mathematics teacher collaboration involves 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. A collaborative group always develops its activity in a given context that provides the elements that justify the need for collaboration as well as the resources to make it happen. At the same time, the context provides constraints that affect the working processes. In this section, we begin by making distinctions between different situations or settings where teacher collaboration can occur.

Looking at different situations, we may understand what they have in common and how they differ, and which specific features may be considered in each kind of teacher collaboration. Each kind of collaboration is illustrated by examples from the study conference papers presented, and for each kind of collaboration, we investigate how the different theoretical perspectives allow us to address different questions related to teacher collaboration. For each situation we begin by making a general description, especially in terms of the features of collaboration; subsequently, we present the research questions that are addressed for this situation; and finally, we describe the theories used to address such questions.

From the papers at the study conference Theme A, a first distinction to make is (a) the collaboration in order to solve a problem or to deal with an issue, that has emerged in a given professional context and is perceived by a group of professionals as necessitating attention and (b) collaboration in the frame of a professional development activity. In this kind of activity, a group of professionals is formed (around a particular issue/problem/question) and typically the group has a ‘minder’ who guides and takes initiative for setting the aims and carrying out the work, as it is the case of LS.

All 18 papers of the study conference Theme A fall under (a) or (b) (with selected ones at the border between the two). Both situations are interesting to discuss as they involve collaborative processes and have a role in enhancing mathematics teachers’ pedagogic practice. However, each paper shows specific features of situation (a) or (b), and each refers to a specific theory or framework, sometimes to several ones, and we pay special attention to these.

4.2 Collaboration to Solve a Problem or Deal with an Issue

In the collaboration to solve a problem or deal with an issue, one may find different kinds of situations in terms of institutional frames that shape mathematics teachers’ collaboration. An important kind is the collaborative project (Sect. 2.4.2.1). A project may be regarded as an activity which aims at a particular outcome and is typically limited in time. When a ‘project’ continues over a prolonged period of time, it tends to become an organisational or institutional activity, which can be classified as another kind of situation, collaborative activity (Sect. 2.4.2.2). In addition, it is possible to speak of collaboration in the frame of an existing organisation (Sect. 2.4.2.3). We can exemplify these kinds of collaborations by selected papers presented at the study conference; when we cannot, we provide examples from the mathematics education literature.

4.2.1 Collaborative Projects

Collaboration-in-time-bound projects have particular features. A collaborative project departs from a ‘problem situation’, that is a situation which demands a solution; this becomes the aim of the project. The tension between the ‘problem situation’ and the desired aim requires a structured organisation of work, including the mobilisation of internal and external resources required to achieve the set aim, and leadership of the project. The issue of leadership, or minder of the project, and relationship between participants is critical in collaborative projects.

An important issue in a collaborative project is the diversity of participants. In some cases, the collaborative group can encompass participants with similar status and experience (e.g. all are teachers with a similar role in their schools, as in Stephens, 2020). In other cases, a collaborative group consists of people with different academic and institutional status (e.g. some are schoolteachers and others are academic researchers, as in Pericleous, 2020). How the participants are organised, how they monitor their work, how they deal with internal tensions and conflicts are important issues to consider.

Another issue is the diversity of aims and outcomes of projects. In a collaborative project, the aim is explicitly set up, and the outcome may be evaluated in the project. These aims and outcomes may vary according to the project (e.g. resources, lessons, teacher learning, research results, diffusion of something, scaling-up). Further, the aim and outcome within a project may be different for different participants (e.g. teachers, researchers, etc.). Theoretical perspectives that enable investigating differing aims and outcomes are valuable for research on mathematics teacher collaboration. Each collaborative project is suited for a specific purpose, and it faces different challenges.

At the study conference, there were examples of papers in this sub-category. For example, in the case of Wake et al. (2020) study, the aim of the collaborative project was to advance understanding on how to organise teaching that holds curriculum coherence. The authors identified as research outcome the roles of the didactical devices/tools that provide connections across topics of conceptual understanding of mathematics and development over time. They studied this using the perspective of Cultural–Historical Activity Theory (CHAT) and the associated notion of boundary object. In this case, the theoretical perspective allowed them to better understand the nature of outcomes, and the different levels of activities the different participants engaged in.

In another project, involving the collaboration of a researcher and a mathematics teacher, Pericleous (2020), also drawing on CHAT, studied the activity of proving in the classroom. The author addressed the collaboration between the two participants, focusing on the design of tasks and lessons and on the classroom implementation in order to gain access to the aims and motivations of the teacher and to understand what drives the teacher decisions during teaching. The paper discusses the conflicts between the two participants that occurred during this collaboration and the role of classroom resources to shape the process of proving.

These two papers use CHAT. In CHAT the unit of analysis is an activity, which is an endeavor directed to an identifiable goal or object (Engeström, 2001). Such activity works at both an individual level (subject, tool, object) and a social level (rules, community, division of labour). The object of a collective activity is constantly evolving, both in its material features and also as a social entity. A main notion in this theory is that of contradictions. These may arise in and between components of the activity system, between different phases of development, and between different activity systems. Contradictions may lead to transformations and expansions of the system, therefore supporting participants’ motivation and learning.

The participation in multiple activity systems leads to the theoretical concept of boundary crossing, as a socio-cultural gap creating discontinuities in the actors’ actions or interactions. Boundary crossing is facilitated by boundary objects, a theoretical notion used in several papers at the ICMI Study conference. Bowker and Star (2000) talk about boundary infrastructures consisting of boundary objects that allow different communities to work together without fully resolving their conflicts or reaching a consensus (Star, 1989, 2010). Historically, Activity Theory has undergone several developments, with the third generation CHAT including the theory of expansive learning (Engeström, 2001), which stresses the role of communities and of transformation of culture, through the construction of new objects and concepts, and the development of new practices.

The notions of division of labor, rules and community seem quite apt to address the composition and activity of collaborative projects. These notions, however, are very general and may be merged to further notions that qualify or modify them, to consider the specific nature of collaborative projects. A similar observation also applies to Meta-Didactical Transposition Framework (MDTF—see later in this section), as both frameworks are well suited to study time-bound projects. Therefore, an open issue is to know how to enrich these frameworks with further notions specifically apt to investigate collaborative projects.

4.2.2 Collaborative Activities

Regarding collaborative activities, the main feature is their flexible nature. In contrast to the collaborative projects, the aims may be more diffuse. Instead of setting a timeline for achieving them, the group might keep progressing as long as benefits are perceived. The aim is not something that, once achieved, empties the need for collaboration, but, on the contrary, is something that may only be achieved by the continuation of the collaboration. In this case, there will be very likely some people that are more central to the collaborative activity than others, but the differentiation of roles is not as stringent as in the collaborative project. An important theoretical issue is to know what kind of bond may keep the activity together, progressing and develo**. At the study conference, there were three examples of papers in this subcategory, one using the theoretical frame of Fractals, already presented in Sect. 2.3 (Suurtamm, 2020), and two others (Hoyos & Garza, 2020; Stephens, 2020) using the Interconnected Model of Professional Growth (IMPG).

In the study of Stephens (2020), the aim was improving the understanding of the processes through which teachers, working in collaboration, integrate new knowledge and improve their practice. The participants were all the teachers, except a novice teacher, of the mathematics department of a U.S. high school. The collaborative activities were rather unstructured and included very varied activities of the mathematics department. The results suggest that collaboration, despite being unstructured, played a very important role in the professional growth of the participating teachers. In the study of Hoyos and Garza (2020), the aim of the study was to build a model for the professional development of mathematics teachers. The participants were 14 middle-school teachers who carried out collaborative activities in relation to a mathematics curriculum reform based on the notion of competence. The results of the study show a change in the participant teachers regarding their conceptions of mathematics teaching.

The IMPG (Clarke & Hollingsworth, 2002) models professional learning through the consideration of interactions of four main domains: (i) the personal domain (e.g. teacher knowledge, beliefs, and attitudes); (ii) the domain of practice (e.g. professional experimentation); (iii) the domain of consequence (e.g. the salient outcomes that are perceived by the teacher); (iv) the external domain (which refers to external sources of information). Between these domains, there are mediating processes of (1) enactment, as participants incorporate new ideas within existing ideas, or carry out a new practice within existing practices and (2) reflection, as participants consider new knowledge, ideas, practices, and outcomes.

The IMPG is intentionally framed in terms of individual teachers. It models the change sequences and growth pathways for individual teachers in relation to the learning experiences they engage in, their knowledge, beliefs and attitudes, their practices, and their perceptions of outcomes. A suggestion made at the study conference was to consider if it would be possible to transpose the model from the level of the individual to the level of the group, to address the idea that collaboration plays an essential role in the professional growth of participants in collaborative groups of mathematics teachers. Whereas in the original model, collaboration would be located in the external domain, in a proposed modified version collaboration might fit in the domain of practice—the practice of the collaborative group.

This model seems especially suited to study professional development because of the ‘external domain’ (the context and activities arranged by the facilitators), which has an important influence in the unfolding of learning processes. In the study conference, it was used in one case with that purpose, and in another case to study a collaborative activity (with the external domain being called upon from the interactions with sources, such as web-based teacher sites). This modification and adaptation of existing models to study new situations may be a fruitful line of theoretical development. At the same time, in this case the model would require further specifications, both to attend to the features of collaborative activities and to effectively model collective rather than individual change and growth.

4.2.3 Collaboration in the Frame of Existing Organisations

Whereas in collaborative projects and in collaborative activities collaboration takes place in settings of informal organisations, collaboration may also take place in settings of a formal organisation. The organisation may have a name, a legal status, working rules, responsibilities for participants, formal procedures for admission, for example. The collaborative group may be a subgroup of a larger organisation, such as a working group of a teacher association. How different levels of the organisation relate to each other, how the work is monitored, how conflicts are handled, and how new members are introduced, are important issues in this kind of collaboration. The Theory of Communities of Practice (Wenger, 1998), which has been framed in the setting of formal organisations, may be useful to study this kind of collaborative activity.

At the study conference, an example of a paper fitting into this subcategory is Pepin and Gueudet (2020). One of the examples presented in this paper concerns the Sésamath Association in France (https://www.sesamath.net/index.php). In this association, volunteer participants, all practicing teachers, have collaborated to write e-textbooks and to create software and other digital curriculum resources (e.g. Gueudet & Trouche, 2012). Another example is the Grupo de Trabalho de Investigação (GTI) of the Associação de Professores de Matemática in Portugal (http://www.apm.pt/gt/gti/), a collaborative group of teachers and researchers that organises multiple activities including the writing of edited books with theoretical essays and practical examples of themes of interest to mathematics teachers (da Ponte, 2008). The first book of this series includes an essay (Boavida & da Ponte, 2002) about mathematics teacher collaboration that provided the blueprint for subsequent work.

4.3 Professional Development Activities

In terms of collaborative professional development activities, there is a variety of situations concerning the role of the different participants. We have 15 examples of this kind of collaboration in the papers presented at the study conference. One of them concerns pre-service mathematics teacher education (Shinno & Yanagimoto, 2020), 14 concern in-service mathematics teacher education.

4.3.1 Pre-service Teacher Education

At the study conference, there was one paper on pre-service mathematics teacher education (Shinno & Yanagimoto, 2020). In pre-service teacher education, collaboration may exist among different participants. However, it is influenced and shaped by the institutional setting, with roles of participants and power relations established by national or institutional norms. In this case, theories and issues about pre-service teacher education should be considered. The aims and processes of the activity should be known, including the negotiation of roles and activities to be carried out, the evaluation of prospective teachers, and the teaching style of the instructors, as well as the contextual affordances and constraints (e.g. institutional rules, previous preparation of participants, time and resources available).

Some pre-service teacher education activities take place in schools, involving the ‘supervising’ mentor (e.g. co-operating subject teacher) and in some cases further actors (e.g. other teachers, parents, etc.). The collaboration may take place in different ways (e.g. different teacher education models), and with different participants. This makes it a very complex but interesting object of study. The main issue here is how to develop and sustain collaborative relationships among participants given the institutional constraints and each participant’s role in this situation.

The study by Shinno and Yanagimoto aimed to understand the planning skills of primary school pre-service teachers and the changes through the experience in LS, which involved researchers and practicing teachers. The LS was carried out outside university at an annual half-day conference with several open lessons centered on mathematics. This conference included primary and secondary school teachers and mathematics education university professors. Interestingly, the pre-service teachers were not really engaged in this LS activity, but worked separately in the university course to write a lesson plan on the same topic used in the open lesson of the conference before participating it. Then after attending the open lesson and the discussion of LS, they had another discussion in the university to re-design a new lesson plan. The results of this study indicate how pre-service teachers adapted their lesson plans in relation to the work carried out by in-service teachers in the LS.

As in the study by Shinno and Yanagimoto, meta-didactic transposition has mostly been used to analyse pre-service teachers’ learning. However, we claim that this framework could also be used to describe collaboration involving in-service teachers and researchers. Indeed, it could be used in a more general way to study collaborations involving two or more different groups of participants. The meta-didactic transposition framework has been described in Arzarello et al. (2014), Robutti (2020) and Aldon (2020). It considers four main concepts: (1) Meta-didactic praxeologies; (2) double dialectics; (3) brokering processes; (4) internal and external components. As mentioned earlier, praxeology is a concept developed within ATD (Chevallard, 2019).

In Shinno and Yanagimoto’s (2020) words, “Didactic praxeologies describe teachers’ didactic activities. Meta-didactic praxeologies describe researchers’ (or teacher educators’) activities related to those of teachers” (p. 175). Concerning double dialectics, the first dialectic occurs in the classroom, with poles on the students’ personal meanings and the scientific meanings. The second dialectic occurs in the interaction between teachers and researchers, and concerns the interpretations of the first dialectic by these two groups of actors.

The brokering processes concern the dialogues that take place between the two groups of actors. The internal and external components refer to relative position of the elements of each block of praxeologies. These brokering processes are at the heart of the interaction between participants in the process (in our case, the members of the collaborative activity) and this framework also uses the notion of boundary object (Star & Griesemer, 1989) to describe those interactions.

Meta-didactic transposition provides a language of description for collaborative activities, which may be useful to identify important elements of the activity that must be considered, in order to understand how the activity started, how it developed, and how it enabled the eventual creation of shared praxeologies between different groups of participants.

4.3.2 In-Service Teacher Education

At the study conference, a variety of in-service collaborative activities were conducted, in which collaboration was assumed to play a prominent role. In this respect it would be important to know about the aims and processes of the activity, as well as the features of the context such as teachers’ motivation to participate and disposition for an active participation, teachers’ previous preparation, time and resources available, style of facilitation, to name but a few. Collaboration may take place among participating teachers, and between teachers and facilitators. How may these collaborative relationships develop? What factors may sustain or inhibit them? What is the potential and what are the limits of such collaborative relationships? We present three examples of theories used to address these processes: (1) the theory of Commognition; (2) the Documentational Approach to Didactics (DAD); (3) the theory of Situated Learning.

(1) The Commognitive theory identifies thinking as communication (Sfard, 2008). This theory is used in the study by Elbaum-Cohen and Tabach (2020). The aim of the study was to know if participating in a professional development program that encouraged the integration of technology in the classroom and reflection led to changes in teaching practice. The authors indicate that the commognitive theory allowed to identify changes in the professional identity of a participant teacher. They suggest that providing such opportunity for reflection may lead to the development of the teachers’ professional identity.

The Commognitive theory claims that thinking is a human capacity that develops through the social activity of communication. It considers learning as becoming a proficient participant in a discourse, which is identified by the changes in the participation in such discourse. It also emphasises the concept of personal identity, defined as the set of all stories that are reified (saying something about what the person is), meaningful (indicating key characteristics), and endorsable (supported with empirical evidence) (Sfard & Prusak, 2005). The identity of a person may be actual, as the stories are told about this person at the present or ‘designated’, if the stories are told about the expected future of the person. This theory distinguishes discourse that concerns mathematical objects (‘mathematising’) and discourse about people that participate in the discourse (‘subjectifying’). The stories related to the identity of a person come from ‘subjectifying’ discourse.

This theory originates in the field of mathematics education, and was initially not intended to study collaborative processes. However, it has been shown that this theory can be beneficial for the study of human interactions and its consequences in participants’ learning. Further, it may be used to study specific phenomena occurring during collaborations, in particular those that take place in professional development processes.

(2) Another theory used to study collaboration in professional development processes is the Documentational Approach to Didactics (DAD) (Hoyos & Garza, 2020; Pepin & Gueudet, 2020). Pepin and Gueudet present three studies that address the use of resources in mathematics teacher collaboration. One of the studies focused on the documentation work of two secondary school mathematics teachers working in collaboration, another studied the production of documents in a group of teachers of the Sésamath association, and the third concerned the documentation work of a Norwegian teacher working in a large European project. The results of the three projects suggest that collective work can promote the evolution of individual schemes.

In DAD, the focus is on teachers’ interaction with resources and on its consequences (Gueudet et al. 2013). The DAD draws on the instrumental approach (Rabardel, 2002) which makes a distinction between artefact and instrument: artefact + utilisation schemes = instrument. In a similar way, the DAD distinguishes between resource and document: resource + utilisation schemes (for a particular goal) = document. The notion of ‘scheme’ (Vergnaud, 1998) is central to the DAD. According to Vergnaud, a ‘scheme’ has four components: (i) the goal of the activity; (ii) rules of action; (iii) operational invariants (concepts-in-action, and theorems-in-action); (iv) possibilities of inferences.

Utilisation schemes can be (i) procedural schemes, regarding the use of a given resource or (ii) mental/cognitive schemes, regarding the knowledge about the resource and strategies how to use it. The DAD has been used to study both individual teachers’ work, as well as teacher collaboration, in terms of their use of resources. In collaborative processes, the group of participants may develop ‘agreed schemes’; in this respect an interesting question is how these schemes develop and how they may influence individual schemes. It is interesting to note that the DAD developed within mathematics education in close relation to theories outside mathematics education, such as instrumental genesis (Rabardel, 2002).

(3) Finally, another theory used to study teacher collaboration is the Situative Theory of Learning (Horn & Bannister, 2020), which is also called Situated Learning (Cobb & Bowers, 1999). Horn and Bannister present two examples of studies drawing on this perspective. One of these projects used principles about teacher collaborative inquiry to create an activity based on video formative feedback aiming to encourage the development of innovative mathematics teaching. The other project used principles about video-based collaborative teacher learning to design what they call a “responsive professional development model” aiming to respond to participant teachers learning needs. The authors suggest that features such as considering the novelty of deep collaboration, working with a shared vision of teaching and provide adequate visions of practice increase the possibilities of mathematics teachers’ learning through collaboration.

Situated Learning (Greeno, 1998; Putnam & Borko, 2000) stands in contrast with cognitive theories that look at learning of individuals; it investigates learning of individuals-in-context. Hence, Situated Learning considers not only individual teachers but also groups of teachers working together, and their social environment. Under this theory, interactionist analysis may address teachers’ professional conversations, looking how these conversations may provide teachers with conceptual resources for their activities. In these conversations, an important distinction is made between interpretative viewpoints that allow teachers to make sense of events, and epistemic stances that refer to claims or statements that teachers consider to be true. In contrast with the other two former theories, Situated Learning had its origin outside mathematics education and is currently applied to the study of general teacher education issues, for example.

Papers dealing with in-service teacher education were, by far, the largest set at the study conference. The sharp distinction between participants in a professional development activity and facilitators would require a careful analysis in terms of its consequences regarding the collaborative activities. What important elements bear in the initial negotiations regarding teachers’ participation in the professional development activity? How do the relationships between participants and facilitators develop during the activity and what factors influence such development? However, these are issues that have not been addressed in the papers and do not stand in a clear way in the theories/frameworks presented in those papers.

4.3.3 Lesson Study

Lesson Study (LS) has been mentioned several times in this chapter: it is one of the professional development activities which is from time to time adopted for the pre-service and/or in-service PD. Several frameworks have been used to study different aspects of LS. At the study conference, six papers discussed LS processes. One paper (Skott, 2020) was already presented in Sect. 2.3, Social Practice Theory/Figured Worlds. Two further papers, already referred to above, concerned the Meta-Didactical Transposition Framework in pre-service teacher education (Shinno & Yanagimoto, 2020) and Cultural–Historical Activity Theory regarding a collaboration between UK and Japanese researchers (Wake et al., 2020).

Another paper using ATD is also referred to in Sect. 2.3 (Otaki et al., 2020). Still, two further papers that present LS as teacher development contexts are White (2020), based on the notion of Professional Learning Community, and Capone et al. (2020), who present several theoretical frameworks. In these papers, LS was regarded as a specific form or model of teacher professional development and not as a particular theory to study mathematics teacher collaboration.

4.4 To Conclude

Each kind of collaboration faces different challenges regarding issues such as: (i) establishing and develo** the aims of the activity; (ii) establishing and develo** the working processes; (iii) establishing and develo** relationships among the participants; (iv) establishing and develo** forms of leadership that support the development and the regulation of the activity; (v) framing the relationship between the collaborative activity and important issues in mathematics, in mathematics education or in mathematics teacher education (such as mathematical focus, curriculum approach, focus on specific didactical issues or materials, focus on the student, and so on).

The examples of theories show powerful ways of looking at specific aspects of teacher collaboration that may be used in collaboration to solve a problem or deal with an issue as well as in professional development. All these theories provide elements for describing teacher collaborations that are useful to understand these processes. However, none of these theories is specifically geared to study teacher collaborations and there are central features of teacher collaboration that are not addressed by any of these theories (e.g. affect in teacher collaboration).

In fact, this analysis shows that there are different kinds of situations. Some theories and frameworks are very general and may be used to study different kinds of social processes that extend beyond collaboration. These theories and frameworks could be enriched with further specific concepts targeted at the study of mathematics teachers’ collaboration. Other theories and frameworks only address selected aspects of the collaborative activities or change in the participants, and these could be networked with other theories and frameworks to study the most essential aspects of mathematics teachers’ collaboration (e.g. those related to the different forms of collaboration, or the participants in the collaboration and the tools used, as addressed in other chapters of this book).

5 Conclusion

In this final section, we summarise the state of the art of our Theme A and the perspectives for future studies. We also discuss what this ICMI Study advances in our understanding of the theoretical and methodological perspectives on the studies in mathematics teacher collaboration.

5.1 Summary of Our Reflections

The analysis of the different theories used in the study of teacher collaboration showed a great diversity of theories, frameworks and models. We identified theories used for understanding teacher collaboration, for designing teacher collaboration, and for both understanding and designing teacher collaboration. We also noted that some theories have their origin in mathematics education, whereas others come from general education, sociology, social psychology, cognitive sciences, to name just a few. In addition, the position of the theories in the research work varies—some present theories to be constructed, others theories to be used or adapted, and still others theories as an object of analysis.

The analysis of the different contexts for teacher collaboration in Theme A papers showed that in many cases collaboration is seen as an element of professional development processes, and more attention is necessary to the work of collaborative groups. We also observed that collaborations are created to solve specific problems, as collaborative projects or activities, or that collaborators develop their activity within existing organisations. Wide-ranging theories, such as Communities of Practice or Activity Theory, have been used to study collaboration, and they could be enriched with further concepts to better address the specific features of the collaborations. In addition, several theories have been used to study some aspects of the work of collaborative groups but leave out of the picture some other aspects of the work or the context of the collaboration. These theories may be further developed to address the work of the collaboration in their complexity.

5.2 Responses to the Initial Questions

In what follows, we respond the four questions initially posed in the discussion document of ICMI Study 25, and addressed in the 18 papers presented in the study conference for Theme A. We provide our answers obtained through the whole discussion on our theme, the theoretical perspectives on studying and mathematics teacher collaboration, in the study conference as well as in this chapter, and show in which ways the studies presented at the conference advanced our knowledge of this theme.

• How do the different theoretical perspectives or networks of theories enhance understanding of the processes of teacher collaboration? How do they enhance understanding of the outcomes of teacher collaboration?

There have been three broad groups of theories addressing teacher collaboration. The first group includes theories developed outside mathematics education that focus on the nature of communities (e.g. CHAT, Communities of practice). These theories facilitate a close look on the activity and working processes of collaborative groups. The second group includes theories, also developed outside mathematics education, that address professional learning occurring inside collaborative groups (e.g. Interactionist perspectives, Enactivism, Zones of enactment). These theories support an understanding of the outcomes of teacher collaboration, especially concerning the learning of participants. The third group includes theories originally developed inside mathematics education (e.g. Meta-didactic transposition, ATD, DAD, and Commognition theory). These theories facilitate a close attention to particular processes and outcomes regarding the activity of collaborative groups that are specific for the work of mathematics teachers.

The outcomes of teacher collaboration can be viewed in terms of the professional learning of the participants, but also in terms of develo** their identity as a group and in terms of their impact within and on the context in which they operate. Whereas those theories that address the professional learning that occurs inside collaborative groups allow an understanding of the participants’ learning, it seems that other kinds of theories, addressing organisational and political issues, will be necessary to study the learning of groups and their impact in the underlying contexts.

• What is illuminated by the different perspectives and methodologies, and what needs further investigation?

The different theoretical perspectives allow the study of processes and outcomes of teacher collaboration as we indicated above. However, some aspects of these outcomes and processes remain largely invisible. These include, for example, the aspects related to the dispositions, motivations and other volitional elements concerning the participants in teacher collaborative groups. Another issue that has not been fully addressed in previous studies is the development of collaboration relative to the context in which it develops. Several papers in the study conference address this issue, in particular in the case of LS, with some theoretical perspectives (e.g. Social Practice theory, ATD). Still another area in need of further study is the content of teacher collaboration—the ‘what question’—especially when the collaboration is carried out to solve problems or to deal with an issue, outside or inside existing organisations. Further investigation is also necessary for other kinds of teacher collaboration by asking questions about, for example, the features of this context that support or hinder the creation and development of collaborative groups; the impact of the work of collaborative groups within or on the context in which they operate.

• What are promising research designs, data collection and analysis methods to study teacher collaboration?

In the study conference, the discussion focussed mainly on the theoretical perspective, whilst methodological aspects have only been scarcely addressed. While some papers referred to the design method of teacher collaboration, research designs did not have significant differences in the methodologies regarding what was found in the ICMI Study 15 Survey. Research designs depend on the object of study (or aspects of teacher collaboration) and the theories within which the study is carried out. Given the high complexity of the phenomena involved in teacher collaboration, it is not surprising that the most common research designs are qualitative.

These designs can be observational, such as participant observation, and case studies of existing collaborative groups; or interventional, such as action research, design-based research, and other developmental designs. Grounded theory studies may also provide important new insights regarding mathematics teacher collaboration, complementing what we may learn based on existing theories. Lesson Study, with its inherent collaborative nature, is a context highly favorable to study teacher collaboration. Regarding data collection and analysis methods, and as well as for many other topics of mathematics education research, an intensive use of technology, such as video recording and data analysis software, may yield interesting and new results that have not been possible with more conventional methods.

5.3 Perspectives for Future Studies

In this chapter, we strived to take the most out of the papers that were presented at Theme A of the ICMI Study conference. The issues that we could address follow from what was presented in the group’s papers; they represent a follow-up from what has already been apparent from the work of the ICMI survey team (Robutti et al., 2016; Jaworski et al., 2017). Regarding wide ranging theories, further theorisation will be necessary to add further concepts to address the issues specific to mathematics teacher collaboration and in turn to support practice. Regarding theories that only address some elements or outcomes of teacher collaboration, further theorisation will be necessary to address the complexity and outcomes of collaborative phenomena. In addition, further theorisation regarding affective, organisational and political issues will be necessary to study issues that so far have been largely invisible in the study of teacher collaboration.