Capturing Collaboration in Mathematics Teacher Education, in Terms of Relevant Actors, Targets and Environments

Abstract

The discussion document Teachers of mathematics working and learning in collaborative groups (IPC, 2019) identifies “Roles, identities and interactions of various participants in mathematics teacher education” as one of four major themes needing further elaboration. In regard to this Theme C, the document indicates, among others, that collaborative groups can include different ‘actors’ in various combinations. These actors can have a variety of roles, which can shift over time. In collaborative interactions, the learning of all participants is also important. The document indicates six ‘actors’ in an exemplary way.

1 Introduction

The discussion document Teachers of mathematics working and learning in collaborative groups (IPC, 2019) identifies “Roles, identities and interactions of various participants in mathematics teacher education” as one of four major themes needing further elaboration. In regard to this Theme C, the document indicates, among others, that collaborative groups can include different ‘actors’ in various combinations. These actors can have a variety of roles, which can shift over time. In collaborative interactions, the learning of all participants is also important. The document indicates six ‘actors’ in an exemplary way.

It is easy to increase this list by including, besides mathematics teachers themselves, other individuals such as teachers of other subjects, lead teachers or teacher leaders, department heads, principals, parents, teacher students, students, critical friends, facilitators, coaches, mentors, mediators, designers, multipliers, mathematics teacher educators, mathematicians, researchers, administrators, superintendents or policy makers—and even this extensive list is not exhaustive at all. In addition, also organisational entities like departments, schools, school boards, districts, committees, ministries or enterprises can be environments relevant to a collaboration: for example, by influencing the goals, processes and results of collaborative activities which take place in projects, programs, teams, communities of practice, networks, study groups, etc. (see, for example, Krainer, 2008). The word initiative sums up different forms of collaborations and is used in this chapter as an umbrella term.

This diversity makes it difficult not only to get an insightful overview, but also to compare initiatives or to grasp their specificity. In preparation for this contribution, we decided to look for selected articles and to analyse them along the following three dimensions:

• the relevant actors of the collaboration (e.g. teachers, teacher educators, researchers, etc.);

• the relevant targets (e.g. aims, goals, etc.) of the collaboration (e.g. improving the knowledge or beliefs of students and/or teachers in geometry, writing a new algebra curriculum, establishing or further developing a school’s emphasis on mathematics teaching, etc.);

• the relevant environments of the collaboration (e.g. a school or a district, a mathematical association, a university, policy makers, curriculum makers, etc.).

Since we focus on Relevant Actors, Targets and Environments, we call our tool RATE. In order to facilitate potential users’ grasping of essential features of each initiative, we visualise selected exemplary initiatives (see the next section, “The selection of articles”) in a diagram (triangle), indicating the number of actors. We then highlight the key intervention of each initiative, its type and duration, and the specificity of the collaboration. Additionally, we provide selected findings on different forms of teacher collaboration presented in the selected articles.

In system theory (Willke, 1999, p. 12; see also Krainer, 2005), noticing a relevant difference (observation) and producing a relevant difference (intervention) are important terms. In teacher education, one kind of difference is of particular interest, namely the status quo of teaching, which often is regarded as unsatisfied, versus the desired target situation (“good” or “high quality” teaching, etc.), marking a possible need for improvement. For example, if nobody (teachers themselves, researchers, educational policy makers, etc.) would see a need to produce a difference (improving teaching), then reforms, professional development courses, etc. would rarely be initiated. However, it is interesting to ask: what is the relevant difference that should be produced? Who defines this? By what means should the relevant difference be achieved? Who are the actors? Who is learning, changing, implementing, improving, etc.? Who supports, who documents, who evaluates, who takes decisions, who controls, etc.? Is collaboration between teachers fostered, are knowledgeable others involved and, if so, in which roles? Such reflections are used to work out the specificity of collaboration in these initiatives and the type of intervention.

The RATE diagram has the form of a triangle with the corners Teachers, Knowledgeable others and Relevant environments. These three corners stand for social entities relevant for the respective initiative.

• 1. Teachers

  This corner represents all participants of the initiative who will become, are or have been members of the teaching profession at a school or a kindergarten (prospective teachers, teachers and retired teachers), aiming at improving teaching (e.g. through planning, implementing, observing, evaluating and investigating teaching, developing curricula or material, etc.). For example, all teachers taking part in an Action Research project or in a Lesson Study group at one school are ‘teachers’, including the department head or lead teacher. Also, a teacher who serves as a teacher educator, multiplier or researcher in another initiative, defining another context, but is a common member in the initiative at hand, is regarded as a ‘teacher’ in this initiative.

• 2. Knowledgeable others

  These participants of the initiative, in most cases, come from outside the teaching profession at a school or a kindergarten, for example, university staff, teacher educators, school developers, researchers and people from school administration or economy. In a few cases, knowledgeable others might be teachers, eventually part-time, if they have gained expertise beyond teaching, for example, in providing teacher education, counselling schools and doing research, and in case they bring in this special expertise in this initiative going beyond their teaching expertise.

• 3. Relevant environments

  These social entities, such as individuals, groups, institutions, etc., are not participants in the initiative, but they have a relevant direct or indirect influence on the initiative. This could be a principal supporting a professional development course and/or the whole school, a superintendent and/or a whole school district, a mathematician and/or a whole mathematical association, a researcher and/or a whole research group, a policy maker or a whole ministry, curriculum makers, etc.

The heading of the diagram indicates the intervention focus and the context of the initiative (in parentheses and in bold), while (uni- or bi-directional) arrows indicate interconnections between the corners, eventually characterised by specific wording. The labelling of the arrows highlights some of the specificities of the collaboration, such as teacher educators bringing in theoretical frameworks for helping teachers better to frame their students’ learning or teachers bringing in their teaching experiences to help teacher educators better to understand the practical needs and constraints. Double arrows indicate, for instance, relevant issues of the different actors’ co-learning. Circle arrows highlight the effect on the self-development of the different actors.

In order to get additional information about communalities and differences between the seven initiatives regarding the specificity of collaboration, we also counted key words across all articles, excluding the references. The key words helped to distinguish the individual collaborations and to identify differences. The results of the searching for key words is part of the description of each initiative. The search comprised key words like activity/ies, belief/s, broker/s, boundary/ies, club/s, collaboration/s, colleague/s, community/ies, decision making, evaluation/s, group/s, intervention/s, learning, Lesson Study, mathematics, mathematician/s, member/s, ministry/ies, parent/s, participant/s, participation, partner/s, project/s, reflection/s, researcher/s, school/s, share/ing, student/s, teacher/s, team/s and university/ies.

2 The Selection of Articles

Acknowledging existing large surveys on mathematics teacher education (Adler et al., 2005; Gellert et al., 2012; Robutti et al., 2016, with a focus on teachers working and learning through collaboration), we focused on a small number of recent publications on teacher collaborations in high-quality journals. Since school systems are very different around the world, entailing different forms of collaboration between stakeholders, our goal was to analyse one or two articles from each continent (Africa, Asia, Europe, North America, Australia and Oceania, and South America), using the same dimensions (aspects of collaboration) for each article to allow a structured comparison. The selection of articles followed three criteria:

• searching for articles with a clear focus on the topic: keywords were linked by logical operators ‘mathematics’ AND ‘collaboration’ OR ‘teacher collaboration’ OR ‘collaborative lesson research’ OR ‘community of practice’ OR ‘community of inquiry’ OR ‘teacher interaction’ OR ‘co-operation’;

• searching for recently published articles (in 2018 and 2019);

• searching in high-quality journals (according to Williams & Leatham, 2017, the focus was directed on the following two ‘very high’ and five ‘high’ quality journals: ESM & JRME; FLM, JMB, JMTE, MTL & ZDM). In order to fulfil the all-continent goal, a hand search for additional quality articles was connected, starting from literature in the discussion document.

When focusing only on titles and abstracts, the systematic search identified 20 articles. The reading of the full texts led to five articles, focusing clearly on the topic: Asia 1, Europe 2, North America 1 and South America 1. Therefore, a hand search was needed, focusing on Africa and Oceania. This led to eight further articles in journals and one article in an anthology. The selected African article is cited as ‘in press’ in the discussion document and is published now, the Australian article is published in a journal very close to ‘high’ ranked, according to Williams & Leatham (2017), and refers to authors cited several times in the discussion document.

A first overview of the seven articles is presented in Table 8.1, indicating the initiatives’ continent, the studied collaboration, the intervention focus and the context in which the implementation occurs, with the research focus including the used method and the research results.

Table 8.1 Overview initiatives

3 Description of the Seven Initiatives

In the following, the seven selected articles are described and visualised using the RATE tool. Additionally, we provide information on the authors of the article, the type of the initiative, the specificity of the collaboration and we present selected findings.

3.1 Initiative 1 (Africa): A Case of Lesson Study in South Africa (Adler & Alshwaikh, 2019)

Relevant Actors

Four secondary mathematics teachers (‘Lesson Study group’) and four researchers (‘project members’, ‘project team’ from university) participated in the project.

Relevant Targets

The targets refer to the two target domains, teachers’ teaching and researchers’ co-learning, thus, “improving the learning and teaching of mathematics in previously disadvantaged secondary schools” (p. 318) and “to systematically research our co-learning” (p. 327). The key intervention, producing a relevant difference, is directed to teachers’ improvement of teaching.

Relevant Environments

One university and three schools from one province in South Africa were involved in the project. Additional partners were the provincial departments of education, while the collaborative work between schools was organised in school clusters.

Authors

The two authors were part of the project team consisting of four researchers. The first author had the role of the project director; the second author, who was relatively new to the project and learning about Lesson Study, was supported by the project director.

Type of Initiative (Duration)

The initiative was a one-year, research-linked professional development, with a Lesson Study group at its core. The key processes of ‘Planning–Teaching–Reflection’ were supported by using an analytic framework (Mathematical Discourse in Instruction and Mathematical Teaching Framework), which serves as a boundary object, moving between being a research tool and a tool for teaching.

Specificity of Collaboration

Small collaborating groups of teachers came from different schools. The authors aimed at “opportunities for teachers and researchers together to learn about teaching and how the tensions and dilemmas we [the researchers] faced were simultaneously opportunities for strengthening the coherence of the community” (p. 326). Although the initiative was regarded as a professional development, those carrying this out did not name themselves as teacher educators or something similar (role), but rather described themselves as researchers (identity). Thus, the ‘actor system’ showed with ‘teachers’ and ‘researchers’ a clear difference in identity and goals (learning in order to teach better; learning in order to generate new scientific insights); however, both had the role of ‘co-learners’. Compared with the six other initiatives, the statistics of key words in the text, excluding references, regarding this initiative shows the highest occurrence of ‘learning’, ‘reflection/s’, ‘group/s’ and ‘Lesson Study’. In combination with the high frequency of use of ‘teacher/s’ and ‘researcher/s’, this confirms the claim of co-learning.

Research Results

The article aims at responding to two questions: “What changes occur in the example set across the lesson plans over a cycle? How do these changes occur?” (p. 326). With respect to the first question, the authors stressed the relevance of a specific framework, theoretically informing change processes. However, regarding the focus of this article on collaboration, the second question was the more important one: “How did the change in plans evolve?” (p. 331). This question involves the issue of interaction within a Lesson Study group where four researchers and four teachers collaborated. The authors show that the changes to the example set were initiated through “example change moments” (p. 335), and that these changes were a ‘collective accomplishment; of both teachers and teacher educators/researchers in a double role. However, within this interaction, teachers’ and teacher educators’ contributions to the change were different: while teacher educators initiated an explicit and theoretically-grounded focus on example sets, teachers brought in their specific concerns about their students’ learning as experienced in the teaching of the lessons and the reflections upon it. The authors stress both the importance of discussing and reflecting as a decisive part of the ‘collective enterprise’ of a Lesson Study group, and the critical role of ‘knowledgeable others’, in particular fostering “structured and theoretically informed observation and reflection” (p. 338) (Diagram 8.1).

Diagram 8.1

RATE 1—Africa

3.2 Initiative 2 (Asia): Mathematicians and Teachers Sharing Perspectives on Teaching Whole Number Arithmetic: Boundary-Crossing in Professional Development (Cooper, 2019)

Relevant Actors

Approximately 20 primary mathematics teachers, two research mathematicians and one broker (“participant-observer researcher”, “I”, “author”, “PhD in mathematics education”) from one university were the stakeholders in the project.

Relevant Targets

The requirement by the Israeli Ministry of Education that primary-school teachers need “to enrol in mathematics PD courses in order to specialise in mathematics” (p. 71) sets the context for the initiative. The author stresses that the project enabled a setting for the primary teachers and the research mathematicians in which, “the two communities could share their perspectives with each other, not only allowing the teachers to benefit from the mathematicians’ perspective, but also providing an opportunity for the mathematicians to attain the sensitive understanding” (p. 70). This indicates that sharing perspectives is the goal of the initiative. The key intervention (producing a relevant difference) is fostering communication and collaboration between all relevant stakeholders regarding mathematics teaching, in particular including mathematicians, thus overcoming “conflicts between the communities of mathematicians and mathematics educators” (p. 69).

Relevant Environments

The article draws on contributions in the realm of ICMI as a powerful international association, which encourages “a link between educational researchers, curriculum designers, educational policy makers, teachers of mathematics, mathematicians, mathematics educators and others interested” (quoted in Cooper, p. 70)—see International Mathematical Union (IMU, 2019). Further involved in the project were the Israeli Ministry of Education and one university.

Authors

The single author of the article was the mathematics teacher educator acting as a ‘participant-observer researcher’ between the participating mathematicians and primary mathematics teachers.

Type of Initiative (Duration)

The initiative reports on a professional development course, co-taught by a Ph.D. student of mathematics and a Master’s student of computer science, lasting one academic year.

Specificity of Collaboration

Two mathematicians provide a professional development course for primary teachers, whereby a mathematics educator serves as a boundary-broker to mediate between the two “communities” or “parties” and pays attention to their sociocultural difference: “I also highlight the role of a participant-observer researcher as a broker in this process, supporting events of boundary-crossing in which the parties came to explicate, and sometimes change, their own perspectives on teaching and learning mathematics with respect to the perspectives of others” (p. 69). A main relevant difference refers to the goals which are mixed in a delicate way. Teachers need to upgrade their mathematical knowledge; at the same time, they are expected to be co-learners with the mathematicians:

Together, these two lesson segments represent two types of PD activity—one “content based”—designed and led by mathematicians and dealing with particular mathematical content—and the other “problem based”—led by teachers and dealing with authentic issues of classroom teaching. In the first, episodes were selected and analyzed in detail to showcase opportunities for learning through “boundary-crossing”. (p. 72)

The frequency of key words like ‘boundary/ies’, ‘sharing’, ‘mathematician/s’ and ‘community/ies’ underline the descriptions above.

Research Results

The article investigates “mechanisms of perspective-sharing” (p. 69) among 2 mathematicians and about 20 primary teachers in a professional development context. In particular, the article discusses what and how the two communities learned from and with each other, “drawing on the notion of boundary as sociocultural differences between communities” (p. 69) and regarding the researcher as a broker in this process, supporting events of boundary crossing. The article highlights three domains in which the two communities “came to explicate, and sometimes change, their own perspectives on teaching and learning mathematics with respect to the perspectives of others” (p. 69):

• (a) when the researcher began to share his analysis of feedback (to teachers) with mathematicians, “they came to appreciate the relevance of this data for their planning and teaching” (p. 78);

• (b) the researchers’ explicit attention (‘listening’) to teachers’ ideas lead to (self-reported) changes by some mathematicians in their university teaching: they dedicate “more attention to students’ ideas” (p. 78), by giving them more time to discuss;

• (c) the primary teachers, by experiencing their “contribution to the mathematical discourse” (p. 79), in particular, “their expertise in the domain of mathematics-for-teaching” (p. 79), were able to build more mathematical confidence.

In all three cases, shared reflections between the two communities—stimulated by the researcher as broker—were the ground for a significant intervention (Diagram 8.2).

Diagram 8.2

RATE 2—Asia

3.3 Initiative 3 (Australia): Boundary Crossing and Brokering Between Disciplines in Pre-service Mathematics Teacher Education (Goos & Bennison, 2018)

Relevant Actors

Six project teams, “comprising at least one discipline academic and one education academic” (p. 256), “mathematicians and mathematics educators” (p. 258) from six universities were involved in the project. Involved were also 23 investigators, that is, “the participants in the research were the mathematicians and mathematics educators who comprised the IMSITE project teams” (p. 261) and six lead investigators.

Relevant Targets

The Inspiring Mathematics and Science in Teacher Education (IMSTE) project contributes to the “improvement in the quality of mathematics and science teachers” (p. 256). This main goal is pursued by “(1) fostering genuine, lasting collaboration between mathematicians, scientists, and mathematics and science educators who prepare future teachers, and (2) identifying and institutionalising new ways of integrating the content expertise of mathematicians and scientists […] with the pedagogical expertise of mathematics and science educators […]” (p. 256). This includes, “identification of principles for fostering new forms of collaboration between discipline academics and education academics” (p. 258). This goal and its ways of fostering it indicate that the key intervention, producing a relevant difference, is a two-fold one: through fostering collaboration between relevant stakeholders in mathematics and science education, the quality of mathematics and science teachers should be improved, assuming that this will improve mathematics and science teaching.

Relevant Environments

The project involves six Australian universities, each working with a “cascade university”, in establishing different “teacher education strategies” (p. 257), funded by an Australian ministry.

Authors

The first author was a mathematics educator and had the role of a project co-leader—the other co-leader was a mathematician. The second author was a project officer and interviewed a larger sample of mathematicians and mathematics educators, including the two project co-leaders.

Type of Initiative (Duration)

Various mainly pre-service, but also in-service, teacher education strategies such as “design courses” in order to improve recruitment and retention, or to initiate innovative curriculum arrangements or “conduct a mathematics pre-service teacher education alumni conference to connect current students, graduates, teachers, teacher educators, and mathematicians in order to promote continuing professional learning” (p. 257), were implemented over 3 years.

Specificity of Collaboration

The collaboration encompasses both initiating and sustaining interdisciplinary collaboration on mathematics and science as subjects, content and pedagogy/education as fields of expertise and inter-university collaboration between universities and between universities and their cascade universities. The collaboration is thought to enable boundary encounters that, “give people a sense of how meaning is negotiated with another practice” (p. 259). Such boundary practices are facilitated by brokering. In this initiative, the boundary broker was the mathematics educator, who mediated expertise of the two fields of mathematics and mathematics instruction.

This complexity is even amplified by integrated foci, that is, primary and secondary schooling, prospective and practicing teachers. Besides the already mentioned goals, the project has also the strategic long-term goal, “to promote strategic change in teaching and learning in the Australian higher education sector” (p. 258), and the ambitious research goal to investigate conditions that enable or hinder sustained interdisciplinary collaboration. This initiative has the highest occurrence of the key words ‘collaboration/s’ and ‘project/s’, indicating the clear focus on bridge building through joint activities. In addition, also the word ‘mathematics’ is used most in this initiative, mirroring the involvement of mathematicians in the project.

Research Results

First, the project initiated collaboration in terms of boundary practices among different communities of practice, involving discipline academics and education academics that focused on curriculum development and community-building activities in teacher education courses. Second, the project contributed an evidence-based classification of aspects hindering or fostering collaboration. For the collaboration within the partner universities, several personal qualities were identified to help building up collaboration between mathematics educators and mathematicians, among them mutual trust, open-mindedness and recognising common or shared problems of preparing pre-service teachers for their future role.

The situation was different for the collaboration between the partner universities and at the cascade universities. Here, collaborating just emerged between the groups of mathematics educators and mathematicians solely. Institutional and cultural barriers were high and made it even difficult for the brokers to bring the two worlds together. Third, the project provided empirical grounding for understanding boundary practices regarding the learning mechanism of transformation at both interpersonal and intra-personal levels, including identifying together shared problems, and developing broking positions to connect disciplinary paradigms (Diagram 8.3).

Diagram 8.3

RATE 3—Australia

3.4 Initiative 4 (Europe 1): Impact of Professional Development Involving Modelling on Teachers and Their Teaching (Maass & Engeln, 2018)

Relevant Actors

Participants were 326 secondary mathematics teachers. In the overall project, more than 1000 primary and secondary teachers for mathematics and science were involved. The ‘course leaders’ from 12 countries, namely teachers, pre-service educators, persons from CPD institutions, were selected and educated by the ‘project partners’ in each country. The ‘we’ in the article refers to the whole project, including all project partners.

Relevant Targets

The goal of the project was to achieve an impact “on teachers and their teaching” (p. 273), in the direction of “implementing innovative teaching” (p. 273) through “inquiry-based learning” (p. 275). Regarding mathematics in secondary school, for example, the target was attaining, “a significant change regarding the implementation of modelling in mathematics teaching” (p. 274). For improving teaching, the project also had the goal of implementing it at a large scale (double-goal). Thus, also the key intervention was doubled: the relevant difference aimed at is innovative teaching, as opposed to current teaching, and this should be implemented and scaled up at many places as opposed to some individual places.

Relevant Environments

The project involved 14 universities (‘project partners’) from 12 countries, participating in a large EU-project (seventh Framework program of the EU which defines the context of the project).

Authors

The first author was a mathematics educator and had the role of the leader of the EU-project and of the international centre for STEM, being primarily responsible for co-ordinating the project. The second author was a science educator from a collaborating research institute and was co-responsible for project-related evaluation and research.

Type of Initiative (Duration)

Continuous professional development (CPD) courses were provided in 12 countries, running, “within a timeframe of 2 years in each country”, from “several weeks” to “the duration of 1 year”. With “about 100 teachers in each country” taking part, the project is relatively large. It is also complex, as reflected in the target groups of both mathematics and science teachers, the diverse cultural contexts of the 12 countries, the heterogeneity of the course leaders’ education and competencies, and their role as brokers, navigating between the goals of the project and the needs of the teachers.

Specificity of Collaboration

In the courses, 7 CPD principles, in the sense of quality criteria, were implemented to “stimulate cooperation between teachers so as to support teachers in the learning-on-job phases” (p. 277). In order to ensure quality across all 12 countries, the project partners “discussed the overall CDP principles and their implementation in the PD course at the biannual project meetings” (p. 278). The research questions reported in the article refer to the secondary level, particularly the mathematics teachers participating in the project, and their students, focusing on the impact of a mathematics course on modelling.

The collaboration among teachers, course leaders and project partners and between them is not a major focus of the article. Given the size of the project and the various contexts of the participating countries, it is likely that the collaboration is manifold and takes very different forms. The size of the project and the large number of teachers dealt with in the research article (macro-perspective) seem to shift the view on concrete collaborations (micro-perspective) to the background. It is not surprising that the occurrences of key words like ‘collaboration/s’ and ‘community/ies’ were low, although those of ‘teaching’ and ‘student/s in regard to the other cases were relatively high.

Research Results

The study, designed as a quantitative pre-post-evaluation using questionnaires, investigates which impact the CPD courses on modelling in 12 countries have on the participating mathematics teachers and their teaching. Two research questions examine to what extent significant changes in teachers’ classroom practice occurred, as perceived by the teachers themselves and by their students. The results, focusing on teachers’ perceptions, show statistically significant pre−/post-differences regarding all three scales of mathematical modelling used—investigative teaching, student-centeredness and authentic connections to students’ life. The results regarding students’ perceptions are quite different: the scales of investigative teaching and student-centeredness show an increase, but not statistically significant. Only the scale of authentic connections to students’ life shows a statistically significant difference, however, as a decrease.

The third research question investigates to what extent students’ and teachers’ perceptions of classroom practices are in agreement. Regarding investigative teaching and student-centeredness, data show a relevant correlation between teachers’ and students’ perception. However, teachers’ perception of the frequency of authentic connections to students’ life is significantly higher than students’ perception. By referring to other research suggesting that authenticity is a social construct that needs to be agreed on in different communities, the authors indicate the importance of sharing what is relevant to students’ (and teachers’) lives, in classrooms and surely also in teacher education and in collaboration among teachers (Diagram 8.4).

Diagram 8.4

RATE 4—Europe

3.5 Initiative 5 (Europe 2): Collaborative Design of a Reform-Oriented Mathematics Curriculum: Contradictions and Boundaries Across Teaching, Research, and Policy (Potari et al., 2019)

Relevant Actors

In sum, thirty-four members participated in a ‘design team’ for developing a reform-oriented mathematics curriculum. Among them were eleven ‘classroom’ teachers from kindergarten to secondary school, fifteen ‘academic researchers’ (two mathematicians and thirteen mathematics educators), and eight ‘policy makers’ (two ministry and six school advisors).

Relevant Targets

The goal is to develop “reform-oriented national mathematics curriculum [that] concerned compulsory education in Greece” (p. 418), in order to establish “the improvement of students’ learning as a common goal” (p. 432). Particularly, “the quality of students’ mathematical thinking and their future citizenship” (p. 430) is stressed. The key intervention is shifting from ‘traditional teaching’ and corresponding resources to ‘research-informed teaching’ and adequate resources. The ministry plays an important role, which “initiated a curriculum reform through the New School act [… focusing on] active engagement of students, openness of the education to society, […] and new roles for teachers as active agents of the curriculum” (p. 421). The commissioned ‘design team’ amplifies this orientation on active engagement by giving key members a voice when generating interview data about the design process.

Relevant Environments

One Ministry of Education commissioned a design team, based on a new policy document (“New School act”, p. 421).

Authors

All authors were mathematics educators and members of the ‘design team’ for the new curriculum. They all carried out interviews and did the analysis. The co-ordinator of the team served as the first author.

Type of Initiative (Duration)

A design team produced a final version of a curriculum (9 months). The article reflects and analyses the design process, grounding it within a specific theoretical framework and on empirical data (e.g. based on interviews with 11 “key design team members”, p. 423).

Specificity of Collaboration

The ministry appointed a co-ordinator, a researcher from a university of a national mathematics curriculum and further members of the design team “based on the coordinator’s recommendations” (p. 422). During the curriculum design process, “the coordinator acted as a broker between the educational policy activity and the designing activity” (p. 421). Thus, this collaboration does not directly take place in a teacher education context, but, in a curriculum context, it sets essential general conditions for future teacher education plans and activities. The social dimension of designing a curriculum and the use of an activity theory perspective is mirrored in a relatively high occurrence of words such as ‘team/s’, ‘community/ies’, ‘member/s’, ‘colleague/s’, ‘broker/s’ and ‘ministry/ies’. Also, words like ‘teaching’, ‘activity/ies’, ‘boundary/ies’ and ‘mathematics’ are used (relatively) often.

Research Results

The research questions centred on the interaction of the design team members, stemming from different communities. Particular attention was paid to emerging contradictions against the background of the three activity systems, that is, within and across the three communities of practice, comprising mathematics teachers, mathematics education researchers and policy-makers. The authors could identify four main contradictions and coded them as: (1) educational innovation versus teaching reality; (2) theoretical versus practice-oriented ideas; (3) research-informed teaching resources versus traditional teaching resources; (4) arithmetic versus algebra in primary school teaching. The first three relate to defining the curriculum objectives, while the fourth one came up as the design team was working on specifying the algebra content and the intended learning outcomes for different educational levels—thus, while taking specific content into account. Additionally, the authors specified for each contradiction the activity systems involved, the boundary objects and the boundary crossings that were found. Regarding the latter, the curriculum structure of algebra was identified as a boundary object. As for boundary crossing, establishing the improvement of students’ learning as a common goal was deemed essential. Also, the role of educational materials and resources for facilitating the boundary crossing was underlined (Diagram 8.5).

Diagram 8.5

RATE 5—Europe

3.6 Initiative 6 (North America): Supporting Secondary Rural Teachers’ Development of Noticing and Pedagogical Design Capacity Through Video Clubs (Wallin & Amador, 2019)

Relevant Actors

Three mathematics teachers “who comprised the entire mathematics department of one secondary school” (p. 515) and one researcher formed a “video club”.

Relevant Targets

The goal is a high level of teachers’ capacity regarding “noticing” of student thinking and “pedagogical design” (p. 515). This means that a key relevant difference (intervention) is teachers’ improved competence. Among others, the research focused on the question whether teachers’ participation in the video club influenced “their view of collaboration” (p. 515). The quotation, “Furthermore it is likely that without the video component of the collaboration process and the coparticipation among the teachers […] of these teachers […] would not have made the degree of growth they were able to accomplish due to their initial beliefs” (p. 534), shows that another relevant difference is seen between teacher collaboration and no teacher collaboration.

Relevant Environments

One school with its entire mathematics department: during the first phase “Introduction to school setting”, a “Meeting with administration” is also mentioned in a rural area and one university participated in the project—the teachers attended university PD courses, etc. led by the first author.

Authors

The first author was a mathematics educator who designed the “video club”, worked with the participating teachers and collected all data. The second author, also a mathematics educator, supported him in analysing the data and writing the article.

Type of Initiative (Duration)

The teachers attended five video club meetings over 1 year. The article reflects and analyses the design process, grounding it in a situated perspective and on empirical data, based on interviews (p. 521).

Specificity of Collaboration

A major part of the collaboration was reflecting on lessons, based on lesson plans and videos. The video club aimed at fostering a “culture of supportive constructive feedback and discourse” (p. 521). “The researcher intentionally selected the video clips for the video clubs himself, as opposed to having the teachers select clips” (p. 523), in order to focus specifically on students’ mathematical thinking: recognising “the value of co-participation, these conversations were informal and mostly directed by the participants, but moderated by the researcher” (p. 523). The school-external member of the video club had a variety of roles, at least comprising researcher, author, teacher educator, club designer, moderator and collaborator. Compared with other cases, the relatively high occurrence of key words like ‘participation/s’, ‘participant/s’, ‘colleague/s’ and ‘school/s’ indicates the collaborative nature of the video club. Often, the used key words, such as instructional ‘decision making’ and teachers’ ‘beliefs’ regarding curriculum, mathematics, tasks, etc. mirror the work on concrete instructional activities.

Research Results

The single-case study on three mathematics teachers aims at answering two research questions (p. 517). (a) How does participation in a video club structure for rural secondary teachers support the development of noticing? (b) How does what rural teachers notice from this experience influence their pedagogical design capacity? Regarding the first question, the study examines—by using a specific framework—the development of teachers’ level of noticing. The findings show that all three teachers, starting from different levels, reached in the fifth and last video club meeting a higher level than in their second meeting. In addition, all three teachers shifted their beliefs regarding curriculum usage and became more comfortable rethinking their current curricular materials. A vignette of one teacher’s growth indicates the process of slowly valuing both student thinking during her lessons and the instruction which promoted it. Thus, this links the two constructs of noticing and pedagogical design capacity.

The findings support other research claiming that teacher noticing is a skill which can be learned. Taking a situated perspective, the authors reflect how the context of the teachers’ interactions may have influenced their growth. They indicate the importance of the video component of the collaboration process and the co-participation among the teachers, being able to reflect on decision-making both of themselves and of their peers. Although none of the teachers taught the same mathematics topics as it is common in urban schools, the process of joint reflection and discussion led to teachers’ growth. Thus, the authors argue that the findings provide evidence of the viability of video clubs to support teachers in rural contexts. The video club experience began to erode professional isolation within the group, fostered their collaboration and led to an increase of the frequency of meetings (Diagram 8.6).

Diagram 8.6

RATE 6—North America

3.7 Initiative 7 (South America): How Teachers Learn to Maintain the Cognitive Demand of Tasks Through Lesson Study (Estrella et al., 2019)

Relevant Actors

Four primary school teachers (‘them’; teachers “with training in mathematics education and who had more than 5 years of experience” (p. 297)) and three researchers (‘we’; “with experience in Lesson Study and teacher training” (p. 297)) worked together in a “Lesson Study group”. There was also a research team, involving six researchers—three of whom had worked in the Lesson Study group, who analysed the implementations.

Relevant Targets

The overall goal was “the improvement of mathematics learning” (p. 293). The specific goal of this study was to investigate “how primary school teachers implement high-level cognitive demand tasks in a data analysis lesson in the context of Lesson Study” (p. 297). Implementing and maintaining a high level of cognitive demand is a special indicator for students’ high quality. The key intervention is directed towards producing a relevant difference in students’ learning, intending a shift from low to high level of cognitive demand. Thereby, Lesson Study is seen as a teaching method “for transforming teaching” (p. 295), by overcoming “the teacher-centered teaching model” which “remains dominant in most schools” (p. 297). The rather general conclusion of the study is that, through “collaborative work among working teachers and researchers in the context of Lesson Study”, it is “possible to design and implement tasks that maintain high cognitive demand in primary school” (p. 305). This indicates that ‘collaboration’ in the context of a Lesson Study is regarded as a key intervention leading to success.

Relevant Environments

Four “Chilean public schools”, where the four teachers work (p. 297) and one university.

Type of Initiative (Duration)

The Lesson Study group had eight two-hour Lesson Study sessions (weekly). During these sessions, “the group prepared the lesson plan and material and discussed the implementation of the lesson and how to improve it” (p. 297). The research team for the analysis of the implementation “met weekly for 2h for 6 months” (p. 298).

Authors

The first three authors were mathematics educators who worked in the Lesson Study group and, being also members of the research team, analysed the data. The fourth author had a master in mathematics education and supported the other authors.

Specificity of Collaboration

One of the researchers’ most important working fields during the Lesson Study group was providing support: “With the support of the researchers, the teachers designed an open-ended task with consideration of the presentation of the data and the context and elements of high cognitive demand tasks for the grade, such as representing and arguing” (p. 297). Another collaborative working field was the professional development of the teachers during the sessions—“The eighth session was dedicated to the teachers’ self-evaluation and reflection on the experience of the Lesson Study cycle, the statistical knowledge acquired and the impact on their professional development” (p. 298). The researchers had also other roles, at least including author, teacher educator, support provider and collaborator. Not surprisingly, this initiative, like initiative 1, has the highest occurrences of the key terms ‘Lesson Study’ and ‘group/s’. In this case, also ‘evaluation/s’, ‘intervention/s’ and ‘reflection/s’ are more-often used words, in contrast to other initiatives with exception of ‘reflection/s’ in initiative 1. This mirrors the teachers’ active and self-critical stance as fostered by the researchers, for example: “The […] session was dedicated to the teachers’ self-evaluation and reflection on the experience of the Lesson Study cycle” (p. 298).

Research Results

For the research presented in this article, the implementations of two out of four teachers were scrutinised. Two research questions were pursued to reveal how these two teachers implemented open-ended tasks for third graders within the context of the Lesson Study project. Particularly, the authors investigated how the teachers maintained the cognitive demand and how it declined, also with the help of the Lesson Study group in view. For the two primary teachers, the results reveal different scores on factors associated with the maintenance or decline of high cognitive demand during lessons. The authors particularly highlight the relevance of the Lesson Study group’s discussion and reviewing of the teachers’ lessons to recognise deficient aspects and to take responsibility for improvements. That is, the collaboration with the researchers helped the teachers to become explicit with respect to obtaining cognitive demand and thus challenging high quality student productions. Also, the reflections initiated in the discussion sessions of the Lesson Study group, helped teachers to understand and improve their deficient actions in the classroom and to build a repertoire of ideas for improving these interactions. The main source for teachers to develop towards the intended goal of the project was the benefit of the co-operation with respect both to the researchers and among the teachers (Diagram 8.7).

Diagram 8.7

RATE 7—South America

4 Comparing the Cases

Comparing seven cases can only be a first step in grasping phenomena. However, it is possible to generate observations (noticing relevant differences), possible hot issues and blind spots. Also, we might create some ideas for developing a tool that can be refined in a larger study, aiming at representation. Before we focus on commonalities and differences of the seven cases, we briefly discuss the seven cases against the background of a survey on research of mathematics teacher education from 1999 to 2003 (Adler et al., 2005). The following two of three claims of this survey are relevant for our comparison.

Claim 1, “Small-scale qualitative research predominates”, can be substantiated with respect to our sample. Since more than the half of our cases (4 out of 7) can be counted as small-scale (N < 20), we have nearly the same picture as in Adler et al. (2005) who found 98 out of 145 empirical articles based on small-scale research. In their study, 135 of 145 empirical articles involved studies with fewer than 100 teachers, in our case the figures are five out of seven. Based on Adler et al., Gellert et al. (2012) conducted a similar survey for the period 2005–2010. Focusing on research methods, methodology and techniques in studies on mathematics teacher education, they reported that Claim 1 is still legitimate (e.g. 89% of studies involved fewer than 100 teachers). A survey for the period 2005–2015 by Robutti et al. (2016; see also Jaworski et al., 2017) focusing on teacher collaboration showed similar results (90% of studies involved fewer than 100 teachers).

Claim 2, “Most teacher education research is conducted by teacher educators studying the teachers with whom they are working” (Adler et al., 2005; similar results can be found in Robutti et al., 2016), gets a very strong confirmation. All seven articles were written by people being involved in the activities of the projects—in six cases directly, in one case as a broker. Presumably, the extreme result is connected to the choice of journals with the scientific community as the major target group. Although, for example, research initiatives with teachers and teacher educators as collaborators (e.g. participatory action research; see Gellert et al., 2012) exist and provide interesting insights, it seems to need time for such initiatives to be presented in the above-mentioned journals.

Another, more general explanation is based on the fact that research on teachers working and learning in collaborative groups is a specific domain within the broad domain of research on teacher education in general, where a focus can be directed on teachers’ actions, beliefs or knowledge without being involved into an intervening activity (besides collecting data). Working with teachers and doing research on their professional growth combines the goals of contributing to the further development of teachers and of the scientific community, thereby also bringing research closer to teaching (see, for example, Cai et al., 2018). Of course, when dealing with research on ‘teacher collaboration’, a research project just might investigate it without intervening. However, how can researchers deeply grasp the phenomenon of ‘collaboration’, when they are not part of the collaboration? More generally, experts in system theory claim that, in order to understand a social system deeply (namely the ways, routines, patterns, hidden rules of collaboration and resistance, etc.), researchers need to intervene in it. This happens automatically when they work with teachers and, thus, when they try to improve something within the system.

Interventions into a social system such as a group, a course, a department, etc. can cause a lot of reactions, in particular towards those intervening, namely the educators, facilitators, designers, etc. The reactions can range from open enthusiasm and collaboration within the social system and with those intervening to open or hidden resistance and tensions, with stark impact on the collaboration, internally and externally. All articles we surveyed reported success, for instance, that teachers’ collaboration improved and that the teacher–researcher collaboration was regarded as a powerful means. No single case reported activities that failed, at least partially. Also, the seven cases do not provide much information on critical aspects of collaborating.

However, collaboration would be an ideal topic to reflect critically on interactions at different levels. For example: (a) the teachers might explicitly be asked to share critical aspects of the collaboration among teachers and with the educators from their point of view; (b) the teachers might explicitly be invited to comment on the results of the research on collaboration—one could even think about inviting one or more teachers to be a co-author of a article, eventually as an additional article to a pure research article; (c) the authors (researchers) could—in addition to presenting their research results on collaboration—also integrate critical reflections on their collaboration as a team and on their activities of fostering teachers‘collaboration. Thus, the above-mentioned claim 2 seems to be particularly relevant when studying teacher collaboration.

In the following, we sketch some observations, using the RATE scheme, in relation to the relevant actors of the collaboration, the relevant targets, the relevant environments and the specificity of the collaboration.

Relevant Actors

The seven initiatives show—apart from mathematics and science teachers in all seven cases—a variety of actors, including educators (six initiatives), researchers (six), heads/principals (five), mathematicians (four), brokers (three), facilitators (three), heads (three), administrators (two), policy makers (two), multipliers (one) and teacher leaders (one). In all initiatives, the key words ‘student/s’ appear, but not ‘parent/s’. As social entities, we find (video) ‘clubs’, different ‘communities’, (Lesson Study) ‘groups’, (project) ‘partners’ and (design and project) ‘teams’. Regarding the number of involved teachers, the seven initiatives include three small ones fewer than ten collaborators (including three or four teachers), two medium ones (20–40 collaborators, including 11 resp. 20 teachers) and two large ones (> 100 collaborators, including 326 teachers resp. not specified).

The relevant actor ‘broker’ needs to be more discussed. We see a ‘broker’ as a particular type of actor. An actor, who is in the role of a broker, could be also a researcher, a principal, etc. In some initiatives, e.g. initiative 2, the mathematics educator did not act as a mathematics educator, but rather as a broker. That is, a broker mediates between actors coming from different communities and supports events of boundary crossing. In this way, an actor in a collaboration can step out of his or her original role and take on that of a broker.

Relevant Targets

In all initiatives, although using different expressions and stating the goal explicitly or more implicitly, one target is the learning of teachers and the improvement of teaching. The improvement of teaching is connected to quite different meanings like ‘innovative teaching’, overcoming ‘teacher-centred teaching’, ‘research-informed teaching’ or supporting ‘inquiry-based learning’, hardly combined with a clear definition of the intended shift in teaching, marking a relevant difference between the status quo and the desired situation. In some cases, other adult learners like mathematicians and researchers are mentioned as co-learners.

Relevant Environments

In all seven analysed articles, universities as working places of researchers play a role. In five cases, also policy-related entities (three Ministries, an EU-program, a scientific association and provincial departments of education) are relevant. In three cases, all related to small initiatives, schools as working places of teachers and places of researchers’ intervention were involved. In the case of one of the seven initiatives, the article indicates that a requirement by a ministry established a distinct context for the initiative. Also regarding other initiatives, it is assumed that it would be interesting to read more about the context, in particular the goals and roles of the stakeholders having an impact on the initiative.

Authors

The number of authors ranges from one to four, with two authors in four cases as the most frequent case. All seventeen authors were involved in the initiative as educator and/or as researcher. Nearly all authors (fourteen) were mathematics educators, one was a mathematician and the two others were a project officer and a science educator. Thus, no teacher or another person of a relevant environment took the role of an author. All first authors had a pivotal role in the initiative, e.g. being a broker, co-ordinator, club designer, director or leader.

Specificity of Collaboration

The ways of collaboration are highly diverse, largely depending on the context of the initiative. For example, the three small and local initiatives (1, 6 and 7, each fewer than ten collaborators) refer to small communities: two Lesson Study groups and one video club where a few researchers collaborated with a few teachers; the two medium initiatives (2 and 5, each 20–40 collaborators) refer to a professional development course and a curriculum design team where mathematics educators, mathematicians and mathematics teachers collaborated; the two large initiatives (3 and 4, each more than 100 collaborators) refer to a national and an international program where researchers and teachers—both from mathematics and science—collaborated, in the context of institutionalised collaborations between universities.

It is not surprising that in the case of the two larger and more complex initiatives, the importance of the cultural context is stressed. For example, due to the collaboration of educators from different countries, it was necessary in initiative five to define common CPD principles. The three small initiatives have in common that they all deal with the impact of researchers’ initiative, finding out that collaboration and reflection are decisive factors for bringing about change. Most initiatives describe extensively their particular approach. In many cases, it would be interesting to read more about similar approaches and what the initiatives have in common or how they differ.

Research Results

As the ways of collaboration reported in the seven articles are diverse, unsurprisingly the research questions and the yielded findings paint a broad picture too. The majority of articles report on generic issues of collaboration, such as the importance of discussions and reflections, the critical role of knowledgeable others, institutional and cultural barriers and the role of brokers, and teachers’ classroom practices seen through the lens of noticing and the design capacity. Some articles integrate aspects specific to teaching and learning mathematics as they, for instance, report on contradictions regarding teaching arithmetic and algebra occurring for the different groups collaborating or on mechanisms of perspective-sharing between mathematicians and primary teachers. Thus, what is specific when researching collaboration in the field of mathematics is not that strongly emphasised in the contributions.

Additional Observations

All seven initiatives report about interventions with a focus on improving mathematics teaching and learning. However, the foci on improvements differ. Three cases (2, 3 and 7) relate to collaborative processes (learning from and with each other; emergence of boundary practices; factors maintaining high cognitive demand tasks in a group); two cases (1 and 6) relate to teachers’ changes in lesson plans and levels of noticing; one case (4) relates to teachers’ and students’ perceptions of implementing modelling or changes in teaching in different countries; one case (5) relates to writing and thus preparing the legislative act for a new curriculum in order to improve students’ learning.

However, the research investigates emerging contradictions occurring during the interaction of team members. The fact that only one case (4) uses data from students as indicators is not surprising, because the corresponding research focuses on collaboration among and with teachers. Like initiative 4, when focusing on students’ perceptions of classroom practices regarding modelling, initiative 7 also refers to the student level in elaborating on high cognitive demand tasks for students. However, in both cases, the focus is not on students’ achievements. In all these three cases, the inexplicitly stated impact chain seems to be as follows: collaborating with a small community leads to changes at teachers’ level, which leads to certain changes at the students’ level (cognitive demand, perceptions), finally leading to a higher quality of students’ learning.

5 Summary

Finally, based on the analysis in this chapter, we formulate some observations related to research on relevant actors, targets and environments of the collaboration.

• Observation 1: Small-scale qualitative research predominates (see Adler et al., 2005). Similar results are reported by Gellert et al. (2012) and Robutti et al. (2016).

• Observation 2: Most research is conducted by teacher educators studying the teachers with whom they are working (see Adler et al., 2005). Similar results can be found in the survey by Robutti et al. (2016).

• Observation 3: Most research focuses on improvements and success: (critical) reflections on teacher educators’ (co-)learning—although focusing on collaboration—are rare. Increasing literature on teacher educators’ growth (see, for example, Beswick & Chapman, 2020; Krainer et al., 2021) might stimulate more publications on teacher educators’ learning through collaboration.

• Observation 4: Most initiatives focus on the learning of teachers and the improvement of teaching. However, the intended shifts in teaching, that is, the marking of a relevant difference between the status quo and the desired goal of the intervention, is rarely defined in a clear way. Also, relating teacher learning explicitly to the collaboration within the projects is rarely in the focus (see findings by Robutti et al., 2016, in their international survey on teachers working and learning through collaboration).

• Observation 5: Only a few initiatives describe the context and relevant environments having a potential impact on the initiative.

• Observation 6: Most initiatives describe their particular approach extensively. In many cases, it would be interesting to get comparisons with similar approaches.

• Observation 7: Most initiatives stress the importance of sharing reflections as a crucial factor for working with teachers and, possibly, with other groups. A similar observation stems from a literature review of PME articles by Llinares and Krainer (2006).

This chapter does not allow the space to discuss about RATE as a tool for reflecting on research on teacher collaboration, for example, regarding its advantages, potentials, challenges and limitations. It is hoped that the description and the comparison of the seven cases and the formulated observations will serve as starting points for discussion and future research. In this chapter, RATE helped to grasp heterogeneity of the collaborations in the seven initiatives. RATE has also provided a framework through which it has been possible to describe the seven initiatives comparatively and to work out the differences using a uniform presentation.