1 Introduction

Most university-based mathematics educators would claim that the aim of their research is to improve the quality of mathematics teaching and learning; yet education research is often criticized for its lack of impact on, and relevance to, classroom practice. This so-called “research-practice gap” has sometimes been explained by reference to the different processes used by researchers and teachers to improve educational practice, and the different forms of knowledge that result. For example, Wiliam (2003) compares the analytic rationality of formal research that seeks to develop generalizations about educational phenomena with the practical inquiry of teachers who need to address immediate day-to-day problems. Thus, the object of research, unlike in teaching, is not to solve problems but to create knowledge that helps us to understand a problem (Labaree 2003). This tension between the aims of formal research and the needs of teachers is also evident in the often unequal relationships between researchers and teachers who participate together in classroom based studies. Breen (2003) argues that true collaboration can only be realized if there is sharing of control and decision-making between the participants. However, he claims that this is an uncommon occurrence as teachers are usually co-opted into the research agendas of university academics because the latter have greater access to power and resources. The result is a “bidirectional disconnect” (Heid et al. 2006, p. 79) where the complementarity of researchers’ and teachers’ knowledge and the ways in which this knowledge develops remain unexplored.

Although the apparent lack of connection between research and practice is an issue that is widely discussed throughout the educational research community, it seems to have particular relevance to mathematics education. International standardized tests of mathematics achievement, such as the OECD’s Programme for International Student Assessment (PISA) and the IEA’s Trends in International Mathematics and Science Study (TIMSS), allow governments to compare national performance with that of other countries and can generate strong pressures to change mathematics teaching practice. Yet it is well documented that classroom practice in many countries remains resistant to the reform approaches promoted by mathematics education researchers (e.g., Gill and Boote 2012). Mathematics education therefore provides a rich and relevant context in which to study relationships between research and practice, and between researchers and teachers who work together to improve practice.

This article examines some of the issues raised above and discusses implications for mathematics teacher educator-researchers who work with teachers to develop their knowledge and expertise. The specific question it addresses is:

How can researchers and teachers work together to develop both theoretical and practical knowledge in mathematics education?

The article begins by outlining the theoretical perspective on learning that informs its purpose. It then introduces a framework for analyzing researcher–teacher relationships in the context of different models for teaching development. In some of these models teachers participate in the research as the objects of inquiry, while in others they engage in some form of inquiry alongside researchers. The framework is used to compare ways in which I, as a university-based researcher, worked with teachers in three different types of research and development projects. The analysis highlights characteristics of successful collaboration and leads to further questions about how university-based researchers and teachers might learn together when working on mathematics education projects.

2 Learning as social practice

Different theoretical approaches exist for studying teacher learning and development (e.g., see the articles in ZDM Volume 45, Issue 5, concerning Theoretical frameworks in research on and with mathematics teachers). This article takes a broadly sociocultural perspective that views learning as changing participation in socially situated practices (Lerman 2001). Here I am interested in using sociocultural theories to understand how mathematics teachers and university-based mathematics educators could work together to develop their professional knowledge and expertise. This investigation draws on Wenger’s (1998) concept of a community of practice. Although originating in the study of learning in apprenticeship contexts (Lave and Wenger 1991), the idea of a community of practice has also been used in research into pre-service teacher education and the professional learning of practising teachers (Llinares and Krainer 2006).

Wenger (1998) describes three dimensions that give coherence to communities of practice: mutual engagement of participants, negotiation of a joint enterprise, and development of a shared repertoire of resources for creating meaning. Mutuality of engagement need not require homogeneity, since productive relationships arise from diversity and these may involve tensions, disagreements and conflicts. Participants negotiate a joint enterprise, finding ways to do things together that coordinate their complementary expertise. This negotiation gives rise to regimes of mutual accountability that regulate participation, whereby members work out who is responsible for what and to whom, what is important and what can safely be ignored, and how to act and speak appropriately. The joint enterprise is linked to the larger social system in which the community is nested. For example, the joint enterprise of teaching mathematics in a particular secondary school is linked to the larger social system of secondary school education in a region or country. Communities of practice have a common cultural and historical heritage, and it is through the sharing and re-construction of this repertoire of resources that individuals come to define their relationships with each other in the context of the community. Based on this description, I would argue that mathematics teachers and mathematics education researchers are members of distinct, but related, communities of professional practice.

Although communities of practice have “insiders” and “outsiders”, there are various ways in which communities may be connected across the boundaries that define them. Wenger (1998) discusses several types of connection, including boundary encounters—discrete events that give people a sense of how meaning is negotiated within another practice. The briefest of these encounters is the one-on-one conversation between individuals from two communities to help advance the boundary relationship. For example, after participating in a one-off professional development workshop, a mathematics teacher might have a conversation with the presenter about some aspect of the workshop that could inform the teacher’s practice. A more enriching instance of the boundary encounter involves immersion in another practice through a site visit. This may occur when a mathematics educator visits a school over a period of weeks or months to collect classroom data, such as lesson observations or interviews with teachers and students, for a research project. However, both of these cases involve only one-way connections between different practices. A two-way connection can be established when “delegations” comprising several participants from each community are involved in an encounter, so that negotiation of meaning takes place within each practice and across the boundary that defines the separate communities. Wenger suggests that if “a boundary encounter—especially of the delegation variety—becomes established and provides an ongoing forum for mutual engagement, then a practice is likely to start emerging” (p. 114). Such boundary practices then become a longer term way of connecting communities in order to coordinate perspectives, or to generate new perspectives in order to resolve problems. Akkerman and Bakker (2011) identify four mechanisms for learning at the boundary between communities: identification, coordination, reflection, and transformation. However, they argue that only the transformation mechanism—involving confrontation with a problem and continuous joint work by partners on either side of the boundary—gives rise to new boundary practices.

According to Wenger (1998), other types of practice can also offer connections between communities that go beyond brief, one-way encounters. The creation of a periphery is one such practice that make the boundaries around communities more permeable. Peripheral experiences allow people who are not intending to become full community members to engage in the less demanding practices of this community, sometimes only via observation. Peripheral experiences nevertheless offer legitimate opportunities for learning by outsiders to a community and by the community with which they interact.

3 Analytical framework

The issue of how mathematics educators may work with practising teachers to develop their instructional expertise has been of interest for some time. For example, a Teachers as Researchers Working Group first met at a conference of the International Group for the Psychology of Mathematics Education (PME) in 1988 (see Zack Mousley and Breen 1997 for work from this group). This was followed by various PME Discussion Groups, a Research Forum and Working Sessions on Teachers researching with university academics from 2005 to 2009. The present article contributes further to this research theme by examining ways in which teachers and university-based mathematics educators might work together to develop theoretical and practical knowledge. It uses an analytical framework developed during the 2007 PME Working Session (Novotná and Goos 2007) from questions and issues identified by participants in discussing their own experiences in research and development work with teachers. The elements of the framework are summarized in Table 1 and discussed below.

Table 1 Framework for analyzing researcher–teacher relationships
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An important question to consider in terms of beginning the researcher–teacher partnership is how teachers enter into this process and who initiates the research. At times, a university-based researcher seeks out teachers to participate in a project that has already been planned. Occasionally a partnership might be initiated by a teacher who seeks out a university-based researcher. Alternatively, teachers could be encouraged or required by a school administration or government education department to enter into a university-based research project. In all these instances it is worth considering “why” as well as “how” such research partnerships are initiated. This element of the framework links to Wenger’s (1998) idea of boundary encounters between communities of practice: the teacher and the researcher are not necessarily seeking to become members of the same community, but instead to establish a relationship that spans the boundary between their separate professional groups.

Other questions relate to how the participants will interact and the purposes of the research. Both of these elements of the framework can be interpreted in terms of Wenger’s (1998) three dimensions of a community of practice, as concerns about the nature of mutual engagement between participants, how they would create a shared repertoire of language and other resources for negotiating meaning, and who is in charge of the (ostensibly joint) research enterprise. The extent to which roles are shared between teacher and researcher is an issue: there may be benefits and disadvantages in either maintaining strong role separation or sharing/swapping roles (teacher-as-researcher or researcher-as-teacher). However roles are determined, expectations need to be made clear from the start as a foundation for building trust and mutual respect. Consistent with Wenger’s ideas, Malone (2000) argued that since teachers and researchers create and act upon two different types of knowledge (practical inquiry in particular contexts versus theoretical inquiry aiming at generalization), they belong to two distinct communities with intersecting interests but asymmetric needs. That these two communities also use and value different forms of language presents a challenge for communicating the findings of research to non-researcher audiences, such as teachers and policy makers. The purposes of the research may depend upon how the partnership is initiated, since this often influences the choice of topic, negotiation of research questions, and realization of any benefits for theory, practice, or policy development.

In the next part of the article the framework sketched out in Table 1 is used to compare researcher–teacher relationships in three contrasting research and teacher development projects I have conducted over the last 20 years. The methodology for selecting and analyzing these cases is described below.

I began by making a list of all externally funded research projects I have carried out since beginning my doctoral study. I then eliminated those that did not involve interaction between the researcher and school teachers; for example, projects investigating learning and teaching in higher education where the teachers were academic colleagues. This left a list of 14 projects from 1994 to 2013, involving teachers in more than 140 schools. I developed an initial analytical matrix that partially categorized these projects along two dimensions of the framework presented in Table 1, “Beginning the partnership” and “Purposes of the research”; that is, I assigned projects to the matrix cells based on how the partnership was initiated and who chose the topic and research questions. The next step was to “fill in” the cells with more detail about the third dimension of the framework, “Participants”. To do this, I described participant roles as either separate, shared, or dual (if the researcher also had a role as a teacher educator or professional developer); developed a proxy indicator of relationships in terms of the frequency and duration of researcher–teacher interactions; and indicated who communicated the findings to research and professional communities in the form of publications and conference presentations.

The resulting matrix is presented in Fig. 1. It identifies four clusters of projects. For the purpose of this article I did not choose to analyze the cluster comprising evaluation projects commissioned by government agencies for which I had no part in defining the topic or research questions. I then selected one case from each of the remaining clusters, where a case might comprise one or more related but separately funded projects (these are underlined in Fig. 1). Hereafter I refer to each case as a “project” to highlight its coherence and boundedness, and label these projects A, B and C.

Fig. 1
figure 1

Initial analytical matrix (projects are numbered 1–14; R researcher; T teacher; R → R and R → T indicate researcher communicates findings to researcher and teacher audiences, respectively)

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Project A was a longitudinal study of the transition from pre-service to beginning teaching, and project B was commissioned by the government to support teachers in embedding numeracy across the middle years curriculum. Project C highlights the development and gradual transformation of a long term collaborative relationship between myself as a university-based researcher and a school teacher as we carried out classroom research together. These projects have been selected as cases for analysis because they exemplify different aspects of the three dimensions that structure the framework for analyzing researcher–teacher relationships shown in Table 1. The first and second projects are also fairly typical of research carried out in pre-service and in-service teacher education, respectively.

The final stage of the analysis involved identifying a minimal set of data sources that would support further elaboration of project features in relation to the three dimensions of the analytical framework. This retrospective analysis relies on different types of secondary data. One source—the research and professional publications and presentations arising from each project—was relevant to all projects. Ethics applications and consent forms were consulted for projects A and B. Other data sources were: research funding application, pre-service student surveys (project A); research contract and final report to the funding body, research team meeting notes, teacher workshop plans and presentations (project B); and for project C the doctoral theses of the researcher and teacher, and a structured audio-recorded conversation between us in which we reflected on our evolving relationship. Each of these data sources was searched for evidence of: how and why the researcher–teacher partnership was initiated; how expectations were communicated; what roles were taken on by the researcher and teachers; how, by whom, and to whom the findings were communicated; the type of relationships that were established; who chose the topic and research questions; and what benefits were experienced by the teachers and the researcher.

The matrix in Fig. 1 and the analytical framework on which it is based are similar to Wagner’s (1997) typology that proposes three forms of researcher–practitioner cooperation: data extraction agreements, clinical partnerships, and co-learning agreements. Data extraction agreements characterize the most traditional form of educational research where university-based researchers describe and analyze the work of teachers, and the roles of researcher and practitioner are separate (cf project A). In clinical partnerships, there is greater cooperation and negotiation of the research process: researchers and teachers work together to improve knowledge and practice, and communicate their findings to both research and professional audiences (cf project B). Wagner describes co-learning agreements as more symmetrical forms of cooperation that also involve reflexive inquiry, in which researchers and practitioners question the nature of educational research and the institutions in which it is conducted (cf project C). The alternative framework offered in this article makes two additional contributions to knowledge. First, it enables further analysis of forms of cooperation in the social design of educational research, as discussed by Wagner, by applying concepts from Wenger’s (1998) social practice theory. Wenger’s concept of a community of practice seems particularly relevant if researchers and teachers are regarded as members of separate, but related, communities. Second, it identifies the dual roles that researchers might take on in some projects if their research is carried out in pre-service or in-service teacher education settings, enabling consideration of the ethical implications of such roles.

Accounts of each case are presented in the next section, but not in chronological order; instead they are sequenced so as to lead towards progressively more collaborative researcher–teacher relationships. Each account is presented in two layers. The first layer provides contextual background by summarizing each project’s aims, analysis methods, and findings. This summary draws on the original research proposal or research contract and published accounts of the findings. The second layer offers a reflective analysis of the researcher–teacher relationship, using the framework provided in Table 1 and the data sources described previously.

4 Project A: research with pre-service and beginning teachers

The first project is typical of research conducted by mathematics teacher educators with their pre-service students. This was a longitudinal study conducted in two waves, from 2000 to 2004 with successive cohorts of my own teacher education students (Goos 2005; Goos and Bennison 2008), with a follow up study from 2006 to 2008 in which some graduates from the first study participated as beginning teachers (Goos 2014).

4.1 Project summary

The aims of this research program were:

  1. 1.

    to analyze processes through which a technology-enriched community of practice is established and sustained in a pre-service teacher education program;

  2. 2.

    to identify and analyze factors influencing pre-service and practising mathematics teachers’ use of technology, and compare ways in which these factors come together to shape teachers’ pedagogical identities.

The research design had two components. The first was a cohort study that identified features of the emerging community of pre-service and beginning teachers. The main source of data was the record of bulletin board discussions within and between two successive cohorts of pre-service teachers over a 2-year period that encompassed their university program and the first year of teaching. The frequency, distribution and content of bulletin board postings were analyzed using Wenger’s (1998) three defining characteristics of a community of practice (see Goos and Bennison 2008). This analysis showed how the pre-service and beginning teachers used the course bulletin board to expand the community of practice through generational encounters with newcomers, define their own professional goals and values, and construct a repertoire of participation structures for maintaining the community after graduation.

The second component of this research comprised longitudinal case studies that allowed snapshots of experience to be captured at developmental stages during practicum sessions and the early years of professional experience after graduation (see Goos 2005; Goos 2014). Data sources included surveys of pre-service and beginning teacher beliefs, lesson observations, and interviews that sought their perceptions of the role of technology in mathematics learning and factors influencing the integration of technology into mathematics teaching practice. Cases were analyzed using a socio-cultural theory of identity development to explain why individuals in a diverse range of professional contexts might embrace or resist technology-related change.

4.2 Analysis of researcher–teacher relationships

In a sense this researcher–teacher partnership was initiated as soon as the pre-service students enrolled in my course and therefore became prospective research participants. I invited all my students to participate in the cohort studies and to submit an expression of interest in participating in the case study component of the project, which would last longer than the course. From those who expressed interest in this extended form of participation I selected a small number for individual case study based on research criteria that involved sampling a range of different practicum school settings, including government and independent schools in capital city and regional locations, with differential access to technology resources. I anticipated that a similar diversity of school settings would be achieved once the students had graduated and found employment as teachers.

There was no sharing of teacher and researcher roles amongst participants, but role boundaries became blurred in another way in that I filled the dual roles of teacher educator and researcher. Adler et al. (2005) pointed out that this personal investment in teaching makes it difficult for teacher educators to take a critical stance towards the research we do with prospective and practising teachers, even though it may assist in building the trust that is needed to establish productive research relationships. Recruitment of students was carried out in accordance with the university’s ethical guidelines, and all participants signed a consent form indicating they understood that participation was voluntary and they were free to withdraw from the project at any time. While the research information sheet explained the benefits of participation in terms of improving professional development and pre-service teacher education programs that address technology usage in mathematics teaching, it is possible that students’ participation may have been motivated by their relationship with me as the teacher–educator. There is evidence that students regarded this relationship as being mutually respectful. For example, during the time frame in which the two studies were conducted, student evaluations of teaching formally mandated by the university indicated strong agreement with the statement “The lecturer treated students with respect” (average scores of 4.8–5.0 on a 1–5 Likert scale), and a similar level of agreement with other statements about the lecturer encouraging students to participate in class activities, communicating enthusiasm for the subject, and knowing the course well. However, students were entering into a different relationship with me as the researcher, and despite the protection provided by ethics approval processes it can be difficult in these circumstances to negotiate the power relationship that exists between the researcher and the researched. Thus, expectations regarding participant roles—beyond the broadly defined activities outlined in the research information sheet and consent form—were never explicitly discussed. With regard to communicating the findings from the research, although I often helped the pre-service teachers to publish their technology integration work in professional journals (e.g., Quinn and Berry 2006), I did not take advantage of opportunities for them to share their research experiences with a wider audience. Instead, I filtered their experiences through my own research perspective when I communicated findings from this project to the research and professional communities.

The funding application for this project notes that the purposes of the research arose from my experiences of teaching previous cohorts and my observations of the potential for technologically knowledgeable pre-service and beginning teachers to act as change agents in schools (Weinburgh et al. 1997). This was motivation for the project from the researcher’s perspective. Thus, the teacher–participants unknowingly influenced the topic and research questions without having any direct input into their formation. Several of these beginning teachers later approached me to volunteer for other research projects, which may indicate they gained some benefit from participation. For at least one of these teachers, participation provided incentive to integrate digital technologies into his lessons in innovative ways (Goos 2014):

This project is good because it gives me the impetus to do something like that [using data loggers to investigate Newton’s Law of Cooling] which … otherwise still might just be a happy thought. (p. 150)

This teacher’s comment suggests that he was becoming aware of what he might gain by participating in the project, and that he saw possibilities for attaining his own goals by working with me, the researcher. Although we had separate goals they proved to be mutually beneficial, since the project provided both impetus for the teacher to pursue technology integration and opportunity for me to investigate factors that influenced his use of technology.

In this case, the researcher–teacher relationship can be characterized as one where the researcher-as-teacher–educator is a more experienced member of the community of professional practice, while the pre-service and beginning teachers are newcomers (Goos and Bennison 2008). Thus, the relationship is concerned with facilitating entry to a professional community of mathematics teaching.

5 Project B: research-based professional development

The second case was a 1 year professional development project that I conducted with two researcher colleagues in 2009. The project was commissioned by a government department of education (see Goos et al. 2011a for an extended report of findings).

5.1 Project summary

From a professional development perspective, the purpose of this project was to support teachers in embedding numeracy across the middle years curriculum. The research aims of the project were to investigate changes in teachers’ instructional practices, personal conceptions of numeracy, and confidence in numeracy teaching.

In Australia, numeracy has a meaning similar to the Organization for Economic Cooperation and Development's (2004) PISA definition of mathematical literacy, as:

an individual’s capacity to identify and understand the role mathematics plays in the world, to make well-founded judgments, and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen. (p. 15)

Numeracy is also regarded as cross-curricular capability that should be developed through the study of all school subjects (Australian Curriculum, Assessment and Reporting Authority 2012). In this project, the researchers introduced teachers to a rich model of numeracy that gives attention to real-life contexts, application of mathematical knowledge, use of representational, physical, and digital tools, and positive dispositions towards mathematics. These elements are grounded in a critical orientation to the use of mathematics. Theoretical support for the numeracy model is provided by the literature on affective issues (McLeod 1992), the role of tools and artefacts in mediating and shaping mathematical thinking (Sfard and McClain 2002), applications of mathematics in real life and other curricular contexts (Noss et al. 2000; Steen 2001), and critical approaches to mathematics education (Jablonka 2003).

Twenty teachers in ten primary and secondary schools participated in the study. Over a full school year, teachers worked through two action research cycles of numeracy curriculum implementation. The research team provided three whole-day professional development workshops throughout the year to support teachers’ planning and evaluation, and visited teachers in their schools on two occasions between the workshops to offer further advice and feedback. Data collected to address the project’s research aims comprised surveys of numeracy teaching confidence, workshop tasks that elicited teachers’ developing understanding of the numeracy model, lesson observations, interviews with teachers and their students, curriculum planning documents, and student work samples. Comparison of Likert-style survey responses at the beginning and end of the project revealed increased confidence in teachers’ knowledge of numeracy and of numeracy planning, teaching, and assessment strategies. Case studies of individual teachers analyzed changes in instructional practices by identifying how well teaching plans and actions aligned with the elements of the numeracy model as the project progressed.

5.2 Analysis of researcher–teacher relationships

This researcher–teacher partnership was initiated by the government education department, which recruited schools to ensure broad representativeness in the sample based on geographical location and socio-economic status. As a result, the research team did not meet the participating teachers until the first professional development workshop that launched the project. As the study progressed, members of the research team inferred from teachers’ behaviours that most were highly engaged participants. Only a minority seemed not to be committed to the project; for example, they were reluctant to design and implement numeracy tasks that aligned with the model of numeracy we had presented.

Clear role distinctions were maintained by participants who were either teachers or researchers. However, the researchers filled dual roles as professional developers who were expected to bring about change in teaching practice. At the start of the first teacher workshop, education department officers who were supporting the project explicitly communicated their expectations that teachers should attend all project meetings and be adequately prepared for school visits. The researchers then explained the iterative action research approach that required teachers to plan, implement, and evaluate at least two units of work. Findings from this project were communicated to the research community via conference presentations and publications (e.g., Goos et al. 2011a; Goos et al. 2011b). The research team also invited four teachers each to serve as lead author of a set of four articles that were published together in a professional journal (e.g., Willis et al. 2012). The project thus gave these teachers a voice and provided them with opportunities to share their practical inquiry with colleagues in other schools.

The broad purposes of the research were determined by the education department and expectations of the researchers in terms of outcomes and deliverables were made explicit through a contractual agreement between the department and the university. However, the pedagogical focus of the project was at the discretion of the researchers and teachers. Thus, although the project’s broad research questions were defined by the researchers, our analysis of teacher interviews and their responses to workshop tasks showed that they identified their own personal goals in exploring different elements of the numeracy model, and these goals shaped their action research cycles of planning, implementation, and refinement of units of work (Goos et al. 2011a). When discussing overall impact more than half of the teachers claimed they now had greater understanding of the cross curricular nature of numeracy. One explained the impact in these words:

During the initial project meeting, where the model was described for what it was to be numerate, exemplar activities were provided that helped me with knowing about numeracy. Returning to school and trying out initial ideas was part of me doing in relation to numeracy. Eventually, though, the continued interaction of my developing knowing and doing led to my present state where my approach to teaching numeracy had become part of my being. I felt that my involvement in the project has changed who I am, both professionally and personally. (Willis et al. 2012, p. 15).

In this second project, the researcher–teacher relationship could be characterized as a series of one-way boundary encounters between two communities, in that the researchers immersed themselves in the practices of the professional teaching community via site visits to schools. However, some teachers also engaged peripherally in publishing practices more common to research communities when they co-authored articles with the researchers in mathematics teaching journals. Nevertheless, although there was some permeability in the boundaries around the two communities, no boundary practices emerged that could connect the researcher and teacher communities beyond the life of the project.

6 Project C: a collaborative research relationship

The third case is a long-term research partnership between myself and a secondary school mathematics teacher (VinceFootnote 1). The partnership began in 1994 with my PhD research, most of which was conducted in Vince’s classroom over a 2 year period.

6.1 Project summary

One of the aims of my doctoral study was to investigate the teacher’s role in creating a classroom culture that supported students’ mathematical thinking and reflected the epistemological values of the discipline (see Goos 2004). I used the concept of a classroom community of inquiry to help me understand how this teacher structured learning activities and social interactions to develop his students’ mathematical thinking. My investigation focused on the detailed practices through which so-called reform approaches were enacted in classrooms.

From analysis of my classroom observation field notes and video-recordings as well as interviews with the teacher and students, I developed a set of five statements that reflected the teacher’s assumptions about mathematics teaching and learning:

  1. 1.

    Mathematical thinking is an act of sense-making, and rests on the processes of specializing, generalizing, conjecturing and convincing;

  2. 2.

    The processes of mathematical inquiry are accompanied by habits of individual reflection and self-monitoring;

  3. 3.

    Mathematical thinking develops through teacher scaffolding of the processes of inquiry;

  4. 4.

    Mathematical thinking can be generated and tested by students through participation in equal-status peer partnerships;

  5. 5.

    Interweaving of familiar and formal knowledge helps students to adopt the conventions of mathematical communication.

Together, the assumptions and teacher actions underpinning them represented a synthesis of evidence from the study as a whole to show how the teacher created a culture of mathematical inquiry.

6.2 Analysis of researcher–teacher relationship

After I completed my doctoral research, Vince enrolled in his own PhD under my supervision. During his candidature he also took an active and high profile role in professional associations, culminating with his election as President of the Australian Association of Mathematics Teachers. In 2005 he embarked on a career change, leaving his job as a school teacher to take up a tenurable position as a university academic. We continue to collaborate on other research projects (e.g., Project B in this article). However, the analysis that follows focuses on the early years of our partnership and draws on an extended conversation that we recorded in preparation for writing a journal article about this researcher–teacher relationship (Goos and Geiger 2006).

Initiation of the partnership came about when Vince and I were introduced to each other by our former pre-service teacher education lecturer, who had become my PhD supervisor. At the time, Vince had recently completed a Masters degree and was motivated to participate in my research by his desire to resume regular professional conversations with someone like his former university supervisor. Thus, there was some equity in the partnership from the start in terms of its initiation and the underlying motivations of the participants.

As participants, although we agreed to keep our roles separate—myself as non-interventionist researcher and Vince as the teacher—the nature and distinctiveness of these roles changed over time as we developed mutual trust. I was a novice researcher as well as a novice teacher, and thus I was conscious of the kind of respectful relationship that needed to be established with this experienced teacher if the research was to be productive. This was especially important in light of Vince’s views about working with university researchers:

I had often been very critical of the way researchers would come into schools and harvest everything, leaving nothing for the teacher except “Thank you” – and we would never see them again. You took a very different approach from that and I really appreciated it. (Goos and Geiger 2006, p. 38)

Vince later explained how he valued my presence as someone who “only ever asked me why I was doing things, you never made any judgmental comments” (Goos and Geiger 2006, p. 37). However, my efforts to understand did eventually lead him towards specific actions so that over time I became more of a participant than a passive observer. For example, our post-lesson discussions about classroom events and my conversations with students often led him to modify his teaching plans for the next lesson. He explained:

The interesting thing for me as a teacher was to think about what made it happen in that way, can we replicate this? … Could we manipulate what was happening to bring about particular types of learning and interaction between students? (Goos and Geiger 2006, p. 38)

As I noted in my PhD thesis, my participation changed in another way when Vince invited me to teach his class for 2 weeks while he attended a conference in the USA. This is a form of role sharing that further demonstrates the mutual trust that had been fostered during the research study.

Vince and I explicitly negotiated issues related to power and what each of us wanted to achieve out of the collaboration as we began to write and present papers together at research conferences. He believed that “teachers’ voices … have to be heard if research is going to make a difference to teaching and learning in schools” (Goos and Geiger 2006, p. 38), and he saw jointly authored publications as acknowledging his equal contribution to creation of the new knowledge reported therein. Likewise, I gained credibility with practising teachers through joint presentations at professional development conferences where Vince was well known because of his leadership and advocacy roles in teacher professional associations. This was how we introduced each other into the distinct sub-cultures of mathematics education to which we separately belonged—the community of educational researchers and the community of teachers—and how we learned to communicate with different audiences using the language of research and the language of practice. Thus, our goals and needs, although different, were mutually recognized and valued.

Although the purposes of the research were determined by my interests in that I proposed the initial topics and research questions, this situation evolved into a more equal arrangement when Vince enrolled in his own PhD under my supervision. In the first chapter of his thesis he identified himself as the teacher at the centre of my PhD study and explained how his own research goals—concerning the role of technology in mediating classroom interactions—reflected a desire to extend the findings of my previous research.

This third example shows how a two-way connection developed between researcher and teacher. As well as immersing ourselves in the practices of each other’s communities, for example, by giving joint presentations at both research and professional conferences, we also created boundary practices that explored the different perspectives of each community. Our conversation about what we learned through working at the boundary is one example of such a practice. Wenger (1998) refers to “delegations” of a number of participants from each community who can establish two-way boundary practices by negotiating meaning within and across the different practices. Although our connection was as individuals rather than delegations, Vince and I took advantage of the leadership roles we held in national mathematics teacher and mathematics education research organizations to share and discuss our boundary practices with members of our respective communities. We were aware of the different types of knowledge that researchers and teachers valued, and also of the complementary knowledge and skills that each of us brought to the partnership. These differences, rather than being a source of tension, strengthened our mutual engagement.

7 Discussion and conclusion: implications for mathematics educator development

The purpose of this article was to consider how researchers and teachers could work together to develop both theoretical and practical knowledge in mathematics education. Taking a broadly sociocultural approach to learning as participation in social practice, the concept of learning in communities of practice was offered as a way of thinking about how researchers and teachers create the types of professional knowledge and expertise most valued by their respective communities. But what happens when members of these two communities come together with the aim of developing knowledge for and about mathematics teaching? What connections between communities are most helpful for building equal relationships and coordinating expertise? Different types of connections between communities were discussed in this article, including boundary encounters that might lead to longer lasting, two-way boundary practices, and peripheral participation by members of one community in some of the practices of another community. In all of these connections, it is important to realize that “dialogical engagement at the boundary does not mean a fusion of the intersecting social worlds or a dissolving of the boundary” (Akkerman and Bakker 2011, p. 152). Boundaries remain markers of difference rather than homogeneity: although researchers and teachers may engage in boundary crossing they remain members of their respective communities. This was the case in all three projects discussed in this article.

Table 2 uses the analytical framework introduced in this article to summarize features of the researcher–teacher relationships in three teaching development projects. My analysis indicated that only in project C was a two-way connection established between the teacher and researcher communities through the development of long-term boundary practices. This project was characterized by mutuality of the researcher’s and teacher’s motivations, roles, and purposes, and complementarity of their expertise and knowledge.

Table 2 Comparison of researcher–teacher relationships in three projects
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This analysis suggests a way of mapping out the research and development space occupied by university-based mathematics educators who work with pre-service or practising teachers to develop their knowledge and expertise. As Wagner (1997) has previously suggested, such a framework could also be used as a template for designing research projects and for negotiating researcher–teacher relationships and the expectations these entail. The framework might also be useful in postgraduate courses for drawing attention to the social design of educational research.

Consideration of the differences between the three projects analyzed in this article could perhaps improve understanding of how to develop more equitable researcher–teacher relationships. While the relationship in project C seems ideal, and possibly idealized, it does offer a glimpse of the possibilities. For example, it is possible that research partnerships with my pre-service students and graduates will continue to evolve over time towards mutually agreed goals, or at least goals that are mutually beneficial in the ways indicated in my analysis of project A. Working towards shared goals and decision-making is more complex in shorter-term projects initiated and funded by education systems that define the research or professional development focus and select the schools that will participate. However, in such circumstances it is possible for the researcher to develop longer lasting relationships with the education system, which may lead to a series of projects with progressively broader scale and focus. This has been the case with project B, where in subsequent projects I have worked with school principals and curriculum leaders to help them develop whole school approaches to numeracy planning and teaching, and with education authorities to develop system-wide curriculum frameworks based on the numeracy model initially tested in project B. This model of teaching development can lead to a beneficial convergence of goals between researcher and policy-makers, but it does not reduce the challenge of achieving mutual understanding between the researcher and teachers who participate in such projects.

The analysis also raises questions about what university-based researchers learn through this work, especially when they are also teacher educators or professional developers in relation to the teachers participating in the research project. Such questions might include those listed below.

7.1 Implications for researchers as teacher educators

  1. 1.

    How can pre-service teacher educators negotiate ethical issues (unequal power) in researching with their own students?

  2. 2.

    How can pre-service teacher educators develop a critical stance (distance and scepticism) towards the research they conduct with their students?

In reflecting on the emerging field of mathematics teacher education Krainer noted that teacher educators have the dual roles of “intervening and investigating … of improving and understanding” (Adler et al. 2005, p. 371). In the same article Adler suggested that in order to fulfil the dual roles of researchers and teacher educators we need to develop effective theoretical languages to distance ourselves from what we are looking at. Perhaps scepticism and distance might also be achieved by subjecting our own practice to the scrutiny of teacher education colleagues in other institutions and countries via collaborative projects.

7.2 Implications for researchers as professional developers

  1. 3.

    Who has the right to “transform” teachers and teaching practice?

  2. 4.

    How can researchers working with teachers balance transformation with critique in ethical and intellectually honest ways?

Each of these questions reflects the challenges of conducting professional development projects where there is an expectation that teaching practice will be transformed for the better. Labaree (2003) pointed out that researchers and teachers speak different languages and work within different paradigms (analytical vs practical), and these cultural differences need to be negotiated with care in order for researchers and teachers to build mutual understanding of, and respect for, each other’s knowledge.

7.3 Implications for linking research and practice

  1. 5.

    In communicating findings from research with teachers, who should speak for whom and to whom?

  2. 6.

    What conditions are needed for researchers and teachers to explore each other’s roles and understand how their respective communities develop generalized versus particularized knowledge of teaching and learning?

The final two questions invite readers to consider how researchers and teachers—representing two different communities with intersecting interests and asymmetric needs—might both develop professional knowledge and “principled practice” (Heid et al. 2006, p. 78). The framework for analyzing researcher–teacher relationships presented in this paper may provide a starting point for this important endeavour.