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The making of theoretical models in mathematics education research
Jean-Franois Maheux, University of Victoria, MaheuxJF@uvic.ca
Nadine Bednarz, Universit du Qubec Montral, nadinebednarz@yahoo.ca
Abstract: During the last years, researchers in mathematics education used numerous frameworks coming from mathematics education itself as well as from other domains, to take into consideration the complexity of the field. These works illustrate new trends in mathematics education research. It is the case particularly for the work developed by researchers to design teaching situations in mathematics education that could support significant learning of mathematics by students. Using our work as a background, we will illustrate these new ways of developing such teaching situations, in a dialogue between theoretical models elaborated by researchers, practical way of thinking mathematics teaching mobilized by teachers, and re-construction of these situations by students. In close interactions between these different perspectives (researcher, teachers and students ones), we show how a teaching situation is emerging through the different contexts of its development, from a first reading of a theoretical model and a literature review, to its experimentation with grade 7 students.
Introduction
New trends in mathematics education research emerged during the last years, creating strong epistemological shift with previous orientations, not only in the stands they adopted towards teaching and learning, but moreover in the way research itself became conceptualized. These new trends, present in different fields of knowledge, aught for more democratic perspective in the research community, bringing forth the necessity to include at various stage of a research the participation of the principal stakeholders, such as practitioners, students or citizens, in phenomenon under investigation. Research thus is understood as conducted with instead of on these participants (Anadon, 2007; Callon, Lacousmes & Barthe, 2001; Desgagn, 2005; Desgagn, 2001; Jenkins et al., 2007), valuing collective form of thinking to turn to account their viewpoints over phenomena that concern them in their daily lives. This participative perspective on research acknowledges the essential role that these stakeholders need to play in the construction of knowledge which directly concern them (Darr, 1999), simultaneously transforming the researchers functioning , as he or she comes to investigate with them what appear as relevant issues for the participants (Desgagn, 1998). These participative researches have been in mathematics education at the origin of new ways to think the design of teaching-learning situations, so that these situations could be viable in the everyday school contexts as well as rich by the opportunities they provide for the students to learn (Bednarz et al., 2001). These researches allowed us not only to better understand how collaborative researches are conducted (Desgagn, Bednarz et al., 2001) and how the interactions between the participants (including the researchers) implicitly happen in the reflexive space created for joint investigation (Bednarz, Maheux & Barry, 2007), but also to document the contribution of practitioners and researchers to the co-construction of classroom activities (Bednarz et al., 2001).
The study presented here situates itself within this emerging trend in the conceptualization of mathematics situation for learning. This design process is based on a mediation between different ways of thinking and doing bring forth by the researchers, the practitioners, and the students. Associated with different culture of practices, the scientific community for the researchers, the professional community for the practitioners, and peers, and family, for example, for the students, their efforts and interests are sometimes difficult to connect. Inasmuch, our interest was to construct knowledge at the intersection of these three worlds, integrating these three perspectives in the process of designing teaching situations. A new knowledge, in terms of process and product, is emerging from this encounter of different worlds.
In this paper, we illustrate, from our two-year study, how the researchers, the practitioners, and the students' perspectives were bring together and had come to influence the design of mathematics situations for learning. Starting from a theoretical framework, based on Wengers model for communities of practice (Wenger, 1998), we show how this model was restructured through different interconnected contexts during our investigation, taking in account three contexts in the design of a classroom situation (Maheux & Bednarz, 2007): First, a researchers context that occurs while the situation was mainly developed by the researchers and for research (while the world of research in mathematics education was at the core, nevertheless, influences from classroom perspectives were already perceptible at this moment); second a teachers context, appearing when a very different version of the situation was designed in close collaboration with an in-service teacher to experiment it in one of her classrooms; and finally a classroom context giving great importance to the students as they now contributes directly in how the situation unfolds, prompting the teacher to reinvent it in the very course of action.
1. The starting point of the process: a research analysis
1.1 Reading Wenger : A first framework intertwining practice, learning and identity
All the process started with a student teacher willing to engage in graduate studies, deeply impressed by a few papers he had read on the situated nature of learning. One of these texts was a book published by Etienne Wenger (1998) seven years after his famous work on situated learning with the social anthropologist Jean Lave (Lave & Wenger, 1991). In this book, he reports how his work as a consultant in several various enterprises led him to develop his theoretical framework, by examining the way participation in a community of practice develop a persons identity. Intertwining learning, identity and participation, he develops a model organized around three modes of belonging and four dimensions of design to support them.
In this model, there is a strong relationship between belonging, participating in a community, and learning. Rooted in the work on situated cognition, participation in a practice is the mean by which learning is made possible, and even more: it constitutes the knowing and the learning itself. Belonging to a community is equally participating into it, being part of its practice. This means that identity as a legitimate member of a community depends on the persons participation within a group, and what is defined as legitimate participation frames what will be learnt and how. In this perspective, students in a mathematics classroom necessarily learn something, day after day: they learn to be members of that group, of that mathematics classroom. The mathematics they will learn depend on the way doing mathematics will be central, among other things, to that classroom practice, for example: sitting quietly, taking notes, remembering mathematical facts, or justifying answers to a problem, looking for alternative ways of thinking, and so on. Briefly outlined, for Wenger, engagement, imagination, and alignment are three modes of belonging in a community of practice by which participants can construct their identity in different ways:
Engagement is an active involvement in mutual relationships in the negotiation of meaning. By their interactions, the participants take part to the action and support each other, and they create the history of their own practice.
Imagination creates images of the world and allows us to see connections through time and space by extrapolating something from our own experiences: Imagination in this sense is looking at an apple seed and seeing a tree (Wenger, 1998, p. 176). Imagination creates differences in the sense of self, of what a person is doing, allowing people to understand the world, others experiences, and the enterprises they share more accurately.
Alignment is the coordination of energy and activities to contribute to broader enterprise. By coordinating enterprises on a large scale, alignment bridges time and space, so that participants can connect to one another, and with other communities.
Following Wengers ideas, supporting engagement, imagination, and alignment in a community of practice allow participants to meaningfully take part in a practice (that practice can be an emerging one), and therefore to construct their identities as members of that community, as well as to learn what that practice entails. Wengers framework takes a step further, and suggests that these three modes of belonging are supported by four dimensions that stress the dialectical nature of the practice of a community. In Wengers view, striving for a dynamic equilibrium among the dimensions and maintaining a to and fro between their two components should enhance the participants' sense of belonging.
Participation and Reification put in light that a practice is structured by reified artifacts, like schedules, plans, tools, and curriculums. On the other side, it recognizes that it is always the participation in activities that goes beyond these reifications and realizes a practice.
Designed and Emergent means that it is what is emerging in response to design that constitutes a practice. Learning cannot be designed, but it can be designed for.
Local and Global looks at how communities of practice exist as local entities with their own practices, while being connected through their participants to other communities. As part of that global network, communities of practices thus affect each other.
Identification and Negotiability deals with the observation that the community provides participants with ways of identifying themselves as members of this community, within its practice, while the participants negotiations of meaning also change this practice.
Wengers model for communities of practice immediately resonates to us, although it was not developed within the field of mathematics education. Considering students experience in mathematics classroom as related to their identity, thinking of mathematics as a particular way to view the world, and doing and learning mathematics in a classroom as a form of participation within a community of practice, opens new perspectives. How could we use this model to develop conditions to support students mathematics learning, providing them positive experiences as participants in their classroom community? How could we use the four dimensions of design to support students belonging, learning through their participation in the community? How could this model affect the way we view and prepare classroom activities for these students?
1.2 A second reconstruction by the researcher: Wengers work from an educational perspective
Wengers work has inspired many researchers in education, including mathematics educators such as Paul Cobb, Joe Boaler, or Stephen Lerman. Reviewing these researches to understand what that model could suggest to us, we have identified several themes in relation to each dimension and mode of belonging (for complete references, see Maheux, 2007).
Researchers in education view Identification and Negotiability as leading students to desire identify themselves as members of their classroom community, especially by given them a form of control over its organization and functioning (Ollila & Simpson, 2004), and by developing a sense of responsibility for the communitys activities (Putz & Arnold, 2001). It is also seen as encouraging the creation of new ways to participate in a group, while keeping in mind what characterizes the practice of that group (Cobb, 1999; Boaler, 2002; Clarke, 2002).
Participation and Reification is conceptualized as the very constitution of a practice through shared stories in relation to a setting (Pallas 2001; Ollila & Simpson, 2004) in which students come to take responsibility of their learning (Putz & Arnold, 2001; Ollila & Simpson, 2004). This duality also asks to disambiguate ???distinguish learning as a process and product, like some learning outcomes (Pallas 2001; Putz & Arnold, 2001), to see how a classroom practice can be both represented and structured in the artifact it produces and in the material it uses (Pallas 2001; Ollila & Simpson, 2004). In the process, this dimension refers also to the fact of providing opportunities to critically reflect on the groups productions (Putz & Arnold, 2001), resulting from the appropriation of cultural knowledge and the making of knowledge within the group. At the core, we find the idea of creating opportunities for the students to participate (Boaler, 2002) in a non-judgmental way, the matter being not to evaluate participation (Clarke, 2003) but to negotiate what is legitimate.
As a third dimension, Design and Emergence was interpreted by educational researchers in relation to the role of the teacher as a guide or a facilitator within the classroom community (Cobb, 1999; Putz & Arnold, 2001), an aspect that offers an interesting contrast with the contexts examined by Wenger (where old-timers and newcomers' interactions are rarely centered on one individual). Accepting the emergent character of practices and knowledge in school settings thus becomes a key aspect which we should try to take advantage of (Putz & Arnold, 2001; Ollila & Simpson, 2004), suggesting that a key component in teaching is to prepare oneself to draw on what happen in the classroom (Cobb, 1999; Cobb et Hodge, 2002), while providing students with clear and visible structures within which they can work (Putz & Arnold, 2001).
Local and Global, with its focus on the classroom and its relationships with other settings, naturally bring forth the making connections with everyday life. This can take place in the type of questions the students investigate in classroom (Cobb, 1999; de Abreu, Bishop & Presmeg, 2002), as well as in situation where students can interact with people outside their community (Ollila & Simpson, 2004): a central aspect to support students sense of belonging to the group (Putz & Arnold, 2001). To develop a specific practice (like doing mathematics) in contrast and in relation with other practices (in or outside school) also appears as a theme, when researchers discuss the movement between local and global aspects of a classroom practice (Cobb et Hodge, 2002; Ollila & Simpson, 2004). Altogether, these elements tend to value the cultural dimension of doing and learning mathematics as a social activity in which, as a society, we believe it is important for students to develop themselves.
In a similar way, educational researchers (from a wide variety of contexts) discussed Wengers three modes of belonging around the ideas listed in the table below:
ModeComponentEngagementTake part in the production of knowledge
Develop a sense of ownership in that production of knowledge
Acknowledge the contributions of others
Showing self-government by taking up ones role in the classroom
Appreciate collaboration as a mean to develop understanding
Explore, discuss, negotiate, validate understandings
Participate according to what is legitimate
Support a free circulation of information
Share questions, ideas, productions, etc.
Be able to use various resources
Support a climate of trust
Appreciate the purpose of the ongoing activities, sustain motivation
Be able to make sense of what is done in the classroom from the outsideImaginationConceptualize oneself as a learner
See oneself as an individual learner and as member of the classroom
Explore new ideas, new ways of seeing things or doing things
Make links between the components of the classroom practice (concepts, procedure, etc.)
Make links with other practices: inside or outside school
Reflect on the ongoing activities and redefine them
Take into account multiple significations, multiple understandingsAlignmentAdopt shared visions and ways of doing
Understand the reasons underlying choices in the functioning of the classroom or on the realization of its activities
Coordinate actions to be able to contribute to broader enterprises
Converge toward shared endeavors
Make sense in adopting standardized ways of doing things
Be guided, accompanied by the teacher
Be in touch with a broader contextTable SEQ Table \* ARABIC 1. Elements associated with Wengers tree modes of belonging by educational researchers
This review of educational literature in relation to the work of Wenger resulted, for us, in a series of observations. The first was linked to the fact that many researchers have found interest into it, have established connections with their previous work, and developed understandings about it that helped us envision its interest for mathematics education. However, while Wengers framework inspired many researchers, including researchers in mathematics education, a clear and integral conceptualization in relation with mathematics education in school was still missing. It seems as if this theoretical framework partially penetrates research in mathematics education, as researchers decided to draw on a concept or another to introduce or to support their own particular perspective. Nine years after its original publication, our review is the first attempt to investigate this model as a whole.
Another observation, that appears to us precisely because we where looking at the different parts of the framework and at how Wenger articulate them, concerns the epistemological stand adopted by the author for his framework. There are many overlaps in Wengers works, a high degree of interconnections between ideas, and a form of recursion between his modes and dimensions, all making it impossible to clearly know what comes first, or what to start with. Forms of belonging are already present in any social setting, but these settings can be designed to support belonging. On the other hand, design is already at play in any social setting (thinking of a classroom, for example), but belonging needs to be developed. In a similar way, modes of belonging or dimensions of design are not sealed tight or mutually exclusive. Shedding light on a landscape was the image that comes to us: Wengers developed his ideas to help us see and understand what is already there and overlaps, therefore, can only be beneficial. Because he believes that learning cannot be designed and that we need to understand the informal yet structured, experiential yet social, character of learning and [translate it] into design in the service of learning (Wenger, 1998 p. 225), Wenger does not attempt to create an articulated structure of learning variables to play with. Instead, he suggests to recursively adopt different perspectives, to take various entry points to look at, to get closer or to touch learning as a phenomenon.
Across what we presented previously, examining the design of classroom situations to revisit the theoretical model developed by Wenger, a first level of analysis emerged. We could use the component identified to each mode and dimension to guide the design of a classroom situation, asking ourselves in which way a given activity reflects these elements and looking for ways to create more room for them. However, the perspective we adopted, considering the practitioners and students in their everyday contexts, will soon confront this analysis, taking us to a different level of understanding.
2. Practical concerns:
Thinking Wengers framework for and in a mathematics classroom practice
To investigate how Wengers model could effectively contribute to mathematics learning and teaching, other voices needed, for us, to be listened to: voices that can only be heard in close interactions with the everyday practical concerns of teaching mathematics in context. The encounter with practice makes it necessary to focus, at least momentary, on an object that serves as a catalyst that crystallizes thinking and operates transactions with others, especially with teachers, to whom theoretical discussions often seems of little interest. For these reasons, and because we wanted to develop our understanding of Wengers framework for practice and in practice, we decided to try to use this framework while designing mathematics learning activities. This exploration would add an additional layer to our theoretical investigation in literature.
We therefore decided to document the design of a mathematics teaching-learning activity, starting from the conception of the activity at the university (by researchers, for research), and then in its adaptation when planning with a teacher up to its implementation in a classroom (by the teacher with her grade 7 students, for learning). We examine all the documents generated in that process (working notes, different versions of the activity and students productions), as well as recordings of the design session and the classroom experimentation, to see what aspects of the situation can connect with Wengers model. The situation we designed (some aspects of which were presented in Maheux & Bednarz, 2007) turned around having students determine the number of participants in a manifestation.
In a first stage, we, as researchers, develop the situation by identifying the mathematics concepts the students could encounter, the strategies they might adopt, the material that could be used, and we even phrase some key prompts to support the students investigations. Being therefore well prepared to talk about it with teachers; we presented the general idea of the situation (with other situations we have developed) in a focus group. Three middle school teachers discuss with us the potential of the situation for their teaching, and one of the teachers volunteered to experiment it. The situation was re-designed with the teacher to fit her needs and interest, ending up with the actual involvement of the students in a manifestationwhere they would be responsible to determine the number of participants. Breaking down the situation in short, yet meaningful, sessions, clearly identifying the objectives underlying each of its aspect and adapting day by day the preparation on the upcoming sessions next to what happen in class then took a central importance. The need to continuously adjust our preparation naturally comes from the transformation of the situation as it was experimented with the students. The interactions with and among the students highlighted different aspects of the situation. Examples are the need to help them see the mathematical thinking embedded in their practical strategies to deal with the real problem of estimating the size of a (moving) crowd, as well as guiding them think about how their mathematic knowledge could be useful to imagine other strategies (more powerful or, simply, more realistic).
To illustrate how thinking of a classroom situation transforms our interpretation of the framework, the following tables, read across the whole design for practice / implementation in practice outline what we have identified as possible features of a teaching-learning situation in association with each mode or dimension of Wengers model.
ModesFeature of a situationEngagementContextualize the problem in a way that will connect with the pupils interests, close to their lives or which concerns them
Offer a situation which goes out of habitual activities
Put the pupils in position to appropriate the situation, so that they can develop by themselves an understanding of its stake
Favor the free flow of information, strategies, etc.
Put the pupils in interactions by making them working in teams or as a whole group
Offer a situation in which the pupils will have as common purpose: the resolution of a problem using mathematics
Solicit volunteer commitment of the pupils in the situation
Make sure that the purpose appears clearly to the students
Accept various degrees of commitment in the undertaking of the situation
Promote students authorship in the resolution path and in the solutions they produceImaginationDraw on a context with which the students will be able to make links with aspects of their lives
Offer a situation in which resolution will have an effect outside of the classroom
Use real information so that the students make links with how they use it and with what is made elsewhere
Give a problem in which resolution will give students a feeling of competence in the world (e.g. resolution of a real problem)
Offer a situation which gives a "practical" sense to the mathematical tools the pupils develop
Offer an open-ended situation to give students opportunities to explore and to be inventive
Encourage students to create their own strategies using their previous mathematical knowledge, making links between concepts
Encourage students articulating a overall understanding of the situation, looking at the mathematical knowledge they used, at their own learning in the situation and the important moments or elements in its realization
Value critical thinking in the situation, discussing the objectives and their achievement
Introduce and discuss different paths to explore the situation, and be open to other possibilities
Raise a questioning on doing and learning mathematics (why do we do mathematics? Why do we use these mathematics particularly?)
Discuss the mathematical approach of a situation as one way to look at it
Encourage a transformation of the mathematical practice of the classroom, and highlight the transformations as they occurAlignment
Offer a problem addressed to the whole classroom as a group
Decide with the pupils the way such a situation will be approached, the general path for its resolution
Make sure the students have a general vision of the situation
Make necessary the adjustment of individual/teams productions (individual > team > whole class)
Support critical thinking over solutions and interpretations, especially by favoring the appearance of contradictory conclusions
Encourage the students to identify the mathematical ideas explored in an open-ended situation, and to make explicit links with the curriculum
Occasionally impose rules for the mathematical approach of a situation (using a given concept, trying out a given procedure, etc)
Validate with the students the concordance between their understanding of concepts and their strategies with standard definitions and procedures
Collectively decide the moment to stop investigating a situation, deciding what is considered as a resolution of the problem
Compare with the students their strategies and understandings to stress what they have in common, their respective advantages, etc.
Have the pupils summarizing their investigation
Encourage students discussion to adopt a limited number of strategies or solution to favorTable SEQ Table \* ARABIC 2. Possible feature of a situation in relation to Wenger tree modes of belonging
DimensionsFeature of a situationIdentification and NegotiabilityConsider different roles for the students and make sure that each one has a role to play.
Allow the students to choose their role, and the members of their team
Define roles in an opened way so that the students can make adjustments
Allow changes in the distribution of roles all along the process
Make every team responsible to produce a result and to have it fit with those of other teamsParticipation and ReificationPlace the students in a problem solving situation
Favor interactions among students, so that they exchange information, discuss their thinking, and share their results or strategies
Have the students record the mathematical ideas they encounter
Allow free access to reified knowledge (such as notebook, textbook, document specially prepared)
Offer a problem the students will have to reformulate; in which they will need to specify what is addressed and how.Local and GlobalSubmit to the students a real real-word problem
Encourage students to communicate their productions, results and strategy outside of the classroom,
Make links with professional practices through the context of a problem or the information to use.
Allow the students to use heuristic coming from their daily life outside of the classroom and help the students recognize their mathematical aspects
Discuss difference between these strategies and what is done in the classroom, draw on them to develop new onesDesign and EmergenceIntroduce the situation in such a way students that they will appreciate, they will see what is at stake
Well choose and prepare the data or information the students are going to work with
Use an open-ended problem with opportunities to adopt several approaches and obtain various solutions
Identify previously possible strategies, solutions, and mathematical concepts at play
Let the students address the situation by means of different mathematical ideas
Plan the general organization of the activity and discuss it with the studentsTable SEQ Table \* ARABIC 3. Possible feature of a situation in relation to Wenger four dimensions of design
This new way of looking at our framework, in contrast to our previous analysis, makes us realize that, during the entire process, we were drawing on resources grounded in three interconnected worlds in an ongoing dialogue between the theoretical and the practical understanding: our researchers world, a teachers and a students worlds.
3. Three worlds in an ongoing dialogue between the theoretical and the practical
In the whole process of developing the framework, the interplay between the theoretical ideas and the practical concerns span across contexts. Wengers framework played a central role in the first version of the design of the situation, by inspiring us in the choice of the problem to be used (an everyday life problem to which students can connect), in the envisioned classroom organization (investigating the problem as a whole classroom, instead of focusing on the individuals, open ended prompts to support students mathematical engagement), and in the choice of the resources that would be made available to the students (data to be examined, factual information, contacts with the organizer of the protest to link local classroom activity to an issue relevant in the global community). During the process, we developed the situation in a constant movement between the theoretical ideas and the practical requirement to design a situation for learning mathematics viable in context. The questions and implications from one realm recursively pulled us back to the other, enriching our understanding of both of them. Analyzing what guided us during the design and the implementation of the situation, we identified six types of influences in which the worlds of researchers, teachers, and students were at play: the intentions, the rationales, the way of doing things, the constraints, the resources, and the role played by the involved participants (for more details, see Maheux & Bednarz, 2007). Each one of these influences unfolded in questions and understandings that had practical and theoretical implications responding to one another. For example, we began the design process with a vision and some key principles grounded in our researchers world that nourished our understanding of Wengers ideas, and then guided us in the design of the situation. At the same time, realizing that this rationale was rooted in our understanding of what it important for mathematics teaching and learning alert us to be sensitive to what was (or might be) identified as such for teachers and students: a practical rational to organize students work in one case, and a relationship to mathematics and mathematics classroom for the others. On the other hand, spontaneously thinking of the resources students would be drawing on (their mathematical knowledge, their peers, their teacher) made us appreciate the resources we, as researchers, were using (including theoretical knowledge and, out of experience, a sensitivity to how a situation is prepared). Our work to develop Wengers ideas toward research in mathematics education and, therefore, mathematics teaching and learning, result from movement from a perspective to another. Moreover, our choice in method clearly supported that movement: the literature review, the preparation with a teacher and the classroom experimentation recursively ask us to take these three worlds in consideration. While the theoretical investigation provides us with a background, the need to connect with teachers and students experiences of the mathematics classroom was already present, guiding our reading through publications coming from the whole educational research community. Respectively, working with the teacher and the students raised questions about the theoretical model itself.
From these questions, Wengers model increasingly appeared to us as addressing issues of different, while interconnected, order: some of its aspects are focused on the individuals experience in the collective, while others are about the collective experience in a broader context. The difficulty to conceptualize the teachers role, when taking the classroom as a community of practice (in what sense is the teacher a member of that community?), and the ambiguity between the design of and the design for a community of practice (the guiding ideas that spread on its whole life term versus its organization to address a given situation) were others. On this point, we realized that classroom situations have indeed a connecting aspect, because they are at the same time designed for a given classroom community (taking in account its actual functioning) and they contribute in the design of that community (in relation to the expectations, the meaning, the kind of place created for and with the students). In an analogous way, the designing process reveals itself as a brokering activity, establishing connections between various, non-isolable, sets of expectations: curriculum requirements, mathematical potential, richness of meaning for the students according to their reality, institutional constraints in relation to design for research (limited time scale for the experimentation, unavoidable lack of familiarity with the students and their context, etc.). Designing community of practice cant be separate from profound considerations about the kind of practice we wish to stimulate: in our case, a certain mathematical practice in which students would be making meaning in and with mathematics.
When the situation was re-designed and carried out together with a teacher, the mutual enrichment between the framework and the practical concerns kept evolving. Dealing with classroom management, the school and board policies and schedules, as well as with the students needs and interests, helped us restructure the situation and revisit our understanding of Wengers framework, identifying constraints and resources, including the teachers practical knowledge and perspective on her teaching. Indeed, this re-construction of the situation with the teacher, contributes to see that thinking of a particular community of practice, even at the limited classroom level, concretely unfolds dealing with individuals that take part in a complex network of other communities. On the other hand, the theoretical framework deepens our appreciation of mathematical teaching and learning as carried out in an everyday classroom context. The sense of belonging to a group, like the students can experience it, effectively appears to be, at least partly, sustained by the mathematical classroom practice, in determining what is a legitimate participation and what is not, or how students come to share common understanding about what they are doing. Then, it shows us why moving toward a different perspective on teaching and learning mathematics can be confronted by the practice in place, as students and teachers might experience discomfort when a situation ask them to think differently. For example, while organizing their work to address the problem that was submitted to them, the teacher and the students often found themselves asking where are the mathematics? And although they sometimes miss opportunities to deepen their mathematical engagement in the situation, because they did not fully recognize them, they were able, by the end, to make sense of their experience of doing and learning within the situation. We also observed the first signs of a change within that community. The students and the teacher were demonstrating thoughtfulness about their everyday classroom practice, as critical thinking and mathematical argumentation emerged as valuable ways to contribute to mathematical activity (thought as one particular way to look at the world, yet not to be taken for granted).
The analysis pointed out, inherent to that type of design, concordances and tensions between the different perspectives (students, teacher and researcher ones): intentions, rationales, knowledge or constraints sometimes work together and sometimes dont. Altogether, the design and the implementation of the situation catalyzed the dialogue between the theoretical and the practical concerns: the design as a process translated in the situation as a product created awareness and understanding. Through the situation as a design-for a certain community of practice, came out the necessity to consider the community already in place, seeing how it influences the situation as enacted with and by the students. We observed struggle to develop new ways to see and do mathematics in the classroom, as presented in the teacher and the students' talks, as well as in their doings. Moreover, it is also this practical sensitivity to the community in place that makes it possible to understand how the situation effectively contributes (or not) to the design-of the specific community we were aiming at. Capturing how these changes can be supported necessarily unfolds from the dialogue between the theoretical and the practical, conceptualizing what we tried to generate and determining what can be done in a specific context.
Using this work as a background, we can now understand how our approach of research in mathematics education contributes to new ways to develop and make use of theoretical models for research, teaching and learning.
Conclusion: Re-reading and re-writing in and across contexts
What have we done during these two years of investigation? Reading and re-reading Wengers works, we have moved across different contexts, developing our framework from the researchers' side (looking at Wengers work as well as other scholars' writings about it) and then with a teacher and her students. Keeping track of the transformation of our understanding, this re-reading also turned to be a recursive re-writing, mapping out, summarizing, expanding, articulating ideas about his modes of belonging and dimensions of design. But not only have we re-read and re-write our emerging framework: it is also researching, teaching, and learning in mathematics education that we recurrently revisited. The reading of our research endeavor, realizing that starting with the idea of exploring Wengers ideas was also becoming an investigation of how we were doing it and why. The reading of what teaching might be about and how it comes to life in terms of supporting a certain practice of mathematical activity in the classroom, but also in connection with people situated outside (for example in dealing with schedule or permission to take the students off the school to join the march). And of course our reading of learning, of the students' activities : ways to relate to each other and to the doing of mathematics in their classroom. Finally, because we were not distant observers of these researching, teaching, and learning activities, we also find ourselves, writing and re-rewriting researching, teaching, and learning in our participation to them. Our readings continuously nurture our doings, and therefore not only span across contexts, but took place in these contexts as well.
Thinking about new trends in mathematics education, this should remind us what research essentially is. Framed by a new way of approaching it, by a dialogue in which not only researchers, but also teachers, and students have a voice, we revisit epistemological foundations and theoretical concepts, and the methodological concerns by developing our framework in a back and forth movement between the theoretical and the practical. The progression in the research was grounded in the everyday classroom experiences of practitioners and students, as well as researchers; a way of doing research in mathematics education that develops it as an endeavor of carrying out a collective theorization to understand and improve teaching and learning mathematics.
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Since over 30 years, the Club 2/3 is a organism that support world-wide solidarity among the youth, and as part of its activity, it organize a annual march in which thousands of youngsters pacifically demonstrate their concern for a more equitable repartition of (www.2tiers.org)
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O[frameworks coming from mathematics education itself as well as from other domains, to take a!!,N,H3!2',2N31!!2ON,2,N,,',22-,22',!,'H,,'"!2N22-!22N,2'2,2,2
AOainto consideration the complexity of the field. These works illustrate new trends in mathematics 22,22'2,!,222,,3N2,3/2!2,!,2=2,',H2!2'2'!,,2-H!,22'2N,2,N,,'2
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education ,22,,222
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'OXteaching situations in mathematics education that could support significant learning of ,,,231'2,22'2N,2,N,,',22,,222,,222'2222!'12!-,2,,!2312!2
ORmathematics by students. Using our work as a background, we will illustrate these N,2,N,,'30'23,2'H'2122!H3!2,',3,-21!2222H,H2'!,,2,',2
?new ways of 2,HH.0'2!2
O\developing such teaching situations, in a dialogue between theoretical models elaborated by 2,2,2221'3,2,-,222'2,22'2,2,212,2,H,,22,3!,,,N23,',,22!,,2402
OYresearchers, practical way of thinking mathematics teaching mobilized by teachers, and reb!,',-!,2-!'2!-,,,I-02!22231N,2,N,-',,,231N22-,230,,,3,!',22",
2
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O>construction of these situations by students. In close interac,22'!2,222!2,','3,22'30'22,2' 2,3',2,!,,82
tions between these different 22'2,I,,22,',2!!-!,22
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O[perspectives (researcher, teachers and students ones), we show how a teaching situation is 2,!'2,,2,'!!,(,,",2,!,,,2-!',22'22,2'22-'!H,'22H22H,,-,221'2,22'2
Obemerging through the different contexts of its development, from a first reading of a theoretical ,N,"1312!23122,2"!,!,2,22,3'2!'2,2,22N,2!!2N,!!'",-2212!,3,2!,-,C2
KO%model and a literature review, to itsN22,,22,,!,2",!-2,H2'F2
K' experimentation with grade 7 students.,32,!N,2,22H21",2,2'22,2'
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O\New trends in mathematics education research emerged during the last years, creating strong H,H!,22'2N,2,O,,',22,,22",',-!,2,N,"1,222!312,,'0-,"',!,,31'!2312
O^epistemological shift with previous orientations, not only in the stands they adopted towards ,2',N221,,'2!H22!,222'2!,2,22'3222022,',22'2.1,222,22H,!2'#2
Oteaching and lea,,,231,22-,2
Prning, but moreover in the way research itself became conceptualized. These new !23123N2!,22,!22,I.0!,(,,!,2',!3,-,N,,23-,22,-,2=2,',2,H@Times--e2
O<trends, present in different fields of knowledge, aught for !,22'2!,',222!",!-2!,2'2!222H,31,-212!2!-42
Wmore democratic perspectivenH2',2-H2,'2,2,''2,,-,-2
in the 22,2
OQresearch community, bringing forth the necessity to include at various stage of a !,',-!,2,2NN2202!2131!2!22,3,-,''022,22,,2,!22''-2,2!, 2
research the ",',-!,22,2
yO]participation of the principal stakeholders, such as practitioners, students or citizens, in 2,!,2,222!2,2!2,2,',2,222,!''2,2,'2!,,22,!''22,2'2!,-,2'22
O[phenomenon under investigation. Research thus is understood as conducted with instead of n22,22N,222222,!22,(1,22C,',-!,222''222,!'222,',2223,,3,H2,2',,22!b2
_O:on these participants (Anadon, 2007; Callon, Lacousmes &,22,2,',3,!,2,2'"H2,2222222C,22<,,22'N,'M:2
_ Barthe, 2001; Desgagn, 2005; C,!2,2232H,'2-12,22222
OdDesgagn, 2001; Jenkins et al., 2007), valuing collective form of thinking to turn to account their H,(1-12,2222(,222',,2222!2,231,2,,2,!2!N2!2223122!22,,,2222,!2
EOaviewpoints over phenomena that concern them in their daily lives. This participative perspective i2,H222'22,!22,22O,2,2,,23,,!22,N22,!2,02,'=2'2-",2,2,2,!'2,,2,52
Oon research acknowledges the22!,',-!,2-,222H,32-'2,e2
< essential role that these stakeholders need to play in the ,'',2,!2,2,3,',',2,222-!'2,,222-022,2
+OTconstruction of knowledge which directly concern them (Darr, 1999), simultaneously ,22'!2,222!222H,32,H2,22!-,0,22,-!22,N!H,!",2222!'N3,2,22'02
O]transforming the researchers functioning , as he or she comes to investigate with them what !,2'!2!N312,!,(,,",3,!!'!23,2221,'3,2"'2,,2N,'222,'2,-H22,NH2,(2
Oappear as relevant ,22,,!,'!,,2,22
9Missues for the participants (Desgagn, 1998). These participative researches e''2,'!2!2,2,!,3,2'!H,(2,13,2222!=3,',2-!-2,2,!,'-,!,3,'2
O\have been in mathematics education at the origin of new ways to think the design of teaching2,2,2-,22N,2,N,-',22,,22,2,2!122!2,HI./'22222,2,'122!,,,231
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OVlearning situations, so that these situations could be viable in the everyday school c,,!231'2,22''22,2,','2,22',2222,2,2,22,,2-#02.0(,222,#2
Hontexts as well 22,3','H,2
jObas rich by the opportunities they provide for the students to learn (Bednarz et al., 2001). These ,'!,2402,2222!22,'2-03!222,!3!2,'22,2'2,,!2"B,23,!-,,2222!=2,',2
O_researches allowed us not only to better understand how collaborative researches are conducted !,',-!,2,',2H,22'2322022,,!222-!',3222H,2,22!,2,!,'-,!,2,',!,,2222,,2R2
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