text:
Anita Valenta
email: anita.valenta@matematikksenteret.no

Kjersti Waege
email: kjersti.wage@matematikksenteret.no

Rehearsals in work with in-service mathematics teachers

Keywords: Rehearsals, in-service teachers, ambitious teaching

An approach that has shown to give pre-service teachers rich opportunities for learning to teach mathematics is through a cycle of enactment and investigation. An important part of the cycle is rehearsal where novices rehearse their plans for enacting particular instructional activity in front of their peer pre-service teachers. The peers and the course instructor take part in the rehearsal as students, and every participant can stop the activity for discussion on different aspects in teaching. We build on the approach developed for pre-service teachers, and work on the adoption and development of the approach for work with in-service teachers in Norway. This paper reports from a pilot that was implemented with a group of in-service teachers. Our research question concerns interactions between in-service teachers and course instructors during the rehearsals and in-service teachers´ opportunities for learning in rehearsals.

Introduction

The aim of mathematics instruction is development of broad mathematical proficiency characterized by conceptual understanding, procedural fluency, adaptive reasoning, strategic competance and productive disposition to mathematics (Kilpatrick, Swafford, & Findell, 2001). This ambitious goal leads to a more demanding, and thus ambitious, conception of mathematics teaching. In this paper we aim to add to the existing knowledge base about how teacher education can support in-service mathematics teachers to learn the work of ambitious mathematics teaching.

In the Mastering Ambitious Mathematics Teaching (MAM) project we develop a course for in-service mathematics teachers in Norway. In designing our intervention, we take our lead from the Learning in, from, and for Teaching Practice (LTP) teacher education project (see [START_REF] Kazemi | New directions for the design and study of professional development: Attending to the coevolution of teachers' participation across contexts[END_REF][START_REF] Kazemi | Developing pedagogies in teacher education to support novice teacher's ability to enact ambitious instruction[END_REF][START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics[END_REF][START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF][START_REF] Kazemi | Learning to teach within practice-based methods course[END_REF]. Central in the LTP-practice based approach is work with specifically designed instructional activities (IAs) in a cycle of enactment and investigation. A key innovative feature of the design is the use of public rehearsals. In a rehearsal, the pre-service teacher is responsible for teaching an IA to a group of peer pre-service teachers acting as students, with the course instructor offering guidance. This paper reports on our work on rehearsals with in-service teachers in a pilot study. We ask: What characterizes the interactions between in-service teachers and course instructors during the rehearsals in the study, and in what ways might rehearsals support in-service teacher´s learning of ambitious teaching?

Rehearsals within a cycle of enactment

Ambitious teaching entails mathematical meaning making, identity building and creating equitable learning experiences for children. It requires teachers to engage deeply with children's thinking -by eliciting, observing, interpreting and responding to student reasoning, language and arguments. Attending to students' experiences and designing instruction to enable each child to do rigorous academic work in school is also a central principle of the approach [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF].

In their work on ambitious mathematics teaching, [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics[END_REF] build on the study of [START_REF] Leinhardt | Seeing the complexity of standing to the side: Instructional dialogues[END_REF] who identified some routines skilled teachers used in leading instructional dialogues and argued that expressing the routines explicit make them teachable for course instructors. [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics[END_REF] use the notion "routines" to denote well-developed practices which have shown useful in teaching, which respect the complexity in mathematics, mathematics teaching and learning. They argue that focus on learning to use these routines/practices can provide novices an opportunity to hold something constant in a process of further learning to teach. The teaching practices that are central in ambitious teaching include aiming toward a mathematical goal, eliciting and responding to students' mathematical ideas, orienting students to each other's ideas, setting and maintaining expectations for student performance, positioning students competently, assessing students' understanding, and using mathematical representations [START_REF] Kazemi | Developing pedagogies in teacher education to support novice teacher's ability to enact ambitious instruction[END_REF]; see also [START_REF] Hunter | Designing opportunities for perspective teachers to facilitate mathematics discussions in classrooms[END_REF]. Teachers who are novices in teaching mathematics ambitiously need to learn to enact the practices in their teaching. They also need to develop the mathematical knowledge needed to teach ambitiously at a particular grade.

Grossmann, Hammerness and McDonald (2009) argue for incorporation of "pedagogies of enactment" and use of "approximations of practice" in teacher education in order to help pre-service teachers develop knowledge, skills and professional identities as teachers. As a type of approximation to practice, Grossmann, Compton at al. ( 2009) suggest use of rehearsals where novices rehearse a particular instructional activity in front of a group of peers. [START_REF] Kazemi | Developing pedagogies in teacher education to support novice teacher's ability to enact ambitious instruction[END_REF] develop instructional activities (IAs) that are designed to be "containers" for the practices, principles and mathematical knowledge that novice teachers need to learn and be able to use in interaction with students (see [START_REF] Kazemi | Learning to teach within practice-based methods course[END_REF] for descriptions of the IAs). The structure of the IAs offers the novices a scaffold in eliciting and responding to student thinking and understanding. The novice teachers learn to teach IAsthrough repeated investigation, discussion, rehearsal, enactment and observation. Each cycle of enactment and investigation consists of six stages [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF], as illustrated in Figure 1: In stage four of the cycle, selected novice teachers publicly rehearse their plans for enacting an IA in front of their peers and with feedback from the course instructor. During the rehearsal, the course instructor or a peer may stop action to ask questions or suggest possible alternative courses of action. The course instructor may also act as a student, by asking and answering questions or by making errors that students are likely to make [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF]. Rehearsals within repeated cycles of enactments and investigation can be considered as an approximation of ambitious teaching. [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF] argue that a rehearsal is an important setting for building novices motivation and commitment to teach ambitiously (p. 239-240). They analyzed 90 rehearsals of IAs by pre-service teachers. The study revealed that rehearsals not only allow pre-service teachers to work on routine aspects of ambitious teaching, but also to attend to more complex aspects of it. The study also showed that rehearsals give the pre-service teachers an opportunity to learn the principles of ambitious teaching while the course instructor guide their progress.

Design of course

The course consisted of seven sessions (each four hours in length) during a period of four months. The sessions were held in a fifth grade classroom of an elementary school 1 .

Session 1: The in-service teachers (ISTs) were introduced to the principles and practices central to ambitious teaching and the instructional activities they would work on during the course.

Session 2-6: In these sessions ISTs were divided into three teams of 4-5, and the teams worked together in planning, rehearsing, enacting, and debriefing course tasks: 1) Teams of ISTs came to class prepared to teach an IA; 2) Teams of ISTs rehearsed the IA under supervision of a course instructor (CI); 3) ISTs observed one of the CIs teach the subsequent session's focal IA to the whole group of fifth graders. This was part of the preparation for the following session; 4) One IST from each team taught a small group of fifth graders the IA that they had come to class prepared to teach. A CI also observed the enactments; 5) After a break, ISTs met in their teams to do a collective analysis of the day´s enactment with their CI; 6) Each team debriefed what they had learned; 7) The CIs prepared the class for the following session's focal IA and, as part of that, shared some reflective comments on the whole group lesson that was taught.

Session 7: The last session was devoted to concluding discussions and try outs.

Method Participants and data

There were 14 in-service mathematics teachers from three different elementary schools participating in the pilot study. The three schools are partner schools of the Norwegian Centre for Mathematics Education. Some of the ISTs in the study had only a few years of experience as mathematics teachers, while others were experienced teachers. A group of six course instructors (including both authors) from the Centre participated in the study. The course instructors had little experience in leading rehearsals.

Rehearsals were carried out in three teams at five of the sessions. All rehearsals were videotaped, but two of the recordings were damaged. Our data is therefore consisting of 13 recordings. Each recording is about 25 minutes.

Coding and data analysis

A rehearsal consists of parts where an IST is teaching the activity, and parts where IST(s) and CI interact. We denote the interactions between IST(s) and CI during the rehearsal as CI/IST exchanges.

To understand what characterizes the CI/IST exchanges during rehearsals, we take a CI/IST exchange as the unit of analysis and we analyze: 1) the substance of exchanges between CIs and ISTs, and 2) the structure of exchanges between teacher CIs and ISTs. In our analysis we used a priori codes adopted from [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF]. Table 1 shows a list of the substance codes and Table 2 shows a list of the structure codes that we built our analysis on. We used Studiocode video-analysis software which allowed for detailed coding of the rehearsals. For each rehearsal, we created a timeline for each video-recorded to capture the substance and structure of exchanges. Coding the video directly allowed for both verbal and visual cues to be considered, such as written representation, gesturing, and movements.

Results

Substance of CI/IST exchanges

In Table 1 we present an overview of frequency of the various substance codes in all CI/IST exchanges in the data2 . The most frequent codes in our data are representation, student thinking, content goals and elicit and respond. These codes were also among the most frequent in the rehearsals by pre-service teachers analyzed by [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF]. Can we stop for a moment? Hmm, this is not so easy.

Substance codes

Description

The student presents her thinking rather imprecisely, and now we need to illustrate it on the image and also write it symbolically. We lose the part about seeing eight as four plus four in the way you represent her thinking. Can we try to represent her idea more accurately?

IST: I can circle four and four… IST3:

But just in the first group. She said that she saw it in the first group and then just multiplied by three. And it is not clear whether she thinks eight times three or three times eight, she says both.3 IST: Yes, right. I tried to make "eight times three or three times eight" clearer by leading to one times eight, and so on. Because it is three times eight.

CI:

Maybe you can ask the student how she would represent it? Or, if you find her explanation too vague, you can ask her to say more?

The IST2 (who plays the role of the student) says that she sees eight as four plus four and she says "eight times three". Later she says "eight three times" which is more in accordance with the image and the convention. The IST takes no notice of the first parts of the utterance, and he grabs hold of the last part which is more in line with his goal. The CI´s first question is about attending to and representation of student thinking. The IST simplifies and changes the student´s contribution through the visual and symbolic representation. Further on, he asks for other suggestions and thus indicates that the discussion is finished. The CI´s second utterance explicitly addresses eliciting and responding.

The combination of content goals and representation of mathematical ideas in the activity appear also often within exchanges in our data, and the combination is illustrated in the following example:

Example 2. The IST´s goal is to use a string of problems to discuss multiplication by ten, hundred and thousand with the students. He starts by four times three, and asks the students for a story that would fit the arithmetic problem. A student (one of the ISTs) suggests four groups of three apples, and the IST draws the illustration as shown in Figure 3. These kinds of exchanges, where representation and the content goal are combined, appear frequently in the data. In Example 2, the main substance discussed is the type of representation that could be appropriate for a given content goal and how to introduce it. In some other exchanges in the data that are coded with these two codes, the discussion is on ways to represent mathematical ideas so that the representation emphasizes the the relations that are targeted in the activity.

Structure of CI/IST Exchanges

The structure codes used in our analyses are the same as those used by [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF]. Table 2 shows the description and frequency of the various codes in our data. Similarly as with substance codes, an exchange can be coded using several structure codes. For instance, In Example 2 above, the exchange starts by "CI facilitates discussion" and develops to "CI gives directive feedback". Considering the structure of the rehearsals, the analysis shows that half of all CI/IST exchanges in our data can be characterized as discussions facilitated by CI (at least partly, in cases where several structure codes are used in the same exchange). In rehearsals analyzed by [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF], the code "CI facilitates discussion" is the least frequently appearing code, whereas "directive feedback" is the most frequent. This indicates that the structures of the rehearsals in the two studies are quite different. One reason can be that our study concerns in-service teachers while [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF] report from their work with pre-service teachers. In-service teachers have more experience with teaching than pre-service teachers, and it is reasonable to expect that their skills in teaching and their identity as mathematics teachers are more developed. A consequence can be that both in-service teachers and course instructors working with them might feel more comfortable in discussions than with CI giving directive/evaluative feedback or scaffolding enactment.

Discussion

Further work with in-service teachers and further data collection will take place in the coming year, and the results presented here are preliminary. However, the pilot study has already yielded a number of valuable insights. The study shows that the interactions between ISTs and CIs during rehearsals are mainly in form of discussions on some central principles and practices of ambitions mathematics teaching -using mathematical representations, aiming toward a mathematical goal, attending to student thinking and eliciting and responding to students' mathematical ideas. More specifically, we have found that multiple substance and structure codes are present within individual rehearsal exchanges, indicating that rehearsals offer in-service teachers the environment and opportunity to work simultaneously on a variety of aspects of practice. The study of [START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF] shows that rehearsals give opportunities for pre-service teachers to learn to enact principles, practices and knowledge entailed in ambitious teaching. The findings in our study indicate that rehearsals function as an approximation of ambitious teaching in work with in-service teachers too, even though the structure of rehearsals is different.

Figure 1 :

 1 Figure 1: Cycle of enactment and investigation

1 )

 1 attending, representation, eliciting and responding to student thinking 2) content goals and representation of mathematical ideas in the activity The following example is representative of the first category above: Example 1. The IST who is teaching during the rehearsal shows the image of three groups of eight dots and asks the "students" how they see it. One of the other ISTs in the team suggest an answer. IST2: I see eight times three. In the first group I saw four plus four, eight. I have eight three times. IST: [Circles three groups of eight. See Figure 2.] So first you have one times eight, so one times eight, so one times eight. [Writes 1x8+1x8+1x8=3x8.] Some other suggestions? CI:

  have been thinking about it. Money can be useful here, when we discuss multiplication by ten, hundred. One can use tenths and talk about 12 tenths in the next step. Same with hundreds. But, another story came up.CI:As a teacher, you have decided what the content goal is, and you have been thinking about what representation would be appropriate. You can ask about a story with money from the start to get the representation you want in the discussion.

Table 1 : Substance codes: description and frequency as percentage of all CI/IST exchanges in the data

 1 Many of the CI/IST exchanges involved more than one substance code, and the same combination of substance codes were frequently found together across different exchanges. For example, student thinking, elicit and respond and representation appear repeatedly in the same exchange. The combination of content goal and representation is also very common, in many cases together with mathematics. The frequency and the combination of the codes indicate that the main substance in CI/IST exchanges consist of

	% of all
	exchanges

Table 2 : Structure codes: description and frequency as percentage of all CI/IST exchanges in the data

 2

Proceedings of CERME10

Due to practical reasons we had to make some changes to the cycles of enactment and investigation proposed by[START_REF] Lampert | Keeping it Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching[END_REF] and illustrated in Figure1.Thematic Working Group

Proceedings of CERME10

Due to the complexity of CI/IST exchanges, an exchange can be coded by several substance codes. As a consequence, the percentages do not sum to 100%.

In Norway, when the multiplication is interpreted as equivalent groups, the meaning of "eight times three" is eight groups of three.


domain: math shs

text:
Andreas Lindenskov Tamborg

Benjamin Brink

Simon Skov Fougt

Morten Misfeldt
email: 4misfeldt@learning.aau.dk

Mapping the logics in practice oriented competence development

Keywords: Teacher training, Action Learning, theory and practice, Arcform

come   L'archive ouverte pluridisciplinaire

Implementing teacher capacity building through collaboration

A crucial aspect of a number of initiatives to improve mathematics education is the ability for teachers to collaboratively question and improve their own teaching [START_REF] Stigler | Video examples from the TIMMS videotape classroom study eighth grade mathematics in Germany, Japan, and the United States[END_REF], sometimes involving resource persons such as researchers or teacher educators. For such in-service training or capacity building to be efficient and scalable it is important that they are anchored in the school organization and not solely dependent on enthusiasts. Initiatives like Lesson Studies and development of own practice (alone or in collaboration with researchers), are examples that requires systemic and organizational attention [START_REF] Lewis | How should research contribute to instructional improvement? The case of lesson study[END_REF]. Systematic approaches often mean that teachers take on certain roles in relation to each other's practice in order to maintain initiatives. Such structures and roles that connect in-service training to practice and build directly on the school organization and culture are important. Research has found that it is very difficult to make sustainable changes with teacher training initiatives (Shear, Gallagher & Patell, 2011;[START_REF] Henriksen | Har projekter et liv efter deadline? Skoleudvikling fra projekt til forankring[END_REF][START_REF] Maurer | Beyond the Wall of Resistance: Why 70% of All Changes Still Fail -And What You Can Do about It[END_REF] has estimated that 70% of teacher training projects fail in changing teachers' practices within the given time frame of the project. This is a major challenge to the mainstream implementation of research findings in mathematics education. According to research literature the difficulties with changing teacher practice through in service training are associated with a lack of connection between training programs and teachers' existing practices [START_REF] Fixsen | Implementation Research: A Synthesis of the Literature[END_REF] as well as with the fact that teacher training programs often lack a focus on establishing active, collegial relations among teachers, which are crucial in order to build sustainable development (Hargreaves 2000, Sølberg, Bundsgaard & Højgaard 2013). When trying to address these concerns certain employees often take on certain roles towards their colleagues in the sense that they advocate for, manage and nurture certain projects. In this paper we explore this challenge by investigating a case where the distance between training activities and day-to-day teaching is very small and where the collegial relations are supported in a direct fashion. We do so by describing an Action Learning case as it has been implemented in a Danish municipality.

Context

In 2012 a municipality in Denmark launched a teachers training program aimed at developing teaching practices and enabling schools in the municipality to develop teacher competencies independently from external resources. The training program involved every school in the municipality and a total of 3500 participating teachers. The program consisted of a combination of a summer university in which teachers where given thematic lectures on a variety of subject-specific and pedagogical topics (implemented at 80% of the schools) and a method called Action Learning (implemented at approximately 20 % of the schools). Action Learning is a teacher-training program developed as an alternative to traditional course based in-service training. It draws on inspiration from Action Research in that it is based on an assumption that solutions to practical problems require practical understandings, which must be gained though iterative attempts to solve the actual problem [START_REF] Plauborg | Aktionslaering. Laering i og af praksis[END_REF]. The "action" in Action Learning refers to a pedagogical or didactical intervention that address real classroom issues as the individual teacher experiences them. While the "problem" is defined by the individual teacher the intervention is developed collaboratively by a group of teachers engaging in a so-called "team-conversation". Local supervisors from each school were designated the responsibility to facilitate professional discussions among the teachers in the team conversations. The local supervisors were teachers from the school who had a mathematics supervisor education, and who provided didactical support on a daily basis. The local supervisors also had the responsibility of anchoring the collaborations among the teachers at the schools to enable their ongoing professional development. The characteristics of Action Learning therefore seem to counter many of the challenges identified in the research literature about teacher training programs; active collegial teams are established, the team conversations are facilitated by the local supervisor in order to maintain an academic focus in the conversations and the starting point is the teachers' existing practices. In this context these initiatives however relied heavily on the local supervisors who were designated a key part in facilitating the team conversations and in anchoring team collaboration at the schools. In our research we therefore investigate the expectations arising to the local supervisors in the implementation of Action Learning and their possibilities meet these expectations (for further information about the program see the full evaluation report [START_REF] Misfeldt | Evaluering af fagdidaktisk kompetenceudviking: Aktionslaeringsforløb og Sommeruniversitet 2013[END_REF].

Method

Our research draws on interviews of key actors in the training program from two schools, namely the participating teachers, local supervisors and school managers. We interviewed 6 teachers, 2 local supervisors and 2 school leaders coming from 2 different schools. Our interviews explored the actors' experiences of the training program and their understandings of the role of the local supervisor in the program. We also collected documents and literature that describes the Action Learning Method and documents from the municipality describing how Action Learning was to be carried out. All of these sources were considered with the goal of identifying how the role of the supervisor was perceived. We analyzed this data by using Clarke's Situational Analysis [START_REF] Clarke | Developing Grounded Theory -The Second Generation[END_REF]. Situational Analysis is rooted in Grounded Theory, but modified according to postmodern assumptions that "boundaries are open and porous; negotiations are fluid and usually ongoing" [START_REF] Clarke | Developing Grounded Theory -The Second Generation[END_REF]. There is no a priori assumption that human actors are of greater importance than either non-human or discursive actors [START_REF] Clarke | Developing Grounded Theory -The Second Generation[END_REF], which allows us to view the Action Learning concept as an actor in itself. In Situational Analysis, Situational Maps provide a methodological approach to organize and visualize empirical data by foregrounding situations [START_REF] Clarke | Developing Grounded Theory -The Second Generation[END_REF]. We initially processed our data by using a sub type of Clarke's Situational Maps: Relational maps (den Outer 2013). Like other situational maps these aim at foregrounding situations rather than individual actors or their actions by mapping all actors (human as well as non-human and discursive) that occur in any situation, but go further than this by showing relations between actors. Relational maps use a type of network notation where actors are represented by labeled nodes and relations are represented by un-labeled lines drawn between the actors/nodes. We drew our relational maps while reading our transcribed interviews and other relevant documents. We began by listing the relevant actors that appeared in our data and their relation to each other. We produced many versions of some maps, modifying them as some actors and relations grew in prominence in our analysis. The messy nature in our data was easily overviewed with the simple structure of relational maps and thus they played an important part in opening up our data and thereby prompted our analyses. However, beyond a certain point they seemed to counteract rather than support overview. Clarke stresses that though situational maps are useful tools for beginning analyses, they are not necessarily an appropriate end-product of analysis (Clarke 2003, 563). We experienced two related problems: Firstly, it was difficult to draw some types of relations between actors and secondly they became difficult to understand/interpret, especially when returning to the analyses after several weeks. To overcome this limitation of our relational maps, we chose to visualize the situations through Arcform notation [START_REF] Allsopp | Introducing Arc Form: Designing a satisfactory highly non-linear alternative to texts for general-purpose idea development[END_REF]. Like relational maps, Arcform maps do not visualize our data, but rather the results of our ongoing analysis. Arcform maps differ from most relational maps, but resemble many network notations by supporting direction and labels on relations (arcs). Thus relations like "local supervisors coach teachers" are clearly visible as an arc labeled "coach" pointing from an actor node labeled "local supervisors" to another actor node labeled "teachers". However, Arcform also differs from most other network notations by allowing arcs to point from or to other arcs. In this way more complex relations like "teachers see local supervisors as coaches" can be drawn as shown in Figure 2.

Results

As our analysis progressed it became clear that the actors in our data articulated their relation to the local supervisor quite differently, and that they had different conceptions of the main job of the supervisors in the action learning project. Though these actors were all engaged in the same project at the same school, their ways of participating and their relation to the supervisor was rather different and seemed at first glance to be related to their role in the school outside the project. Besides being a part of a project, the actors were respectively also teachers, supervisors and school managers, and this fact seemed to be of importance. Our maps also revealed that this meant that the actors had different expectations on the supervisor and that these expectations could intersect with problematic consequences. In order to refine our analysis of these preliminary results we decided to use a notion of cultural logics developed by [START_REF] Nielsen | Teamsamarbejdets Dynamiske Stabilitet. En Kulturhistorisk Analyse Af Laereres Laering i Team[END_REF], which we will introduce below.

In a study on teachers' learning from collaboration in teams, Nielsen develops a view of teacher collaborations as having a dynamic stability [START_REF] Nielsen | Teamsamarbejdets Dynamiske Stabilitet. En Kulturhistorisk Analyse Af Laereres Laering i Team[END_REF]. It is dynamic because it involves numerous ongoing activities that are oriented towards one or more objectives. It is stable because it involves a perceived regularity in actors' actions suggesting a stable understanding underlying these activities. Such logics effect peoples' objectives and can be difficult for externals to change, because they reflect the every-day phenomena which are experienced as urgent by the actors involved. For example, although teachers most likely find the learning processes of students an important objective to orient their collaboration towards, so too may they find the practicalities that make a well-settled lesson [START_REF] Nielsen | Teamsamarbejdets Dynamiske Stabilitet. En Kulturhistorisk Analyse Af Laereres Laering i Team[END_REF]. In situations where there are multiple cultural logics we can expect actors sometimes to be caught in a tension between these logics.

The notion of cultural logics is highly useful in our context as the Action Learning training program is a project in which several actors', who occupy diverse positions, participate. Viewed this way the role of the supervisor is at risk of being caught in a tension between multiple cultural logics. As the local supervisor is a key actor in implementing and anchoring the Action Learning method, such tensions and their implications are of particular interest in this study. We identified three dominant logics. We refer to these as the workplace logic, the curriculum logic and the project logic. The cultural logics are characterized by situations in which certain aspects of the training program are foregrounded over others which translate into a set of expectations on the local supervisor. In brief the logics translate in to the following expectations:  In the workplace logic the supervisors are expected to manage the project and to avoid delays in the project.

 In the curriculum logic the supervisors are expected to be willing and able to guide the teachers academically in their professional development.

 In the project logic the supervisors are expected to initiate and support the teachers' professional development in a coaching-manner where an equal relation between supervisor and teacher is crucial.

The map below illustrates how the role of the supervisor is formed by the different cultural logics.

Figure 2. An Arcform map showing how two actors (school managers and teachers) see local supervisors as three different roles (project managers, academic beacons and (equal) coaches) with three different cultural logics (workplace logic, curriculum logic and project logic).

The goal in the project logic are progress and development of the school, while the curriculum logic foregrounds the quality of teaching. Though these logics can be complementary, our mapping revealed that the supervisors are faced with a tension due to a collision between the project logic and the workplace logic. In the following two sections we will therefore further describe the dynamic stabilities of the two latter cultural logics and unfold the tension emerging from here.

The project logic

The project logic concerns the cultural logic of the Action Learning training program as it occurs in documents describing the Action Learning concept and the expectations to the role of the supervisors emerging from it. In the Action Learning concept, the primary priority is the competence development of the teachers participating. In this, the supervisors are first and foremost expected to have the will to develop the school and the teachers and to do so as an equal coach rather than as a managerial authority. The supervisor is expected to initiate the Action Learning collaboration and to support the teachers in their development -not to lead/manage them. This is crucial as it is an acknowledgement that it is the teachers themselves who are experts on their own practicesthe role of the local supervisor is therefore to facilitate conversations that creates the best setting for this knowledge to be shared [START_REF] Plauborg | Aktionslaering. Laering i og af praksis[END_REF]). The statement below from a local supervisor illustrates her view of the Action Learning project suggesting that she embraces the project logic and that she is capable of seeing the potentials in the method.

Local supervisor: (…) there were some 3 rd grade teachers who said: "We have already tried this method. Why do we have to go through it again?" And my argument was that even though we have tried the method before, it is not implemented at our school. We don't use it as a method as things are now.

The statement indicates that the supervisor views competence development as ongoing and Action Learning as a way to enable such ongoing developments. She therefore argues to her colleagues that Action Learning is not a syllabus which you only have to read once and then move onrather, Action learning is a concept that involves specific ways of collaborating which are not implemented at the school. The statement thereby demonstrates a will to develop the school that resonates with the expectations embedded in the Action Learning concept. It also tells us that the supervisor has the skill needed to spot and to articulate that the crux is to integrate Action Learning as a way of collaborating.

The workplace logic

The workplace logic concerns the main objective of the training program from the school managers' view and their expectations to the local supervisors' role in the project. From the interviews with school managers the training program appears as a project among many other projects in which the main priority is to safely navigate the school through it and to avoid any delays. Though the school managers presumably also have an interest in developing the competencies of their teacher staff, safely getting through the project appears as the dominant cultural logic. Interviews with school managers show that this logic translates into an expectation that the local supervisors will be managers of the project due to high trust of the professionalism of the supervisors. The statement below from a school manager illustrates how the supervisor is referred to through the workplace logic.

School manager: I highly trust my supervisor's skills. Our supervisor is very professional and she is currently going through a pre-leader course. (…) and I thought that she therefore was better qualified to manage the project than I was.

In what appear as an acknowledgement of a supervisor's skills, this supervisor is given the responsibility to manage the project. At this particular school a group of teachers refused to participate in the training program due to short notice and discomfort about having to be observed as a part of the project. As the school managers had distributed the responsibility to manage the project, this became an issue for the local supervisor to handle. Consequently, the supervisor was obliged to "persuade", as she puts it, another group of teachers to participate in the project.

Local Supervisor: I didn't lure them but… I just told them that it wasn't optional. They just had to do it, you know.

As the responsibility of managing the project was designated to the supervisor through the workplace logic, these project management issues become a task for the supervisor to handle. This implies that the supervisor is required to draw on a formal leadership mandate by reminding the teachers that participation in the project is mandatory. As the supervisor describes in the following excerpt, this incident resulted in an uncertainty among the teachers about the role of the supervisor:

Local Supervisor: I think that this made it very unclear for the teachers what my part in this project was. Am I here to check if they are doing a bad job? Will I go to my manager and say: "That teacher does a bad job. She is really bad at teaching math". Or whatever it might be.

The supervisor's task of managing the project is not necessarily problematic in itself. But as a group of teachers refuse to participate in the training program, this is an issue that becomes a task for the supervisor to handle. In order to handle this issue, the supervisor is obliged to find another group of teachers that are willing to participate. As no other teachers were willing to participate in the project the local supervisor was obliged to emphasize to a specific group of teachers that they were obliged to participate as this was necessary for the Action Learning project to carry on.

Implementing research findings in practiceemerging problems and prospects

Our analyses have identified three cultural logics, two of which we have unfolded above. Each of these cultural logics produces a certain set of expectation to the supervisor in terms of how he or she adequately should participate in the project. What becomes evident from our analyses is that the local supervisors are met by mutually exclusive expectations as a consequence of these logics; the project logic expects the supervisor to support the teachers as an equal peer whereas the workplace logic expects the supervisor to manage the project as a superior. This has at least two consequences. Firstly, the supervisor's delegated management role triggers an uncertainty among the teachers of the intentions of the supervisor and raises the questions of whose errands he or she is running. Is the supervisor's main task to support the teachers in their professional development or to monitor their work on behalf of the management? This uncertainty makes it difficult to draw on the supervisor as an equal facilitator. A key component in the Action Learning concept is the joint observations of each teacher's practice, which subsequently are meant to be the starting point for a conversation aiming to develop the teachers' understandings of their own practices. Such an uncertainty among the teachers in respect of the supervisor's role represents a substantial barrier in creating a safe environment in which the teachers can learn from their own practices. Secondly, the coexisting logics cause a tension on the supervisor as he or she is expected to fill many roles at the same time.

Each of the cultural logics influence the actors' expectations to the role of the supervisor according to their own dynamic stability, thus tying the supervisor to different, incompatible priorities at the same time. As the potential for anchoring the Action Learning concept is closely connected to the role of the supervisor, there seem to be little chance that the supervisors are capable to do so under such difficult circumstances.

Our analysis also points to more general issues related to implementation of research findings in practice. Though Action Learning addresses what seem to be the main challenges in gaining longterm results from teacher training programs, the different expectations arising on the local supervisor complicates the implementation and the anchoring of the training program. The training program investigated in this paper exemplifies how many actors are at play in a school setting and that each participating actor may have different agendas in and around such projects. Though this perhaps is no surprise, our research suggests that difficulties in implementing research informed training programs can be the result of the differing actors' agendas outside the project. In Action Learning, as in many other training approaches, some actors are of immense importance in order to harvest the potential results of research informed approaches. The professional development of teachers involves and affects many others than the participating teachers and enters the professional lives of actors, which may have different priorities, agendas and available resources. A main problem about the issues identified in our research is that the co-existing cultural logics and the expectations arising to the supervisor thereof largely remain tacit. Though different agendas and the effects of such cannot be eliminated by simply making them explicit, an increased awareness and joint management of expectations would most likely be a step towards overcoming such hurdles.

Figure 1 .

 1 Figure 1. The sentence "Teachers see local supervisors as coaches" expressed in Arcform.

Proceedings of CERME10


text:
Per Nilsson
email: per.nilsson@oru.se

Andreas Ryve
email: andreas.ryve@mdh.se

Maria Larsson
email: maria.larsson@mdh.se

Characterizing theories aimed at supporting teachers' mathematical classroom practices

Keywords: Theoretical frameworks, theories for actions, mathematical classroom practice

 on productive classroom practice to contribute to the research on the nature of theories for action in mathematics education. By relating the results from the review to theories and literature on educational policy research, professional development research and implementation research we construct a framework for categorizing theories aiming at supporting teachers' actions in mathematical classroom practices.

Introduction

The development of frameworks and theories 1 that aim at guiding the actions of teachers have rendered much recent attention in educational research (e.g., [START_REF] Mckenney | Conducting educational design research[END_REF][START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF]. Within research areas such as curriculum material [START_REF] Davis | Designing educative curriculum materials to promote teacher learning[END_REF], planning and implementing whole-class discussions of cognitively demanding tasks (Smith & Stein, 2011), professional developments programs [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF][START_REF] Desimone | Improving Impact Studies of Teachers' Professional Development: Toward Better Conceptualizations and Measures[END_REF], and improvement of mathematical instructions at scale [START_REF] Cobb | Analyzing Educational Policies: A Learning Design Perspective[END_REF] cumulative work has been conducted to establish theories for action. Further, frameworks for supporting teachers' actions and thinking such as curriculum materials, theories and models have been put forward as essential components of effective professional development [START_REF] Cobb | Analyzing Educational Policies: A Learning Design Perspective[END_REF] and for establishing productive mathematical classroom practices [START_REF] Franke | Mathematics teaching and classroom practice[END_REF]. However, we need to know more about how theories should be designed to facilitate implementation; to be used and do real work in supporting and constructing teachers' actions [START_REF] Mckenney | Conducting educational design research[END_REF]. In this paper we relate result from a recently published systematic literature review [START_REF] Ryve | Kartläggning av forskning om formativ bedömning, klassrumsundervisning och läromedel i matematik[END_REF] on productive classroom practice to literature on educational policy, professional development and implementation research in order to construct a framework for understanding and facilitating the implementation of theories and research results aiming at supporting teachers' mathematical classroom practices.

Relevant research

The approach of the present study is based on the assumption that 'theory matters' for teachers, to enhance their ability to develop rich mathematical classroom practice [START_REF] Charalambous | Teacher knowledge, curriculum materials, and quality of instruction: Unpacking a complex relationship[END_REF]).

1 There are many ways to denote theories serving the purpose of guiding actions and this is further discussed below.

For instance, by adopting and making use of theoretical tools teachers are supposed to enhance their ability to establish productive routines in their classroom practice [START_REF] Franke | Mathematics teaching and classroom practice[END_REF], develop and continuously adjust a learning trajectory and the means to support that trajectory [START_REF] Cobb | Design experiments in educational research[END_REF], become more sensitive to notice instructional opportunities in the moment and be methodical without being mechanical. However, [START_REF] Burkhardt | Improving educational research: Toward a more useful, more influential, and better-funded enterprise[END_REF] argue that most theories that have been applied to education are quite broad, lacking the specificity that helps teachers to guide and understand the design and analysis of learning activities. [START_REF] Cobb | Design experiments in educational research[END_REF] adhere to this view, claiming "General philosophical orientations to educational matterssuch as constructivismare important to educational practice, but they often fail to provide detailed guidance in organizing instruction" (p. 10). So, there is this dilemma; theoretical constructs are supposed to enhance teachers' capacity to teach but, to do such work, theories need to be of a certain kind.

Perspectives on theories

diSessa and [START_REF] Disessa | Ontological innovation and the role of theory in design experiments[END_REF] detail the nature of different theories relevant for research in mathematics education. They distinguish between grand theories, orienting frameworks, frameworks for action, domain-specific instructional theories and ontological innovations. Skinner's behaviourist theory provides an example of a grand theory. Even if grand theories have a prominent position in educational research, they appear to be too general to provide guidance for explaining and supporting the learning of mathematics. Orienting frameworks, such as constructivism [START_REF] Von Glasersfeld | Radical Constructivism: A Way of Knowing and Learning[END_REF] or communities of practice [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF], provide general support for specifying issues of learning, teaching and instructional design whereas frameworks for actions concern analytical constructs of a more or less general prescriptive character (diSessa [START_REF] Disessa | Ontological innovation and the role of theory in design experiments[END_REF]. Domain-specific instructional theories are also of a prescriptive nature as they are typically specific to a domain or even learning trajectory of certain content and the means by which this trajectory can be supported. An ontological innovation is descriptive in nature. It is about developing analytical categories by which aspect of a phenomenon can be discerned. The framework of Socio-mathematical norms [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF] exemplifies an ontological innovation.

diSessa and [START_REF] Disessa | Ontological innovation and the role of theory in design experiments[END_REF] categorization not only labels the nature of different frameworks, it also points to the descriptive, explanatory, predictive and prescriptive purposes of different theories. Firstly, theories could be used to describe the world and many theories and frameworks within mathematics education serve such a purpose. The contribution to research in engaging in describing or characterizing objects or processes as certain phenomena could be understood in terms of new or unconventional lenses for viewing the world. Secondly, a further purpose of theories is to explain relations between phenomena and as mentioned above this purpose is often stressed as absolutely central for theories. A prerequisite for explaining those relations is to explicitly characterize each phenomenon. Therefore, theories used for explanatory purposes build upon or encompasses descriptive theoretical contributions. Thirdly, in a similar vein predictive theories necessitate explanations and clear descriptions of phenomena. Predictions include foreseeing effects of certain actions under certain conditions. Finally, prescriptive theories are used to identify and articulate productive ways to make decisions and performing actions. This kind of theory integrates descriptive, explanatory and predictive knowledge to guide actors in constructing and establishing interventions. Within design research prescriptive theories are often denoted design principles but neither the term nor the nature of those design principles are settled [START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF].

Theories for action

In this paper we are particularly interested in theories for actions and what [START_REF] Mckenney | Conducting educational design research[END_REF] denote the prescriptive role of theories. Both the characteristics of theories of actions and prescriptive theories and ways of denoting them are not settled in educational research as indicated above (cf. [START_REF] Mckenney | Conducting educational design research[END_REF]. As becomes apparent in [START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF], the relation between the terms used to denote theories for action is not just connected to neutral ways of denoting the same phenomenon but instead accentuates particular features and characteristics of such prescriptive theories. For instance, [START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF] shortly muse about the relation between the design tools they introduce and design principles. They suggest that the conceptual set up of grand theories, intermediate frameworks and design tools introduced in [START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF] stresses theoretical underpinnings for sensitizing researchers to critical issues while design principles from US often prescribe certain course of actions and are typically more loosely anchored in theoretical perspectives. One may ask, should theories for actions prescribe and sensitize teacher? In general, the development and understanding of design principles is weakly developed and in summarizing the most urgent issues for educational design research [START_REF] Mckenney | Conducting educational design research[END_REF] suggest "a worthy challenge facing educational design researchers is to further the development of predictive and prescriptive theories" (p. 212). We want to add to this research.

Method

The design was framed by a ten-step process for systematic literature reviews [START_REF] Gough | Learning from research: systematic reviews for informing policy decisions: a quick guide[END_REF]: (1) Need, (2) Review questions, (3) Scope, (4) Search, (5) Screen, (6) Code, (7) Map, (8) Appraise, (9) Synthesize, and (10) Communicate. In the project we engaged in processes 1-7 and 10. Our review questions were: (a) What characterizes research on classroom teaching practices, teaching approaches and teaching methods in mathematics? (b) What characterizes research on teachers' instructional strategies used to establish classroom practices in mathematics? and (c) What does research tell about teaching for the learning of mathematical competencies?

We searched in title, keywords and abstract in Web of Science 2 . Search strings 3 were iteratively developed while reading some abstracts. In total, we had 622 hits that we screened for relevance according to our scope defined by our inclusion criteria 4 . The screening was made in two steps based on: (1) title, keywords, and journal name, and (2) abstracts. Uncertain cases were discussed and decided upon collectively among the three researchers. After the two screening steps, 242 articles remained potentially relevant for the scope of the review. Simultaneously as the screening of 2 We searched Web of Science Core Collection as a way to focus on high-quality journal articles. We limited our search to the year span 2008-2014 and to the document types "article" and "review". 3 See Appendix A in [START_REF] Ryve | Kartläggning av forskning om formativ bedömning, klassrumsundervisning och läromedel i matematik[END_REF] for the exact search strings. 4 For an article to be included in the review, the article must be: (1) about mathematics teaching/learning/education, (2) related to compulsory school (grade 1-9), and (3) about the teacher's role.

Thematic Working Group 23 abstracts, the articles that remained relevant were coded on Object of study, Method, Number of participants, Context, Results, and Implications for practice. When needed, we also read other parts of the articles apart from the abstract. 201 articles remained relevant for the review based on the inclusion criteria. The next step of the mapping was to structure and characterize trends and interests within the discourse of research in mathematics education that focus on teaching methods, classroom practice and teacher's role in classroom practice. Therefore, we looked closer at the abstracts of the 201 articles, categorizing them in relation to object of study. See [START_REF] Ryve | Kartläggning av forskning om formativ bedömning, klassrumsundervisning och läromedel i matematik[END_REF] for detailed descriptions of steps and rationales in the processes.

Results

Of the 201 articles, we found 168 to be structured according to an analytical relationship between an outcome variable and a design variable. In the remaining 33, the object of study did not follow such a structure. The design variables we distinguished are gathered in the left column of the matrix (Figure 1), while the outcome variables we found are categorized in the top row of the matrix. In elaborating on the mapping we intend not to go through the entire mapping. We highlight and provide examples on some specific findings, which we then follow up on in the discussion. In the discussion we construct a new framework, to be used for categorizing theories to be implemented in mathematics classroom practices. On this account, and due to space limitation, we are not backing up all claims and findings by references from the mapping.

Elaborating on theories within the mapping

Student knowledge

Our review reveals a clear bias towards research that focuses students' product knowledge. Mathematical products relate to conventions, symbol systems, concepts and calculation techniques of mathematics. Looking at studies that emphasized a product view of mathematics and those emphasizing a process view we also notice a methodological difference. In the product view knowledge is expressed in the language of mathematical products and students understanding are profiled and ordered in accordance to the mathematics itself. Connected to such a conceptualization of knowledge, students' performances are often measured by standardized tests (Desimone, Smith, & Phillips, 2013). In the process view, qualities and progressions are not explicitly elaborated on. These frameworks are descriptive in nature, specifying a set of analytical categories, which is used to sensitizing [START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF] the analyst on some certain characteristics of, for instance, students' ability to communicate and reason in and with mathematics.

Interactional strategies

Studies belonging to this category explicitly refer to teachers' moves and actions. The teacher takes an active role in these studies; how he/she acts in interaction with the students, is central to the investigation. The focus is on how teachers communicate and engage with their students, and what role the communication and engagement play in students' learning of mathematics. It could be about, for instance, how a teacher uses gestures and questions [START_REF] Shein | Seeing With Two Eyes: A Teacher's Use of Gestures in Questioning and Revoicing to Engage English Language Learners in the Repair of Mathematical Errors[END_REF] and follow up on students' ideas in order to develop the mathematical classroom practice [START_REF] Akkus | Following Students' Ideas: How Much to Let Go?[END_REF].

Teaching approaches

Teaching approaches refer to studies taking a broad perspective on classroom teaching in mathematics. The teachers' and interactional behavior are not the main object of investigation. The teacher may be important, but it is the more general and overall structures of teaching that are the object of study. In our review we found different examples of teaching approaches, such as technology-based teaching, mathematical games, problem-based teaching, and contrasting ways of using textbooks in mathematics teaching.

Learning material (task design)

Some studies focus on how a specific artifact or design principle can support or challenge students' learning in mathematics. In these studies, the teacher takes a passive role. Focus is on the students' interaction with the learning material and the role of the teacher is basically to execute the lesson.

Studies belonging to this category may investigate the role of visualization or simulation in the learning of mathematics [START_REF] David | The role of visual representations for structuring classroom mathematical activity[END_REF]. In this group we also include issues of task design; types and sequences of tasks [START_REF] Hattikudur | Learning about the equal sign: Does comparing with inequality symbols help[END_REF] and instructions for solving mathematics tasks [START_REF] Orosco | Word Problem Strategy for Latino English Language Learners at Risk for Math Disabilities[END_REF].

Background variables

Several studies did not connect classroom practice and students' learning to any didactical design variable. These were studies giving accounts of personal attributes such as teachers' beliefs, attitudes and knowledge in order to explain classroom practice and students' performance.

Characterization

In 33 articles of the articles, the object of study did not follow the structure of an analytical relationship between two didactical variables. In this group of studies, to describe a certain practice or teaching approach is the focus in itself. The goal is to provide descriptive accounts of analytical categories of a teaching/learning phenomenon, which can be used to sensitizing researchers and teachers to critical issues of the phenomenon in question [START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF]. It may concern the characterization of curriculum material [START_REF] Sherin | Curriculum strategy framework: investigating patterns in teachers' use of a reform-based elementary mathematics curriculum[END_REF], the orchestration of math-talk with interactive whiteboards [START_REF] Beauchamp | Interactive whiteboards and all that jazz: the contribution of musical metaphors to the analysis of classroom activity with interactive technologies[END_REF], mapping the mathematics in classroom discourse [START_REF] Herbel-Eisenmann | Mapping Mathematics in Classroom Discourse[END_REF], or profiling students' understanding or strategies of specific subject matter content [START_REF] Wagner | Feeling number: grounding number sense in a sense of quantity[END_REF].

Discussion

By relating the results of our mapping to literature on educational policy, professional development and implementation research we construct a framework for understanding and facilitating the implementation of theories and research results aiming at supporting teachers' mathematical classroom practices.

In studying the papers it is apparent that teachers are ascribed different roles in different research studies. While quite a few studies within the category of learning material position teachers as administrators of tasks and computer programs other studies highlight the role of expert and orchestrator of classroom practices. Within the latter categories of articles, the role of teachers is central in asking questions, explaining content and acting formatively to support and challenge students' mathematical thinking. In understanding the implementation of theories and results aimed at improving classroom practices and students' mathematical learning it seems essential to consider how theories construct the role of teachers in classrooms. [START_REF] Ruthven | Design tools in didactical research: Instrumenting the epistemological and cognitive aspects of the design of teaching sequences[END_REF] notice the distinction between theories that prescribe teachers' actions and theories that aims at sensitizing teachers to essential aspects of classroom practices. In a similar vein, while some theories and studies in our review are clearly prescriptive towards teachers (e.g., theories belonging to Instructional strategies and Learning material) and what they should do in classroom practices, others aims at sensitizing and empowering teachers (theories belonging to the Characterization category). We are not normative about these different ways and suggest that both could be productive for different teachers in different context. Further, we hypothesize that these two strands are correlated with research methodologies in that many studies within mathematics education taking an educational design perspective aims at empowering teachers while studies taking a stricter experimental approach prescribe and praise clear prescriptive instructions.

However, to what extent and in which ways theories prescribe or sensitize teachers seem relevant to consider for anybody collaborating with teachers.

Cobb and Jackson (2012) stress that tools and frameworks within educational policies play a prominent role. When it comes to designing and using tools Cobb and Jackson suggest that it is important that the tools can be used by agents immediately in that they are easy to access, but at the same time harmonize with the planned reorganization of the practices. In addition, in developing frameworks, theories and tools it is essential to consider the amount and type of learning that are required for teachers to develop in order to use them in an appropriate and reliable way. Of course, such using requires good mathematical knowledge for teaching [START_REF] Ball | Content knowledge for teaching what makes it special[END_REF]. However, there is also reason to believe that certain types of frameworks are easier then other to use and apply in a mathematical teaching practice. For instance, in our survey we notice how the product perspective dominates research focusing on student knowledge. That the product perspective has a long tradition in the field is probably the major reason for this. However, taking a closer look to the mapping we can also understand this dominance as if there is a higher degree of transparency in how to use product knowledge theories compared to process-knowledge theories.

From this we learn that, in implementing theories to school practice, we need to consider how userfriendly, accessible and transparent different types of theories are to teachers. In other words, we need taking into account to what extent and which kind of teacher learning is necessary for productively implementing theories or frameworks to mathematical classroom practices? [START_REF] Desimone | Improving Impact Studies of Teachers' Professional Development: Toward Better Conceptualizations and Measures[END_REF] put forward coherence as a critical feature of professional development programs. The concept of coherence refers to the relation between the PDP and teachers' knowledge and beliefs. This raises questions about the extent to which theories should be coherent with teachers' knowledge and beliefs. In other words, should theories strengthen teachers' knowledge and beliefs or should it challenge their knowledge and beliefs? Should theories aim at strengthening classroom practice or should theories aim at reorganizing classroom practices [START_REF] Cobb | Analyzing Educational Policies: A Learning Design Perspective[END_REF]? The reorganization of practice could include working with new types of mathematical problems, new roles for students and teachers, and the establishment of new classrooms. Hence, in examining and choosing theories, frameworks and models, mathematics educational researchers working with teachers should consider whether the aim is to strengthen or reorganize ongoing practices and, consequently, consider how frameworks are supportive for such endeavors.

To conclude, as a complement to categorize theories for actions in terms of content areas we suggest it is productive for researchers working with teachers to consider theories in terms of: the positioning of teachers in classroom practices; the positioning of the teacher as a receiver of the theory; the amount and type of teachers' learning required; and if theories primarily function to strengthen or to reorganize practices.

Figure 1 .

 1 Figure 1. Categories and analytical relationships of the mapping

Proceedings of CERME10


text:
Lena Lindenskov
email: lenali@edu.au.dk

Katrine Kirsted

Theoretical constructs for early intervention programs in mathematics: Who cares? -A Danish example

Keywords: Early intervention programs, teaching principles, theoretical constructs

There is nothing so practical as a good theory". The statement from Kurt Lewin is frequently cited, also in mathematics education. The statement invites for and requires close cooperation between different agents, whatever their specific relation to practice and theory is. It is not a straightforward endeavour. One reason is that the term theory as well as the term practice may very well be given different meanings by different agents. This variation is, in our view, to be considered in "implementation research" and Lewin's statement ought to be qualified by two questions: "Who cares about a good theory?" and "What makes a good theory good for whom?" This paper explores the variation of how theory is perceived by mathematics teachers and by mathematics researchers involved in a developmental project on early intervention in mathematics education in Denmark. The paper exemplifies how agents' different work conditions and work requirements seem to constitute qualitatively different needs for theoretical constructs, despite some common interests.

Background for the early mathematics intervention project

We noticed a long tradition of integrating early mathematics intervention programs into compulsory education practice like, e.g. Mathematics Recovery (MR) [START_REF] Wright | Teaching Number: Advancing Children's Skills and Strategies[END_REF][START_REF] Wright | Developing pedagogical tools for intervention: Approach, methodology, and an experimental framework[END_REF] and Extending Mathematical Understanding Intervention Program (EMU) [START_REF] Gervasoni | Longitudinal Progress of Australian Six-Year-Old Students Who Participated in a Mathematics Intervention Program[END_REF], in Australia, Ireland, the UK and the USA. A similar tradition exists in Denmark for early reading intervention programs, as such programs are implemented at a regular basis in all schools in Denmark. Either as part of a municipal policy or a matter of choice, schools launched early (from the first grade) intervention processes in reading to support individual children, who show signs of reading difficulties.

At the ministry level in Denmark concerns were raised about pupils failing at mathematics in the National Official Guidelines in the 2003 National Mathematics Curriculum from the Ministry of Education (UVM, 2003). In 2004 the need to support failing pupils in mathematics in the first school years was emphasised (Mortimore et al.) and the national official guidelines to the revised 2009 national curriculum (UVM, 2009) described, for the first time, the issues in detail. Still, no intentions of integrating programs for early mathematics intervention into compulsory education practice were seen in Denmark until recently [START_REF] Lindenskov | State-of-the-art Danmark 2005 + Litteratur op til marts[END_REF].

In 2009 the material Early Intervention in Mathematics [Danish: Tidlig Indsats i Matematik, TIM], written to primary school mathematics teachers was published and used in some places. Just before, in 2007 local politicians and school authorities in the municipality of Frederiksberg in the capital area, decided to give priority to mathematics teaching and learning in their 9 public schools in the period 2007-2013. Priority was given to a development project on early mathematics intervention for their 9 schools in collaboration with the researchers Lena Lindenskov and Peter Weng. It soon became clear that the existing intervention framework and written materials solely focused on numbers and arithmetic, which were insufficient to comply with the Danish Mathematics Education Philosophy and Curriculum. Also, approaches in the existing intervention framework and written materials were insufficiently inquiry and problem based to correspond with the curriculum. Finally, teachers' freedom and responsibility to adapt materials to their own students were too limited in the existing frameworks and materials.

With this background a research-based developmental project with four design cycles was prepared in order to develop a Danish program for early intervention in mathematics that would fit into the Danish Mathematics Education Philosophy and Curriculum.

Perceptions of what is good theoryby mathematics school teachers

Our research question is, what is a good theory for whom -teachers as well as for researchers? Our analyses draw on our communication with the 18 teachers involved in the design cycles of the project. In the following these teachers are called pilot teachers. All pilot teachers were chosen by their school principal as among the most qualified and motivated mathematics teachers at the school. Some also were 'Mathematics Counsellors' with a one-year diploma course.

It is the use of theoretical constructs in the four cycles, which is analysed in the following. We analyse interactions between teachers and researchers. The data come from written materials and recorded minutes and notes from seminars1 , training sessions, coaching sessions and e-mails Generally, Danish teachers have a relatively high self-confidence and a strong wish to influence. It is shown, for instance, that Danish teachers, more than British teachers, prioritise their students' personal development and see their students' mathematical development as a means for personal development [START_REF] Kelly | Comparing pedagogy in mathematics in Denmark and England[END_REF]. Because of the way the 18 pilot teachers were chosen by their principals, we anticipated that the teachers would be involved, to a high degree, in the project's four design cycles. The specific choice of mathematical and other aspects for the framework and written materials for the early intervention was actually made in dialectic processes involving the researchers and the 18 pilot teachers. Further descriptions of the cycles are found in [START_REF] Lindenskov | Early mathematics intervention in a Danish municipality: Theory and teachers' reflections in the pilot project[END_REF].

The framework and materials developed through four design cycles

(1) From January 2009 to September 2009, Weng and Lindenskov developed the first draft material, based on theory, empirical results and their knowledge about mathematics in life and in primary and lower secondary schools in Denmark. They initially doubted whether the teachers would find it relevant to study the rationales and theory behind the choice of mathematical areas, materials and evaluating and teaching principles. That is why only a few theoretical constructs and justifications were in the first draft communicated to the teachers.

But, as the structure and each part of the draft material were critically explored and discussed during the teacher training sessions from 14 -18 September 2009, this expectation of the teachers' perceptions of their needs for theory was wrong. The teachers endorsed the underlying ideas, but actually asked for further explanation of rationales and theoretical constructs. The teachers also asked for an extensive introduction to the program as such. The time teachers were expected to use in the development processes did not include reading articles, so the researchers presented articles orally and provided printed extracts or copies of some articles as handouts. The main printed materials were newly developed diagnostic test materials and the problem solving materials to be used with their students, plus general introduction and justification for the choice of mathematical areas and instruction approach.

At the end of the week, the teachers suggested that measurement as a mathematical area and the use of measurements in other mathematical areas should be expanded in the next draft.

(2) Lindenskov and Weng developed a second draft of material based on the pilot teacher feedback and feedback from the research assistant. This meant that for more mathematics areas further justification for and explanation of rationales and theoretical constructs were included. The second draft was sent to each school October 2009 for experimenting. Each pilot teacher tried out specific parts of the material in the fall of 2009. The distribution of the parts to each school was decided through discussions among all pilot teachers. Each pilot teacher was requested to try out two or three activities with as many pupils as possible. The age of the pupils was not important. If possible, more material was to be attempted. The pilot teachers were given a specific task in order to evaluate the materials: they were asked to document in as much detail as possible -by writing in premade tables -how each mathematics task and each mathematics and attitude question led to pupil-teacher conversations which could indicate the pupil's thinking. The experiments were concluded with a seminar on 3 December 2009, where each pilot teacher presented results. Anything that had particularly surprised the teachers was also presented and some common concerns were then discussed. It was put forward that the pages with descriptions of justifications and theoretical constructs were helpful, 'or else we do not know why the chosen mathematical areas, concepts and competences are important to focus on.'

(3) Based on these results Weng and Lindenskov developed the third draft material and introduced it at a seminar on 28 January 2010. Justifications and theory were included for all mathematical areas. In the following months, each pilot teacher tried out parts of the material with a number of pupils. This time all the pupils were in the second grade. The aim was to allow the pilot teacher to experience the structure of the material and to practice pupil-teacher conversation. Peter Weng visited and coached every teacher once and the teachers were invited to contact the researchers at any time during the pilot study.

At a midway seminar on 9 March 2010, the teachers described their general impression as positive and generally considered the material adequate. Several pilot teachers said they found it motivating to work with the material together with the pupils and that they had heard from the pupils' ordinary mathematics teachers that the intervention seemed to have a positive impact on the pupils' learning process.

The individual schools' prioritization of subjects was also discussed: how to decide between pupils' participation in a class excursion or a TMF session?

At the seminar a representative from the Egmont Foundation was present, as the Foundation had decided to fund the project. In the developmental project, the choice of pupils was left to the schools and the criteria differed between schools. The Egmont representative was particularly interested in the discussions on ethical issues: Whether pupils with very weak home support should be chosen over pupils with better support from home, who would probably benefit more? It is well known from research that socioeconomic factors are important for pupils' learning and development. Maybe particular pupils need this intervention the most, but are they really the ones chosen?

Issues regarding the scope and range of the material were discussed, for instance how to prioritise between presentations of many mathematical aspects or assuring success in fewer mathematical areas. The risk that the material put severe strain on teachers, especially when they were unfamiliar with it, was also discussed. To illustrate this discussion, we have listed three pilot teacher transcripts and one researcher transcript below:

Teacher 1: I feel pinned down by the material. I feel like, 'Now I must do this, then I must do that,' and you have to look for concrete material yourself. It is very restraining. While I look for extra material, I give the pupils small tasks on the computer to work with, OK.

Teacher 2:

The material could be constraining. But the material is important as a database of ideas. The material gives me ideas. It supports my own inspiration process and it helps me to include everything in my practice.

Teacher 3:

The material is useful, when I prepare the intervention sessions.

Researcher Weng:

Try to think about the material as something that provides you with opportunities and inspiration. We invite you to a flexible adaptation to specific pupils.

(According to the minutes, authors'translation)

The final seminar on 27 May 2010 discussed organisational and psychological issues in detail. The teachers wanted organisational and psychological aspects of individual pupils' learning and instruction to be emphasised as equally as the mathematical aspects.

Also the teachers again asked for more geometry and measurement in future versions, as well as a compiled list of recommended materials, but they did not mention any further need for justification and theory. (According to the minutes, authors' translation)

(4) The fourth draft was developed by 12 August 2010 and was to be used from 2010 onwards in the regular TMF for individual second grade pupils in all of Frederiksberg's public schools. The research assistant, Tina Kjaer, examined the material and ensured that the teachers' suggestions were taken into account. Strongly supported by the pilot teachers' feedback, organisational and psychological aspects of individual pupils' learning and instruction were included as just as important as mathematical aspects.

Example of how researchers' theoretical understanding is communicated to teachers

As an example of how the researchers communicated their theoretical understanding underpinning the developmental project to the pilot teachers, we have chosen the mathematics area "Basic Strategies for Numbers in Addition and Subtraction". The table below shows in the left column four of the theoretical constructs chosen by the researchers to underpin the project. The right column shows how the constructs were being communicated and discussed between researchers and teachers. The right column consists of citations from the final written materials, which was published in 2013 and meant to be studied and discussed among teachers involved in intervention projects.

Researchers' choice of theoretical constructs and justifications

Citations from the published intervention materials (Lindenskov & Weng, 2013)

Relational understanding (RU) and instrumental understanding (IU): Although IU in its own context is often easier to understand and gives correct answers with less knowledge involved, RU is more adaptable to new tasks and easier to remember. [START_REF] Skemp | Relational understanding and instrumental understanding[END_REF][START_REF] Skemp | Relational understanding and instrumental understanding[END_REF].

When the pupil experiences a productive development in his/her basic strategies in addition and subtraction, it opens the pupil's possibilities of becoming capable in doing relevant addition and subtraction and to use it in many contexts. Also, potentially this experience will contribute to another highly relevant math competence: good estimating skills for big numbers.

Constructivist teachers' primary activity is communicating with students. In the constructivist view, teachers should continually make a conscious attempt to "see" both their own and the children's actions from the children's points of view. [START_REF] Cobb | The Constructivist Researcher as Teacher and Model Builder[END_REF].

Some teachers might, for the last decades, have misunderstood the core of constructivism. Some teachers might have been inclined not to interfere when the pupils calculated and developed their calculation skills and strategies. Some teachers might have believed that the pupils by themselves would develop at the pace that was most optimal for them individually. But we know, it is a risky affair.

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Pupils who engage in strategy development decisively perform better in the long run than pupils who do not. [START_REF] Ostad | Children With and Without Mathematics Difficulties. Aspects of Learner Characteristics in a Developmental Perspective[END_REF].

Pupils, who from an early age, start developing his/her strategies, tend to continuously improve existing strategies and increase the number of strategies. In contrast, pupils, who stick to their strategies, tend not to start improving them until later on. It is shown that pupils, who stop developing their strategies, will toil hard and will still lag behind.

Strategies, strategy development and teaching strategies should be the core of fundamental mathematics instruction and learning. [START_REF] Ostad | Strategier, strategiudvikling og strategiundervisning med fokus på den fundamentale matematiklaering[END_REF].

Conclusion

This paper has shed light on what is a good theory for whom -teachers as well as researchers, how to explain theory and justifications to the pilot teachers in a meaningful way and how to develop material in collaboration between researchers and teachers? During the four development phases the pilot teachers endorsed the underlying ideas of the intervention project and asked for the rationale behind every included aspect to be explicitly communicated. They encouraged more extensive introduction and to expand the included measurement aspects into two measurement aspects.

The teachers explicitly endorsed the theoretical construct and justifications in the material, as they said it helped them to acknowledge many opportunities to help the pupils and to identify pupils' potentials and motivation while exploring and developing their mathematical needs. The teachers appreciated that the material gave a firm frame and at the same time invited and inspired the teachers to adapt and further expand the materials to the specific learning situations with the pupils.

The teachers recommended the material to be expanded with more mathematical concepts and competences, which are considered relevant in the Nordic contexts [START_REF] Dalvang | Med compass mot mestring -et didaktisk perspektiv på matematikkvansker. [With a compass towards mastering -a didactical perspective on mathematical difficulties[END_REF]Niss & Højgaard, 2011) and by the teachers as potentially troublesome for the weaker pupils, and to expand the materials on measurements and part-whole.

The teachers asked for further ideas and materials which could be used as they were or could be adapted in order to fit their own pupils' needs and motivations. The teachers did not suggest more clarified theoretical constructs and justifications underpinning the program than were communicated to them already.

For the educational researcher the task was to find and select theoretical constructs to underpin the intervention and communicate these to the teachers, as it is further described in [START_REF] Lindenskov | Theories to be combined and contrasted: Does the context make a difference? Early intervention programmes as case[END_REF]. It could not be communicated as in scientific journals, but as justified practical advices. Both theorists and practitioners care for theory, but in very different ways.

  The private Danish fund Egmont showed interest in a Danish early mathematics intervention program (personal communication), and together Egmont, Frederiksberg Municipality and Aarhus University assured the budget for this project. The project was named Early Mathematics Intervention at Frederiksberg with the Danish abbreviation TMF [Danish: Tidlig Matematikindsats Frederiksberg].

Proceedings of CERME10

Mathematics Recovery Programme (MR) is a source of inspiration for the developmental project, seeWright et al.,

This is why we included a teacher seminar with Ms. Noreen O'Loughlin from Mary Immaculate College, University of Limerick focusing both on some hypotheses and issues in MR and on specific concerns at Frederiksberg. Thematic Working Group 23 Proceedings of CERME10