Uffe Thomas Jankvist

Mario Sánchez Aguilar

Jonas Dreyøe

Morten Misfeldt
email: misfeldt@learning.aau.dk

Morten Misfeldt Adapting Implemen-

Mario Sánchez Aguilar

Adapting implementation research frameworks for mathematics education

Keywords: Implementation research, innovation, theory of change, value

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Implementation research has received an increase in attention within mathematics education during the last few years (e.g. [START_REF] Cai | Making classroom implementation an integral part of research[END_REF][START_REF] Jankvist | Introduction to the papers of TWG23: Implementation of research findings in mathematics education[END_REF]. But the discussion of implementation is neither new in educational research in general, nor in mathematics education in particular. Already [START_REF] Fullan | Research on curriculum and instruction implementation[END_REF] remarked that "implementation is not simply an extension of planning and adoption processes. It is a phenomenon it its own right. " (p. 336). Even if the discussion of implementation has been around for some time, both the work presented and carried out during CERME 10 and the recent discussions in JRME (e.g. [START_REF] Cai | Making classroom implementation an integral part of research[END_REF] illustrate clearly that there is still a large potential indeed in addressing initiatives within mathematics education from an implementation research point of view. However, it is not at all clear how exactly to do this. Hence, we first need to ask ourselves which existing implementation research frameworks that are applicable to which initiatives in mathematics education as well as how and why? The purpose of this paper is to take some initial steps towards addressing this big question. As a reference point we take implementation frameworks from other research fields to try to sketch an implementation research framework in mathematics education. Then we "test" it on an illustrative example of the decentralized implementations of the Danish mathematical competencies framework-KOM-in selected mathematics programs of the Danish educational system.

Implementation research frameworks

Within the context of health science, [START_REF] Nilsen | Making sense of implementation theories, models and frameworks[END_REF] proposed a taxonomy of three aims for the use of theoretical approaches in implementation science, and five categories of theories, models and frameworks. The first aim of theoretical approaches is to describe and/or guide the process of translating research into practice. Process models lay out specific steps to implement research into practice, and thus provides practical guidance in planning and carrying out implementation. The second aim concerns understanding and/or explaining what influences implementation outcomes. Three different types of frameworks and theories are found to supports this aim. The first type is the determinant frameworks that identifies barriers and enablers, which influence the implementation outcomes as well as specifying the relationships between barriers and enablers. The second type is the classic theories, which are theories that originate from fields outside of implementation science, but may be applied to understand some aspects of implementation. The last type related to this aim are actual implementation theories, which are those developed with the sole purpose of explaining aspects of implementation. The third and last aim concerns evaluating implementation. This aim can be supported by the category of evaluation frameworks that may provide a structure for evaluating implementation.

In their review of implementation research within education, [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] offer the following (working) definition: Implementation research, by our working definition, is the systematic inquiry of innovations enacted in controlled settings or in ordinary practice, the factors that influence innovation enactment, and relationships between innovations, influential factors, and outcomes. Thus, frameworks that inform the organization of implementation research address two main concerns-how to conceptualize and describe the innovation itself, and how to identify and organize the contexts, conditions, and characteristics that influence innovation enactment (influential factors). These two fundamental concepts-(a) characteristics of the innovation and (b) influential factors-are basic elements of varied theories of change and a key part of most recent research syntheses or metaframeworks depicting innovations in context. (p. 181) This definition builds on two other terms, the meaning of which will have to be explained as well; namely those of innovation and change. [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] operate with a definition of innovations as "programs, interventions, technologies, processes, approaches, methods, strategies, or policies that involve a change (e.g., in behavior or practice) for the individuals (end users) enacting them" (p. 170). More traditionally, an innovation is oftentimes considered to be new ideas creating economic value [START_REF] Darsø | Innovationspaedagogik: Fremtidens Laering in Innovation i Skolen [Innovation pedagogy: The future's learning in innovation in school[END_REF], or as the middle part of the CIE (Creativity, Innovation and Entrepreneurship) model for value creation and change [START_REF] Paulsen | Innovation & laering: Filosofiske og Kritiske Perspektiver[END_REF]. However, the dependence on economic bottom line thinking makes this definition difficult to apply to educational situations. Nevertheless, if we instead consider innovation as changes of practices and technologies for the better, then we set ourselves free from the problem of economy as the core goal for innovation. This however leaves open what kind of normative change for the better that we should value in innovation. We should ask what (kind of) value and for whom. The definition of educational innovations provided by [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] focuses on the change for end users (e.g. teachers and students) and simply views innovations as the object of implementation research. While this is indeed an important perspective, we suggest that the notions of value and stakeholders are critical; an innovation is only an innovation if it creates value for some stakeholders [START_REF] Krainer | Teachers as stakeholders in mathematics education research[END_REF], but the stakeholders do not necessarily have to be the end users. Value can be for the society in general, for educational administrations, school management, etc. We thus define educational innovations as change in educational practice that are valued by some stakeholders. However, in order to conceptualize any change, it is necessary to acknowledge that change may take place at different levels (at the individual level, classroom level, school level, etc.), but also that change is mediated by people and institutions who can adopt, modify or even reject the intended innovation implementation. Thus, as pointed out by [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF], a question that lies at the heart of the implementation research is: "what does it take for people, organizations, and systems to change?" (p. 178).

To address a question like that in the context of mathematics education, we need to draw on a wide range of conceptual and methodological tools from within and outside the field. For example, to try to identify and organize the conditions that influence innovation enactment at the classroom level, it would be important to understand the intentions and individual motives that a mathematics teacher has to adopt or not an educational innovation. Research on curricular implementation in mathematics education has shed some light on this type of issues (e.g. [START_REF] Remillard | Conceptualizing the curriculum enactment process in mathematics education[END_REF]. If we refer to change beyond the classroom and on a larger scale, say at the institutional level, it could be useful to resort to studies on organizational change from different disciplines such as medicine [START_REF] Rohrbach | Type II translation: Transporting prevention interventions from research to real-world settings[END_REF] and management science [START_REF] Burnes | Complexity theories and organizational change[END_REF], which can provide us with theories and methodological approaches to help us understand how to produce large-scale change behaviors in school systems. Here studies on mathematics education with an institutional perspective could also be useful (e.g. [START_REF] Castela | Institutions influencing mathematics students' private work: A factor of academic achievement[END_REF]. On a wider scale, [START_REF] Rogers | Diffusion of innovations[END_REF] has developed a frequently cited theory describing how innovations are spread across a social group (see for example [START_REF] Koichu | Implementation enterprise through the lens of a theory of diffusion of innovations: A case of online problem-solving forums[END_REF]. He distinguishes several groups of adopters; innovators (adopting a new technology instantly), early adopters, early majority, late majority and laggards (who adopts very late). Rogers shows that the distribution of adoption is similar to a normal distribution with the bulk of people in the early and late majority category. Implementation is obviously related to how people adopt. If you aim for mainstream implementation of an innovation, when it is only adopted by innovators, you are bound to fail.

Five factors that influence implementations

The identification of the factors that influence innovation enactment is a fundamental component of implementation research. For years educational researchers have tried to identify the variables that influence the implementation of educational innovations; these variables can be classified in the following five spheres of influence [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF].

Characteristics of the individual users:

The change that an educational innovation is aimed at generate, it is mediated by the people involved in the implementation process. Hence, it is important to know their individual characteristics. We distinguish between (a) characteristics of the individual in relation to the innovation (mathematical background, experience using the materials or resources involved in the innovation, etc,) and (b) characteristics of the individual that exist independently of the innovation (willingness to try new teaching methods, attitudes towards new artefacts in the classroom, etc.).

Organizational and environmental factors:

In the case of an innovation implemented in a mathematics classroom, organizational factors refer, on the one hand, to the characteristics of the setting itself (number of students, characteristics of the physical space, access to material resources, etc.), and on the other hand to the collective beliefs and behaviors of the members of the class (identity, sociomathematical norms, didactic contract, etc.). Environmental factors refer to those outside the organization, but which have an influence on how an innovation is adopted and implemented (economic conditions, educational policies, priorities of government agencies, etc.).

Attributes of the innovation:

The attributes of the innovation can influence its implementation, however, it is important to distinguish between the actual attributes of the innovation (objective characteristics) and the perceived attributes of the innovation (subjective characteristics perceived by the user). Of course, the perceived attributes may vary from user to user.

Implementation support strategies: It is important that an innovation initiative comes accompanied by an intentional and planned support for the final users and their institutions. Such support strategies can be professional development, specific resources, etc.

Implementation over time: Another factor that influences the implementation of an innovation is time. Thus, it becomes relevant to study innovation endurance over time: how can we promote that an innovation, besides being adopted, is preserved over time until it is routinized? It is in this branch of the implementation research where longitudinal studies will become essential to answer questions like the one previously stated.

The KOM frameworkan "implementation story"

As an illustrative example of an implementation, we take a historical and chronological look at the Danish mathematics competency framework, referred to as KOM, which was first published in Danish (Niss and Jensen, 2002), and later in an English translation (Niss and Højgaard, 2011). This framework has heavily influenced mathematics education in Denmark, where it has been implemented in primary and lower secondary school through the so-called "Faelles Mål" (common goals) (Undervisningsministeriet, 2014), and in both the technical stream (htx) and business stream (hhx) of upper secondary school-to a lesser extent in the classical stream (stx)-in the mathematics teacher education program as well as in some of the mathematics-related programs in tertiary education. On an international level, KOM's competencies descriptions were an integral part of the PISA assessment framework for mathematics from approx. year 2000 through 2018. The description has also been influential in mathematics programs in several countries' (see Niss and Højgaard, in progress). Now, in itself the KOM framework constitutes a normative text, which is not directly translationable/implementable in the various mathematics programs. But KOM's competencies description, with its eight distinct yet interconnected mathematical competencies, was never meant as a standalone. It must be implemented together with a curriculum for the mathematics program at a given educational level. One distinct feature of the KOM approach is its matrix thinking that links specific competencies to various mathematical areas, i.e. competencies in the rows and mathematical areas, e.g. algebra, geometry, in the columns of a matrix. From an implementation perspective, however, KOM makes up an interesting case.

When published in 2002, KOM was supplemented with an implementation support strategy of meetings and seminars debating specificities of the framework and the value of placing development of mathematical competencies as the key feature. But besides this, the implementation support strategies must be characterized as ad hoc and rather scattered. Still, KOM's competency description was adopted by a large part of the teacher educators in Denmark. It was implemented in the national standards for compulsory school, partly between 2003 and 2006 and more thoroughly in 2009. Furthermore, it was implemented in the teacher education standards in 2012 and in upper secondary school in 2013 and again in 2017. In 2012 the impact of the competencies framework was evaluated as part of a general evaluation of the national standards (Danmarks Evalueringsinstitut, 2012), showing that teachers were neither using the standards nor the eight competencies to a very large extent. In 2014 the K-9 curriculum was reformed towards an outcomeoriented curriculum, and the competencies were now embedded in this structure ("Faelles Mål"). Furthermore, the number of competencies was changed from eight to six. A transformation that was heavily criticized by the people behind KOM (see e.g. [START_REF] Niss | Målstyret matematikundervisning?! [Goal oriented mathematics teaching?[END_REF]. As of now, the overall structure of the output-oriented curriculum is being debated again and will be partly rolled back. Nevertheless, for the eight competencies this transformation actually appears to have increased their impact and the outreach of the KOM framework. These days more teachers do seem to know more about the competencies than before; partly due to timing in terms of adoption and partly because the 2010 reform made everyone aware of the curriculum/standards and hence the competencies. As mentioned above, such awareness concerning KOM does not seem to be present at the classical stream of upper secondary school, stx, which is by far the largest of the three upper secondary school programs. Why is this? The answer to this question-we believe-is to do with environmental (and institutional) factors as well as the characteristics of the different individual users.

The characteristics of the individual users in the various institutions are quite different. Danish K-9 mathematics teacher educators are mainly educated at the Danish School of Education, where they are exposed to and thus become accustomed to an educational approach that relies heavily on the competency framework (e.g. [START_REF] Højgaard | Educating mathematics teacher educators: A competencybased course design[END_REF]. In their future professions at the university colleges, these teacher educators thus often come to act as enablers for the implementation of KOM, not least in relation to the teachers that they educate. The K-9 mathematics educational system, however, is a big ship to turn since it is comprised of teachers of all ages educated under various different educational paradigms and reforms. Hence, as mentioned above, the prevalence of awareness of the eight competencies was a lengthy process. One indicator of this is the fact that many teachers reacted negatively against the PISA 2012 assessment in mathematics, despite the fact that the 2012 PISA mathematics framework was perfectly aligned with KOM and thus also the national standards of 2009. Yet, as part of these standards, an explicit matrix structure between the competencies and mathematical areas was developed for each grade level. This has played a major role in the implementation of KOM in K-9.

When it comes to the upper secondary stx mathematics teachers, they oftentimes hold a master's degree in mathematics from a university mathematics program. Their teachers at the university typically hold a PhD in some area of mathematics, and hence are not necessarily very well versed in the area of mathematics education research, including the competency approach. This of course serves as a barrier for the implementation of KOM in stx. The reason that the situation is a bit different in the two other streams of upper secondary school, as mentioned above, is most likely to do with the fact that the mathematics teacher population there has a more varied background, e.g. at htx several teachers have a background as engineers. Another identified barrier is to do with the lack a matrix structure in the stx curricular documents. Although these do mention a selection of KOM's eight competencies, then mainly do so on a rhetorical level, i.e. the various competencies are not linked to actual mathematical areas and concepts. To a much larger extent this is done in the curricular documents of both htx and hhx (for a further discussion, see Niss and Højgaard, in preparation).

Implementing the KOM framework has taken time. As described above different stakeholders have adopted the framework to different degrees. Even though all these stakeholder groups are distributed across Rogers' (1962) different adoption-types, it is still possible to distinguish groups that overall are moving faster than others. The teacher educators, for instance, have adopted KOM much faster than the K-9 teachers. However, in the case of the K-9 teachers time seems to work for the implementation of KOM, even though adoption is slow. The 2014 K-9 reform was timewise in a place where the innovators and early adopters was already using the competencies framework. This is to say that the timing was working for the competencies, and the curriculum reform thus acted as a catalyst for the early and late adopters to embrace the framework. In the case of the stx teachers, we are less sure to what extent the adoption and implementation will progress over time. KOM has the learning of mathematics at its core, and hence it should be relevant for this population of teachers as well. Still, the lack of an implemented matrix structure in the curricular documents of stx, which, as described above, is an essential attribute of the innovation, does not hold a lot of promise. Due to the stx curricular documents' use of the competency terms only on a rhetorical level, the actual attributes of KOM and the stx teachers' perceived attributes may simply not be aligned.

Discussion and analysis

Referring back to Nilsen's framework as presented above, the analysis of the implementation of KOM in Denmark mainly addresses the second aim of understanding and explaining what influences implementation outcomes. Here both aspects of what Nilsen refers to as determinant frameworks are in play, as the project identified certain barriers and enablers which influence the implementation outcome. [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] description of factors that influence implementation (see above) was useful alongside Rogers ' (1962) focus on stakeholders' adoption. Hence, these implementation research frameworks have provided us with some general perspectives of where to look and focus our attention, but in themselves they are not detailed enough for cases such as KOM -or any case of mathematics education, we predict. In particular in relation to Nilsen's first and third aim, the description of the implementation process and the evaluation of implementation, one thing that the general frameworks often cannot supply us with is a theory of change. Which on one hand can guide the process of implementation and on the other hand is used to evaluate the success of an implementation. The parameters on which to measure change may oftentimes be found within theoretical constructs from mathematics education itself. Let's exemplify.

In mathematics education there are of course many different kinds of innovations, but for the sake of simplicity let us distinguish two. One type concerns that of implementing normative frameworks such as KOM, which in a sense call for a philosophical or cultural change in mathematics programs as to what it means to master the subject. Another type are the grand scale initiatives as for example the Swedish Boost for Mathematics (BM), which offered a further education program to the majority of Swedish mathematics teachers. Although the BM did include a framework, involving e.g. didactical contract, socio-mathematical norms, etc., it was in itself not a framework such as KOM is. This means that when assessing qualitative aspects of the implementation of the innovation, KOM becomes its own theory of change, whereas BM does not. We are not claiming that this necessarily is a problem; only that we should be aware of this when discussing implementations in mathematics education. Another difference is that the implementation of the BM was top-down, whereas many of the local implementations of KOM were more bottom-up. Also, the difference in scale of the two implies that the evaluations of them may be of different nature. While the BM was often evaluated on quantitative terms, e.g. how many schools and teachers participated, etc., KOM to a larger extent was only evaluated on qualitative terms. The qualitative theory of change parameters of both KOM's and BM's evaluative frameworks come from within mathematics education research, while the quantitative parameters of BM's evaluation do not necessarily do so. [START_REF] Nilsen | Making sense of implementation theories, models and frameworks[END_REF] framework also includes theories which focus on change from other fields than implementation science, the category of classic theories. However, this category concerns something different than what we call a theory of change, since we regard a theory of change to be something that is locally developed in conjunction with the framework being implemented. The benefits of the close relation between innovation and evaluative framework is that we set us free from binary and simple evaluative categories. In a sense, it is like choosing between relevance and accuracy/circular inference (begging the question/circular argument) in the evaluative frameworks. [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] provided us with a definition of implementation research that has been useful in our effort to begin to explore how implementation research in the field of mathematics education might look. However, this conceptualization of implementation research is not exempt from criticism, nor from competing visions in the conceptualization of this area of research. For instance, one could point out the fact that the conceptualization made by Century and Cassata does not take into consideration-at least explicitly-the role of adopters in the implementation process of an innovation. A conceptualization like this runs at risk of positioning innovation as an imposition for the adopters, who may perceive it as an external element and alien to their own didactical ecosystems. To address this issue, we could consider complementing such conceptualization with other already existing theoretical frameworks from the field of mathematics education. For example, a documentational approach could help us to consider the process through which an innovation can be reshaped and transformed by the adopters.

Concluding remarks

Thus, although implementation frameworks can be useful to focus our attention on key elements of an implementation process and even to conceptualize them, in the case of mathematics education some of these key elements will be of a general nature, while others may be of a more specific nature and connected to mathematics education. Thus, it is feasible to suggest that the implementation research in mathematics education needs to be based on a bricolage of different constructs and theoretical frameworks typical of research in mathematics education, in combination with constructs from implementation of research in other disciplines with a longer research tradition in this regard. However, there is a lot of work to be done to achieve such integration of theoretical frameworks and constructs. A natural step in this effort may be to begin to identify and organize the already accumulated research in the field of mathematics education that addresses aspects of implementation.


Dorte Moeskaer Larsen

Mette Hjelmborg

Bent Lindhart

Jonas Dreyøe

Claus Michelsen

Morten Misfeldt
email: misfeldt@learning.aau.dk

Designing inquiry-based teaching at scale: Central factors for implementation

Keywords: Implementation, inquiry-based mathematics, teacher development

In this paper, we describe and discuss a recent attempt to implement inquiry-based mathematics teaching in Danish compulsory school. After a description of the project and an example of the teaching sequences involved, we will introduce a framework for understanding implementation issues in mathematics education, and use this framework to discuss the problems and potential that our project might encounter when it is considered part of a larger implementation of inquiry-based teaching in Danish mathematics education. We found at least three critical factors for implementation: stakeholders, teacher resources, and the balance between well-prepared and more open teaching materials.

Introduction: Implementation and inquirya story of potential without impact

This paper addresses the implementation of inquiry-based teaching in mathematics. The idea of developing an investigative and problem-oriented approach to teaching mathematics is not new. Even though the discourse around inquiry-based mathematics teaching only dates back to the late 1990s (National Research Council, 1996), the goal of refocusing mathematics education to practice where students are actively investigating and developing mathematical content at play dates back much further (e.g., [START_REF] Dewey | Logic: The theory of inquiry[END_REF][START_REF] Freudenthal | Revisiting mathematics education. China lectures[END_REF][START_REF] Papert | Mindstorms. Children, computers and powerful ideas[END_REF]. Crucial for bringing about changes to mathematics education is dissemination of experiences and processes to a larger group of schools and teachers. This calls for large scale and long-term capacity building providing pedagogical support for teachers so that they can develop the repertoire of skills and understandings required to teach for inquiry-based mathematics. Furthermore, a sustainable impact depends on a good balance of internal and external resources and support [START_REF] Krainer | Inquiry-based learning for students, teachers, researchers, and representatives of educational administration and policy: reflections on a nation-wide initiative fostering educational innovations[END_REF][START_REF] Schoenfeld | A US perspective on the implementation of inquirybased learning in mathematics[END_REF].

More recently, several European development projects have focused on articulating, testing, and implementing inquiry-based approaches to teaching mathematics, science, and engineering (http://www.primasproject.eu, http://www.fibonacci-project.eu).

These projects have generally reported positive results showing the potential of inquiry-based approaches to teaching mathematics. However, despite the fact that Denmark participated in a number of these projects, the findings have not given rise to significant changes in the teaching of mathematics at scale [START_REF] Mogensen | Point-driven mathematics teaching[END_REF][START_REF] Østergaard | Inquiry-based science educationhar naturfagsundervisningen i Danmark brug for det? [Inquiry-based science educationis it needed for Danish STEM teaching?[END_REF]. With this in mind, the Danish government has initiated a three-year project called KiDM (Quality in Danish and Mathematics Education) focused on developing the quality of teaching in both mathematics and first language (Danish).

One of the goals in the KiDM project is to identify and investigate the enablers and constraints for a large-scale implementation of inquiry-based mathematics teaching. This will be addressed in the present paper, where we describe the inquiry-based activities from KiDM and report on how they are received by the teachers, in order to plan and support the implementation. Using a framework for implementation research [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF], we then try to pinpoint important stakeholders and factors that support, hinder, and alter the implementation of the activities of KiDM. In order to discuss problems, potentials, and relevant foci for KiDM as a large-scale implementation project.

Description of "Better Quality in Danish and Mathematics" (KiDM)

KiDM is a three-year, design-based research and development program named "Kvalitet i Dansk og Matematik" (Better Quality in Danish [first language] and Mathematics); its aim is to make the teaching in Danish compulsory school more inquiry-based. The program involves (1) surveying the literature on inquiry in teaching mathematics and Danish, (2) developing inquiry-based teaching activities for a four-month mathematics teaching approach for 4 th and/or 5 th grade implemented at 107 schools, and (3) testing the effect of the intervention with a Random Controlled Trial. The control schools and intervention schools are randomly selected with respect to geography, size and ethnicity. They participated with 2-4 classes each, the intervention concerns all lessons in mathematics for a whole semester. The control schools only participated in the tests.

Initial investigations and design principles

The literature survey [START_REF] Dreyøe | Inquirybased learning in mathematics education: Important themes in the literature[END_REF] as a preliminary investigation of KiDM, was conducted to gain insight into the most important issues and main concerns associated with Inquiry Based Mathematics Education (IBME). A systematic search of six of the highest-ranked journals led to five important themes/issues: 1) The literature survey stresses that communication in the mathematics classroom should be facilitated as open, inquiring, and related to the students' activities with a starting point in the students' prior knowledge. 2) Mathematical skills and competences are critical to participation in inquiry-based teaching. The modeling view of problem activities especially holds the greatest learning potential for students in inquiry-based teaching. Inquiry-based teaching has a positive impact on students' mathematical creativity. 3) The students should be allowed to move in and out of the mathematical domain, and hence it is important to use a wide spectrum of activities in teaching. This requires students to be flexible thinkers and prepares them to cope with situations outside of school. Furthermore, the literature survey contained examples of and knowledge about both 4) tools and resources for planning and implementing inquiry-based learning, and 5) professional development and collaboration. In relation to planning (4), the literature survey suggested that teachers should try to predict the students' answers and prepare general and specific questions to scaffold, extend, and promote the students' ability to generalize mathematical ideas. Regarding development and collaboration (5), it is worth noting that any change to a teacher's approach is a lengthy process. By utilizing pre-planned teaching units, one can contribute to a teacher's ability to reflect on how IBME differs from his/her own approach to teaching.

Apart from surveying the literature, we also interviewed six supervisors in mathematics prior to developing the specific designs in KiDM [START_REF] Michelsen | Research based knowledge about inquiry-based mathematics teaching[END_REF]. Diplomas (60 ECTS) in mathematics supervision have been available since 2009, and many schools have a local supervisor in mathematics who initiates changes and supervises fellow teachers. The supervisors are organized in a national network. The interviews revolved around similar themes and to the need for supporting teacher-teacher and student-student collaboration -"We need time to cooperate inventing and exploring".

From the above, we formulated three principles which implications were important for the didactical intervention of the design for KiDM:

 Principle 1: An exploratory, dialogical, and application-oriented teaching method with room for student participation increases the effect of the student's understanding of mathematical concepts and develops appropriate ways of working.  Principle 2: In order to enhance motivation and learning, we prioritize that the students' experience of the teaching and the content should be meaningful both from an internal mathematical perspective and from the perspective of the situation of application/inquiry.  Principle 3: An exploratory, dialogical, and application-oriented teaching approach with room for student participation increases the possibility of implementing mathematical competences.1

To assess the students' development of mathematical concepts and mathematical competences related to inquiry-based teaching, a pre-and post-test was used in intervention schools and control schools; student and teacher surveys were also conducted focusing on changes in the students' motivation and experiences and the teachers' extent of implementation and experiences, respectively.

The intervention

The inquiry-based mathematics intervention was developed in collaboration with school teachers, supervisors in mathematics, teacher educators, researchers, and professors in an iterative process during which the intervention was tested in a number of cycles in special development schools. The intervention was then tested in a pilot study in 14 schools and adjusted before the actual program began. The program runs for three subsequent semesters in the years 2017-2018. The results for the entire program are expected in 2019.

The KiDM intervention consists of a website that contains a detailed teacher's guide and accompanying student pages with various student activities that take into consideration students' prior experiences, students' opportunity for participation and appropriate ways of working with exploration, applications and mathematical competences especially communication, reasoning, and modeling (principle 1, 2 and 3). In addition to activity descriptions, the teacher's guide also contains specific instructions on how the activities should be introduced, what hints can be given along the way, and how the collective discussion should be designed. The activities are designed to be open for the pupils o engage in mathematizations on a number of different levels. One example of such an activity is to count the number of books in the local library, where the pupils are invited to develop strategies for addressing this question in a structured manner. In Figure 1 we see the activity "How many knots?", where students produce their own data sets by tying knots on a string. The students start by tying as many knots as they can in one minute. Afterwards, every student must count how many knots each have made. In groups of 3-4, students must make charts of the numbers. The assignment is: "If a man comes through the door and asks how many knots a student in this class can tie in one minute, what could you tell him?" In the above example, the students' participation is crucial; they create and use their own data. In a meaningful, explorative way of working they get the opportunity to reason about the data, communicate their result by developing their own model.

All supervisors and teachers in the intervention are required to participate in three project meetings during the semester; they are also required to participate in three group meetings with all school mathematics teachers. All the meetings are clearly facilitated in detail on the website, where short video clips are viewed and discussed in the group meetings. Finally, there is an evaluation sheet that must be completed after each meeting. The group meeting focuses on the establishment of IBME teaching, the action phase in IBME teaching, and the validation phase in IBME education. For instance, the teachers and the supervisors reflected on the validation phase of the lessons they have tried out. The supervisor in mathematics is essential for the implementation by arranging, facilitating and evaluating these structured meetings. Lastly, the intervention teachers, supervisors, and, if possible, the headmasters participate in an all-day introductory meeting focused on the intervention, including both didactical and practical organization of the project, and an all-day evaluation meeting at the regional university. During the intervention, one teacher educator or one researcher will visit the school; this visit may focus either on observation of lessons or participation in a group meeting or project team meeting. The schools receive a stipend for each class participating in the project to cover expenses.

KiDM as an implementation project

In the following section, we will provide some concepts from implementation research and try to use them as a lens to investigate KiDM as an implementation project. We build on the definition of implementation research from the work of [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] as: "systematic inquiry regarding innovations enacted in controlled settings or in ordinary practice, the factors that influence innovation enactment, and the relationships between innovations, influential factors, and outcomes" (p. 170). [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] provided a description of important factors for the implementation of educational innovations. These factors are related to (1) the users, (2) the broader organization, (3) the actual innovation, (4) the strategies supporting its implementation, and (5) the timewise dimension of the implementation. In relation to the individual end users, we should be aware that the changes that educational innovation aims for are mediated by the people involved in the implementation process and aimed at their "change". [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] distinguished the characteristics of individuals that build on their relation to the innovation (such as prior experience with the approach) from the characteristics of individuals that exist independently of the innovation (such as willingness to try new things). What [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] referred to as organizational and environmental factors are both characteristics of the specific setting (e.g., the classroom, the colleagues at the school) and to the broader ecology it sits in (e.g., the municipality or school district). Furthermore, these factors can also be completely outside the specific school or district (e.g., national policies). We can also distinguish between the actual and the perceived attributes of the innovation. This distinction highlights the difference between the explicit blueprints and detailed plans and the subjective (and often diverse) experiences of using these plans. The last factor that [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF] discussed is implementation over time. Time is always a factor in the implementation and diffusion of practices and innovations.

Methods and data

The supervisors where responsible for the local implementation of the KiDM project and they were in charge for the evaluation sheets developed by the project, concerns implementation of KiDM, e.g. the explication of the teacher guide and inquiry-based teaching in general, e.g., key issues and challenges. To understand the status and challenges of KIDM as an implementation project we use the lens from [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF]. In the following section, we use the five factors from above to analyze our project. By reviving and coding the free-text information from the supervisors' evaluation sheets that the local supervisors wrote as part of the documentation of the group meetings and project meetings. The sentences (i.e., citations) where categorized first individually by the first and second author of this paper according to the five factors but also in connection to enablers and constrains in implementation. Afterwards the coding was discussed in the whole group.

Characteristics of individual end users of KiDM

The end users of the KiDM project are teachers of mathematics, supervisors in mathematics at the participating schools, and students. Intervention schools were randomly selected according to a number of parameters, and therefore, we assume that we consider all types of mathematics teachers. However, there was a request that teachers had mathematics as their main subject and teaching experience in mathematics. This request was not met for all schools. Another request was that the school provide a coordinator who is a supervisor in mathematics. Some schools were unable to fulfill this, and their coordinators were participating teachers or the headmaster. We take into consideration the heterogeneity of the teachers by being very explicit with detailed plans in the teacher's guide. However, in terms of explication, there were still challenges, "It is very hard for a teacher who doesn't have math as their main subject to read the teacher's guide". In a similar vein, "Preparation of the activities is hard work. We are only two educated math teachers in a group of six teachers, so we have to do most of the work". Others found the material to be very explicit, "The didactic description is very thorough. A teacher who doesn't have math as their main subject would be able to teach if he had the time and opportunity to read the description".

The teachers are a critical part of the end users and the data we currently have suggest that one important aspect of the project's ability to satisfy this group is to provide enough information so that the instructions are clear (since inquiry-based approaches are new to some of the teachers) while not providing so much information that the workload becomes too great.

Organizational and environmental factors

The project is funded by the Danish Ministry of Education and supported by the Danish Union of Teachers. The school receives a stipend, which covers extra time for preparation and participation in meetings; this is important for the participants, "We have spent a lot of time reading the teacher's guide. If there had been no extra time given in the project, we would not be able to use such an extensive teacher's guide".

The intervention is planned in accordance with the national curriculum, yet some of the participants questioned the extent of this, "Do we meet the needs for the National Test?" and "Are the curriculum in play equivalent to half a year's work?"

As described above, the supervisors play an important role at most of the participating schools since they organize the meetings and local progress in the project. The project meetings are important for cooperation and for shift in collective attitudes: "We have cooperated far more than we have ever done", "Wonderful to 'nerd' with mathematics at the meetings", and "It has been liberating to be able to engage in discussions about mathematics and didactics at the group meetings".

The important organizational factors are economic resources/time, legislative support, and collaborative structures. Furthermore, we can see that the supervisors in mathematics becomes critical as the person who plans and facilitates the activities.

Attributes of the innovation

The degree of specification of what teachers and students must do in KiDM is high. The operationalization is not left to the teachers to perform by themselves. This tight scaffolding can be seen as critical to the success of KiDM. On the homepage, the teachers have access to a teacher guide that includes specific questions to ask in all phases of the activities, the aims of the activities, and agendas for all project meetings and group meetings. The agenda includes questions for discussion, links to short videos to watch and discuss, and tasks to solve and discuss. The teacher's guide has some flexibility in the material, for example, to differentiate between 4 th and 5 th grade or to include students who are especially talented in mathematics.

The teachers in the project have varying attitudes toward this detailed and tight operationalization; some teachers said that having a very strict teacher manual helped them in the beginning, but at the end of the intervention, some of the teachers did not follow the manual entirely.

Implementation support strategies

The implementation strategies in KiDM are very broad. One important strategy is the meetings held at the local schools, where the focus is on discussion, operational planning, and professional development among mathematics teachers at the school. There were various comments about the idea that the teachers in the project must present their work at the larger mathematical group meeting at the local schools -"It does not work. You cannot involve the entire school in something that three teachers participate in"but most of the teachers responded very positively about these meetings -"We have discussed teaching and reflected about teaching and learning much more than we usually do. We hardly ever say, 'How about page 87?'" The supervisor has an important role in these meetings because he/she must facilitate these meetings. If these meeting are not successful, it will affect the implementation at the school. The supervisor in mathematics can therefore be seen as a change agent. In addition, the KiDM teachers are given more time than usual for their lesson preparation and their extra meetings, which can also be considered an implementation strategy in the hopes that this additional resource can support further development of teachers.

Implementation over time

It is not yet possible to determine whether the teachers' methods of inquiry-based teaching have been implemented, as the project is still ongoing. At this point, it is not possible to determine whether the teachers are at a specific developmental stage (from awareness to initial adoption to sophisticated innovation), but some teachers have stated that they already have ideas for how they will implement an inquiry-based approach in the future: "We have applied for a workday where all the math teachers will try to formulate a strategy for how to implement inquiry-based mathematics" and "All classes from the 1 st to the 5 th grade receive one extra hour per week. This hour must be used for inquiry-based mathematics". However, there are other variables that also come into play, such as the involvement of school leadership: "It is a turning point, the head of school is interested", or "We don't have a final agreement for the continuation of the work with the head of school yet". The involvement of the mathematics group at the school is also important over time: "Despite the skepticism at the start (from the teachers not participating in the project), everybody is much more positive about the project as well as discussing didactics topics at the meetings".

Implementation as core stakeholders, resources, and an act of balancing

In answering our question about enablers and constraints for KiDM considered as a large-scale inquiry-based project in Denmark, we see at least three factors that are critical, namely core stakeholders, balance, and resources. The first is the existence of core stakeholders in this change; Denmark is lucky to have a relatively newly educated group of supervisors in mathematics that can act as such. In a survey initiated by the Danish Ministry of Education (Mogensen, Rask, Lindhardt, Østergaard, & Rostgaard, n.d.), supervisors seem to be very influential in implementing government initiatives. The supervisors and group meetings are crucial organizational factors in KiDM. Hence, the supervisors in mathematics have the potential to act as change agents in future large-scale implementations. The second factor that we see as critical is the existence of time and money to support the work. This is stressed repeatedly by the participants. However, rather than just providing more money to the schools, we can see that the visibility of the project resources for the teachers involved is critical, at least in our case. Therefore, necessary resources should be available to the end users (in this case, the teachers). The last factor is that of balance between well-prepared material/structures and the freedom to change and modify. In this project, we have been quite

precise about what has been prepared, and this has worked particularly well when teachers and supervisors adopted the approach and made it their own. These three themes have been in the way that we describe the envisioned output and evaluation criteria of the project and hence as a part of the theory of change of the project.

Figure 1 :

 1 Figure 1: Strings from the activity "How many Knots?"

Mathematical competences(Niss & Højgaard,

2011) is a key concept in the Danish curricular standards.


Ioannis Papadopoulos

Nafsika Patsiala
email: npatsiala@eled.auth.gr

Capturing Problem Posing landscape in a grade-4 classroom: A pilot study

Keywords: Problem posing, primary school, mathematical complexity

In this pilot study, grade-4 students in a Greek primary school are invited to pose problems in three different situations (no feedback, peers' feedback, teacher's suggestions). The aim was to capture the problem posing landscape of the classroom in terms of the complexity of the problems the student pose in these situations. The posed problems were examined in terms (a) of mathematical complexity in the sense of the number of semantic relations they involve, and (b) of linguistic complexity. The findings give evidence that as the students become progressively aware of the initial problem's attributes, they start to pose more complex problems by increasing both the number of problems that involve at least 2 semantic relations and the number of linguistic instances.

Introduction

Problem posing has been recognized as an important intellectual activity in school mathematics and integral part of a balanced mathematics curriculum [START_REF] Hansen | Problem posing from a modelling perspective[END_REF]. The truth is however, that in the school setting students are merely asked to solve problems rather than pose [START_REF] Stoyanova | Extending students' understanding of mathematics via problem posing[END_REF], and therefore they often face difficulties in posing mathematical problems [START_REF] Silver | An analysis of arithmetic problem posing by middle school students[END_REF]. Even though research findings support that students are capable of posing interesting and important mathematical problems [START_REF] Cai | Problem-posing research in mathematics education: Some answered and unanswered questions[END_REF] most of these problems are mainly cognitive undemanding and textbook-like [START_REF] Crespo | What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems[END_REF]. This contradiction makes apparent the potential role of an intervention and many research studies [START_REF] Crespo | Learning to pose mathematical problems: Exploring changes in preservice teachers' practices[END_REF][START_REF] English | Children's problem posing within formal and informal contexts[END_REF] seem to agree that a proper intervention might help students to learn posing problems given that they already have experience in solving problems similar to those they are asked to pose and actually this highlights also the teacher's role in this process.

In this paper, we make a preliminary effort to capture the problem posing landscape in a Greek grade-4 classroom in terms of the complexity (mathematical and/or linguistic) of the generated problems in a variety of situations. The findings will feed the design of a year-long intervention aiming to foster the students' problem posing abilities. Thus, our research question is: How the level of complexity in the problems posed by primary school students varies according to the setting in which the problem posing takes place?

Literature Review and Theoretical Framework

'Are students able to learn to pose problems?' If yes, what is the proper way to achieve that? [START_REF] Brown | The art of problem posing[END_REF] claim that problem posing can be taught. For each problem students can ask what kind of information the problem gives us (known), what kind of information is unknown and what kinds of restrictions are placed on the answer. Then many new problems can be generated if the solvers remove, change or loosen the restrictions. In essence, solvers are asked to list the attributes of the problem, to take an attribute and ask "What-If-Not" that attribute.

Exploring students' problem posing performance is not new [START_REF] English | Children's problem posing within formal and informal contexts[END_REF][START_REF] Silver | An analysis of arithmetic problem posing by middle school students[END_REF]. Research findings reveal that even primary school students are able to pose good/meaningful mathematical problems. [START_REF] Lowrie | Designing a framework for problem posing: Young children generating openended tasks[END_REF] claims that through a supportive and motivating learning environment, grade-1 students increasingly generated more sophisticated problems that were openended in nature. [START_REF] Gade | From problem posing to posing problems via explicit mediation in grades 4 and 5[END_REF] worked with grade-4 and grade-5 students who posed problems in three different stages (formulating written questions, problem posing in dyads and posing problems to one another) using slips of papers. Through these three stages, remarkable progress was made so as in the problems posed by the students as well as in the way they used the slips of papers. [START_REF] Chen | Enhancing the development of Chinese fifthgraders' problem-posing and problem-solving abilities, beliefs, and attitudes: A design experiment[END_REF] implemented a training program aiming to develop grade-4 Chinese students' problem-posing abilities. After the implementation, the problems generated from the students were significantly better proving that elementary students can be taught to pose better and meaningful problems. [START_REF] Cifarelli | Problem posing as reformulation and sense-making within problem solving[END_REF] connect the development of the problem posing abilities with successful problem solving. By focusing on episodes of two grade-4 students they explain how both problem posing and solving coevolve during the solution activity and also how the development of problem posing abilities contributed beneficially in the students' problem-solving abilities, too.

According to the literature the problems posed by students can be more interesting when they have been previously solving similar problems themselves, when the posed problems are addressed to people outside the classroom and when they are prompted by informal instead of formal symbolic contexts [START_REF] Crespo | Learning to pose mathematical problems: Exploring changes in preservice teachers' practices[END_REF][START_REF] English | Children's problem posing within formal and informal contexts[END_REF]. [START_REF] Winograd | Ways of sharing student-authored story problems[END_REF] working with fifth-grade students found that they were highly motivated to pose problems that their classmates would find interesting or difficult without losing their interest during the process of sharing posed problems. Problem posing is rather redefining the way students learn mathematics since they are actively engaged in the learning process by being encouraged to set questions and generate problems instead of being passive receivers of knowledge [START_REF] Brown | The art of problem posing[END_REF].

The problems posed by the students can be evaluated on the basis of three criteria [START_REF] Silver | Assessing students' mathematical problem posing[END_REF]: Quantity, Originality and Complexity. Quantity refers to the number of the generated problems. More specifically, the number of the mathematically correct problems may be indicative of the progress the students accomplished. Originality refers to the number of unusual responses and thus rareness is a way to measure it (they refer to these answers as atypical or original). The third criterion, complexity, can be examined at least from two different perspectives: the mathematical and linguistic complexity. Linguistic complexity focuses on linguistic structures (presence of an assignment, relational and conditional propositions in the statement of the problem). Mathematical complexity is related to the mathematical structure found in the posed problems and one plausible way to measure it is by enumerating distinct semantic relations using a classifications scheme of arithmetic word problems developed by [START_REF] Marshall | Schemas in problem solving[END_REF]. This includes five categories of relations: Change, Group, Compare, Restate, and Vary. The Change schema applies when an initial quantity changes over time (increase or decrease). The Group schema relates to a situation where a number of small quantities are combined into a larger one. The Compare schema involves situations where two things are contrasted with an emphasis on the relation between them (greater, smaller, more, less, etc.). Restate schema involve situations where there is a relation between two variables at a given frame only (e.g., exchange rates). Finally, Vary schema involves a fixed relationship between two variables that persists over time (e.g., relation between euros and cents).

Mathematical complexity has attracted the interest of several researchers on problem posing. According to [START_REF] Silver | An analysis of arithmetic problem posing by middle school students[END_REF] the complexity of the problems posed by the students tend to increase when students generate sequences of chained problems using the results of the simpler problems to pose more complicated ones. [START_REF] Ellerton | Children's made up mathematics problems: A new perspective on talented mathematicians[END_REF] comparing the problem posing abilities of high-and low-ability students found that high-ability students generated more complex problems. Finally, it seems that certain interventions influence the quality of the generated problems [START_REF] Crespo | Learning to pose mathematical problems: Exploring changes in preservice teachers' practices[END_REF][START_REF] English | Children's problem posing within formal and informal contexts[END_REF]. English (1997) conducted an intervention to grade-5 students and found that after the program the students posed a greater number of problems that were also more complex.

Design of the study

This study is the first in a series of small pilot studies aiming to capture the landscape of Greek primary students' problem posing abilities. The students were asked to pose problems in three different settings to compare the variation on the complexity of the problems posed in each setting. Apart from the first setting, the following two settings were developed during the implementation taking into consideration the students' answers. Eighteen grade-4 students from a private school participated in this study. The intervention took place in parallel to the normal teaching of mathematics in the sense that an hour per week is dedicated in math tasks aiming to develop students' mathematical thinking. The students had no prior experience on relevant problem posing activities.

In the first phase, the students were given the starting sentence of a potential problem: "Peter has 75 cents…" and they were asked to complete the problem in as many as possible different ways without having to solve their problems afterwards. [START_REF] Silver | Assessing students' mathematical problem posing[END_REF] consider this kind of activities appropriate for assessing students' problem posing abilities. The teacher did not give any clue about the operations or the mathematical concepts the students could involve in the produced problems. This session was followed by a whole class discussion and the students had the chance to present their examples. One of the students presented an example that initiated the second phase of the study. In his example he actually dropped an attribute of the problem using a "What-If" approach and produced a problem that attracted the attention of his peers. The attribute was the kind of currency and his question was "What if instead of euros the currency was lev?" (the currency of the adjacent country of Bulgaria).

Based on his suggestion the students were asked to pose new problems in the same spirit using the same "What-if" technique of their classmate even though they did not have similar experience before (2 nd phase). No discussion on the notion of listing and dropping attributes took place. Again, a whole discussion followed the session and the students presented their problems.

Finally, during the third phase, certain attributes were explicitly provided to the students by the teacher. More precisely, the teacher asked them to pose problems by considering certain attributes such as the total number of coins, the kind of the coins/currency and the value of coins.

The problem in its incomplete form makes apparent one attribute in an explicit way: the total amount of money. Therefore, students are left to involve and negate more attributes that could potentially be part of the problem such as the number or the kind of coins.

The students' worksheets with the problems they generated during the three phases constituted the data for this study. The written responses were analyzed based on the assessment tools developed by [START_REF] Silver | Assessing students' mathematical problem posing[END_REF] and [START_REF] Marshall | Schemas in problem solving[END_REF]. According to these tools, the responses are organized in three major categories depending on whether the questions included in the problem were mathematical, nonmathematical or whether there was merely a statement in the problem rather than a question. Then, the mathematical questions were evaluated on their solvability (solvable and nonsolvable). Finally, the solvable ones were examined for indications of mathematical and/or linguistic complexity. We determined how many of the five semantic structural relations [START_REF] Marshall | Schemas in problem solving[END_REF] co-existed in each problem. Problems that involved a larger number of semantic relations are considered more mathematically complex compared to those that involve less relations. A comparison followed across all the three phases. All the responses were evaluated independently by the two authors and validity and reliability were established by comparing sets of independent results, clarifying categories until agreement.

Results and Discussion

Phase One

As already mentioned, during the first phase of the study, the students were given the first sentence of a problem and were asked to pose as many as possible problems without feedback neither from their classmates nor their teacher. Each student posed one or two problems. Table 1 summarizes the results of the first phase. All the mathematically complex problems are distributed further according to whether they involved 1, 2 or 3 semantic relations. Moreover, sometimes linguistic and mathematical complexity might co-exist in the same problem and this explains why the partial sums in Table 1 are not in agreement with the total number of solvable problems. As it can be seen, in this phase most of the responses (19 out of 23) were solvable problems exhibiting to a certain degree mathematical and/or linguistic complexity. The problems more or less copy the problems the students are familiar with and are included in their mathematics textbooks. Mainly they were problems including 1 or 2 relations that could be solved in one or two steps using more the addition and/or subtraction operations (instead of multiplication or division). One example with 1 relation was: Peter has 75 cents. He gave 30c to one poor child and later he gave 7c to another. How much are left? (1 relation, Change). Another example including 2 relations was: Peter had 75c. His grandmother gave him 95c but later took 25c from him. His grandfather gave him two more euros but took 125c from him. How many euros will Peter have? (Change, Vary). This problem combines addition/subtraction to find partial results (Change) but at the same time the concurrent use of cents and euros involves converting cents to euros and vice versa (Vary) which means that multiplication and/or division (depended on how the problem will be solved) are also necessary. Another example of mathematically complex problem which exhibited three semantic relations (change, group and vary) exhibiting at the same time linguistic complexity was: Peter has 75c. At New Year's Eve his dad gave him 87€ and 46c. In March he bought 8 gums per 27c each. In April he bought an ice cream for 36€. Two weeks ago, he bought a skateboard for 19€ and 29c. a) How much money did he spend on February in order to have 75c now? b) How much € did all the gums cost? and c) How much did he spend for all of them? The solver has now to interpret the existing information. 'New Year' refers to January and the chain of expenses refer to the period from January to May (the 'two weeks' is related to the time the session took place in school. It was during May). However, February is not mentioned. The initial amount of money is interpreted as the money Peter owns and the end. So, it is a multistep problem involving almost all the four operations and requires thought and interpretation of the information provided by the statement of the task.

There was finally an instance (1 out of 22) the students made a statement. That was: Peter had 75 cents. He wants to buy a toy for 100€. It is obvious that the student did not pose any question.

During the whole classroom discussion one student posed the question (addressing it rather to the teacher than his classmates): 'What if instead of euros the currency was lev? Would you be able to solve the problem?'. As already has been mentioned this question initiated the next phase.

Phase Two

During Phase 2 the students were invited to generate new problems in the spirit of the 'What-if' approach exhibited by their classmate in the previous phase. There was not any explicit reference to the list of the problem's attributes and their negation. The results of this phase are summarized in Table 2. Most of the students' responses were solvable problems (16 out of 21). There was only one problem asking a question that was considered nonmathematical: Peter has 75 cents. What if he loses all of his money and cannot buy anything? Furthermore, no linguistic complexity was detected in the students' responses in this phase. What is interesting is that almost all the students followed the instruction to use the 'What-If" approach but actually they copied the problem of their classmate in the sense that they just focused and/or negated the attribute of currency. For example: Peter has 75 cents. What if Peter had a friend from Japan and he wanted to trade yen with 75c? (restate). Or in another example: Peter has 75 cents. What if his money was alien money. If an alien cent has the same value as 9€ and wants to buy a wallet which costs 49€, how much change will he get? (restate and change). In these responses, the students engage the potential solver with currency conversions (which means true statements for a short period of time), and the solution of the posed problem relies heavily on the relationships between the two currencies. During the whole classroom discussion, the teacher shifted the focus on identifying and negating more attributes of the problem and this initiated the third phase of the study.

Phase 3

During this phase of the study, certain problem's attributes were provided to the students such as the number, value and kind of the coins and they were asked to consider them in order to pose new problems. The results of this phase are summarized in Table 3. An example of a 'What-If' approach considering the kind of coins was: Peter has 75 cents. If he only has 5c coins, how many coins are in his pocket? (vary). It looks like a common problem. However, the choice of division as the necessary operation to solve the problem is not met frequently in the students' responses. An example of a moderate complex problem in this phase is: What if there were also 7€ and 16€ coins and Peter has in his pocket sixteen 7€ coins and three 16€ coins. He spent 5.27€ and then lost 3.25€? How many euros are left? How many cents are they? (Change, Vary, Group). The solver must be familiar with this new element of the weird coins that is not common at our linguistic repertoire and then must relate euros with these new coins and depending on the strategy there will be necessary to use multiplications and/or divisions for the interplay between euros and cents. The students' awareness of the attributes of the problems seems to stimulate both the number and the quality of the posed problems. There was a noticeable increase in the number of the problems (almost 1,5 times more). More precisely, the number of problems that include at least 2 relations was 20 out of 29 problems. This increased number of semantic relations [START_REF] Silver | An analysis of arithmetic problem posing by middle school students[END_REF] indicate the presence of mathematical complexity. Furthermore, in this phase, there was a significant number of instances (8 problems) exhibiting a sense of linguistic complexity: Peter has 75 cents. Is it possible to have only 2c coins? The linguistic element in this problem is that its question differs significantly from the ones posed in the first two phases. The word "possible" must be interpreted correctly given that the participants are grade-4 students and a certain language competency is required to negotiate phrases such as "at least", "possible", "the more" that are included in mathematical problems.

Conclusion

The purpose of this (first among others that will follow) pilot study was to capture the landscape of problem posing in a Greek grade-4 classroom in order to setup a year-long intervention in the classroom aiming to develop the problem posing abilities of the students. The students were invited to pose their problems in three different setting and the results give evidence that given certain circumstances grade-4 students can pose interesting mathematical problems. Almost all the students were able to pose mathematical questions. Only a couple of statements or nonmathematical questions were identified. Moreover, the complexity and the sophistication of the generated problems tend to increase along with the evolution of the sessions.

During the first phase of the study, students posed without any external influence a series of solvable problems. Most of these solvable problems included one or two semantic relations resembling textbook like problems dealing with the four operations and specifically addition and subtraction which is in accordance with [START_REF] Crespo | What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems[END_REF]. There were also a few instances of three relations or linguistic complexity signs that were identified in some of the answers.

In the next phase of the study, a "What-if" technique was used by one of the students who negated the currency attribute of the problem. This was an opportunity to invite students posing new problems under the light of this new approach. The situation was more or less similar to the previous one, and the posed problems were merely mimicking the one posed by their classmate. So, the students did actually drop the attribute of currency not as an action of negating the structural elements of the task but as a way to follow an example that seemed to be successful. No signs of problems' linguistic complexity were identified.

During the final phase of the study, the students became aware of certain attributes that were highlighted by the teacher. This awareness seemingly helped the students to increase the number of posed problems increasing at the same time the number of the involved semantic relations per problem. Also, more evidence of linguistic complexity emerged.

Given the lack of any relevant experience on problem posing and its strategies the students were able to take advantage of the feedback provided during the whole classroom discussion either from their peers or their teacher. This feedback actually shifted their attention to specific attributes of the task. It is very interesting that the students' answers determined the design of the next two phases. This study, being a pilot one, feeds the design of a future intervention aiming to create a class environment that will encourage students to raise questions and pose problems. This small-scale study encourages the idea that a systematic teaching on problem posing may develop students' posing abilities which is in alignment with English (1997).

Table 1 : Results of Phase-1

 1

		Total number of Responses: 23
	Nonmath	Math Questions:	Statements:
	Questions:		22	1
	0
		Solvable:	Nonsolvable:
		19		3
		Math Complex.	Ling Complex.
		19	2
	1 relation	2 relations 3 relations
	8	8	3

Table 2 : Results of Phase-2

 2

		Total number of Responses: 21
	Nonmath	Math Questions:	Statements:
	Questions:		18	2
	1
		Solvable:	Nonsolvable:
		16		2
		Math Complex	Ling Complex
		16	0
	1 relation	2 relations 3 relations
	8	7	1

Table 3 : Results of Phase-3

 3

		Total number of Responses: 32
	Nonmath	Math Questions:	Statements:
	Questions:		31	0
	1
		Solvable:	Nonsolvable:
		29		2
		Math Complex	Ling Complex
		29	8
	1 relation	2 relations 3 relations
	9	16	4