Rikke Maagaard Gregersen

Sine Duedahl Lauridsen

Uffe Thomas Jankvist

Operationalizing implementation theory in mathematics education researchidentifying enablers and barriers in the Swedish "Boost for Mathematics"

Keywords: Implementation research, enablers and barriers, policy dimensions, degree of merit

In this paper, we identify a number of implementation enablers and barriers in the Swedish 'Boost for Mathematics' (BM). Through an analysis of two selected BM modules' use of sensitizing and prescriptive theoretical constructs from mathematics education research, we attempt to link the existence of enablers and barriers to identified policy dimensions of the BM. Besides the two modules on the web portal (the Lärportal), the study's empirical data also consist of surveys of four involved teachers and interviews with two involved researchers and module designers. The analyses also make use of the innovation's novelty and value, the degree of merit in how it is imparted and Rogers' three determinant factors of an innovation.

Introduction

This paper looks at one of the largest initiatives on improving mathematics teaching and learning in the Nordic countries in recent times, namely the Swedish Boost for Mathematics (from now on referred to as BM), which was implemented from 2012 to 2016. The BM was a national professional development initiative with the goal of enhancing students' mathematics performance in Swedish primary and secondary school. The initiative was a reaction to continuously disappointing result in international tests, such as TIMSS [START_REF] Bentley | TIMSS 2007 Swedish pupils' mathematical knowledge[END_REF] and PISA (Utbildningsdepartementet, 2012). BM was designed and developed as a collaboration between the Ministry of Education and Research, the Swedish National Agency for Education, and the Swedish national resource centre for mathematics. The BM involved 260 municipalities, 37,000 mathematics teachers, and had a cost of Skr. 649,000,000 [START_REF] Nyström | Presentation: Matematiklyftetutveckling av kompetensutvecklingskultur och undervisningskultur [Boost for Mathematicsdevelopment of competency development culture and teaching culture[END_REF]. The BM has been evaluated in a handful of reports (e.g., Österholm, Bergqvist, Liljekvist, & Bommel, 2016;[START_REF] Ramböll | Slututvärdering: Utvärderingen av matematiklyftet 2013-2016[END_REF]. In this paper, however, we consider the BM from an implementation research perspective, meaning that we apply theoretical constructs from implementation research to address selected aspects of the implementation. process. From such theory, it is well known that the success of an implementation process to some extent depends on its enablers and barriers, i.e., people or factors that either enhance aspects of the implementation or somehow diminishes them (e.g., [START_REF] Nilsen | Making sense of implementation theories, models and frameworks[END_REF]. Like many other initiatives concerning the improvement of mathematics instruction and learning, the BM drew on and sought to implement research results and findings from mathematics education research. We thus consider the BM as an implementation program in a mathematics education context. The research question that we ask is: What enablers and barriers for implementing mathematics education research results and findings in practice can be determined in the Swedish Boost for Mathematics initiative? The study reported in this paper and the empirical data that it builds upon stem from a master's thesis in mathematics education at the Danish School of Education, Aarhus University [START_REF] Lauridsen | Implementation of mathematics education research results in mathematics education: A case study of two modules from the Swedish Matematiklyft[END_REF].

Theoretical constructs related to implementation research

First, we elaborate on our understanding of implementation and innovations, and then we introduce the terms and constructs applied in the analysis.

"Implementation takes place when an individual puts an innovation into use" (Rogers, 2003, p. 20). But what is an innovation in the context of mathematics education? If we view the BM as an implementation process, mathematics education research results are the innovations. Hence, a clarification of innovation is in order. In the original definition by [START_REF] Schumpeter | The fundamental phenomenon of economic development[END_REF][START_REF] Schumpeter | The fundamental phenomenon of economic development[END_REF], an innovation entails a valuable change in practice. According to [START_REF] Rogers | Diffusion of innovations[END_REF], an innovation, on the other hand, has to be considered novel, not necessarily in common understanding, but for the individual doing the implementation. We shall draw on both positions and define an innovation in the context of mathematics education as: Mathematics education research results that are subjectively novel and bring valuable changes into practice once implemented [START_REF] Lauridsen | Implementation of mathematics education research results in mathematics education: A case study of two modules from the Swedish Matematiklyft[END_REF] taking into account that what is to be considered valuable is of course also to some extent subjective.

To determine enablers and barriers (see [START_REF] Nilsen | Making sense of implementation theories, models and frameworks[END_REF] within the implementation program, we draw upon parts of Rogers' ( 2003) theory of diffusion of innovations. Rogers claims that there are three determinant factors for an innovation to be implemented: the awareness-knowledge of the existence of the innovation; how-to knowledge, which is knowing how to apply the innovation; and experience of relevance of the innovation. Each factor is influenced by an undefined number of variables, which either enable or barricade the implementation [START_REF] Nilsen | Making sense of implementation theories, models and frameworks[END_REF]. The aim of analyzing enablers and barriers is to obtain insight, nuance and challenge the determinant factors. We also use Rogers ' (2003) term principles-knowledge, which refers to understanding how and why the innovation works. Principles-knowledge is not a determinant factor but is seen as a valuable contributor to a successful implementation, hence in this context it can be seen as an enabler for implementation. To gain further insight into barriers and enablers, the BM's theoretical content is viewed as either prescriptive or sensitizing [START_REF] Nilsson | Characterizing theories aimed at supporting teachers' mathematical classroom practices[END_REF]. Prescriptive theories supply the teacher with actions that should be applied in practice, whereas sensitizing theories supply insights into learning processes. Sensitizing theories can be seen as a way to obtain theoretical principles-knowledge, and the prescriptive theories as a way to obtain theoretical how-to knowledge. This comparison allows us to connect theoretical content to the knowledge framework from implementation theory.

To classify the BM, we adopt the notion of policy dimensions by [START_REF] Trouche | Technology-driven developments and policy implications for mathematics education[END_REF]. They use this notion to classify the policies concerning implementations of technological developments in mathematics education. Our use differs slightly, however, as we consider implementations of mathematics education research results. [START_REF] Trouche | Technology-driven developments and policy implications for mathematics education[END_REF] define three policy dimensions: top-down/bottom-up referring to the origin of the implementation process initiative; offering access/supporting integration referring to the extent to which the implementation process is supported beyond supplying access to the research results; and collective/individual referring to whether the process of implementation is conducted with colleagues or not. [START_REF] Trouche | Technology-driven developments and policy implications for mathematics education[END_REF] claim that bottom-up, supporting and collective policies promote implementation, yet in our understanding it must be the variables connected to a policy that act as enablers/barriers.

Educational background, research methods and empirical data

In the BM, groups of mathematics education researches and mathematics teacher educators were asked to author modules for an online open-access resource platform, the so-called Lärportal (https://larportalen.skolverket.se/#/), which now contains more than 30 modules. A module contains eight segments, each working through a four-phase process: (A) individual preparation, where participating teachers read texts and watch videos; (B) collegial work, discussing text and preparing lessons/activities; (C) individual execution of lessons/activities; and (D) collegial evaluation of learnings and classroom experiences. Four didactical perspectives are common to all modules: Classroom norms/socio-mathematical norms; teaching by the mathematical competences, assessment of learning and teaching in mathematics; and routines and interaction in the classroom (Österholm et al., 2016). School leaders and counsellors were educated to support and guide teachers in the process [START_REF] Skolverket | Lärportalen[END_REF]. All Swedish mathematics teachers were granted attendance of two modules. After that, it was up to the individual schools to finance further participation.

The method of the paper is a case study of two selected modules within the BM program: "Algebra module grades 7-9" and the "Problem solving module, grades 7-9" [START_REF] Skolverket | Lärportalen[END_REF]. The main theoretical construct behind the algebra module is that of variation theory [START_REF] Marton | Phenomenography -Describing conceptions of the world around us[END_REF], although it also contains a range of topic specific constructs concerning the learning and teaching of equations, variables and formulas. The main theoretical constructs of the problem solving module are the "five step model" [START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF] for planning and conducting problem solving lessons and the "four phased problem solving process" [START_REF] Skolverket | Lärportalen[END_REF] for organizing students' problem solving.

The empirical data on which this study relies was gathered as part of the previously mentioned master's thesis [START_REF] Lauridsen | Implementation of mathematics education research results in mathematics education: A case study of two modules from the Swedish Matematiklyft[END_REF]. The data was obtained by three different methods:

 Coding of the module content in regard to whether texts and videos were communicated primarily in a prescriptive or sensitizing manner [START_REF] Nilsson | Characterizing theories aimed at supporting teachers' mathematical classroom practices[END_REF].  Interviews with two representative researchers who participated in the development of the algebra module (Researcher 1) and problem solving module (Researcher 2), respectively.  A questionnaire answered by four available teachers, three who had completed the problem solving module (Teachers 1, 2, 3), and one who had completed the algebra module (Teacher 4).

Policy analysis and coding of module content

In regards to the three policy dimensions [START_REF] Trouche | Technology-driven developments and policy implications for mathematics education[END_REF] the dominant policies of the BM are top-down, supportive, and collective, which of the top-down policy is not considered favourable in an implementation process. Despite the top-down policy, the teachers' workflow has been highly organized in consideration to teachers' needs, in terms of influence and practical relevance. We also see this in the developers' (Researcher 1 and 2) awareness on imparting content in the modules towards teachers. This indicates that a top-down policy under certain conditions can support and organize several enablers. The supportive policy of the BM appears, for example, in the financial contribution to participating schools and in the teachers' access to counsellors. The financial support is limited to two modules, which lessens the extent of the support dimensions of the future use of the Lärportal. The BM was primarily influenced by a collective policy, which was central to the design of the Lärportal, with a few individual elements [START_REF] Lauridsen | Implementation of mathematics education research results in mathematics education: A case study of two modules from the Swedish Matematiklyft[END_REF].

The coding of the content in the algebra module determines that it mainly contained sensitizing theoretical constructs and only a few elements of prescriptive theoretical constructs. On the contrary, the problem solving module mainly contained prescriptive theoretical constructs and only a few sensitizing constructs. These conclusions were confirmed by Researchers 1 and 2 [START_REF] Lauridsen | Implementation of mathematics education research results in mathematics education: A case study of two modules from the Swedish Matematiklyft[END_REF]. Please note that the article 'Programming and programming process' has been added to the Algebra module April 2018 and is not included in the coding. When comparing the questionnaires answered by the teachers, some interesting statements about their learning outcomes become apparent. Teacher 4 stated that she mainly developed how she teaches the concept of equality and the meaning of the equality sign. This is particularly notable as the main theory in the algebra module was variation theory [START_REF] Marton | Phenomenography -Describing conceptions of the world around us[END_REF]. For this reason, we analyzed the algebra module for content on the equality sign and the conformity of the presented and used theoretical constructs. We found that there were only a few mentions of actual research results on students' difficulties and misconception on the concept of equality [START_REF] Lauridsen | Implementation of mathematics education research results in mathematics education: A case study of two modules from the Swedish Matematiklyft[END_REF]. [START_REF] Kieran | The learning and teaching of school Algebra[END_REF]

Implementation analysisenablers and barriers

In this section, we analyze the BM for influential variables related to the three determinant factors of considering them as enablers or barriers, and further relate them to the policy dimensions as well as the categorized module content.

Firstly, we analyze for enablers and barriers in relation to awareness-knowledge. Following [START_REF] Rogers | Diffusion of innovations[END_REF], access to mass media is a certain source to knowledge of the existence of the innovations. In the BM, the Lärportal represented a mass media where the teachers were able to achieve awarenessknowledge on a range of mathematics education research. In continuation of this, it may be of a certain importance that the content was conveyed to teachers in a way that was understandable and therefore also accessible. Both Researchers 1 and 2 stressed that it had been an important part in the development of the modules that communication was targeted the teachers as receivers in the Lärportal. This indicates that the ways in which innovations are imparted to the receiver may be considered an enabler of awareness-knowledge. In the section above, we noticed the importance of imparting innovations with a certain degree of merit, to support guidance of teachers. The question is whether the teachers really do achieve awareness-knowledge, if the presentation of the innovation is not communicated with a certain degree of merit, e.g., that the key findings are stressed when research is referred. Hence, the lack of merit of the innovation conveyed, may be regarded as a barrier of the awareness-knowledge and hereby also the following implementation process in the BM.

Secondly, concerning how-to knowledge, teachers may obtain such throughout the implementation process. The Lärportal is a major source of how-to knowledge. Teacher 1, 2 and 4 described using the Lärportal, as they searched for videos and written suggestions for classroom activities to find inspiration during and after the BM. Consequently, the access to a mass media, containing how-to knowledge, can be considered an enabler in the BM. When comparing the types of knowledge in the algebra and the problem solving modules, the latter module mainly contains how-to knowledge (cf. above). Teacher 1 and 2 both implied that they now use the five-step model [START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF] as well as the four-phased process [START_REF] Skolverket | Lärportalen[END_REF] in their practice (cf. above), i.e., the main theoretical constructs of the module. This suggests that mass-media access to how-to knowledge such as task suggestions and videos of classroom examples may be considered enablers. Teacher 4 had not, in the same manner, implemented the main theory, variation theory [START_REF] Marton | Phenomenography -Describing conceptions of the world around us[END_REF], in the algebra module, but she had gained a new perspective on planning classroom presentations focusing on students' misconception. This implies that the teacher had gained some how-to knowledge through the planning and execution of the experimental practice in phase B and C as a way of implementing the principles-knowledge gained in phase A (cf. above). On this account, the experimental practice can also be understood as an enabler for how-to knowledge about the gained principles-knowledge and hereby for implementation. According to [START_REF] Rogers | Diffusion of innovations[END_REF], the more complex an innovation is, the more how-to knowledge is needed. This might be the case for Teacher 4. Complex theoretical constructs, such as variation theory, may become a barrier, when not supported with sufficient how-to knowledge. Another example of this is found in the lack of development of socio-mathematical norms [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF], which is documented in an evaluation of the BM (Österholm et al., 2016). Researcher 2 confirms this:

Researcher 2: [...] it is very easy to say that these norms are important, but how should you act as a teacher to change them? The norms are not something you can get to grips with. My guess is that teachers have been presented with too few concrete examples on how to work with these norms.

Recall that the BM can be characterized as a collective policy. This entails that for the majority of the process, the teacher has access to an interpersonal source of how-to knowledge. This gives teachers access to colleagues' experimental how-to knowledge. [START_REF] Trouche | Technology-driven developments and policy implications for mathematics education[END_REF] argue that collective policies can be a support for implementing a technology. The same argument seems to withstand when it comes to implementing research findings in mathematics education into practice. The collective processes in the BM may, therefore, be viewed as enablers for how-to knowledge.

Thirdly, we analyze enablers and barriers regarding the last determinant factor for implementation [START_REF] Rogers | Diffusion of innovations[END_REF], i.e., the subject's experience of relevance of the innovation. We do so based on two examples of the teachers' work with the two modules. The first example is based on that we previously made it clear how Teacher 4 continued the use of her new knowledge of the equality sign after having ended her work with the BM. Following [START_REF] Rogers | Diffusion of innovations[END_REF], this continuous use of the mathematics didactical knowledge is a sign of a successful implementation. In relation to relevance of the theoretical constructs, Teacher 4 expressed her experience of relevance of the didactical research concerning the equal sign: "...as it is so important and frequently used in mathematics" (our translation). Our second example is from Teacher 3, who verbalized:

Teacher 3: I do not think the Boost for Mathematics was a good "boost". The information it provided was for example the kind of material that we had used in our school for several years [...]. There was no new and inspiring tasks to use, which was sad in the view of it all. (our translation)

We regard this as an expression of a lack of experience of relevance. Teacher 3 was already aware of the presented strategies and theories in her current practice. This is of course not the same as her considering the content of the problem solving module irrelevant, only it was not relevant to her. This indicates that if the teachers are not presented with anything new, i.e., no innovation, it can become a barrier for their experience of relevance of the BM. In our further analyses of Teacher 3's questionnaire answers, we see no indication of a use of theories from the BM. Hence, for Teacher 3 the BM must be considered an unsuccessful implementation.

To understand the enablers and barriers of the teachers' possibilities of having an experience of relevance, we examine the primary differences at the two modules. Recall from the previous section that the theoretical content in the algebra module was mainly sensitizing while it was mainly prescriptive in the problem solving module. It is relevant to consider the extent to which this might influence enablers or barriers in the teachers' experience of relevance. Previously we illustrated the connection between prescriptive theory and how-to knowledge and sensitizing theory and principles-knowledge, respectively. In the second example, Teacher 3 primarily met prescriptive theoretical constructs in the module and thereby descriptions of phases and steps about how to teach problem solving. 'Unfortunately' Teacher 3 already had an existing practice built on these phases and steps. According to [START_REF] Rogers | Diffusion of innovations[END_REF], a principles-knowledge can contribute to a better understanding of why something works. This understanding of 'why' can qualify 'how' one does things in practice. Therefore, a sensitizing theory might possibly have contributed with an understanding as to why Teacher 3 already did as she did in her practice, and thus support further development. In this way, the principles-knowledge can become an enabler of the teachers' experiences of relevance in the BM. In the first example, Teacher 4 primarily met principlesknowledge. This gave her an opportunity to understand why the teaching of the concept of equality is important and not just how she should teach it. This again indicates that principles-knowledge is an enabler for the teachers' experience of relevance in the BM. As mentioned earlier, one of the policies of the BM was a top-down policy. In contrast to a bottom-up policy, this increases the probability of one encountering already known innovations that might already part of one's daily practice. Based on the analyses of Teachers 3 and 4 and in the context of BM's top-down approach, it might be an advantage that the teachers also acquire principle-knowledge, since principlesknowledge may act as an enabler of experience of relevance in the top-down policy context.

Conclusion

In the analysis, we have identified a number of enablers and barriers in the BM and we have attempted to link the existence of these to the identified policy dimensions of the BM as well as the modules' use of sensitizing and prescriptive theoretical constructs from mathematics education research.

In particular, the main findings concern implementation programs with a top-down policy. Within this policy we see that the experience of relevance is challenged if the content is mainly how-to knowledge. We support this claim by the example of Teacher 3 not experiencing anything new in the BM. To accommodate this we argue that principles-knowledge can support the teachers' further insight into the innovation. Such understanding of 'why' may upgrade the quality of 'how' the teachers do things in practice, and make the content relevant for the teachers. Following this argument, we find that principles-knowledge enables the experience of relevance, which is determinant for implementation. With this in mind, we suggest that future mathematical education implementation programs with a top-down policy consider the content balance of how toknowledge and principles-knowledge. Furthermore, it could be an object of investigation in the research community to understand the balance and synergy of knowledge-types in implementation programs.

In addition, the degree of merit that innovations are imparted with influence all three determinant factors. It can be argued that the importance of selecting relevant innovations and how they are conveyed is fundamental to avoid misguidance of teachers' practice. It becomes clear that success of implementation in a mathematics education context is not limited to Rogers' (2003) definition of an innovation merely being put into use; the implementation of the innovation must lead to a change for the betternot least as perceived by the end uses, in our case the teachers. We must take into account [START_REF] Schumpeter | The fundamental phenomenon of economic development[END_REF][START_REF] Schumpeter | The fundamental phenomenon of economic development[END_REF] claim that an innovation should entail a valuable change in practice. This supports the need for defining innovations as subjectively novel, which bring about valuable changes in practice when dealing with implementation processes in mathematics education. Before determining content knowledge in implementation programs, conducting reviews and using solid findings (Education Committee of EMS, 2011), may be one feasible way to accommodate valuable changes.

  is cited, but the key findings on the need for experiences of complex arithmetic of the form ax + b = cx + d is left out. This fact leaves you wondering to what extent the changes in Teacher 4's practice may have been supporting her students' concept formation of equality. It thus becomes a question of merit concerning the accuracy in which research results are imparted to teachers, with which the definite findings and conclusions of the research are referenced in the text.


Manuel Bächtold

David Cross

Valérie Munier Lirdef

How to assess and categorize teachers' views of science? Two methodological issues

Keywords: nature of science, scientific inquiry, philosophy of science, teacher knowledge

This paper addresses two methodological issues related to the assessment of teachers' views of science. The first concerns the distinction between the "nature of science" (NOS) and the nature of "scientific inquiry" (SI): should the points related to NOS and those related to SI be analyzed independently? The second concerns the categorization of teachers' views: is it relevant to analyze them according to a grid of predefined epistemological conceptions? Alternatively, does an empirical approach allow the emergence of epistemological profiles with an overall consistency? To investigate these issues, an empirical study has been performed based on a questionnaire on teachers' views of science (called the "VOS questionnaire") which has been submitted to 160 in-service primary teachers. With respect to the first issue, the analysis of the outcomes brings to light implicative relations between teachers' views on NOS-type items and their views on SI-type items. This calls into question the alleged necessity of separating NOS and SI when assessing teachers' epistemology. Concerning the second issue, a multiple correspondence analysis leads us to identify clusters of teachers with various levels of expertise concerning different epistemological points and without an overall consistency from the point of view of philosophy of science. This suggests that teachers' consistency has to be found not in their view of science taken in isolation but considered in combination with other kinds of knowledge (e.g., pedagogical content knowledge) and their practices of science teaching.

Introduction: two methodological issues

Students' understanding of the "nature of science" (NOS) has been advocated as an "essential part of any science education" (National Research Council, 2012, p. 78): it is assumed to raise students' interest in science, to facilitate learning of science content, and to favor informed decision-making on socioscientific issues (Hodson, 2014). To improve students' understanding of NOS, "explicit" epistemological discussions guided by the teachers about various aspects of science seem to constitute the most efficient approach [START_REF] Akerson | Teaching nature of science explicitly in a first-grade internship setting[END_REF]. In this regard, a necessary condition for teachers to be able to implement such an approach is that they hold themselves an informed view of science [START_REF] Lederman | Nature of science: past, present, and future[END_REF]. This is the first reason why research related to NOS in science education has been concerned with the assessment of teachers' views of science. Another reason is that teachers' view of science is part of a whole set of beliefs and knowledge which might affect their teaching practices, or be affected by them [START_REF] Guerra-Ramos | Teachers' ideas about the nature of science: a critical analysis of research approaches and their contribution to pedagogical practice[END_REF][START_REF] Waters-Adams | The Relationship between understanding of the nature of science and practice: the influence of teachers' beliefs about education, teaching and learning[END_REF]. A better understanding of these complex relationships may be of interest in building more efficient training programs for teachers [START_REF] Anderson | The nature and influence of teacher beliefs and knowledge on the science teaching practice of three generalist New Zealand primary teachers[END_REF].

The very question of how to assess teachers' views of science has been investigated for several decades. In particular, there has been extensive discussion on the epistemological points that should be assessed (Allchin, 2011;Hodson, 2014;Kelly, 2014;Lederman et al., 2002;Matthews, 2012;[START_REF] Mccomas | The principal elements of the nature of science: dispelling the myths[END_REF]. Some authors (Lederman et al., 2002;[START_REF] Mccomas | The principal elements of the nature of science: dispelling the myths[END_REF] have put forward a set of consensual epistemological points which deserve to be studied in the frame of science teaching: for example, the empirical nature of scientific knowledge, its tentativeness, the fact it is theory-laden, or its social and cultural [START_REF] Nott | Probing teachers' views of the nature of science: how should we do it and where should we be looking?[END_REF], teachers are not "professional" philosophers of science; they do not aim at studying scientific activities and build a global and coherent conception of science. As a consequence, it seems disputable to interpret teachers' views of science in terms of the welldefined epistemological conceptions endorsed by the philosophers of science. Alternatively, several researchers like [START_REF] Lederman | Nature of science: past, present, and future[END_REF]their colleagues (2002, 2014) have proposed to characterize teachers' views of science by distinguishing a set of epistemological points and assessing their levels of expertise on each of these points taken in isolation. Such an approach, as a matter of fact, is not subject to the former objection. No global a priori categorization is applied to teachers' views of science. This approach is partly empirical: it allows for the emergence of new epistemological profiles on the basis of teachers' views on a whole set of points. Considering the fact teachers are not professional philosophers of science, a new question then arises: do teachers' epistemological profiles, identified with such an empirical approach, correspond to epistemological conceptions with an overall consistency?

In this paper, we present an empirical study designed to investigate both methodological issues. Our research questions can be recalled as follows:

(1) To what extent does the combined analysis of teachers' views on NOS and SI enable a deeper understanding of their view of science? (2) To what extent do teachers' epistemological profiles, identified by means of an empirical approach, correspond to epistemological conceptions with an overall consistency? To study these two questions, we built a questionnaire including questions on NOS and SI and submitted it to 160 in-service primary teachers. In the following sections, we will first present the epistemological points we chose to assess by means of our questionnaire, before describing the methods for building the questionnaire, for coding teachers' answers, and for validating the questionnaire. We will then present the outcomes gathered with our sample of in-service primary teachers and discuss their implications with respect to both research questions.

Selecting a set of epistemological points to be assessed

The VNOS and the VASI questionnaires are two meaningful and valid tools allowing assessing teachers' views respectively on NOS and SI. In particular, the VNOS questionnaire has been used in the frame of numerous studies, either relative to NOS instruction in the classrooms or relative to teachers' training on NOS. Accordingly, these two questionnaires seem to be suited tools to investigate the first research question, that is, to assess teachers' views on NOS and their views on SI before looking for possible relations.

However, when taking a closer look at the epistemological points assessed by VNOS and VASI (Tables 1 and2), some of these points appear to be very similar.

NOS1: The empirical nature of scientific knowledge NOS2: Observation, inference and theoretical entities in science NOS3: Scientific theories and laws NOS4: The creative and imaginative nature of scientific knowledge NOS5: The theory-laden nature of scientific knowledge NOS6: The social and cultural embeddedness of scientific knowledge NOS7: Myth of the scientific method NOS8: The tentative nature of scientific knowledge Table 1. The epistemological points assessed by VNOS (Lederman et al., 2002) SI1: Scientific investigations all begin with a question but do not necessarily test a hypothesis SI2: There is no single set and sequence of steps followed in all scientific investigations SI3: Inquiry procedures are guided by the question asked SI4: All scientists performing the same procedures may not obtain the same results SI5: Inquiry procedures can influence the results SI6: Research conclusions must be consistent with the data collected SI7: Scientific data is not the same as scientific evidence SI8: Explanations are developed from a combination of collected data and what is already known Table 2. The epistemological points assessed by VASI [START_REF] Lederman | Meaningful assessment of learners' understandings about scientific inquiry: the views about scientific inquiry (VASI) questionnaire[END_REF] Consider first SI6, which states that "research conclusions must be consistent with the data collected." This amounts to saying that the construction of scientific knowledge needs to take account of the observations or experiments that have been performed on the subject under study. In this respect, SI6 can be viewed as a specific statement of NOS1, namely "the empirical nature of scientific knowledge." Note that Lederman and colleagues make very similar statements when explaining the content of NOS1 and SI6: concerning NOS1, they write that "science is at least partially based on observation of the natural world" (2002, p. 499); concerning SI6, they explain that "scientific knowledge is empirically based" (2014, p. 70).

Let us look now at SI4, which states that "all scientists performing the same procedures may not get the same results." A possible explanation of this point is that scientists may interpret the data in the frame of different theoretical frameworks, as Lederman and colleagues emphasize: "Students need to understand that scientific data do not stand alone, can be interpreted in various ways, and 'that scientists may legitimately come to different interpretations of the same data'" (2014, p. 69). This point can be viewed as a direct consequence of NOS5, that is, "the theory-laden nature of scientific knowledge."

The similarity between both questionnaires can also be found in the case of NOS7 and SI2. NOS7, the "myth of the scientific method," is described by Lederman and colleagues as "the belief that there is a recipe-like stepwise procedure that all scientists follow when they do science" (2002, p. 501). This is precisely what is disputed by SI2, which states, "There is no single set and sequence of steps followed in all scientific investigations."

The design of VNOS and VASI as two distinct questionnaires was motivated by the alleged necessity of making a clear distinction between NOS and SI. Paradoxically, these questionnaires do not manage to isolate two independent sets of epistemological points. To some extent, the sets of points which are investigated by means of the two questionnaires are overlapping. This shows how difficult it is to make a sharp distinction between the points related to NOS and those related to SI.

Besides this overlap problem, the choice of the epistemological points made for the VNOS questionnaire has been challenged by several authors (Allchin, 2011;Hodson, 2014;Matthews, 2012): some of these points are disputable from the point of view of philosophy of science, other appear to be too subtle and not essential in the context of NOS education, while several important epistemological points are missing. As a consequence, although we decided to take the VNOS and VASI questionnaires as a starting point of our study, we examined and revised the two sets of epistemological points associated to them (see Tables 1 and2): we avoided the overlap problem by putting aside redundant points (NOS1 being redundant to some extent with SI6, as well as NOS5 with SI4, and NOS7 with SI2); we discarded those which are disputable from the point of view of philosophy of science (NOS3 and NOS4) or which are not essential in the NOS education context (SI5 and SI7); and added several points considered by philosophers of science as depicting essential aspects of science among which three are consensual (i.e., scientists develop models that are distinct from the empirical reality; errors may play a constructive role in the development of science; interactions between scientists actively contributet to the construction and validation of scientific knowledge) and two controversial (i.e., realism versus instrumentalism with respect to the role of models; relativism or rejection of relativism concerning the influence of the socioeconomic and cultural context on the construction of scientific knowledge). Eventually, we selected eleven points (see Table 3; for details on the selection process of these points, see Online resource 1).

Each point is described as referring either to NOS or to SI based on the following criteria: the point under consideration refers to NOS if it concerns the status of knowledge which is already produced and stabilized; the point refers to SI if it concerns the process of developing new knowledge, which therefore is not completely produced and stabilized yet.

Methods

Since we chose a new set of epistemological points, we also had to build a new questionnaire, which we labelled the "VOS questionnaire" (VOS for "View Of Science"). Let us present the method for building and validating it.

Choosing the kinds of questions

This questionnaire was developed in the frame of the research project FORMSCIENCES. Some of the choices made during the construction of this questionnaire resulted from two constraints of the project: the number of teachers involved in the project (over 150) and limited time for answering the questions (between 20 and 30 min). Accordingly, we had to develop a questionnaire to identify the views of science of a large number of teachers in a very short time. We chose to combine open-ended and closed questions. Open-ended questions allow teachers to answer freely on concrete cases; they are assumed to provide a more in-depth understanding of their individual views of science on some points. Closed questions ensure a better rate of answers and are faster to answer and easier to analyze. Note that many studies have brought to light several recurrent answers concerning teachers' views of science so that the state-of-the-art is sufficiently developed to legitimate closed questions on some points. Besides, we included both general and context-specific questions (atomism, astronomy). Each question of the VOS questionnaire provides information for either one or two epistemological points. Conversely, the epistemological points are examined by means of one, two or three questions.

The VOS questionnaire includes 9 questions, some with sub-questions. The relations between the questions and the VOS items are described in Table 4. We present two examples below (the complete questionnaire can be found in Online resource 2). The first example is a context-specific open-ended question dealing with atomism. This question (combined with other questions) is intended to assess teachers' views on two points: reference to empirical support and the role of models.

Q2. Some philosophers in Ancient Greece proposed to describe matter as consisting of atoms. This description was later abandoned for several centuries. A description in terms of atoms was eventually endorsed by scientists at the beginning of the 20 th century. What are the possible reasons, according to you, to explain why this description was accepted at the beginning of the 20 th century? Note that no scientific and/or historical knowledge is required to answer this question.

The second example is a context-dependent closed question that intends (with other questions) to assess teachers' views on several points: reference to empirical support, tentativeness of scientific knowledge, and the role of models.

Q9. (A) In ancient times, Greeks thought that the Sun turned around the Earth (geocentric description). We believe today that it is the Earth that turns around the Sun (heliocentric description). With which one of the following claims do you agree the most (only one possible choice): □ The geocentric and heliocentric descriptions are both models that aim to describe the world, but the heliocentric model is closer to reality. □ The geocentric and heliocentric descriptions are both models that enable the explanation and prediction of observable phenomena, but the heliocentric model enables the explanation and prediction of more observable phenomena. □ The geocentric description is only a model, whereas the heliocentric description is in line with reality. □ None. In this case, explain why:

Method for coding the answers

For each epistemological point, the participant's answers are examined in light of several possible views, either in line with expert views of philosophers of science or in line with novice views indicated in previous empirical studies [START_REF] Abd-El-Khalick | Embedding nature of science instruction in preservice elementary science courses: abandoning scientism[END_REF][START_REF] Cobern | Investigation of pre-service elementary teachers' thinking about science[END_REF][START_REF] Lederman | Nature of science: past, present, and future[END_REF][START_REF] Windschitl | Folk theories of "inquiry:" how pre-service teachers reproduce the discourse and practices of an atheoretical scientific method[END_REF]. For the consensual points, the participant's view is then coded as "novice," "rather novice," "rather expert," "expert," or "depending on the context." For the two controversial points, a specific coding is proposed. We describe the coding method for one epistemological point, namely for VOS1: "the construction of scientific knowledge is based on empirical support" (for the other points, see Online resource 3). The respondent's view on this point is inferred from her/his answers to Q2 and Q9B. In the case of Q2 (which is related to atomism), we distinguish three possible answers: (a) answer with no reference to experience; (b) answer associated with naïve realism (i.e., with reference to experience but without the idea that the atomist description is a model and with the idea that experience can verify or reject with certainty the atomist description);

(c) answer associated with balanced empiricism (i.e., with reference to experience and with the idea that the atomist description is a model, and/or fulfills an explicative or predictive functions, and/or is a tentative description of matter). In the case of Q9B (which is related to the geo-and heliocentric representations), we distinguish three possible answers: (a) answer associated with naïve empiricism (at least answer #2); (b) answer associated with balanced empiricism (answer #1 or answers #1 and #3); (c) answer not interpretable in terms of naïve or balanced empiricism (answer #3). The view of the respondent concerning VOS1 is then coded as follows: "novice" if {Q2: a or b} and {Q9B: a}, "rather novice" if ({Q2: a or b} and {Q9B: c or NA}) or ({Q2: NA} and {Q9B: a}), "rather expert" if {Q2: c} and {Q9B: c or NA}, "expert" if {Q2: c} and {Q9B: b}, "depending on the context" if ({Q2: a or b} and {Q9B: b}) or ({Q2: c} and {Q9B: a}).

Validity of the questionnaire

To ensure that teachers did not assign a different meaning to the words of the questions, the latter was first discussed with 3 in-service teachers during interviews with an average duration of 1 h. We modified some terms that were ambiguous and could be misinterpreted by teachers. A new version of the VOS questionnaire was submitted to 24 pre-service primary teachers during a course session that aimed to introduce some key ideas of philosophy of science. A collective discussion was conducted to ensure good understanding of the questions and to propose some supplementary modifications of formulations.

We then submitted this questionnaire to 10 experts in the field of philosophy of science and science education. For the consensual epistemological points, at least 8/10 of these respondents were identified as being "expert" or "rather expert." This result confirms that the assessed points are indeed consensual among experts and provide validation of the questionnaire.

Participants

We administered the VOS questionnaire to a sample of 160 in-service primary teachers, mostly female (~ 70%), coming from 4 different regions of France, and teaching in grades 3, 4, or 5. Their average declared time for teaching science in their classrooms was approximately 1.3 h per week. They had been teaching in primary school for 16 years on average (with an average age of 47 years old). Approximately 40% of them had a degree in science.

Reliability of the coding process

To ensure reliability of the coding process for the open-ended questions, every answer from the 160 teachers was analyzed either by two or three researchers simultaneously. For each answer, the researchers first performed their own analysis before checking whether they agreed on the coding. In case of disagreement, these answers were discussed collectively until agreement and confidence on the coding were reached. Moreover, we presented the coding method to a researcher external to the project, with expertise in philosophy of science. He coded the answers of 20 teachers randomly chosen among our sample. A high degree of interrater agreement (86%) was obtained.

Outcomes

We first provide the distribution of teachers' views for each epistemological point. We then present the outcomes of a statistical analysis intended first to determine if teachers' views on NOS-type items and their views on SI-type items are independent or not, and second to investigate the overall consistency of teachers' epistemology.

Distribution for each epistemological point

The distribution of teachers' views for each VOS item is given in Table 5 Concerning most consensual epistemological points, we observe that teachers' views are distributed in a relatively balanced way between novice and expert views. Concerning one point, VOS1, most teachers are (rather) novice. On the contrary, concerning two points, VOS3 and VOS5, most teachers are (rather) experts.

We note also that a significant proportion of teachers provide answers depending on the context under consideration: the views of some teachers on VOS1, VOS4, VOS7, and VOS9 differ if the context is related to the atomist description or to the geo-and heliocentric descriptions.

Considering the two controversial epistemological points, we observe that half of teachers are naïve realists with respect to the role of models (VOS10), while the others are either balanced realist or instrumentalist on this point. It appears also that more than half of teachers are (rather) strong relativist with regard to the influence of the socio-economic and cultural context on the construction of scientific knowledge (VOS11).

Statistical analysis

To further analyze the teachers' answers to the VOS questionnaire, we performed a statistical analysis of the data. To avoid circularity in the outcomes of this analysis, we combined items VOS 4, 7, and 10, which are associated with the same series of questions, into a unique indicator called VOS "Model." These items were submitted to a multiple correspondence analysis with hierarchical clustering on principal components [START_REF] Greenacre | Multiple correspondence analysis and related methods[END_REF].

The results show that teachers' answers can be assembled into four clusters, labelled Models 1, 2, 3, and 4, which correspond to four levels of expertise: from less expert (Model 1) to most expert (Model 4).

We then performed an implicative statistical analysis [START_REF] Gras | Implicative statistical analysis[END_REF], which allows answering the following question: given two binary variables a and b within a given population E, what is the measure according to which "if a is true" then "b is also true"? (Figure 1). The numbers corresponds to a measure of implication. White bubbles are NOS-type items, and black bubbles are SI-type items.

This statistical analysis brings to light several relations between NOS-type items and SItype items. For instance, we can observe implications between Model (NOS-type item), on the one side, and VOS3 or VOS5 (both SI-type items), on the other side. Likewise, there is an implication between VOS6 (NOS-type item) and VOS11 (SI-type item). More precisely, teachers who believe that the socio-economic and cultural context has a strong influence on the construction of scientific knowledge are likely to acknowledge the theory-ladenness of empirical data. Such an implication, which is meaningful from the point of view of It is also noteworthy that VOS6 and VOS11, which appear here to be linked statistically, are consensual and controversial epistemological points, respectively. This result shows the potential insight that questions on controversial aspects of science might provide to understand teachers' views of science.

Besides, we observe implications between some of the items which display a consistency in terms of the level of expertise. For instance, teachers who are rather expert concerning the nature and role of models are very likely to be rather expert concerning the interactions between science and the socio-economic and cultural context; or teachers who are novice concerning the nature of questions and problems are very likely to be novice concerning the nature and role of models. However, the results also show that some teachers can answer as experts concerning one item but as novices concerning another item. For example, teachers who are rather expert on VOS 9 are very likely to be novice on VOS 1, and teachers who are rather novice on VOS 9 are very likely to be expert on VOS 5. This last example may appear incoherent to some extent from the point of view of philosophy of science. Indeed, both VOS5 and VOS9 refer to a social dimension of the development of science, even though these social dimensions do not lie at the same level (VOS5 being at the level of the global interaction between science and society, whereas VOS 9 being at the level of the social interactions between scientists). This part of the results argues against a consistency within teachers' views of science from a strictly epistemological point of view.

Discussion

Let us consider our first research question 1: to what extent the combined analysis of teachers' views on NOS and SI enables a deeper understanding of their view of science? A preliminary remark concerns the questionnaires designed to asses these points: it appears that the sets of epistemological points assessed by the VNOS and VASI questionnaires (designed by Lederman and colleagues) are overlapping to some extent; in other words, it is difficult in practice to draw a sharp distinction between the points related to NOS and those related to SI. Beyond this conceptual difficulty, the statistical analysis of our data gathered with in-service primary teachers reveal the existence of several relations between their views on NOS and their views on SI. For instance, teachers who consider that the construction of scientific knowledge is strongly influenced by the socio-economic and cultural context (SI-type item) also tend to view the empirical data as theory-laden (NOS-type item). This calls into question the alleged necessity of separating NOS and SI when assessing teachers' epistemology. Furthermore, some of the relations identified empirically appear meaningful from the point of view of philosophy of science. This can be explained by the close relation, emphasized by Hodson (2014), between the nature of scientific knowledge and the methods for constructing this knowledge. As a consequence, our results show that assessing and analyzing teachers' views jointly on NOS-type and SI-type items may favor a deeper understanding of teachers' views of science.

The outcomes of this empirical study also shed light on the second research question: teachers' views of science, taken globally, cannot be located on a scale of expertise nor can they be associated with well-defined profiles with an inner consistency from the point of view of philosophy of science. As Guerra-Ramos (2012, p. 648) puts it: "generally speaking, [teachers] hold eclectic or mixed views which do not mirror consistently a single particular view or a particular philosophical position."

For all that, teachers are not incoherent individuals. Recall that they are not "professional" philosophers of science [START_REF] Nott | Probing teachers' views of the nature of science: how should we do it and where should we be looking?[END_REF]; they do not aim at developing a global and consistent conception of science for itself; they are concerned primarily with science teaching, and that, in various domains which are possibly disconnected. Consistency might therefore be found not in teachers' views of science taken in isolation but in their views of science considered in association with their teaching practices, and accordingly with other kinds of knowledge required for teaching, such as subject matter knowledge or pedagogical content knowledge [START_REF] Shulman | Those who understand: knowledge growth in teaching[END_REF]. In this regard, another interesting outcome of our study is that teachers' views on some VOS items appear to be context dependent. For instance, for some teachers, their view on the nature of models is different when considering either the atomist model or the heliocentric model. Note that the atoms which are microscopic elements of matter can never be observed directly, unlike the Earth and the Sun which are macroscopic and hence observable systems. This ontological difference between the systems of both models may influence teachers' answers. Another part of the explanation of the contextdependency might be the fact that many primary teachers of our study are used to teach the heliocentric model in their classrooms (in accordance with the official curriculum), which is not the case regarding the atomist model. Accordingly, this outcome suggests that teachers' views of science are situated to some extent in their teaching practices. This finding provides support for the idea of a mutual influence between teachers' views of science and their teaching practices, as suggested by former studies [START_REF] Guerra-Ramos | Teachers' ideas about the nature of science: a critical analysis of research approaches and their contribution to pedagogical practice[END_REF][START_REF] Waters-Adams | The Relationship between understanding of the nature of science and practice: the influence of teachers' beliefs about education, teaching and learning[END_REF].

In light of these outcomes, we argue for an assessment method of teachers' views of science which combines a theoretical approach that makes use of the literature in philosophy of science to identify a whole set of possible epistemological points related both to NOS and SI, and an empirical approach that allows the emergence of possibly unexpected epistemological profiles for which the consistency must be considered in relation to the teaching practices and to other knowledge. We believe that this relationship is complex and difficult to grasp and therefore requires investigation using a qualitative and contextualized approach. Shedding light on this complex relationship and the ways it is built might be of great interest for improving teachers' training programs by making them more suited to both teachers' current knowledge and practices.

Online resource 1 Selection process of epistemological points for the VOS questionnaire

The epistemological points of the VOS questionnaire were selected as follows: we examined the two sets of epistemological points associated to the VNOS and VASI questionnaires and avoided the overlap problem by putting aside redundant points; we discarded those which are disputable from the point of view of philosophy of science or which are not essential in the NOS education context; and added several points considered by philosophers of science as depicting essential aspects of science and put forward by some researchers in the NOS literature.

From the set of epistemological points associated to the VNOS and VASI questionnaires, we maintained the six following points: "scientific knowledge is based on empirical support" (VOS1) which is identical to NOS1; "there is no single scientific method" (VOS2) which is similar both to NOS7 and SI2; "scientific investigations are motivated and guided by questions and problems" (VOS3) which is similar to SI3 and refers partly to SI1; "scientific knowledge is always tentative" (VOS4) which is identical to NOS8; "science interacts with the socio-economic and cultural context" (VOS5) which is similar to NOS6; "The empirical data on which scientific knowledge is based are always theory-laden" (VOS6) which is similar to NOS5 and may be related to SI4 (see above).

We did not maintain two points of VNOS (NOS3 and NOS4) because we considered them as disputable from the point of view of philosophy of science. NOS3 puts forward a distinction between "scientific theories and laws," which is based on the definition of laws as "descriptive statements of relationships among observable phenomena" (Lederman et al., 2002, p. 500). However, this definition holds merely for phenomenological laws, such as Boyle's law mentioned by Lederman and colleagues, but not for theoretical laws [START_REF] Cartwright | How the laws of physics lie[END_REF], that is, laws with a wider scope that are the result of a more theoretical reasoning and are only indirectly related to experience (e.g., Newton's laws of motion or Schrödinger's equation). Such laws may constitute the core of a theory. Hence, for such laws, it seems difficult to make a strong distinction between laws and theories. The second disputable point concerns "creativity" (NOS4), presented as a feature of science contrasting with "rationality" (Lederman et al., 2002, p. 500). Some usual kinds of reasoning in science have a creative power, such as analogical reasoning [START_REF] Hofstadter | Surfaces and essences: analogy as the fuel and fire of thinking[END_REF]. Making an analogy, which amounts to describe new phenomena by referring to well-established knowledge, may be considered as a rational process. This type of creativity in science cannot be opposed so easily to rationality. Therefore, we did not select these two points in our questionnaire.

We discarded two points of VASI (SI5 and SI7), because we considered them as too subtle and specific in the context of NOS education. SI5 states that "inquiry procedures can influence the results," what can be related to an idea emphasized by the philosophers of science belonging to the new experimentalism movement: the instruments used by scientists do not merely test theories but have an active role insofar as they determine the way we interact with the world and because they may produce new phenomena [START_REF] Hacking | Interpreting and intervening: introductory topics in philosophy of natural science[END_REF][START_REF] Pickering | The mangle of practice: time, agency, and science[END_REF]. Even if the acknowledgement of this idea was as an important step in the field of philosophy of science, it seems quite difficult to make sense of it in the context of science education. According to SI7, "scientific data are not the same as scientific evidence." Although scientific evidence may imply some further interpretation not implied in the scientific data, both of them are to some extent theory-laden. In this regard, the difference between data and evidence is quite subtle.

Besides, in line with Allchin (2011), Hodson (2014), Matthews (2012), we chose to include five other epistemological points which correspond to essential aspects of science. These points may be of importance with respect to NOS education and are possibly relevant for investigating the relationships between teachers' views of science and their science teaching practices. Three of them are consensual in philosophy of science. The first point concerns the nature of models which recently have drawn much attention both in philosophy of science [START_REF] Frigg | Models in science[END_REF][START_REF] Hesse | Models and analogies[END_REF][START_REF] Varenne | Théorie, réalité, modèle: épistémologie des théories et des modèles face au réalisme dans les sciences[END_REF] and science education [START_REF] Gobert | Examining the relationship between students' understanding of the nature of models and conceptual learning in biology, physics and chemistry[END_REF][START_REF] Gogolin | Students' understanding of the nature and purpose of models[END_REF]. Among the various epistemological aspects of models, we chose to assess the following one: "Scientists develop models that are distinct from the empirical reality" (VOS7). Note that this point can be viewed as a specific instance of NOS2. It seems worthwhile that the various models studied in science teaching are not taken by students as mere copies of the empirical reality (Harrison & Treagust, 2000), but identified as intellectual constructs which can be empirically tested and may be revised or replaced by new models (Gilbert, 2004).

The second point concerns the role of errors: "Errors may play a constructive role in the development of science" (VOS8). This point has first been emphasized by [START_REF] Bachelard | La formation de l'esprit scientifique: contribution à une psychanalyse de la connaissance[END_REF][START_REF] Bachelard | La formation de l'esprit scientifique: contribution à une psychanalyse de la connaissance[END_REF]) who described the advancement of science as a constantly renewed process of identification of errors within the current knowledge and "rectification" of these errors. Similarly, [START_REF] Kuhn | The structure of scientific revolutions[END_REF] has provided many examples of new theories (e.g., quantum mechanics) of which the development has been stimulated by "anomalies" (e.g., the dark body spectrum) identified in the frame of former theories. More recently, [START_REF] Kipnis | Errors in science and their treatment in teaching science[END_REF] and [START_REF] Allchin | Teaching the nature of science through scientific errors[END_REF] have stressed that errors deserve to be studied with students since they are natural and unavoidable part of scientific process: in particular, they display the "variety of ways for scientific development" (Kipnis, 2011, p. 681), and "convey the tentativeness of science" (Allchin, 2012, p. 906).

A third point, which has been studied in depth in sociology and philosophy of science but which is missing in VNOS and VASI, concerns the role of the interactions between the scientists: "Interactions between scientists actively contribute to the construction and validation of scientific knowledge" (VOS9). Without considering in details the complex process of social interactions between scientists as brought to light both by sociologists and philosophers of science [START_REF] Longino | The social dimensions of scientific knowledge[END_REF]Pestre, 2006), we refer here to the practices of construction and validation of new knowledge based on argumentation, evaluation and critique which are developed in the frame of oral or written interactions between scientists [START_REF] Duschl | Supporting and promoting argumentation discourse in science education[END_REF][START_REF] Jiménez-Aleixandre | Epistemic practices and scientific practices in science education[END_REF]Kelly, 2014).

The nine points formulated above are consensual in the field of philosophy of science. Following the suggestion of Matthews (2012) and Hodson (2014), we propose to include two other points which concern controversial aspects of philosophy of science. Recall that in our study we are looking for the consistency of teachers' views of science; in this regard, we can wonder if their views on controversial points might not be instructive. This deserves to be tested by means of our questionnaire. We chose two controversial aspects which are related to the former epistemological points. The first one (VOS10) concerns the role of models and brings into opposition the two following views: the models aim at describing the world (realist view of models); or they aim at explaining and predicting observable phenomena (antirealist or instrumentalist view of models) [START_REF] Frigg | Models in science[END_REF][START_REF] Hesse | Models and analogies[END_REF][START_REF] Varenne | Théorie, réalité, modèle: épistémologie des théories et des modèles face au réalisme dans les sciences[END_REF]. The second one (VOS11) develops the question of the interaction between science and the socio-economic and cultural context (VOS5): does this context have an influence on the construction of scientific knowledge (e.g., on the formulation of a scientific hypothesis or on the interpretation of an empirical outcome) [START_REF] Longino | The social dimensions of scientific knowledge[END_REF]Pestre, 2006)? On this question, one may support or reject a "(strong) relativist" view.

Online resource 2

The questions of the VOS questionnaire Q1. Two groups of scientists dealing with the same question gather the same data. Do you think they will come to the same conclusions? □ No, not necessarily, because in some cases the data are not sufficient. But with complementary data, the scientists would come to the same conclusions. □ Yes, because if the data are the same, the scientists must come to the same conclusions. □ No, not necessarily, because however large the set of data may be, there are always several possible conclusions. □ I don't know.

Q2. Some philosophers in Ancient Greece proposed to describe matter as consisting of atoms. This description was later abandoned for several centuries. A description in terms of atoms was eventually endorsed by scientists at the beginning of the 20 th century. What are the possible reasons, according to you, to explain why this description was accepted at the beginning of the 20 th century? Note that no scientific and/or historical knowledge is required to answer this question.

Q3. Do you think a scientific approach involves: always often sometimes never I don't know emitting hypotheses

□ □ □ □ □ measurements □ □ □ □ □ a preliminary step of observation □ □ □ □ □ the statement of a problem □ □ □ □ □ tests of hypotheses □ □ □ □ □ building a model □ □ □ □ □ an analysis of the current works □ □ □ □ □ data processing □ □ □ □ □ a theoretical construction □ □ □ □ □ experiments □ □ □ □ □ computer-assisted simulations □ □ □ □ □ asking questions □ □ □ □ □ Q4.
Ideally, an experimental approach is closer to approach 1 or 2? to approach 1 □ to approach 2 □ there is no ideal experimental approach □ I don't know Q9. (A) In ancient times, Greeks thought that the Sun turned around the Earth (geocentric description). We believe today that it is the Earth that turns around the Sun (heliocentric description). With which one of the following claims do you agree the most (only one possible choice): □ The geocentric and heliocentric descriptions are both models that aim to describe the world, but the heliocentric model is closer to reality. □ The geocentric and heliocentric descriptions are both models that enable the explanation and prediction of observable phenomena, but the heliocentric model enables the explanation and prediction of more observable phenomena. □ The geocentric description is only a model, whereas the heliocentric description is in line with reality. □ None. In this case, explain why: (B) The geocentric description has been abandoned because (you can check one or several answers): □ it is unable to account for some phenomena in a simple manner.

□ it has been shown to be false. □ Copernicus and Galileo brought new ideas. □ other

Online resource 3

The coding method of the VOS questionnaire (a) answer suggesting the respondent is unaware of the diversity of the kinds of investigations and believes that making science always involves experimentation ("always" is checked for "measurements" and/or "experiments") (b) answer suggesting the respondent is or may be aware of the diversity of the kinds of investigations ("always" is checked neither for "measurements" nor for "experiments") Q3':

(a) answer associated to naïve inductivism, i.e., suggesting the respondent believes in the primacy of observation ("always" is checked for "a preliminary step of observation") (b) answer suggesting the respondent is not endorsing naïve inductivism ("always" is not checked for "a preliminary step of observation") Q4:

(a) answer suggesting the respondent believes that making science amounts to follow a well-defined and ordered set of steps (answer #1) (b) answer suggesting the respondent acknowledges that scientists can follow various sets and orders of steps (answer #2 or #3) (c) answer #4 (I don't know)

-novice if answer (a) to all three questions -rather novice if (answer two times (a) answer suggesting the respondent is unaware of the role of questions and problems in science ("always" is not checked neither for "the statement of a problem" nor for "asking questions") (b) answer suggesting the respondent is aware of the role of questions and problems in science ("always" is checked for "the statement of a problem" and/or for "asking questions")

-novice if {Q3'': a} -expert if {Q3'': b} -NA 20 VOS4 Scientific knowledge is always tentative Q2':
(a) answer suggesting the respondent believes that the atomist description has been proven definitively, and/or especially that experience can verify with certainty the atomist description (e.g., with the use of words like "verify," "prove," or "demonstrate") (b) answer suggesting the respondent is aware of tentativeness of the atomist description, and/or especially of the limited role of experience (e.g., with the use of words like "support," or "corroborate") Q9A:

(a) answer suggesting the respondent believes that the heliocentric description has been proven to be true (answer #3) (b) answer suggesting the respondent is aware of the tentativeness of the heliocentric description (answer #1 or #2) (c) answer #4 (None) (a) answer suggesting the respondent is unaware of the interaction between science and society (answer "no" or "I don't know") (b) answer suggesting the respondent is aware of this interaction (answer "yes, a little" or "yes, strongly") Q8A:

(a) answer suggesting the respondent is unaware or hardly aware of the kinds of interactions that may exist between science and society (answer "no" or "I don't know" for every item, or "yes" to only one item) (b) answer suggesting the respondent is aware of the kinds of interactions that may exist between science and society (answer "yes" at least for two items) Q8B:

(a) answer suggesting the respondent is unaware of the kinds of interactions that may exist between science and society (answers "no" or "I don't know" for every item) (b) answer suggesting the respondent is aware of the kinds of interactions that may exist between science and society (answer "yes" at least for one item)

-novice if answer (a) to all three questions -rather novice if answer (a) to one or two among the three questions -rather expert if answer (b) to one or two of three questions and NA to the other(s) -expert if answer (b) to all three questions -NA

VOS6

The empirical data on which scientific knowledge is based are always theoryladen Q1:

(a) answer suggesting the respondent is unaware of theory-ladeness (answer #1, #2, or #4) (b) answer suggesting the respondent is aware of theory-ladeness (answer #3) -novice if {Q1: a} -expert if {Q1: b} -NA

VOS7

The models built by the scientists are distinct from the empirical reality Q2'':

(a) answer suggesting clearly the respondent is taking the atomist description as the reality (b) answer which does not suggest clearly the respondent is taking the atomist description as the reality, nor he is aware that the atomist description is a model (e.g., with no use of words like "model" or "representation") (c) answer suggesting the respondent is aware that the atomist description is a model (e.g., with the use of words like "model" or "representation") Q9A:

(a) answer suggesting the respondent takes the heliocentric description as the reality (answer #3) (b) answer suggesting the respondent is aware that the heliocentric description is a model (answer #1 or #2)

-novice if ({Q2'': a} and {Q9A: a or NA}) or ({Q2'': b or NA} and {Q9A: a}) -rather expert if {Q2'': b or NA} and {Q9A: b} -expert if {Q2'': c} and {Q9A: b} -depending on the context if ({Q2'': c} and {Q9A: a}) or ({Q2'': a} and {Q9:b}) -NA if {Q2'': b} and {Q9A: NA} 21 VOS8 Errors may play a constructive role in the development of science Q5 (multiple choice question): (a) "never" or "I don't know" is checked (b) "sometimes," "often," or "always," is checked Q5' ("Explain why"): (a) answer associated to strong relativism ("yes" and "always" are checked) (b) answer associated to moderate relativism ("yes" and "sometimes" or "often" are checked) (c) answer associated to rejection of relativism ("no" is checked) (d) "I don't know" or NA Q8A':

(a) answer associated to strong relativism ("yes" is checked at least for "on the interpretation of the outcomes" and/or "on the choice of the hypotheses") (b) answer associated to moderate relativism ("yes" is checked only for "on the choice of the topics being investigated" and/or "on the allocation of human and financial resources") (c) answer associated to rejection of relativism ("no" is checked for all item)

-strong relativism if answer (a) to both questions -rather strong relativism if ({Q7: b, d or NA} and {Q8A': a}) or ({Q7: a} and {Q8A': NA}) -rather moderate relativism if ({Q7: a or d} and {Q8A': b}) or ({Q7: b} and {Q8A': NA}) -moderate relativism if {Q7: b} and {Q8A': b} -rejection of relativism if ({Q7: c or d} and {Q8A': c}) or ({Q7: c} and {Q8A': NA}) -NI if (Q7: c} and {Q8A': a or b) -NA

Figure 1 .

 1 Figure 1. Implicative graph displaying relations between teachers' views on the VOS-items.The numbers corresponds to a measure of implication. White bubbles are NOS-type items, and black bubbles are SI-type items.

  (a) and one time (b), NA or (c) to the set of questions Q3, Q3' and Q4) or (answer one time (a) and one time (b) and one time NA or (c) to the set of questions Q3, Q3' and Q4) -rather expert if answer one time (a), NA or (c) and two times (b) to the set of questions Q3, Q3' and Q4 -expert if answer (b) to all three questions -

  (a) answer which puts forward the negative role of errors(b) answer which claim the constructive role of errors in science but are not able to explain why (c) answer which claim the constructive role of errors in science and are able to explain why (d) not interpretable -novice if ({Q5: a} and {Q5': a, d or NA}) or ({Q5: NA} and {Q5': a}) -rather novice if ({Q5: a} and {Q5': b or c}) or ({Q5: b} and {Q5': a, b or NA}) or ({Q5: NA} and {Q5': b}) -rather expert if {Q5: NA} and {Q5': c} -expert if {Q5: b} and {Q5': c} -NI if {Q5: b} and {Q5': d} -"not so important," "not important,' or "I don't know" is checked (b) "very important," or "important" is checked Q6' ("Why?"): (a) no answer or answer with none of the two following ideas (b) answer with the idea that these interactions enable to communicate knowledge or that they enable cooperation or confrontation for the construction of knowledge (c) answer with both ideas Q9B: (a) answer suggesting the respondent believes that the evolution of knowledge is due to isolated genius (answer #3 only) (b) otherwise (including NA) -novice if {Q6: a} and {Q6': a or NA} (whatever the answer to Q9B) -rather novice if ({Q6: a} and {Q6': b or c}) or ({Q6: b} and {Q6': a or NA}) (whatever the answer to Q9B) or ({Q6: NA}, {Q6': NA} and {Q9B: a}) -rather expert if ({Q6: b}, {Q6': b} and {Q9B: b}) or ({Q6: NA}, {Q6': c} and {Q9B: b}) -expert if {Q6: b}, {Q6': c} and {Q9B: b} -depending on the fact the question is contextualized or not if {Q6': b or c} and {Q9B: a} (whatever the answer to Q6) -NI if {Q6: NA}, {Q6': NA} and {Q9B: b} answer associated to naïve realism (answer #3) (b) answer associated to balanced realism (answer #1) (c) answer associated to instrumentalism (answer #2) -naïve realism if {Q9A: a} -balanced realism if {Q9A: b} -instrumentalism if {Q9A: c} -

Table 3 .

 3 The epistemological points selected for the VOS questionnaire

	VOS 1	Scientific knowledge is based on empirical support	NOS
	VOS 2	There is no single scientific method	SI
	VOS 3	Scientific investigations are motivated and guided by questions and problems	SI
	VOS 4	Scientific knowledge is always tentative	NOS
	VOS 5	Science interacts with the socio-economic and cultural context	SI
	VOS 6	The empirical data on which scientific knowledge is based are always theory-laden	NOS
	VOS 7	The models built by the scientists are distinct from the empirical reality	NOS
	VOS 8	Errors may play a constructive role in the development of science	SI
	VOS 9	Interactions between scientists actively contribute to the construction and validation of scientific knowledge	SI
	VOS 10 Realism versus instrumentalism with respect to the role of models	NOS
	VOS 11	Relativism or rejection of relativism concerning the influence of the socio-economic and cultural context on the construction of scientific knowledge	SI

Table 4 .

 4 The VOS items assessed by the questions of the VOS questionnaire

Table 5 .

 5 . Distribution of teachers' views for each VOS item in terms of percentages

		VOS1	VOS2	VOS3	VOS4	VOS5	VOS6	VOS7	VOS8	VOS9
	Novice	25	10	9	14	0	42	51	5	1
	Rather Novice	55	32	na	32	0	na	na	41	36
	Rather Expert	5	38	na	20	47	na	39	1	18
	Expert	0	20	91	2	53	58	1	50	2
	Context dependent	8	na	na	11	na	na	5	na	25
	No answer	1	1	0	21	0	0	5	3	19
	Answer not Interpretable	6	na	na	na	na	na	na	0	0
	("na" stands for "not applicable")
		VOS10
	Naive Realist	51
	Balanced Realist	15
	Instrumentalist	28
	No answer	6

  □ Some ideas and theories have been put aside throughout history (e.g., the idea that Earth has the shape of a plate, that spermatozoids entail tiny human beings already formed, that bleeding makes fever decrease…). Do you think that these ideas and theories have contributed to the development of knowledge?

	Q5. always	often	sometimes	never	I don't know
	□	□	□	□	□
	Explain your answer. You can provide one or two examples.
	Q6. According to you, with respect to the development of science, interactions between scientists are:
	very important important not so important not important	I don't know
	□	□	□		□	□
	Why?
	Q7. According to you, scientific researches are dependent on the socio-economic and cultural context?
	no	yes, a little	yes, strongly I don't know
	□	□	□	□
	If yes, is it:
	sometimes	often	always
	□	□	□
	Q8. (A) According to you, on which aspects of research can the socio-economic and cultural context have an
	influence?
					yes	no	I don't know
	on the choice of the topics being investigated		□	□	□
	on the interpretation of the outcomes		□	□	□
	on the allocation of human and financial resources	□	□	□
	on the choice of the hypotheses			□	□	□
	other:
	(B) According to you, what are the socio-economic and cultural aspects that can have an influence on research?
						yes	no	I don't know
	financial					□	□	□
	political					□	□	□
	intellectual currents (religions, philosophical conceptions…)	□	□	□
	other:
	Approach 1:
		problem	hypotheses	experiments	outcomes
	Approach 2:
		problem	hypotheses	experiments	outcomes

Acknowledgments

We would like to thank Luz Martinez and Karine Molvinger for their contribution to the design of the VOS questionnaire, as well as Suzanne Bellue and Kévin de Checchi for their work concerning the coding process.

Funding Information

This study received financial support from Agence Nationale de la Recherche (ANR, France) -Project FORMSCIENCES

Electronic supplementary material

Online resource 1: Selection process of epistemological points for the VOS questionnaire Online resource 2: The questions of the VOS questionnaire Online resource 3: The coding method of the VOS questionnaire