text:
Annette Rouleau
email: annetterouleau@gmail.com

Simon Fraser

Tensions in the role of mathematics coaches

Keywords: Tensions, mathematics coaches, professional development

The use of mathematics coaches as a means of professional development for teachers is an increasing phenomenon in North American schools. The research presented here identifies tensions experienced by mathematics coaches and how they cope with those tensions. Utilizing a framework that characterizes tensions as dichotomous pairings, the results indicate that there are tensions that are unique to mathematics coaches. This adds to a growing body of research into the role of mathematics teachers.

Introduction and background

In their search for school-wide models that support improvements in the teaching and learning of mathematics, districts are, "embracing coaching as a model of authentic professional development wherein teachers can learn in the context of their schools and their instructional practice" (Campbell & Malkus, 2014, p. 213). Underlying this development is the recognition that schools need to become places where both students and teachers can learn [START_REF] Hawley | The essentials of effective professional development: a new consensus[END_REF]. With this in mind, districts have begun placing mathematics specialists in their schools to work directly with practicing teachers [START_REF] Anstey | Leading and supporting mathematics teacher change: The case of teaching and learning coaches[END_REF].

Bearing a variety of labels, such as mathematics specialist, numeracy specialist, lead teacher or learning coach, a mathematics coach is generally a highly knowledgeable teacher hired to support the improvement of mathematics teaching and learning within a district [START_REF] Anstey | Leading and supporting mathematics teacher change: The case of teaching and learning coaches[END_REF]). An effective mathematics coach would have a deep understanding of mathematical content combined with pedagogical expertise and strong interpersonal skills. Usually they are former classroom teachers, recognized for their abilities and promoted from within [START_REF] Campbell | The impact of elementary mathematics coaches on student achievement[END_REF].

Mathematics coaches are responsible for providing ongoing professional development of the inservice teachers in their districts by "advocating for their change, nurturing their performance, advancing their thinking, increasing their mathematical understanding, and saluting their attempts" (Campbell & Malkus, 2011, p. 459). To reach this goal, mathematics coaches' work varies from modeling mathematics lessons in a teacher's classroom to observing and supporting a teacher as they teach [START_REF] Campbell | The mathematical knowledge and beliefs of elementary mathematics specialist-coaches[END_REF]. This is a varied, demanding role that [START_REF] Campbell | The impact of elementary mathematics coaches on student achievement[END_REF] suggest "the profession does not understand and is only beginning to examine" (p. 449).

The tensions experienced by mathematics coaches is one such unexamined area. A review of the literature reveals little information about the dilemmas mathematics coaches face. Literature regarding generalist coaches is much richer and suggests several common tensions. For instance, [START_REF] Neufeld | Coaching: A strategy for developing instructional capacity[END_REF] identified a series of tensions experienced by generalist coaches that begins with a lack of time. This is a frequently experienced tension, whether it is a lack of time to conference with teachers or lack of time to prepare for working with teachers. Leaving the classroom environment causes tensions for some generalist coaches, as does the switch to working with adults. Tensions also occur when teachers are slow to uptake change or are actively opposed to its implementation. Finally, [START_REF] Neufeld | Coaching: A strategy for developing instructional capacity[END_REF] suggests tensions for generalist coaches arise from working with uncooperative school cultures or administration and from a lack of opportunities for personal professional development. While there are perhaps some commonalities between these tensions experienced by learning coaches in general and those experienced by mathematics coaches in particular, it would be of benefit to identify whether there are any tensions specific to mathematics coaches. [START_REF] Jones | Making the transition: Tensions in becoming a (better) mathematics teacher[END_REF] suggests that "members of the mathematics education community, whether in schools, colleges, or universities, have a responsibility to help one another recognize and deal with tensions in a productive way" (p. 233). The intent of this paper then, is to identify some of the tensions experienced by mathematics coaches.

Theoretical background

Endemic to the teaching profession, tension encompasses the inner turmoil teachers experience when faced with contradictory alternatives for which there are no clear answers [START_REF] Adler | Teaching mathematics in multilingual classrooms[END_REF][START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF]. Building on the work of [START_REF] Berlak | The dilemmas of schooling[END_REF] who identified sixteen dilemmas that illuminated the relationship between everyday school events and broader social, economic, and political issues, it was [START_REF] Lampert | How do teachers manage to teach? Perspectives on problems in practice[END_REF] who first suggested the notion of teachers as dilemma managers who accept conflict as useful in shaping both identity and practice.

For the purposes of this study, I turn to the work of [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] whose self-study of tensions in her role as a teacher-educator resulted in a binary categorization of tensions. Seeking to depict the inner turmoil she experienced from the competing pedagogical demands in her practice, she proposed a framework for both identifying and understanding tensions. Isolating the following six pairs of interconnected tensions, Berry used these as a lens to examine her practice: (1) Telling and growthbetween informing and creating opportunities to reflect and self-direct (2) Confidence and uncertainty-between exposing vulnerability as a teacher educator and maintaining prospective teachers' confidence in the teacher educator as a leader (3) Action and intent-between working towards a particular ideal and jeopardising that ideal by the approach chosen to attain it (4) Safety and challenge-between a constructive learning experience and an uncomfortable learning experience (5) Valuing and reconstructing experience-between helping students recognise the 'authority of their experience' and helping them to see that there is more to teaching than simply acquiring experience 6) Planning and being responsive-between planning for learning and responding to learning opportunities as they arise in practice (Berry, 2007, pp. 32-33).

Although initially used as a framework to isolate tensions in the work of teacher education of preservice teachers, [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] framework has been used in other contexts as well. As part of a larger, ongoing project, of which this paper is a part, Liljedahl, Andrà, Di Martino, and Rouleau (2015) applied Berry's tensions framework to a fictional composite of a mathematics teacher that comprised a collection of data sets. Their work expanded Berry's framework by identifying new tension pairs and they concluded that some tensions may be the driving force behind a teacher's pursuit of professional development by fueling a desire for change in practice.

While considered relative newcomers, mathematics coaches are part of a mathematics community that includes both pre-service and inservice teachers. Given that developing a shared understanding of the tensions teachers face gives them the power to shape the course and outcomes of their teaching practice [START_REF] Adler | Teaching mathematics in multilingual classrooms[END_REF]; it is likely the same would be true for mathematics coaches. Bringing the challenging aspects of their work to light would offer mathematics coaches the opportunity to recognise, talk about, and act on the tensions in their practice.

My goal then, in this article, is to isolate some of the tensions experienced by mathematics coaches. Specifically, using [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] framework, I will identify and describe the tensions they face and how they cope with them. Thus, my research questions are as follows: (1) What are some of the tensions experienced by mathematics coaches? (2) How do mathematics coaches cope with those tensions?

Context and method

This study is part of an ongoing research project regarding tensions in teaching. In particular, it is the first look into the tensions experienced by mathematics coaches. This is a small scale, qualitative study that involves only three participants. As such, I am focusing on proving the existence of a phenomenon rather than its prevalence. It is important to note, however, that I chose to report only on those tensions that were experienced by more than one participant. While aware that a single instance of a tension can be as revealing as multiple instances, it is less likely to be seen as representative of a generalizable pattern. In keeping with that, the data corpus comprises interviews with three mathematics coaches working in three separate school districts.

Tara is employed by a small urban school district that employs 430 teachers in 17 schools. She has been in the role of K-12 District Numeracy Coordinator for 4 years. Prior to that, she worked as an elementary classroom teacher for 18 years. Having had negative experiences as a learner of mathematics, Tara's interest in math was only ignited 14 years ago after attending a mathematics professional development workshop. During the ensuing years, she attended every mathematics professional development opportunity offered and developed a passion for the teaching and learning of mathematics. She is about to begin a Master's degree program with a numeracy focus.

Pam works for a small rural school district with 23 schools and 320 teachers. Employed as a classroom teacher for over 30 years, she has taught all grades K-7. Pam completed a Master's degree with a numeracy focus in 2011 and left the classroom in 2012 to take on the role of Numeracy Helping Teacher. She has always enjoyed and had a passion for math.

Ray is employed by a large urban school district that employs over 1000 teachers in 49 schools. He worked for his district for 18 years as a secondary math teacher before taking on the role of Math and Science Program Consultant 4 years ago. During his time in the classroom, Ray completed a Master's in secondary mathematics education and was involved with his province's math teachers' association. Ray went on to serve a term as the association's president, while working as his district's math consultant. Like Pam, Ray has always enjoyed math and wants to recreate that experience for the teachers and students in the classrooms he supports.

Data was collected from the participants during semi-structured interviews that were transcribed in their entirety. The data corpus was then scrutinized using [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] framework as an a priori frame for identifying and coding tensions. To begin this was done by searching the interview transcripts for evidence of tensions. In particular, I looked for evidence of utterances with negative emotional components such as "I think what's been difficult…" or utterances that conveyed doubt or uncertainty such as "I wasn't 100% sure, but…". The identified tensions were then grouped according to the pairings described by Berry. Additionally, the framework was extended to encompass a tension that did not fit within her established framework.

Analysis

In the following analysis, [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] framework will be used to identify and analyze the tensions experienced by the three participants in their roles as district mathematics coaches. During the analysis, the following four tensions pairs were evident.

Safety and challenge -Unwelcome in the classroom

All three of the participants mentioned the conflict they experienced between their desire to be working with teachers in their classrooms and not having that support seen as threatening. They describe the teachers as uncomfortable in having someone observe them and therefore are unable to utilize this valuable learning opportunity. A tension arises for the mathematics coaches who, like [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF], want the teachers to feel safe, but who also recognize the value in challenging the teachers to open their doors. This is evident in the following excerpts: Ray: And I think that teachers are a little reluctant to have people in their classroom and do, sort of team teaching or have someone observe them… that hasn't happened as much as I kind of thought it would or as much as I'd sort of like.

Pam: The kind of biggest piece, I think, for us, is how do you support those teachers that are too nervous or too anxious about having someone come in? Tara: If I get invited in, I'm in, I go. Absolutely. But unless I'm invited in, it doesn't, like, I don't just, well, I shouldn't say I don't just show up…. but to be actually modelling in a classroom and doing observations, that's all by invite.

As a rationale for the teachers' reluctance, both Pam and Ray offer related possibilities. Pam suggests that the teachers' reluctance stems from a fear of being evaluated even though she feels she makes it clear that her role is one of mentorship and has no evaluative elements stating, "They haven't shifted away from that fear yet, that I'm there to judge. I'm not, I'm there to support them." Ray suggests that the teachers are concerned with the overall quality of their lesson, which then becomes a barrier to observation, "When it comes down right to it, you know, we're all a little bit unhappy with every lesson we ever do so I don't really want you seeing me because, you know, it's got its warts and all that stuff. And so, a lot of good people, but not necessarily wanting people in their classrooms." For all three, this appears to be an unresolved, ongoing tension in that none have successfully found ways to make classroom visits an accepted part of their roles. Pam, in particular, mentions that this tension leads her to consider ways of presenting this opportunity to learn as risk-free noting, "Well, I'd really like to be in more rooms and influencing more teachers. I'm trying to think of ways I can do that to support them."

Valuing and reconstructing experience -Resistance to change

Another of the tensions that was apparent for all three participants was similar to [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] tension of valuing and reconstructing experience. The mathematics coaches experienced a dilemma between acknowledging the authority of the teachers' experience and helping them to see that there is more to teaching than simply having acquired a requisite amount of experience. This is best exemplified by Ray in the following excerpt: Ray: I think the biggest barrier tends to be, as teachers, we've gone through a system a certain way that we can visualize how it looks in the classroom. We've taught that way and we see successes in that, in either ourselves or some students, and we hang onto those successes as sort of validation for doing what we do. And we tend to say, 'Well, those other kids just aren't being successful. They're not working hard enough. They're not trying hard enough. They need to do things differently. They need to change.' And I don't think a lot of teachers are as good at saying, 'Well, what do I need to do differently? What do I need to do to change?'

The mathematics coaches value the experience their teachers have, but know that experience can always be broadened and improved. None of data from the participants suggest they use a deficit model approach to coaching teachers, but rather they believe there is always room for growth. This belief perhaps stems from their own experience with life-long learning. They want the teachers they work with to consider which areas of their practice would benefit from further learning and support. As Ray suggests, "And, I think if teachers just come out a little bit more with the willingness that, you know, as good as I am, (laugh) I probably could be a lot better. That would be very helpful."

The data suggests that all three mathematics coaches see this tension as a resistance to change and this manifests in different ways. For Tara, who described her own career in terms of ongoing growth and change, the tension stems from the assumption that her colleagues would be open to similar experiences. She finds it difficult to accept that change is slow stating, "So I made the assumption that was once other teachers kind of have these a-ha moments [as she did], they would just fly and I've come to realize that's not the case."

Ray also experiences tension in slow change, but notes that, while "teachers can be very confident about some things and don't necessarily challenge themselves as much as they could", it is possible that "as much as we sometimes want to change, it's a lot of work to change and people only have so much time in the day so they sometimes just don't even get started." Ray's view suggests that outside influences play a role in teachers' readiness or willingness to change.

Pam views the resistance to change as more of a readiness factor. Her tension lies in the fact that the teachers she works with are not as ready to reconstruct their teaching experiences as she would like them to be. She recognizes that she "wants them to try more than they're capable of trying" and is aware that she's "not giving them time to slowly implement what they're comfortable with". She values the experience they bring, but struggles to encourage them the consider new practices. This too appears as an unresolved tension that all three mathematics coaches deal with on an ongoing basis. Pam was the only participant to offer a partial solution, albeit unsatisfactory to her. She approaches this tension with perseverance tinged by frustration saying, "Well, I think, you kind of got to persist, but it can kind of get a little frustrating at times."

Confidence and uncertainty -Questioning role and ability

A tension that surfaced for both Tara and Pam correlates closely with [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] tension pairing of confidence and uncertainty. Both coaches mention having had colleagues question their role and their qualifications. This created a tension between the necessity of exposing their vulnerability and maintaining the confidence of the teachers they mentor. Tara mentions, "You get the naysayers in the room that might, you know, question you on things. The biggest thing I get is what are your qualifications to do this job. That's what I get all the time." And Pam adds: Pam: I've had people that have said to me, I know enough that I don't really need you and I don't understand why the district is wasting money on your job. It's the senior math people, the 10, 11, and 12, that are the hardest to influence and they don't want to be influenced by me. I've been told many times by them that I have not the experience.

For both Pam and Tara, this appears to be a managed tension. Although the questions regarding their qualifications continue to be asked, neither seem to regard it as an ongoing source of tension. Both admit to limiting their role to elementary and junior high school and, for Pam, holding a Master's in Numeracy was perhaps sufficient to manage any remaining tension. Tara chose two other methods, which appear to offer the credibility she needs to answer any questions-she outlines her credentials and acknowledges the research behind best practice in mathematics:

Tara: So what I started to do more of after that was, whatever I was giving a recommendation for, I always had research to back up my recommendation. So I was always presenting what the research was saying. Always. […] I lay out what courses I've taken, the journey I talked to you about, and why it's a passion. They seem to be a little better once they hear that story.

Initiative and systemic barriers -Working with learning assistants

This is a tension pairing that extends [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] framework as it does not have a counterpart within her original set of tensions. It surfaced when the mathematics coaches were asked what they would like to implement in their role but have not been able to as of yet. Both Pam and Ray mentioned working with learning assistants. A strong desire to support every adult involved in the learning of students in their district drives them to want to work more closely with the learning assistants. Yet to do so would disrupt an existing functioning system. Ray expresses this clearly in the following excerpt:

Ray: I've got a few things that are sort of happening, but not as deeply as I'd like. One of them is the learning support group in our district. They all work a little bit differently and it's kind of hard to connect with them the way we're set up in the system.

Their initiative meets with resistant in the form of systemic barriers. In Pam's case, it is a result of an administration system that limits her contact with colleagues to only classroom teachers. For Ray, it is a result of different priorities. Like Pam, the learning assistants in his district provide both numeracy and literacy supportand that support tends more towards literacy. Ray notes, "They tend to be very heavily focused on reading recovery/writing kind of stuff over the years and they just haven't had a lot of time to get together and talk about anything around math."

Both Pam and Ray mention wanting to circumnavigate the systemic barriers and provide professional development to the assistants, who, in their respective districts, tend to work one-on-one in pullout environments with students. Ray wants the opportunity to offer more effective resources. Pam agrees, adding, "They are sending these support people out to work with students, but they're working same old, same old. The child gets the same kind of repetitive practice over and over again and it never moves them forward." This is an unresolved tension that has both mathematics coaches searching out solutions. Ray offers the vague "hope" that he will be able to connect with the learning assistants this coming school year, but does not go deeper into his plan. Pam plans on speaking with her assistant superintendent to seek her assistance in convincing the learning assistant teachers that she is capable of providing them with support.

Discussion and conclusion

The first goal of this study was to identify tensions experienced by mathematics coaches. Three tensions emerged that closely aligned with the tension pairings in [START_REF] Berry | Tensions in teaching about teaching: Understanding practice as a teacher educator[END_REF] framework. The fourth was a tension the mathematics coaches experienced in their desire to support learning assistants. With no obvious parallel in Berry's work, likely due to her role as a pre-service teacher educator, the presence of this tension requires the framework to be extended when considering tensions experienced by mathematics coaches.

The findings also revealed tensions that could be considered unique to mathematics coaches, as there were two tensions they experienced that were not included in the list of tensions identified by generalist coaches. The first finding suggests that mathematics coaches may experience tension regarding their qualifications. This could be explained by the expectation that a specialist in one subject would be expected to have specific skills that a generalist, who works across all subjects, would not be expected to have. Additionally, given that many of the mathematics coaches are pulled from teaching positions within their districts [START_REF] Campbell | The impact of elementary mathematics coaches on student achievement[END_REF], their former colleagues might question their abilities. Interestingly, Ray, who was a secondary mathematics teacher has never experienced this tension. Despite having no elementary experience, he stated that he has always been "well received" by the elementary staff. His status as a high school mathematics teacher appears to offer him credibility across all grade levels. The second tension experienced by mathematics coaches, but not generalists, was working with learning assistants. This might be the result of the relative newness of the role of mathematics coaches [START_REF] Anstey | Leading and supporting mathematics teacher change: The case of teaching and learning coaches[END_REF]. Districts are still in the process of determining the scope of the responsibility of the mathematics coaches in their employ. As Tara suggests, "As for the job itself, it was pretty much I just had to build the airplane as I was flying it."

The second goal of this study was to identify how the mathematics coaches coped with the tensions they experienced. The findings suggest that they appear to fit [START_REF] Lampert | How do teachers manage to teach? Perspectives on problems in practice[END_REF] image of dilemma managers who accept and cope with continuing tension. This means that the mathematics coaches initially manage the tensions that surface while never fully resolving their competing conflicts. What was interesting was the managed tension that Pam and Tara, both of whom are elementary trained teachers, experienced regarding questions about their role and qualifications. Their method for managing this tension was avoidance of interactions at the high school level. Similarly to the finding in [START_REF] Liljedahl | Teacher tension: important considerations for understanding teachers' actions, intentions, and professional growth needs[END_REF], this suggests that this tension is managed on some levels but there is a possibility it could resurface at some point. While both are only required to work with teachers who volunteer and are willing, both of their roles encompass grades K to 12.

While the small number of participants in this study may limit its generalization, the findings do indicate the presence of tensions experienced by mathematics coaches. This language of tensions could be useful as a means for discussion and reflection on the practice of mathematics coaches. Whether managed or unresolved, identifying and describing these tensions will contribute to a small, but growing body of research into mathematics coaching. If employing mathematics coaches in schools is to be a viable complement to professional development, more study will be necessary.

Proceedings of CERME10


domain: math shs

text:
Rita María

Viviana Carolina Otero
email: rotero@exa.unicen.edu.ar

Marcelo Llanos

María Paz Arlego
email: marlego@exa.unicen.edu.ar

Gazzola

Carolina

María Rita Otero

Viviana Carolina Llanos
email: vcllanos@exa.unicen.edu.ar

Marcelo Arlego

María Paz Gazzola
email: mpgazzola@exa.unicen.edu.ar

Co-disciplinary mathematics and physics Research and Study Courses (RSC) within two groups of pre-service teacher education

Keywords: Inquiry based teaching, pre-service teachers training, modelization; research and study course

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Anthropological Theory of the Didactic (ATD) has proposed the Paradigm of Research and Questioning the World [START_REF] Chevallard | Théorie Anthropologique du Didactique & Ingénierie Didactique du Développement[END_REF](Chevallard, , 2013 a) a) advocating an epistemological and didactic revolution [START_REF] Chevallard | Théorie Anthropologique du Didactique & Ingénierie Didactique du Développement[END_REF] of the teaching of mathematics and school disciplines, where knowledge should be taught by its usefulness or potential uses in life. The present work shows results obtained in two courses of pre-service mathematics teacher education (N=25) during a teaching inspired in the paradigm of questioning the world, by means of a Research and Study Course (RSC). To learn what an RSC is, and which kind of teaching is involved in, the trainee teachers (TT) must deeply experience a genuine RSC. Thus, a physics and mathematics codisciplinary RSC is designed, implemented and analyzed with the students. Co-disciplinary means that in this case, physics does not only trigger the study of mathematics, but rather that both disciplines play a central role, being necessary to study both as well. The starting point of the RSC is the question Q0: Why did the Movediza stone in Tandil fall? Which, to be answeredin a provisional and unfinished way-needs study Physics and Mathematics jointly. The rationale of the paper is to describe the trainee teachers' activities and their difficulties when they must experience a genuine RSC and to face a strong question. Some reflections on the ecology an economy of this kind of didactic devices for the pre-service teachers training are performed.

The research and study courses (RSC)

The ATD defines the RSC as devices that allow the study of mathematics by means of questions. The RSC establish that the starting points of mathematical knowledge are questions called generative questions, because its study should generate new questions called derivative. Teaching by means of RSC is complex and demands rootle changes in the roles of the teacher and students. The RSC are defined by the developed Herbartian model [START_REF] Chevallard | Analyses praxéologiques: esquisse d'un exemple[END_REF]:

[S(X;Y;Q){R ◊ 1, R ◊ 2, R ◊ 3,…, R ◊ n., Qn+1,…, Qm+1,…,Om, Om+1,…,Op }] R ♥ Thematic Working Group 18 Proceedings of CERME10
Where Q is a certain generative question; S is a didactical system around of the study of Q. S is formed by a group of people trying to answer the question (X) and by people helping the study (Y). In classrooms of mathematics, X represent the students and Y represent the teacher and other instruments helping in the search of answers to Q. S has to build a didactic medium M to study Q, whereas M is composed by different knowledge, expressed by R ◊ i, Qj and Ok. The R ◊ i are any existing answer or "socially accepted answer", the Qj are derivative questions of Q, and the Ok are any other knowledge that must be studied developing the answers. Finally, R ♥ is some possible and partial response to Q given by S. In the a priori analysis stage, the specific and didactic knowledge which could be involved within an RSC is set up and the Praxeological Reference Model (PRM) is elaborated. The researchers analyze the potential set of questions which the study and the research into Q might encompass together with the knowledge, mathematics and physics in this case, necessary to answer those questions (Chevallard, 2013). The PRM underlies the whole of the teacher, student and researcher´s activity, being always likely and desirable to identify and clarify it, emphasizing the dynamic nature of the PRM.

Methodology

This is a qualitative and exploratory research that aims to carry out inquiry based teaching as it is proposed by the ATD, in a mathematics teacher training course at the University. The RSC was implemented in a state university, in the city of Tandil, Argentina, in a discipline which is part of the didactic studies within the Mathematics Teaching Training Course, in which the researchers are also the teachers, where N=12 and N=13 students from the last year (4 th ), aged 21-33 took part in it. The students had studied the ATD in two Didactic courses; however, they had problems to understand what an RSC is, and how it works? To emphasize the inquiry dimension of the RSC, the lessons were carried out in the University Library, given the wide availability of books and internetbased searching, during 10 weeks (the half of the course extension), with a total of 7 weekly hours provided in two lessons. Six work groups were organized with approximately 4 members each. During the lessons students and teachers interacted permanently. In a RSC, the generative question Q0 has to be pointed out by the teacher, and this was made in the first lesson. Then, the students started their research in the library, by selecting some texts, documents etc. as possible R ◊ i. At the end of this class, each group presented and discussed with the teacher and the other groups their findings and possible ways to face Q0. In the second class, many emergent questions Qi were made explicit, and the teacher and students groups selected which questions Qi and their related knowledge Ok were to be studied. This was the regular dynamic during the RSC. Recordings of each meeting were obtained and the students' productions were digitalized and returned in the subsequent meeting. The data analysis was performed by using the categories provided by the developed Herbartian model (Chevallard, 2013) summarized before, and all derivative questions Qi, all "socially accepted answers" R ◊ i found by the students, together with the Ok studied were described and analyzed.

The Praxeological model of reference (PMR) and the RSC

The starting question Q0 is: Why did the Movediza Stone in Tandil fall down? This enormous basalt stone has remained the city's landmark, providing it with a distinctive feature. Many local people and national celebrities visited the place to watch closely the natural monument. It was a 248-ton rock, sitting on the top of a 300-meter-high hill (above sea level), which presented very small oscillations when disturbed in a non-arbitrary spot, as shown in Figure 1. Unexpectedly, on February 28, 1912, the stone fell down the cliff and fractured into three pieces, filling the town with dismay at the loss of their symbol. For over 100 years the event produced all kinds of conjecture, legends, and unlikely scientific explanations for the causes of the fall. Within the two groups where the RSC has been performed, there existed a certain curiosity and interest in finding a scientific answer to this question.

Once in contact with the available information, the question evolved into: What conjectures are about the causes the Movediza Stone fall, and which is the most likely from a scientific viewpoint? Considering that the fall can be explained by means of the Mechanical Resonance phenomenon, several questions Qi arose which are linked to the physical and mathematical knowledge necessary to understand and answer Q0. If we consider that the real system is an oscillating system, the study can be carried out within the Mechanic Oscillations topic, starting from the spring or pendulum models which are ideal at the beginning. In this case, frictionless systems are used, in which the only force in action is the restoring force depending in a linear way on the deviation respect to the equilibrium position, and which produces oscillating systems known as harmonic, whose motion is described by a secondorder linear differential equation, called by the same name. In the case of the pendulum, the restoring force can be considered depending on the oscillating angle (for small angles).

Progressively, the system becomes more complex. If friction-produced damping is considered, it provides a new term to the differential equation connected to the first derivative of the position (speed). Finally, it is possible to study systems that apart from being damped, they are under the action of an external force, and therefore called driven systems. Whenever this given force is periodic and its frequency coincides with the natural (free of external forces) frequency of the oscillating system, a maximum in the oscillation amplitude is produced, generating the phenomenon known as mechanical resonance. By increasing the complexity of the model, it is possible to consider a suspended rotating body, instead of a specific mass. This leads to the study of the torque and the moment of inertia of an oscillating body. Here again, the linear system is for small amplitude oscillations and the damped and driven cases can be also considered, corresponding to the same mathematical model, but in which the parameters have a different physical interpretation.

However, as it refers to a suspended oscillating body, this is not a suitable physical model for the Movediza stone system. Since that the base of the Stone was not flat, it is necessary to consider more precise models of the real situation. This leads to the mechanics of supported (and not hanging) oscillating rigid solids. In this case, we consider a rocker-like model in which the movediza stone base is curved and it lies on a flat surface, where the oscillation is related to a rototranslation motion (Otero, Llanos, Gazzola, Arlego, 2016 a, b). The application of Newton Laws to the rocker model of the movediza stone leads to a differential equation of the type harmonic oscillator, where the parameters are now specific of the movediza system: mass, geometry, inertia moment, friction in the base, external torque, etc. It leads to the following effective Harmonic oscillator mathematical model of the movediza physical system:

) cos( ) / ( 0 2 0 wt I M w          
(1)

The solution to equation ( 1) is

       wt t M cos ) (
, being the amplitude M  and the phase 

2 2 2 2 2 0 0 ) (   w w w I M M               2 2 0 1 w w w tg   The maximum of M  is for 2 2 0    w w
. The parameters: M0 (external torque), I (inertia moment), w0 (natural oscillation system frequency) and  (friction coefficient), must be estimated. Detailed data about the shape, dimensions and center of mass position of the movediza stone are available [START_REF] Peralta | Proyecto estructural de la réplica de la piedra movediza: comportamiento estático y dinámico[END_REF] after a replica construction and its relocation in 2007 on the original place (although fixed to the surface and without possibility to oscillate). These data bring us the possibility to fix some parameters in our model, as e.g. mass, inertia moment, and the distance of 7.1 m, from which the external torque could be exerted efficiently by up to five people (per historical chronicles) to start the small oscillation. By using these values, it is possible to study the behavior of the ) (w M  function for w0 in a range of frequencies between 0,7 Hz and 1 Hz, historically recognized as the natural oscillation frequencies in the movediza stone system and calculate for each case the maximum amplitude

) ( m M w  that occurs for 2 2 0 m γ w w   .The Stone would fall if ) ( m M c w    , being   I w M w m M 0 0 / ) (  the maximum value of the amplitude function, that is to say ) / ( 0 0   I w M c 
. The value of c  can be determined by an elementary stability analysis, which according to the dimensions of the base of the stone and the center of mass position is estimated to be approximately of 6°. In the present model, we cannot estimate . If we adopt "ad doc" for this parameter, a magnitude order 2 10    , we obtain in the various situations considered with different torques and the interval frequencies previously mentioned, that all the scenarios support the overcoming of the critical angle, i.e., predict the fall. Later, in search of a more appropriate approximation of the physics model for the damping that is clearly not due to air, we can consider the stone as a deformable solid, where the contact in the support is not a point, but a finite extension. Therefore, the normal force is distributed on such a surface, being larger in the motion direction and generating a rolling resistance, manifested through a torque contrary to the motion due to friction. The rolling resistance depends on the speed stone, giving a physical interpretation to the damping term. Therefore, the physics behind the damping is the same that makes a tire wheel rolling horizontally on the road come to a stop, but in the case of the stone, the deformation is much smaller. Although the deformable rocker model has extra free parameters, tabulated values of rolling resistance coefficient for stone on stone, which are available in the specialized literature, allowed us to estimate and justify the damping values that we incorporate otherwise ad-hoc in the rigid rocket movediza model.

Some results obtained in the two implementations

During the implementations, the students aim at answering how and why the stone fell down the cliff. At the beginning, the TTs searched in the elementary physics textbooks for an "already-made" mathematical and physical model, which allowed them to solve a differential equation in a specific way. In both implementations, several physical and mathematical questions arose; the main preoccupation of the TTs was to study the oscillations subject, because it was a new knowledge to them. The implementation was carried out in parallel with a Differential Equations course, and none of the groups seemed to have difficulties with the underlying mathematics. In both implementations, the TTs tried to find a physical model suitable for the situation and they decided on the physical pendulum model initially, whose mathematical model might be adequate to the problem, although physically inadequate. However, the path performed in the first implementation was different from the second. In the first case, the TTs did not question the physical model, and spent most of the time to the study of inertia moment concept and their calculation for regular solids, which would result in an appropriate model for the irregular shape of the stone. Several interesting questions were therefore generated, for which the answers were provided by the teacher and the students together. After, the students calculated by themselves the characteristic frequency of the system, making use of the moment of inertia previously obtained. Thus, only a parameter resulted undetermined: the damping. But in the end, the physical pendulum model became an obstacle, because the stone was a supported body, and not hanged. On the other hand, the damping they considered was due to airfriction, whereas in the case of the movediza stone the main source of friction is the contact with the support surface.

At this point, the external torque (there were different trials to analyze and estimate it) and the solution of the equation remained unstudied. Until this moment, the solution for the differential equation did not seem to present any obstacles to the students, who considered they were facing an initial value problem. Once they had obtained the parameters, which they considered fixed, the solution seemed simple. However, they had problems to arrive at a final solution, even though this can be found in the physics textbooks (without its deduction). For this reason, it was discarded and they decided to do the calculation on their own. This event complicated the quantification they aimed to obtain, as well as the physical interpretation. Some groups in this cohort removed the term of damping, to reduce degrees of freedom; thus, the stone would have been in perpetual motion. This did not create any contradiction to them. Other ones adopted a damping value due to airfriction, which also led to wrong results. In summary, instead of adopting and adapting the solution that was presented in physics books, the TTs in this cohort dismissed it and did not interpret the answer in the texts concerning the Stone. The decision of the teachers to delay their intervention was with the purpose to make students live the dynamics of progress and drawbacks typical of the research and study courses. In addition the TTs had problems to understand the utility and necessity of mathematical models, due to an epistemological conception close to pure or formal mathematics. The TTs did not understand how to use the mathematical models, neither the role that the parameters could play, which were considered as fixed and universal. In consequence, they failed to establish different sets of parameters and did not generate the feasible families of functions and values, whose compatibility with the physical situation could have been analyzed. These difficulties were considered for the second implementation of RSC. In the second cohort (TT2), the teachers had already perceived that the fundamental problem seemed to be in the models and in the modelization. For this reason, it was decided to devote 8 sessions to the development of two intramathematical RSCs [START_REF] Chappaz | Il était une fois…. La boite du pâtissier[END_REF][START_REF] Ruiz | La modelización funcional con parámetros en un taller de matemáticas con Wiris[END_REF], that the TTs could experience by themselves, therefore emphasizing the role of the modelization and the use of devices as spreadsheets and plotters. Besides, in this case, the teachers intervener as soon as the students proposed the physical pendulum and spring models. One group studied the AMS for the simple pendulum, the spring, and the physical pendulum, another group studied the spring model in all its possibilities and the third one did not develop further than the AMS in simple pendulum and spring. The synthesis stage corresponding to that class allowed the production of a complete answer for the three models and its possibilities, from which the TTs and the teacher arrived at the conclusion that the same mathematical model represented (9) nine different physical systems (Figure 2). A large amount of time was devoted to pondering on the differences and similarities between the mathematical and physical models and their connection with the real system we aimed at modeling. Then, the answers to the equations presented in the books were checked out. In both cohorts, as a fixed route that is inevitably set by the books, the TTs came across the physical pendulum. However, in the second cohort some students presented strong objections to the possibility of using it in the case of the stone, not so much in relation to a body that is supported but as an "inverted" pendulum. This drew the discussion once more towards the real system and the standing point, so that the path went through the models which refer specifically to the system and that are not, usually in elementary books, like a rocker.

Firstly, the equilibrium was analyzed and the critical angle was calculated, and then, the model of the base of the movediza stone was sophisticated. For the study of the rigid solid physical model, the teacher proposed to the students a little text, as a new Ok that could be introduced into the didactic medium M. Finally, the students and the teacher calculated and estimated the parameters of the differential equation solution, and the classroom elaborated an answer that allows the explanation, by means of a model, of the plausibility of the fall.

Conclusions

Despite of the difficulties, the TTs experienced a genuine RSC within its means. There is a visible initial reluctant attitude on the part of the TTs: Why physics should be studied if we are teachers of mathematics? Later, it was gradually understood that the idea was to experience a genuinely codisciplinary RSC, to analyze it and comprehend the teaching model supporting an RSC.

Even though the TTs had studied the ATD and other didactic theories, they did it in a traditional way comparable to the traditional training they get. This is reflected in the difficulties they have to understand and to use both physics and mathematical models. It was not expected that the TTs developed the models by themselves, but it was expected that they used the mathematical answers presented in the physics textbooks in a pertinent and exoteric manner. This fact did not occur in the first group and improved in the second one from the didactic decision to make a previous incursion into mono-disciplinary RSC particularly suitable for evidencing the role of the functional modelization. In addition, this allowed teachers to discuss the relationship between the mathematical model and the physical model and the meaning and role of the parameters. The TT's behavior is interpreted from the fact that although they have experienced four years of "hard" university studies, the utility of the science they aim at teaching had never been visible. The epistemological conception about the mathematics produced by the traditional paradigm is so ingrained, that it is complex to reverse it. This would be, in our view, the most relevant drawback to permit the TTS at least understand what an RSC is and how the modelization activity works? However, it is important to notice that the sporadic incursions in the modelization activity do not seem enough to allow the TTs develop such school practices. Although the predominant teaching is mainly traditional, the TTs will face increasing demands for a change to a mathematic teaching based on the research, questioning and modeling. It is unlikely that a teacher whose training has been answers-based teaching can teach by means of questions. Then the training of teachers has to change profoundly.

Figure 1 :

 1 Figure 1: Photography of the Movediza Stone (Photo Archivo General de la Nación Argentina, available in: http://bibliocicop.blogspot.com.ar/2012/02/piedra-movediza-100-anos-de-su-caida.html)

Figure 2 :

 2 Figure 2: Protocol of the student E17. Implementation 2

Proceedings of CERME10


text:
Maria Mellone
email: maria.mellone@unina.it

Roberto Tortora
email: rtortora@unina.it

Arne Jakobsen
email: arne.jakobsen@uis.no

Miguel Ribeiro

Prospective teachers interpret student responses: Between assessment, educational design and research

Keywords: Teacher education, prospective secondary teachers, interpretation of students' answers, arithmetic and algebra

We present a study about the development of interpretative skills in prospective teachers. In particular we discuss a kind of tasks designed by us for teacher education, containing the request of interpreting students' answers. The task in this study was built on an item concerning the sum of powers of 10 and was proposed to a group of prospective secondary teachers who were attending a Math Education course. The task was first faced individually by them and then discussed in group. We present the interpretations proposed by two prospective teachers before and after the collective discussion, in order to reflect on the differences in terms both of mathematical knowledge put in play and of attitudes exhibited.

Introduction

In their daily practice, teachers are required to continuously interpret students' responses and productions. This not only assists them in evaluating their difficulties and achievements, but also allows to plan the next steps of the teaching activities. Therefore, this "interpretation activity" is one of the most crucial (and often most difficult) tasks teachers perform. Empirical evidence suggests that the true quality of a mathematics teacher stems largely from his/her ability to interpret students' productions, along with a flexible attitude to redesign the teaching approach based on them. However, the ability to make sound and accountable interpretations is rarely recognized as a crucial goal in teacher training. Moreover, in a previous research study [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF], we observed that teachers do not naturally develop this ability as they gain work experience. So, presently, our research questions are the following. Is it possible for a teacher to acquire these interpretation skills or is it a matter of innate talent? In the first case, how can a prospective teacher develop this ability? Should we engage prospective teachers in mathematical discussions concerning the interpretation of students' reasoning? Guided by our conviction that it is possible for a teacher to develop this ability, as a part of a joint research project, we explored a particular type of tasks we have conceptualized. In one part of these tasks, we asked the prospective teachers to interpret some students' responses to a problem and reflect on possible feedback they could provide to each student. This exercise had a twofold aim: to support prospective teachers in developing the skills required for interpreting and commenting on student work, and to investigate to what extent the particular mathematical knowledge and skills possessed by the (prospective) teachers support or hinder them in their interpretations and "constructive" reactions. Our analysis shows that the abilities to interpret and to design an educational activity based on students' productions are inhibited in prospective teachers with a poor mathematical knowledge, due to their limited understanding of the subject and lack of appreciation of various ways that problems can be solved [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF]. In particular, this is true for prospective primary teachers, many of whom have poor mathematical knowledge. However, we have also observed that prospective secondary mathematics teachers-who have studied more advanced mathematics during the three years of their Bachelor in Mathematics degree [START_REF] Jakobsen | Discussing secondary prospective teachers' interpretative knowledge: A case study[END_REF])-also struggle with this kind of work. This led us to posit that the ability and knowledge to interpret student work depends not only on (prospective) teachers' mathematical knowledge and its components, but also on their attitudes and beliefs toward mathematics and its teaching.

The tasks described herein are presently used in our Mathematics Education courses in three different modalities. First, we ask prospective teachers to individually solve the problem, before interpreting and reflecting on some selected students' productions, and finally engaging in group discussions on the mathematical aspects involved in these students' productions. Given that the prospective teachers (both primary and secondary) have difficulties in interpreting and in giving meaning to some students' answers, it was necessary to first assess their ability to solve problems that these students are given. This was informative, as some prospective teachers struggle with providing constructive feedback to the students even when they do not encounter any difficulties in solving the given problems for themselves. The findings yielded by this first phase of our research were utilized in the subsequent mathematical discussions of students' solutions and corresponding teachers' interpretations. These group discussions were helpful to most prospective teachers, as they were able to gain new perspectives on students' work and strategies that can be employed in teaching.

In this paper, we present the interpretations, given by a group of Italian prospective secondary teachers, of students' responses concerning a problem where sums of powers of 10 are involved. (see [START_REF] Jakobsen | Discussing secondary prospective teachers' interpretative knowledge: A case study[END_REF] for details). Here, we present analysis of the interpretations teachers gave before and after the mathematical discussions, in order to document their progress, as most demonstrated more sophisticated attitudes and greater mathematical knowledge following group discussions.

Theoretical framework

In order to characterize and study the features of teachers' interpretations of students' productions, in some of our previous work (see for example Ribeiro, Mellone, & Jakobsen, 2016), we have introduced the notion of interpretative knowledge, framed within the general Mathematical Knowledge for Teaching (MKT) framework [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. We define interpretative knowledge as the knowledge that allows teachers to give sense to pupils' answers, in particular to "non-standard" ones, i.e., adequate answers that differ from those teachers would give or expect, or answers that contain errors. We posit that interpretative knowledge is closely related to the ability of teachers to support the development of pupils' mathematical knowledge, starting from their own reasoning, even if students' ideas are incomplete or non-standard. Some similar ideas are implied in the notion of discipline of noticing [START_REF] Mason | Researching your own practice: The discipline of noticing[END_REF]. In particular, our construct encompasses the idea of teachers working "on becoming more sensitive to notice opportunities in the moment, to be methodical without being mechanical" (Mason, 2002, p. 61). The development of pupils' mathematical knowledge starting from their own reasoning is, in our view, only possible if the teacher activates a real process of interpretation, shifting from a simple evaluative listening to a more careful hermeneutic listening [START_REF] Davis | Listening for differences: An evolving conception of mathematics teaching[END_REF].

In this sense, the notion of interpretative knowledge incorporates into the MKT framework the idea that errors and non-standard reasoning are considered as learning opportunities [START_REF] Borasi | Reconceiving mathematics instruction: A focus on errors[END_REF]. Moreover, the content of interpretative knowledge shapes teachers' ability to make informed choices in contingency moments (as defined by [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF], in order to respond to and deal with non-planned situations. In that sense, we felt the need to incorporate the role beliefs and attitudes pertinent to the use of mathematical knowledge [START_REF] Carrillo | Determining specialised knowledge for mathematics teaching[END_REF].

With the goal of better understanding (prospective) mathematics teachers' act of interpretation, we characterized their interpretations of students' productions and attitudes, using the following three categories: (i) Evaluative interpretation: a process through which the teacher determines congruence between pupils' productions and the mathematical scheme of correct answers he/she has; (ii) Interpretation for the educational design: the manner in which the teacher designs educational steps based on the work produced by the students; (ii) Interpretation as research: teacher's willingness and ability to revise his/her mathematical formalization in order to ensure that it is coherent with students' productions (even when these seem in conflict with the traditional mathematics taught in school).

In Webster dictionary, "interpretation" is defined as "The act of interpreting, explanation of what is obscure"; however, it is also defined as "An artist's way of expressing his thought or embodying his conception of nature." This last definition stresses the potential creative nature of the act of interpreting that is in our context perceived as the potential new mathematical knowledge that can be developed owing to the process of analyzing students' productions.

Context and method

For several years, we have been studying the nature of (prospective) teachers' interpretative knowledge (e.g., [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF][START_REF] Jakobsen | Discussing secondary prospective teachers' interpretative knowledge: A case study[END_REF]) by exploring the manner in which prospective teachers respond to specific interpretation tasks. In this design study the tasks are developed after the typical cycles of redesign of the design study method [START_REF] Cobb | Design experiments in educational research[END_REF]. In their present form, essentially consist of three steps: (i) the teachers are initially required to solve a mathematical problem by themselves; (ii) they are given several students' productions in response to the same problem, some containing errors and some mathematically valid but following less standard procedures, which they are asked to interpret; and (iii) teachers are prompted to provide what they deem would be appropriate feedback to these students based on their solutions. The teacher trainees are asked to address these requests individually and in paper format (they are usually given 90 minutes to complete all three steps). In the next phase of the study, the educator engages all prospective teacher participants in a collective mathematical discussion (which again typically lasts about 90 minutes). The framework of the mathematical discussion is based on that proposed by [START_REF] Bussi | Mathematical discussion and perspective drawing in primary school[END_REF], as the aim is to allow the group of prospective teachers create a polyphony of articulated voices on the mathematical object starting from the interpretation of a student's production. Upon completion of the group discussion, the prospective teachers are asked to provide in writing a new individual interpretation of the students' productions, allowing the researchers to determine if any progress has been made.

The task utilized in the present study is depicted in Figure 1, and was adopted from the annual Italian national assessment (2010-2011) for grade 10 released by INVALSI (Istituto Nazionale per la VALutazione del Sistema educativo di Istruzione e di formazione). A group of 34 fourth-year master students of mathematics enrolled in a Mathematics Education course took part in this investigation. Since most of these students are going to become secondary school teachers, we consider them prospective secondary teachers.

In our previous study, we focused on the interpretation these prospective secondary teachers gave to their students' productions [START_REF] Jakobsen | Discussing secondary prospective teachers' interpretative knowledge: A case study[END_REF]. Our analysis revealed that they experienced problems in mobilizing their mathematical knowledge for interpreting students' work. Indeed, while they were able to "see" some of the mathematical aspects involved in the solutions to the problems their students proposed, they seemed unaware of many important aspects relevant to mathematics teaching and problem solving. In the next section, we will present two out of seven students' productions that were included in the task given to the study participants (for their selection, see [START_REF] Mellone | Different ways of grasping structure in arithmetical tasks, as steps toward algebra[END_REF]). This will be followed by the interpretations of these students' productions, provided by two prospective teachers-to whom we refer as Rossella and Gennaro (pseudonyms)-before (BF) and after (AF) the mathematical discussion.

Interpretation of two students' productions

The following brief analysis of the students' productions included in the task aims to elucidate our reasoning behind the decision to deem these two students' productions effective for exploring prospective secondary teachers' ability to interpret the work of others.

Emanuela (Figure 2a) obtained the correct result, despite making three errors in her work: the first and the last can be described as lack of use of parenthesis and the second can be seen as a wrong application of linearity. Ciro (Figure 2b) arrived at the right answer using the arithmetical algorithm of the arrangement of the decimal representation of the numbers in column. In his responses, we can also recognize his perception of the algebraic structure connected with more general ideas implicit in calculus. Indeed, Ciro's use of the ellipses reveals the potentiality of generalization of his Thematic Working Group 18

Proceedings of CERME10

response to other two consecutive powers of ten, and not only to the particular case given (as one of the mathematics students that took part in our study noted during the collective mathematical discussion).

(a) -From Emanuela's protocol (b) -From Ciro's protocol

Reflections on Emanuela's work

When individually interpreting Emanuela's response, prospective teachers seemed to experience difficulty in trying to understand the steps she used in arriving at the solution (see [START_REF] Jakobsen | Discussing secondary prospective teachers' interpretative knowledge: A case study[END_REF]. Here, we focus on the interpretations given by a secondary prospective teacher -Rossellabefore (BF) and after (AF) the discussion1 :

The second prospective teacher, Rossella, shared the following:

Rossella (BF):

There is no application of rules; it is pure invention. I don't know what I would say to the girl, but I would think that she had copied the solution and then tried to invent a justification.

Rossella (AF): Even if

Emanuela's answer is correct, her arguments are far from being mathematically founded. Still, we can observe an interesting aspect in them, namely that if we repeat the steps with two powers having the base different from 10 and the exponents differing by one, we get the correct result. I followed this approach using different numbers, just to test this reasoning, which allowed me to assert that Emanuela's thought process appeared to work. More specifically, when you change the bases and use two consecutive numbers as exponents, or even not consecutive, her logic gives the correct result.

ae è ç ö ø ÷ 3 + - 1 10 ae è ç ö ø ÷ 4 = - 1 10 ae è ç ö ø ÷ 3 + - 1 100 ae è ç ö ø ÷ 3 = - 9 100 ae è ç ö ø ÷ 3 = 9 10 ae è ç ö ø ÷ - 1 10 ae è ç ö ø ÷ 3 2 -3 +2 -2 = 2 -3 +4 -3 = 6 -3 = 3  2 -3 x n + x n+1 = x n + (x 2 ) n = (x+x 2 ) n = ((x+1)x) n = (x+1)x n
In other words, the above three errors, made in sequence, yield a correct result that does not depend on the particular numbers used. This observation has prompted our reflection on how mathematics is usually managed. We are used to judge, without hesitation, arguments such as those mentioned above as wrong and completely invented. However, after our discussions, we were of view that errors should be seriously considered and, if possible, exploited as a stepping stone toward the construction of new knowledge. This is exactly what I tried to do in order to bring out something new from Emanuela's reasoning. Indeed I also tried to build a new system of rules for powers according with her reasoning. Several attempts have convinced me that this is not possible. Thus, my conclusion is that no new knowledge about the rules of powers can be derived from Emanuela's suggestions. However, her errors can be an invaluable tool to stimulate discussions and to highlight the need for a true comprehension of the rules of powers, which are often hard to grasp for students.

Reflecting on Rossella's BF and AF words, we can observe a change in the attitude when interpreting student's work. The first interpretation is an evaluative one-there is no effort to understand the rationality of Emanuela's steps. This results in Rossella's bias towards a solution like that Emanuela offered (referring to an her possible unfair behavior), which can be due to fear of moving toward an educational path she cannot (immediately) control.

The interpretation given by Rossella after the discussion is markedly different. She not only made an effort to derive a generalization from the errors in Emanuela's production, but further concluded that the three steps Emanuela used in solving the problem will give the right answer with other bases and exponents. Rossella thus went beyond the simple observation that Emanuela's steps are not mathematically sound, as she investigated the possibility to build "a new system of rules for powers according with her reasoning." For this reason, her second interpretation can be considered a form of interpretation as research.

Reflections about Ciro's work

Gennaro (BF): Ciro reached the correct answer by a more practical method than those employed by his peers. In addition, the formalism seems original. He appears to have a strong expertise in the calculations with powers of 10, which highlights their significance and the importance of handling them correctly. Still, his method seems limited to powers of 10. It would be interesting to see how Ciro would proceed if presented with a different base. I think that Ciro's protocol could be used as an opportunity to explore differences between the properties of powers of 10 and those of other bases.

Gennaro (AF): Ciro's argument is of an arithmetic character. Nonetheless, it allows us to appreciate some deep algebraic insights. Moreover, although it seems confined to powers of 10, it can actually be generalized to any base, if one represents the number in the base of the power. Hence, from Ciro's production going further, it would be possible to study the tables of operations in different bases, or even the divisibility rules in bases other than 10.

In his first interpretation of Ciro's protocol, Gennaro appreciates the originality of his method, while noting that it is limited to powers with base 10. Based on this observation, Gennaro proposed possible questions and issues that could be explored with Ciro and the rest of the students in the classroom, starting from his production. For this reason, Gennaro's first interpretation is aimed at educational design.

As with Rossella, whose interpretations we analyzed previously, the comments Gennaro gave on Ciro's protocol after the discussion shifted in focus. First, there is a subtle distinction between arithmetic and algebra that could be investigated and debated endlessly. Moreover, Gennaro's comments reveal his awareness that Ciro's method can be applied to other bases (indeed, 100…0 always represents the n-th power of the base). This fact was observed during the collective discussion by another prospective teacher, and for Gennaro, this discovery was so important that it became part of his new written interpretation. In other words, Gennaro's knowledge and interpretation benefitted from the mathematical discussion on Ciro's production. He reconceived the systems of representing numbers in different bases, which motivated him to explore the true meaning of digits, as well as of strings of digits. For this reason, Gennaro's second interpretation is perceived as interpretation as research.

Conclusive remarks

We started this paper by asking if mathematics teachers can develop the ability to interpret their students' productions in order to flexibly redraw the mathematical learning path, or if this should be considered as an innate talent. We are convinced that it is not only possible to develop this skill but is highly desirable. The observed difficulties these prospective teachers experienced when giving sense to student productions, along with the findings yielded by extant studies, indicate that the development of this ability requires a special attention in teacher education. These first results about our proposed method of working with prospective teachers appear to support its effectiveness. It stimulates prospective teachers' interpretive and critical skills and increases knowledge they must possess in order to teach effectively, taking into consideration the specificities of such knowledge. The value of our method stems from the nature of interpretive tasks involving student productions, as well as subsequent discussions among peers under the guidance of an expert on these interpretation tasks.

Our analysis of the interpretations given by two prospective teachers, before and after the collective discussion led by the educator, clearly demonstrates changes in terms of both their attitude and mathematical knowledge or awareness. We can hypothesize that the collective discussion mobilized mathematical knowledge that was previously present, but probably not put in play, and it also support the development of new mathematical knowledge, like for Gennaro. However, the improvements we witnessed were also due to the change in attitudes and beliefs supported by the discussion, and of course by the attitudes and beliefs incorporated in the educator's practice.

Still, our work leaves many questions to be answered in future research. It would be interesting to evaluate the sustainability of these changes, for example, by following the work of these prospective teachers in their future educational practices. Moreover, analysis of the mathematical discussion on interpretative task needs to be developed in order to clarify its features and dynamics.

Figure 1 :

 1 Figure 1: Item given to students to solve

Figure 2 :

 2 Figure 2: Two out of seven students' productions Prospective teachers' comments on students' productions before and after mathematical discussion

Proceedings of CERME10

Of course, the students' words were translated from Italian into English. We tried our best to retain the exact expressions, including some errors, but some nuances are inevitably lost.


text:
Viktor Isaev
email: isaev@mathematik.uni-kassel.de

Andreas Eichler
email: eichler@mathematik.uni-kassel.de

Measuring beliefs concerning the double discontinuity in secondary teacher education

Keywords: Beliefs, double discontinuity, prospective teachers, teacher education

In this paper, we discuss a research project to measure prospective mathematics teachers' beliefs towards their perceived disagreements between mathematics at university level and school mathematics that is known as double discontinuity. Firstly, we introduce the double discontinuity problem. Secondly, we refer to the construct of beliefs as the main part of our theoretical framework. Afterwards, we outline our method including a brief discussion on our approach of bridging the double discontinuity problem with so-called teacher-oriented tasks that are appropriate to illustrate connections between university mathematics and school mathematics. Furthermore, our results of pilot studies aiming to measure prospective teachers' beliefs are provided.

Introduction

Over 100 years ago, Felix Klein coined in the preface of his textbook "Elementary Mathematics from a Higher Standpoint" the term of a "double discontinuity" (Klein, 1908, p. 1). This notion embodies the challenges of transitions in the mathematical socialization of mathematics teachers. The first discontinuity, i.e. the transition from secondary to tertiary education, became a main topic in research on university mathematics education during the last decades (e.g. [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF][START_REF] Thomas | Key Mathematical Concepts in the Transition from Secondary School to University[END_REF]. A challenge that accompanies the second discontinuity is the transformation of academic mathematics gained at university into educational forms of school mathematics [START_REF] Prediger | Unterrichtsmomente als explizite Lernanlässe in fachinhaltlichen Veranstaltungen: Ein Ansatz zur Stärkung der mathematischen Fundierung unterrichtlichen Handelns[END_REF][START_REF] Winsløw | Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach to calculus[END_REF]. As a consequence of both discontinuities, teachers may lose sight of academic mathematics after university studies and, thus, teach on the basis of experiences from their own schooldays [START_REF] Bauer | Schnittstellenmodule in der Lehramtsausbildung im Fach Mathematik[END_REF][START_REF] Hefendehl-Hebeker | Doppelte Diskontinuität oder die Chance der Brückenschläge[END_REF].

Although mathematics instruction in schools as well as teacher education in universities changed considerably since Klein's claim, the phenomenon of a double discontinuity still seems to exist and prospective teachers nowadays frequently believe that the topics of university mathematics do not meet the demands of their later profession in school (cf. Ableitinger, Kramer, & Prediger, 2013;[START_REF] Hefendehl-Hebeker | Doppelte Diskontinuität oder die Chance der Brückenschläge[END_REF]. The prospective teachers' perception of the mentioned relationships between school mathematics and university mathematics could be regarded as part of teachers' beliefs [START_REF] Eichler | Disagreements between mathematics at university level and school mathematics in secondary teacher education[END_REF]. Considering this background, we primarily rely on teachers' beliefs as the main construct of our theoretical framework in the next paragraph. Subsequently, we outline the method for our research project with a specific focus on pilot studies. Finally, we use the results of these pilot studies to explain the development of our instrument aiming to measure prospective teachers' beliefs concerning the double discontinuity.

Theoretical framework

Prospective teachers' beliefs

On the construct of beliefs we refer to the definition of Philipp (2007, p. 259), who defines beliefs as "psychologically held understandings, premises, or propositions" affecting an individual's view of the world or a special part of it, such as filters for receiving information. Beliefs can be considered as a component of teachers' mathematics related affect [START_REF] Hannula | Exploring new dimensions of mathematics-related affect: embodied and social theories[END_REF], which itself can be defined as "a disposition or tendency or an emotion or feeling attached to an idea or object" (Philipp, 2007, p. 259). Thus, in our research we regard prospective teachers' beliefs as their propositions concerning the relationships between school mathematics and university mathematics. Following [START_REF] Calderhead | Teachers: Beliefs and Knowledge[END_REF], teachers' beliefs are understood not only to impact on teachers' professional knowledge but also to potentially have an effect on teachers' classroom practices (cf. also [START_REF] Skott | Contextualising the notion of 'belief enactment[END_REF]. Therefore, it seems essential to tackle the issue early and to investigate, how beliefs concerning the double discontinuity develop in secondary teacher education.

In related literature, beliefs are partially seen and sometimes even defined as relatively stable dispositions. In contrast, several studies provided a change of beliefs due to interventions, especially concerning prospective teachers (e. g. [START_REF] Decker | The relationship between quality of discourse during teacher induction classes and beginning teachers' beliefs[END_REF]. In this paper, we aim to provide an instrument in order to investigate, whether and how prospective mathematics teachers change their beliefs concerning the double discontinuity phenomenon based on our intervention project. From this background, we address beliefs being changeable and consider change as "a natural part of the development of beliefs and the reaction of beliefs in the face of experiences" (Liljedahl, Oesterle, & Bernèche, 2012, p. 35).

Related Research

Although numerous projects and institutions across the world aimed to overcome the perceived gap between school mathematics and university mathematics in recent years, not much research has been done on the perception of prospective teachers' concerning the double discontinuity problem. [START_REF] Winsløw | Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach to calculus[END_REF] distinguished three dimensions of Klein's double discontinuity, which are not independent but important to separate: the institutional context (i.e. school vs. university), the difference in the subject's role within the institution (i.e. student at university or school vs. teacher of mathematics), and the difference in mathematical contents (i.e. elemantary vs. advanced). In our research, we primarily refer to the content aiming to figure out possibilities of "building bridges" (Winsløw & Grønbaek, 2014, p. 64) and to investigate the effect of bridging activities on the prospective teachers' beliefs referring to the double discontinuity. [START_REF] Becher | Beliefs on benefits from learning higher mathematics at university for future secondary school teacher[END_REF] used narratives in order to ask prospective secondary teachers in their third year of university studies about what benefits they see in learning university mathematics for their future career as a school teacher and which aspects are articulated by the prospective teachers in their evaluation of benefits of university mathematics. The results revealed a wide range of prospective teacher's beliefs on benefits of learning university mathematics with regard to school mathematics. Most of the statements can be matched with one of four levels of mathematical content knowledge based on [START_REF] Krauss | Mathematics Teachers' Domain-Specific Professional Knowledge: Conceptualization and Test Construction in COACTIV[END_REF], i.e. "A deep understanding of the content of the secondary school mathematics curriculum (e.g., 'elementary mathematics from a higher standpoint,' as taught at university)" (Krauss et al., 2013, p. 155). Taking these studies into account, we developed a questionnaire for measuring prospective mathematics teachers' beliefs towards their perceived disagreements between mathematics at university level and school mathematics.

Method The institutional frame

Referring to this theoretical framework, the main target of our project is to investigate changes in prospective teachers' perception of a double discontinuity that could be explained by our approach of building bridges. Prospective secondary mathematics teachers in Germany are usually enrolled in the same mathematics courses as mathematics majors (e.g. analysis), particularly in the first semesters. A big challenge for all students in the initial phase of the studies is the task to complete a range of exercises every week as homework [START_REF] Ableitinger | Lernen aus Musterlösungen zur Analysis und Linearen Algebra: Ein Arbeits-und Übungsbuch[END_REF]. These tasks can be solved on the basis of the plenary lectures (by usual four hours per week) and are reviewed in additional small courses (by two hours per week) which are organized by student assistants. Our aim is to develop and establish the desired bridges in these introductory mathematics courses for prospective secondary teachers. Our focus is here to enrich the set of tasks for homework with socalled "teacher-oriented tasks" that are appropriate to illustrate connections between university mathematics and school mathematics to prospective secondary teachers.

Teacher-oriented tasks

We conceptualize specific tasks which potentially demonstrate bridges between school mathematics and university mathematics to a model of domains of teacher knowledge according to [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF]. More precise, we differentiate our teacher-oriented tasks referring to the subdomains of specialized content knowledge (SCK), knowledge of content and students (KCS), knowledge of content and teaching (KCT) as well as curriculum knowledge. One example that represents specialized content knowledge (SCK) within the notion of mathematical tasks for teaching is provided below. In this exercise, "Evaluating the plausibility of students' claims" and "Giving or evaluating mathematical explanations" (Ball et al., 2008, p. 400) are requirements which can be used well to describe the setting.

In the subsequent task from a mathematics contest for students ("Känguru der Mathematik 2009") the participants were asked to solve which of the following figures is the greatest one.

(A) √2 -√1 (B) √3 -√2 (C) √4 -√3 (D) √5 -√4 (E) √6 -√5
A student in grade 12 chose answer (E) and stated: "√6 -√5 is the greatest figure, because roots are monotone. So, the greater is x, the greater is f(x). Thus, their difference is the greatest, as well (by going more to the right)."

1. Analyze the student's answer. Where do you see problems with the argumentation? 2. Provide an own student-oriented answer to this topic.

Show in general: lim

The design

In order to gain empirical evidence for the efficiency of our method, prospective teachers in the relevant mathematics courses are assigned at random to a treatment group and a control group. While the control group is taught traditionally, the treatment group gets an extra teacher-oriented task for homework every week that focuses on bridging mathematics at university level and school mathematics. Our main research question is whether and how prospective teachers change their beliefs about the double discontinuity phenomenon based on our intervention in the introductory mathematics courses. In this paper we only refer to pilot studies where prospective teachers got homework including an extra teacher-oriented task on a trial basis. The main aim of these pilot studies was to develop an instrument for measuring teachers' perception of a double discontinuity.

During the winter semester 2015/16, two basic mathematics courses at the University of Kassel were selected in which the mentioned teacher-oriented tasks bridging mathematics and mathematics education was administrated: "principles of mathematics" and "analysis". Prospective mathematics teachers attend these courses usually in the first or in the third semester of their university studies. Three prospective teachers from the third semester were interviewed at the end of the semester. The data was analyzed by qualitative content analysis. We also developed a questionnaire with 16 items for measuring prospective teachers' beliefs concerning their perceived disagreements between mathematics at university level and school mathematics. The questionnaire was piloted in a mathematics course for prospective secondary teachers (N = 60) and is outlined in the following paragraph.

Discussion of results

The analyses of the interviews revealed that all prospective teachers took the contents dimension into account when reflecting their university studies. For example, the first prospective teacher (PT_1), believes, that in university mathematics, there are too little relations to school with regard to the content.

PT_1:

"There are not many connections -direct contents connections, so -as well as in the other lectures. And also in analysis I notice, that it -that the university stuff of the mathematics lecture -almost simply goes beyond school and it is more or less by chance, if there are contents, which fall together with school mathematics -I have the feeling."

Further, two of the prospective teachers also mentioned another aspect of the double discontinuity problem, i.e. their beliefs concerning the relevance of learning university mathematics with regard to school mathematics, such as can be found in the proposition of PT_2.

PT_2: "If I finished school, I would have the same status, which I want to teach the students. And if now deeper questions arise, I would not be able then to answer them, for instance, because I myself have never had this and then -such a teacher one also did not want formerly, who could only tell, what he has just done."

Indeed, [START_REF] Klein | Elementarmathematik vom höheren Standpunkte aus. Teil I: Arithmetik, Algebra, Analysis[END_REF] addresses both aspects in the double discontinuity phenomenon since on the one hand the problems at university may not suggest the things at school, and on the other hand, university studies may remain only a memory with no relevance upon teaching.

The collected data from the interviews also supported the development of our current questionnaire. Our first version (cf. [START_REF] Eichler | Disagreements between mathematics at university level and school mathematics in secondary teacher education[END_REF]) contained 9 items using a 6-point Likert scale to assess students' beliefs about the double discontinuity problem. The questionnaire was piloted in a mathematics course for prospective secondary teachers (N = 60) and seemed to provide good internal consistency (Cronbach's alpha 0.782). Interestingly, a higher reliability (Cronbach's alpha 0.831) was achieved when regarding only the prospective teachers in the course (N = 35) and not all university students in the mathematics course. As a possible reason for this phenomenon, we identified two theoretical discernable domains in our questionnaire. On the one hand, we asked items which contained a personal statement including the words "I" or "me" like "I think that I require a deep understanding of mathematics in order to teach mathematics in school." On the other hand, we provided a few statements such as "University mathematics has mostly little relation to school mathematics", which were rather matter-of-fact. These items might have led to a different extent of identification within the different groups of university students. To provide useful information, we applied both dimensions in our current questionnaire containing 16 items.

Taking into account the four levels of mathematical content knowledge based on [START_REF] Krauss | Mathematics Teachers' Domain-Specific Professional Knowledge: Conceptualization and Test Construction in COACTIV[END_REF], we derived further items from the interviews such as "By the use of university mathematics, gaps are filled in the mathematical knowledge that is required in school". We grouped all items into three subscales which we identified through our development process: "contents relationship", "relevance for profession" and "higher standpoint". In a further pilot study with prospective secondary teachers in higher semesters (N = 24), we approved these subscales to be internal consistent1 with a total reliability value of Cronbach's alpha .911. Moreover, we used our qualitative approach in order to validate our survey afterwards.

Contents relationshipbeliefs concerning the connections between university mathematics and school mathematics on the contents dimension (4 items; Cronbach's alpha .821):

4.

University mathematics offers many parallels to school mathematics with regard to contents.

9.

School mathematics and university mathematics are applied to each other in contents.

11. * In university mathematics, there are too little relations in contents to school.

12. * School mathematics and university mathematics are two different worlds with regard to contents.

Relevance for professionbeliefs concerning the relevance of university mathematics for the later profession as a school teacher (6 items; Cronbach's alpha .814):

1.

University mathematics is very useful for the teaching profession.

3. * I will hardly ever need university mathematics after studying.

5.

By the use of university mathematics, I am well prepared to the job profile of a mathematics teacher.

10. Without university mathematics, I could hardly teach mathematics in school.

15. * Learning mathematics at university is not so important for the teaching profession.

19.

The relevance of university mathematics for the teaching profession is2

Higher standpointbeliefs concerning the usefulness of university mathematics as a higher standpoint for elementary mathematics (6 items; Cronbach's alpha .818):

2. University mathematics helps me to get deeper into school mathematics.

6.

By the use of university mathematics, gaps are filled in the mathematical knowledge that is required in school.

7.

By the use of university mathematics, I gain a deeper understanding of concepts in school.

8.

By the use of university mathematics, I understand relationships within school mathematics much better.

14.

Learning mathematics at university promotes me to be in thinking "one step ahead" of the students.

16.

As a mathematics teacher, an in-depth mathematical content knowledge is required. The possible range of scores for each component is between 1 and 6. Higher scores correspond to more positive beliefs (by reversing the responses to the negatively formulated items indicated with an asterisk*).

Concluding remarks

The main topic of this paper was to discuss our approach of measuring prospective mathematics teachers' beliefs towards their perceived disagreements between mathematics at university level and school mathematics that is known as double discontinuity. In order to be able to investigate changes in prospective teachers' beliefs referring to the double discontinuity problem, we chose a mixed methods design and developed a questionnaire including 16 items that actually seems to measure these beliefs. Grounded on our preliminary results, the following steps of our research will be a comparison between two groups of prospective teachers -one group in a traditional course and one group in a course using the mentioned teacher-oriented tasks to prove if the type of the course has an effect of the prospective teachers' beliefs. Since a variety of other factors may be related to our outcomes, we also collect among others data to study interest [START_REF] Schiefele | Der Fragebogen zum Studieninteresse (FSI)[END_REF] and study satisfaction [START_REF] Dargel | Zielbindung und Zielplanung: Entwicklung und Überprüfung eines Interventionsprogramms zur Steigerung der Zieleffektivität[END_REF] in a pretest and a posttest as well as additional items referring to relevant demographic and academic background information.

Figure 1 :

 1 Figure 1: Task "roots" for prospective secondary mathematics teachers

Table 1 : questionnaire for measuring prospective teachers' beliefs concerning a double discontinuity

 1

Proceedings of CERME10

7 Personal items with Cronbach's alpha .845 and 12 non-personal items with Cronbach's alpha .858.Thematic Working Group 18Proceedings of CERME10

Whereas in the previous items the prospective teachers may choose an option in a scale from "strongly disagree" to "strongly agree", the last item refers to a scale from "very low" to "very high".

Acknowledgment

The related project is supported by the Federal Ministry of Education and Research (BMBF).