domain: math shs

text:
Ghislaine Gueudet
email: ghislaine.gueudet@espe-bretagne.fr

Veronica Parra
email: vparra@exa.unicen.edu.ar

Teachers' collective documentation work: A case study on tolerance intervals

Keywords: Documentation work, resources, teachers' knowledge, tolerance intervals, variability

In this paper we use the documentational approach to investigate teachers' collective work. We follow two teachers, preparing together a lesson on tolerance intervals for grade 11. We identify Mathematical Knowledge for Teaching (MKT) that influences the use of resources by the teachers. We evidence that their collective work fosters important documentation work; but we observe significant differences between the documents developed by the two teachers.

Introduction

Teachers interact with curriculum resources in and out-of-class (TWG22 call for papers). In previous works we have identified the importance of these interactions in terms of teachers' professional development (Gueudet, Pepin & Trouche 2012), and we have evidenced that teachers often work collectively with resources. In this paper we study further this collective work of teachers with resources and its consequences.

The work presented here takes place within the French national project REVEA1 (Living Resources for Teaching and Learning). We consider the case of two mathematics teachers at upper secondary school in France teaching sampling variability in statistics in grade 11. We firstly expose the theoretical perspective we use and our methods. Then we present the data we collected, and the context of teaching sampling variability in France. Finally we expose our analyses of the teachers' work and of its links with teachers' knowledge in particular.

Investigating teachers' documentational work: Theory and method

We use for our research the theoretical and methodological perspective of the documentational approach (Gueudet et al. 2012). Mathematics teachers interact in their work with a large range of resources [START_REF] Adler | Conceptualising resources as a theme for teacher education[END_REF]. Resources designed for teaching purposes like textbooks or software, resources coming from the students, e-mails exchanged with colleagues etc. Teachers choose resources, transform them, use them in class; we call this work "teachers' documentational work". In previous research, we have evidenced that this work is closely linked with teachers' professional knowledge. The choice of resources by teachers, the way teachers modify and use the resources is driven by their professional knowledge (and this is called an instrumentalisation process, drawing on Rabardel's instrumentation theory, [START_REF] Rabardel | Les hommes et les technologies: approche cognitive des instruments contemporains[END_REF]. In a reverse way, the features of the resources used modify teachers' knowledge (in an instrumentation process). In the documentational approach, we consider that from a set of resources teachers develop a document: transformed resources associated with a scheme of use [START_REF] Vergnaud | Toward a cognitive theory of practice[END_REF]. A scheme of use comprises the aim of the activity, rules of action and professional knowledge. The development of a document is called a documentational genesis.

Teachers' Communities of Practice (CoP, Wenger 1998) have a shared repertoire that the documentational approach interprets as shared resources. In previous works [START_REF] Gueudet | Collective design of an e-textbook: teachers' collective documentation[END_REF] we have investigated the documentation work of a CoP (Sésamath, an association of teachers in France) designing online resources and identified the development of shared documents. Here we study a more "ordinary" CoP, composed of two teachers working together for the preparation of their courses. We are interested in particular in the commonalities and differences in professional knowledge within the documents developed by these teachers. The research question we investigate can be formulated as:

How do professional knowledge and resources interact in the collective design and implementation of a lesson? Concerning teachers' professional knowledge involved in the schemes, we are especially interested in the identification of Mathematical Knowledge for Teaching (MKT, [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]: professional knowledge linked with the mathematical content to be taught.

The documentational approach is associated with a specific method, called "the reflexive investigation method". Documentational geneses are long term processes; moreover documentational work can take place everywhere and at any time. Thus we follow teachers over long periods of time; and involve them actively in the collection of data. These data are interviews of the teacher; videos of the teacher's work in class and out-of-class (videos of collective work, if teachers work together); resources chosen and transformed by the teacher.

For analyzing the data, we start with the transcribed interviews of the teachers. We identify in them the aims of the teacher's activity. For each aim, we search in the data for the resources used, and the other components of the document (rules of action and professional knowledge) . We submit these elements to the teacher who corrects and complements them if needed. We present in the next section the data collected for the case we study here.

Data collected and context

We follow since 2014 two mathematics teachers in an upper secondary school of a middle-sized town in France: Valeria and Gwen. They are both very experienced: Valeria teaches for 34 years, Gwen for 36 years, they both regularly follow teacher education sessions and are trainers for new teachers in their school. They also regularly work together, we consider them as a CoP [START_REF] Wenger | Communities of practice. Learning, meaning, identity[END_REF]. In 2015-2016, they decided to take two grade 11 classes called "economics and science", a specialty they taught for the first time. We followed their work for these classes, in particular for a chapter entitled: "sampling variability" (that they both used to teach grade 11 "science", with a similar content). For this chapter, we video recorded their common preparation (one hour), their individual courses (four hours each), and for each of them an individual post-teaching interview. We collected all the resources they used and produced, and the students' productions for the final assessment of the chapter. For both teachers we identify the professional knowledge/beliefs, the possible origin of these beliefs, the consequences in terms of the activities/resources produced and the resources used.

Sampling variability is taught in France since 2010. This teaching starts in grade 10 where the idea of sampling variability is introduced using material like coins and dice and simulations on the calculator and on the spreadsheet. In the curriculum in France, the concept of tolerance interval is central in the teaching of sampling variability. In grade 10 the students have to learn how to identify the population; the sample and its size n; the probability p of a given feature in the population, and the frequency f of this feature in the sample. A first tolerance interval is introduced, without justification: [p-1/√n, p+1/√n]. If f does not belong to this interval, the students learn to reject the hypothesis "the sample follows the population's law" with a 5% risk level. At grade 11 (scientific or economics and scientific) the binomial distribution is introduced; it provides another tolerance interval, which can be found using the table of the binomial distribution produced for example with a calculator or a spreadsheet. The chapter we followed concerns the introduction and use of this interval. At grade 12, the normal distribution is presented; this leads to a new interval (the asymptotical tolerance interval).

Many research works have investigated teaching variability; they emphasized the specific nature of reasoning in probability [START_REF] Steinbring | The Theoretical Nature of Probability in the Classroom[END_REF] and the need for particular knowledge to teach this subject [START_REF] González | Conceptualizing and assessing secondary mathematics teachers' professional competencies for effective teaching of variability related ideas[END_REF]. [START_REF] Eckert | Contextualising samplingteaching challenges and possibilities[END_REF] used the notion of Mathematical Knowledge for Teaching Probability (MKTP) for characterizing the specific knowledge needed by the teachers in probability and statistics, and for sampling variability in particular. Finding situations in relevant contexts; emphasizing the idea of variability, using the different kinds of possible representations all require specific knowledge from the teacher. Our aim here is to investigate how the interactions with resources are shaped by, and contribute to MKTP.

Results

In this results section, we firstly consider the two teachers' documentation work, during the common preparation session, the lessons taught by each teacher and finally during the design of a common assessment. Then we present our analyses of the most important aspects of this documentation work (in terms of MKT involved), focusing on the collective-individual articulation.

A collective documentation work

During the common preparation, Valeria and Gwen used several textbooks (9 different textbooks). They did not actually chose resources together, but drew on exercises and problems in the textbook to illustrate their declarations. They also talked about resources they intended to use: exercises, software (GeoGebra, spreadsheet) and the calculator.

Their discussion during the common preparation started by stating a difference: Valeria intended to use the spreadsheet from the beginning and during the whole lesson. The students have learned, in the "binomial law" chapter to produce with the spreadsheet and read tables displaying the value of P(X=k) and P(X≤k), when X is a random variable following a binomial law of parameters N and p. Valeria wanted to recall this, then to introduce the binomial law tolerance interval and the method for finding it, using the P(X≤k) table produced with the spreadsheet. Gwen, in contrast, intended to use only the calculator, and no other software. After introducing the binomial law tolerance interval, she wanted to ask students to write and implement on their calculator a program producing the tolerance interval.

In the other aspects of the common preparation, Valeria and Gwen agreed on all the points they discussed. They mentioned in particular the need to recall the grade 10 tolerance interval and to compare it with the new interval introduced.

All these aspects discussed during the common preparation are present in the lessons actually taught by Valeria and Gwen. We analysed these lessons drawing on the observations and videos in class, the resources collected and the post-lesson interview.

Valeria started indeed by recalling how to produce and read the binomial law table with the spreadsheet. She also recalled the grade 10 interval with exercises chosen in a textbook's "revision section". Then she introduced the new interval through a problem concerning overweight in USA. This problem came from another textbook, and she modified it in particular by suppressing the table giving P(X≤k), because she wanted the students to produce it themselves with their calculator. She presented how to find the interval from the table P(X≤k). Valeria insisted on the need to formulate very precisely the decision rule. At the end of the chapter, she worked with her students on the algorithm: the students implemented it on their calculator, but this program was actually not used as a tool to find the interval in exercises.

Gwen started with a problem that she built herself, about red-haired people in Scotland (inspired by a textbook problem with a different context). The first part of the problem recalled the grade 10 interval, and more generally the idea of sampling variability. The second part of the problem introduced the new interval. Just after this session, Gwen worked with the students on the production of an algorithm and its implementation on the calculator to find the binomial law interval. Afterwards this program was always used to find the interval. Gwen said that she found the binomial law interval too technical, she did not want her students to learn how to find it. She preferred to use it as an opportunity to work on algorithms. She distributed a sheet to the students presenting the interval and a diagram. Then she proposed different exercises about decisions; in particular one exercise with samples of different sizes.

The final assessment of this chapter was also the final assessment of the year for the two Grade 11 ES classes. Valeria and Gwen wrote it together; we analyse their documentation work drawing on the resources they used and produced; the e-mail they exchanged and their interviews. The control text comprised one exercise on tolerance intervals. This exercise, with an introductory text (figure 1) and three questions, concerned the rate of twins in India; it came from a textbook. Valeria and Gwen modified the initial text which was, in their opinion, too long and complex.

" India: Kodinji, the mysterious twins village In the state of Kerala (south-west India), there is an amazing village. The rate of twins is much higher than the national average. 440 twins live indeed in this town for 14600 inhabitants. This average is outstanding, since the national average is 16 twins for 1000 births". Extract of an article (Courrier International, 2009) Let X be the random variable counting the number of twins in a 14600 Indians sample. They modified the first question: in the textbook the parameters of the binomial law followed by X were given; they wanted the students themselves to find the parameters. They also changed the second question, where the students are asked to produce the tolerance interval, to display the two different methods expected for each class: use of a table (given in the text) for Valeria's class, use of the calculator's program in Gwen's class. Question 3 was left unchanged. 42 students were present at the final assessment. In the first question, 35 students justified that X followed a binomial law of parameters n and p; 29 students determined correctly the value of n, but only 15 the value of p. In the second question, 19 students determined correctly the endpoints of the tolerance interval. In the third question, 19 students justified correctly the rejection of the hypothesis ("the Kodinji village follows the national figures").

Valeria and Gwen documentation work and use of resources: General statements

We can notice that like other research works using the documentational approach (Gueudet et al. 2012) this description evidences that Valeria and Gwen are designers of their own resources. They use various curriculum resources, but transform most of them. Only some exercises texts are left unchanged, and one textbook extract presenting the binomial law interval (for Gwen). Moreover, textbooks (on paper, they do not use digital textbooks) are central resources, used as a reference to discuss the lesson plan, and to choose exercises for practicing the new methods, for the assessment/test, or to find an introductory problem (for Valeria). More surprisingly, they did not search the Internet for resourcesthis can be a consequence of their common preparation: Valeria and Gwen sometimes search the Internet for preparing their lessons, but always at home. For the common preparation they were at school with no Internet access. Another result already well known in the documentational approach is that the observation of students (their written texts, or oral discussions in class) constitute a very important resource for the teachers, leading to a constant modification of the resources produced along the documentation work. Both Valeria and Gwen intend to modify this lesson on tolerance intervals next year, because they consider that the students made too many mistakes in the final assessment/test.

Documents developed by Valeria and Gwen

In this section we analyse our data in terms of documents developed by Valeria and Gwen. Since our focus is on MKT, we do not give a complete description of each scheme of use but only mention the aim of the activity and the MKT involved. We have chosen three examples of documents, corresponding to different situations in terms of similarities or differences.

Valeria and Gwen had a shared aim that can be described as "Recalling previous knowledge needed for the binomial law tolerance interval". They both considered that this new chapter must start by recalling the grade 10 interval, because they knew from their observation of students during the year that "many students do not remember this interval" (Gwen even added that some students perhaps never saw it, since some colleagues keep this content for the end of the year and run out of time). This shared MKT lead however to two different documents for this aim, because of the teachers' different resources: Valeria used a lot the classroom textbook, and thus proposed revision exercises coming from this textbook, while Gwen wrote her own problem text.

Valeria and Gwen also shared a general aim that can be presented as: "Teaching how to find a tolerance interval with a binomial law". During the previous years, Valeria has developed for this aim a document including various resources: the spreadsheet (as software or in the calculator), exercise texts, the illustrating diagram, and MKTP: "The students must learn to find the endpoints of the interval by reading the table". This knowledge can have different sources; we claim that it comes in particular from institutional texts (the official curriculum) and from textbooks. For the same aim, Gwen has developed a different document, including: the calculator, an algorithm, the illustrating diagram, exercise texts as resources; and MKTP like: "The binomial law interval is too technical"; "there are no questions about the binomial law interval at the baccalaureate2 "; and the MKT "it is important to work with students on algorithms". This knowledge comes from reading the texts of the baccalaureate, and from a personal mathematical difficulty: Gwen declared that she "cannot remember [herself] how to find the endpoint of the interval". Moreover she considered that this grade 11 curriculum is only a transition between the grade 10 interval and the grade 12 interval (with the normal distribution) while algorithms are always present in the baccalaureate texts. Valeria and Gwen discussed this difference during the common preparation. Valeria integrated in her lesson the programming of the algorithm on the calculator, but she did not want her students to use it, because she feared that the students do not really understand the algorithm and use their calculator as a "black box".

For the aim: "Assessing the students' ability to determine and interpret a tolerance interval", Valeria and Gwen used shared resources: they wrote the assessment text together drawing on the same textbook exercise. The choice of this exercise was guided by MKT firstly expressed by Gwen, and adopted by Valeria: "the students must learn to find information in a text". Nevertheless, the text produced was also transformed to incorporate the use of two possible methods, because of their different MKT concerning how to find the binomial law tolerance interval.

In Table 1 below, we synthesise these three documents, evidencing the common and different elements.

Aim

Conclusion

In this paper we investigated the research question: "How do professional knowledge and resources interact in the collective design and implementation of a lesson?" in the case of a lesson on tolerance intervals for Grade 11 students in France. In the frame of the documentational approach, investigating how professional knowledge and resources interact means investigating the documents developed by teachers. For each of the two teachers we followed, we observed that they developed an important design work, choosing resources, associating and modifying them. This work was guided by their professional knowledge, in particular MKT and MKTP. We observe that this MKT is mostly of the type: "Knowledge of Content and Students"; deepening the analysis in terms of types of MKT is an interesting perspective for further work. In a reverse way, resources influenced the development of MKT, and this MKT can be different for each individual teacher. For example the official curriculum influenced Valeria and Gwen in different ways: while Valeria aligned with the curriculum about tolerance intervals, Gwen attached more importance to the algorithms. She developed a personal interpretation of the official curriculum, not focusing on the chapter she taught but taking into account the whole year.

In previous works (Gueudet et al. 2012) we evidenced that collective work is present in many aspects of teachers' activity and that it is a stimulator of documentation work, especially when it takes place within CoPs. In [START_REF] Gueudet | Collective design of an e-textbook: teachers' collective documentation[END_REF] we analysed the common documentation work in a CoP: an association of teacher designing an e-textbook. We evidenced that they developed common documents, drawing on their individual documents. In the present study we investigated a CoP composed by two teachers preparing their courses together. We evidenced that, in spite of the collective work the documents developed by the two teachers for the same aim are never completely identical. The consequence of the collective work is that these documents sometimes share common elements. The main reason for the differences seems to be the long experience of both teachers: they already developed in previous years documents for the same aims, and thus have MKT or MKTP associated with these aims and also specific resources. The collective work can bring new resources (the algorithm on the calculator for Valeria) or new knowledge (the students must be able to find information in a text, for Valeria again), but the previous knowledge developed during interactions with resources over many years is still present and produces differences in the documents.

These teachers will go on working together; with a longer common work, evolutions may take place, and we will try to analyse these evolutions. We also hypothesize that evolutions of practice are more likely to take place in teachers' CoPs when the members of the CoP are involved in a common design activity (for example in the Sésamath case, [START_REF] Gueudet | Collective design of an e-textbook: teachers' collective documentation[END_REF]

Figure 1 :

 1 Figure 1: Introductory text of the control exercise (our translation)

Table 1 : Synthetic presentation of documents developed by Valeria and Gwen. Shared elements are in italics.

 1

			Resources used		MKT/ MKTP
	Recalling		Valeria: Revision exercises in the	"Many students do not remember the
	previous		classroom textbook		grade 10 tolerance interval"
	knowledge		Gwen: Her own problem text
	Teaching how to	Valeria: Problem and exercises texts	Valeria: "The students must learn to
	find a tolerance	from	different	textbooks,	the	find the endpoints of the interval by
	interval with the	spreadsheet,	algorithm	on	the	reading the table"; "they must not use
	binomial law		calculator (coming from the collective	the calculator as a black box"
			work)
			Gwen: Problem composed herself,	Gwen: "it is important to work with
			exercises from different textbooks,	students on algorithms" "the binomial
			algorithm on the calculator	law interval is too technical"
	Assessing	the	Shared assessment text written	"The students must be able to identify
	students' ability	together from a textbook exercise, but	information in a text" (shared)
	to interpret find tolerance interval and a	integrating two possible methods.	+ MKT/MKTP described in the above line

  in the context of professional development, Pepin & Miyakawa 20016). We intend to investigate this hypothesis in further research.

https://www.anr-revea.fr/ Thematic Working Group

Proceedings of CERME10

Proceedings of CERME10

In France, the final secondary school assessment, at the end of Grade 12.Thematic Working Group 22Proceedings of CERME10


text:
Audrey Daina
email: audrey.daina@hepl.ch.

From textbook to classroom: A research on teachers' use of pedagogical resources in the context of primary school in the French speaking part of Switzerland

Keywords: teachers' practice, textbooks, primary school, cross-analysis

, which allows us to describe teachers' choices as a coherent system that does not depend only on learning objectives but also on characteristics of the profession and on certain constraints.

Introduction

School textbook is a field of research which has been widely increasing since the last three decades as described by Fan, Zhu and Zhenzhen (2013) survey study for the special issue of ZDM "Textbook Research in Mathematics Education". The authors conclude this survey by suggesting future directions in this field of research: it is necessary for researchers to establish a more solid fundamental conceptualization and theoretical underpinning of the role of textbooks and the relationship between textbooks and other variables not only in curriculum, teaching and learning but also in a wider educational and social context" (Ibid,p. 643) The large-scale cross national research study TIMSS allowed us to have a wider vision on the role of textbook in connection with social or political issues as presented by [START_REF] Valverde | According to the book: Using TIMSS to investigate the translation of policy into practice through the world of texbooks[END_REF] who analyze textbooks in terms of cross-national differences in educational opportunities "the configuration of social, political and pedagogical conditions to provide pupils chances to acquire knowledge, to develop skills and to form attitudes concerning school subjects" (Ibid, p. 6).

Our contribution goes in this direction by presenting the results based on our PhD theses [START_REF] Daina | Utilisation des ressources: de la préparation d'une séquence à sa réalisation en classe de mathématiques[END_REF], which aimed to describe and analyze:

 the social and political context in which the uniform textbooks are designed in French speaking part of Switzerland, known as "Moyens d'Enseignement Romand pour les Mathématiques"(MERM);  how five teachers in Geneva use this resource for mathematics, considering their "ordinary" practices.

Our research brings a different focus on the question of the connections between social and political issues and teachers practices by considering a qualitative methodology and a theoretical framework which allowed us to understand the practice we observe as a complex but coherent system that is dynamic and does not depend only on learning objectives but also on characteristics of the profession and on certain constraints we can infer from the study of the context [START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An exemple in a French 10th-grade class[END_REF].

Research design

Our research is a case study, which is based on interviews with teachers and classroom observation. The data were collected from April to June 2009 in five classrooms in Geneva in two different schools. In every class, we collected the following data: an interview before the teaching sequence; the observations in class with video recordings of the various activities of the teaching sequence on the theme of area measure; an interview at the end of the teaching sequence.

To understand the political and social issues connected to our context of study, we collected selected documents, which showed different aspects of the official textbook design process (institutional requirements, project design, review report, etc.)

Theoretical frameworks and research questions

Our project aimed to study how teachers use the textbook and prepare their lessons which is an "invisible" part of the teachers' practices. In addition, we wanted to observe ordinary practices, with all the complexity implied. Therefore, to build our theoretical framework and our methodology we lean on two theoretical approaches: Margolinas' model (2005) and the model of Robert and Rogalski (cross-analysis of the teacher's activity). We did not combine this two framework, instead we asked different independent questions relating to these two theoretical frameworks which we study using tools and methodology from the relevant framework. In this presentation, we will focus on the second model and the results related to it.

The model of Robert and Rogalski: Cross-analysis of the teacher's activity

This model combines two different approaches. Firstly, a didactic approach: teacher's practices are linked with the learning objectives. In this sense, the knowledge content of the teaching and the way the teacher organizes his teaching are analyzed using a didactic theoretical framework. Secondly, a psychological approach from cognitive ergonomics: the teacher is considered as a professional whose practices are subject to a professional contract, with particular goals, repertories of action, representations of mathematical objects and their learning, and, more generally, personal competencies which determine his activity.

Robert and Rogalski ( 2002) determined five dimensions to analyze teacher's activity.

-The cognitive and mediation dimensions which concern the set of teacher's choices about the content and the organization of the knowledge before (cognitive dimension) and during the class-time (mediation dimension).

The combination of this two dimensions allows us to trace what "kind" of mathematics is proposed in the class « la fréquentation des mathématiques qui est installée, ce qui est valorisé par les scénarios et leur accompagnement et ce qui pourrait manquer » (Robert & Rogalski, 2002, p. 514).

This first part relates to the description of the teacher activity, then we would like to interpret and highlight what determines these practices. We refer to three dimensions:

-The social, the personal and the institutional dimensions which permit to define the constraints and the personal aspects of each teaching project.

These three dimensions are studied based on interviews, deduced from the observed teaching sequence and also from the study of the institutional and local context.

This bring us the following research questions:

-What type of mathematics is promoted by various teachers, according to different scenarios and their execution in different contexts? Which kind of logic of action can we observe? Are these practices compatible with the didactic and pedagogical choices of the MERM resource designers?

-What hypotheses can be formulated concerning the dimensions (social, personal, institutional) that determine the teachers' practices of our study? How do the MERM influence the observed practices?

Method of analysis

The analysis was realised according to the following stages:

-A transcription of the lessons we observed was made using Transana, a software which allowed us to have permanently the video and the transcription on the same screen and give the possibility to introduce time codes and keywords. Referring to methodology used in various research using the double approach framework, we divide each lesson in temporal phases, which we call an episode and corresponds to a content unity.

-The scenario of the whole teaching sequence was then reconstituted and analysed to clarify on the one hand what kind of mathematical content is presented during the teaching (cognitive dimension) and, on the other hand, the dynamics in which the content appear during the classes (mediation dimension). The episodes we defined in the first stage of the analysis were coded according to the mathematical content and the teaching strategies (for more details, see Daina 2013).

we wrote a report based on the interviews with teachers which allowed us to define the "profile" of each teacher, a synthesis of all the information we collected, which give information, among other aspects, on social, personal, institutional dimensions.

Selected results

French speaking Switzerland context and pedagogical resources

Switzerland has a highly decentralized system with no federal or national Ministry of Education. Each of the 26 cantons which composes the country has its own education legislation. However, four Regional Conferences, including French-speaking Switzerland which is represented by the CIIP (http://www.ciip.ch), have led to some effective coordination since the last five decades, drawing up common curricula, publishing material, jointly managing institutions and recognizing qualifications and admissions.

In the 1970s in all French speaking Switzerland a common official set of pedagogical resources, the MERM, was designed by the CIIP appointed group of experts and teachers from the different cantons. This first edition originates from a double necessity: a will of « inter-cantonal » coordination of education (mathematics but more widely all the disciplines) and the introduction of a new curricula (CIRCE I), linked with the reform of the "modern mathematics ". In the 1990s, modern mathematics were abandoned and the pedagogical resources have been renewed in the 1990s according to a new educational paradigm based on problem solving, strongly influenced by the socio-constructivist approach.

Switzerland has therefore a long tradition of diverse cantonal educational policies but also the willingness to coordinate the educational system in order to facilitate in particular communication and student mobility. The MERM are the "symbol" of this process, especially in mathematics because they were the first to be done. In fact, the MERM are the result of a long process of discussions and compromises because they have to be approved and accepted by all the cantons of the French speaking part of Switzerland. It is necessary to allow three or four years to realize the MERM for one degree.

We have to take into account this complex context in our observation and analysis of this resource (institutional dimension). The MERM must for example be compatible with all the plans of studies. They cannot thus be too prescriptive and require an opening.

Besides, the pedagogical resources are central to the reforms and innovations regarding mathematics education and more than just simple resources, they have the role of promoting innovation, in particular thanks to the teacher's textbook which describes the didactic and educational choices. They have to introduce the changes and harmonize the practices. This is also a critical element we have to take into account in our analysis (institutional dimension).

The MERM have been thought as a set of resources, therefore they mostly consist of a succession of activities for class, regrouped in 6 to 8 main themes, without hierarchy. Contrarily to textbooks in other countries, they do not give a day-to-day organised plan for teaching, which remains the teachers' responsibility. To do this, the teacher must provide a considerable amount of preparation prior to the lesson and our PhD theses aimed to study the ordinary practices in connection with this resource in order to make visible this essential part of the work of the teacher.

Case study: Mathilde and Sophie

To exemplify our methodology and present some results of our research, we will now present two examples of our case study. We will first provide the chosen information about their "profile", which gives information about the personal and social dimensions. Then we will present and compare their teaching sequences.

Mathilde and Sophie were both young teachers (4 and 5 years of experience) and they worked in the same school. The year of our experiment, they taught the same degree (6P) and they met twice a month and collaborated to develop mathematics lesson plans. Looking at the exchanges between the teachers during this meeting, we saw that they spoke little about the teaching objectives which seemed to be implicitly known and shared.

Mathilde said that the textbook was the reference and did not feel the need to talk more about the objectives. She specified during the interview that her colleagues "trust" her concerning the choice of the activity because it's impossible, according to her, to choose the activities in the textbook without assistance. Sophie also speaks about "trust" during the interview what shows the importance of the collaboration in the preparation, by filling what is felt as a "lack" of the resource.

However, the study of Sophie's and Mathilde's profiles allows us to highlight that the objectives of both teachers are very different. For Sophie the calculating procedures in particular the introduction of the techniques of calculation of areas for triangles and parallelograms represent an important aim in the sequence. For her part, Mathilde considers that the main objective concerns the understanding of the notion of area and the formula to calculate the area of a square. Even though their objectives are different, this point remains implicit throughout their collaboration.

The scenarios analysis and their carrying out in class will allow to see how will evolve the educational projects of the two teachers whose starting point, by the way, is the same list of activities. The analysis of the series of activities proposed in Sophie's classroom show a split teaching sequence where activity from the official textbook are often mixed with improvised tasks on the blackboard. We represented a structure in four parts: an introductory activity, a series of activities on the area of square and rectangle, an introduction in the measure of areas of the other regular polygons and a session of revision.

Although Sophie and Mathilde based there teaching sequence on the same common project, we see well how both scenarios evolved. While the starting point is almost identical, the gap is widening in the course of the sessions, leading to differents mathematics promoted in each class (considering the cognitive dimension, Daina 2013). It is nevertheless interesting to note that in the interviews, Mathilde and Sophie did not seem conscious of these differences.

The analysis of the first session, while Sophie and Mathilde proposed the same activity, "Fraction of a field", allowed to see more in details this difference.

Fraction of a field

Father Joseph has a square field. He splits it using three ropes passing through the vertices or the midpoints of the sides. One of his sons, Francis, will inherit the grey part of the field.

What fraction of the field will he receive?

The table below synthesizes the succession of the phases during the carrying out of the activity in Sophie's and Mathilde's class. Here we see very clearly that, although the same activity was proposed to the pupils, its management in class is totally different between the two teachers (mediation dimension).

In the class of Mathilde the number of phases is limited. The instruction is very short. During the pooling, the interactions testify of a discussion between the teacher and the pupils who have an active participation in advancing the discussion. Corrections are made individually.

In the class of Sophie we observe at first a longer instruction time. Sophie testifies of a will to make sure that the pupils understand well « what they have to do ». The progression of the project is managed collectively. Sophie makes regular pooling in the course of which, she directs the pupils on a strategy of resolution which is going to become common. However, the task is also diverted on secondary tasks which are in connection with knowledge bound to calculate the area of parallelograms or triangles which are central objectives for her. Sophie's more personal project enters thus in tension with the progress of the main activity which becomes a material medium to introduce new knowledge in a lecture style of teaching on whiteboard.

We can thus observe a big variability in the practices of these two teachers. What is really questioning is that these differences do not seem to worry them and in spite of their various ways of functioning, the teachers find an interest to prepare together the teaching sequences.

The space provided in this paper only allows to give a limited insight into our research result but some analysis made permit to identify tensions bound to the use of MER. The first cause of tension results from the "shape" of this resource. Indeed, the quantity of the activity proposed is too important and a novice teacher can't know them all and make a real choice. As said in the introduction, there is no supplied sequence and not much piece of information given about each activity. This leads the teachers to follow what make other colleagues, even if sometimes they did not share the same objective and teaching strategy. Furthermore, as we show in the first part, the specific status of this resource will act as a strong constraint on the teacher who will use it to be in "conformity", even if they do not follow the socio-constructivist approach and will finally distract from the goal of the activities.

Figures 1

 1 Figures 1 and 2 allow to have a global vision of Mathilde's and Sophie's scenarios. Mathilde provides a teaching sequence we analysed in three parts: an introductory session, three sessions dedicated to the area of square and rectangle and a session of introduction of the area of other polygons. This corresponds to what is proposed in the textbook.

Figure

  Figure 1 Mathilde's scenarios

Figure

  Figure 2 Sophie's scenarios

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