text:
Annalisa Cusi
email: annalisa.cusi@unito.it

Francesca Morselli
email: morselli@dima.unige.it

Cristina Sabena
email: cristina.sabena@unito.it

Designing and analysing the

Keywords: Formative assessment, technology, digital resources, task design. Thematic Working Group 21

Introduction

This contribution stems from the European Project FaSMEd ("Improving progress for lower achievers through Formative Assessment in Science and Mathematics Education"), aimed at investigating the role of technologically enhanced formative assessment (FA) methods in raising students' attainment levels.

Within FaSMEd, FA is conceived as a method of teaching where evidence from learning is used to adapt both teaching and learning. Wiliam andThompson (2007, in Black &[START_REF] Black | Developing the theory of formative assessment[END_REF]) focus on three central processes in learning and teaching, which represent the aims related to the collection, interpretation and exploitation of these learning evidence: (a) Establishing where learners are in their learning; (b) Establishing where learners are going; (c) Establishing how to get there.

In the model developed by Wiliam and Thompson these three central processes are connected to the three main agents that intervene (the teacher, the student, the peers) and to the FA key-strategies that could be activated: (A) Clarifying and sharing learning intentions and criteria for success; (B) Engineering effective classroom discussions and other learning tasks that elicit evidence of student understanding; (C) Providing feedback that moves learners forward; (D) Activating students as instructional resources for one another; (E) Activating students as the owners of their own learning.

A theoretical model to analyse the use of technology in FA practices has been elaborated within FaSMEd (Aldon et al., in print, and Cusi, Morselli & Sabena, 2016). The model extends Wiliam and Thompson's model, taking into account three main dimensions: (1) the five FA key-strategies described by Wiliam and Thompson (ibid.); (2) the three main agents that intervene; (3) the functionalities of technology. The third dimension -the functionalities of technologywas added within FaSMEd to focus on the ways in which technology can support the three agents in developing the FA strategies: (a) Sending and sharing, that is the ways in which technology supports the communication among the agents of FA processes; (b) Processing and analysing, that is the ways in which technology supports the processing and the analysis of the data collected during the lessons; (c) Providing an interactive environment, that is when technology enables to create environments in which students can interact to work individually/in group on a task or to explore mathematical/scientific contents.

We argue that the FaSMEd three-dimensional framework may represent a useful tool for:

 designing digital materials for technology-enhanced FA practices and the corresponding methodology, and  analysing how these materials are implemented in the classroom.

In this paper we will refer to the study carried out in Italy, where we added the fundamental assumption that, in order to raise students' achievement, FA has to focus not only on cognitive, but also on metacognitive factors [START_REF] Schoenfeld | Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics[END_REF]. For this reason, our design is aimed at i) fostering students' ongoing reflections on the teaching-learning processes, and ii) focusing on making thinking visible (Collins, Brown & Newmann, 1989) through students' sharing of their reasoning with the teacher and the classmates, by means of argumentative processes.

In the following we present the design of digital materials and of the methodology for their implementation in the classroom. We will then focus on the specific case of "helping worksheets" and analyse its implementation in a case study. The analysis will be based on the FaSMEd threedimensional framework. The analysis of FA strategy C (Providing feedback that moves learners forward) will be deepened with reference to the four major levels of feedback introduced by Hattie and Temperley ( 2007): (1) feedback about the task; (2) feedback about the processing of the task;

(3) feedback about self-regulation; (4) feedback about the self as a person.

Design of the digital materials and of the methodology for their implementation

The research developed within FaSMEd has been built on the model of design-based research [START_REF] Cobb | Design experiment in Educational Research[END_REF], so it is based on successive cycles of design, observation, analysis and redesign of classroom sequences. The design experiments we carried out in Italy were characterised by three subsequent cycles of design. The first two cycles were carried out in March-May 2015 and in September-December 2015. The third cycle started in May 2016 and has not been completed yet. The results we present in this paper refer to the first two cycles of the design.

In tune with the theoretical assumptions presented in the previous paragraph, we chose to use a technology that supports the students in sharing, discussing and comparing both their written productions and the strategies developed to carry out the different tasks. Specifically, we explored the use of a connected classroom technology (CCT), which creates a network between the students' tablets and the teachers' laptop, allowing the students to share their productions, and the teacher to easily collect the students' opinions and reflections: IDM-TClass.

The design experiments involved 25 classes (from grade 4 to grade 7) from three different clusters of schools located in the North-West of Italy. Each school was provided with tablets for the students and computers for the teachers, linked to IWB or data projector. In order to foster collaboration and sharing of ideas, students were asked to work in pairs or in small groups on the same tablet.

During the first two cycles of design, we carried out about 450 hours of lessons. The researcher was in the class as both an observer and a participant (to support the teacher in the use of the technology and in the implementation of the digital resources). In some cases, also Master students were present as observers. The corpus of data is constituted by video-recordings of the lessons, written transcripts, field notes taken by the observers, teachers' interviews after sequences of lessons, students' written questionnaires and groups of students' interviews during a Q-sorting activity (questionnaires and Q-sorting data were collected at the end of the design experiments).

The use of IDM-TClass was integrated within a set of activities on relations and functions, and their different representations (symbolic representations, tables, graphs). These activities, in line with the aims of the FaSMEd Project, where adapted starting from existing research-informed materials.

For each activity, we have prepared a sequence of different worksheets, to be sent to the students' tablets or to be displayed on the IWB (or through the data projector). The worksheets were designed according to four main categories: (1) worksheets introducing a problem and asking one or more questions (problem worksheets); (2) helping worksheets, aimed at supporting students, who meet difficulties with the problem worksheets, through specific suggestions (e.g. guiding questions); (3) worksheets prompting a poll between proposed options (poll worksheets); (4) worksheets prompting a focused discussion (discussion worksheets).

Usually the activity starts with a problem worksheet, sent from the teacher's laptop to the students' tablets. Students work in pairs or small groups of three. After facing the task and answering the questions, the pairs/groups send back to the teacher their written productions. The teacher can decide to send helping worksheets to some groups, or the groups can ask for them.

After all groups have sent back their answers, the teacher sets up a classroom discussion in which the students' written productions are shown and feedbacks are given by the teacher and by classmates. The discussion is engineered starting from the teacher's selection of some of the received written answers, to be shown on the IWB, and aims at highlighting: (a) typical mistakes; (b) effective ways of processing the tasks; (c) the comparison between the different ways of justifying. During the part of the discussion focused on these aspects, therefore, the criteria for success could be clarified through the analysis and comparison of the different written productions.

The teacher can also display the discussion worksheets or poll worksheets, if she realises that some specific aspects were neglected, to support the class discussion during different parts of the lessons. It is also possible to create polls on the spot to check students' understanding, or their awareness about what has been developed during the activity, or their attitudes toward the activity.

In the next paragraph we illustrate the design of the helping worksheets and the corresponding implementation, and present the analysis of an example from a case study. The example has been chosen because it is a paradigmatic one, which enables to highlight how the implementation of helping worksheets, through the support of CCT, fosters the activation of FA strategies and the dynamics between them.

The design and implementation of the helping worksheets: Analysis of an episode

Helping worksheets are conceived to support students in facing the tasks posed through the problem worksheets and are sent to selected students during the problem-solving phase, when: (a) they ask to receive a help; (b) the teacher realises that they are stuck; (b) the answers they send to the teacher highlight mistakes or difficulties. Moreover, helping worksheets may be sent to all groups, after they sent their answers to the teacher, as a checking tool for their work.

Usually we design sets of differentiated worksheets, according to the possible difficulties students could meet when facing a problem worksheet. Since our activities are adaptations of existing research-informed materials, the hypothesis about students' difficulties and the corresponding feedback that could be provided are drawn also from these materials.

We focus on helping worksheet 1A (see figure 1), which is matched to problem worksheet 1 within an articulated activity on time-distance graphs. The activity, which is our adaptation of some materials from the Mathematics Assessment Program, developed at the University of Nottingham (http://map.mathshell.org/materials/lessons.php), starts with the interpretation of a given timedistance graph and develops through the matching between graphs and stories and the construction of graphs associated to specific stories. We adapted the tasks in order to propose them to students from grade 5 to 7. To ground the time-distance graph on a meaningful activity, we designed an introductory activity on the use of a motion sensor (a device connected to a graphic calculator, showing, in real time, the Cartesian representation of a produced motion).

Worksheet 1 introduces a task on the interpretation of a time-distance graph representing the journey of a student, Tommaso, from home to the bus-stop. The worksheets' sequence connected to this task was conceived to gradually lead students in the interpretation of the graph, focusing their attention on the meaning of ascending, descending and horizontal traits of the graphs. Students are also asked to focus on the reasons supporting the correct interpretation of a time-distance graph, with the aim of making them, on one side, reflect on their thinking processes and share these processes with their classmates and, on the other side, consolidate their competencies in justifying and analysing their answers. The question on worksheet 1 (within the white box, see fig. 1) requires students to interpret the meaning of a descending line within the graph. Students have to highlight that in the period of time from 50s to 70s the distance from home decreases, so Tommaso is going back for a while. Helping worksheet 1A (fig. 1) first of all makes students focus on the word "straight" to help them to abandon the idea that the graph could represent the drawing of the road. Moreover, it aims at fostering a correct interpretation of the descending line in the graph, making students look at two specific points within the graphs, that is (50, 100) and (70,40), to highlight that the distance from home is decreasing.

Sending a helping worksheet to a specific group of students is a way to activate FA strategy C, because students are provided with feedback about the task (if they receive this kind of worksheets, they realise that their answers should be completed and/or corrected) and feedback about the processing of the task (the suggestions and the guiding questions on the helping worksheets are aimed at supporting the students in facing the problem). Moreover, giving feedback represents a way of making students activate themselves as owners of their learning (FA strategy E).

Journey to the Bus Stop

Every morning Tom walks along a straight road from his home to a bus stop, a distance of 160 meters.

The graph shows his journey on one particular day. During our design experiments, some students (in particular, low-achieving students) face difficulties also in interpreting the purpose of the provided helping worksheets, as supports to face the tasks. This is a manifestation of their lack of metacognitive control. For this reason, after the first cycle of design experiments, we introduced the displaying and collective meta-level analysis of helping worksheets as a fundamental characteristic of the methodology for their implementation.

As an example of this specific implementation of helping worksheets, we present and analyse an excerpt from a discussion on helping worksheet 1A, which was carried out in a 5th grade class. The discussion was aimed at making students aware of the goal of the helping worksheet and at pointing out specific mathematical aspects related to the task, namely to make them: (1) look at points within time-distance graphs as bearers of two linked information (the distance from home and the time spent); (2) interpret the variation of the distance in terms of moving away/approaching home; (3) avoid the typical mistake of interpreting the graph as a drawing. We remind that the researcher takes part in the discussion as both an observer and a participant.

220 Researcher: The first ones who are going to speak are those who did not receive this helping worksheet. Let's read the help that is given and try to say why, in your opinion, it is an help…what it helps you to do… The main question to be answered is still this one (she indicates question 1, presented in Worksheet 1).

The help says (reading) "Remember that Tommaso is walking on a straight road. What is his distance from home after 50s? What is his distance from home after 70s?" 221 Teacher: Why do the suggestions focus on this? 222 Researcher: What do these questions help to do? Several students raise their hands. 223 Carlo: Because they help you to understand the distance in the period between 50s and 70s.

Because, at 70s, he is nearer… 224 Researcher: So you are saying that it enables to look at the distance, aren't you?

This discussion was planned with the objective, on one side, of eliciting evidence of students' understanding at a metacognitive level (strategy B), and, on the other side, of activating some students as resources for their classmates (strategy D). In fact, the researcher (lines 220, 222) and the teacher (line 221) are fostering a meta-reflection, involving the students that did not receive the help in clarifying the reasons why the questions posed on helping worksheet 1A could provide help in answering questions 1. Their aim is, therefore, to activate FA strategy C at the peer's level. Specifically, Carlo's intervention (line 223) represents a feedback about self-regulation because he highlights that the questions in worksheet 1A enable to focus on the change in Tommaso's distance from home, during that period of time. Through this comparison with their classmates, students can therefore become aware of the kind of support that helping worksheets could give and can also develop new tools to face similar activities in an effective way.

Then, the teacher focuses students' attention on a part of the helping worksheet that was not mentioned by Carlo: Again the teacher and the researcher focus students' attention on the suggestions contained in helping worksheet 1A to make them become aware of its role in supporting the resolution of the task (strategy C). The teacher (line 225) focuses on the first suggestion given in worksheet 1A (Remember that Tommaso is walking on a straight road) and the researcher (lines 227) aims at making students reflect on the possible misinterpretations that this suggestion wants to prevent. Students are, in this way, provided with both feedback about the processing of the task and feedback about self-regulation, because they can become aware of the possible mistakes that could be done in the interpretation of this kind of graphs, learning how to monitor their work. Also the teacher (line 231) provides a feedback about self-regulation because she is making the students notice how the previous experience has influenced their answer to the current question. Carlo (line 226) and Arturo (line 228) are activated as instructional resources for their classmates (strategy D).

Discussion

In this paper we referred to the theoretical lenses provided by the FaSMEd framework to present and discuss the design of digital resources and the corresponding method of implementation, with a special focus on the helping worksheets. The analysis we developed, on one side, shows that the FaSMEd three-dimensional framework represents a useful tool for both designing digital materials for technology-enhanced FA practices and the corresponding methodology, and analysing how these materials are implemented in the classroom. On the other side, this analysis provides a validation of the design and implementation because the FaSMEd framework offers some important criteria according to which the digital worksheets and the methodology for their implementation can be evaluated as effective tools to foster FA processes: (1) the activation of different FA strategies; (2) the involvement of all the agents; (3) the evolution of the FA strategies (in particular toward strategy E, which should constitute a constant objective of the activities); (4) the different levels of feedback provided; (5) the support provided to the three fundamental FA processes.

At the same time, the analysis of the design and implementation of the helping worksheets in the chosen episode enabled us to highlight a pattern that characterises the evolution of FA strategies when helping worksheets are implemented. In fact, the use of the helping worksheet, combined with the sending and displaying functionality of technology, turned into the activation of several FA processes, with the involvement of all the agents. During the group-work phase, by sending the helping worksheet to the students, the teacher is activating FA strategy C with the aim of activating also strategy E. After the group-work phase, a meta-level discussion devoted to the sharing and analysis of helping worksheet is planned by the teacher (strategy B). As a result of the design based process, two different ways of fostering students' meta-level reflections have been identified: initially, the students who did not receive the worksheets are asked to reflect on the possible role played by the provided help (becoming instructional resources for their classmates, strategy D); then, the students who did receive the help (mostly low achieving students) are asked to discuss on the ways in which they used it, making their reasoning explicit and being activated as the owners of their own learning (strategy E). During the discussion, all the students receive feedback from the teacher and their classmates (strategy C) and are provided with the opportunity to clarify the learning intentions associated to the worksheet (strategy A).

We think that this pattern, since it is recurring throughout our corpus of data, represents an important validation of the design of helping worksheets, because it highlights the effectiveness of these resources and their implementation in fostering the development of FA strategies and the fruitful involvement of all the agents.

Other elements of validation can be highlighted if we interpret the results of the activities carried out through the helping worksheets in terms of the three fundamental FA processes that are supported: (a) the students are supported in establishing where they are in their learning when they use the help as a feedback to assess their own answer; (b) the teacher is supported in helping students clarify where they are asked to go when, during the class discussion, the characteristics of given answers are analysed and discussed; and (c) the teacher and the students are supported in establishing what needs to be done to get there when, during the class discussion, the helping worksheets are analysed to highlight in what ways they could help and what kind of suggestions they give.

We are now developing a similar analysis to highlight, referring to these criteria, how the other categories of worksheets are used to foster the activation of FA strategies through the support provided by technology. This will enable us to identify the connections and mutual support between the different worksheets and the methodologies through which they are implemented during the lessons.

1.Fig. 1 :

 1 Fig. 1: The helping worksheet

  225 Teacher: And why does it [the help] suggest that Tommaso is moving on a straight road? 226 Carlo: Because it wants to make us reason on the fact that he is going back. 227 Researcher: What mistake couldn't be done if I remember that the road is straight? … (Silence) If I don't know that the road is straight, what could I think? Anna mimes a curvy road with her hands. 228 Arturo: I could think that the sensor initially indicates a direction, then he goes on the right… (Arturo is referring to the introductory activity with the motion sensor) 229 Teacher: So a change in the direction. 230 Researcher: That we are zigzagging, in a strange way. 231 Teacher: It is the reason why it remembers us that the road is straight. You recalled, with your memory, what we experimented last time. If we hadn't worked with the sensor, you, maybe, would have proposed different answers.

Proceedings of CERME10

Acknowledgment Research funded by the European Community's Seventh Framework Programme fp7/2007-2013 under grant agreement No [612337].


domain: math shs

text:
Maud Chanudet
email: maud.chanudet@unige.ch

Teachers' formative assessment practices: The case of an IBMEcentered course

Keywords: Formative assessment, summative assessment, problem solving, IBME, teachers' practices

In Geneva canton (Switzerland), a special course has been designed to deal with problem solving. Within one 45-minute period per week, teachers have to improve students' problem solving competencies and to assess them frequently. In order to assess students' problem solving competencies, most teachers use grids of criteria with a summative purpose. The global objective of our research is to find out if and how using such a tool can also foster formative assessment processes. In this paper, we present an exploratory study focused on teachers' formative assessment practices. In order to do so, we study the practices of a teacher giving this IBME-centered course and show how she uses formative assessment in her teaching practices.

A course dealing with problem solving

In French-speaking Switzerland, the shared curriculum for compulsory education insists on the importance of problem solving in mathematics education in order to make students familiar with inquiry based mathematics education (IBME). The aim is to promote students' scientific processes of thought. But as Dorier and Maass say "inquiry based mathematics education remains quite marginal in day-to-day mathematics teaching" (2014, p. 303). That is why in Geneva canton, a special course called mathematics development has been created to focus on and develop students' problem solving competencies. Students aged 13-14 years old (grade 8) with a scientific profile are involved. Within one 45-minute period per week, teachers have to improve the students' problem solving competencies and at the same time, assess them frequently. This course is subject to many constraints and raises two fundamental questions in mathematics education: how to foster and how to assess students' problem solving competencies? Thus it is necessary to identify what problem-solving competencies are and consequently what students are expected to learn and to know.

IBME and problem-solving competencies

In the French mathematics teaching tradition, problem solving has been seen for many years as a means to develop specific mathematical content and knowledge [START_REF] Brousseau | La théorie des situations didactiques[END_REF]. For the past couple of years, however, many countries, and especially European countries, have been emphasizing problem solving in mathematics and inquiry-based mathematics and science education (IBMSE) as a learning goal for its sake. The European Rocard's report [START_REF] Rocard | Science Education Now: A Renewed Pedagogy for the Future of Europe[END_REF] promotes a wider implementation of IBMSE in classrooms as a tool to make sciences and mathematics more attractive to students. Nevertheless, this increasing interest in IBMSE has not been followed by a concise and commonly shared definition [START_REF] Dorier | Challenges and opportunities for the implementation of inquirybased learning in day-to-day teaching[END_REF]. If we are to summarize, it refers to a student-centered paradigm of teaching mathematics and science, in which students are invited to work in ways similar to how mathematicians and scientists work. This means they have to observe phenomena, ask questions, look for mathematical and scientific ways of how to answer these questions (like carrying out experiments, systematically controlling variables, drawing diagrams, calculating, looking for patterns and relationships, making conjectures and generalizations), interpret and evaluate their solutions and communicate and discuss their solutions effectively. (Dorier & Maass, 2014, p. 300)

Intended learning outcomes of IBME

The goal of IBME is to make students work in a way similar to the one of mathematicians and to make students familiar with a scientific approach to solve problems. For [START_REF] Hersant | Recherche et résolution de problèmes dans l'enseignement des mathématiques : une étude didactique pour identifier les savoirs et les apprentissages possibles[END_REF], a scientific approach cannot be considered as a relevant learning goal, especially because it is unclear, non-unique and too ambitious. Thus the intended learning outcomes of IBME are not so easy to interpret and implementing IBME in classrooms remains a crucial issue. If we look at institutional instructions of the mathematics development class, teachers are invited to propose open-ended problems [START_REF] Arsac | Problème ouvert et situation-problème. IREM de Lyon[END_REF] to students, which is, in France and in French speaking Switzerland, a traditional way of introducing students to IBME. An open-ended problem is a problem which has a short text, has no obvious solution and method, deals with students' familiar conceptual domain and enables students to make the problem their own. Facing such a problem, students should learn different strategies. It aims more generally at both establishing scientific debate rules and developing a scientific approach following the pattern of try -conjecture -test and prove. But according to [START_REF] Hersant | Empirisme et rationalité au cycle 3, vers la preuve en mathématiques[END_REF], what gives this approach a scientific dimension is not only the existence of trials, conjecture and proof but the articulation among these. She also emphasizes that there is no unique scientific approach. The first goal is not so clear, neither is the second. Debate rules can indeed refer to logical rules (several examples don't prove a proposition, a counter example is sufficient to disprove a conjecture, etc.) or to social rules (listen to the others, etc.). Consequently, curriculum and instructions about open-ended problems do not seem to be sufficient to help teachers to define what is institutionally expected about students' problem solving competencies.

Identifying what we want students to learn and to know about problem solving is still a problematic issue. The identification of the intended learning outcomes from IBME is by no means obvious even for teachers, and the danger is that students might not be aware of what they are supposed to learn and to know. That is why IBME learning goals should be at the midst of specific discussions with students in class. Even though such discussions should also be encouraged when acquiring a more classical mathematical knowledge, it is all the more important in the case of IBME.

Problem solving narration activity

To assess students' problem solving competencies, teachers have to be able to access what students did in order to solve the problem and especially what solving strategies they used. That is why the problem solving narration activity [START_REF] Bonafé | Les narrations de recherche du primaire au Lycée[END_REF] has been institutionally chosen as a means to assess students. It can be defined as a new contract between students and teachers in which students have to explain the best they can, how they solved, or tried to solve, the problem (including mistakes, wrong ways, dead-ends, help they received…) and teachers have to assess students on these and only these points and especially not take into account the fact whether students found the right answer or not. With this activity, the fact that students have to explain all the strategies they tried and all the ideas they had to someone else, presupposes that they are capable to do so firstly to themselves. They have to reconstruct their reflection and make a synthesis of which strategies were effective, which one were wrong ways or led to dead-ends, etc. In that sense, it can emphasize students' reflection about what solving problems in mathematics means, about their own problem solving competencies and it can encourage the development of para and proto-mathematical knowledge. Problem solving narration activity as a scheme used principally for summative assessment can also foster students' problem solving competencies and assume a formative function. This last observation leads us to consider the assessment of problem solving competencies, not only with summative purpose, but also with a formative purpose.

Assessing students' problem-solving competencies

According to [START_REF] Allal | Evaluation des apprentissages[END_REF] assessment is summative as soon as a synthesis of the competencies and knowledge learnt by the student at the end of his curriculum is established. Thanks to the distinction made by [START_REF] Scriven | The methodology of evaluation[END_REF] and then by [START_REF] Bloom | Learning for Mastery[END_REF] between summative and formative assessment, Black and Wiliam give the following definition:

Practice in a classroom is formative to the extent that evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers, to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have taken in the absence of the evidence that was elicited (Black & Wiliam, 2009, p. 9).

The notion of feedback is a key component of formative assessment. Formative assessment contains "all those activities undertaken by teachers, and/or by their students, which provide information to be used as feedback to modify the teaching and learning activities" (Black & William, 1998, pp. 7-8).

Another key component of formative assessment is that students understand the target of their work and that they grasp what is expected [START_REF] Harlen | Assessment & Inquiry-Based Science Education: Issues in Policy and Practice[END_REF]. But it means that "students need to have some understanding of the criteria to apply in assessing their work" (Harlen, 2013, p. 17). Once again, the necessity of specific discussions with students about assessment criteria and about what they are expected to learn is emphasized.

To classify classroom formative assessment, [START_REF] Shavelson | On the role and impact of formative assessment on science inquiry teaching and learning[END_REF] are using a continuum, that ranges from formal embedded assessment to informal, on the fly formative assessment. It means that formative assessment does not take a unique form but that it can be planned or not, it can refer to formal tools to collection of data or not, etc. Adopting this point of view, formative assessment can be considered as a practice integrated within the learning process [START_REF] Lepareur | L'évaluation dans les enseignements scientifiques fondés sur l'investigation : Effets de différentes modalités d'évaluation formative sur l'autorégulation des apprentissages[END_REF]. Referring to formative assessment about IBME is all the more relevant that the practice of formative assessment, through teachers and students collecting data about learning as it takes place and feeding back information to regulate the teaching and learning process, is clearly aligned with the goals and practice of inquiry-based learning. (Harlen, 2013, p. 20) These definitions of summative and formative assessment enhance that identifying assessment according to when it occurs (after a phase of teaching vs within a teaching activity for instance) or how it occurs (paper-pencil test vs worksheet for instance) seems less relevant than distinguishing assessment according to its function. But it does not mean that these two principal functions of assessment (summative and formative) cannot coexist. Thus some researchers [START_REF] Allal | Assessment for learning » culture in the classroom… and beyong[END_REF][START_REF] Harlen | On the relationship between assessment or formative and summative purposes[END_REF][START_REF] Shavelson | On the role and impact of formative assessment on science inquiry teaching and learning[END_REF] argue that they can coexist in what [START_REF] Earl | Assessment As Learning: Using Classroom Assessment to Maximize Student Learning[END_REF] calls assessment for learning. The same assessment activity can serve to summative and formative purpose. It means that data collected by the teacher can be used to give students a mark but also to improve learning and teaching. On the other hand, for Shavelson et al. "formative assessment could serve summative needs" (2008, p. 298). In our research, we deal with teachers' practices in the mathematics development class. Our objective is to find out if and how using an assessment tool as a grid of criteria, firstly with summative purpose, can also foster formative assessment processes. In this paper, we focus only on teacher's formative assessment practices. For that, we study the practices of a teacher giving this IBME-centered course.

Teachers' formative assessment practices

Context of the research

The teacher whose practice we are going to analyze, was a member of a one year commission, created in September 2015, and gathering another teacher and ourselves. The purpose of this commission was to give teachers of mathematics development classes a common tool to assess students' problem solving competencies with both summative and formative purpose, and consequently to ensure common expectations about IBME (from teachers, and more globally from schools). To do so, we have been working for one year to elaborate a grid of criteria aiming to assess students' problem solving narration activity. This development of the tool was mainly based on teachers' expertise. Indeed, teachers have implemented the grid in their class, and according to their experiences, we adjusted, removed and added some criteria. Nevertheless, we dealt with an existing tool elaborated by the Geneva team in the wake of the PRIMAS1 project and were careful to take into account some research results (as those of [START_REF] Hersant | Empirisme et rationalité au cycle 3, vers la preuve en mathématiques[END_REF]). The grid in its final version summarizes criteria related to five dimensions of such an activity: presentation, narration, research, technique and modelling. These dimensions induce ways to look at the students' production according to expected qualities. For instance, the modelling dimension is characterized by two criteria: "Appropriation of the problem: rephrase the problem in French and/or express it with drawings, diagrams, tables" and "Use of pertinent mathematical tools and theories, strategies". So, this teacher whose practice we are going to analyze has been reflecting on the intended learning outcomes of this IBME-centered course and on the summative assessment of problem solving competencies, for one year, thanks to the meetings with the other members of the commission. She used the grid elaborated by the commission, in her class, principally with a summative purpose. However, thanks to our hypothesis that summative and formative assessment can co-exist, we would like to see if she also referred to formative assessment, thanks to specific discussions with students about criteria, and feedback related to their production. That is why we analyzed her formative assessment practices.

Theoretical framework

To characterize teachers' formative assessment practices, we referred to criteria elaborated by Lepareur (2016). She defined five strategies 2 : eliciting goals and criteria (S1); managing discussions and activities which can produce some evidence of effective learning (S2); giving feedback to students which make them progress (S3); helping students to be responsible for their learning (S4); helping students to be a resource for their peers (S5). Even though her research dealt with science and mathematics teachers' formative assessment practice, in our case, we only study mathematics teachers' practices. In that sense, we made some adaptations about key words and sub-strategies she defined. The table 1 is the grid we used to characterize the mathematics teacher's formative assessment practices. The T is used for the teacher, S for the students.

Methodology

We video recorded two consecutive periods of mathematics development given by this teacher, member of the commission, at the end of the school year. The nine students of the class were working in four groups (3 groups of 2 students, 1 group of 3). They were working on two problems related to the introduction of algebra. At the end of the second period, students had to give a narration about the problem they were working on to the teacher. Consequently half of the second period was devoted to the narration and students were invited to use office software to write their research down. To interpret data and make it relevant with our theoretical framework, we transcribed all interactions occurred in class for both lessons (about 67 minutes) and classified interactions according to strategies and substrategies defined in the table 1.

2 Translated from [START_REF] Lepareur | L'évaluation dans les enseignements scientifiques fondés sur l'investigation : Effets de différentes modalités d'évaluation formative sur l'autorégulation des apprentissages[END_REF].

Strategy

Key words Code Description

S1

Goals

S11

T explains the goals of the activity.

Criteria S12

T explains the intended learning outcomes, what will be assessed.

S2

Progress in activity S21

T collects information about students' progress in the activity. Strategies S22 T collects information about strategies used by S. Understanding S23 T questionnes S about their understanding of the goals. Knowledge S24 T takes information about previous S' knowledge.

Self assessment S25 T helps S to situate themselves in relation to assessment and success criteria.

S3

Feedback (what students have to do) S31 T gives an information to make explicit what S have to do left. Feedback (how students can do it) S32 T provides explicit information about how S have to do it, to move on.

S4

Responsabilisation S4

T emphasizes S' ideas, gives them independancy to access resources.

S5

Interactions (group) S51 T encourages S to discuss with others members of their group. Peers are seen as a resource.

Interactions (class) S52

T integrates S' propositions and encourages others to react on. For instance, during the following interaction between the teacher and a student, we can see that the teacher tries to make the student explains his strategy.

Teacher: Yes but how did you find this? Student:

I made a lot of stuff. Teacher:

But try… what did you do? It's interesting to know how you were thinking. Student:

I made all of this but like everything in reverse. Teacher:

Yes so the first step. What did you do at the first step? Student: 24 minus 7. Teacher:

Yes, you went backwards in your calculations. Yes. It's a good idea. Doing backward calculation is in fact a lead.

In that case, we identify a formative assessment practice, according to the strategy S2 "managing discussions and activity which can produce some evidence of effective learning" and more specially the strategy S22 "collecting information about strategies used by students".

Results and analysis

Figure 1: Strategies of formative assessment

We found out 44 episodes when formative assessment strategies occurred thus we can say that this mathematics teacher refers frequently to formative assessment in her practice. We identified 4 times the strategy S1; 25 times strategy S2; 10 times strategy S3; 3 times strategy S4 and 2 times strategy S5. The figure 1 illustrates the percentage of apparition of each strategy. To summarize, we can notice that the teacher refers to every strategy (S1, S2, S3, S4 and S5).

On top of that, she uses principally the strategy S2: "managing discussions and activities which can produce some evidence of effective learning". It represents more than half of the strategies of formative assessment used by this teacher. The strategy S3 which refers to "give feedback to students which makes them progress" is also used frequently, about one time out of four.

But if we look deeper, we can see that each sub-strategy is not used with the same frequency (figure 2).We can see that for the strategy S1 related to the goals and criteria, the teacher only explains the goals of the activity (S11) but not the criteria of assessment or the intended learning outcomes (S12). This lesson occurred at the end of the year so we can make the hypothesis that by then students knew well what they were expected to do.

The most represented sub-strategy related to "discussions and activities which can produce some evidence of effective learning which is used frequently" (S2) is "collecting information about strategies used by students" (S22). It appears 19 times. It is also the strategy that occurs the most, all categories taken into account. The only other significant strategy dealing with S2 used by the teacher is "taking information about students' progress in the activity" (S21). So the teacher focuses on where students are in the activity and what they have done to get there.

Then, for strategy S3 about feedback, the two ways; "provides information to make it explicit what is left for the students to be done" (S31) or "how they have to move on" (S32) are represented, but the second one more than the first one. The idea is that this teacher helps students both knowing what they have to do but even more how they can continue. For the last strategy (S5) dealing with interactions, the teacher focuses on discussions within the groups. The absence of strategy S52 about "the integration of students" can easily be explained by the fact that during these two lessons, students only worked in group, without any collective classroom discussion.

Conclusion

We can say that this mathematics teacher refers frequently to formative assessment in her practices (44 times during a 67 minute-lesson). She uses a very large set of formative assessment tactics; eliciting goals and criteria; managing discussions and activities which can produce some evidence of effective learning; giving feedback to students which make them progressing; helping students to be responsible for their learning; helping students to be a resource for their peers. But she uses principally formative assessment to manage discussions and activities which can produce some evidence of effective learning, and especially, she collects information about strategies students use. The feedback she provided to students is mainly about how they can continue, how they can do what they have to do to solve the problem. The only strategy which does not appear is this related to explanation of criteria and expected learning outcomes. We can make the hypotesis that it has been at the core of a discussion in the first part of the schoolyear. In that sense, it should be interesting to focus on what happens at the beginning of the schoolyear and to study how teachers explain and negociate the intended learning outcomes with students.

To conclude this paper, we can say that this exploratory study shows that formative assessment seems to be relevant for this teacher in order to foster her teaching practices in the case of an IBME-centered course. It is, nevertheless, necessary to enlarge the study, in order to compare and expend or not the results, and to have more information about how teachers refer to formative assessment. On top of that, we can imagine that working with teachers about these strategies could foster their formative assessment practices.

Figure 2 :

 2 Figure 2: Strategies of formative assessment according to key words

Table 1 : Grid of analysis of formative assessment practices, adapted from Lepareur (2016)

 1

Proceedings of CERME10

Available at http://www.primas-project.eu/fr/index.do Thematic Working Group

Proceedings of CERME10