Sghaier Salem Béchir
email: sghaierb@yahoo.fr

Fabrice Vandebrouck²

Teaching and learning continuity with technologies

Keywords: teaching and learning of analysis and calculus, novel approaches to teaching, continuity, digital technologies

We developed a digital tool aiming at introducing the concept oflocal -continuity together with its formal definition for Tunisian students at the end of secondary school. Our approach is a socioconstructivist one, mixing conceptualisation in the sense of Vergnaud together with Vygotski's concepts of mediation and ZPD. In the paper, we focus on the design of the tool and we give some flashes about students' productions with the tool and teachers' discourses in order to foster students' understanding of the continuity.

The definition of continuity of functions at a given point, together with the concept of continuity, remains a major difficulty in the teaching and learning of analysis. There is a dialectic between the definition and the concept itself which make necessary the introduction of the two aspects together.

The definition of continuity brings FUG aspects in the sense of [START_REF] Rabardel | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF]. This means first that it permits to formalize (F) the concept of continuity. But it also allows to unify (U) several different images (or situations) of continuity encountered by students: in [START_REF] Tall | Concept image and concept definition in mathematics, with special reference to limits and continuity[END_REF], several emblematic situations of continuity are established (see below) and the definition aims at unifying all these different kinds of continuity. Moreover, the definition of continuity allows generalisations (G) to all other numerical functions, not already encountered and not necessarily with graphical representations, or more general functions inside other spaces of functions. As [START_REF] Rabardel | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF] stresses for the definition of limit of sequences, notions which bring FUG aspects must be introduced with a specific attention to mediations and especially the role of the teacher.

Our ambition is then to design a technological tool which allows on one hand students activities concerning the two aspects of continuity and, on the other hand, allows the teacher to introduce the concept of continuity with its formal definition, referring to the activities developed on the technological tool. As it was noticed in the first INDRUM conference, papers about introduction of technologies in the teaching of analysis remain very few.

We first come back to well-known concept images and concept definitions of continuity. Then, we explain our theoretical frame about conceptualisation and mathematical activities. This theoretical frame leads us to the design of the technological tool which brings most of the aspects we consider important for the conceptualisation of continuity. Due to the text constraints, the results of the paper are mostly in term of the design itself and the way the tool encompasses our theoretical frame and our hypotheses about conceptualisation (with tasks, activities and opportunities for mediations). Then, we can give some flashes about students' activities with the software and also teachers' discourses to introduce the definition of continuity, based on students' mathematical activities on the software.

CONCEPT IMAGES AND CONCEPT DEFINITIONS OF CONTINUITY

No one can speak about continuity without referring to Tall and Vinner's paper about concept images and concept definitions in mathematics, whose particular reference is about limits and continuity [START_REF] Tall | Concept image and concept definition in mathematics, with special reference to limits and continuity[END_REF]. Tall considers that the concept definition is one part of the total concept image that exists in our mind. Additionally, it is understood that learners enter their acquisition process of a newly introduced concept with preexisting concept images. [START_REF] Sierpinska | On understanding the notion of function[END_REF] used the notion of epistemological obstacles regarding some properties of functions and especially the concept of limit. Epistemological obstacles for continuity are very close to those observed for the concept of limit and they can be directly relied to students' concept images, as a specific origin of theses conceptions (El Bouazzaoui, 1988). One of these obstacles can be associated to what we call a primitive concept image: it is a geometrical and very intuitive conception of continuity, related to the aspects of the curve. With this concept image, continuity and derivability are often mixed and continuity means mainly that the curve is smooth and have no angles. Historically, this primitive conception leads Euler to introduce a definition of continuity based on algebraic representations of functions. This leads to a second epistemological obstacle: a continuous function is given by only one algebraic expression, which can be called the algebraic concept image of continuity. This conception has led to a new obstacle with the beginning of Fourier's analysis. Then, a clear definition is necessary. This definition comes with Cauchy and Weierstrass and it is close to our actual formal definition.

We also refer to [START_REF] Bkouche | Points de vue sur l'enseignement de l'analyse : des limites et de la continuité dans l'enseignement[END_REF] who identifies three points of view about continuity of functions which are more or less connected to the epistemological obstacles we have highlighted. The first one is a cinematic point of view. Bkouche says that the variable pulls the function with this dynamic concept image. The other one is an approximation point of view: the desired degree of approximation of the function pulls the variable. This last point of view is more static and leads easily to the formal definition of continuity. These two points of view are also introduced by [START_REF] Rabardel | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF] when she studies the introduction of the formal definition of limit (for sequences). A third point of view is also identified by Bkouche that is the algebraic point of view, which is about algebraic rules, without any idea of the meaningful of these rules.

At last, we refer to more recent papers and specifically the one of Hanke and Schafer (2017) about continuity in the last CERME congress. Their review of central papers on concept images about students' conceptions of continuity leads to a classification of the eight possible mental images that are reported in the literature: I : Look of the graph of the function : "A graph of a continuous function must be connected" -II : Limits and approximation : "The left hand side and right hand side limit at each point must be equal" -III : Controlled wiggling : "If you wiggle a bit in x, the values will only wiggle a bit, too" -IV : Connection to differentiability : "Each continuous function is differentiable" -V : General properties of functions : "A continuous function is given by one term and not defined piecewise"-VI : Everyday language : "The function continues at each point and does not stop" -VII : Reference to a formal definition : "I have to check whether the definition of continuity applies at each point" -VIII : Miscellaneous

We can recognize some of the previous categories, even if some refinements are brought. Mainly, concept images I, II, IV and VI can be close to the primitive concept image whereas VII refers to the formal definition and V seems to refer to the algebraic approach of continuity.

CONCEPTUALISATION OF CONTINUITY

We base our research work on these possible concepts image and concepts definition of continuity. However, we are more interested in conceptualisation, as the process which describes the development of students' mathematical knowledge. Conceptualisation in our sense has been mainly introduced by [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF] and it has been extended within an activity theoretical frame developed in the French didactic of mathematics. These developments articulate two epistemological approaches: that of mathematics didactics and that of developmental cognitive psychology as it is discussed and developed in [START_REF] Vandebrouck | Activity Theory in French Didactic Research[END_REF].

Broadly, conceptualisation means that the developmental process occurs within students' actions over a class of mathematical situations, characteristic of the concept involved. This class of situations brings technical tasksdirect application of the concept involved -as well as tasks with adaptations of this concept. A list of such adaptations can be found in [START_REF] Horoks | Tasks Designed to Highlight Task-Activity Relationships[END_REF]: for instance mix between the concept and other knowledge, conversions between several registers of representations [START_REF] Duval | Sémiosis et pensée humaine: registres sémiotiques et apprentissages intellectuels[END_REF], use of different points of view, etc. Tasks that require these adaptations of knowledge or concepts are called complex tasks. These ones encourage conceptualisation, because students become able to develop high level activities allowing availability and flexibly around the relevant concept.

A level of conceptualisation refers to such a class of situations, in a more modest sense and with explicit references to scholar curricula. In this paper, the level of conceptualisation refers to the end of scientific secondary school in Tunisia or the beginning of scientific university in France. It supposes enough activities which can permit the teacher to introduce the formal definition of continuity together with the sense of the continuity concept. The aim is not to obtain from students a high technicity about the definition itselfstudents are not supposed to establish or to manipulate the negation of the definition for instance. However, this level of conceptualisation supposes students to access the FUG aspects of the definition of continuity.

Of course, we also build on instrumental approach and instrumentation as a sub process of conceptualisation [START_REF] Rabardel | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF]. Students' cognitive construction of knowledge (specific schemes) arise during the complex process of instrumental genesis in which they transform the artifact into an instrument that they integrate within their activities. [START_REF] Artigue | Learning mathematics in a cas environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF] says that it is necessary to identify the new potentials offered by instrumented work, but she also highlights the importance of identifying the constraints induced by the instrument and the instrumental distance between instrumented activities and traditional activities (in paper and pencil environment). Instrumentation theory also deals with the complexity of instrumental genesis.

We also refer to Duval's idea of visualisation as a contribution of the conceptualisation process (even if Duval and Vergnaud have not clearly discussed this point inside their frames). However, the technological tool brings new dynamic representations, which are different from static classical figures in paper and pencil environment. These new representations lead to enrich students' activitiesmostly in term of recognition -bringing specific visualization processes. Duval argues that visualization is linked to visual perception, and can be produced in any register of representation. He introduces two types of visualization, namely the iconic and the non-iconic, saying that in mathematical activities, visualization does not work with iconic representations [START_REF] Duval | Representation, vision and visualisation: cognitive functions in mathematical thinking. Basic issues for learning[END_REF].

At last, we refer on Vygotsky (1986) who stresses the importance of mediations within a student's zone of proximal developmental (ZPD) for learning (scientific concepts). Here, we also draw on the double approach of teaching practices as a part of French activity theory coming from [START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An example in a French 10th-grade class[END_REF]. The role of the teacher' mediations is specifically important in the conceptualisation process, especially because of the FUG aspects of the definition of continuity (as we have recalled above).

First of all, we refine the notion of mediation by adding a distinction between procedural and constructive mediations in the context of the dual regulation of activity. Procedural mediations are object oriented (oriented towards the resolution of the tasks), while constructive mediations are more subject oriented. We also distinguish individual (to pairs of students) and collective mediations (to the whole class).

Secondly, we use the notion of proximities [START_REF] Bridoux | Les moments d'exposition des connaissances : analyses et exemples[END_REF] which are discourses' elements that can foster students' understanding and then conceptualisation -according to their ZPD and their own activities in progress. In this sense, our approach is close to the one of Bartolini [START_REF] Bartolini Bussi | Semiotic Mediation in the Mathematics Classroom: Artifacts and Signs after a Vygotskian Perspective[END_REF] with their Theory of Semiotic Mediations. However, we do not refer explicitly at this moment to this theory which supposes a focus on signs and a more complex methodology than ours. According to us, the proximities characterize the attempts of alignment that the teacher operates between students' activities (what has been done in class) and the concept at stake. We therefore study the way the teacher organizes the movements between the general knowledge and its contextualized uses: we call ascending proximities those comments which explicit the transition from a particular case to a general theorem/property; descending proximities are the other way round; horizontal proximities consist in repeating in another way the same idea or in illustrating it.

DESIGN OF THE TECHNOLOGICAL TOOL

The technological tool called "TIC-Analyse" is designed to grasp most of the aspects which have been highlighted above. First of all, it is designed to foster students' activities about continuity aspects in the two first points of view identified by Bkouche: several functions are manipulatedcontinuous or notand for each of them, two windows are in correspondence. In one of the window, the cinematicdynamical point of view is highlighted (figure 1) whereas in the second window the approximation-static point of view is highlighted (figure 2). The correspondence between the two points of view is in coherence with Tall's idea of incorporation of the formal definition into the pre-existing students' concept images. It is also in coherence with the importance for students to deal with several points of view for the conceptualisation of continuity (adaptations). In second, the functions at stake in the software are extracted from the categories of [START_REF] Tall | Concept image and concept definition in mathematics, with special reference to limits and continuity[END_REF]. For instance, we have chosen a continuous function which is defined by two different algebraic expressions, to avoid the algebraic concept image of continuity and to avoid the amalgam between continuity and derivability. We also have two kinds of discontinuity, smooth and with angle.

There is an emphasis not only on algebraic representations of functions in order to avoid algebraic conceptions of functions. Three registers of representations of functions (numerical, graphical and algebraic) are coordinated to promote students' activities about conversions between registers (adaptations). The design of the software is coherent with the instrumental approach mostly in the sense that the instrumental distance between the technological environment, the given tasks, and the traditional paper and pencil environment is reduced. However the software produces dynamic new representationsa moving point on the curve associated to a numerical table of values within the dynamic window; two static intervals, one being included or not in the other, for the static windowoccurring non iconic visualisations which intervene in the conceptualisation process. The software promotes students' actions and activities about given tasks: in the dynamic window, they are supposed to command the dynamic point on the given curvecorresponding to the given algebraic expression. They can observe the numerical values of coordinates corresponding to several discrete positions of the point and they must fill a commentary with free words about continuity aspects of the function at the given point (figures 1, 3). In the static window, they must fill the given array with values of α, the β being given by the software (figures 2, 4). Then, they have to fill a commentary which begins differently according to the situation (continuity or not) and the α they have found (figures 4, 5).

As we have mentioned in our theoretical frame, students are not supposed with these tasks and activities to get the formal definition by themselves. However, students are supposed to have developed enough knowledge in their ZPD so that the teacher can introduce the definition together with the sense and FUG aspects of continuity.

STUDENTS ACTIVITIES AND TEACHER'S PROXIMITIES

The students work by pair on the tool. The session is a one hour session but four secondary schools with four teachers are involved. Students have some concept images of continuity but nothing has been thought about the formal definition. The teacher is supposed to mediate students' activities on the given tasks. Students are not supposed to be in a total autonomy during the session according to our socio constructivist approach. We have collected video screen shots, videos of the session (for each schools) and recording of students' exchanges in some pairs. Students' activities on each tasks are identified, according to the tasks' complexity (mostly kinds of adaptations), their actions and interactions with computers and papers (written notes), the mediations they receive (procedural or constructive mediations, individual or collective, from the tool, the pairs or the teacher) and the discourses' elements seen as "potential" proximities proposed by the teacher.

It appears that the teacher mostly gives collective procedural mediations to introduce the given tasks, to assure an average progression of the students and to take care of the instrumental process. Some individuals mediations are only technical ones ("you can click on this button"). Some collective mediations are most constructive such as "now, we are going to see a formal approach. We are going to see again the four activities (ie tasks) but with a new approach which we are going to call formal approach...". The constructive mediations are not tasks oriented but they aim at helping students to organize their new knowledge and they contribute to the aimed conceptualisation according to our theoretical approach.

As examples of students' written notes (as traces of activities), we can draw on figure 3 and4. A pair of students explains the dynamic non-continuity with their words "when x takes values more and more close to 2 then f(x) takes values close to -2,5 and -2. It depends whether it's lower or higher" (figure 3) which is in coherence with the primitive concept image of continuity. The same pair of students explains the non-continuity in relation to what they can observe on the screen: "there exists β positive, for all α positivealready proposed by the tool in case of non-continuitysuch that f(i) not completely in j… f is not continuous". We can note that the students are using "completely" to verbalize that the intersection of the two intervals is not empty. However, the inclusiveness of an interval into another one is not expected as a formalized knowledge at this level of conceptualisation. Their commentary is acceptable. Students are expressing what they have experimented several times : for several values of β (β = 0,3 in figure 4), even with α very small (α = 0,01 in figure 4), the image of the interval ]2-α, 2+ α[ is not included in ]-2,5-β, -2,5+ β[. Concerning a case of continuity, the students are also able to write an acceptable commentary (figure 5) "for all β positive, their exists α positivealready proposed by the tool in case of continuitysuch that f(i) is included in j."

Students' activities on the given tasks are supposed to help the teacher to develop proximities with the formal definition. It is really observed that some students are able to interact spontaneously with the teacher when he wants to write the formal definition on the blackboard. This is interpreted as a sign that the teacher's discourse encounters these students' ZPD. Then the observed proximities seem to be horizontal ones: the teacher reformulates several times the students' propositions in a way which lead gradually to the awaited formal definition, for instance "so, we are going to reformulate, for all β positive, their exists α positive, such that if x belong to a neighbour of α … we can note it x 0 -α, x 0 + α…."

Of course, it is insufficient to ensure proof and effectiveness of our experimentation. The conceptualisation of continuity is an ongoing long process with is only initiated by our teaching process. However, we want to highlight here the important role of the teacher and more generally the importance of mediations in the conceptualisation process of such a complex concepts. We only have presented the beginning of our experimentation. It is completed by new tasks on the tool which are designed to come back on similar activities and to continue the conceptualisation process.

Figure 1 :

 1 Figure 1: two windows for a function, the dynamic point of view about continuity

Figure 2 :

 2 Figure 2: two windows for a function, the static points of view about continuity

Figure 3 :

 3 Figure 3: example of commentary given by a pair of students in the dynamic window

Figure 4 :

 4 Figure 4: example of commentary given by a pair of students in the static window

Figure 5 :

 5 Figure 5: example of commentary given by a pair of students in the static window


Fabrice Vandebrouck

ACTIVITY THEORY IN FRENCH DIDACTIC RESEARCH

Keywords: Mathematics, Tasks, Activity, Mediations, Technologies

The theoretical and methodological tools provided by the first generation of Activity Theory have been expanded in recent decades by the French community of cognitive ergonomists, followed by a sub-community of researchers working in didactics of mathematics. The main features are first the distinction between tasks and activity and second the dialectic between the subject of the activity and the situation within which this activity takes place. The core of the theory is the twofold regulatory loop which reflect both the codetermination of the activity by the subject and by the situation, and the developmental dimension of the subject's activity. This individual and cognitive understanding of Activity Theory mixes aspects of Piaget and Vygotsky's frameworks. It is first explored in this paper, associated with a methodology for analyzing students' mathematical activites. Then we present findings that help to understand the complexity of student mathematical activities when working with technology.

Introduction

Activity Theory is a cross-disciplinary theory that has been adopted to study various human activities, including teaching and learning in ordinary classrooms, where individual and social levels are interlinked. These activities are seen as developmental processes mediated by various contextual elementshere we consider the teacher, the pair and the artefact (Vandebrouck et al., 2012: 13). Activity is always motivated by an object; a characteristic that distinguishes one activity from another. Transforming the object into an outcome is another key feature of activity. Subject and object form a dialectic unit: the subject transforms the object and at the same time is him/herself transformed. This framework can be adapted to describe the actions and interactions that emerge in the teaching/learning environment, and that relate to the subjects, the objects, the artefacts and the outcomes of the activity [START_REF] Wertsch | The concept of activity in soviet psychology: an introduction[END_REF].

Activity Theory was originally developed by, among others, [START_REF] Leontiev | Activity, consciousness and personality[END_REF]. A well-known extension is the systemic model proposed by Engeström and al. (1999), called third generation of Activity Theory. It expresses the complex relationships between the elements that mediate activity in an activity system. In this paper, we take a more cognitive and individual perspective. This school of thought has been expanded over the course of the past four decades by French researchers working in the domain of occupational psychology and cognitive ergonomics, and has since been adapted to the didactics of mathematics. The focus is on the individual as a cognitive subject and an actor in the activity, rather than the overall systemeven if individual activity is seen as embedded in a collective system, and cannot be analysed outside the context in which it occurs.

An example of this adaptation is already well-established internationally. Specifically, it refers to the distinction between the artefact and the instrument, which is used to understand the complex integration of technologies into the classroom. The notion of instrumental genesis (or instrumental approach) was first introduced by [START_REF] Rabardel | Les hommes et les technologies, approche cognitive des instruments contemporains[END_REF] in the context of cognitive ergonomics, then extended to didactics of mathematics by [START_REF] Artigue | Learning mathematics in a cas environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF] and it is concerned with the subject-artefact dialectic of turning an artefact into an instrument. In this paper, we draw upon and try to encompass this instrumental approach.

1 -2 First, we describe how Activity Theory has been developed in the French context. These developments are both general and focused on students' mathematical activity. Next, we present a general methodology for analysing students' mathematical activity when working with technology. Then we develop an example of application, and we describe our findings. Finally, we present some conclusions.

Activity theory in French context

The first notable feature of Activity Theory in the French context is the distinction between tasks and activity [START_REF] Rogalski | Theory of Activity and Developmental Frameworks for an Analysis of Teachers' Practices and Students' Learning[END_REF]. Activity relates to the subject, while tasks relate to the object. Activity refers to what the subject engages in to complete the task: external actions but also inferences, hypotheses, thoughts and actions he/she decides to take or not. It also concerns elements that are specific to the subject, such as time management, workload, fatigue, stress, enjoyment, and interactions with others. As for the taskas described by [START_REF] Leontiev | Activity, consciousness and personality[END_REF] and extended in cognitive ergonomicsthis refers to a goal to be attained under certain conditions [START_REF] Leplat | Regards sur l'activité en situation de travail[END_REF].

Activity Theory draws upon two key concepts: the subject and the situation. The subject refers to an individual person, who has intentions and competencies (potential resources and constraints). The situation provides the task and the context for the task. Together, situation (notably task demands) and the subject codetermine activity. The dynamic of the activity produces feedback in the form of twofold regulatory loop (Figure 1) that reflects the develomental dimension of Activity Theory [START_REF] Leplat | Regards sur l'activité en situation de travail[END_REF]. The concept of twofold regulation reflects the fact that the activity modifies both the situation and the subject. On the one hand (upper loop), the situation is modified, giving rise to new conditions for the activity (e.g. a new task). On the other hand (lower loop), the subject's own knowledge is modified (e.g. by the difference between expectations, acceptable outcomes and the results of actions).

More recently, the dialectic between the upper and lower regulatory loops (shown in Figure 1) has been expanded through a distinction between the productive and constructive dimensions of activity [START_REF] Pastré | Apprendre des situations[END_REF][START_REF] Samurcay | Modèles pour l'analyse de l'activité et des compétences: propositions[END_REF]. Productive activity is object-oriented (motivated by task completion), while constructive activity is subject-oriented (the subject aims to develop his or her knowledge). In teaching/learning situations, especially those that involve technologies, the constructive dimension in the students' activity is key. The teacher aims the students to develop constructive activity. However, especially with computers, students are mostly engaged in producing results and the motivation of their activity can be only towards the productive dimension. Then the effects of their activity on students' knowledge -as it is stipulated by the dual regulatory loop -are mostly indirect with less or without any constructive aspects.

The last important point to note is the fact that French Activity Theory mixes Piagetian approach of epistemological genetics, together with Vygotsky's socio-historical framework to specifiy the developmental dimension of activity. As Jaworski (in Vandebrouck, 2013) writes, "the focus on the individual subjectas a person-subject rather than a didactic subjectis perhaps somewhat more surprising, especially since it leads the authors to consider a Piagetian approach of epistemological genetics alongside Vygotsky's sociohistorical framework". Rogalski (op. cit.) responds with "the Piagetian theory looks from the student's side at epistemological analyses of mathematical objects in play while the Vygotskian theory takes into account the didactic intervention of the teacher, mediating between knowledge and student in support of the students' activity".

The dual regulation of activity is consistent with the constructivist theories of Piaget and Vygotsky.

The first author [START_REF] Piaget | The equilibration of cognitive structures: The central problem of intellectual development[END_REF] provides tools to identify the links between activities and development, through epistemological analyses. [START_REF] Vergnaud | Cognitive and developmental psychology and research in mathematics education: some theoretical and methodological issues[END_REF][START_REF] Vergnaud | La théorie des champs conceptuels[END_REF], expands the Piagetian theoretical framework regarding conceptualisation and conceptual fields by highlighting situation classes relative to a knowledge domain. We therefore define the students' learningand development -with reference to Vergnaud's conceptualisation.

On the other hand, Vygotsky (1986) stresses the importance of mediation within the student's zone of proximal developmental (ZPD) for learning (scientific concepts). Here, we refine the notion of mediation by adding a distinction between procedural and constructive mediations in the context of the dual regulation of activity. Procedural mediations are object-oriented (oriented towards the resolution of the task), while constructive mediations are more subject-oriented. This distinction can be seen as an extension to what Robert [START_REF] Robert | Why and how to understand what is at stake in a mathematics class?[END_REF]) calls teachers' procedural and constructive teacher's aids. A more detailed exploration of the complementarity Piaget/Vygotski can be found in [START_REF] Cole | Beyond the individual-social antinomy in discussions of Piaget and Vygotski[END_REF].

General methodology for analysing students' mathematical activities

Following Activity Theory, we postulate that students' learning depends directly on their activity, even though other elements can play a part -and even if activity is partially inaccessible to us and differ from one student to another. Students' activity is developed through the actions that are carried out to complete tasks. Through their actions, subjects aim to achieve goals, and their actions are driven by the motivation for the activity. Here, we draw upon the three levels originally introduced by [START_REF] Leontiev | Activity, consciousness and personality[END_REF]: activity associated with a motive; actions associated with goals; and operations associated with conditions. Activity takes place in a specific situation, such as the classroom, at home, or during a practical session. Actions, involved by the proposed precise tasks, can be external (i.e. spoken, written or performed), or internal (e.g. hypotheses or decisions) and partially converted in operations. As [START_REF] Galperine | Essai sur la formation par étapes des actions et des concepts[END_REF] and [START_REF] Wells | Reevaluating the IRF sequence: A proposal for the articulation of theories of activity and discourse for the analysis of teaching and learning in the classroom[END_REF] note, the three levels are relative and, for instance, operations can be considered as actions that have been routinized.

Here, we use the generic term mathematical activities (rather than activity), to refer to students' activity on a specific mathematical task in a given context. Mathematical activities refer to everything that surrounds actions and operations (also non actions for instance). They are a function of a number of factors (including task complexity, but extending to the characteristics of the context and all mediations that occur as tasks are performed) that contribute to regulation and intended development in terms of mathematical knowledge.

Two methodological levels can be adopted from the dynamic of activity within the twofold regulatory loop. First of all, regulations can be considered at a local level as short-term adjustments of activities to previous actions and as procedural learning (also called functional regulations, upper loop in the figure 1). Secondly, at a global level, regulations are mostly constructive ones (also called structural regulations) and they correspond to the long-term development of the subject (in link with conceptualisation).

2.a The local level

At the local level, the analysis focuses on students' activities in the situation, in the form of tasks, their context, and their completion by students with or without direct help from the teacher. The initial step is an a priori analysis of the tasks given to students (by the teacher, the computer…), which is closely linked to the situational context (e.g. the students' academic level and age). We use [START_REF] Robert | Outil d'analyse des contenus mathématiques à enseigner au lycée et à l'université[END_REF] categorisation to characterise these tasks.

First, we identify the mathematical knowledge to be used for a given task: the representation(s) of a concept, theorem(s), definition(s), method(s), formula(s), types of proof, etc. The analysis aims to answer several crucial questions: does the mathematical knowledge to be used already exist for students or is it new? Do students have to find the knowledge to be used by themselves? Do the task only require the direct application of this knowledge without any adjustment (technical task), or does it require adaptations and/or carrying out subtasks? A list of such adaptations can be found in Horoks and [START_REF] Robert | Tasks Designed to Highlight Task-Activity Relationships[END_REF]: mix of knowledge, the use of intermediaries, change of register [START_REF] Duval | Sémiosis et pensée humaine: registres sémiotiques et apprentissages intellectuels[END_REF], change of mathematical domain or setting [START_REF] Douady | Jeux de cadre et dialectique outil-objet[END_REF], introduction of steps, choices, use of different points of view, etc. Tasks that require the adaptation of knowledge are referred to as complex tasks and encourage conceptualisation, as students become able to more readily and flexibly access the relevant knowledge, depending however on the implementation in the classroom.

The a priori analysis of tasks leads us to describe what we have called the intended students'activities associated with the tasks. Here we draw upon [START_REF] Galperine | Essai sur la formation par étapes des actions et des concepts[END_REF] functions of operations, and adapt them to mathematical activities. Galperine distinguishes three functions: orientation, execution and control. Next, we use three "critical" mathematical activities that are characteristic of complex tasks [START_REF] Vandebrouck | Proximités en acte mises en jeu en classe par les enseignants du secondaire et ZPD des élèves : analyses de séances sur des tâches complexes[END_REF].

 First, recognizing activities refer mainly to orientation and control. They occur when students have to recognize mathematical concepts as objects or tools that can be used to solve the tasks they are given.

Students may also be asked to recognize modalities of application or adaptation of these tools.

 Second, organizing activities refer mainly to orientation: students have to identify the logical and temporal steps in their mathematical reasoning, together with any intermediaries.

 Third, treatment activities refer to all of the mathematical activities associated with execution on mathematical objects. Students may be asked to draw a figure, compute, substitute, transform expressions (with or without giving the steps), change registers, change mathematical domains, etc.

Following Vygotsky, we supplement our local analysis of intended students' activities by developing ways to analyse classroom teaching (a posteriori), and to approach effective students' activities as functions of the different mediations that occur. For this, we use videos and observations in the classroom. We also record students' discussions, teacher's discourses and writings, and capture students' computer screens to identify observable activities. The data that is collected concerns how long students spend working on tasks, the format of their work (the whole class, in small groups, by pairs of students etc.), its nature (copying, reading, calculation, investigation, written or oral, graded or not, etc.) and all elements of the context that may modify intended activities. This highlights, at least partially, the autonomy given to students, the nature of mediations, and opportunities for students to show initiative, in relation to the adaptation and availability of knowledge.

Multiple aspects of mediations are analysed with respect to their assumed influence on student activities. Some relate to their format (interactions with students, between students, with teacher, with computers, etc.), while others concern the specific ways of taking into account the mathematical content (mathematical aids, assessment, reminders, explanations, corrections and evaluations, presentation of knowledge, direct mathematical content, etc.).

Two types of mediations have already been introduced, depending on whether they modify intended activities, or whether they add to activities (effective or at last observed). The first are object-oriented; here we use the term procedural mediations. These mediations modify intended activities and correspond to instructions given by the teacher, the screen or by other students, directly or indirectly, before or during task completion. They are often seen in open-ended questions form the teacher such as 'What theorem can you use?' They can be given by the computer giving feedbacks which transform the task to be performed or with some limitations in the provided tools which give indirect indications to students about the way to achieve the task. These procedural mediations may lead to the subdivision of a complex task into subtasks. They usually change knowledge adaptations in complex tasks and simplify the intended activities in such a way that it becomes more like technical tasks (for instance students having to apply a contextualized method).

The second type of mediations are more subject-oriented, here we use the term constructive mediations. They are designed to add something to the students' activities and the knowledge that can emerge from these activities. They can take the form of a simple summary of what has been developed by students, an explanation of choices, a partial decontextualization or generalisation, assessments and feedbacks, a discussion of results, etc. On some computers, the way a geometrical figure has been achieved by a student can be replayed to recall him the order in which instructions have been given without any wrong ways.

It should be noted here that our framework leads to the hypothesis that there is an internal transformation of the subject in the learning process: constructive mediations aim to contribute to this process. However, the mediations can be constructive for some students and remain procedural for others. On the contrary, some procedural mediations can become constructive for some students, for instance if they extract by themselves a generalisation from a local indication. Moreover some constructive mediationbut also perhaps productivecan belong to some students' ZPD in the sense of Vygotski or they can remain out of the ZPD. When they belong to the ZPD, they can be identified to appreciate the explicit links between the expression of the general concepts to be learned and their precise applications, in contextualised tasks, according to the necessary dynamic between them. Distinguishing between the kinds of mediations and the way they belong or not to some students' ZPD can be very difficult.

2.b The global level

The local level can be extended to a global level that takes into account the set of mathematical activities, the link with the intended conceptualisation (long term constructive loops), and teaching practices in the long term. We link mathematical students' activities to the intended conceptualisation of the relevant mathematical notion, establishing a "relief map" of this mathematical notion. This relief map is developed from an epistemological and mathematical analysis of the notion, the study of the curricula, and didactical analyses (e.g. students'common difficulties). This global analysis focuses on the similarity between students' activities (intended, observed, effective) and the set of activities that characterise the intended conceptualisation of the relevant notion.

However, the didactical analysis of one teaching session is insufficient. It is necessary to take into account, on a day-to-day basis, all of the tasks students are asked to complete, and teachers' interventions. We use the term scenario to describe a sequence of lessons and exercises on a given topic. The global scenario could be understood as a long term "cognitive road" [START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An example in a French 10th-grade class[END_REF].

Example of application: the 'shop sign' situation

To illustrate the utilisation of our Activity Theory, this section presents an example of a situation that aims to contribute to students' conceptualisations of the notion of function. Then we outline some limitations of the methodology at the global level.

The example relates in fact to the GeoGebra 'shop sign' family for learning functions. This family refers to mathematical situations that lie at the interface between two mathematical domains: geometry and functions.

There are many examples of shop sign situations, but they share the idea that a coloured area is the lit area of a shop sign [START_REF] Artigue | The challenge of developing a European course for supporting teachers' use ICT[END_REF] which depends on some moving variables in the shop sign. The task is set for grade 10 students (15 years old). One solution is to identify DE as an independent variable x. Then f(x), the sum of the two areas, is equal to x² (for the square) plus 4(4-x)/2 (for the triangle): equivalent to x²-2x + 8. In the French curriculum at grade 10, the derivative is not known and students must compute and understand the canonical form (x-1)² + 7 as a way to identify the minimum 7 for the distance DE=1 (which is the actual position on the figure).

Students are working in pairs on computers. They have already worked with functions in the traditional pencil and paper context, and they also have manipulated GeoGebra for geometrical tasks that do not refer to functions. In this new situation, GeoGebra helps them to begin the task by making conjectures about the minimum. Students can also trace the graph of the function, as shown in Figure 6. Then, in the algebraic register, they can find the canonical form of the function f(x) and the characteristics of the minimum.

We first identify the relief map on the notion of function and the intended conceptualisation. Then we give the a priori analysis of the task and the intended students' activities. We finish with the observation to two pairs of students to identify observable and effective activities.

3.a The global level: relief map on the notion of function and intended conceptualisation

The function is a central concept in mathematics and links it to other scientific fields and real-life situations. It both formalises and unifies [START_REF] Robert | Why and how to understand what is at stake in a mathematics class?[END_REF] a diversity of objects and situations that students encounter in secondary school: proportionality, geometrical transformations, linear, polynomial growth, etc. A diversity of systems of representations (numerical, graphical, algebraic, formal, etc.) and a diversity of perspectives (pointwise, local and global) are frequently combined when working with them [START_REF] Duval | Sémiosis et pensée humaine: registres sémiotiques et apprentissages intellectuels[END_REF][START_REF] Maschietto | Graphic Calculators and Micro Straightness: Analysis of a Didactic Engineering[END_REF][START_REF] Vandebrouck | Points de vue et domaines de travail en analyse[END_REF]. As it is summarized by [START_REF] Artigue | Mathematics thinking and learning at post-secondary level[END_REF], the processes of teaching and learning of function entail various intertwining difficulties that reinforce one another in complex ways.

Educational research [START_REF] Bergen | A theory of mathematical growth through embodiment, symbolism and proof[END_REF][START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF]Hitt and Gonzalez-Martin, 2016) shows that an efficient conceptualisation of the notion requires a rich experience that illustrates the diversity illustrated above, and the diversity of settings in which functions are used [START_REF] Douady | Jeux de cadre et dialectique outil-objet[END_REF]. It also means that functions are available as tools for solving tasks, and can be flexibly linked with other concepts. There must be a progression from embodied conceptualisations (where functions are highly dependent on physical experience) to proceptual conceptualisations (where they are considered dialectically and work both as processes and objects), paving the way for more formal conceptualisations [START_REF] Tall | Thinking through three worlds of mathematics[END_REF][START_REF] Bergen | A theory of mathematical growth through embodiment, symbolism and proof[END_REF].

At grade 10, the intended conceptualisation can be characterized by a set of tasks in which functions are used as tools and objects. They can be combined and used to link different settings (including geometrical and functional), numerical, algebraic and graphical representations, and the dialectic between pointwise and global perspectives. The shop sign task is useful in this respect, as students have to engage in such mathematical activities. A priori, optimisation tasks in geometrical modelling help to build the intended functional experience, and link geometrical and functional settings.

Technology provides a new support for physical experience, as the modelling process provides new systems of representation and helps to identify the dynamic connections between them. It also offers a new way to approach and connect pointwise and global perspectives on functional objects, and supports the building of rich functional experiences. A famous contribution is the one of [START_REF] Arzarello | Approaching functions through motion experiments[END_REF], who use sensors to introduce students to the functional domain. The framework is already an activity theoretical framework, together with more semiotic approaches, but it is not in a context of dynamic geometry. Many experiences exist about learning functions throught dynamical geometrical situations. For instance, [START_REF] Falcade | Approaching functions: Cabri tools as instruments of semiotic mediation[END_REF] study the potential of a didactical engineering with Cabri-Géomètre. The authors take a Vygotskian perspective about semiotic mediations which is more precise than our adaptation of Vygotsky inside Activity Theory, but which is also more restrictive in the sense that they don't consider deep connexions between given tasks and mathematical activities. Moreover, it doesn't concern ordinary classrooms. More recently, [START_REF] Minh | Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level[END_REF] analyze students' activities on functions using Casyopée. This solftware is directly built for the learning of functions and the authors adopt the model of Mathematic Working Spaces [START_REF] Kuzniak | Mathematical Working Spaces in Schooling[END_REF]. They built on three important challenges for students in the learning of functions: to consider functional dependencies, to understand the idea of independent variable, and at last to make sense of functional symbolism. The aims of the 'shop sign' family is consistent with such a progression which is closed to Tall's one introduced above.

3.b The local level: a priori analysis of the task and intended students'activities

The task is to identify the position of E on [DC] in order that the sum of the areas DFGE and AGB are minimal (Figure 2). It requires actual knowledge about geometrical figures and functions. However, it assumes that the notion of function is available, i.e. students have to identify the need for a function by themselves.

In a traditional pencil and paper environment, students first draw a generic figure. They can try to estimateby geometrical measurementssome values of the areas for different positions of E. They can draw a table of values but this kind of procedure is usully not enough to obtain a good conjecture of the minimum value. Moreover such a procedure can reinforce the pointwise perspective because it doesn't bring the continuous aspects of the function at stake. Usually, the teacher quickly ask students to produce algebraic expressions of the areas. Students try themselves to introduce an algebraic variable (DE=x), or the teacher gives them procedural aids.

In the example given here, the teacher provided students with a sheet of paper showing a figure similar to the one given in Figure 2, and the instructions as summarized in Figure 3. Figure 3 shows that the overall task is divided into three subtasks. Organizing activities are directed by procedural mediations (functional regulation), which is a way to ensure that most students can engage in productive activity.

A priori analysis of the first subtask: the construction of the figure

In the geometrical subtask students have to identify the fixed points (A, B, C, D), the free point (E) on [DC], and the dependent points (F and G). The order of construction is crucial to the robustness of the final figure, but are not important in the paper and pencil environment. Consequently, organizing activitiesthe order of instructionsare more important in the GeoGebra environment.

The subtask also requires students to make choices. It is possible to draw either G or F first, and the sequence of instructions is not the same. Moreover, there are other choices that have no equivalent in the paper and pencil environment: whether to define the polygons (the square and triangle) with the polygon instruction, or by the length of their sides; whether to use analytic coordinates of fixed points or a geometrical construction; whether to use a cursor to define E; etc. These choices refer not just to mathematical knowledge, but also to instrumental knowledge (following the instrumental genesis approach). This means that treatment activities include instrumental knowledge and are more complex than in the traditional environment. Once the construction is in place, students can verify its robustnessa treatment that is also specific to the dynamic environment.

A priori analysis of the second subtask: the conjecture

There is no task really equivalent to this subtask in the paper and pencil environment. This again leads to specific treatment activities. These are engaged with the feedback provided by the software, which assigns numerical values of the areas DFGE and AGB, according to the position of E. However, students are required to redefine DFGE and AGB as polygons if they have not already used this instruction to complete subtask 1 (Figure 5). They also have to create in GeoGebra environment a new numerical value that is the sum of the two areas in order to refine their conjecture. It is not clear in what extent these specific treatment activities refer to mathematical knowledge, and we will return to this point later.

A priori analysis of the third subtask: the algebraic proof

This subtask appears similar to its equivalent in the paper and pencil environment. However, as students already know the value of the minimum, the motivation for activity is different and only relates to the proof itself. The most important step is the introduction of x, as a way to pass from the geometrical setting to the functional setting. This step brings recognizing activities (students must recognize that the functional setting is needed), which is triggered by a procedural mediation (the instructions given on the sheet).

Students have to determine the algebraic expression of the function. Existing knowledge about the area of polygons must be available. They also have to recognize a second order polynomial function associated to specific treatments. Treatment activities remain to obtain the canonical form (as students have not been taught about derivatives, they must be helped in this by the teacher). At last, the recognition of the canonical form as a way to obtain the minimum of the area and the position of E which correspond to this minimum is correlate to the importance of the dialectic between pointwise and global perspectives on functions.

3.c A posteriori analysis: observable and effective activities

Students worked in pairs. The teacher only intervened at the beginning of the session (to ensure that all students were working), and at the end (to summarise the session). Students mostly worked autonomously, although the teacher helped individual pairs of students. The following observations are based on two pairs of students: Aurélien and Arnaud, and Lolita and Farah.

Analysis of the first pair of students'activities: Aurélien and Arnaud

This pair took a long time to construct their figure (more than 20 minutes). They began with A,B,C,D, in sequence, using coordinates, and then drawing lines between pairs of points. This approach is closest to the paper and pencil situation, and while it is time-consuming it is not crucial for global reasoning. They then introduced a cursora numerical variable j that took a value between 0 and 4in order to position E on [D, C]. However, the positioning of F (at 0, 3), was achieved without the cursor, which leads to a wrong square (Figure 4). G was drawn correctly. After they had completed their construction, they moved the cursor in order to verify that their figure was robust; an operation which revealed that it was not (Figure 4). This mediation from the screen is supposed to be a constructive mediation: it does not change the nature of the task and it is supposed to permit a constructive regulation of students' activities (lower loop in reference to Figure 1). However, the mediation doesn't encounter the students' ZPD and it is insufficient for them to regulate their activity by their own. The mediation supposes in fact new recognizing activities, specific to dynamic geometry on computers, that these students are not able to develop.

In this case, the teacher makes a procedural mediation and helps the students to rebuild their figure ("You use the polygon instruction to make DFGE […] then again to make the polygon ABG"). Once the two polygons have been correctly drawn, the values of their areas appear in the numerical window of GeoGebra (called poly1 and poly2, shown on the left-hand side of the screens presented in Figure 5). In the conjecture phase (second subtask, 8 minutes), the students made the conjecture that the sum is always 8 ("Look, it's always 8…"), by computing poly1+poly2 in their mind. The numerical window of GebGebra now shows 18 different pieces of information, including the areas of DFGE (poly1) and ABG (poly2). Students must introduce another numerical variable (e.g. poly3) that is equal to the sum of poly1+poly2. However, this requires new organizing activities that GeoGebra does not help with. In fact, there is already too much information in the numerical window. Here again, the teacher provides direct procedural assistance ("introduce poly3=poly1+poly2").

In the algebraic phase (third subtask, 20 minutes), the students are unable to express the areas DFGE and ABG as functions of x. Analyses reveal that again new recognizing activities are awaited to switch from the computer environment to the paper and pencil environment. These new recognizing activities are not an evidence for the students. They suppose both mathematical knowledge and instrumental knowledge about the potentialities of solftware and the mathematical way of proving the existence and the values of the minimum. Then, students' attempt to implement DE=x in the input bar leads to feedback from GeoGebra (in the form of a syntax error), which informs them that their procedure is wrongbut does not provide any guidance about what to do instead.

It is difficult to know whether to categorise this kind of mediation as procedural or constructive as it does not add any mathematical knowledge.

The teacher asks the students to try to find a solution with pencil and paper (procedural assistance). However, the introduction of x, which is linked to the change of mathematical setting (adaptation of knowledge), seems very artificial. The students start working on their algebraic formula by looking at their static figure, with E positioned at (1, 4). The base of the triangle measures 4 and its height is 3. One of the pair suggests that "it depends on x" means that each algebraic expression ends in x, as the following dialogue between the two students shows:

"This is 4x At this point, the teacher provides another direct procedural assistance. This once again shows that although the mediation of GeoGebra help students to discuss and progress, it is insufficient for them to correctly regulate their activity. Without procedural assistance from the teacher, they are unable to find the formula for the area of triangle. In the end, the students don't have enough time to finish the task by themselves.

At end of the session, the teacher gives a procedural explanation to whole class of how to find the canonical form (as "x²-2x+8 = (x-_)²+_"). Although Aurélien and Arnaud write it down, they do not make the link between it, and their classroom work. Consequently, they do not understand the motivation for the activity and cannot make sense of the explanation of the canonical transformation given by the teacher.

Then the teacher gives constructive explanation about the meaning of the coefficients in the canonical form and the way they give the minimum and the corresponding value of x. But according to Aurélien and Arnaud's activities, it is too early and they do not make the link with their numerical conjecture. In other words, the collective mediation of the teacher seems too far from the students' ZPD and it is not at all constructive for this pair of students.

Analysis of the second pair of students' activities: Lolita and Farah

Lolita and Farah are better students and quickly draw their robust figure. Their numerical conjecture is correct and the teacher gives them another subtask: to find a graphical confirmation of their conjecture. The procedural instruction is to find a new point, M, whose abscissa is the same as E and ordinate is the value poly1+poly2. However, Lolita and Farah do not recognize this. One says "this is not a curve" and then "the minima, we have seen this for functions but here…".

They only recognize the trace as a part of a parabola (geometrical setting) and associate its lowest point with the value of the minimum area.

The graphical observation confirms to Lolita and Farah that their numerical conjecture was correct. However, this is a proof for them and they do not understand the motivation of the third subtask which does not make sense to them. Although they succeed in defining the algebraic expression of the function and they find the canonical expression, they do not make the link with their graphical observation.

Here again, the teacher's summary of how to obtain the canonical form of the function, the value of the minimum and the corresponding value of x is not useful for this pair as it is not the problem they encountered.

A constructive intervention about the motivation for the third subtask and how the canonical form was linked to the conjecture would have been a mediation closer to their ZPD.

What does this tell us about students' mathematical activities?

The main result concerns complex activity involving technology: here the complexity is introduced by mathematical activities that require either mathematical or instrumental knowledge, particularly knowledge about the real potentialities of technologies in contrast with what is supposed to be solved within the paper and pencil environment. This leads also to new treatment activities (e.g. in the construction and conjecture subtasks) and new recognizing activities, New, onscreen, representations appear, typically dynamic, and students must recognize them as mathematical objects (or not). The example of Aurélien and Arnaud shows how difficult it was for them to recognize a robust figure, and dynamic and numerical representations of variations in areas. Similarly, it was difficult for Lolita and Farah to recognize the trace of M as a special part of the graph of a function.

The second main result concerns the increase in recognizing activities and the new balance between the three types of critical activities. While in a traditional session, the teacher can point out the mathematical objects to use, the screen presents far more information to students, meaning that they have to recognize what is most important in their treatment activities. Organizing activities also increase, both before treatment activities related to construction, and during conjecture. For instance, Aurélien and Arnaud failed in the conjecture task because they were not able to introduce a third numerical variable by themselves. Classroom observation [START_REF] Vandebrouck | Proximités en acte mises en jeu en classe par les enseignants du secondaire et ZPD des élèves : analyses de séances sur des tâches complexes[END_REF] has led to the idea that most of effective students'activities are treatment activities as the teacher must make productive interventions before most students can begin the task.

Recognizing and organizing activities are mostly activities for best students. These students often have an idea of how to begin the resolution of the task, they are able to adapt quickly their knowledge, and they develop all three types of critical mathematical activities, whereas weaker students find it difficult to engage in the task waiting for any procedurale assistance of the teacher. In classroom sessions that use technology, students are confronted alone with all of these critical activities, which may help to explain the difficulty of weaker students.

A further finding concerns mediations. In such sessions, teacher's mediations are mostly procedural and clearly aim to foster productive activity. Onscreen mediation leads to specific, new recognizing activities (dynamism) but is insufficient for students (not only weaker) to regulate their own activity. It appears that most of the time this mediation is not procedural or constructive enough, leading to more teacher intervention. Moreover, it seems that onscreen mediation is always associated with treatment activities and does not help students in their recognizing or organizing activities.

The last point concerns constructive mediation and the heterogeneity of the students' knowledge (and ZPD).

Student activities in classroom sessions that use technology are difficult for the teacher to evaluate. Even if he/she tries to manage the best "average" constructive mediations for all students, our examples show that this is very challenging. This raises the question of what is the real impact of such sessions with respect to the intended conceptualisation. The availability and recognition of functions as tools to complete such tasks was not really investigated, in the sense that the independent variable x was given to students (on paper), and none of them returned to the geometrical setting as in the traditional modelling cycle -in reference to Kaiser and Blum [START_REF] Maas | What are modelling competencies?[END_REF]. Moreover, Aurélien and Arnaud did not explore the dynamic numerical-graphicalalgebraical flexibility, which was one of the aims of the session; on the other hand Lolita and Farah did, but lacked the constructive mediations needed to complete the cycle.

Conclusion

We have presented Activity Theory in the context of French didactics, notably the dual regulation found in the activity model, which was first developed in ergonomic psychology and then adapted to didactics of mathematics, for studying students' activities. Other works, which we have not discussed here, have looked at teachers' practices in some different ways [START_REF] Robert | A didactical framework for studying students' and teachers' activities when learning and teaching mathematics[END_REF][START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An example in a French 10th-grade class[END_REF]). An important component of this model is the impact of activity on subjects, which represents the developmental dimension of students'activity. This focus highlights the commonalities and complementarities of the constructivist theories of Piaget (extended to Vergnaud's conceptual fields) and Vygotsky. The connection between Activity Theory, the work of Piaget and Vygotsky, and didactics of mathematics, provides a theoretical foundation for a dual approach to students' activity from the viewpoint of mathematics (the didactical approach) and subjects (the cognitive approach).

Our analysis does not provide a model of students' activity (or teachers' practices). However, it leads to the identification of similarities and differences in terms of the relations between subtasks, students' ways of working, mediations, mathematical activities, and compares this complex task with the traditional, paper and pencil environment. One of the specificity of our approach is the deep connection between the students' activities analysis and the a priori tasks' analysis, including mathematical content. But we do not look for the teacher's own intention, unlike what is done in some English research (for instance, [START_REF] Jaworski | Bridging the macro-and micro-divide: using an activity theory model to capture sociocultural complexity in mathematics teaching and its development[END_REF]. Moreover, we do not attempt to raise the global dynamic between individual and collective interactions and learning. We should take now a threefold approach to the investigation of students' practicesdidactical, cognitive and socio-cultural. As [START_REF] Radford | The epistemic, the cognitive, the human: a commentary on the mathematical working space approach[END_REF] argues, with respect to Mathematical Working Space [START_REF] Kuzniak | Mathematical Working Spaces in Schooling[END_REF], the individual-collective dynamic remains difficult to understand in both our Activity Theory and MWS which are discussed together. This represents a new opportunity to better investigate the socio cultural dimension of Activity Theoryespecially the one developed by Engestrom -and integrate it into our didactical and cognitive approach.

Figure 1 :

 1 Figure 1: Codetermination of activity and twofold regulatory loop

Figure 2 :

 2 Figure 2: Shop sign

First

  Figure 3: Main instructions given to students

Figure 4 :

 4 Figure 4: Exploring the robustness of the shop sign

Figure 5 :

 5 Figure 5: Exploration of varying areas by moving the point E on [DC]

Figure 6 :

 6 Figure 6: The shop sign task showing part of the graph of the function