text:
Frode Rønning
email: frode.ronning@math.ntnu.noheidi.stromskag@math.ntnu.no

Heidi Strømskag

Entering the mathematical register through evolution of the material milieu for classification of polygons

Keywords: Register, language of nearness and distance, polygons, adidactical situation, milieu

This paper is based on classroom observations from Norway with 7-8 years old children working on geometrical shapes. The intention is that the children shall classify different polygons according to their number of edges. The observations are part of a teaching sequence that is designed using principles from Brousseau's Theory of Didactical Situations (TDS). From the teaching sequence we identify certain challenges in the children's development of scientific terms and the observations allow us to conclude that these challenges to some extent are connected to specific semantic features of the Norwegian language. The use of TDS is instrumental in revealing the challenges that occurred and explaining what was changed to overcome them.

Introduction

This paper reports on a teaching sequence within the project Language Use and Development in the Mathematics Classroom (LaUDiM)-an intervention study carried out in collaboration between researchers at the Norwegian University of Science and Technology and two local primary schools in the period 2014-2018. The main objective of the project is to study pupils' development and use of mathematical language in order to gain knowledge that will help the teachers to develop their teaching-aimed at pupils' increased proficiency in expressing mathematical ideas, mathematical reasoning, arguing and justification. Teaching sequences are designed in collaboration between researchers and teachers, where the design is guided by principles from the theory of didactical situations [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990[END_REF].

In this paper we study a teaching sequence at one of the project schools including pupils from Grade 2 (7-8 years old) and their teacher. The main aim of the teaching sequence-consisting of three sessions-is that the pupils shall develop their language use about polygons, with the specific aim that they shall be able to classify polygons based on seeing visual images and that they can discern different parts of a polygon (vertices and edges). The classroom observations (recorded on video) give insights into pupils' evoked concept images [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limit and continuity[END_REF] of vertex and edge, and how discrepancy between these and the scientific definitions of them constrains the teacher's goal of the first session. Further, we show how this is resolved by the teacher in the subsequent sessions.

Theoretical framework

According to Halliday a register is "a configuration of meanings that are typically associated with a particular configuration of field, mode and tenor" (1985, pp. 38-39). Halliday compares register to dialect and states that "dialects are saying the same thing in different ways, whereas registers are saying different things" (Halliday,p. 41). So changing between registers can mean that the same word gets a different meaning. The mathematical register is characterised by the property that words have very precise meanings and sometimes the same word may be used in the mathematical register and in the register of everyday language but with different meaning. This feature is language specific in the sense that for a given word it can be present in one language but when this word is translated to another language it may lead to one word in the everyday register and another word in the mathematical register. [START_REF] Koch | Sprache der Nähe-Sprache der Distanz. Mündlichkeit und Schriftlichkeit im Spannungsfeld von Sprachtheorie und Sprachgeschicht[END_REF] make a distinction between language being conceptually oral or conceptually written. They refer to the first category as a language of nearness, where the interlocutors are in direct contact and can comment on each other's utterances and directly refer to the given situation, for instance by using gestures. The second category they refer to as a language of distance, where the sender and the receiver are not necessarily in contact and the language therefore has to be more precise. The mathematical language is in its nature conceptually written because of the way it strives for precision and unambiguity. However, much of the communication in the mathematics classroom has many of the features characterising a conceptually oral language, a language of nearness. In particular, when working with young children the communication is characterised by dialogue, face-to-face interaction and a desire to avoid complexity, features characterising a language of nearness. However, one of the aims of schooling is to develop the mathematical language into a language characterised by greater precision, compactness, density of information, features characterising a language of distance Koch et al., 1985, p. 23. The theory of didactical situations in mathematics, TDS [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990[END_REF] is a scientific approach to the problems related to teaching and learning of mathematics, where the particularity of the knowledge taught plays a significant role. Its methodology-for a targeted piece of mathematical knowledge-is based on creating a situation with a problem to be solved, where the knowledge aimed at is the optimal solution to the given problem. In the following, based on [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990[END_REF], we explain some concepts of TDS that are relevant for our analysis.

An adidactical situation is a situation in which the student takes a mathematical problem as his own and tries to solve it without the teacher's guidance and without didactical reasoning (i.e., not trying to interpret the teacher's intention with it). The milieu models the elements of the material and intellectual reality on which the students act when solving a problem-these elements are conditions for the students' actions and reasoning. The milieu may comprise: the problem to be solved; material or symbolic tools provided (artefacts, informative texts, data, etc.); students' prior knowledge; other students; and, arrangement of the classroom and rules for operating in the situation (determinative of who is supposed to interact with whom). The milieu of an adidactical situation is called an adidactical milieu. An appropriate adidactical milieu provides feedback to the students, whether their responses are adequate with respect to the knowledge at stake.

After devolution, a phase where the teacher has (temporarily) transferred responsibility for solving the problem to the students, four situations (or phases) follow: Situations of action, formulation, and validation are (intentionally) adidactical situations, whereas the situation of institutionalisation is a didactical phase. The situation of action is where the students engage with the given problem on the basis of its inner logic, without the teacher's intervention. The students construct a representation of the situation that serves as a "model" that guides them in their decisions. This model is an example of relationships between certain objects or rules that they have perceived as relevant in the situation.

The situation of formulation is where the students' formulations are useful in order to act indirectly on the (material) milieu-that is, to formulate a strategy enabling somebody else to operate on the milieu. In this situation the teacher's role is to make different formulations "visible" in the classroom. The situation of validation is where the students attempt to explain some phenomenon or verify a conjecture. In this situation the teacher's role is to act as a chair of a scientific debate and (ideally) intervene only to structure the debate and try to make the students use more precise mathematical notions. The situation of institutionalisation is where the teacher connects the knowledge built by the students-through adidactical interaction with the milieu-to the scholarly and decontextualised forms of knowledge aimed at by the institution.

Methodical approach

Each teaching sequence in the project starts with a planning session where teachers and researchers work together to plan the activities for two classroom sessions, and in particular set the learning goals for the classroom sessions. Activities and actions are planned according to the phases of TDS, devolution, action, formulation, validation, and institutionalisation [START_REF] Brousseau | The theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990[END_REF]. Some days later, the first classroom session takes place, immediately followed by a reflection session, where experiences from the first classroom session are discussed and adjustments are made for the second session, taking place yet a couple of days later. Researchers are also present in the classroom. Observations from all sessions (planning, classroom implementation and reflection) are recorded on video, and additional audio recording is used to secure the quality of the sound. In the classroom sessions selected pupils working in groups (2-3) are video recorded, as is the teacher in whole-class sessions. Tasks given to the pupils in the observed sessions and written material produced by the pupils are also data sources. After completing a cycle of planning, reflection and classroom sessions, teachers and researchers meet to watch parts of the video recordings from the classroom. This represents the first step in analysing data, where interesting sequences from the classroom are identified. In the planning session and the video session, teachers from both schools are present. This paper is based on a teaching sequence on geometrical shapes, consisting of three classroom sessions1 . Data from the teaching sequence form the basis for answering the following research question: What conditions enable or hinder pupils' opportunities to categorise polygons according to their number of edges?

The utterances reproduced are excerpts from a transcript of the video recorded whole-class discussion in the second session. The camera faces the teacher at the board and it captures the dialogue between the teacher and the 14 pupils who are sitting in a semi-circle close to the board. Parts of the dialogue between teacher and pupils have been transcribed and translated from Norwegian into English. In cases where it is important for the analysis to emphasise the meaning of a particular word in Norwegian, the Norwegian word is included in square brackets in the transcript.

Our analysis is based on ethnomethodological conversation analysis, focusing on the thematic development of an interaction rather than on its structural development [START_REF] Holstein | Interpretive practice and social action[END_REF]. This gives the possibility to analyse the relationship between language and the figures with their components while teachers and students negotiate mathematical meaning [START_REF] Fetzer | The interplay of language and objects in the mathematics classroom[END_REF].

The teaching sequence analysed here is chosen because: (1) it illustrates how the phenomenon of words having different meanings in the mathematical and everyday registers constrains pupils' conceptual development; and (2) it illustrates how an evolution of the milieu gives a rationale for using the target knowledge.

Analysis of the teaching sequence

First session-classification

In the first session of the teaching sequence, pupils work in pairs on sheets of paper showing 12 shapes, as presented in Figure 1 (one pupil has blue, the other has red figures). The task they get is that each pupil shall (individually) sort the figures into groups (cutting the individual figures from the sheet) and give a name to each group (ACTION). Then they are supposed to compare (in the pairs) how they have sorted the figures and agree on a way to sort them and also agree on a name for each group (FORMULATION). The final result from each group is a sheet of paper on which the pupils have glued on figures from the same group and with the text "These are <__> because <____>" and the pupils have filled in the blanks (VALIDATION). After the session the teacher collects the worksheets and she uses them as background for a whole-class discussion in the second session (INSTITUTIONALISATION).

In Norwegian, polygons are named literally after the number of edges, using the standard Norwegian number words, so that a triangle is called a "three edge" (trekant), a quadrilateral is called a "four edge" (firkant), and similarly for the others. An accepted name for the generic concept polygon is 'mangekant' which literally means "many edge". Learning names of polygons, and understanding the reason for the names, is therefore not considered to be a challenge for Norwegian students. This is in contrast to the situation in English where it is not obvious from the everyday language that for instance a pentagon is a shape with five edges. The teacher has seen from the collected worksheets that all groups have given names to the shapes based on the number of edges and they have written for instance "these are 'five edges' because they have five edges". However, from the discussion in pairs she has observed that even if all the pupils talk about edges (kanter), the way they point at the figures indicates that some counts the edges but others count the vertices. The Norwegian language has no precise scientific word for vertex, the word which is used is 'hjørne', which (also) means corner.

Second session-the meanings of edge and corner

In the institutionalisation phase the teacher asks pupils to come to the board and explain their reasoning. She has observed Oliver and Amelia counting the vertices and Thomas, Daniel and Sophie counting the edges. Among the figures is a quadrilateral with three acute angles and one reflex angle (Figure 2), where we have inserted the letter A for reference in the dialogue. Although Oliver and Amelia have grouped this among the quadrilaterals, A Oliver expresses some doubt when he is called to the board to explain how he and Amelia have thought.

Oliver: If we had pulled this out a little (pointing to vertex A with the reflex angle) it would have been a "four edge" [firkant]. Teacher:

OK, but still you have grouped this among the "four edges". Oliver:

One, two three, four (pointing to the vertices).

Teacher:

So what is an edge? Oliver:

That is the pointed parts [spissene].

When Thomas is called to the board he uses a rectangle as his example and clearly points to the edges, counting "one-two-three-four". The teacher has in many sessions talked about "what mathematicians do" and that they for instance decide and agree on what names to give to mathematical objects. At this stage the teacher says that now we have to agree on something-as the mathematicians do-so that we have a common understanding of what an edge is. The teacher has also observed that some pupils use the word 'hjørne' and she draws their attention to this. One pupil, Jessica, says that they had talked about 'hjørne' but they did not know what it was, so they had written 'kanter'. The teacher asks the pupils to explain what a 'hjørne' is and encourages William to come to the board.

William:

That inside is a corner and those outside are edges. Teacher:

Can you show us? William:

This is a corner (points to vertex A with the reflex angle in Figure 2) and that is an edge (points to one of the acute angles).

Teacher:

But what about this (points to the rectangle)? Oliver:

Edge, edge, edge, edge (points to each of the four vertices).

Then the pupils continue to discuss the inner and the outer angle at a vertex, and that one is a 'hjørne' and the other is a 'kant'. Thomas says that "the corners are inside and the edges are outside" and Chloe agrees that the corner is inside but she refers to the outside as the "pointed parts" (spissene). William is making a distinction between the vertex at the reflex angle of the nonconvex quadrilateral (Figure 2), which he refers to as a 'hjørne', and the vertices at the acute angles, which he denotes by the word 'kant'. Oliver, using the rectangle as his reference context, refers to all the vertices by the word 'kant'.

To get the pupils to agree on one name for the same object the teacher brings in a reference context from their everyday life, a mini-pitch. The picture in Figure 3 is shown on the whiteboard. Using this picture as the reference context, the teacher asks questions like "If I say that you should place yourself on the edge of the mini-pitch, where would you be standing?" or "…place yourself at the corner, where would you go?" There is still some confusion among the pupils, so the teacher says that she will tell them "what the mathematicians have decided". She holds up a rectangular sheet of paper (A4) and says:

Teacher: Corner (vertex), that is where two sides meet. When we talk about edge, we can also call this the side-edge [sidekant], and where two edges meet, that is a corner (vertex). There is the corner (points to a vertex of the sheet).

The last part of the session was completed at the mini-pitch in the schoolyard, where they played the game of the teacher telling where to go-by using the concepts of edge and corner-and the pupils went to a place which (supposedly) fulfilled the teacher's command.

The teacher's reference to "work like mathematicians" and that this entails giving precise definitions indicates that she intends to introduce her pupils to the mathematical register. However, the communication is hindered by the fact that some of the words have different meanings in the mathematical register and the everyday register. In particular this is the case for the word 'hjørne' which can mean vertex (mathematical register) as well as corner (everyday register). In everyday language the corner is a spacious area, somewhere you can stand, but in mathematics it is a point, the intersection between two lines. The word 'kant' has the same connotation as English edge, and in everyday language this is used as something that is sharp. This may explain why 'kant' is used to denote both the side, or edge, and the vertex when it is approached from the outside.

Third session-a milieu that affords feedback

As a follow up, a game with 12 tiles was developed. On one side of the tile was depicted a polygon where the edges had one colour and the vertices were marked with another colour. On the reverse side was written "<name of shape> with <colour> edges" or "<name of shape> with <colour> vertices". An example is shown in Figure 4. On the back of this was written "Pentagon (femkant) with blue edges". This game was played in pairs of pupils both having the full set of tiles. One pupil reads the text and the other one is supposed to pick the correct shape. After picking he/she can turn the tile and read the text to see if the correct shape has been picked.

Discussion

The target knowledge of the teaching sequence was that the pupils should develop the scientific language for naming 2D shapes and become aware that these names are based on the number of edges in the shape. To know the difference between edges and corners (vertices) will then also be part of the target knowledge. A condition that hinders pupils' opportunity to categorise polygons according to their number of edges, is the ambiguous use of the concept of edge. Many of the pupils thought that edge ('kant') referred to corner/vertex, and since the number of edges equals the number of vertices it gave meaning to classify polygons according to the number of corners.

The material milieu in Session 1 did not have an adidactical potential for categorisation according to the number of edges, since it was possible to solve the task apparently correct, without the pupils having a common understanding of what is an edge and what is a corner/vertex. There was no feedback from the milieu that could have told them whether they used the desired concept to classify: In action they counted either edges or corners (which gave the same answer); in formulation they compared their categorisations (and if they had a figure that was categorised differently, they used either edges or corners as a basis for categorising jointly and agreeing); in validation, if they reasoned on the basis of different attributes (edges or corners), they concluded that it did not matter which attribute to use.

During Session 1, the teacher realised that the pupils had other connotations of edge and corner/ vertex than the scientific ones. In institutionalisation (Session 2), the teacher let the different connotations be displayed, and-with reference to mathematicians-she introduced the scholarly meaning of the concepts, in the mathematical register. Further, she connected them to the pupils' everyday register, through the mini-pitch context.

Based on results from Sessions 1 and 2, the teacher designed a new material milieu (tiles) that has an adidactical potential (see Figure 4). The game will produce a win if the pupil uses the scholarly meaning of edge and corner, and a loss if not. Hence, the pupil will need the target knowledge to act on the milieu-a principle at the core of TDS' instructional design. The evolution of the milieu described here is a condition that enables the pupils' opportunity to categorise polygons according to their number of edges.

The teacher's desire to introduce precise mathematical terms also points to introducing a language of distance. However, the situation is such that the pupils are able to express themselves clearly using gestures together with oral language, thereby using a language of nearness. However, in the game with the tiles it is necessary to use a language of distance in order to pick the correct tile. Hence, the intended language development is stimulated by the activity's adidactical potential.

It has been observed earlier that Norwegian children focus on the vertices when naming polygons [START_REF] Rønning | Language and concept development in geometry[END_REF] but the observations made in this paper show that they may use different words depending on whether they approach the vertex from the inside or from the outside. We have also seen that they may use different words within the same shape, as with the non-convex quadrilateral in Figure 2. This shape is also interesting in the sense that it is not really accepted by Oliver as a quadrilateral but it would have been if "we had pulled this [the vertex with the reflex angle] out a little". We interpret this as Oliver's inclination to distinguish between convex and non-convex polygons. In future learning of geometry, the concept of a convex polygon will be introduced and this example indicates that early exposition to non-convex shapes can be important for making pupils familiar with these shapes.

Distinguishing between edges and vertices can also be seen to be important for future learning. For polygons, the number of edges is equal to the number of vertices so to name a polygon one may just as well count the number of vertices instead of the number of edges. However, for polyhedra the number of vertices, edges and faces are not the same, and the naming is based on the number of faces.

The results presented here are relevant for mathematics teachers and teacher educators: They present challenges and affordances related to teaching of properties of polygons-with emphasis on language and characteristics of the milieu with which the pupils interact when solving a problem.

Figure 1 :

 1 Figure 1: Shapes to be classified

Figure 2 :

 2 Figure 2: Non-convex quadrilateral

  difference between what Oliver did and what Thomas did? Megan: Thomas counted the lines [strekene] and Oliver counted the pointed parts [spissene]. Teacher: So actually we did not quite agree on what an edge really is. The dialogue above reveals that there are different opinions among the pupils as to what the word 'kant' means. All the pupils claim that they are counting the edges but when asked to explain what they have counted, Oliver points to the vertices and Thomas points to the edges. Megan makes the observation that the two boys have actually counted different parts of the polygon.

Figure 3 :

 3 Figure 3: The mini-pitch

Figure 4 :

 4 Figure 4: Pentagon with blue edges

Proceedings of CERME10

The analysed teaching sequence consists of three sessions (instead of two which is common in the project). The third session involves pupils' interaction with a milieu designed so as to give feedback in the devolved adidactical situation.Thematic Working Group 09Proceedings of CERME10


domain: math shs

text:
Peter Ludes
email: peter.ludes@tu-dresden.de

Marcus Schütte

Fostering of interdisciplinary learning through basic education in computer science in mathematics in primary education

Keywords: Computer science, interaction, mathematical competences, digitalization

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Computer Science, digitalization or technology are only three out of many other terms concerning the digital age that ubiquitous in today's society. Nearly every professional branch has to deal with technology and digitalization at some point. Employees have to learn how to deal with these digital structures mainly on their own, because they never experienced a fundamental education in computer science. Many schools in Germany offer Computer Science only as a subject in secondary school although coding is seen as the language of the 21 st century. Whoever is able to "speak" a programming language will be understood all over the world. Many IT unicorn businesses have their roots in a teenager's room or the parents' garage. Often, these businesses came to be because exceptional skill and auto didactical training come together. The masses only seem to use the technology that emerges from these businesses without understanding the basic underlying structures that enable the creation of a product. No matter which idea came to be, it has almost certainly been realized through programming. The choice of programming language is rather secondary as they all follow the same basic rules. Children, from first grade onwards, know about digital technology and computers in different shapes and formats, but when they enter school, these digital tools often cease to exist. Therefore, a responsible use of these technologies and learning in computer science is not fostered in primary schools. Tablets, laptops, smartphones and gaming consoles are at least present once in every household. According to the KIM study, most families own more than one of these devices and use them more than once a week (e.g. the percentage of smartphone/laptop ownership is 97%). In fact, 41% of children from 6-10 own a smartphone themselves. Children learn from an early age how to use e.g. tablets. The functions of these devices are often self-explanatory. The underlying structures, e.g. how is software developed, what rules does a computer follow or how to troubleshoot problems, are not that clear to most children although these structures are relatively similar across all digital devices, this also is true for most teachers, who did not grow up with technology and are now supposed use it in the classroom and teach its responsible use. Compared to mathematics this would be similar to learning specific calculations with specific numbers or materials but not understanding how to transfer and apply e.g. long division to other tasks. The paper does not aim to defy media competences and how they can be accomplished, it is rather a claim to go further and to embrace computer science itself into primary education and see where and when interaction and negotiation of meaning between children occurs and how it fosters the learning of computer science and mathematics competences. This would empower children to accomplish much more with their knowledge about media and their use (knowledge, they undoubtedly gained while growing up in a digital world).

Status Quo: Media competences, a watered-down term

Computer Science is a topic that is present in almost every secondary school, either as a mandatory or a facultative subject. Either way, the subject is taught by teachers, who have a higher education degree in computer science. To become a primary school teacher in Germany, as in many other countries, the selection of specific subjects is not necessary, as the primary teacher is trained to teach all subjects that are part of the primary school curriculum (although many universities offer the opportunity to choose a core subject (e.g. mathematics) that will give the student the ability to enjoy an even more focused education in this specific area. Computer Science as a subject is not present in either primary school or primary school teacher education. Future teachers rather can archieve something that is often called media certificate or media training. Mostly, this is integrated into the courses that are already taken at university. E.g. one course would include a topic like "use of digital media in geometry in primary school". This is neither standardised, nor does it aim at specific competences. It is rather a subject specific realization of the use of technology for a certain topic. Today, most schools decide themselves how and when to use digital media and often one feels that it is mostly aimed at helping teachers to keep up with what children already know. If they decide to do so it is often done under the term of media competency. [START_REF] Krauthausen | Digitale Medien im Mathematikunterricht der Grundschule[END_REF] mentions that the term "media competences" is colorful and can be interpreted very differently according to specific interests. For some schools media competency means that the teacher uses e.g. an iPad, for others that the children learn how to use word processing software. Other projects try to foster the use of specific types of media such as podcasts to facilitate learning process for mathematics (cf. [START_REF] Schreiber | Podcasts selbst erstellen? Na klar: PriMaPodcasts! Grundschulunterricht Mathematik[END_REF]. This is far away from being similar to what computer science would request children to learn. Computer science is considerably more than just knowing about the functions of a computer. It is learning about logic, about algorithms, about programming & robotics and (late-breaking) cryptography. Questions like: What is an algorithm? What is logic? What technology can I use to facilitate my work? can be answered by young children to improve their learning. But not only will the acquisition of such skills foster the expert knowledge, it will also nurture competences that can be interdisciplinary used. This is important, as we have to ask whether our traditional cultural competences are still sufficient in a more and more digitalized world or whether we need to teach computer science as well. Some projects try to accomplish this, e.g. [START_REF] Herper | Informatische Bildung im Primarbereich[END_REF] through computer science education in primary school and [START_REF] Weigand | Algorithmik in der Grundschule[END_REF] with his project: "Algorithms in primary school" (title translated by the author). Weigand especially shows that to work on basic principles of computer science (in this case algorithms) a computer itself does not necessarily have to be available, as he uses pen and paper to work on algorithmic processes. Primary schools do not offer the subject of computer science and it will not be easy to implement it into the curriculum as an independent subject. To achieve this goal, a back door has to be found to sneak computer science into primary school. Then, once its benefits have become obvious the step from being part of another subject to being an independent subject is only a small one. The answer to the question what subject should be used to include computer science in its contents could not be answered more easily. Mathematics seems to be the ideal candidate, as competences in both subjects are similar right up to identical. We will now focus on these similarities in more detail.

Competences in Mathematics and Computer Science

The German core curriculum provides two types of competences for the subject of mathematics: the general mathematical competences and the content-related mathematical competences (KMK Bildungsstandards, 2005). General mathematical competences include arguing, problem solving, communicating, modelling and the presentation of mathematics. Although these competences will equally be taken into account, the main focus here will be placed onto the content-related mathematical competences, which are numbers and operations, space and shape, pattern and structures, sizes and measurement and data, frequency and probability. These content-related mathematical competences provide the framework for the contents of the learning environments. To justify these contents, a side by side comparisons of some chosen content-related mathematical competences and where to find them in computer science will be done. The mathematical competences will be presented as written down in the German core curriculum, then computer science content that also matches these competences will be provided.

Mathematics Computer Science

Understand the relationship between and the representation of numbers, Understand and master calculations Algorithms use calculations, Loops have to be counted, Sorting algorithms, Types of variables integer, float, double This selection of competences from the core curriculum shows that to each content related mathematical competence a related topic in computer science can easily be found. This is highly interesting, as it suggests that mathematical competences are similar to those that are required to perform tasks in computer science. Thus, mathematics can fulfil the requirements to integrate elements of computer science into its curriculum. To prove this claim, it would hardly be possible to modify the existing mathematical lessons to include this content. Rather, specific learning environments have been developed to show that learning computer science topics nurtures the competences that are necessary for both mathematics and computer science.

The pilot project

Partner school

The search for partner schools was far from easy. Many schools were not interested. One school on the other hand was immediately willing to participate and upon further information managed to interest 19 children from grade four to take part in the project. As we did not expect such an overwhelmingly large number of participants from one single school, we cancelled all further efforts to acquire more schools and decided to work exclusively with the just one partner school. The school was very open to our project and supported us from day one. The first feedback suggests that all children are highly enthusiastic and motivated. A claim that we can only support after our first three weeks. In agreement with the school, we chose two time slots of 90 minutes in the afternoon twice a week (Tuesday and Thursday) over the course of six weeks. During the time slots the children worked on the learning environments in pairs (some tasks required two groups to come together to discuss) supervised by a student from the seminar or Peter Ludes.

Learning environments

To examine whether certain competences are used during working with tasks, specific learning environments have been developed that have a computer science topic as a core topic. The learning environments have been developed during one of Peter Ludes' empirical seminars for future elementary school teachers. The students developed the first draft of the learning environments on their own and after a review process finalized them collectively during the seminar. The main topics are: logic (general and propositional), algorithms, cryptography and programming/robotics. The main challenge for the students was to develop learning environments with core topics that are not (yet) part of their actual studies, as computer science is not a topic that is taught in elementary school. Therefore, an extensive introduction into the field of computer science has been necessary.

A second demand that had to be kept in mind during the development has been a focus on interaction. The tasks should be designed in a way that -at least partially-fosters interactional processes between the participants as we focus on learning through collective argumentation and participation [START_REF] Krummheuer | Representation of the notion "learning-as-participation" in everyday situations of mathematics classes[END_REF]. The first learning environment focused on the topic of logic. The primary task always aims to get a first impression of what the participants already know (or think to know) about the concept. This very open question (e.g. What does logic mean? or When does a person have to think logically?) provides a wide variety of possible answers without any pressure for right or wrong.

The supervising students are always advised to let the children speak freely as much as possible, unless a lively discussion does not occur, general questions or guidance is to be avoided. Although the content is important, the actual focus for this pilot project is not to survey content learning, but rather the learning that occurs between individuals whilst discussing and arguing about the specific topic. Learning in primary school is initially dialogical learning. That is, learning is seen as a dialogical process in contrast to learning as a monological process, which would be rooted in the individual (cf. [START_REF] Miller | Kollektive Lernprozesse[END_REF]. The learning of mathematics can be seen as an increasing autonomous participation in collective argumentations that are produced and nurtured collectively by the group itself [START_REF] Krummheuer | Lernen mit" Format[END_REF][START_REF] Voigt | Thematic patterns of interaction and sociomathematical norms[END_REF]. This idea of learning can be transferred to the learning of basic competences in computer science. To build upon this concept, all ways of communication and interaction between the children has to be supported as much as possible. To ensure an efficient way to videotape the children working on the learning environments, we designed the environments to be worked on in pairs or groups of three, sometimes with a closing task, that included a larger group discussion in groups of up to six children. Working in smaller groups for us provides the advantage, that single children are not able to extract themselves from group tasks or discussions but rather encourage them to participate. Every learning environment is designed to cover three timeslots of 90 minutes. Here, two sessions of 90 minutes are planned for the actual content-related activities and one session of 90 minutes reserved for documentation and evaluation of learning processes though tools like learning diaries, wikis and storyboards. Learning diaries can e.g. be used to recapitulate the learning process, correct misconceptions and enables the child to visualize what learning progress it has made. We chose the storyboard as an adequate tool for the programming/robotics learning environment. This environment will be realized with LEGO Mindstorms EV3 Educations sets. The main goal after developing and building the robot itself will be to program its specific actions. These actions should be planned beforehand because the children can choose from a variety of actions and sensors with endless combinations. For this task, a storyboard is the ideal candidate as it enables the children to structure their thoughts and plan the movements and sequences that the robot has to fulfil. It also makes trouble-shooting rather easy as the children can always compare their plan to the robot's actual movements and actions.

Analysis

The analysis of the results of the work on the learning environments is based on methods of interpretive classroom research, such as interaction analysis. As we place much importance onto interaction, cooperation and discussions this method seems most appropriate to us. It helps to find more suitable tasks that help children to collaborate in a productive way. The negotiation of meaning is a key element that has to be focused (cf. [START_REF] Krummheuer | Lernen mit" Format[END_REF]. How does the negotiation of meaning in collective processes of argumentation occur and how can it be supported thought the task itself? According to our perception, learning does not occur inside the individual but during the interaction between individuals, whilst discussing, talking, arguing and also justifying the own answers and the answers of others (cf. [START_REF] Krummheuer | Lernen mit" Format[END_REF]. The underlying concept is founded on the ideas of the symbolic interactionism (cf. [START_REF] Blumer | Symbolic interactionism[END_REF] that will allow us to examine learning as the increasing autonomous participation in collective argumentation in computer science discourse. Possibly this could lead to the definition of participation profiles (cf. [START_REF] Brandt | Kinder als Lernende -Partizipationsspielräume und -profile im Klassenzimme: Studie zur Partizipation im Klassenzimmer Frankfurt[END_REF] for different students, specifically tailored to the computer science classroom. This could not only benefit primary education, but also computer science classrooms in secondary schools. If a specialized and standardized computer science education in primary school was mandatory, secondary schools would have a contact point and a profound basis on which they could build and focus their curriculum.

First impressions

The first impressions of the pilot project are consistently positive. One major question during the development process was whether children could be unchallenged or overwhelmed with the tasks as they work with topics that are not being taught in primary school and are therefore unfamiliar. The individual knowledge of the children concerning this specific topic could therefore be developed very differently. The children engaged immediately with the tasks and lively discussions occurred. In the very beginning, a certain insecurity was noticeable. This was expected as the children naturally try to give correct answers. This was not possible as most questions are designed in a way that multiple answers and also very individual answers are possible. After the first sessions, the children got more and more used to the types of tasks and felt more comfortable when joining the discussions.

Interestingly, mathematical discussions occurred very often, whether they were planned or not. One example where a rather simple question led to a lively discussion occurred during the learning environment: logic. The children had to decide whether a statement is right or wrong. First, S1 states that he is not convinced that this statement is true and he tries to find a shape that is pointy and round. He therefore draws a shape that has corners but is not just one straight line, similar to a part of a polygon. S2 then looks at the drawing and defines the shape as pointy because it has corners. S1 then again discusses the word round and seeks for a better word to describe his drawing and proposes the word roundly for something that is not a straight line but follows the form of a circle part although it has corners. S1 and S2 clearly have a dissent in what qualifies as being round and then shift this disagreement into a consensus through the introduction of a new vocabulary. It is rather interesting that the S1 and S2 discuss to find a consensus and the questions remains, what would happen if they could not find one. Here, the material itself proposes a research question: How do consensus and dissent influence cooperative learning. Is learning through collective argumentations also possible on the basis of a dissent or does a consensus have to be reached in order to complete/move on with the task itself? The structures in cooperative learning opportunities could be fundamentally different if the necessity of reaching an agreement would not be mandatory. This question will be focused during further analysis as the concept of computer science learning environments provides a perfect frame: It has clear and visible connections to mathematics but is fundamentally new and different, so that children can learn a new topic in which they do not have prior knowledge to build on.

Prospects

The learning environments with the topics cryptography, algorithms, programming/robotics are also completed by now. The videos and writings of the children are being extensively analysed using methods of interpretive classroom analysis. The analysis will enable us to look at the negotiation processes in more detail and determine, where the learning of individual computer science concepts and understandings occurs and how meaning is negotiated during the task itself also in regard to cooperative learning and the underlying structures. Through this, it will be possible to rework the learning environments to tailor them even more specifically to their purpose. The reworked learning environments will then be used in a more extensive main study to examine our claims and to strengthen the position of computer science as a key part of a profound and forward-thinking education that will not only benefit children's abilities in this specific subject but also strengthen their mathematical competences

Table 1 : Competences in Mathematics and CS: Numbers and Operations Mathematics Computer Science

 1

	Spatial orientation	Program robots, Define an area of
		movements,Plan with obstacles and
		predict motion sequences

Table 2 : Competences in Mathematics and CS: Space and Shape Mathematics Computer Science

 2

	Recognize and characterize	Structure and plan algorithms, Plan
	regularities	processes and translate them into a
		programming language, logic,
		Sorting algorithms

Table 3 : Competences in Mathematics and CS: Pattern and structures Mathematics Computer Science

 3

	Have the ability to imagine sizes,	Determine the step range of a robot,
	Have the ability to use sizes in	Determine run time
	specific situations

Table 4 : Competences in Mathematics and CS: Size and Measurement

 4

	Mathematics	Computer Science
	Understand the relationship	Algorithms use calculations, Loops
	between and the representation of	have to be counted, Sorting
	numbers, Understand and master	algorithms, Types of variables
	calculations	integer, float, double

Table 5 : Competences in Mathematics and CS: Data, Frequency and Probability

 5

	(KMK Bildungsstandards, 2005)

  One of the statements claimed: If something is round it is not pointy. Right or wrong? The children then engage into a discussion, where they have to decide what qualifies as being round:

	S1:	Like this and so on and on \ [points at the drawing in his folder]
	S2	Yes, but if something is round [draws a circle into the air with his pen] so this here
		[points at S1's drawing] that is not round\ that is pointy\
	S1:	Yes, it is roundly but fi it is round then you are right\
	S1:	If something is round it is not pointy\ [reads]
		That does not have to be true\
	S2:	(It is not\) [laughs].
	S1:	That does not have to be true maybe it is so to say a pointy circle\
	I:	How does a pointy circle work?\
	S1:	So . wait (4) [draws into his folder]
	S2:	[looks into S1's folder]

Proceedings of CERME10

Thematic Working Group 09Proceedings of CERME10


text:
Christina M Krause
email: christina.krause@uni-due.de

DeafMath: Exploring the influence of sign language on mathematical conceptualization

Keywords: Sign language, deaf learners, gestures, mathematical discourse and social interaction

Sign languages are performed in a modality other than spoken languages, using the entire body in a spatial-visual-somatic way. With reference to spoken language, performance of the language in terms of articulation, but also perception and interpretation, changes in the medium of sign language as a visual means of expression. Considering mathematical discourse and social interaction as an important factor in the learning of mathematics, this paper discusses theoretical approaches of a research program, currently underway, that aims at getting a better understanding of how the use of sign language may influence the learning of mathematics. From this a more profound basis shall be derived for developing didactical strategies responding to the special needs of deaf learners and understanding the role of bodily language in mathematical conceptualization.

Introduction

Research in the area of Deaf studies and Deaf education points at the special challenges deaf students face when learning mathematics. Their lack of basic mathematical skills-deaf children lack several years on average behind hearing peers [START_REF] Nunes | Teaching mathematics to deaf children[END_REF][START_REF] Traxler | Measuring up to performance standards in reading and mathematics: Achievement of selected deaf and hard-of-hearing students in the national norming of the ninth edition Stanford Achievement Test[END_REF]-is considered to be mainly caused by social and linguistic aspects. Deaf children do not 'pick up' informal knowledge [START_REF] Ginsberg | The development of children's mathematical thinking: Connection between research and practice[END_REF] about mathematical concepts in early childhood as easily as hearing children do, due to growing up in an environment that is primarily aligned to auditive social experience [START_REF] Nunes | Is hearing impairment a cause of difficulties in learning mathematics[END_REF]. For example, everyday phrases of "mathematical conversation" [START_REF] Gregory | Mathematics and deaf children[END_REF] just as 'It is five to twelve' or 'Turn right in three quarters of a mile' can provide a first contact to numbers that is not accessed 'en passant' by deaf children. Not necessarily growing up in a deaf community, they may also lack everyday interaction with peers that may initiate first instances of problem solving in playful situations, e.g. dividing a quantity in equal parts. Furthermore, deaf learners struggle with reading, understanding and processing written word problems [START_REF] Hyde | Deaf and hard of hearing students´ performance on arithmetic word problems[END_REF]. Their challenges are partly explained by a decreased short term memory in serial recall of linguistic material, by a problematic comprehension of certain language structures like conditionals, comparatives, inferentials and lengthy passages [START_REF] Rudner | Using standard tests with the hearing impaired: The problem of item bias[END_REF], and by the semantic understanding of the written language as a second language [START_REF] Barham | Mathematics and the deaf child[END_REF][START_REF] Traxler | Measuring up to performance standards in reading and mathematics: Achievement of selected deaf and hard-of-hearing students in the national norming of the ninth edition Stanford Achievement Test[END_REF].

Hence, and probably as no surprise, language is considered a main factor influencing the learning of mathematics for deaf learners. However, language has mostly been considered a problematic condition that impedes deaf students' learning rather than investigated as an integral part of the learning process itself. As a spatial-visual-somatic language, the sign language used by the Deaf provides access to mathematical ideas different than that of spoken language. But what exactly does this mean for the learning of mathematics? And what can we learn from looking at how learning under these special conditions takes place?

Thematic Working Group 09

Proceedings of CERME10

The approach presented takes into account the specificity of sign language to encounter the peculiar characteristics of mathematical discourse and social learning processes in the deaf classroom. Furthermore, I support the claim that the modality of language not only affects how mathematics is learned, but that it also influences how mathematical ideas become conceptualized by impacting the structure and process of thinking [START_REF] Healy | Hands that see, hands that speak: Investigating between sensory activity, forms of communicating and mathematical cognition[END_REF]. This contribution therefore outlines theoretical approaches and possible implications of a new research program that aims at developing a better understanding of how mathematics is learned using the medium of sign language.

Sign language(s) and gestures

Sign languages are visual languages that are formed by several components such as the configuration, movement and orientation of the hands and their location in space, body posture, facial expression and the viseme (or 'mouthing': the movement of the mouth). These aspects shape what is considered the utterance in sign language and are, just as spoken language, more or less conventionalized. These conventions distinguish the manual expression from the gestures defined in the style of McNeill. While he defines co-speech gestures as "idiosyncratic spontaneous movements of the hands and arms accompanying speech" (McNeill, 1992, p. 37), I adapt this definition for an understanding of co-sign gestures as being 'idiosyncratic spontaneous movements of the hands and arms' accompanying the signed discourse. Signers use non-conventionalized gestures in addition to the signs and both types of gestural expression can hardly be distinguished (see also [START_REF] Healy | Mathematics in the hands of deaf learners and blind learners: Visual-gestural-somatic means of doing and expressing mathematics[END_REF]. Being performed in the same visual-gestural modality, signs and gestures are deeply intertwined in their use and in their interpretation, probably even more intertwined than in the case of spoken language.1

Cognitive aspects of the influence of sign language on the learning of mathematics

Embodied cognition

Following the theory of embodied cognition, our (mathematical) thinking is deeply influenced by how we experience the world as physical beings [START_REF] Lakoff | Where mathematics comes from: How the embodied mind brings mathematics into being[END_REF]. How we act in and perceive the world structures our thinking and shapes to large extent our conceptual understanding: Human concepts and human language are not random or arbitrary; they are highly structured and limited, because of the limits and structure of the brain, the body, and the world. (Lakoff & Núñez, 2000, p. 1) A slightly more cautious claim is stated by Wilson and Foglia in the embodiment thesis:

Many features of cognition are embodied in that they are deeply dependent upon characteristics of the physical body of an agent, such that the agent's beyond-the-brain body plays a significant causal role, or a physically constitutive role, in that agent's cognitive processing. (Wilson & Foglia, 2011, paragraph 3) More precisely, Wilson and Foglia distinguish three roles the body can play in cognition: It can constrain cognition, distribute cognitive processing and regulate cognitive activity (Wilson & Foglia, 2016, paragraph 3). In sum, "such determinate forms of the Embodiment Thesis can ascribe the body either a significant causal role, or a physically constitutive role, in cognition" (Wilson & Foglia, 2016, paragraph 3).

However, the "body as constraint" is not to be understood with a merely negative connotation as one may get at first sight, taking into account two further implications provided by [START_REF] Wilson | Embodied cognition[END_REF]:

 Some forms of cognition will be easier, and will come more naturally, because of an agent's bodily characteristics; likewise, some kinds of cognition will be difficult or even impossible because of the body that a cognitive agent has.

 Cognitive variation is sometimes explained by an appeal to bodily variation. (paragraph 3)

This view on embodied cognition is coherent with the approach taken by Healy and colleagues who understand bodily organs as tools in the sense of Vygotsky, influencing structure and process of thinking [START_REF] Healy | Hands that see, hands that speak: Investigating between sensory activity, forms of communicating and mathematical cognition[END_REF]. As instrumental tools, the sensory organs can be substituted among each other, which "is expected to cause a profound restructuration of the intellect" (p. 299).

Such a substitution comes into play for deaf learners, where the lack of auditive perception becomes substituted by other sensory experiences. In the hearing classroom, information and ideas are shared to a large extent verbally while deaf students acquire information and interact by means of visual modes of expression, just as sign language. Following the theoretical approaches laid out, such a variation concerning the process of learning mathematics should alter cognitive structures and thinking processes, perhaps also leading to differences in conceptualization of mathematical ideas.

Features of sign language

Research in the field of Deaf Studies in fact indicates that deaf people 'think differently' [START_REF] Grote | Denken Gehörlose anders? Auswirkungen der gestisch-visuellen Gebärdensprache auf die Begriffsbildung[END_REF][START_REF] Grote | Modality relativity? The influence of sign language and spoken language on semantic categorization[END_REF]. Grote emphasizes that the modality of language-whether it is communicated in vocal language or in sign language-influences processes of conceptualization. She identifies two features of language modalities with such influences: Articulation and iconicity.

While information is strung together sequentially and linearly in vocal language, sign language offers the possibility to represent different aspects of the utterance simultaneously. This can compensate for the greater time required by spatial articulation in sign language over that of verbal articulation [START_REF] Bellugi | A comparison of sign language and spoken language[END_REF][START_REF] Grote | Modality relativity? The influence of sign language and spoken language on semantic categorization[END_REF]. However, sign language can represent only those concepts simultaneously that stand in a syntagmatic relationship, that is, concepts that consist of several aspects connected through linguistic contiguity. Signs that bundle these aspects by using a particular handshape to express additional information are sometimes called polycomponential signs [START_REF] Grote | Modality relativity? The influence of sign language and spoken language on semantic categorization[END_REF], classifier predicates, or depicting verbs [START_REF] Liddell | Grammar, gesture and meaning in American Sign Language[END_REF]. In contrast to this stand the representation of concepts from the same paradigm, e.g. concepts that are connected in hierarchy (see Fig. 1). These paradigmatic (or 'associative' [START_REF] Saussure | Course in general linguistics (R. Harris, Trans.)[END_REF]) relationships need to be articulated linearly, just as in verbal language [START_REF] Grote | Denken Gehörlose anders? Auswirkungen der gestisch-visuellen Gebärdensprache auf die Begriffsbildung[END_REF][START_REF] Grote | Modality relativity? The influence of sign language and spoken language on semantic categorization[END_REF]. [START_REF] Grote | Denken Gehörlose anders? Auswirkungen der gestisch-visuellen Gebärdensprache auf die Begriffsbildung[END_REF] claims that this may lead to a preference for communicating those ideas that stand in a syntagmatic relationship and gives empirical evidence that this preference may engender the establishment of a stronger link between these relations over paradigmatic ones.

Furthermore, gestures often show a certain resemblance with what they signify; they evoke an iconic relation to its referential object. This relationship, however, needs to be established since it is not self-evident. Related to the process of conceptualisation, Grote claims that assuming that epistemic processes are processes inherently mediated by signs, the similarity that forms the relationship between icon and referential object is constituted actively. This means that in the process of iconisation, there is a focus on specific features of the semantic concept which probably become stronger linked and get an exposed position in the semantic net. (Grote, 2010, p. 312, translated by the author)

When conducting verification tests, she found remarkably shorter reaction times for those pictures that showed the feature that was iconically reflected in the sign. This pointed to a stronger semantic link between this feature and the signed concept and provided evidence that "those features that are reflected in the iconic moment of sign language get a specific relevance for the whole semantic concept" (Grote, 2010, p. 316, translated by the author).

So what might this mean for the learning of mathematics?

Learning mathematics is not perceived as a purely cognitive phenomenon but can be understood as a social process in which individuals co-construct mathematical meaning and knowledge within the social interaction that is constituted by the use of signs. These signs can be of written, spoken, or gestural form or anything else that can be considered a semiotic sign, performed in any modality. In this sense-and taking into account the embodied approach outlined earlier-learning is understood "as a multimodal process" (Arzarello, 2006, p. 1), influenced by production and perception of signs within social interaction. The use of sign language plays part in both, production and perception.

Based on this, possible issues that can arise are the following:

 A preference of communicating syntagmatic relationships may lead to place special emphasis on these when carrying out social epistemic processes in social interaction and therefore, may lead to make syntagmatic relations conceived as being more important for the related mathematical concept.

 Knowledge about which relations are 'linked' linearly and which simultaneously can influence teaching methods. While in the learning of deaf students there needs to be emphasis on developing paradigmatic inner-mathematical relations, the use of co-speech gestures may support strengthening syntagmatic links also in the regular classroom.

Theoretical foundations for such an approach are provided by the results on gestural specification of the verbal utterance in processes of constructing mathematical knowledge in social interaction, as described in [START_REF] Krause | The mathematics in our hands: How gestures contribute to constructing mathematical knowledge[END_REF].

 Providing 'mathematical signs' as nonverbal terms to students, it needs to be noted that the iconicity of the sign may lead to an exposed position of the aspects that become visually reflected in it. Oftentimes, official and conventionalized 'mathematical signs' do not exist or are not known so that a 'suitable' mathematical sign may develop hand in hand with the knowledge during the learning process in the mathematical classroom (see also [START_REF] Fernandes | Algebraic expressions of deaf students: Connecting visuo-gestural and dynamic digital representations[END_REF][START_REF] Krause | Embodied geometry: Signs and gestures used in the deaf mathematics classroomthe case of symmetry[END_REF]). To support the conceptualization of mathematical ideas, it is therefore important to take a closer look at which aspects of a mathematical idea are reflected iconically in a mathematical sign, and how meaning develops in the respective signs in a process of iconization while the ideas become encountered. Within this process, the iconicity of the gesture may inform about the signer's current conceptionalization of the mathematical idea. This may be used for the purpose of assessment and fits the development of the 'associated gestures' found in hearing learners' social processes of constructing mathematical processes [START_REF] Krause | The mathematics in our hands: How gestures contribute to constructing mathematical knowledge[END_REF].

 Many mathematical concepts are shaped metaphorically so that the mathematical concepts are understood through something familiar or more illustrative [START_REF] Lakoff | Where mathematics comes from: How the embodied mind brings mathematics into being[END_REF]. These metaphors cannot be represented iconically in a direct way, the developing sign/gesture rather refers to an 'underlying' meaning (see again [START_REF] Fernandes | Algebraic expressions of deaf students: Connecting visuo-gestural and dynamic digital representations[END_REF]. Gestures developed by deaf students while constructing mathematical knowledge in social interaction may therefore indicate possible approaches to these ideas and concepts. Knowledge about these approaches can also help in cases of learning mathematics in a second language since linguistic approaches to metaphors may not be accessible.

The research program "DeafMath"

These considerations motivate my research program in which I investigate the influence of sign language on the conceptualization of mathematical ideas, focusing on two main aims:

 Contributing to the development and further elaboration of a theory on the role of the body in the conceptualization of mathematical ideas,  Providing theoretical foundations for developing didactical methods and strategies that involve the body in processes of teaching and learning.

Furthermore, another goal lies in the development and evaluation of methodological approaches that take into account the specificity of the research setting when working with deaf children. The crucially different characteristics of sign language as a visual-gestural language, as well as the students' difficulties with written language, demand an adaption of methods for collecting, preparing, and analysing data. This becomes especially important with respect to qualitative studies that follow interpretative and reconstructive methods since the holistic representation in sign language cannot be captured merely in written form that can only reflect linear and segmented language. A (more) suitable methodological approach might place a greater emphasis on the coordination of written transcripts, pictures, and videos for means of analysis, but also for the documentation of the results.

Potential long-term goals with respect to implications on teaching methods and strategies concern the following aspects:

 The identification of challenges that are specific to deaf students and countering them in their core: Is one challenge grounded in their understanding of (some) mathematical concepts as deviating due to the deviating modality of their language?

 Understanding the inclusion of deaf learners and their way of communicating as actual surplus in the inclusive classroom. Results gained from these studies can point out how an actual inclusion of hearing-impaired students can enrich the entire classroom.

 Using representational gestures in a goal-directed way as didactic means. In [START_REF] Krause | The mathematics in our hands: How gestures contribute to constructing mathematical knowledge[END_REF] I describe how the use of representational gestures can influence the collective formation of mathematical concepts in a beneficial way by its various representational functions. Results derived from the here described study may give insights in how these representational gestures may look like.

This program therefore considers 'barriers and chances': While the different kind of communication may lead to specific challenges when learning mathematics, taking into account these differences entailed by the spatial-visual-somatic and embodied medium of sign language might help to "become better able to respond to their particular needs, but also build more robust understandings of the relationships between experience and cognition more generally" (Healy, 2015, p. 289).

Fig. 1 :

 1 Fig. 1: Example for paradigmatic and syntagmatic relationships (following Grote 2013, p. 313)

This also becomes a methodological issue. It is almost impossible to translate from sign language to written language, even if using lexemes for the notation. Gestures contribute naturally to the interpretation of the utterance such that the analytical distinction between which aspects are signed and which are gestured cannot be made as clear as analytical distinctions between the spoken and the gestured. Neither can be considered separately.Thematic Working Group 09Proceedings of CERME10

Proceedings of CERME10


text:
Faten Khalloufi-Mouha
email: fkhalloufi@yahoo.fr

Constructing mathematical meaning of the cosine function using covariation between variables in a modeling situation in Cabri

Keywords: Verbal signs, semiotic mediation, covariation, collective discussion, Cabri

Constructing mathematical meaning of the cosine function using covariation between variables in a modeling situation in Cabri

Faten Khalloufi-Mouha

Introduction and research problem

The notion of trigonometric function is related to three mathematical domains: trigonometry, algebra and function. Understanding the idea of a trigonometric function requires students to realise the relationship between the length of an arc and the measure of the angle that is subtended by the arc, the relationship between the real line and the unit circle, the covariation between a point moving on the unit circle and its projection on one of the axes, and the idea of the graph of a function [START_REF] Khalloufi-Mouha | Etude du processus de construction du signifié de fonction trigonométrique chez des élèves de 2ème année section scientifique[END_REF][START_REF] Khalloufi-Mouha | Constructing mathematical meaning of a trigonometric function through the use of an artifact[END_REF].

From a historical perspective, the idea of the functional relationship between the real line and the unit circle is related to the notion of motion as a variation in the space and time. [START_REF] Falcade | Approaching functions: Cabri tools as instruments of semiotic mediation[END_REF] So, this idea could be understood from a dynamic perspective, referring to the wrapping of a line around the unit circle and to the variation of a moving point. Many studies (e.g., [START_REF] Laborde | Core geometrical knowledge for using the modelling power of Geometry with Cabri-geometry[END_REF][START_REF] Laborde | Grounding the notion of function and graph in DGS[END_REF] claim that the key to understand the dynamic aspect of function is the notion of trajectory. According to [START_REF] Laborde | Core geometrical knowledge for using the modelling power of Geometry with Cabri-geometry[END_REF] the graph of a function could be considered as the trajectory of a moving point, representing the dependent variable according to the variation of a variable point on the axis of abscissas, so, the covariation of these two points becomes a relationship between two variations depending on time. Adopting a teaching approach based on the idea of covariation in the semantic field of space and time, [START_REF] Laborde | Grounding the notion of function and graph in DGS[END_REF] and [START_REF] Falcade | Théorie des Situations, médiation sémiotique et discussions collectives, dans des séquences d'enseignement avec Cabri-géomètre pour la construction des notions de fonction et graphe de fonction[END_REF] assume that the teaching and learning of the notion of function should start in a dynamic environment. They claim that the use of Cabri allows access to the ideas of variation, functional relationship and covariation through the movement of a point on the screen. According to the Vygotskian (1978) perspective of semiotic mediation, [START_REF] Laborde | Grounding the notion of function and graph in DGS[END_REF] and Bartolini Bussi and [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF] argue that some of Cabri's features can be viewed as potential tools of semiotic mediation carrying mathematical meanings. Cabri's tools can be thought as external signs referring to a specific mathematical meaning, and may become tools of semiotic mediation [START_REF] Laborde | Grounding the notion of function and graph in DGS[END_REF].

Following these studies, we propose a teaching experiment integrating the technological artifact Cabri, aiming to introduce the trigonometric function cosine as covariation.

Theoretical framework

According to [START_REF] Vygotsky | Mind in society. The development of higher psychological processes[END_REF], human intelligence is defined by the ability to use various types of tools with signs playing the role of mediators. Vygotsky distinguishes between tools (technical tools) and signs (psychological tools). Tools are externally oriented and aimed at controlling the process of nature. Signs are internally oriented and aimed at mastering the individual's behavior and cognitive processes. Vygotsky claims that, through the process of internalization, a tool may be transformed in a sign, and then may function as a tool of semiotic mediation. Vygotskian theory supposes that the development of behavior and cognitive processes is the product of activities practiced in the social institutions of the culture in which the individual grows up. For Vygotsky, language is the most important semiotic mediator allowing the passage from the interpsychological level to the intrapsychological level. The theory of semiotic mediation (TSM), developed by Bartolini Bussi and Mariotti (see, for instance, Bartolini Bussi & Mariotti, 2008), is situated in the Vygotskian perspective. Two key notions support this theory, the zone of proximal development and the internalization. Vygotsky introduces the concept of the zone of proximal development as "the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers" (Vygotsky, 1978, p.86). This concept models the learning process through social interaction. In fact, the collaboration between one individual, whose cognitive attitude has the potential to facilitate change, and another individual (or group) who intentionally cooperate to accomplish a task or to pursue a common aim may generate a possible development [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF]. The transposition of this concept in the school context is possible thanks to the intrinsic asymmetry of the relationship between students and teacher in terms of knowledge. Therefore, the teacher's actions should be within this zone for achieving a didactic objective through the use of the signs and tools as semiotic mediators. In this zone of proximal development, cognitive development is modelled by the process of internalisation.

According to the TSM, the social use of an artifact in the accomplishment of a task leads to the emergence of signs expressing students' personal meanings, i.e. "the conscious sense created by each individual to express the direction of his or her action" (Bartolini Bussi, 1998, p.69). The evolution from personal meanings towards mathematical meaning, i.e, what describes the properties of the concepts under scrutiny inside a system of theoretical knowledge (Bartolini Bussi, 1998, p. 81), is an educational aim that can be realized promoting the evolution of signs, expressing the relationship between artifact and tasks, into signs expressing the relationship between artifact and knowledge [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF]. The teacher can guide this evolution through collective discussions.

Rationale for the teaching experiment

Our teaching experiment aims at constructing the meaning of the cosine function and of its graph using the notion of covariation [START_REF] Khalloufi-Mouha | Constructing mathematical meaning of a trigonometric function through the use of an artifact[END_REF] The experiment has four parts. The first two parts use the situation A rope on a wheel (Genevès, Laborde & Soury-Lavegne, 2005), which is a modelling situation in Cabri: a thread that can be wound around a wheel through the Dragging tool. The experiment involves description and prediction tasks intended to provide students with opportunities to realise that in the unit circle the length of an arc is equal to the measure (in radian) of the angle that it subtends. In addition, wrapping the thread on to the wheel aims to make students construct, for a given real x, the point M on the unit circle with the arc 𝐼𝑀 ̂= 𝑥. Thus, they could realise the functional relationship between the real line and the unit circle. Mathematically, this relation is interpreted by the existence of a surjective group homomorphism between the set of real numbers and the unit circle. Throughout these activities the role of the teacher is crucial. He has to establish strategies fostering the development of meanings of functional relationships and variation. The third and fourth parts of the experiment aim to analyse how the Dragging tool, the Measurement tool, and the Trace tool function as semiotic mediators for the ideas of variation, the functional relationship and covariation. The cosine function is introduced as a relationship between two variations depending on time, and its graph as the trajectory of a moving point.

The teaching experiment involves different typology of activities aimed to develop different components of the complex semiotic process: working in pairs and collective discussions generally initiated and guided by the teacher. When working in pairs, students use the artifact to accomplish the given tasks. This type of activities promotes social exchange through the use of verbal signs (both oral and written), gestures, drawing... Consequently, this provides students with opportunities to construct personal meanings linked to the mathematical target concepts. Students' personal meanings may evolve to the shared mathematical meanings through collective discussions. These discussions involve all students and allow the confrontation between their personal meanings and the mathematical meanings. These collective discussions may reach the status of mathematical discussions, in the sense of Bartolini [START_REF] Bartolini Bussi | Mathematical discussion and perspective drawing in primary school[END_REF], and can also involve phases of institutionalization [START_REF] Brousseau | Théorie des situations didactiques[END_REF] and the introduction of new formal notations. The analysis of the verbal signs used by the students and the manner in which the teacher exploits them can highlight the evolution process from the students' meanings towards mathematical meanings. According to the TSM, the status of the signs belonging to the different categories vary in the evolution process, they can be used as an index of the move from personal sense to mathematical meaning. [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF].

Methodology

The sample of this study consisted of 16 students from a class of 2 nd year (16-17 years) in a high school in Bizerte, Tunisia. The students were familiar with analytic trigonometry and they regularly used computer software in class. The teaching session was carried out in the computer laboratory. Students were grouped in pairs and asked to produce a shared written answer on a worksheet. The discussions within pairs and the collective discussions have been audio-recorded.

The analysis is based on several kinds of data that were collected: protocols produced from the student working in pairs (audio-recording, texts, and drawings), audio-recordings of collective discussions and the teacher's and observer's notes.

The impact of the use of Cabri on the construction of mathematical knowledge was analysed by identifying the development of the students' and the teacher's verbal signs related to the target mathematical concepts. We distinguish the two types of signs identified by [START_REF] Falcade | Théorie des Situations, médiation sémiotique et discussions collectives, dans des séquences d'enseignement avec Cabri-géomètre pour la construction des notions de fonction et graphe de fonction[END_REF]: simple signs and complex signs. Simple signs are "easily recognizable by representations of almost atomic type (words or specific formulations)" (p.202). For example, the word "circle" is a simple sign. Amongst the simple signs [START_REF] Falcade | Théorie des Situations, médiation sémiotique et discussions collectives, dans des séquences d'enseignement avec Cabri-géomètre pour la construction des notions de fonction et graphe de fonction[END_REF] distinguishes three types, i.e. artifact-signs (arising directly from the use of the artifact, their meanings are personal and commonly implicit. The artifact-signs are strictly related to the use of the artifact to accomplish the task), mathematical-signs (referring to the culture of mathematics and constitute the goal of the semiotic mediation process.), and pivot-signs which play a pivotal role between the semantic and mathematical fields. "The characteristic of these signs is their shared polysemy, meaning that, in a classroom community, they may refer both to the activity with the artifact; in particular they may refer to specific instrumented actions, but also to natural language, and to the mathematical domain." [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF].

The complex signs refer to relationships between families of relatively 'simpler' signs. For example, the sentence "the set of points that are equidistant from a given point" is a complex sign that refers to the mathematical-sign circle. Complex signs are subdivided into four categories: characterisations, definitions, interpretations and instantiations. Characterizations tend to highlight some characteristics that could be interpreted in mathematical terms. Nevertheless, characterizations are not real definitions because in the mind of the speaker the statement is not precise. A definition for an object provides a "boundary in words" which was until then unknown or little known. They are not definitions in the mathematical sense, but can be considered as being part of the process towards a mathematical definition. The interpretations concern explicit links between two families of signs which belong to two different semantic fields. Instantiations are signs which concern the establishment of an interpretative link between artifact-signs and mathematical-signs. [START_REF] Falcade | Théorie des Situations, médiation sémiotique et discussions collectives, dans des séquences d'enseignement avec Cabri-géomètre pour la construction des notions de fonction et graphe de fonction[END_REF] elaborates that "Instantiations are of the same nature as the interpretations. However, the latter concern universal, while instantiations refer directly to the specific activity in Cabri" (p.210).

In our analysis of the development of students' and teacher's verbal signs, all the relevant mathematical notions and all the different signs used were identified. The classification of Falcade allows us to analyse the development of students' personal meanings. Every evolution from an artifact-sign to a pivot-sign will be interpreted as a step towards the construction of personal meanings. The use of complex signs of the type characterization or instantiation will be interpreted as an attachment to the artifact. The use of the complex sign interpretation will be interpreted as a step in the process of semiotic mediation. Finally, the complex sign definition will be interpreted as a step in the process of internalisation of a mathematical definition of the object.

Findings and discussions

The first two parts of the teaching experiment focus on the situation "a rope on the wheel".

Figure 1: The situation A rope on the wheel

From the first task, when working in pairs, students used a large number of artifact-signs related to the ideas of motion and numerical domain, such as "to unwind", "to turn", "to move", "to increase", and "to decrease". For some students, we identified initial pivot-signs related to variation such as "to change" and "to vary", and related to the meaning of functional relationship such as "depends on" and "being a function of". The pair discussions indicate that the use of the artifact facilitated students' use of complex signs of the instantiation type: "If the length of the arc changes, the point N will change", and of the interpretation type: "Then the length of the arc varies as a function of the angle". These complex signs show the construction of personal meanings of the relationship between the length of an arc, the measure of the angle and the radius of the wheel. The collective discussions initiated by the teacher led to the use of these signs in the process of constructing mathematical meanings. The role of the teacher was crucial in the collective discussion. Taking into account individual contributions, the teacher engaged students through juxtaposing their personal meanings, and encouraged them to disregard the artifact and then to focus on the mathematical concepts.

The second part of the experiment aimed at exploring the metaphor of the winding of the real line around the unit circle. During the working in pairs, we observed artifact-signs such as "put the point", "move the point", and pivot-signs as "length of the arc". This characterizes a first step related to the recognition of the numerical and geometric variation. Students used also mathematical-signs "For every x, we can construct a point" and interpretations of the functional relationship between the real line and the unit circle. This can be interpreted as a second step: the identification of the covariational relation at a perceptual level. During the collective discussion, the teacher used the rope as a semiotic mediator to introduce the idea of the functional relationship between the set of real numbers and the points of the unit circle. The collective discussion highlights the use of mathematical-signs such as "for every real x in ..", "symmetry", "absolute value" and "a function of". In addition, we also observe an interpretation of the idea of the functional relationship: "For every positive real x we can construct a point on the unit circle". It seems that students found themselves in a familiar mathematical environment and had no difficulties in generalising the use of the artifact. They were engaged spontaneously in the intended process of mathematical meaning construction.

A definition of the cosine function

The third part of the teaching experiment focused on the ways the Cabri-tools Dragging, Measurement and Trace function as semiotic mediators for the development of the definition of the cosine function as a covariation, (i.e. the relationship between two variations depending on time), and the construction of its graph as a trajectory of a moving point.

Students were asked to construct a point N on the x-axis (1,0) with abscissa x. Using the Measurement tool, they should then construct the point M of the unit circle such that 𝐼𝑀 ̂= 𝑥. Finally they should construct the point K on the x-axis(1,0) with the same abscissa as M, and the point H on the y-axis (0,1) with the same ordinate as M. Using Dragging tool, students were asked (a) to describe the relationship between the points N, M, H and K when N is moving, (b) to determine the values taken by the abscissa of M when the point M is moving on the unit circle and (c) to identify the relationship between x and the abscissa of M. The evolution of the verbal signs elaborated and used by the students and the teacher attest the existence of four steps towards the definition of the trigonometric function cosine.

st Step: Recognition of numerical and geometric variation

The use of the artifact to accomplish the task promoted the emergence of verbal signs related to the idea of motion. The first step is characterised by an important use of atifact-signs stressing the attachment to the activity with the artifact. In fact, students used verbs of action as "to move", "to vary", "to turn", "to change".... and used expressions as "we draw the point M on the unit circle and we move it" or "we can choose different values for x..."

nd

Step: Identification and recognition of covariation at a contextual level This step is characterized by the emergence of complex signs "interpretation" linking two variations. In these signs the reference to space and time was eliminated and we noticed the use of expressions related to the indirect variation and to the simultaneity of variations such as "When X varies then Y varies" or "If X varies then Y varies" 3 rd Step: Interpretation of covariation as a functional relationship. This step is characterized by the use of pivot-signs "depend on", "a function of" and "relation between" to replace the expressions "When X varies then Y varies". Students identify the functional relationship as a relation between two variations. This was interpreted as an evolution of students' personal meanings related to the notion of function.

In the collective discussion the teacher used the artifact to promote the emergence of the students' simple and complex signs related to the ideas of functional relationship and covariation "T : Well. Now what is going to change if you move N?" The artifact is used as a semiotic mediator supporting the transition of signs expressing the artifact-tasks relationship into signs expressing the artifactknowledge relationship. As a result of the guidance of the teacher, students expressed mathematicalsigns such as "the function which associates M to N", and others related to the idea of covariation.

th Step: The mathematical definition of the notion of cosine function

When exploring the relationship between x and the abscissa of M through the use of Dragging, we observed students' interpretation of the movement of point K as a variation of the abscissa of M and the description of the functional relationship between M and its abscissa.

The use of mathematical-signs related to the meaning of the cosine function illustrates the development of personal meanings related to the mathematical meaning of the cosine function as covariation. The collective discussions, guided by the teacher, allowed for the construction of the accepted mathematical meaning, i.e. the mathematical definition of the cosine function. [START_REF] Khalloufi-Mouha | Etude de l'évolution des pratiques d'un enseignant lors d'une séquence d'enseignement intégrant un artifact technologique[END_REF] The main goal of the teacher's actions in the collective discussion is managing the students' discourse, in order to support them to move from the artifact and place them in a mathematical environment which allows the recognition of the cosine function. The discussion shows a large commitment of students to the process of meaning construction. By using many mathematicalsigns, they succeeded in linking their mathematical and physical knowledge. In this case, we can say that the artifact was a powerful resource for the construction of the mathematical meaning based on the activity with the artifact.

Conclusion

The epistemological importance of introducing the notion of trigonometric function as covariation to make the link between trigonometry and trigonometric functions led us to design and experiment a teaching sequence integrating the artifact Cabri. The analysis of the process of constructing the mathematical meaning of the cosine function, through the analysis of the verbal signs, allowed us to identify that this process begin by a first step related to the recognition of the numerical and geometric variation. This step is characterized by an important use of the artifact-signs and an attachment to the task with the artifact. The second step is the identification of the covariational relation at a perceptual level. The third step marks a mathematical interpretation of the activity with the artifact following the collective discussions orchestred by the teacher. The process is carried out following the interventions of the teacher, by the mathematical definition of the notion of cosine function. The teacher's role is very important in the process of constructing mathematical meaning among students using a technological artifact. The teacher has to use the semiotic potentialities of Cabri and orchestrate the discussions in order to guide this process towards the construction of mathematical meaning. For this reason, further investigations into the role played by the teacher are required for a better description of the process of construction of mathematical meaning among students using a technological artifact.

Figure 2 :

 2 Figure 2: The figure related to the third part of the teaching experiment.

Proceedings of CERME10