Núria Planas
email: nuria.planas@uab.cat

Marie Therese Farrugia
email: marie.t.farrugia@um.edu.mt

Jenni Ingram
email: jenni.ingram@education.ox.ac.uk

Marcus Schütte
email: marcus.schuette@ifs.uni-hannover.de

Introduction to TWG09: Transforming language-sensitive mathematics education research into papers and posters

Keywords: Research communication, conference formats, written and oral languages, languagesensitive mathematics education research, representational challenges

In pursuing the writing and presenting of 'autonomous' conference texts -i.e., conference texts functioning in terms of meaning making on their own-, all authors deal with issues of reduction, transformation and representation of the concepts and contexts of initiation and development of their research. In this introduction, we point to challenges tied to the writing and presenting of language-sensitive mathematics education research for communication in conference formats. We discuss some ideas for the improvement of current guidelines and standardised decisions relative to processes and texts produced within the CERME culture. Drawing on experiences provided through our roles as co-leaders of the 'Mathematics and Language' thematic working group (TWG09), along with the insights gained from the TWG09 set of papers and posters on the occasion of CERME11, each of us brings a focus to the discussion of challenges and changes that might be feasible and worthwhile.

Introduction

In conference papers, but also in journal articles, we often read sentences like "because of space restrictions, we do not show the totality of the interview transcript", or "the lesson data transcribed has been translated into English", "the lesson data translated into English have been transcribed". It is also common to find written research without explanation of, or reference to, the original languages involved in data collection and/or the ways of addressing the transformation and reduction of data into certain written formats. These types of omissions and their implications at many levels deserve discussion in any domain of study. Such discussion is even more relevant in mathematics education research that is largely language-sensitive, regardless of the analytical traditions and theoretical standpoints taken. Indeed, throughout meetings of the 'Mathematics and Language' Thematic Working Group of ERME (TWG09), including the most recent meeting at CERME11 in Utrecht, many participants have expressed interest in challenging and improving some of the tacit typical practices when dealing with lesson and interview oral data in papers and posters.

In the following sections, we identify four major challenges around fostering a more languageresponsive communication of research work, and propose specific changes and directions that could help in this respect. The two guiding questions that we pose to ourselves are as follows:

-What are some of the challenges of communicating language-sensitive mathematics education research in written and oral conference formats, particularly the CERME format? -What changes in both the tacit positions of researchers and the written/oral products could improve the communication and understanding of the research reported?

Drawing on the experiences provided by our roles as co-leaders of TWG09, along with the insights gained from the TWG09 set of 28 papers and 2 posters on the occasion of CERME11, each of us brings a different focus to the discussion of challenges and changes that might be feasible and worthwhile. We hope that the sections below will give a sense of how research in the domain of mathematics education and language can inform important reflections for the whole field. Although there is increased understanding of the relevance of language in the various domains of mathematics education research, the point at which language issues are considered in written and oral conference formats is inconsistently reported and sometimes treated as a minor question. In our discussion below, each challenge is based on and arises in a number of papers and posters presented at CERME 11, however, each paper and each poster is only used once to illustrate the points being made.

The challenge of original language information

The papers of Azrou, of Mizzi, of Prediger, Uribe and Kuzu, and of Salekhova and Tuktamishov are examples of work conducted in multilingual classroom settings addressing issues of mathematics education and language diversity. Mizzi provides two-column transcripts with the bilingual lesson data in the source languages, Maltese and English, in one column and the target English language in the other, hence doubling the length of the transcript in terms of space taken in the paper. When confronted with text formats of eight pages, doubling transcripts may require about two-thirds of the totality of the report for data presentation. In the papers of Azrou, of Prediger, Uribe and Kuzu, and of Salekhova and Tuktamishov, transcripts of the Arabic-French, Turkish-German and Tartan-Russian source versions of bilingual talk in the respective lessons are not shown, with all the written pieces of data translated into English, its alphabet and writing direction. Similarly, Ranges and Eikset refer to Norwegian students' home language, while presenting transcripts in English, and Rønning and Strømskag present data of conversations in a planning meeting with teachers of Norwegian schools in English. The use of Norwegian is limited to translating technical vocabulary and occasional incorrect terminology. In their presentations, some of these authors claimed practical reasons for the option chosen and particularly mentioned their willingness to share the original data with those participants who had further interest in their works.

Looking back at the whole set of papers and posters across ERME thematic working groups compiled in the CERME10 Proceedings (Dooley & Gueudet, 2017) and given the space restrictions, it is not surprising that the presence of original languages tends to be avoided with the resulting monolingual bias in the perception, representation and communication of the contexts of research. Any option in this respect is problematic. For those papers that incorporate information of original language with translated language text, accomplishment of space restrictions is not trivial. On the other hand, for all papers that have languages other than English in their contexts of research, translation between languages is not trivial either. At CERME11, in his paper, Sträßer refers to the many challenges of translating data and ideas from a source language to a target language (see also Geiger & Sträßer, 2015). Overall, we find several examples of papers dealing with the decision of how to communicate in English what is said in the source language. In Chesnais' paper, the discussion refers to French grammatical features of the students' interactions, while the presentation of interaction transcripts and of their interpretation is done in English. This can also be found in Erath's paper, where the focus is on lexical and syntactical means produced in German but are presented and written in English. At the most recent meeting of TWG09, only the papers of Ingram and Andrews and of Postenilcu report work developed in a situation in which the source language of data is English exclusively.

Despite the difference of options being used, over these years the discussions in TWG09 clearly reveal that groups of authors, and in general all participants, consider data in the original languages as relevant evidence, not a technical distraction or tangential to the work we do. Reflections in this respect can be found in Planas, Chronaki, [START_REF] Planas | Challenges and research priorities in the context of mathematics and language[END_REF], [START_REF] Planas | Introduction to the papers of TWG09: Mathematics and language[END_REF], [START_REF] Planas | Mathematics and language. Lessons and directions from two decades of research[END_REF], and [START_REF] Rønning | WG9-Language and mathematics: Its origin and development[END_REF], among other CERME presentations and publications. The belief that untranslated source data is worthwhile, and that access to it is key, are strongly shared and rooted in our working group. We are very aware that untranslated source material retains a great deal of linguistic information that is of interest and can disappear in translation, with important implications for data analysis, interpretation and results. Importantly, the source languages provide vital information for gaining insight and understanding in multilingual contexts of mathematics education research, as well as for developing critical awareness of the monolingual emphasis under which most research products are shaped in our field.

Furthermore, the body of CERME Proceedings brings up paradoxical situations such as having a paper with the source data in French that is only read in its English translation by researchers who are French speakers or who are proficient in this source language and could therefore have a more comprehensive and direct understanding of the research reported in its original language. For each paper or poster, a number of researchers have the expertise and are able to review material in the original languages. With that in mind, ERME could possibly consider some changes oriented to facilitate access to the source languages without the necessity for a dramatic reduction of the space available for discussion in papers. In the age of technological developments and virtual environments, new arrangements could be made to include a multilingual site for registered access to repositories of source data matched with printable texts in the conference proceedings. Such arrangements would be similar to the practical experiences of what some journals do in the electronic version. This way, after consultation, authors could choose the option of linking their papers to the protected site containing information regarding the materials and texts that can be made available in the data's original languages. By using this option, authors could improve the communication of the research reported in the conference format.

The challenge of choosing what data is relevant for the paper

The current practice of CERME papers is to offer authors who are transcribers the choice of two transcript styles. Either they may use Transcript or Numbered Transcript if the author needs to refer to specific lines in the text. However, the template does not number lines, it numbers turns. In language and interactional research within mathematics education, there is often a need to be more precise. For example, it is frequently interesting to consider interactions where two or more people are speaking at once, or when their turns overlap, particularly when considering classroom interactions where there are multiple speakers and the data is often messy, such as in the work by Friesen, Schütte and Jung. Gestures, gazes and movements, such as pointing, that occur in social interaction are also often highly relevant to the analysis and these often occur concurrently with the spoken discourse. In these situations, line numbers rather than turn numbers are more appropriate, as well as the potential to use specifically designed transcription systems. Conforming to the transcription styles often results in a reduction of the data that members of TWG9 can present, and in some cases results in the loss of that data in the written presentation, as authors need to rely on their short presentation to share certain features of their data. More flexibility in the transcription formats is necessary if we are to offer closer representations of the language data that we are researching.

Greater flexibility in the transcription formats also implies some reference in the template for authors who choose to emphasize the multimodal nature of the data they work with, and who particularly do not produce verbal representations of gestures, body movements and uses of physical space and material objects, but opt for multimodal transcription systems. With the increasing use of video recording in our research field and the development of newer visual methods and analytical frameworks such as digital ethnography, the question of inserting photos and images in the transcript is very timely. Much of the research in TWG09 combines gestural, visual and verbal data, and hence goes beyond the purely verbal mode. This is the case of the papers by Fyhn and Hansen on how meaning of pattern changes depending upon the objects with which students are familiar and interact with, and by Götze on meaning-related language for understanding multiplication where the language is tightly associated with visual representations. The idea of communication 'mode-switching' [START_REF] Sindoni | Through the looking glass: A social semiotic and linguistic perspective on the study of video chats[END_REF], which relates to an important feature of multimodal data, is also present in the paper by Peters regarding learners' reactions and responses to auditory material from radio podcasts.

Some theoretical traditions (e.g., conversation analysis, corpus linguistics) have a specific transcription system of annotation and coding associated with them where not only line numbers are required. There is a specific font and way of spacing transcripts to illustrate features such as pausing, overlapping speech and concurrent gestures (see the topic of transcription decisions and related representational differences and analytical implications in [START_REF] Ingram | Research methods for classroom discourse[END_REF]. These systems often come with complex notation to indicate not only what was said, but also how it is said and this notation needs to be explained in the paper in order for the reader to make sense of the transcripts, which is another challenge to face in meeting the length requirements. Many of the papers in TWG09 are focusing on how students make sense of the mathematics they are engaging with, and this involves a combination of spoken interactions, written materials, and ways of interacting with objects. The current traditions around the written presentation of papers limit how this data can be shared as well as how much data can be shared. As suggested above, the repository of source data could enable the sharing of video or audio data that is often used during the presentations. This might also be a solution to the issue around the need to share transcription conventions, as well as offering an opportunity to share longer (multimodal) transcripts or additional transcripts for the interested reader.

The challenge of comprehensibility of theory development

The last two challenges differ in nature to the previous ones since they focus on how to make theory and conceptual development more accessible to the readers and the audience. In the domain of mathematics and language, many researchers are guided by a qualitative-reconstructive methodology and they use methods of qualitative social research in their inquiry design. The aim of many research endeavors is to reconstruct underlying structures and patterns of specialized verbal processes of negotiation through extensive analysis of data and the help of a comparative approach ("constant comparative method" in Strauss & Corbin, 1994, p. 273). Generally, research gaps can be identified and based on these gaps components of local theories or theories with limited scope but high comprehensibility are developed using the processes of conjecture and abduction [START_REF] Peirce | Schriften zum Pragmatismus und Pragmatizismus[END_REF]. In the conference proceedings and oral presentations of TWG09 -presumably similar to what happens in other working groups of CERME-only a small portion of the data can be communicated in form of a transcript. However, a transcript excerpt does not usually display, nor even illustrate with precision, the complex process of theory development. Rather, the process emerges across many structural comparisons within an analysis of constant selection of features, decontextualization to a certain degree, and theoretical abduction. In oral presentations and conference proceedings authors can usually only show the best possible visualization, explanation and synthesis of the developed theories, which are mostly striking examples from instances of data materials.

Some of the papers declare and address the challenge of presenting moving from concrete instances of data to theory development in the short space available. The papers by Sauerwein, by Keuch and Brandt, by Alfaro, by Bednorz, and by Umierski and Tiedemann all include some mention to the complexity of making theoretical development explicit and precise by means of the choice and use of selected empirical material. In some of these papers, the examples mirror the process of gradual theory development with local scope in a limited way and are not as comprehensible as originally intended or expected. The empirical material in the transcripts does not, and cannot, provide a sufficiently clear illustration of the theoretical argument. Cohors-Fresenborg and Mackay, Thurlings, Schüler-Meyer and Pepin, during the time for oral presentation of their respective posters, were reflective about the limits of communicating theory development and conceptual abstraction through the examination of examples. For understanding the concepts in the poster of Cohors-Fresenborg, for example, the relationship between what is observed in the lesson data and the scope of application of the concepts introduced by this author is very important.

For compatibility reasons with future research endeavors, especially from early career researchers, it would be preferable if publications and presentations could give a deeper insight into the cumulative process of theory development. Following the line of suggestions in the previous sections, this could also be made possible through a protected area, where researchers could provide further material for the working group. Thus, the gradual process of theory development for a certain period of time could be demonstrated and shared. This would include transcripts that are not yet published but were crucial for the forming of the developed theory components in the course of the constant comparative method. Additionally, there would be an opportunity to allow any author including early career researchers who have completed their PhD, to present their research process at the conference and give them the opportunity to publish a longer paper in the protected area. Anyone interested could hand in an 8-page paper, from which some could then be selected to be offered the opportunity to write a longer 'master-paper'. All this would certainly contribute to the generation of more activity and transparency in theory building.

The challenge of meeting informed audiences

The struggle of TWG09 participants to find new ways of expressing an updated understanding of what research on mathematics and language means today, and what the identity features of the domain are, frames the context for this last challenge. On the occasion of CERME11, due to the increase of the group size, a division of TWG09 into participants attending either Group 09A or Group 09B for all sessions was necessary. There were no 'subtitles' for the subgroups; rather, papers and posters were grouped into (usually) sets of threes by the TWG co-leaders, based on a perceived commonality among them. The organizational need for a practical division due to the number of participants, however, brought up a more profound conceptual discussion. Half of the papers had multilingual settings in common and hence, the recognition that a number of papers shared some multilingual specificity and fitted with each other was used in the final grouping. In the end, one language group consisted mostly of papers that considered language, interaction or mathematics within contexts that are (at least overtly) monolingual, as was the case for the paper of Theens (Swedish), and of Farrugia, which describes a context wherein English was used as lingua franca. In these studies, the languages of the speakers are (generally) not taken into account, unless the speakers themselves make it relevant. The second language group was then mostly constituted by papers wherein the presence of two or more languages in the mathematics classroom is a key element of the discussion, as were the papers of Schüler-Meyer, Prediger and Weinert (German-Turkish), of Ní Ríordáin and Flanagan (Irish-English), and of Chico (Catalan-Spanish).

While the arrangement appeared to make sense and to 'work', and participants' collaboration ensured that things ran smoothly, during the last day it was suggested that multilingual specificity and interactional foci had been central to both Group 09A and Group 09B. In fact, throughout the working sessions the emergent discussion of the multilingual specificity in the papers presented in each subgroup became somehow problematic. The characteristic of 'multilingual' as being restricted to group languages was unintentionally suggested at some points in the discussions of the two subgroups and hence, the broader interpretation of switching between vocabularies, grammatical constructions, intonations, gestures, and so on was less attended to in one of the subgroups but considered in parallel to multilingualism in the other. For a long time, our group has discussed studies that focus on linguistic challenges faced by students that are not necessarily related to a diversity of 'natural' language systems but generally involve multilingualism in the form of (mode) switching between ways of communicating in mathematics lessons. In his paper at Utrecht, for example, Gíslason presented the linguistic and communication repertoires and challenges faced by students who are low attaining in mathematics. The determination of operational criteria for splitting TWG9 is not trivial at all and what is difficult indeed is to find labels of conceptual distinctiveness that do not diminish the opportunities for all participants to share their research with the interested audience.

In particular, the challenge of the group size with the subsequent splitting of TWG09 leads to the challenge of allocating the papers and posters for their presentation to the most appropriate audience. The conference format of short presentations does not allow much time for clarification of approaches and meanings that participants in the audience must then infer. What can be a rather comprehensible presentation for an audience, may not be equally comprehensible for an audience unfamiliar with specific theoretical stances and research emphases. With respect to this situation, it seems again important to offer authors the possibility of uploading additional material of their work in a protected site so that other participants can more easily follow the ideas and reasoning both in advance and after the working sessions. It is important, for instance, to know what is involved in the use of common terms like 'languaging' that may refer to very different stances and emphases [START_REF] Shohamy | Language policy: Hidden agendas and new approaches[END_REF]. As can be attested by the variety of topics addressed by the papers and posters in TWG09, studying language in mathematics education may take many forms of languaging, some of which involve different 'natural' languages or language systems, while some others are related to the alternation of discursive practices necessary for participation in mathematical activity. The papers by Fetzer and by Albano, Coppola and Ferrari show a kind of languaging focused on written practices of argumentation, with shifts between 'natural', symbolic and pictorial languages within one language system (German/Italian). Other papers with multilingual lesson data show a kind of languaging placed in the newer tradition of 'translanguaging' as the act of utilizing one's full linguistic repertoire [START_REF] García | Translanguaging with multilingual students: Learning from classroom moments[END_REF]. Participants also need to know what is involved in the use of the term 'multilingual'. TWG09 participants are aware that attempting to talk about group languages in the classroom context as separate (or separable) systems is artificial and hence, unhelpful for research purposes, but there may be substantially different ways of consider multilingual specificity in data. Of course, the use of group languages overlaps with the use of classroom-or subject-specific discursive practices, so that 'language in the mathematics classroom' is a complex area of study with a number of stances and emphases that cannot be introduced in short presentations.

We anticipate that the group size will require organizational measures in the near future. Due to the overlap between general language or discursive elements and the 'media of instruction' of the classroom, but also to questions regarding the precise meaning for contested terms in the domain, it would be important that the two subgroups find the time to have a joint session. There is a lot of commonality in the theoretical and methodological approaches that all members of the group draw upon or consider in their work, and thus identity distinctiveness should be addressed very cautiously. It will be an internal decision amongst TWG09 participants as to how to utilize the separate and joint sessions, in order to encourage fruitful and deeper discussion; however, it will be equally crucial that the CERME organization allows for the flexibility needed with regard to the organization of sessions as well as the provision of additional information and materials.

More challenges in what comes next

The idea for this introduction paper began as a response to guiding questions posed to ourselves regarding challenges and revisable options of communicating language-sensitive mathematics education research in the written and oral CERME formats. Attention to these questions has been of help to uncover a number of theoretical and reflective issues behind the intricacies of improving the representation and sharing of our work within the CERME community. In our role of co-leaders we have done our best to become openly critical in bringing together some of the challenges that emerge when addressing the writing and presenting of language-sensitive mathematics education research for communication in conference formats. Despite the focus on, and discussion of, conference formats, we can also learn something important about the heterogeneous nature and theoretical complexity of the activity that goes in TWG09. Some of these many different challenges and approaches to mathematics education research on language will be investigated more closely in the diverse papers and posters that constitute this chapter of the proceedings. Through these papers and posters, we will be able to find out what the authors themselves consider the challenges to be that make a strong case for their studies and professional development as researchers in the domain. The creative dialogue with all their points of view will enrich what we have written in more specific and unique ways.


Helen Alfaro Víquez
email: helen.alfaroviquez@tuni.fi

Taking advantage of the different types of mathematical languages to promote students' meaningful learning

Keywords: Mathematical languages, university mathematics education, languaging exercises

The low performance in mathematics of non-mathematics majors has forced higher education institutions to implement different measures to address the problem. Many of these measures have focused on curriculum modifications. This study presents a methodological way of approaching the problem, using written exercises, which combine symbolic, natural and pictorial languages to improve the mathematical learning of university students. These exercises promote the development of essential mathematical skills to achieve successful mathematical learning. In this paper, I describe one exercise and analyze the solutions of 28 students of Calculus 1 course, at the University of Costa Rica. The results suggest that the exercises allow exploring the benefits of different mathematical languages, so that the students can make connections between knowledge and theoretical concepts.

Introduction

During the last few years, improving the mathematics performance of university students has been an important issue. Special attention has been paid to students of non-mathematics majors in the transition process from school to university [START_REF] Goodchild | Teaching mathematics at higher education[END_REF], since the students' mathematical background is not strong enough when they enter university. They may reach the levels of reproduction of procedures, but without understanding the mathematical significance of the contents involved [START_REF] Winsløw | Research on university mathematics education[END_REF]. Thus, the students do not have the level of mathematical reasoning, abstract thinking and rigor required at university [START_REF] Gruenwald | Reducing the gap between the school and university mathematics: university lecturers' perspective[END_REF]. This situation is reflected in the alarming failure and dropout rates presented in the initial courses, from many students who have mathematics in their academic programs [START_REF] Biza | Research on teaching and learning mathematics at the tertiary level: State-of-the-art and looking ahead[END_REF]. This gap in the mathematical knowledge of students has led universities to implement several measures to improve the problem. For instance, peer work, bridging courses, mathematical support centers, interactive lectures, videos, digital assessment, among others [START_REF] Mustoe | Mathematics for the European engineer: A curriculum for the twenty-first century[END_REF]. [START_REF] White-Fredette | What is mathematics? An exploration of teachers' philosophies of mathematics during a time of curriculum reform[END_REF] highlights that the actions taken for facing this situation should consider the instructional level. Similarly, [START_REF] Gruenwald | Reducing the gap between the school and university mathematics: university lecturers' perspective[END_REF] suggest that teachers should look for effective ways to help students to "understand the abstract concepts, master the formal language, follow rigorous reasoning, get a good feeling for the mathematical objects and acquire so-called mathematical maturity" (p. 12). In a nutshell, attention should be paid to the students' understanding of the mathematical concepts and the need to develop their mathematical thinking.

Considering this need, I present the written languaging exercises as a teaching resource to improve the understanding of mathematical concepts by students, using different languages. The exercises ask students to provide written explanations or justifications using symbols, drawings or their own words. In this way, they must organize their thoughts and review the reasoning that led to their solution, being aware of the knowledge and concepts used, and the connection between them. In Finland, the languaging exercises applied in university engineering mathematics [START_REF] Joutsenlahti | Developing learning and teaching in engineering mathematics with and without technology[END_REF] and honor mathematics courses [START_REF] Silius | What can be done to bridge the competency gap between upper-secondary school and university mathematics[END_REF] have showed promising results. In this paper, I present the outcomes of applying languaging exercises in a Calculus I course for non-mathematics majors in Costa Rica.

Theoretical background

As literature indicates, it is necessary to promote conceptual understanding in students [START_REF] Engelbrecht | Interventions to improve teaching and learning mathematics in first year courses[END_REF], teach them how to make connections between concepts [START_REF] Nardi | The novice mathematician's encounter with mathematical abstraction: tensions in concept-image construction and formalisation[END_REF] and how to deal with the abstract nature of mathematical concepts and the complexity of mathematical thinking [START_REF] Biza | Research on teaching and learning mathematics at the tertiary level: State-of-the-art and looking ahead[END_REF] required at the university level. Those actions may help students to experience a successful learning of mathematics and facilitate the process of transition from school to university. The mathematical proficiency theory offered by [START_REF] Kilpatrick | Adding it up: Helping children learn mathematics[END_REF], proposes the development of key mathematical skills that help in this purpose. The theory suggests five main competences that are necessary to accomplish effective mathematics learning: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. These competences promote, among other, the ability to identify connections between concepts; to understand and provide justifications and reasons for procedures; to perform procedures flexibly, accurately and efficiently, knowing how, when and why to do it; to think logically, to represent, formulate and solve mathematical problems in different context; and to consider different strategies of solution. All the strands are therefore interwoven and should be practiced equivalently.

The mathematical proficiency competences can be developed by means of languaging exercises, which are designed based on the languaging theory. Languaging is defined as the students' expression of their mathematical thinking using different languages [START_REF] Joutsenlahti | Developing learning and teaching in engineering mathematics with and without technology[END_REF], including mathematical symbolic language (SL), natural language (NL) and pictorial language (PL). In this way, the languaging written exercises combine models and tasks, which aim to promote different mathematics competencies. As well, the exercises use different languages to access the characteristics of the mathematical objects and students' mathematical thinking.

Languages play an important role in mathematics communication. Following the semiotic approach, it is a tool for representation, communication, thinking and constructing knowledge [START_REF] Schleppegrell | Language in mathematics teaching and learning: A research review[END_REF]. [START_REF] Lemke | Mathematics in the middle: Measure, picture, gesture, sign, and word[END_REF] argues that the integration and cross-referring of NL, SL and PL languages "form a single unified system for meaning-making" (p. 215) and is the combination of them that make possible the mathematical reasoning [START_REF] Schleppegrell | Language in mathematics teaching and learning: A research review[END_REF]. In addition, research evidence that for students, the use of the three languages facilitates the understanding of concepts and mathematical exercises (e.g. [START_REF] Alfaro | Appealing multimodal languages to access first year university students' understanding of mathematical concepts in Costa Rica[END_REF][START_REF] Joutsenlahti | Developing learning and teaching in engineering mathematics with and without technology[END_REF]. The use of different languages allows the exploration of more properties of a mathematical object than using only one [START_REF] Dreher | Why use multiple representations in the mathematics classroom? Views of English and German preservice teachers[END_REF], because each one shows specific features and connotations (O' Halloran, 2015).

The choice of written languaging exercises is based on research which suggests that by writing, students have to organize their thoughts, review and clarify the mental processes they went through in the solution of a task [START_REF] Morgan | The place of pupil writting in learning, teaching and assesing mathematics[END_REF]. Furthermore, they must try to express it in a clear and concrete way, so that readers can understand their mathematical thinking [START_REF] Morgan | The place of pupil writting in learning, teaching and assesing mathematics[END_REF]. According to [START_REF] Kline | Procedural explanations in mathematics writing: A framework for understanding college students' effective communication practices[END_REF], this process improves students' understanding.

Context and method

Due to the high rates of failure of non-mathematics majors in Calculus 1 course in the University of Costa Rica, the School of Mathematics decided to introduce the pre-calculus course, in order to provide students with the necessary knowledge for studying mathematics at university level. However, the high failure rates simply transferred to this new course, and the problem remains unsolved. Therefore, in this study, I suggest a different way of approaching the problem, with a resource that can be introduced in classes for students to have meaningful learning, by analyzing their solution processes when they have to write or explain them.

These languaging exercises were applied to 28 voluntary participants of non-mathematics major taking Calculus 1 course at the University of Costa Rica. There were 17 languaging exercises (see [START_REF] Alfaro | Appealing multimodal languages to access first year university students' understanding of mathematical concepts in Costa Rica[END_REF], for details) that were used in class or as homework during the study of the derivative. The exercises were designed combining different tasks, including: explain with your own words, complete missing steps, identify mistakes, argumentation of the solution, organizing solution steps, and follow given solutions; combined with the use of the three languages. The purpose of the exercises is to promote the different competences of the mathematical proficiency theory, especially procedural proficiency, conceptual understanding and adapting reasoning; and to allow students to experience the use of different languages to express their thoughts.

The aim of this paper is to answer the research question: how the students' understanding of the cases where the function is not derivable, can be evidenced by SL, NL and PL. For that purpose, I describe one exercise (number 3) which exemplifies the use of the three languages to make different representations of a mathematical knowledge. The intention is to provide evidence of the languaging exercises as an effective teaching resource for improving students understanding of mathematical concepts. For the analysis of students' solutions, I did a qualitative analysis based on the stablished knowledge of derivatives for the Calculus I course, studies about students' difficulties with derivatives (e.g., [START_REF] Asiala | The development of students' graphical understanding of the derivative[END_REF] and my teaching experience.

Description of the exercise

Exercise three (Figure 1) consists of a table that presents three cases in which a function is not derivable. Each case is exemplified with one language: symbolic, natural or pictorial, and the students must complete the empty boxes with examples in the missing languages respectively, as shown in Figure 1. The use of the three languages allows students to explore different characteristics and properties of each case. Case I is described in NL with the phrase "At points where the curve presents peaks, since the lateral derivatives would be different." This statement has several characteristics. First, it does not refer to a particular function; therefore, students are not limited to the examples they can provide. In addition, it emphasizes the pictorial features by mentioning the graphical form (sharp points) of the function where the derivability requirement is violated. Finally, it refers to the theoretical aspect that fails (the lateral derivatives are different as shown in Figure 1).

For the case II, the example is given in SL, and refers to the situation in which the function has a vertical tangent line, at a point. This example refers to a specific function. However, the students must interpret from the symbolic expression what case it refers. It means that the student has to recall the definition of derivability to identify the characteristic that makes the function not derivable. Finally,

A) B)
case III shows a graph of a function that presents a discontinuity in the point x 0. As in the previous case, the student must identify which case is presented in order to express it in NL.

The objective of the exercise is to observe if students understand the concepts and rules involved, in such a way that they can interpret them from any of the given representations and can express them in different ways. In the next section, I will present some excerpts of the students' solutions, as an evidence of the different uses of the languages they made, the different ways in which the students expressed the cases in their own words and some errors of interpretation and formality.

Results and discussion

Case I: Statement in natural language

For this case, students have to offer examples in symbolical and pictorial language. In the column of SL, they wrote diverse function samples such as absolute value and piecewise functions, with criteria of minor and greater complexity (Figure 2). As well, some included general expressions such as , and calculated the values of the lateral derivatives. What are the possible cases in which a function is not derivable? Give examples of each of them using the three types of language.

Mathematical symbolic: numbers, symbols.

Natural Language: written words.

Pictorial Language: drawings, graphs, etc.

I

At points where the curve presents peaks, since the lateral derivatives would be different.

II III

It is important to note that all the examples chosen by the students were correct and represented the given case, which means that they were able to interpret correctly the sentence in NL. In addition, by combining the SL and PL columns, it was possible to evaluate the students' abilities to graph functions correctly, pointing out asymptotes and points of intersection, as shown in Figure 3. Finally, some errors of rigor can be observed when writing in SL, as in example B of Figure 2, where the student writes the limit without indicating the function involved.

Case II: Example of the criteria of a function in SL

In this case, the most interesting results were presented in the NL column. In the PL column, most of the students drew the plot of the given function and a few drew the tangent line. The results in NL, however, show that the students were not sure about how to explain this case. Among the expressions, there were students (n=11) who could not even identify what was happening, arguing that the function was indefinite at that point, was discontinuous, had vertical asymptote or was constant. Nevertheless, there were cases (n=8) in which the students seemed unable to express their ideas in a mathematically correct way. Examples as such are when referring to a vertical line as the function: "where the function is a vertical line" or when associating the derivative with the line instead of the slope: "where the derivative is vertical."

Within the phrases they used to explain the phenomenon in case two, we can identify different connections between concepts that students used to justify their claims. Some made references to the calculation of the limit of the derivative at that point and others properly to the relationship between the derivative and the slope. Examples can be found in Table 1. Although in these sentences one can identify some conceptual errors, such as the idea of a "vertical function", they show that the students had an idea of what was happening in the given case.

When the tangent of the point is vertical, as in this case, it is considered that the function is not derivable at that point. (S3)

In the points where there are vertical lines, because this has no slope and therefore has no derivative. (S15)

In the points where the derivative tends to , since this would mean a perpendicular tangent line, which does not exist. (S10)

When solving the limit results in , then it is a vertical function. (S5) The task of the students in this case was to complete the SL and NL columns. In the SL column, responses with different characteristics were presented. Some students wrote piecewise functions in At points where the curve presents peaks, since the lateral derivatives would be different.

which, as in case one, they forgot to point out the point of discontinuity. However, in other cases, the students, in addition to the criterion of the function and the point where the function was not derivable, also added the calculations of the conditions of continuity: lateral limits and the value of image in the point (Figure 4A). This was also evidenced in answers as in figure 4B. Nevertheless, in this case they used a more general form. Regarding the answers in NL, students were able to express the case of discontinuity recurring to the graphical feature of the "jump" or "breaks" (n=2) and others mentioned conceptual aspects as the fact that the lateral limits were different or the that limit does not exist at the point (n=9), the discontinuity of the function (n=15) and the fact that if a function is not continuous in x = a, then it is not derivable in x = a (n=6). Considering the different answers, it is possible to observe that students were able to recall different knowledge about continuity and the conditions of derivability to correctly explain what happened in the point.

Final considerations

As stated at the beginning, when students begin their university studies, they lack mathematical skills and knowledge required for university level. I suggest languaging exercises as a methodological tool to address those issues and promote students' meaningful learning. The written languaging exercises offer an option for boosting students' understanding of mathematical concepts and noticing the connections between concepts, the rules and properties that justify the procedures and the different representations. The use of the three languages reinforces different strands of mathematical proficiency, such as conceptual understanding, adaptive reasoning and strategic competence. For example, from the results it is possible to conclude that for solving the exercise students must understand, identify and verbalize the connections between concepts, as well as represent mathematical situations in different situations. These actions are associated with conceptual understanding. It is important to highlight that in each case, in order to complete the empty boxes, the students had to interpret the given example, and from that point they were already connecting between representations. The adaptive reasoning is evidenced in the justifications and explanations that the students provide in natural language, and the strategic competence is present since the task presented to the students is not common for them, so they must show a flexible approach to solve this novel situation. All these actions helped them to experience meaningful learning and think about the concepts involved in the exercises instead of solving them mechanically.

The different languages allow studying different characteristics of the mathematical concepts involved. NL evidenced the theoretical knowledge involved and, how the connections between the ideas were made. SL showed aspects related to the correct use of the symbols and specificities in relation to the examples, as in the mention of the point where the derivability was violated. In PL, all A) B)

the features were combined and represented. In addition, the use of different languages makes it possible to observe the gaps in knowledge, misconceptions or difficulties the students have. As evidenced in case 2, where most of the students could make the graph from the interpretation of the symbolic expression, but they faced problems when trying to explain the situation in NL, for which deeper knowledge of the subject was required.

This experience shows the potential of the use of different languages to improve the mathematical learning of the non-mathematics major students and to promote their competences to become mathematically proficient. However, more studies are required to explore various ways to integrate different languages in mathematics learning at the university level.

Figure 1 :

 1 Figure 1: Languaging exercise #3

Figure 2 :

 2 Figure 2: S9 and S15 examples in SLIn most cases, students (n=10) did not make explicit in which point of the function the derivative does not exist, neither in the SL, nor in the PL. From this situation the questions of whether the students are aware of what is important in their example is the specific point where the derivative does not exist and what happens in it, can be raised. In the solutions in which it was possible to associate the example in SL with PL, one could verify if the student knew in what point the function described was not derivable by referencing the drawing. However, in the others, it was not clear. There were students' examples in which the function presented two cases where the derivative does not exist, and if they did not mark the point, one cannot know if they understood the case under discussion.

Figure 3 :

 3 Figure 3: Function with two cases of no derivability (S10)

Figure 4 :

 4 Figure 4: S12 and S4 examples in SL

Table 1 : Students' answers in NL Case III: The graphic of a discontinuous function

 1


Nadia Azrou
email: nadiazrou@gmail.com

Learning mathematics in French at the undergraduate level in Algeria

Keywords: French as second language, learning mathematics, students' difficulties

In this paper, some students' difficulties, related to learning mathematics in a second language (French) when entering university, in Algeria, are examined. I present these difficulties as part of other difficulties caused by the transition from high school to university. The analysis of students' interviews, using Cummins' theoretical framework, revealed that low mastery of French as natural language and low mastery of Arabic as academic language impeded several students from performing well in mathematics and expressing them in French academic language.

Introduction

While mathematics is often seen as language free, in many ways learning mathematics fundamentally depends on language [START_REF] Barwell | ESL in mathematics classroom[END_REF][START_REF] Planas | One speaker, two languages: Learning opportunities in the mathematics classroom[END_REF]. The relationship between language and mathematics has been considered within a range of perspectives: political perspective, social and cultural perspectives, epistemological perspective and cognitive perspective. Learning mathematics in a second language at different school levels has also been a subject of many investigations.

On the other hand, Ní Ríordáin and O'Donoghue (2007) expose two opposite positions about learning mathematics in a second language with many references; studies which argue about positive aspects and satisfying achievement (immersion programs) and studies which argue about the underachievement of pupils (submersion programs). Research about teaching and learning mathematics in a second or a third language is very popular at many school levels, but little research has investigated this issue at the undergraduate level so far.

The present study has been motivated by a PhD research about undergraduates' difficulties with proof and proving in Algerian universities. The findings indicated that the weak mastery of French as a second language for learning mathematics was one of the reasons for third year university student' difficulties with writing a proof text. Even though I acknowledge that low students' proficiency in languages is associated with underachievement in mathematics [START_REF] Cummins | Language, power and pedagogy: Bilingual children in the crossfire[END_REF], I tried to understand better how this really works. In this paper, I studied this relationship particularly with students at the beginning of the university, where they learn mathematics completely in French for the first time. I focused on the mathematical difficulties that are originated in the relationships between French language, dialect and Arabic considered as tools to approach, develop and report mathematical thinking; I also investigated students' behaviours when they make mathematics in French and their related opinions.

Linguistic overview of Algeria

Algerian population communicate in different languages, according to different regions, ethnic groups, institutions and circumstances. The most spread languages are Classical Arabic, Dialect, Berber language and French (first foreign language). Classical Arabic is shared by Arabic countries.

The Dialect is the common language; it is the most used spoken language in daily life in Algeria; it's a mixture of classical Arabic, French and other languages (Spanish, Turkish, Italian…). In Algeria, people speak in their families either Berber language (indigenous language) or dialect, which is the same in all the country with slight differences in some words and accents. The language of instruction in schools is classical Arabic, so children learn from primary school on in a different language. Only Arabic language is used for writing, but for the oral, the situation has declined; around thirty years ago, Arabic language was used also for the oral, but dialect has been used more and more. Actually, it's rare for teachers and students to speak only Arabic in the school. French is the first foreign language in Algeria, it is taught from grade 3 to grade 12. With the exception of French language, all courses in all school levels are taught in classical Arabic. Teaching French has been affected during the dark decade of the civil war (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000) by the lack of teachers in many regions; consequently, many students have not had French courses for many years. On the other hand, the technological development and the increasing use of English for computers, cell phones and social media have pushed students to be distant from French. At the end of high school level it is quite rare to find students who master correctly the speaking and the writing of this language, in spite of ten years of learning French.

Languages of teaching mathematics in Algeria

The teaching of mathematics in Algeria has shifted from French to classical Arabic (oral, writing and mathematical symbols) for the three school levels (primary, middle and secondary school) since the 70's. Teaching mathematics in classical Arabic (including Arabic symbolism) has lasted more than thirty years, till the beginning of the 21 st century, when it shifted again. In 2005, there was a big change in teaching mathematics: in school formulas and symbols are written from left to right with Latin alphabet (that stands for French), while comments, terminology and verbal expressions are maintained in classical Arabic (from right to left). This is an example to see how to read and write in both senses: the area of the square ABCD is equal to 20 (arrows show the direction to read).

المربع مساحة ABCD

تساوي 20 m 2

This reform has been undertaken to make the transition to university mathematics (taught in French) less difficult. However, this situation has resulted in a mess and caused some difficulties for students at different school levels. The language of higher education, in Algeria, has always been French for scientific and technical disciplines (medicine, mathematics, architecture, computer science, engineering sciences, etc.) with French textbooks; while the other disciplines (history, Arabic literature, humanities, psychology, etc.) are taught in Arabic.

Literature review

The interest is in investigating undergraduates' difficulties when they learn mathematics in a second language. [START_REF] Durand-Guerrier | Challenges and opportunities for second language learners in undergraduate mathematics[END_REF] pointed out delicate points related to language, which might present difficulties for students. In particular, same words in two languages have different mathematical meanings when we translate from a language to another (decimal in English and decimal in French); and even mathematical notations that "contain a lot of information, so that being able to decode or unpack such information is an important part of mathematical proficiency" (Durand-Guerrier et al., 2016, p. 87), like gcd (greater common divisor). The difficulty of interpreting formal statements in their native language that undergraduate face might be worsened when learning in a second language. Students might have problems when they learn new mathematical words they already know in the natural language (like group or distance) "learning the new vocabulary that is centrally mathematical may be easier than learning the technical meanings for words that students already know in other contexts" (Schleppegrell, 2007, p. 142). Students should be encouraged to express mathematics using language, otherwise, their learning would be sterile, "at college level, there are few opportunities to put into practice long term activities aimed at improving linguistic skills. Requiring high degrees of correctness to students with a poor linguistic background just means inducing them to learn by heart or to use stereotyped expressions with no understanding" (Ferrari, 2004, p. 389). When students do not master the second language they fail to understand their teachers communication (spoken and written), mostly the part that is expressed in a highly academic language; "the language of textbooks and instruction frequently calls for a high degree of familiarity with words, grammatical patterns, and styles of presentation and arguments that are wholly alien to ordinary informal talk (Fillmore, 1982, p. 6)" (Cuevas, 1984, p. 135); what Cummins (1980, p. 43) has identified as the difference between basic interpersonal communicative skills (BICS) and cognitive academic language proficiency (CALP) as "the manipulation of language related to literacy skills in academic situations" Another major issue is the assessment of mathematics achievement of students when made in the second language, many researchers argue that one principal reason for inappropriateness of tests is the use of language that students do not understand [START_REF] Cuevas | Mathematics learning in English as a second language[END_REF].

Method Theoretical framework

Given that this research concerns difficulties of undergraduates when learning in a second language, it is important to examine how do students shift from Arabic to French and how do they behave with their teachers talking, and with their written courses. I will use, for the analysis, Cummins' theory. Cummins distinguishes between conversational and academic languages, as being respectively basic interpersonal communicative skills (BICS) and cognitive academic language proficiency (CALP). The latter is defined as "the extent to which an individual has access to and command of the oral and written academic registers of schooling" (Cummins, 2000, p. 67), in other words, it is the ability to think in and use a language as a tool for learning, by using abstractions in a sophisticated manner. What is important to note here is that, while second language learners may pick up oral proficiency (BICS) in their new language in as little as two years, it may take up to seven years to acquire the decontextualized language skills (CALP) necessary to function successfully in a second language classroom [START_REF] Cummins | Cognitive/academic language proficiency, linguistic interdependence, the optimum age question and some other matters[END_REF]. Later, Cummins expanded these concepts to include two different types of communication, depending on the context in which it is involved: context-embedded communication and context reduced communication; both concern conversational fluency and academic language proficiency. Context-embedded communication provides supports to the listener or the reader (as gestures, images or basic math computations) in order to make the information comprehensible. Context-reduced communication provides less support to understanding (as a phone conversation or lecture with few illustrations). Moreover, Cummins makes the difference between cognitively demanding and cognitively undemanding communications. And considers that BICS (in both contexts) are cognitively undemanding, while both contexts of CALP are cognitively demanding, as shown in his model, illustrated in Figure 1. This theory is furthered by his Interdependence Hypothesis, which proposes that the greater the level of academic language proficiency developed in the first language, the stronger the transfer of skills across to the second language [START_REF] Cummins | Language, power and pedagogy: Bilingual children in the crossfire[END_REF]. I suppose that, for many students, BICS in French language are poor, which would effect the development of their CALP, and the few students who could have developed BICS in French, need much more time to develop their CALP. Moreover, I hypothesize that students did not develop university mathematics in an academic Arabic language in high school because school mathematics are reduced to algorithms and management of formulas that do not require such a level of academic language proficiency. Consequently, students are faced with the difficulty of going from context-embedded tasks (B quadrant, Figure 1) to context-reduced tasks (D quadrant), which is the most difficult for students.

Formulation of research questions

I would like to answer the principal research question: How undergraduates' difficulties with mathematics are related to learning in French as a second language?

According to literature and my PhD work, I made two research sub-questions, each question related to specific difficulties, about how students deal with the teaching and learning of mathematics in French during the first university year, taking into consideration Cummins' theory. Being inspired by the research cited before and observing students and interacting with them as a teacher made us aware of some facts that stand for first hypotheses.

1. Students confuse some French words as they have close pronunciation in French, for example: 'ou', 'où' and 'd'où' (same pronunciation) which are 'or', 'where' and 'hence or thus'. Example: x=1 ou x=0, où x 0, d'où x=1, this might mean 'x=0' for a student who considers all of them as the same as 'ou (or)', (because 'P or Q' is true if either P or Q is true). Students miss the meaning of the uniqueness when we use 'le' or 'la' (as 'the') with the use of 'un' or 'une' (the article 'a') -a difficulty which might not depend on mastery of French only. When we talk about bases in vector spaces, we say B is 'une' base and 'la' dimension because we can have more than one base for a vector space but only one dimension. This meaning packed in these articles is not clear for students. RQ1. Why do students not perform well in French as an academic language? Is it only about second-language performance? H1. I suppose that their weak BICS is a major reason for their problems in CALP. But also their CALP in Arabic has not been developed, which prevents them to make the transfer to French CALP. 2. Most of the time, students are not able to read parts of their course notes, written by them, because they might not know what is the meaning of what they wrote or even to spell exactly what they wrote (written sometimes wrong). In general, I perceive students' big difficulties when I observe their behaviours when dealing with CALP with context reduced tasks (and communication). Taking into consideration that teachers write in French, explain in French and in Arabic (both classical and dialect), while students write in French -but several of French words are unknown to them. RQ2. Why do students meet difficulties in managing their mathematics activities (reading, memorizing and revising their notes from lectures and exercises' sessions)? H2. I suppose that even if teachers would simplify their explanation in Arabic (or dialect), students face the difficulty of shifting between three languages when copying, revising the courses and writing their own responses in French.

Interviews and students' responses

In order to confirm my hypotheses and find out more about the difficulties of students when they learn mathematics in a second language, I designed semi-structured interviews for first university year (18-19 year old) students in the University of Medea (Algeria). The statistics show that more than half of the students (first year) fail, but only one third repeat the year. Eighteen students were chosen (half of them are male) within the discipline of mathematics and computer science. This discipline has two main courses, at the first year, algebra and analysis, where students manifest many difficulties. I included two students with very few difficulties and good marks, ten students with some difficulties and average marks, and six students who are remaking the first year (had many difficulties and low marks last year: RPT student). Interviews have been audio-recorded in May 2018. I have categorized the questions according to the two stated hypotheses as follows:

Q1. If teachers write and speak only in French, would it be better? H1. R1. 18 students said 'no'. In the first year, some student are bad in French language, that's why they give up when teachers speak and write only in French'; 'no, it depends on the level of students in French language but for me, this method did not help me' (4 RPT students). Q.2. If teachers translate orally into Arabic (classical and dialect), would it be better? H1 R2. 18 students said, yes. 'We could at least follow what teachers is dealing with, otherwise, we are lost'; 'if the teacher explains in Arabic, yes it would help us, because we memorize the words better' (a RPT student).

Q3. When you make mathematics in French, do you translate to Arabic or do you try to understand in French language? H2 R3. 17 students said they always translate in Arabic then try to understand. 'I always translate to be able to understand in Arabic language' (a RPT student).

Q4. How do students copy their courses in French? What if some words are not clear? H2 R4. 14 students said, they write everything even if they do not understand. 'we should copy, in case a word is not clear we copy it as it is'. Students say they should copy any unclear words because they are part of the course: 'If the teacher wrote them, they are part of the course and must be important, so we copy them as they are, even if we have to draw them'. 'Students do not ask questions about the unclear words, because they do not understand them and because they cannot speak French very well.' One student said that he writes just what he needs (a good student), two students said, they write and try to understand, one student said 'when I write I try to understand what I am writing, and if I can't understand I ask the teacher to help, because if I try to translate maybe it's not the same meaning as the teacher's' (a good student).

Q5. What happens when a student writes the solution, in French, on the exam-sheet? H1, H2 R5. All the 18 students declared to have difficulties with the use of words when they write their solution on the exam paper. Four students said, they wrote in classical Arabic if they could not write in French language. 'I am not good at French language so, if I can't write a solution in French I write in classical Arabic' (a good student); 'Naturally, I translate the question in Arabic in order to get the solution but often when I try to shift into French language it is difficult' (a RPT student); 'If I write something wrong the teacher would not understand it'; 'I had always difficulties in writing the solution, this is why I'm always scared' (a RPT student).

Q6. In which language do you think when you solve an exercise? H1, H2 R6. All students said, they think in Arabic language first, then they write their solution in French language. 'I think in Arabic to understand the question then I write in French language' (a RPT student). Q7. There is almost no student speaking good French and understanding perfectly French, but how did students who succeeded do, according to you? H1 R7. 10 students responded that students, who cannot speak French and succeeded, rely on memorization (3 RPT students). Among the responses 'I think those students learn with memorizing'. 8 students said that students rely on mathematics formulas without using words, 'maybe they are good in mathematics and so they don't need language, they would use only symbols'.

Results of the analysis of the interviews

All students do not accept that teachers speak only in French, they need teachers speaking in Arabic to better understanding, according to responses to Q1 and Q2. Students who failed last year declared that they are lost if they hear explanations only in French; according to them, Arabic would help them also to memorize the words they need often in mathematics (as mathematical terms or as verbs and words that are used regularly). Responses to Q3 and Q5 show that almost all students translate into Arabic language to understand mathematics (the exception was for one good student). In lack of developed CALP in Arabic language, translating in Arabic is not of much help too. In fact many confusions, and misunderstandings can occur. This validates H1 about weak mastery of natural language that affects logical aspects of mathematics.

Answers to questions Q4, Q5, and Q6 show that students have difficulties with the writing, most of them copy down what teachers wrote on the blackboard as it is, even without deciphering the letters. Because students believe that it is their job and cannot ask questions about unclear words (literally or semantically) as their understanding is very limited and also, because of their weak mastery of French language (to make a clear question). This impedes them to revise their course notes at home. The difficulty of writing is showed more during the exams, when students have to translate from Arabic (used to think), to French (used to write). Most students have doubts about their written texts to be not understood by the teacher when correcting them; consequently, they feel powerless and unconfident. These responses show the big difficulty when they deal with tasks involving context-reduced communication with a high cognitive level, as expected in H2. Responses to Q7 reveal two criteria that were mentioned by students: memorization and focusing on symbols and formulas. Students' conception of mathematics from their long school experience dictates them the fact that language is not so important. They believe that formulas or symbolic expressions are more important and so, language has a secondary role. In doing so, their BICS do not improve and so will be their CALP. This sustains H2.

Conclusion

Students depend completely on translation, they expect teachers to talk in Arabic, explain in Arabic and translate all the words given in French. They also think in Arabic, translate in Arabic and then try to translate again to French for the writing. This fact shows that students' BICS in French is weak and will persist to be so. This fact persists longer as sustained by students' beliefs about language as being not so important in learning mathematics that are reduced to formulas and calculations. I also think that this belief is strengthened by the way they learnt mathematics before (formulas in French and verbal expressions in Arabic). They repeat the same experience: as mathematics in high school focus mainly on calculation, the role of Arabic language is minimal, consequently, students behave similarly with university mathematics whose formulas are clear for them (as they are written as before), and verbal language (in French) might not be important, as before. Thus, students would ignore verbal expressions and focus only on mathematical formulas; consequently, they might memorize a chain of formulas as a procedure for solving some exercises as they are used to do in high school. This is caused, according to me, by two important factors; the first one is due to their mathematical conception from school; that presents to students mostly exercises that are solved by a chain of formulas based on calculation rules. The second factor is the loss of the logical structure in mathematical statements and the lack of knowledge about the process of solving problems. This suggests the validity of H1. If students manage what teachers say orally in Arabic, when it comes to writing, students are in front of a difficult process. When they copy down from the board, they might lose the ideas they got before (in the listening process) because the words (in French) are meaningless for them, that is why they copy them incorrectly or just transcript them as they are presented. Under the didactic contract, students think that they have to copy down what teachers write in the boards, even if it means nothing for them. Moreover, they hesitate and do not ask questions because their French is so weak. The problem of writing rises again, when students have to write by their own, the solution of the tasks of the exams. Students face serious difficulties with CALP with reduced context tasks. This validates H2.

Figure 1 :

 1 Figure 1: BICS and CALP (Cummins, 2000)


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

Carolin Gehrke
email: carolin-gehrke@web.de

Katharina Böcherer-Linder
email: boecherer-linder@math.uni-freiburg.de

Markus Vogel
email: vogel@ph-heidelberg.de

A training in visualizing statistical data with a unit square

Keywords: Bayesian reasoning, visualization of statistical data, unit square, school training study

Recent research yielded empirical evidence for a unit square being a useful visualization of Bayesian situations. However, most of the studies in the research field of visualizing Bayesian problem situations were conducted in well-controlled experimental settings with university students as participants. Therefore, we focus in this paper on a training study with 38 eleventh graders in school about visualizing statistical data with a unit square for coping with Bayesian problem situations. Firstly, we outline some theoretical and empirical basics concerning research about Bayesian situations and a unit square as a facilitating visualization tool. Afterwards, we present a short training sequence in using the unit square effectively. We report on methods, implementation and results of a pilot study in school. The promising results were discussed at the end.

Introduction

According to [START_REF] Arcavi | The role of visual representations in the learning of mathematics[END_REF] it is an important task of mathematics education research to investigate visualizations as specific graphical representations of mathematical objects in a theoretical and empirical way. Thus, we focus in our ongoing research about the unit square as a powerful visualization of Bayes' formula on theoretical and empirical insights. The unit square is shown in Figure 1 visualizing a Bayesian situation which are situations where the Bayes' formula could be applied.

After travelling to a far country, you learn that on average 10% of the travelers contracted a new kind of disease during their trip. The disease proceeds initially without any clear symptoms, therefore you don't know whether you had been infected or not.

You learn that a medical test was developed which has the following characteristics: 80% of infected people (I) get a positive test result (sensitivity of the test; +). 15% of not infected people ( ) get also a positive test result (specificity of the test, +).

Finally, you decide to carry out the test and get a positive test result. What is the probability that you had been infected actually? Given for example a Bayesian situation as described in Figure 1 the unit square would look like as displayed in Figure 1 in the right side. All essential information is represented in a numerical and a geometrical way. Thus, a unit square facilitates via different strategies the computation of the essential probability, that is, the probability of being infected given a positive test result .

Focusing on facilitating strategies in Bayesian situations in detail is the main goal of this paper: We refer to certain visualization features of a unit square which makes it an effective facilitator for coping with Bayesian situations and for adequately applying Bayes' rule which is a fundamental model for dealing with risk in situations of uncertainty. For this we provide first a brief literature review to discuss the advantages of the unit square in comparison to further visualizations of Bayesian situations. Afterwards we refer to the design and the results of a quasi-experimental training study with 38 students in grade 11 which compares the performance of a treatment group vs. a control group.

Our question for this step in an ongoing research project was whether it is possible to improve students' dealing with Bayesian situations by a brief training that is based on representing Bayesian situations by natural frequencies and by a unit square. Implications and conclusions are discussed at the end of the paper.

The unit square as facilitator of Bayesian situations

Despite the Bayes' rule's relevance for real life situations, psychological and educational research gained evidence that people often fail when applying Bayes' rule [START_REF] Diaz | Teaching independence and conditional probability[END_REF][START_REF] Gigerenzer | How to improve Bayesian reasoning without instruction: Frequency formats[END_REF]Kahneman, Slovic, & Tversky, 1982). In their meta-analysis [START_REF] Mcdowell | Meta-Analysis of the Effect of Natural Frequencies on Bayesian Reasoning[END_REF] concluded that only about 5% of participants in various studies were able to solve a problem as exemplarily given in Figure 1 without any facilitating strategy.

A widely accepted strategy to increase people's performance in Bayesian situations is representing the statistical information via natural frequencies instead of using percentages (e.g. [START_REF] Gigerenzer | How to improve Bayesian reasoning without instruction: Frequency formats[END_REF]. In Figure 2, the Bayesian situation given in Figure 1 is displayed by using natural frequencies. The meta-analysis of [START_REF] Mcdowell | Meta-Analysis of the Effect of Natural Frequencies on Bayesian Reasoning[END_REF] provides empirical evidence that by using natural frequencies as the only information format people's performance in coping with Bayesian situations increases from about 5% to about 25%.

After travelling to a far country, you learn that on average 100 out of 1000 travelers contracted a new kind of disease during their trip. The disease proceeds initially without any clear symptoms, therefore you don't know whether you had been infected or not.

You learn that a medical test was developed which has the following characteristics: A further facilitating strategy is to provide additional visualizations (e.g. [START_REF] Brase | Pictorial representations in statistical reasoning[END_REF][START_REF] Mcdowell | Meta-Analysis of the Effect of Natural Frequencies on Bayesian Reasoning[END_REF]. However, there are ambiguous research findings about which visualization properties could effectively facilitate dealing with Bayesian situations.

From a theoretical point of view, three properties could be suggested to have a facilitating effect. The first is to use icons as "real, discrete and countable" objects (Cosmides & Tooby, 1996, p. 33). However, from a practical point of view (in terms of time costs) this strategy turns out to be hardly realizable in regular classroom instruction where students have to create visualizations. Further-more, results of our own research suggest that icons actually have an additional, but small facilitating effect compared to a unit square as shown in Figure 2 (Böcherer-Linder & Eichler, 2019).

The second facilitating strategy is to make the nested-sets structure transparent which constitutes fundamentally every Bayesian situation [START_REF] Oldford | Probability, problems, and paradoxes pictured by eikosograms[END_REF]. Sloman, Over, Slovak, and Stibel (2003, p. 302) stated that "any manipulation that increases the transparency of the nested-sets relation should increase correct responding". Actually, our own research revealed empirical facts that the unit square allows exactly for such a manipulation (Böcherer-Linder & Eichler, 2017[START_REF] Böcherer-Linder | How to Improve Performance in Bayesian Inference Tasks: A Comparison of Five Visualizations[END_REF]. The subsets that have to be identified for successfully solving a task with the Bayes' formula are directly neighboured in the unit square (cf. Figure 1) and, by this means, they are transparently visualized.

In particular, in different experimental studies, we yielded empirical evidence that the unit square makes the nested sets-structure of a Bayesian situation more transparent than the commonly used tree diagram and showed further that the unit square is more effective to facilitate dealing with Bayesian situations (Böcherer-Linder & Eichler, 2017). Actually, compared to only using natural frequencies the additional use of a unit square could further increase people's performance in coping with Bayesian situations from about 25% (cf. above) to about 65%-70%.

A third facilitating strategy is to use area-proportional representations of statistical information (e.g. [START_REF] Talboy | Improving Accuracy on Bayesian Inference Problems Using a Brief Tutorial[END_REF]. Going beyond pure numerical information a unit square provides additionally the statistical information by displaying exact graphical proportions of the rectangular subplots. When comparing a unit square with a 2x2-table which differs from a unit square regarding the missing area-proportionality we found evidence for the effectiveness of the area-proportionality (Böcherer-Linder & Eichler, 2019). Nevertheless, understanding Bayesian situations is not fully reached by only solving tasks as shown in Figure 1 [START_REF] Borovcnik | Multiple perspectives on the concept of conditional probability[END_REF]. Moreover, understanding also includes the adequate influence-judging of parameters in these Bayesian situations, that is, the base rate or the conditional probabilities that are called sensitivity and specificity in a medical diagnosis situation. For this purpose, we also investigated the facilitating effect of a unit square compared to a tree diagram by regarding the influence of changing parameter values in Bayesian situations (Böcherer-Linder, [START_REF] Böcherer-Linder | The impact of visualization on flexible Bayesian reasoning[END_REF]. In this research, we found a significant supremacy of a unit square compared to a tree diagram. Although we have not yet compared a 2x2-table and a unit square concerning these kinds of changing parameter tasks in Bayesian situations, we hypothesize based on theory [START_REF] Eichler | Die (Bild-)Formel von Bayes [The (picture-)formula of Bayes[END_REF]) that a unit square is of advantage by means of additional visualizing changing proportions.

The research discussed so far is mostly conducted in well controlled studies without a systematic training. However, training should be understood as a fourth strategy which provides facilitating effects additional to the effective strategies mentioned before. However, these very few training studies reported so far were not focused on facilitating properties of visualizations and, they were mostly carried out with university students as participants (e.g. [START_REF] Sedlmeier | Teaching Bayesian reasoning in less than two hours[END_REF][START_REF] Talboy | Improving Accuracy on Bayesian Inference Problems Using a Brief Tutorial[END_REF]. For this reason, we conducted a training study that addressed learning how to adequately deal with Bayesian situations based on using natural frequencies and, in addition, on using a unit square. Referring to a sample of university students, the results of a pilot study suggest a positive effect of the training with a unit square (Böcherer-Linder, [START_REF] Böcherer-Linder | Visualizing statistical information with unit squares[END_REF].

In this paper, we present a training study with students of grade 11. The main question in this study was whether students' performance in Bayesian situations as well as students' understanding of Bayesian situations in terms of judging the influence of changed parameters could be improved in a brief training based on representing Bayesian situations by natural frequencies and a unit square.

The training

Step 1: Choice of the sample size For solving the problem, we first consider the question of what the probabilities mentioned in the text imply for a concrete group of travelling people. We choose a sample size of 1000 people.

We represent the group of 1000 people by drawing a unit square.

Step 2: Construction of the frequency representation Since 10% of the travelling people contracted the disease during their trip, 100 out of the 1000 people are expected to be infected. 900 out of the 1000 people are expected to be uninfected.

Thus, we divide the unit square in vertical direction for "infected" and "uninfected" at the ratio of 100 to 900. At the bottom of the narrow rectangle we write "100" and at the bottom of the broader rectangle we write "900".

Since 80% of the infected people get a positive test result, 80 out of the 100 infected people are positively tested. Accordingly, 20 out of the 100 infected people are tested and result negative.

Therefore, we subdivide the narrow rectangle horizontally into two parts and write "80" and "20" into the resulting areas.

Since 15% of the uninfected people get a positive test result, 135 out of 900 uninfected people get a positive result after testing (because 15% of 900 is 135). Accordingly, the other 765 uninfected people get a negative test result.

Therefore, we subdivide the broader rectangle horizontally into two parts for "positive" and "negative" and write the numbers into the resulting areas.

What needs to be found is the probability that a person with a positive test result is actually infected. Thus, we have to calculate which proportion of the people resulting positive is actually infected. For this aim, we surround all positive people with a dashed line in the unit square and emphasize with grey color all of them, that are infected.

We read out the following numbers:

Number of infected and positive to the test: 80

Number of all positive to the tested: 80+135=215

We calculate:

This proportion corresponds to the requested probability of 37%. Due to an ecological validity, we presented the problems in the probability-text-version (see Figure 1) and trained the students to translate the probabilities into natural frequencies and, secondly, to graphically represent the statistical information in the process of problem solving. The training had 0,37 "picture formula" two phases: phase 1 (10 minutes) contained three steps. It was a worked example showing how to solve the problem of Figure 1 with the help of the unit square (Figure 3; translated from German).

Phase 2 (10 minutes) included an exercise that was structurally identical to the worked out example (Figure 2 and Figure 3) but had another context (Trisomy). An oral explanation and a presentation of the correct solution were given. Even though it was possible to draw a unit square true to scale in this case, we only showed a rough drawing of a unit square, since we wanted to enable the participants to work with the visualization as a thinking tool for problem solving which did not necessarily imply a precise drawing.

Finally, we added a brief exercise (5 minutes) on the change of parameters. For this, we only show the instruction referring to the change of the base rate (Figure 4). The design of the quasi-experiment is shown in Figure 5. Since there were not all of the students present in the three phases shown in Figure 5, our analysis is based on 16 students (treatment group) and 13 students (control group).

Both the pre-test and post-test consisted of two Bayesian situations. One of the situations in both tests was the same, the other situation differed to prove whether repeating a task has an effect on performance in Bayesian situations. In the pre-test, one situation included only a performance-task and the second situation included both a performance task and a task in which the influence of changed parameters was to be estimated. In Figure 6 we show the task that was identical in both tests and include both forms of tasks (translated from German).

There are tests for diagnosing people if they have infectious diseases like measles or scarlet. Concerning such an infectious disease and the corresponding test the following information is given:

The probability of having the infectious disease is 2%. Given there is a patient having the disease, the test yields in 90% of all cases a "positive" result, which means it indicates correctly the infectious disease. Given there is a noninfected patient, the test shows in 5% of all cases also a "positive" result, which means it indicates the infectious disease by mistake.

a) What is the probability of a patient having actually the disease given a "positive" test result? b) How changes the probability of section a) if the probability of having the disease is higher?

Results

Regarding the few items the values of Cronbach's alpha of 0.63 (pre-test) and 0.85 (post-test) appear to be good. Thus, we summarized the scores in each test referring to the performance test-items a). We applied a mixed ANOVA (within-factor: performance in the pre-test and post-test; between factor: group) to investigate the training effect in the treatment group compared to the control group. According to the robustness of an ANOVA [START_REF] Schmider | Is It Really Robust?[END_REF], we did not regard non-normality and differences in variances. In a descriptive way, Figure 7 (left) reports the results referring to the pre-test and the post-test. Figure 7 (right) visualizes a clear effect of the treatment referring the means. As expected, the ANOVA yields a significant effect ( =20,733, =0,000) of the treatment with a strong effect ( 2 =0,434). Regarding item b) the results were not as expected (Figure 8). Actually, also the control group has an increased average of correct solutions and the differences between the treatment group and the control group are not significant concerning the ANOVA ( =0,424, =0,520, 2 =0,015). By contrast, there is a significant main effect for both groups regarding the difference between the pre-test and the post-test ( =24,938, =0,000, 2 =0,480).

Discussion

Regarding the results of the performance tasks it could be stated that a training with the unit square turns out to be helpful for students in school. This finding is in line with the results of our recent study in school investigating the effect of the visualization (cf. Vogel & Böcherer-Linder, 2018). Furthermore, this is particularly emphasized within a practical point of view, a unit square's effective use can be trained within a very short time (in Germany 45 minutes corresponds to one lesson). Regarding the tasks of estimating the effect of changing parameters it should be mentioned that these kinds of tasks were estimated to be more difficult by going beyond the demand of only building a ratio as the Bayes problem's solution. However, most of the students in both groups were able to find the correct solution, even in the control group and, thus, the significant difference between pre-and posttest indicates a learning progress. Particularly the missing difference between the treatment group and the control group could firstly be explained by the high guessing probability of one third, or even one half, if the question "How changes" is interpreted as either increasing or decreasing. Moreover, maybe the students intuitively guessed that the conditional probability would be higher if the base rate increased without analyzing the parameter dependence. However, we expect tasks which necessitate an analysis of the parameter dependency to be trainable as successfully as the performance tasks. This effect is focused on in subsequent studies being conducted currently.

Figure 1 :

 1 Figure 1: Bayesian situation and visualization with the unit square

Figure 2 :

 2 Figure 2: Bayesian situation with natural frequencies

Figure 3 :

 3 Figure 3: First intervention in the training study ("picture-formula", cf. Eichler & Vogel, 2010)

3.

  Change of the base rateThe base rate indicates the proportion of an individual having trisomy (5% of the children of women of age 45). An increase of the base rate results in an increase of the area of the thin rectangle on the upper left side of the unit square and, accordingly, a decrease of the area of the rectangle on the upper right side.If the base rate increases to 10%, there would be 90 true-positive people and 90 false-positive people. The change of the base rate changes the proportion as shown below:Actually, the result shows a considerable change of the probability.

Figure 4 :

 4 Figure 4: Part of the training to changed parameters

Figure 5 :

 5 Figure 5: Design of the training study

Figure 6 :

 6 Figure 6: A test item (translated from German)

Figure 7 :

 7 Figure 7: Descriptive results of the training concerning item a)

Figure 8 :

 8 Figure 8: Descriptive results of the training concerning item b)

test result. What is the pro- portion that you actually have the disease?

	Finally, you decide to carry out the test and
	get a positive
	80 out of 100 infected people (I) get a positive test result (sensitivity of
	the test; +).
	135 out of 900 not infected people ( ) get a positive test result (specifici-
	ty of the test, +).