Timo Dexel
email: t.dexel@wwu.de

Ralf Benölken
email: benoelken@uni-wuppertal.de

Marcel Veber
email: marcel.veber@uni-osnabrueck.de

Diversity, inclusion and the question of mathematics teacher education -How do student teachers reflect a potential-related view?

Keywords: Diversity, inclusive mathematics education, teacher education

come

Introduction

According to [START_REF] Ainscow | Taking an inclusive turn[END_REF], inclusion is seen "as a reform that supports and welcomes diversity amongst all learners" (p. 3). Its aim is to analyze barriers to participation and learning experienced by students within school systems. In some countries, inclusive education is seen as teaching children with disabilities within general education, traditionally by special educators. [START_REF] Slee | Excluding the included: A reconsideration of inclusive education[END_REF], who express their skepticism about inclusion as co-location of pupils, criticized this understanding earlier. There is a similar discourse in Germany: [START_REF] Hinz | Inklusion: Von der Unkenntnis zur Unkenntlichkeit![END_REF] states that inclusion moved from ignorance to being unrecognizable, reminding us that inclusion had been ignored by educational science and politics, and now the scientific discourse shifts its attention from inclusion to de-segregation: Instead of focusing on barriers, the individual problems of pupils with special needs and their location in regular schools are discussed. This causes consequences for teacher education: Mathematics teacher education for inclusion is often realized by implementing elements of special education [START_REF] Werner | Inklusiver Mathematikunterricht aus sonderpädagogischer Perspektive. Konsequenzen für die Lehrerbildung[END_REF], or inclusive mathematics education is reduced to the question of how pupils with special needs learn and what can be done to compensate for their deficits [START_REF] Krause | Diagnose im Fach inklusive[END_REF]. Three problems can be found in this approach: Empirical research showed that marking pupils with special needs is highly influenced by socio-economic status or migration, leading to negative consequences for further schooling and to the potential danger of stigmatization [START_REF] Sturm | Lehrbuch Heterogenität in der Schule[END_REF]. Secondly, a focus on categories might neglect other children who do not belong in a predetermined category and, therefore, become marginalized [START_REF] Messiou | Research in the field of inclusive education. Time for a rethink[END_REF]. Finally, concentrating on pupils' individual deficits ignores the role of the social and political environment in mathematics education and individualizes failure [START_REF] Veber | Individuelle Förderung in inklusiver Bildung: Eine potenzialorientierte Verortung[END_REF]). An alternative approach and the framework of this paper is given by viewing inclusion not limited to deficits or categories like gender or social class, but by perceiving individuals' differences as an advantage. We will outline that addressing pupils' differences in this sense of diversity, which we call potential-related, understanding inclusion as a concept welcoming diversity, and a focus on barriers to participation are suitable frames for mathematics teachers' education [START_REF] Benölken | Das Projekt 'Inklusiver Mathematikunterricht': Konzeptuelle Ansätze für Unterricht und Lehrerbildung[END_REF]. In this paper, we address the learning and reflection of student teachers in the context of inclusive mathematics education. Therefore, we do not only describe our attempts on inclusive teacher education but focus on the students attending these courses. The following questions will be investigated: How can a seminar for student teachers be conceptualized into inclusive mathematics education from a potential-related perspective? How do student teachers reflect on barriers in inclusive mathematics teaching before and after taking part in the seminar? First, brief overviews of the theoretical frameworks will be given. On these bases, the seminar concept will be outlined. Finally, the design and results of an exploratory study analyzing learning-maps in a pre-postcomparison by applying methods of pedagogic-iconological image interpretation will be subsumed and discussed.

Theoretical framework: A brief overview

Inclusive education demands a positioning in the understanding of 'inclusive'. While most of German mathematics education research regards the concept in the sense of special education, an alternative approach is a potential-related view. [START_REF] Benölken | Das Projekt 'Inklusiver Mathematikunterricht': Konzeptuelle Ansätze für Unterricht und Lehrerbildung[END_REF] identified three axioms of inclusion: (1) The rejection of labels such as 'disabled' or 'dyscalculia' for pedagogical actions is an inherent part of inclusive approaches. It means that differences are produced in contexts and not determined in a static way. Most pedagogical categories emphasize negative aspects and, therefore, reduce people to one dimension of their individuality. Vice versa, a potential-related view focuses on the resources of learners. (2) The traditional German schoolsystem is based on homogeneity, assuming pupils learn better if they visit schools for their specific needs. For example, the Gymnasium for supposedly 'intelligent' children or the Förderschule for 'disabled' children. Inherent is the assumption that heterogeneity poses an obstacle for successful learning. In contrast, a potential-related view demands to accept and to appreciate differences, which is represented in the term diversity. (3) Inclusion has a political approach. It is based on human rights, and it demands for changes in society [START_REF] Slee | Excluding the included: A reconsideration of inclusive education[END_REF]. The research status on inclusion in German mathematics education is characterized by proposals for inclusive lessons (Häsel-Weide & Nührenbörger, 2017), often using the concept of natural differentiation. There are few concepts that include other aspects such as co-teaching, diagnostics and beliefs [START_REF] Käpnick | Verschieden verschiedene Kinder[END_REF]. There tends to be a lack of a theoretical discussion about the principles of inclusive mathematics education. Often, the needs of pupils with difficulties in learning mathematics are discussed to design learning environments; the role of mathematics education itself as a producer of inequality is hereby neglected. The international research on inequality and exclusion has given useful suggestions on reflecting discriminating structures and the influence of mathematics education and its teachers (e.g., [START_REF] Bishop | Diversity in mathematics education: Towards inclusive practices[END_REF]. Consequently, we think that it is not sufficient to add special needs courses to mathematics teacher education. We suggest reflexive inclusion as a promising theoretical construct [START_REF] Budde | Reflexive Inklusion[END_REF]. Reflexive inclusion means both to perceive differences when it is needed and to make discrimination visible, but at the same time a rejection of implicit norms and institutional categorization. In this understanding, reflexive inclusion is an interdisciplinary task of mathematics education, special education and pedagogy. In contrast to other concepts, it is not reduced to the dimension of disability, but framed intersectionally, that is, several dimensions of diversity act in combination. This understanding meets our axioms and provides the theoretical background of a potential-related view. Thus, it is relevant for student teachers to learn that differences need to be deconstructed when possible, and to recognize discrimination. Furthermore, specific knowledge about pedagogical diagnostics and categories of social inequality such as disability, gender, milieu and such is necessary. Therefore, it is of interest to scrutinize seminar designs, and to explore the development of student teachers' reflection, knowledge or perhaps beliefs on, for example, ex-and including factors.

Survey of a seminar conception

We will outline an exemplary teaching concept (for further details see [START_REF] Benölken | Das Projekt 'Inklusiver Mathematikunterricht': Konzeptuelle Ansätze für Unterricht und Lehrerbildung[END_REF]. It has proven itself in implementations at the Universities of Münster, Osnabrück, Halle and Kassel as well as at the Pedagogical University Tyrol. It aims at students who have already acquired their first subject-didactic and subject-scientific knowledge. Thus, it requires a level of ambition that enables students to expand and network knowledge already acquired with a focus on the organization of inclusive mathematics lessons. Against the background of interdisciplinary networks, the aim of the seminar is to enable students to plan, reflect and analyze inclusive mathematics lessons. The overarching questions of the seminar sessions can be identified as follows: Which influencing factors must be considered in the design of inclusive mathematics lessons? How can concrete mathematics didactic implementations be identified, and, conversely, do they meet the requirements of inclusive education? The seminar concept approaches these questions with a potential-related understanding of inclusive education. Consequently, various facets that are discussed in the relevant literature concerning the realization of lessons for all children without exclusion are taken into account: Attitudes and experiences, professional competencies in relation to various diversity facets (including facets relevant to special education), diagnostic and methodical spectra, educational framework conditions, social relationships as well as cooperative forms of teaching. In summary, the complex of all presented aspects mirrors a holistic approach to student teachers' education, and they explore single facets of the complex as well as interrelated patterns by research-based learning, that is, by raising relevant questions, developing corresponding concepts and sometimes examining them in greater depth in bachelor and master theses. The organization of the seminar combines usual demands of higher education didactic such as different methods and both traditional and digital media, multifaceted combinations of theoretical contents and authentic practical examples. Additionally, there are supplements of expert lectures (often as 'best practice' information), school visits and workshops, in which student teachers can network and deepen their knowledge. A recurring highlight are discussions about both potentials and possibilities of implementing developed ideas, concepts and findings in inclusive mathematics teaching. In each case it is scrutinized whether work products meet both mathematics educational and inclusive pedagogical postulates of organizing teaching-learning processes.

The study

The study focuses on the question how student teachers reflect on barriers in inclusive mathematics teaching before and after taking part in the seminar. The sample consisted of 124 participants (101 f., 23 m.), who were primary and secondary student teachers and took part in the study during the semester in which they attended the seminar. The data originated from implementations of the seminar concept in Münster (between the winter semester 2015/2016 and the summer semester 2017), Osnabrück, and Innsbruck (each in the summer semester 2018). Mostly, the participants were in their first year of undergraduate studies. The study has an explorative character, that is, generalizations were not intended, but existential propositions [START_REF] Lamnek | Qualitative Sozialforschung[END_REF]. Thus, a qualitative design was advisable. As to the method, qualitative data were generated by applying learning maps in a pre-post-comparison, which were anonymized by codes to ensure unbiased interpretations. In the maps' header, the student teachers were given the impulse to craft their way between their current status and their future work at schools: "Inclusive education is currently considered as the greatest challenge for teachers. The question arises as to how teachers can structure their specialized instruction in such a way that (really) all children can learn together in order to reach their respective zone of proximal development. This joint learning of all children without exclusion might arise opportunities, challenges, stumbling obstacles and much more. What does this mean to you personally? Which way have you covered or which way will you have to cover in the future? Please lay out your way" (translated from German). All participants designed the maps for the first time, and the maps were created at the beginning of a semester and its end. As to the analysis, the pre-and post-maps were compared by a reconstructive pedagogic-iconological image interpretation, following its characteristic steps: (1) Discussion of previous history and selection of key motives, (2) image description and analysis (with regard to the factual, expressive and form-related sense), (3) context analysis, and (4) comparative analysis [START_REF] Schulze | Bildinterpretation in der Erziehungswissenschaft[END_REF]. The evaluation procedure was piloted in a research workshop with student teachers in the summer semester 2016; afterwards, the analyses were conducted in a research group.

Results

As to the map's previous history, the reflections' interpretations must be seen in the context of the seminar, which attempts to combine school requirements in dealing with diversity and university teaching. For example, led by theory, the student teachers develop concrete task formats in the seminar, which provides one of the topics in the context of methodical spectra. This framing implies that the student teachers receive a view of school practice from the university perspective between the pre-and post-drawings, which produces impacts on the post-maps' design. Additionally, the seminar's evaluation was conducted in a mixed-method design, where the maps provide one constituent. Regarding key motives and their description, main types were drawing both (1) paths (or a systems of paths), and (2) obstacles. Subsequently, we focus on spotlights of the reconstructions of one example shown by the Figures 1 and2, which reflect typical features (the figures show facsimiles with translations from German; the original data can be requested from us).

As to the factual sense of the pre-map (Figure 1), the most important symbols are a path and a mountain. With regard to the sense of form, the path begins and ends relatively narrow, and it winds directly before the finish. The mountain is on the path. Additionally, some persons at the end of the path smile and stretch their arms into the air. As to the sense of expression, another noteworthy detail is that the remarks are often written in questions. The expression is rather sober. As to the factual sense of the post-map (Figure 2), the spotlights are basically repeated. Additionally, at the beginning of the path is a large suitcase, which is taken along: Small suitcases are carried by a person walking on the path. With regard to the sense of form, the path is not winding, but it leads straight to the destination without detours. The suitcase is swung back and forth by the person. The mountains stand beside the path in the background. As to the sense of expression, most of the remarks do not represent questions, but statements. The bigger person smiles and stretches the arms upwards. Colors are used to highlight single aspects.

As to a context analysis of the pair of maps, the feet drawn in the pre-map (Figure 1) seem to indicate that the creator of the map reflects that he or she has started the journey to inclusive teaching, but there are several aspects showing ambiguity like the real starting line is not reached yet or the annotations are phrased as questions. The creator still seems to expect to gain experience, knowledge and the like as shown by both the words on the path and the envelopes, especially the big one with the question mark, but his or her bases are "empathy" and "experience" (which is not explained in detail here). With regard to the central symbol 'mountain', an ambivalence must be noted: On the one hand, mountains can block the view, can be a hurdle. On the other hand, mountains represent challenges that require effort but offer opportunities in relation to the goal being achieved, and both the 'path is the goal' and the 'goal is the goal'; climbing and reaching the summit can cause a 'flow'. The latter can justify the previous effort. Mountaineering can also be influenced by different weather conditions and the like, that is, different conditions that require special equipment. Thus, the symbol 'mountain' can stand for different chances but also challenges and dangers (on the part of the teachers, but not only there) and this requires a specific preparation. As indicated, in the pre-map, the mountain is drawn on the path, which seems to reflect that the creator can imagine to cope with inclusive teaching at school in principle (which is also indicated by the cheerful group of people at the school), but he or she still perceives barriers that will be difficult and challenging to overcome. This interpretation is confirmed by the fact that a curve is drawn between the mountain and the school. The comparison with the post-map (Figure 2) suggests that the creator now reflects to perceive a direct access to inclusive teaching: The big suitcase seems to indicate that he or she can classify own experiences in a more differentiated way. Experiences and likeness are reflected to be more comprehensive than it was the case in the pre-map. The potential-related perspective can be assumed as an essential impulse for this fundamental change (e.g., "knowledge about different children and their diverse needs"). Questions posed in the premap (e.g., "who?", "how?", "what?") turned to concrete intentions and planning steps (as expressed by the phrasings on the path or the "wishes" of the bigger person on the right). Especially, as to the central symbol 'mountain', in the pre-post comparison two aspects appear central: the position of the mountains in relation to the path and their number. As to the position of the mountains, it should be noted that, at the beginning, the mountain stands in the way as a hurdle and it makes it necessary to overcome it. Those who want to walk the path cannot avoid overcoming the mountain. In the end, the position of the mountains has shifted significantly, as the path now leads less curvy in front of the mountains: climbing changes from a compulsory to an optional experience. In connection with the altered position, the increasing number of mountains indicates that in the end the ambivalence of mountains is perceived to a much greater and more intensive extent. However, these opportunities and risks no longer appear to be hurdles that cannot be circumvented. Thus, the symbol 'mountain' allows exemplary illustrations of a deeper interpretation of the observable changes considering the previous history of the maps: The student teacher experiences a paradoxical situation, since a synchronization of university theory and school practice provokes numerous difficulties (mountain peaks) within the framework of inclusion. However, the mountains are no longer obstacles (i.e., a key motive was reinterpreted). This interpretation is confirmed by the facts that the path is drawn continuously and no longer winding. Furthermore, there are different impressions reflecting a higher level of confidence like the swinging of the small suitcases, the use of colors (e.g., the 'sun' on the right) or the big person, whom one may assume to be the creator him-or herself. The synthesis of his or her size and his or her cheerful expression might be interpreted as a symbol of prospective knowledge and experience, that is, he or she can now well imagine coping with diversity at school and experiences him-or herself better equipped for the journey (e.g., by the suitcases).

Based on these typical examples, a comparative analysis of all pre-and post-maps suggests that both the paths of student teachers and their experience of walking on the paths, thus, the reflections of barriers in inclusive mathematics teaching, changed to more confident characteristics. In sum, these changes can be described by three reflexively connected levels: key motives, lines of development, and their antitheses or contrasts. For example, the symbol 'mountains' clearly shows that the student teachers change their image of inclusive mathematics teaching in the course of the seminar. On the one hand, it becomes more differentiated and extensive, since in their opinion it is no longer sufficient to simply overcome a hurdle and the goal is already in front of their eyes. It rather seems that inclusion in mathematics teaching offers numerous challenges. These offer opportunities, but also dangers for teachers. The given pre-post-map-example represents a main type of all pairs of maps, the 'expectant expert'. A feature of this type is that knowledge about inclusion increases and the person develops confidence in inclusive teaching. Another type shows similar characteristics, but in a lower level of contrast, which we named the 'optimistic novice'. In both cases, we reconstructed an increase of knowledge and reflection, so we assumed deeper understanding ofor at least insight inthe possibilities and challenges of inclusive education. Finally, the type 'interested layman' was reconstructed showing much lower levels of confidence's increase (forming the subtypes 'positively stagnating' and 'no process'), but it was rarely found.

Discussion

The results indicate that participating in a seminar which takes a potential-related perspective on inclusive education and which considers the idea of research-based learning as well as intertwining theory and practice in a university context contributes sustainably to student teachers' reflections of removing barriers in inclusive mathematics teaching. Thus, beyond the context of mathematics education, the results suggest the hypothesis that the seminar's foundation provides an appropriate approach to student teachers' education in the field of diversity, which is, for example, reflected by the main type 'expectant expert'. Of course, the study's character is explorative and it has obvious limitations, for example, the reconstructions were conducted within a group of researchers, but it remains uncertain if a consensus view is the right one [START_REF] Lamnek | Qualitative Sozialforschung[END_REF]. The interpretations reported in this article focus on the most important spotlights, especially regarding the characterization of the previous history of the images' geneses, the connection between motives emerging from the previous history and the central symbols applied in the maps' drawings, or the introduction of different types. As to the sample, all data collected between 2015 and 2018 were included in the analyses in order to get both various and sustainable impressions, although single conceptual improvements were conducted during this period because of feedbacks from the single evaluations. However, the basic principle of a potential-related perspective always remained the same. Subsequent research might focus on a deeper clarification of the evaluation of the seminar's theoretical framework in the context of both mathematics education and education in different subjects. Another step could be adapting contents of the seminar to different types of learners we found in the comparative analysis.

Figure 1 :Figure 2 :

 12 Figure 1: Example of a pre-map


Oğuzhan Doğan
email: oguzhan.dogan@yeditepe.edu.tr

A farewell to mathematics: A personal choice or social exclusion?

Keywords: Mathematics education, farewell to mathematics, exclusion, social media, ekşisözlük

One of the dimensions that the socio-political understanding of mathematics education brings to mathematics education research is the study of how students are excluded from mathematics education. Social media could be a fruitful site to examine students' exclusion from mathematics. Therefore, the purpose of this study is to analyze how some social media users describe the moments of their farewell to mathematics and discuss what these moments imply to mathematics education researchers, specifically to the critical ones. Content analysis was utilized to analyze social media users' description of their own farewell moments to mathematics. From the analysis of these descriptions, it was concluded that almost all stages of the formal education system from the beginning of elementary school to the very last day of university, and almost all mathematics topics from multiplication to differential equations could be the moments of farewell to mathematics.

Introduction

Understanding mathematics education as socio-political construct has been finding increasingly more place in the mathematics education research community in the last 20 years. Straehler-Pohl, Pais, and Bohlmann (2017) portrayed this increase prominently:

Established conferences like ICME and CERME now incorporate in their programmes working groups exclusively dedicated to sociopolitical studies; the "Mathematics Education and Society" conference series has become an inherent part of the field; on a regular basis, themes like "equity", "diversity", "social justice" and "critical education" are problematized in edited collections and special issues of the most renowned journals; and one of the four sections that compound the very recent Third International Handbook of Mathematics Education is dedicated to "Social, Political and Cultural Dimensions in Mathematics Education". (p. 2) Although educational conferences, journals, edited collections or handbooks provide particular spaces for the social and political dimension of mathematics education, it is questionable whether or not such understanding reflects on educational policies and public opinion. For example, for the case of Turkey, 'equity' is still only a mathematics topic but not a policy identifier [START_REF] Doğan | Neoliberal and nationalist discourses in Turkish elementary mathematics education[END_REF].

One of the dimensions that this socio-political understanding of mathematics education brings to mathematics education research is the study of how students are excluded from mathematics education. The social, cultural, class-, gender-and race-based explanations of this exclusion are clearly presented in critical mathematics education literature [START_REF] Jablonka | Theories for studying social, political and cultural dimensions of mathematics education[END_REF][START_REF] Jorgensen | Structural exclusion through school mathematics: Using Bourdieu to understand mathematics as a social practice[END_REF][START_REF] Kollosche | Auto-exclusion in mathematics education[END_REF]. However, we still have limited knowledge of how this exclusion is explained by the excluded themselves.

In the last decade social media has become one of the important places that individuals use to communicate with each other, explain their point of view, share their ideas, discuss different perspectives and so help the construction of public opinion. With the awareness of the common popularity of social media and its growing potential, it could be fruitful to examine social media shares to understand their relations with mathematics. Considering this potential, this paper will examine a popular social media site (eksisozluk.com/sourtimes.org) in Turkey in terms of how social media users describe their exclusion from mathematics education.

Exclusion in Mathematics Education

Similar to what [START_REF] Marx | The communist manifesto[END_REF] indicated about the written history of mankind, "the history of all hitherto existing society is the history of class struggles" (p. 14), it could be argued that the history of all hitherto existing mathematics classrooms is the history of exclusion. In the first handbook of research on mathematics teaching and learning, [START_REF] Secada | Race, ethnicity, social class, language, and achievement in mathematics[END_REF] highlighted this history as such:

For over 40 years, we have been confronted with an ever-growing body of research documenting that the American educational system is differentially effective for students depending on their social class, race, ethnicity, language background, gender and other demographic characteristics. This differential effectiveness has been found in mathematics as well as in many other academic subjects. (p. 623) Following comprehensive reviews on mathematics education research [START_REF] Jablonka | Theories for studying social, political and cultural dimensions of mathematics education[END_REF][START_REF] Bishop | Issues in access and equity in mathematics education[END_REF]) also emphasized that a wide range of students (from different racial, ethnic and religious minorities, working-class students, female students and students with disabilities) had suffered from conflicts with mainstream mathematics education. It can be concluded from this consistent body of literature that disadvantaged groups of students are excluded from mathematics education on both micro level (in mathematics classrooms) and macro level (national tests, higher education entrance, job preference etc.). Mathematics has an important role in the intellectual selection, preparation and guidance of students to enter higher education studies and has used to help select those who will occupy different social positions. Thereby it serves as a critical filter. Understanding these exclusion processes can help us to understand the role of mathematics education in the reproduction of social inequalities. These exclusion processes or moments are investigated under the roof of different theoretical frameworks. Within a traditional Marxist perspective, it can be argued that working class children have fewer resources (economic opportunities, family support etc.) to be successful in mathematics and are subjected to be a member of future working-class thorough education/schools [START_REF] Bowles | Schooling in capitalist America[END_REF]. With a Žižekian contribution to this perspective, it can be argued that failure or exclusion in mathematics education is not an observed singularity that can be solved through better mathematics classrooms with hard-working mathematics teachers, but a necessary feature of existing schooling and mathematics education [START_REF] Pais | A critical approach to equity[END_REF][START_REF] Straehler-Pohl | To participate or not to participate? That is not the question![END_REF]. As Pais (2012) influentially stated "the 'slogan' or 'motto' 'mathematics for all' is used as an exemplary case for the social fantasy that is concealing the crude reality that, as any mathematics teacher knows, mathematics is not for all" (p. 58).

In addition, in a Bourdieusian sense, it can be argued that working class students or students with limited cultural baggage do not have the necessary cultural capital to 'play the game' that is involved in the learning of mathematics and are less likely to understand the mechanisms required to succeed in mathematics [START_REF] Lareau | Moments of social inclusion and exclusion: Race, class, and cultural capital in family-school relationships[END_REF][START_REF] Doğan | Neoliberal and nationalist discourses in Turkish elementary mathematics education[END_REF]. Similarly, based on Bernstein's theory of the language and codes used in educational settings, it could be argued that some context or language use in mathematics lessons can results in mathematical exclusion [START_REF] Helenius | Mathematical exclusion with the every day[END_REF].

Moreover, from a Foucauldian perspective, the concept of 'power' can be an expressive tool to understand moments of exclusion. For example, Valero (2007, p. 226) indicated that viewing power as a "capacity to participate by taking and defining the positions and conditions for engaging in social practice" offers investigation of the microphysics of power in mathematics education practices. According to her, such an analysis clarifies the mechanisms through which different actors get positioned as more or less influential participants and thereby get included and excluded.

These perspectives can provide mathematics education researchers a supplementary toolbox for understanding and portraying the processes and moments of exclusion. On the other hand, how these processes and moments were interpreted by the excluded themselves needs to be clarified. Therefore, the purpose of this study is to analyze how some social media users describe the moments of their farewell to (or exclusion from) mathematics and discuss what these moments imply to mathematics education researchers, specifically to the critical ones.

Methodology

In line with the purpose of the study, the research questions are: (i) How do social media users describe the moments of their farewell to mathematics? (ii) What are the themes that emerge as a portrayal of these farewell moments? and (iii) What are the implications of these farewell moments for mathematics education researchers?

To answer these questions, one of the most popular social media sites in Turkey, Ekşisözlük (Sour Dictionary) was analyzed. Ekşisözlük can be considered a unique website, somewhat similar to Wikipedia, in which each member of the dictionary writes their own definition/description of a specific concept, people, event etc. For example, you can find users' own definitions of lasagna, Deep Purple, Utrecht or the latest comments of political figures. In 2018, there are almost 500.000 active members writing in Ekşisözlük and it (www.eksisozluk.com) is the 15th most visited website in Turkey.

The research methodology was content analysis. The content of the users' definitions (entries) of 'the moments of farewell to mathematics' was investigated in this study. There were 945 different entries under this heading (https://eksisozluk.com/matematige-veda-edilen-an--4614088), which means that 945 different users describe their own farewell moments to mathematics. As an example, the last entry was "In the second year of university, when I passed the differential equations. I got rid of mathematics, but then I faced more terrible things".

The 945 entries under the heading 'the moments of farewell to mathematics' were retrieved from the website in June 2018 with the help of NVivo 11 (NCapture for NVivo 11). Each entry was read in the beginning of data analysis. 236 of the entries were not related with the title and did not include any meaningful definition, so they were omitted in the further analysis. After this first reading, the possible themes for the farewell moments were constructed, such as the year of the farewell and the specific reason of the farewell. Each entry was coded according to these initial themes with the help of NVivo 11, and through the analysis the themes were combined, separated or re-constructed. Finally, the data (709 entries) were re-coded with the final version of themes, subthemes and codes.

Findings

To clarify how I examined Ekşisözlük users' description of their farewell moments, I will present some of the representative entries and try to explain how I interpreted these entries. One of the entries under the heading 'the moments of farewell to mathematics' was this: Entry 1: I think, for everyone, there is a moment when they say goodbye to mathematics. Some go away when they see 'x' or 'y', some wait for the derivatives. The moment that I feel I did not understand mathematics at all was the first grade where we saw the multiplication with 7. I am still mad at this multiplication table.

As it can be seen, the user pointed out the first grade of primary school and multiplication table as the moment of farewell. S/he did not identify any specific reason for the farewell but only indicated the mathematics topic (multiplication with 7). There were a lot of similar entries indicating specific mathematics topics and grade levels, for example:

Entry 2: Math was my favorite lesson in primary school. However, when the letters came into stage I fell out of love with mathematics.

Entry 3: My body was fumbling with functions, and the last blow was trigonometry.

The writer of Entry 2 pointed out the 'letters' as the farewell moments, so it can be inferred that the mathematics topic was algebra and the time was elementary school (although there is no specific grade level, algebra is one of the main mathematical concepts of elementary schools (5 th to 8 th grade) in Turkey). Entry 3 pointed out two mathematics concepts (functions and trigonometry) from high school as the farewell moments.

In addition, some of the entries gave further reference to the users' reasons of farewell. For instance:

Entry 4: With the start of limit-derivative-integral and because of the teachers who cannot explain what these topics mean.

While the user specifically indicated the topic (calculus) and so also the grade level (High school, 12 th grade), s/he also pointed out the teachers and their lack of explanation as the reason of farewell.

On the other hand, some of the entries did not specify any mathematics topics or reasons of farewell but indicated a national exam. As an example:

Entry 5: It is the moment that you give up hope from KPSS (Exam for Civil Servant Selection).

Five main themes have emerged as a result of content analysis of the farewell moments; the grade level of the farewell, the specific mathematics topic of the farewell, the specific national exam of the farewell, the reasons of the farewell and no farewell. The number of entries that have specific reference to these main themes is presented in Table 1. When the grade level theme was examined, it can be seen that the users identify all stages of education from elementary school to university graduation as a farewell moment. While high school and university grades were the most referenced grades, a significant number of users emphasizes the farewell moments almost at the start of their education career, primary school (41; 18% of grade level references).

In addition, when the mathematics topic theme is examined, it is understood that the users have highlighted a variety of mathematics topics from multiplication table to calculus as a farewell moment. Calculus (43; 26% of mathematics topic references) and algebra (31; 18% of mathematics topic references) were the most referenced mathematics topics.

Moreover, when the reasons of the farewell are examined, two main reasons emerge: the teacher and poor grades. In addition, it should be noticed that most of the entries did not include any specific reason of farewell: only 64 entries (9%) of 709 entries pointed out a specific reason.

Lastly, 47 (7%) of the entries indicated that it is not possible to say goodbye to mathematics and so there can be no such farewell moment. Two examples were these:

Entry 6: I think there is no such moment, math is life.

Entry 7: Only death can separate us.

To sum up the findings of this content analysis, it can be argued that Ekşisözlük users portrayed their moments of farewell by drawing mostly on specific mathematics topics and grade level but do not identify socio-political reasons of these farewells. When they indicated a specific reason, it was either the teacher or their poor grades.

Conclusion and Discussion

From the analysis of the Ekşisözlük users' description, it can be concluded that almost all stages of the formal education system from the beginning of elementary school to the very last day of university, and almost all mathematics topics from multiplication to differential equations could be the moments of farewell to mathematics. It could be re-stated after these findings that the history of all hitherto existing mathematics classrooms is the history of exclusion. Additionally, when the grade level and mathematics topics are considered together, it can be concluded that multiplication table in primary school, algebra in elementary school, functions and trigonometry in high school and calculus in university emerged as the key moments of farewell to mathematics.

Furthermore, another conclusion can be drawn from what was absent in these descriptions. Although Ekşisözlük users identified (i) mathematics itself (they mostly identify specific mathematics topics as the farewell moments), or (ii) the difficulty of mathematics (poor grades as the farewell moments), or (iii) teachers as the reasons their farewell, they did not give any reference to social, cultural, political, gender-or class-based reasons for their farewell. While identifying the reasons of their farewell, social media users did not cross the wall of mathematics classrooms, therefore they oscillated between their teachers and themselves.

From this absence of socio-political reasons among the reasons of farewell, it can be inferred that although understanding mathematics education as socio-political construct finds increasingly more space in the mathematics education research community, it seems that this understanding could not get across through the walls of mathematics education research to reach public opinion. While we, as (critical) researchers in mathematics education, examine and emphasize that one of the most important variables affecting mathematics education is socio-economic status, that gender is both a factor in achieving mathematics and in selecting mathematics-based studies/jobs, that cultural capital is responsible for mathematics scores, our students mostly emphasize the individual or mathematics-specific factors as the reason of their farewell to mathematics.

Following these conclusions, the implications for mathematics educators and researchers are twofold; firstly, there are strong personal reasons for farewell to mathematics, such as mathematics being perceived as difficult and unachievable (reference to poor grades), complex and meaningless (referenced the specific mathematics concepts). Therefore, the inside of the classroom should be organized to make mathematics more meaningful and achievable for students. Additionally, as [START_REF] Kollosche | Auto-exclusion in mathematics education[END_REF] emphasized "auto-exclusion is not merely the result of psychological dispositions of the individual, which could then be changed by pedagogical intervention, but that auto-exclusion is created in the interplay of the individual and the social environment" (p. 2). So, counter-acting auto-exclusion from mathematics is as much an issue of changing the social environment outside the classroom as it is an issue of changing the mathematics classroom itself.

Secondly, social, cultural, political and gender-and class-based factors that divorce students from mathematics need to get more reference in the public sphere. In spite of the socio-political turn within mathematics education, specific working groups in educational conferences and special issues in high stake journals, it could be argued that the social, cultural, political and class-based exclusion in mathematics education is not publicly echoed adequately. Therefore, as critical mathematics educators, it would be better to demand more to get more public spaces so as to not confine ourselves to special issues or handbook chapters.

Table 1 . The number of entries that have specific reference to main themes

 1

	Theme	Code


Daniel Heldt
email: dheldt@math.tu-berlin.de

Kolja Knauer
email: knauer@math.tu-berlin.de

Torsten Ueckerdt
email: ueckerdt@googlemail.com

On the bend-number of planar and outerplanar graphs

The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.

Introduction

In 2007 Golumbic, Lipshteyn and Stern defined an EPG 3 representation of a simple graph G as an assignment of paths in the rectangular plane grid to the vertices of G, such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid edge, see [START_REF]Edge intersection graphs of single bend paths on a grid[END_REF]. EPG representations arise from VLSI grid layout problems [START_REF] Brady | Stretching a knock-knee layout for multilayer wiring[END_REF] and as generalizations of edge-intersection graphs of paths on degree 4 trees [START_REF] Golumbic | Representing edge intersection graphs of paths on degree 4 trees[END_REF]. In the same paper Golumbic et al. show that every graph has an EPG representation and propose to restrict the number of bends per path in the representation. There has been some work related to this, see [START_REF] Asinowski | Edge intersection graphs of systems of paths on a grid with a bounded number of bends[END_REF][START_REF] Biedl | On edge-intersection graphs of k-bend paths in grids[END_REF][START_REF]Edge intersection graphs of single bend paths on a grid[END_REF][START_REF] Heldt | Edge-intersection graphs of grid paths -the bend-number[END_REF][START_REF] Ries | Some properties of edge intersection graphs of single bend path on a grid[END_REF]. A graph is a k-bend graph if it has an EPG representation, where each path has at most k bends. The bend-number b(G) of G is the minimum k, such that G is a k-bend graph. Note that the class of 0-bend graphs coincides with the well-known class of interval graphs, i.e., intersection graphs of intervals on a real line. It is thus natural to view k-bend graphs as an extension of the concept of interval graphs and the bend-number as a measure of how far a graph is from being an intervalgraph. Intersection graphs of systems of intervals together with a parameter counting how many intervals are needed to represent a vertex of a given graph G have received some attention. Popular examples are the interval-number, see Harary and Trotter [START_REF] Harary | On double and multiple interval graphs[END_REF] and the track-number, see Gyárfás and West [START_REF] Gyárfás | Multitrack interval graphs[END_REF]. Extremal questions for these parameters like 'What is the maximum interval-/track-number among all graphs of a particular graph class?' have been of strong interest in the literature. Scheinermann and West [START_REF] Scheinerman | The interval number of a planar graph: three intervals suffice[END_REF] show that the maximum interval-number of outerplanar graphs is 2 and of planar graphs is 3. Kostochka and West [START_REF] Alexandr | Every outerplanar graph is the union of two interval graphs[END_REF] prove that the maximum track-number of outerplanar graphs is 2 and Gonçalves and Ochem [START_REF] Gonçalves | On some arboricities in planar graphs[END_REF] prove that the maximum track-number of planar graphs is 4. In [START_REF] Biedl | On edge-intersection graphs of k-bend paths in grids[END_REF] Biedl and Stern show that outerplanar graphs are 3-bend graphs and provide an outerplanar graph which has bend-number 2, see Fig. 1. They conjecture that all outerplanar graphs are 2-bend graphs. We confirm this conjecture in Theorem 1, showing the stronger result that graphs of treewidth at most 2 are 2-bend graphs. The major part of this paper is devoted to planar graphs. Biedl and Stern [START_REF] Biedl | On edge-intersection graphs of k-bend paths in grids[END_REF] show that planar graphs are 5-bend graphs but the only lower bound that they have is 2, given by the graph of Fig. 1. In Proposition 1 we provide a planar graph of treewidth 3 which has bend-number 3, thus improving the lower bound of the class of planar graphs by one. Indeed in Theorem 2 we show that every planar graph with treewidth 3 is a 3-bend graph. The main result of this article is Theorem 3. We improve the upper bound for the bend-number of general planar graphs from 5 to 4.

Preliminaries

We consider simple undirected graphs G with vertex set V (G) as well as edge set E(G). An EPG representation is a set of finite paths {P (u) | u ∈ V (G)}, which consist of consecutive edges of the rectangular grid in the plane. These paths are not self-intersecting (in grid edges), and P (u) ∩ P (v) = ∅ if and only if {u, v} ∈ E(G), i.e., only intersections at grid edges are considered. A bend of P (u) is either a horizontal grid edge followed by a vertical one or a vertical edge followed by a horizontal one in P (u). Let us now introduce some terminology: The grid edges between two consecutive bends (or the first (last) bend and the start (end) of P (u)) are called segments. So a k-bend path consists of k + 1 segments, each of which is either horizontal or vertical. A sub-segment is a connected subset of a segment. In an EPG rep-resentation a set of sub-segments (eventually from different segments) is called a part. Furthermore, a vertex u is displayed, if there is at least one grid edge, which is exclusively in P (u) and in no other path. We then say that this grid edge is a private edge of P (u). An edge {u, v} ∈ E is displayed, if there is at least one grid edge in P (u) ∩ P (v), which is only contained in P (u) ∩ P (v) and not element of any other path. This grid edge is then called a private edge of P (u) ∩ P (v). We also say that a part (or sub-segment) displays the corresponding vertex or edge if it consists only of grid edges which are private edges of the respective vertex or edge. Finally, two horizontal sub-segments in an EPG representation see each other if there is a vertical grid line crossing both sub-segments. Similarly, two vertical sub-segments see each other if there is a horizontal grid line crossing both.

EPG representations of graphs in terms of treewidth

In this section we consider graphs of bounded treewidth. Therefore we denote the treewidth of a graph G with tw(G) and make use of the fact, that every graph G with tw(G) ≤ k is a subgraph of a k-tree as well as the fact, that every k-tree admits a construction sequence, starting with a (k + 1)-clique and iteratively stacking a new vertices into k-cliques. For further definitions and more about construction sequences of graphs of bounded treewidth we refer to [START_REF] Hans | A partial k-arboretum of graphs with bounded treewidth, Theoret[END_REF].

Theorem 1. For every graph G with tw(G) ≤ 2 we have b(G) ≤ 2.
Proof. Let G be the 2-tree which contains G and (v 1 , . . . , v n ) be a vertex ordering of G implied by G's construction sequence. We construct a 2-bend representation of G along the building sequence G 2 ⊂ . . . ⊂ G n = G, where we add vertex v i to G i , such that the two neighbors of v i in Gi form a 2-clique in Gi (and not necessarily in G i ). We maintain that Γ i is a 2-bend representation of G i , such that every 2-clique in Gi satisfies one of the two invariants in Fig. 2 a), i.e., for i = 2, . . . , n and {u, v} ∈ E( Gi ) we have a sub-segment p u ⊆ P (u) and a part p v ⊆ P (v) such that one of the following sets of conditions holds: (i) p v is a sub-segment, p u and p v see each other, p u displays u, and p v displays v as depicted in the top row of Fig. 2 a), (ii) p v consists of two consecutive sub-segments p v,1 , p v,2 such that there is a bend between them, p v,2 displays v, p u \p v,1 displays u, and p u ∩p v,1 displays {u, v} as depicted in the bottom row of Fig. 2 a).

The two possible starting representations for G 2 are shown in Fig. 2 a). Now for i ≥ 2, let Γ i be a 2-bend representation of G i that satisfies our invariant.

Let {u, v} be the 2-clique that is the neighborhood of v i in Gi . If v i has an edge in G i+1 with both, u and v, we introduce the path for v i as illustrated in Fig. 2 b) depending on the type of the invariant for {u, v}. The parts, displaying the new 2-cliques {u, v i } and {v, v i } in Gi+1 , are highlighted in dark gray and the part, which displays {u, v}, is highlighted in light gray. If v i has an edge in G i+1 with u and no edge in G i+1 with v, we introduce the path for v i as illustrated in Fig. 2 c) depending on the type of invariant we have. Again the parts, which display the new 2-cliques {u, v i } and {v, v i }, and the old 2-clique {u, v} in Gi+1 are highlighted in dark gray, and light gray, respectively. If v i has an edge in G i+1 with v and no edge in G i+1 with u, the roles of u and v are basically exchanged. In case, v i neither has an edge in G i+1 with u nor with v, we introduce the path for v i as illustrated in Fig. 2 d) depending on the type of type of invariant for {u, v}.

⊓ ⊔

Since every outerplanar graph G has a treewidth ≤ 2 we obtain the following corollary.

Corollary 1. For every outerplanar graph G we have b(G) ≤ 2.

This confirms a conjecture of Biedl and Stern [START_REF] Biedl | On edge-intersection graphs of k-bend paths in grids[END_REF], who show that the graph in Fig. 1 has no 1-bend representation. Therefore this bound cannot be further improved.

Theorem 2. For every planar graph G with tw(G) ≤ 3 we have b(G) ≤ 3.

Proof. By a result of El-Mallah and Colbourn [START_REF] Ehab | On two dual classes of planar graphs[END_REF] G is a subgraph of a plane 3-tree G. So there is a vertex ordering (v 1 , . . . , v n ), such that G3 is a triangle {v 1 , v 2 , v 3 } and Gi is obtained from Gi-1 by connecting v i to the three vertices u, v, w of a triangle bounding an inner face of Gi-1 . The triangle {u, v, w} is then not bounding an inner face of Gi anymore and hence no second vertex may be attached to it. We build a 3-bend representation of G concurrently with the building sequence

G 3 ⊂ • • • ⊂ G n of G w.r.t. the vertex ordering (v 1 , . . . , v n ).
We maintain the following invariant on the 3-bend representation Γ i of G i , for i ≥ 3:

(a) Every vertex u ∈ Γ i has a horizontal and a vertical sub-segment displaying u, and (b) every facial triangle {u, v, w} of Gi has two vertices, say u and v, of one of the following two types: (i) there is a sub-segment, which displays the edge {u, v} as in the left example in Fig. 3 a), (ii) there is an entire segment s of P (v) displaying v and a sub-segment of P (u) displaying u and crossing s, see Fig. 3 a).

Moreover, we require that all the displaying parts above are pairwise disjoint and that the entire segment in every cross in (ii) cannot see displaying parts from (a). However, we need this assumption only in the case in Fig. 3 c).

PSfrag replacements It is not difficult to find a 3-bend representation Γ 3 of the subgraph G 3 of G, which satisfies invariant (a) and (b). Indeed, by introducing three artificial outer vertices we can assume w.l.o.g. that G 3 is an independent set of size three. For i ≥ 4, the path for vertex v i is introduced to Γ i-1 according to the degree of v i in G i and the type of invariant for the facial triangle {u, v, w} in Gi-1 that v i is connected to. Fig. 3 b)-f ) shows all five cases and how to introduce v i , which is illustrated by the bold path. Consider in particular the case that v i has an edge with each of u, v, and w, and moreover the triangle {u, v, w} has a cross, see Fig. 3 c). Here we use that the entire v-segment cannot see the partial w-segment in order to get a horizontal displaying sub-segment for the new path corresponding to v i .

u u u u u v v v v v v v v w w w w w vi vi vi vi vi a) invariants for {u, v} of type (i) and (ii) b) deg G i (vi) = 3, {u, v} of first type c) deg G i (vi) = 3, {u, v} of second type d) deg G i (vi) = 2 e) deg G i (vi) = 1 f ) deg G i (vi) =
In the figure the parts from invariant (b) displaying edges of the new facial triangles {u, v, v i }, {u, v i , w} and {v i , v, w} in Gi are highlighted in dark gray. Additionally, every path, including the new path for v i , has a horizontal and a vertical sub-segment, which displays them. Moreover they can be chosen such that they do not see any entire segment of a cross. For example, consider the case that v i is not adjacent to u, v, or w in G i , see Fig. 3

(G) = 3.
The graph G is depicted in the left of Fig. 4 and it is constructed in the following way: Two vertices u and v together with 50 white vertices form an induced K 2,50 subgraph, i.e., a complete bipartite graph on 2 + 50 vertices. Any two consecutive white vertices together with 50 black vertices form another induced K 2,50 . Finally, between any two consecutive black vertices the graph H is suspended, which is the 29-vertex graph depicted in the right of Fig. 4. This graph is clearly planar. Furthermore one can find a planar 3-tree containing G, i.e., G is planar and has treewidth at most 3.

PSfrag replacements Now we will show that 2 bends do not suffice for an EPG representation of G.

G = H = H H
Proof. It is known [1] that b(K 2,n ) = 2 if n ≥ 5. Hence b(G) ≥ 2.
For the sake of contradiction let us assume that b(G) = 2 and consider a 2-bend representation of G together with the induced 2-bend representations of all induced K 2,50 subgraphs. Furthermore, it is known [START_REF] Biedl | On edge-intersection graphs of k-bend paths in grids[END_REF] that in any EPG representation of K 2,n at most 12 vertices of the n-bipartition class represent their two edges using the same or consecutive segments. Moreover the 2-bend paths of u and v together have at most 12 endpoints of segments. Thus at most another 12 white vertices contain an endpoint of u or v in the interior of a segment. Hence there are two consecutive white vertices, say u ′ and v ′ , that establish their edges with u and v with two distinct segments of the same direction, say vertically. Both these segments are completely contained in the corresponding segment of u and v.

Now the same argument can be applied to the K 2,50 subgraph induced by u ′ , v ′ , and 50 black vertices. Thus there are two consecutive black vertices, which we denote by x and y, that establish their two edges with u ′ and v ′ with two distinct segments of the same direction, each completely contained in the corresponding segment of u ′ and v ′ . Moreover, x and y intersect u ′ and v ′ on the same segment of u ′ and v ′ , respectively. We conclude that the paths of u ′ , v ′ , x, and y are positioned as the black bold paths in Fig. 5. It follows that the horizontal segment of g within the x-segment is completely covered by a 2 and a 3 and hence lies strictly between a 1 and a 4 . In consequence, the six white vertices that are adjacent to x and g, but not to a 2 or a 3 , intersect g on its second horizontal segment. Let us denote the set of eight white vertices, which are common neighbors of g and x by W and the horizontal line supporting at least six edges between g and W by ℓ. Evidently, the segment of g on ℓ crosses only one of {a 1 , a 4 }, but not both, say it does not cross a 4 , see Fig. 5.

PSfrag replacements u ′ v ′
The vertex in W that is adjacent to a 2 , respectively a 3 , is denoted by w 2 , respectively w 3 . One can check that for i = 1, 2 the edge between w i and its white neighbor is established on ℓ and moreover that w 2 lies to the left of w 3 on ℓ. This implies that the white vertices inside the triangle {x, g, w 3 } intersect the horizontal x-segment to the right of a 4 . In order to establish their adjacency with g all these three paths contain the part of ℓ between a 3 and a 4 , making them pairwise adjacent, which in G they are not -a contradiction. ⊓ ⊔

4-bend representation for planar graphs

We show that the bend-number of every planar graph is at most 4, improving the recent upper bound of 5 due to Biedl and Stern [START_REF] Biedl | On edge-intersection graphs of k-bend paths in grids[END_REF]. Our proof is constructive and can indeed be seen as a linear-time algorithm to find a 4-bend representation of any given planar graph. We use the folklore fact that every plane triangulation can be constructed from a triangle by successively glueing 4-connected triangulations into inner faces. In every step our algorithm constructs a 4-bend representation Γ ′ of a 4-connected triangulation G ′ and incorporates it into the already defined representation. The construction of Γ ′ is based on a wellknown representation of subgraphs of 4-connected triangulations by touching axis-aligned rectangles. These are the basic steps of the algorithm:

1.) Fix some plane embedding of the given planar graph and add one vertex into each face to obtain a super-graph G that is a triangulation. If we find a 4-bend representation for G, removing the paths from it that correspond to added vertices, results in a 4-bend representation of the original graph. 2.) Construct a 4-bend representation of the outer triangle of G so that invariant I, presented ahead, is satisfied. 3.) Let Γ ′ and G ′ denote the so far defined 4-bend representation and the graph that is represented, respectively. The graph G ′ will always be a plane triangulation, which is an induced subgraph of G. We repeat the following two steps 4.) and 5.) until we end up with a 4-bend representation Γ of the entire triangulation G. 4.) Consider a triangle ∆ which is an inner face of G ′ , but not of G. Let G ∆ be the unique 4-connected triangulated subgraph of G, which contains ∆ and at least one vertex lying inside of it. (No vertex in G ∆ , except those in ∆, is represented in Γ ′ .) 5.) Construct a 4-bend path for every vertex in G ∆ \∆ and add it to the representation Γ ′ so that all edges of G ∆ are properly represented, i.e., Γ ′ is a 4-bend representation of G ′ ∪ G ∆ . Additionally, we ensure that our invariant is satisfied for every inner facial triangle of G ′ ∪ G ∆ .

The algorithm described above computes a 4-bend representation for every given planar graph. Moreover, steps 1.) to 3.) can easily be executed in linear time. For an efficient implementation of step 4.) we first identify all separating triangles of G, which can be in linear time [START_REF] Itai | Finding a minimum circuit in a graph[END_REF]. The graph G ∆ then arises from G by removing all vertices in the exterior of the triangle ∆ and all vertices in the interior of every remaining separating triangle different from ∆. This can for example be done in

O(|V (G ∆ )|) if G is stored with adjacency lists.
The crucial part is step 5.), i.e., the construction of a 4-bend representation of the 4-connected triangulation G ∆ . Our construction relies on a well-known geometric representation of proper subgraphs of 4-connected plane triangulations. Actually, we need the slight strengthening in Lemma 1 below. A plane non-separating neartriangulation, NST for short, is a plane graph on at least four vertices without separating triangles such that all inner faces are triangles and the outer face is a quadrangle. For example, if G is a plane 4-connected triangulation and e = (u, w) an outer edge of G, then G\(u, w) is an NST. Additionally, for every v ∈ V there is a vertical segment t v from the bottom to the top side of R(v), such that the following holds:

(d) The segment t v lies to the right of t w for every v ∈ V and R(w) touching the top side of R(v).

Proof. It is well-known that every NST has a representation with touching rectangles, i.e., items (a)-(c) hold [START_REF] Koźmiński | Rectangular dual of planar graphs[END_REF][START_REF] Lai | An algorithm for building rectangular floorplans[END_REF][START_REF] Ungar | On diagrams representing maps[END_REF]. Moreover, those rectangles can be computed in linear time. We take such a rectangle representation and apply two modifications to it as illustrated in the top and bottom of Fig. 6, respectively. The modifications operate on the incidence relation of segments in the representation, that is, the inclusion-maximal horizontal and vertical lines. However, what follows is a more easily interpreted graphic image of the situation.

For the first modification we define for every inner rectangle R(v) a line ℓ, like the dashed one in Fig. 6. To be precise, ℓ consists of two vertical rays, one starting at the contact between R(v) and the leftmost rectangle touching the bottom side of R(v), and one starting at the contact between R(v) and the rightmost rectangle touching the top side of R(v). Those vertical rays are connected inside R(v), e.g., by a straight line segment. Then the plane is cut along ℓ and pulled apart to the left and right while extending every rectangle that has been cut by ℓ. As soon as the lower ray in the right part lies to the right of the upper ray in the left part we define the segment t v to lie between those two rays.

Having applied this modification to every inner rectangle, each segment t v is spanned between the rightmost rectangle touching the top of R(v) and the leftmost rectangle touching the bottom of R(v), for every v ∈ V .

PSfrag replacements For the second transformation another line ℓ ′ is defined for every pair of a rectangle R(v) and its rightmost rectangle R(w) touching the top side of R(v). Again ℓ ′ contains two rays which both start in R(v) ∩ R(w). The ray emerging downwards starts to the left of t v and the ray going upwards starts to the right of t w . Both rays are connected by a straight line segment contained in R(v) ∩ R(w).

R(v) R(v) R(v) R(v) R(w) R(w) tv ℓ ℓ ′
Similarly to the first modification, the plane is cut along ℓ ′ and pulled apart (extending every rectangle that is cut by ℓ ′ ) until t w lies to the left of t v .

Having applied this second modification to every pair of a rectangle R(v) and its right-most top neighbor R(w), condition (d) in Lemma 1 is fulfilled. Moreover, operating on the combinatorics of segments only, each modification can be done in constant time. Hence we obtain a linear-time algorithm, which completes the proof.

⊓ ⊔

Remark 1. There is an alternative proof of Lemma 1 that, however, involves some concepts. It is known that rectangle representations are in bijection with so-called transversal structures [START_REF] Fusy | Combinatoire des cartes planaires et applications algorithmiques[END_REF]. Given a transversal structure, one can split a vertex v into two adjacent vertices v ′ and v ′′ , distribute the neighbors of v among v ′ and v ′′ to have a triangulation again, and carry over the transversal structure to the new graph. And all this can be done in such a way that in the rectangle representation corresponding to the new transversal structure the right side of R(v ′ ) coincides with the left side of R(v ′′ ). In other words, R(v ′ ) ∪ R(v ′′ ) is a rectangle with a vertical segment t v . This attempt directly gives a lineartime algorithm to compute a rectangle representation satisfying items (a)-(d) in Lemma 1.

We will refer to a set of rectangles satisfying (a)-(d) in Lemma 1 as a rectangle representation of G. Rectangle representations are also known as rectangular duals. It is easily seen that every inner (triangular) face ∆ = {u, v, w} of G corresponds to a unique point in the plane given by R(u) ∩ R(v) ∩ R(w), i.e., the common intersection of the three corresponding rectangles. Now we describe our invariant mentioned in steps 2.) and 5.) above.

Invariant I: Let G ′ be a plane triangulation. A 4-bend representation Γ ′ of G ′ is said to satisfy invariant I if there are mutually disjoint regions, i.e., polygonal parts of the plane, associated with the inner facial triangles of G ′ . For each inner facial triangle ∆ = {u, v, w} we require that inside the region for ∆ there are segments s u , s v , s w , and s uv displaying u, v, w, and the edge {u, v}, respectively, such that (up to grid-symmetry) each of the following holds. See Fig. 7 for an illustration. The regions are illustrated by the dashed line in Fig. 7. It is important that the regions in Fig. 7 may appear reflected and/or rotated by 90 degree.

Our algorithm to find a 4-bend representation of a planar graph executes steps 1.) to 5.) as described above. It is not difficult to find a 4-bend representation Γ ′ of a triangle in step 2.) which satisfies invariant I, see Fig. 8. Hence we only have to take care of step 5.), which boils down to the following.

Lemma 2. Let {P (u), P (v), P (w)} be a 4-bend representation of a triangle ∆ = {u, v, w} which satisfies invariant I, and G be a 4-connected triangulation whose outer triangle is ∆. Then {P (u), P (v), P (w)} can be extended to a 4-bend representation Γ of G which satisfies invariant I, such that every new 4-bend path, as well as the region for every inner face of G, lies inside the region for ∆.

Moreover, such a representation can be found in linear time.

Proof. For convenience we rotate and/or flip the entire representation such that the region for ∆ is similar to one of Fig. 7, i.e., s u and s uv are the leftmost segments and s uv lies above s u . Define G ′ = G \ {u, w} if the region for ∆ is of type (i), and G ′ = G \ {v, w} if the region for ∆ is of type (ii). By Lemma 1 G ′ can be represented by rectangles so that conditions (a)-(d) are met. We put this rectangle representation into the part of the region for ∆ which can be seen by each of s u , s v , and s w . This part is highlighted in Fig. 7. Now we explain how to replace the rectangle by a path for every vertex i = u, v, w in G ′ . We start with a snake-like 4-bend path P (i) within the rectangle corresponding to i. To be precise, P (i) starts at the bottom-left corner of R(i), goes up to the top-left corner, where it bends and goes right up to the segment t i , which is defined in Lemma 1. Then P (i) follows along t i down to the bottom side of R(i), where it bends and goes to the right up to the bottom-right corner and then up to the top-right corner. Afterwards, every path P (i) is shortened at both ends by some amount small enough that no edge-intersection is lost. See Fig. 9 for an illustration. Note that if R(j) touches the top side of R(i), then the paths P (i) and P (j) do have an edge-intersection because t i lies to the right of t j .

PSfrag replacements ti P (i) Fig. 9. The 4-bend snake-like path P (i) within the rectangle R(i) corresponding to i.

It is not difficult to see, that these snake-like paths form a 4-bend representation of G ′ \{u, v, w}. The remaining edges in G ′ , i.e., those incident to u, v or w, are established by extending the paths corresponding to the neighbors of u, v and w in G ′ . What follows is illustrated with an example in Fig. 10. Let x be the unique 4 neighbor of u and v in G. For every neighbor i = x of v the upper horizontal segment of P (i) is shifted up onto the segment s v (extending the left and the middle vertical segment of P (i)). Similarly, for every neighbor i = x of u the leftmost vertical segment of P (i) is shifted to the left onto the segment s u (extending the upper horizontal segment of P (i)). For the case i = x, the upper horizontal segment of P (x) is shifted up onto some horizontal line through s uv . Then the leftmost vertical segment of P (x) is shifted to the left onto the segment s uv , and shortened such that it is completely contained in s uv . The paths of neighbors of w are extended according to the type of the region for ∆. If the region is of type (i), the rightmost vertical segment of P (i) for every neighbor i of w is shifted to the right onto s w . If the region is of type (ii), the lower horizontal segment of P (i) is shifted down onto s w . Let us again refer to Fig. 10 for an illustration. Including the already given paths P (u), P (v), and P (w) gives a 4-bend representation Γ of G. It is easy to see that Γ is indeed representing the graph G and that the above construction can be done in linear time.

It remains to show that Γ satisfies invariant I. We have to identify a region for every inner facial triangle of G, such that each region lies within the region for ∆ and all regions are pairwise disjoint. Therefore let ∆ ′ = {u ′ , v ′ , w ′ } be such an inner facial triangle. First, assume that none of {u ′ , v ′ , w ′ } is a vertex of ∆. This is, ∆ ′ consists of inner vertices only. Then the intersection R(u

′ ) ∩ R(v ′ ) ∩ R(w ′ )
of the corresponding rectangles is a single point in the region for ∆. Moreover, exactly one of the three rectangles has its bottom-left or top-right corner at this point. Hence, the paths P (u ′ ), P (v ′ ) and P (w ′ ) locally look like in one of the cases in Fig. 11. In the figure, for each case a region for ∆ ′ = {u ′ , v ′ , w ′ } is highlighted. Note that these regions can be chosen to be pairwise disjoint. PSfrag replacements

P (u ′ ) P (u ′ ) P (u ′ ) P (u ′ ) P (v ′ ) P (v ′ ) P (v ′ ) P (v ′ ) P (w ′ ) P (w ′ ) P (w ′ ) P (w ′ )
Fig. 11. The four possibilities for an inner face ∆ ′ = {u ′ , v ′ , w ′ } of G with at most one vertex from ∆. The region for ∆ ′ is highlighted.

Similarly, if exactly one of {u, v, w} is a vertex in ∆ ′ , then there is a point on the corresponding segment (s u , s v or s w ) where the paths P (u ′ ), P (v ′ ) and P (w ′ ) locally look like in the previous case. Hence in this case we find a region for ∆ ′ , too. Finally, exactly three inner facial triangles in G contain exactly two outer vertices. The regions for these three faces are defined as highlighted in Fig. 10. More formally, the region for the triangle {u, v, x} contains the segments s ′ u , s ′ v , s ′ x , and s ′ uv which are contained in s u , s v , the middle vertical segment of P (x), and s uv , respectively. The region for the triangle {u, w, y} contains the segments s ′ u , s ′ w , s ′ y , and s ′ wy , which are contained in s u , s w , the lower horizontal (type (i)) or the middle vertical (type (ii)) segment of P (y), and P (y) ∩ s w , respectively. Finally, the region for the triangle {v, w, z} contains the segments s ′ v , s ′ w , s ′ z , and s ′ wz , which are contained in s v , s w , the lower horizontal (type (i)) or the rightmost vertical (type (ii)) segment of P (z), and P (z) ∩ s w , respectively. All these regions are pairwise disjoint and completely contained in the region for ∆. We remark that even in case5 x = y = z the above definitions of 4-bend paths and regions meet the required properties, see Fig. Since G does not contain separating triangles, x, y, and z either all coincide or are pairwise distinct, which completes the proof of the lemma.

⊓ ⊔

Lemma 2 gives a construction as required in step 5.) of our algorithm. Hence, we have proved our main theorem.

Theorem 3. The bend-number of a planar graph is at most 4. Moreover, a 4-bend representation can be found in linear time.

Putting Theorem 3 and Proposition 1 together we have shown the following.

Theorem 4. In an EPG representation of a planar graph 4-bend paths are always sufficient and 3-bend paths are sometimes necessary, i.e., the maximum bend-number among all planar graphs is 3 or 4.

Conclusions

Although we could raise the previously known lower and upper bound, it remains open to determine the maximum bend-number of planar graphs.

Conjecture 1. There is a planar graph G, such that every EPG representation of G contains at least one path with four bends.

Another interesting class of representations is as edge-intersection graphs of arbitrary polygonal paths. It is straight-forward to obtain a 4-bend representation from a representation of touching ⊤s and ⊥s [START_REF] Hubert De Fraysseix | On triangle contact graphs[END_REF]. What is the right answer?

Fig. 1 .

 1 Fig. 1. A 2-bend graph and an EPG representation. If a grid edge is shared by several paths we draw them close to each other.

Fig. 2 .

 2 Fig. 2. Invariants for a 2-clique and insertion rules for the path for the new vertex vi (drawn bold). a) pu and pv for the two types of the invariant. b) Vertex vi has an edge with both, u and v. c): Vertex vi has an edge with u and no edge with v. d): Vertex vi has no edge with u or v.

Fig. 3 .

 3 Fig.3. Building a 3-bend representation of a planar graph with tree-width 3, a vertex vi is attached to the facial triangle {u, v, w} in Gi-1 : In a) the two types of invariant for {u, v} are shown. In b)-f ) it is shown how to insert the new vertex vi (drawn bold) depending on its degree in Gi and the invariant of {u, v}. The invariants for the three new facial triangles {u, v, vi}, {u, vi, w}, and {vi, v, w} in Gi are highlighted.

Fig. 4 .

 4 Fig. 4. A planar graph G with tw(G) = 3 and b(G) = 3.

Fig. 5 .

 5 Fig. 5. A detail of a hypothetical 2-bend representation of G.

Lemma 1 .

 1 Let G = (V, E) be an NST. Then G can be represented as follows: (a) There is an axis-aligned rectangle R(v) for every v ∈ V . (b) Any two rectangles are either disjoint or intersect on a line segment of non-zero length. (c) The rectangles R(v) and R(w) have a non-empty intersection if and only if (v, w) is an edge in G.

Fig. 6 .

 6 Fig. 6. Top: The modification at an inner rectangle and the positioning of the segment tv. Bottom: The modification at a pair R(v), R(w), with R(w) being the rightmost rectangle touching the top of R(v).

( a )Fig. 7 .

 a7 Fig.7. The regions of type (i) and type (ii) for a triangle ∆ = {u, v, w} (up to gridsymmetry). The part of the region that can be seen by each of su, sv, and sw is highlighted.

Fig. 8 .

 8 Fig. 8. An EPG representation of a triangle ∆ = {u, v, w} satisfying invariant I of type (i).

Fig. 10 .

 10 Fig. 10. Examples of the 4-bend representation Γ contained in the region for ∆. The new regions of the three facial triangles that contain two vertices from ∆ are highlighted.

12

 12

  .

Fig. 12 .

 12 Fig. 12. The case x = y = z for a type (i) region and a type (ii) region in the proof of Lemma 2.

  f ). Here the new facial triangles {u, v i , w} and {v i , v, w} have a cross of an entire segment of P (v i ) and a sub-segment displaying w and v, respectively. The new facial triangle {u, v, v i } has the edge {u, v}, which still satisfies both parts of the invariant.

⊓ ⊔ We conclude this section showing that the bound in Theorem 2 is tight. This confirms what Biedl and Stern strongly suspected in [2]. Proposition 1. There is a planar graph G with tw(G) = 3 and b

  Now consider the subgraph H which is suspended between x and y. W.l.o.g. we assume that the segments which display x and y are horizontal. The graph H contains, aside from x and y, a set A of four black vertices denoted by a 1 , a 2 , a 3 , and a 4 , and a set B of six gray vertices, see the right of Fig.4. Every vertex in A ∪ B is adjacent to x or y, but neither to u ′ nor to v ′ . Hence each edge of such a vertex with x or y is established horizontally. From the adjacencies between vertices in A ∪ B it follows that the vertical segments of a 1 , a 2 , a 3 , and a 4 appear in order -w.l.o.g. from left to right. Moreover the edge {a 2 , a 3 } is not established vertically since a 2 has a private neighbor. W.l.o.g. assume that {a 2 , a 3 } is established within the x-segment, see Fig.5. Let us denote the gray vertex adjacent to a 2 , a 3 , and x by g.

This happens if and only if G = K4.