Alicia Sánchez
email: asanchezb@ub.edu

Vicenç Font
email: vfont@ub.edu

Adriana Breda
email: adriana.breda@gmail.com

Secondary school preservice teachers' references to the promotion of creativity in their master's degree final projects

Keywords: Preservice teachers, creativity, master's degree final project, didactic suitability

What do preservice teachers think about creativity and the ways to promote it in their classes? In this study, we analyze the references to creativity that appear in the master's degree final projects of a group of secondary school preservice teachers of mathematics. The projects hold the preservice teachers' reflections on their own practice, but it is important to note that the master's degree does not include any specific training in creativity. In our analysis, we first searched for explicit references to creativity in the projects and registered them. Then, we classified the comments depending on the elements of the teaching and learning process that are related to creativity in each case. We observe a good variety of comments and more than half of the preservice teachers mention creativity, although some of them just do it superficially.

Introduction

In recent years, creativity has become an important focus of interest of social sciences. Creativity could be related to the development of other abilities such as critical thinking, problem solving, communication and the use of new technology, current challenges in the globalized society [START_REF] Pásztor | Technology-based assessment of creativity in educational context: the case of divergent thinking and its relation to mathematical achievement[END_REF]. In education, creativity has also gained importance and its development is a goal in modern curricula [START_REF] Pásztor | Technology-based assessment of creativity in educational context: the case of divergent thinking and its relation to mathematical achievement[END_REF]. In addition, many authors [START_REF] Mann | Creativity: the essence of mathematics[END_REF][START_REF] Silver | Fostering creativity through instruction rich in mathematical problem solving and problem posing[END_REF][START_REF] Sriraman | Are giftedness and creativity synonyms in mathematics[END_REF] claim that creativity should be fostered at school by inclusive activities, working with every student and not only with those considered gifted students. However, mathematics lessons at school are commonly associated with repetitive procedures and regardless of students' creativity.

Research on the promotion of students' creativity by improving the teachers' training [START_REF] Hosseini | The effect of a teacher professional development in facilitating students' creativity[END_REF][START_REF] Panaoura | Teachers' awareness of creativity in mathematical teaching and their practice[END_REF] and research on teachers' conceptions of creativity [START_REF] Lev-Zamir | Creative mathematics teaching in the eye of the beholder: focusing on teachers' conceptions[END_REF] disregards the following question: how do the teachers consider the promotion of students' creativity in their learning sequences, when they do not receive specific training in this subject? In order to answer this question, in this study, we focus on how preservice teachers conceive the promotion of students' creativity when they analyze their own practice and propose some changes to improve it, in their master's degree final projects.

Theoretical framework

There are several definitions and conceptualizations of creativity [START_REF] Kampylis | Redefining creativity -Analyzing definitions, collocations, and consequences[END_REF][START_REF] Kaufman | The International Handbook of Creativity[END_REF]. [START_REF] Kaufman | Beyond big and little: the four C model of creativity[END_REF] propose a model with different types of creativity: Big-C, which refers to eminent people whose creations have had an impact on a field; little-c, which is associated with everyday creativity which appears in daily activities; Pro-C, which is present in those professional activities that are generally considered specially creative; and minic, which they define as "the novel and personally meaningful interpretation of experiences, actions, and events" (Beghetto & Kaufman, 2007, p.73). Mini-c creativity is related to learning processes, since it implies a personal interpretation of new facts based on previous experiences and knowledge. In our research, we consider the development of creativity at secondary school level, so we use the term in accordance with the definition of mini-c creativity.

Then, a question emerges: how is creativity generated in a teaching and learning process? Regarding this question, the first step is assuming that creative thinking is a complex process that can be studied as a process that emerges from others, as [START_REF] Malaspina | The role of intuition in the solving of optimization problems[END_REF] explained with the intuitive process. These authors propose that a method to investigate complex processes is decomposing them into more simple processes and they use the metaphor of the vector space: the more complex process, creative thinking in this case, can be understood as a linear combination of the basis vectors (other processes). This approach to creativity as a process emergent from other more simple processes, was also considered in the European project A Computational Environment to Stimulate and Enhance Creative Designs for Mathematical Creativity (MC2) [START_REF] Sala | Contribución del EOS en la construcción de una herramienta de evaluación del pensamiento matemático creativo[END_REF]:

Then, when we concluded that the way to conceptualize the creative process in mathematics is decomposing it into other processes, we considered, together with the MC2 team, that the best way to go further in its study was adopting the method proposed by the Ontosemiotic Approach [START_REF] Malaspina | The role of intuition in the solving of optimization problems[END_REF]] …but introducing an important variant. Instead of using a "basis" of processes defined a priori by the theoretical framework, we would use the "basis" that would be the result of a brainstorm performed by the participants in the project …MC2 when each of them explained what they understood by "creative process"…. As a consequence, the following aspects were revealed: a) When the members …[participants in the project] aim at determining the criteria of design and assessment, they do not directly refer to the concept of CMT [Creative Mathematical Thinking], but they consider its decomposition into different dimensions, or processes, of mathematical activity, whose integration, through task design, [participants] think that would help to develop the CMT. b) There is a shared assumption that "the mathematical creative thinking will emerge from the interaction and integration" of these different dimensions or processes (p. 4-5).

On this line, in the master's degree final project -a document where the preservice teachers propose some changes to improve their own practice-, we could expect to find: 1) comments about creativity, since enhancing creativity is usually regarded as positive; and 2) in these comments, creativity would appear as a consequence of working other aspects that improve the instructional process.

Several studies on teachers' training suggest the reflection on the own practice as a key strategy to professional development and the improvement of instruction processes. On this line, the construct of didactic suitability of the Ontosemiotic Approach [START_REF] Breda | Meta didactic-mathematical knowledge of teachers: Criteria for the reflection and assessment on teaching practice[END_REF] has been used as a tool to structure the teachers' reflection in some teachers' training programs in Spain, Chile, Panama, Argentina and Ecuador [START_REF] Breda | A noção de idoneidade didática e seu uso na formação de professores de matemática[END_REF].

The theory of didactic suitability originates from the need to have an instructional theory to help teachers to make decisions in designing, implementing and assessing their practice. Didactic suitability [START_REF] Breda | Meta didactic-mathematical knowledge of teachers: Criteria for the reflection and assessment on teaching practice[END_REF][START_REF] Sullivan | The relationships between task design, anticipated pedagogies, and student learning[END_REF] comprise six types of suitability. The epistemic suitability deals with the quality of the mathematics that are taught, and how the explained content is representative of the complexity of the mathematical object. The cognitive suitability is about, on the one hand, the adequacy of students' previous knowledge for learning the intended content and, on the other hand, the relation between what students should learn a priori and what they have learnt indeed after the implementation. The mediational suitability has to do with the availability of time and material resources necessary in the teaching and learning process. The interactional suitability is the effectiveness of interactions (teacher-students or between students) to identify and solve conflicts of meaning and improve students' autonomy. The affective suitability refers to the students' involvement in the teaching and learning process. Finally, the ecological suitability is the adequacy of this process to the school's educational project, the curriculum and the social environment. In order to use these criteria as an assessment tool, some observable components and characteristics are associated with each criterion. They were used in the design of a practical rubric that can be found in [START_REF] Breda | Meta didactic-mathematical knowledge of teachers: Criteria for the reflection and assessment on teaching practice[END_REF]. As an example, we show the components and characteristics of epistemic suitability in Table 1.

Components Characteristics

Errors Practices considered mathematically incorrect are not observed.

Ambiguities

Ambiguities that could confuse students are not observed; definitions and procedures are clear and correctly expressed, and adapted to the target level of education; explanations, evidence and demonstrations are suitable for the target level of education, the use of metaphors is controlled, etc.

Diversity of processes

Relevant processes in mathematical activity (modelling, argumentation, problem solving, connections, etc.) are considered in the sequence of tasks.

Representation The partial meanings (constituted of definitions, properties, procedures, etc.), are representative samples of the complexity of the mathematical notion chosen to be taught as part of the curriculum.

For one or more partial meanings, a representative sample of problems is provided.

The use of different modes of expression (verbal, graphic, symbolic…), treatments and conversations amongst students are part of one or more of the constituents of partial sense. In our study, we distinguish which components of the different types of suitability preservice teachers relate to creativity in their comments, while they use the criteria to analyze their own practice.

Context

The program of the master's degree in teaching in secondary school (specialization in Mathematics) includes a work placement in a secondary school divided into two periods. In the first period, that lasts two weeks, preservice teachers attend classes of their supervisors and know the group of students with which they will work later. The second period lasts six weeks, and the preservice teachers should implement a learning sequence that they have prepared. After the work placement, in the subject of Innovation and research in Mathematics education, the didactic suitability criteria [START_REF] Breda | Meta didactic-mathematical knowledge of teachers: Criteria for the reflection and assessment on teaching practice[END_REF], with their components and characteristics, are introduced. Then, preservice teachers should use them in the analysis of the learning sequence that they implemented. The master's degree final project (MFP) includes this analysis and, based on it, the preservice teacher's proposal with some changes in the learning sequence that could help to improve the suitability of the teaching and learning process.

Research questions and method

The aim of this study is to examine preservice teachers' ideas about promotion of students' creativity, while they analyze their own practice and justify some changes of the learning sequence to improve it, in their MFP. Preservice teachers do not receive any specific training in how to develop students' creativity during the master's degree. Our research questions are: 1) Do preservice teachers frequently include comments about the promotion of creativity in their MFP? 2) Which aspects of the teaching and learning process are related to the promotion of creativity in the preservice teachers' comments?

We considered the 198 MFP from the years between 2009-2010 and 2014-2015. First, we made a register with some information of each MFP (name of the preservice teacher, year, title of the learning sequence, level). We searched for explicit references to creativity or other words of the same word family (creative, creation, creator, create) in the MFP and made a second register with only those projects that include these comments (we used a reduced list of keywords in comparison with other studies, such as the one of [START_REF] Joklitschke | Theories about mathematical creativity in contemporary research: A literature review[END_REF]). Registered data of these MFP include the extracts with the explicit references to creativity, as shown in Table 2.

Then, we read the MFP with references to creativity and tried to find common patterns or characteristics in the comments in order to infer a classification of comments. Since the preservice teachers use the didactic suitability criteria to structure their analysis, we also used them to distinguish aspects that are associated with the promotion of creativity in the comments. It means that, in this first phase, we used the didactic suitability criteria as previous categorization to analyze the preservice teachers' discourse. Finally, in a second phase, we classified the comments into the different categories that we had formed inductively from the data.

For example, in the MFP of [START_REF] Fontalba | Treball final de màster[END_REF] we found two explicit references to creativity (shown in Table 2). In her analysis of the implemented learning sequence, the epistemic suitability is one of the worst valued. When she analyzes this suitability, she remarks that the learning sequence should include activities that enhance creativity (first extract in Table 2). Then, in the new proposal, she adds an activity in which the students have to create their own problems (second extract in Table 2).

We consider that this comment is related to the diversity of processes (a component of the epistemic suitability) and, in particular, to the process of posing problems.

Title and year

Geometry in space. 2013 -2014 Preservice teacher's name F.D.

Level 3 rd course of Secondary School (14-15 years old)

Branch of mathematics Geometry

References to creativity Yes

Number of references 2

Extract Diversity of processes. The mathematical quality of the learning sequence could be improved and one of the key components is the diversity of processes. The student should make, should learn through trial and error, manipulation, experimentation and exploration. There is an excess of algorithmic activities, to calculate and execute procedures, therefore activities that hold curiosity, creativity, imagination and discoveries should be included. (Fontalba, 2014, p. 4) Competence 4. Generating mathematical questions and posing problems. Clearly, this competence is developed since the goal of the contest is the creation and resolution of problems. (Fontalba, 2014, p. 11) Comments She refers to the development of creativity in the assessment of epistemic suitability, and she says that the diversity of processes should increase. Then, in the new proposal, there is a task in which the students have to create problems.

Results

We found explicit references to creativity in 119 of the 198 MFP of the master's degree in teaching in secondary school from years between 2009-2010 and 2014-2015: in 2009-2010, 9 of 15 MFP include references to creativity; in 2010-2011, there are 16 out of 21 MFP;[START_REF] Lev-Zamir | Creative mathematics teaching in the eye of the beholder: focusing on teachers' conceptions[END_REF]-2012, 20 out of 34;in 2012-2013, 14 out of 24;in 2013-2014, 25 out of 47;andin 2014-2015, 35 out of 57. In the MFP with references to creativity, we found different elements of the teaching and learning process associated with creativity. In the following lines, we describe the main categories that we distinguished and then we focus on one of them. The development of creativity is related to: a) Tasks with notable processes (epistemic suitability). In some projects, the preservice teachers explain activities where students have to do some hypothesis, plan a resolution method, justify or put in practice other processes that are especially important while learning mathematics. In this category, we include those comments in which creativity is related to this type of activities. These preservice teachers expect to develop creativity using it to solve the activities. At the same time, epistemic suitability improves, since so does the diversity of processes, which is one of its components. 61 MFP include comments in this category.

b) The use of manipulatives and technology (mediational suitability). In this case, preservice teachers explain activities where students' creativity could be fostered by the use of computers and other material resources. Sometimes, preservice teachers do not refer to a mathematical creativity, but a plastic or artistic creativity that students practice in making a certain object that would be used in a mathematical activity later. 13 MFP include comments in this category.

c) The development of other skills that are useful in the current society (ecological suitability). In some MFP we found comments about a responsible use of creativity. Preservice teachers relate creativity to critical thinking or the development of the students' social competence. We associated these comments with the ecological suitability because they refer to some skills that are useful for living in society nowadays. 3 MFP include comments in this category.

d) Cooperative tasks (interactional suitability). In other MFP, preservice teachers explain activities in which cooperation between students plays a key role. Students' cooperation is necessary to solve the task and it would help to develop their creativity, while sharing their ideas in the resolution. 5 MFP include comments in this category.

e) The assessment procedures of the learning sequence (cognitive, interactional and affective suitability). In this category, we include the references to creativity that appear in the assessment criteria of the learning sequence or in the objectives of a certain activity that the preservice teacher proposes. Usually, in the corresponding MFP there are some changes in the assessment tools. For example, preservice teachers design new rubrics. They often oppose the traditional understanding of evaluation, proposing a competency based assessment and sharing the responsibility with the students through auto-evaluation and co-evaluation. These changes relate to several components: the learning, as a component of the cognitive suitability; the autonomy and the formative evaluation, as components of the interactional suitability; and attitudes and emotions, as components of the affective suitability. 18 MFP include comments in this category.

General comments. In some comments, we could not identify a clear connection between creativity and a particular element of the teaching and learning process, so we refer to them as general comments. In this category, there are references to the creative nature of the mathematical activity, references to the teachers' creativity when designing tasks to their students, and references to the role of creativity in the constructivism paradigm.

We observed that in 66 MFP there are more than one reference and, in some cases, several aspects of the learning sequence are related to the development of students' creativity. Most of the comments that are associated with a certain suitability criterion are in the category in which creativity is related to the diversity of processes. Therefore, we focused on this category, distinguishing different characteristics of the activities that the preservice teachers propose.

With regard to the comments in the first category, we found that in 14 MFP the development of students' creativity is associated with the resolution of problems, especially open-ended problems. Sometimes, creativity is specific to the phase of devising a plan to solve the problem. In 12 MFP, preservice teachers mention creativity when they ask the students to invent problems, think on significant contexts or create different representations. In 12 MFP, preservice teachers consider that the students' creativity can be fostered by working with real situations. 5 MFP include comments about developing creativity through project based learning. Five preservice teachers propose mathematical activities in the form of games that could enhance students' creativity. In the same category, there are some references to processes that preservice teachers relate to creativity: creation of arguments or justification (in 6 MFP), intra and interdisciplinary connections (in 5 MFP), creation of formulas and hypothesis (in 6 MFP), creation of mathematical models (in 5 MFP).

Conclusions

Even when the preservice teachers do not receive a specific training in how to promote students' creativity during the master's degree, more than half of the MFP include references to creativity. First, we conclude that a significant number of preservice teachers in our sample implicitly consider that creativity can be developed in the mathematics lessons. There seems to be an implicit general agreement on the positive effect of the development of students' creativity on mathematics learning.

Second, preservice teachers' comments evidence that creativity is a complex topic and there are different ways to understand and promote it. We observe various elements or aspects of the teaching and learning process that are somehow related to the development of creativity in the preservice teachers' comments. Note that preservice teachers not only refer to mathematical creativity, their comments cover a wide view of creativity. Categories of the classification that we present here could change and evolve with the incorporation of new results from the analysis of the MFP of the following years of the master's degree. We conclude that preservice teachers consider that creativity is something that emerges from the interaction and integration of other aspects (especially different mathematical processes).

This study can contribute with results to other research on preservice teachers' education, showing evidence of the preservice teachers' perspective on creativity and its promotion in their learning sequences, when they do not receive any specific training in how to promote students' creativity. It could be interesting to observe how specific training in fostering students' creativity would affect the analysis that preservice teachers do in their MFP. We could also consider the possibility of using more keywords related to creativity to find comments in the MFP. As [START_REF] Joklitschke | Theories about mathematical creativity in contemporary research: A literature review[END_REF] explain, there are several conceptualizations of creativity and it is important to be aware that other words could be used to describe or conceptualize creativity. Then, we should consider other words that preservice teachers could use as synonyms of creativity. On the other hand, sometimes, the use of the word "creativity" does not seem to be relevant to the meaning of the whole comment or it is ambiguous (for instance, it is not clear whether it refers to teacher's or students' creativity). A possible solution would be revising the comments and rating their validity concerning their ambiguity.

Table 1 : Components and characteristics of epistemic suitability (Breda et al., 2017, p. 1903).

 1

Table 2 : Example of the data of a MFP with some explicit references to creativity

 2

Acknowledgment

This work is part of the research project REDICE18-2000 (ICE-UB). Alicia Sánchez is funded by the grant FI 2018.


Ann-Kristin Tewes
email: ann-kristin.tewes@tu-dresden.de

Elisa Bitterlich

Judith Jung

Keywords: Heterogeneity, group discussion, teacher beliefs, collective orientation, teacher education

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

"In school you notice the performance gap and how different it is between the students" -Student teachers' collective orientations about the learners' heterogeneity in mathematics Introduction

Heterogeneity is a central term of current debates regarding education and school system in Germany. In the educational context, it is associated with various categories of difference such as language, culture, gender or (dis)ability, and it is perceived to express "difference as a challenge to be dealt with actively" (Sliwka, 2010, p. 213). In Germany, discussions on heterogeneity in classrooms have recently been stimulated by educational policies like the ratification of the UN Convention on the Rights of Persons with Disabilities and social and demographic changes, e.g. increased linguistic-cultural differences because of a higher number of children with an immigrant background [START_REF] Decristan | The interplay between class heterogeneity and teaching quality in primary school[END_REF]. Due to changes in the main areas of attentionwhich are mostly oriented towards current political and social debatesdifferent facets of heterogeneity and difference have been at the centre of focus (for some periods) lately. The impetus for a renewed focus on heterogeneity in Germany has been provided by the results of international comparison studiesin particular PISA, 2000/2009 [START_REF] Klieme | PISA 2009. Bilanz nach einem Jahrzehnt[END_REF] which have highlighted especially the sizeable differentiation in student achievement, the alarmingly high number of very lowachieving pupils, and a close relationship between social background and academic success [START_REF] Döbert | Conditions of school performance in seven countries[END_REF][START_REF] Trautmann | Heterogenität in der Schule. Eine kritische Einführung[END_REF].

The HeLeA project addresses this central debate about heterogeneity. The main assumptions of the project are that teachers' and student teachers' orientations, knowledge and attitudes play an important role in creating effective learning environments for all learners and in the development of an inclusive school that considers the learners' heterogeneity as something positive and normal [START_REF] Booth | Curricula for the common school: What shall we tell our children?[END_REF][START_REF] Reynolds | Education for inclusion, teacher education and the teacher training agency standards[END_REF]. The quality of education depends to a high degree on the teaching staff, who play a key role in preparing their learners to take their place in society [START_REF] Savolainen | Responding to diversity and striving for excellence: The case for Finland[END_REF]. Through qualitative (group discussion) and quantitative methods (questionnaire), the project aims at the reconstruction of student teachers' ways of thinking, speaking and feeling concerning the heterogeneity of learners in school, especially in mathematics. On the basis of the survey results, the project intends to design concepts for the education of student teachers in order to sensitize them to different facets of heterogeneity. The goal is to prepare them for encountering heterogeneity among their prospective pupils and to equip them with approaches for dealing with it in mathematics and also other school subjects. In this article, the focus is on the qualitative approach from HeLeA, as it provides interesting and emotional insights into the current state of teachers' opinions. Based on group discussions with high school mathematics students, the focus of this paper is on: "What collective orientations about the learners' heterogeneity in mathematics do student teachers have?".

Theoretical background

The pedagogical discourse indicates multiple unresolved problems, and there is some criticism of previous approaches dealing with heterogeneity in school, like the homogenisation of learning groups through selection and forms of external differentiation [START_REF] Trautmann | Heterogenität in der Schule. Eine kritische Einführung[END_REF]. At the same time, however, there is no lack of existing ideas and concepts for improving these approaches. Didactic-methodological concepts for internal differentiation have been discussed since the 1970s [START_REF] Sliwka | From homogeneity to diversity in german education[END_REF][START_REF] Trautmann | Heterogenität in der Schule. Eine kritische Einführung[END_REF]. Strikingly, however, for most student and practising teachers the question of how to deal with heterogeneity nevertheless represents an important problem area in planning and teaching lessons. [START_REF] Schönknecht | Heterogenität aus der Perspektive von Studierenden oder der Wunsch nach Normalität [Heterogeneity from the perspective of student teachers or the desire for normality[END_REF] point out that in describing heterogeneity, student teachers often seem influenced by polarisations and dichotomisations, as well as a limited perspective focusing on supposed "problem children". Many studies emphasise the individual and personal views of teachers and student teachers about heterogeneity (cf. [START_REF] Bitterlich | This has to be done without bending over backwards': Collective orientations and experiences of student teachers in relation to heterogeneity in school[END_REF]. Most of these studies focus on teachers' beliefs and stereotypes2 [START_REF] Winheller | Professionalisierung von Lehrkräften für einen reflektierten Umgang mit Ethnizität und Geschlecht in der Grundschule[END_REF][START_REF] Zobrist | On dealing with heterogeneity in secondary school using the example of mathematics lessons[END_REF]. We focus on student teachers' collectively shared orientations concerning the learners' heterogeneity. In contrast to studies about teachers' and student teachers' beliefs and stereotype-threat, studies about collective orientations reconstruct those experiences and types of knowledge that can be considered as shared within the group of teachers. In this regard, collective orientations could be seen as "socially-agreed-upon knowledge base [...]. The key assumption behind them is that the members of the respective groups share a more or less common experience of enculturation into these groups" [START_REF] Gellert | Routines and collective orientations in mathematics teachers' professional development[END_REF].

Following [START_REF] Mannheim | Structures of thinking[END_REF], teachers tend to have the same or similar experiences and opinions because they belong to a 'conjunctive space of experience' (konjunktiver Erfahrungsraum).

Those who have biographic experience in common, have commonalities in their history of socialization and, thus, have a common or conjunctive experiential space, understand each other immediately insofar as these biographical commonalities become relevant in interaction and discourse. (Bohnsack, 2010, p. 105) Teachers as well as student teachers represent a professional group whose conjunctive experiences materialise, on the one hand, via practical experience and, on the other hand, via conceptualtheoretical involvement with teaching and the didactic handling of heterogeneity [START_REF] Sturm | Praxeologische Unterrichtsforschung und ihr Beitrag zu inklusivem Unterricht [Praxeological education research and its contribution to inclusive education[END_REF]. The interweaving, or double structure, of their conceptual-theoretical conjunctive space of experience based on their own practical experiences, allows their participation in the social practice of teaching, and indeed creates it [START_REF] Bohnsack | Praxeologische Wissenssoziologie [Praxeological sociology of knowledge[END_REF].

In this context, [START_REF] Gellert | Routines and collective orientations in mathematics teachers' professional development[END_REF] emphasises that teaching should be perceived as a cultural practice that takes place in communities rather than an individual and independent practice that takes place in isolation behind closed classroom doors. In his findings, he emphasizes that the mathematics primary school teachers' collective orientations about their own professional development sometimes work as obstacles against development. "Mutual validation can turn experience into law.

[...] Collective teaching experience can be blind to alternative conceptions of teaching" [START_REF] Gellert | Routines and collective orientations in mathematics teachers' professional development[END_REF]. [START_REF] Schieferdecker | Orientierungen von Lehrerinnen und Lehrern im Themenfeld Heterogenität. Eine rekonstruktive Analyse[END_REF] carried out group discussions with several groups of teachers about heterogeneity in society. The focus of his research is on the reconstruction of collective orientation patterns of teachers that can contribute to a broader understanding of the perception and management of heterogeneity in the educational practice of teachers. In this regard, the aim is to identify structures (which he assumes to be collectively shared) that teachers use in order to cope with the notions of social heterogeneity and heterogeneity in pedagogical practice. He reconstructs a tension between the conception of feasibility and the experience of powerlessness. On the one hand, teachers see themselves as solely responsible for the learning success of their students. On the other hand, they fail to 'pick up every student where he or she stands'. But the reasons for their failure are suspected outside school (e.g. with parents). Even experienced teachers tend to see learners' heterogeneity as a problem of increasing complexity which complicates successful teaching processes. To handle learners' heterogeneity, teachersand this is something that is collectively sharedhomogenise them and create polarisations (e.g. high-and low-achieving learners) [START_REF] Schieferdecker | Orientierungen von Lehrerinnen und Lehrern im Themenfeld Heterogenität. Eine rekonstruktive Analyse[END_REF].

The aim of this paper is to reconstruct the collective orientations of student teachers in relation to mathematics. The performance heterogeneity 3 of the students is of great importance in Germany, as the school system aims for homogenisation [START_REF] Trautmann | Heterogenität in der Schule. Eine kritische Einführung[END_REF]. We therefore believe that the factor of performance differences in mathematics is particularly important to student teachers when talking about their experience in teaching math and when reflecting their own teacher education. We suspect that in a subject such as mathematics, the differences in performance become particularly clear. In this context, [START_REF] Thompson | Teachers' beliefs and conceptions: A synthesis of the research[END_REF] describes that American teachers consider mathematics as something static that contains a set of rules and procedures. To teach mathematics and to get a correct result, these have to be used, no matter if they have been understood. Since mathematics is often considered as a subject with 'clear answers', formal procedures and easily comparable results, student teachers might hold the (collective) orientation that especially in mathematics differences in achievement are more noticeable and with a higher weight than, e.g., in language-based subjects were different opportunities exist to express something or to write an essay.

Methodology and procedures

One way to access collective and action-leading orientations of prospective teachers is through group discussions (e.g. Przyborski & Wohlrab-Sahr, 2014). Group discussions can help to identify and analyse the implicit or tacit knowledge of the participants while they talk about a specific topic (e.g. heterogeneity). The self-dynamic of the discussion process, without any interruptions by the researcher, is important to discover conjunctive spaces of experience, which become visible through 'focusing metaphors' in which the group adjusts itself to those specific topics that are most relevant in its members' common experience [START_REF] Bohnsack | Documentary method and group discussion[END_REF].

Concerning group discussions, the immanent meaning comprises that stock of knowledge which is made explicit by the participants themselves. This has to be distinguished from knowledge of experience, which is so much taken for granted by the participants that it must not and often cannot be made explicit by themselves. The participants understand each other because they hold common knowledge without any need to explicate it for each other. (Bohnsack, 2010, p.103) The documentary method is a suitable method to analyse data from group discussions to identify and reconstruct collective experiences and common (used) ways of acting [START_REF] Weller | Documentary method and participatory research: Some interfaces[END_REF]. In the identification, description and reconstruction of the (future) teachers' collective perceptions, we see one possible way to gain an impression of the current perspectives of teachers and student teachers on dealing with the learners' heterogeneity, especially in mathematics.

The documentary method offers […] an access to the pre-reflexive or tacit knowledge, which is implied in the practice of action. Asking for the documentary meaning can […] be understood as asking for how: how is practice produced or accomplished. (Bohnsack, 2010, p. 103) So far, six group discussions have been held during which teacher students spoke about the general question: What kind of experiences have you made concerning the difference of learners in school and in class? The groups were homogeneous in terms of the participants' school type. In each discussion, between four and seven student teachers spoke together for around one hour about their experiences in dealing with heterogeneity in the school context. Each discussion was video recorded and transcribed. This paper focusses on statements on the performance heterogeneity of students in relation to mathematics. We compare transcript scenes from two different group discussions (number two and four), because there the student teachers are from the same school type (high school) and all are teaching mathematics. As we described above, the teachers, in our case mathematic high school teachers, tend to have the same or similar experiences and opinions. Because of their academic background they belong to a 'conjunctive space of experience'. To answer the research question "What collective orientations about the learners' heterogeneity in mathematics do student teachers have?" the scenes are analysed with a reflective interpretation. Within the framework of the documentary method, the stage of reflective interpretation is particularly promising for the identification and reconstruction of the 'conjunctive space of experience' [START_REF] Mannheim | Structures of thinking[END_REF].

Results

From the very beginning of both group discussions, the focus was less on education in general than on mathematics, as the student teachers seem to have a 'conjunctive space of experience' concerning this topic. Differences in the learners' achievement was a meaningful aspect of the discussion. Following the research question, we could identify the collective orientation that, especially in mathematics, performance differences are visible and challenging. We will illustrate this by showing short transcript extracts of group discussion number two and four, each followed by a brief analysis of the scene.

Group number two

Wiebke: I think in school you notice the performance gap and how different it is between the students. I think, especially in the subject math you always have these,

[laughs] that are bomb in math and those, who do not understand it at all. To arrange the lessons in a way that all can somehow follow or are not underchallenged is quite difficult.

Natalie:

Yes. What is also always a topic are differentiated tasks, in a way, that either the tasks get more difficult in the end or the top-performing pupils get more tasks. And at the beginning the easier tasks. That is also addressed in teacher education.

And also, that you should have extra tasks for those who are faster so they still have something to do or talk about. That happens all the time.

Wiebke:

But I think it is only about these differentiated tasks and for the faster pupil new tasks. I had a seminar, not in math but in English, where another heterogeneity, like language differences, was mentioned. And this could be transferred to math. Maybe a child is super good at math but just does not understand what's in the task. I think, you don't learn anything about this at university.

The first reaction on the impulse of the discussion concerned the differences in the learners' performance. Wiebke argues, that in mathematics there are 'always' pupil who are extremely high achieving, what she illustrates with the metaphor of a bomb (which is a common expression in Germany) while there is also 'always' a group of pupils who do not understand mathematics at all. Through this dichotomous distinction, she describes from personal experience that it is 'quite difficult' to arrange lessons. The metaphorical use of 'bomb' for something exploding and powerful illustrates that such 'bomb' students are not necessarily considered by her as something positive. Natalie seems to understand Wiebke and argues that in university they 'always' heard something about how to use differentiated tasks and how to make sure they work. This could be seen as something positive. But in contrast to Natalie, Wiebke is not satisfied with this one-sided preparation with the focus on achievement and tasks. In her opinion, it is also important to 'learn' something about other facets of heterogeneity, like language differences, which she emphasizes with the comparison to her studies in English. She notes that it is important for students to understand the tasks in order to solve them because linguistic competences are often needed to show mathematics performance. But her claim, to learn 'nothing' about this in university, could also be seen as an exaggeration.

Group number four

Konrad: Especially during internships, I have seen great differences in the performance and skills. Niklas: Yes, I also noticed that. Either in middle school or in 'pure' high school classes.

In one such pure high school it was such a monotony and all students were wellmannered and from suburban areas and tagged along even during the worst lessons. But even there you recognized achievement differences in mathematics like in middle school classes. Even in urban areas you have the same differences.

As in group discussion two, in discussion number four the reaction on the impulse for the discussion is about differences in achievement. Konrads statement shows, that it seems easier to 'see' the learner's heterogeneity when you are not the person teaching in front of the class. Konrad seems to protect teachers as they are not able to 'see' differences when they have to concentrate on the lesson itself. Perhaps, he himself also made the experience of not being able to perceive so many achievement differences and to adequately respond to them when teaching himself. Niklas enlarges this topic by comparing different types of school and different environments. Either in (lower-achieving) middle school classes or in (higher-achieving) 'pure' high school classes, he claims that in mathematics the same differences in achievement exist. This also applies for urban and suburban areas. He points out one class of well-mannered students, which is a 'pure' high school class in a suburban area, which tolerates even the worst and most boring lessons.

Discussion

Especially in the group discussion with high school student teachers for mathematics, differences in achievement was a central point of the discussion. In both conversations we could reconstruct the view, that especially in mathematics differences in achievement are suspected to be 'visible' and that teachers have to react to them through (differentiated) tasks. This aspect could therefore be seen as the most important for student teachers when they consider the heterogeneity of their prospective class in mathematics. It is noticeable, that it is less discussed, how differentiation of the content or the social formation in the class could look like. Differences in achievement seem to be important for the student teachers but possibly they don't know how to deal with this heterogeneity. The predication of Nathalie shows that theoretical knowledge of how to deal with heterogeneity and differentiated tasks has apparently been taught in the university. However, the two excerpts show that this knowledge cannot be applied in practice. This result also coincides with the four other group discussions of the HeLeA project. Similarly to [START_REF] Gellert | Routines and collective orientations in mathematics teachers' professional development[END_REF] and [START_REF] Schieferdecker | Orientierungen von Lehrerinnen und Lehrern im Themenfeld Heterogenität. Eine rekonstruktive Analyse[END_REF], tensions were reconstructed which have to do with the problem of making pedagogical specialist knowledge (learned in university) compatible with the practice of teaching a heterogenic learning group. In all six group discussions, it was possible to reconstruct discontinuities between the everyday discourse of student teachers and the academic discourse on the subject of heterogeneity.

One result of the HeLeA project is that the theoretical knowledge seems to be incompatible with the student teachers' experiences in school practice. To ensure that future teachers are more capable to apply the theory they have learned in practice, we have to give them tools to reflect their own lessons and train their ability to interpret situations. Personal practical experiences should be constantly analyzed and reflected in professional lifeone possibility for this is team teaching. The results collected so far already provide initial insights into mathematics student teachers' collective orientations and guiding ideas on heterogeneity in mathematics (c.f. [START_REF] Bitterlich | This has to be done without bending over backwards': Collective orientations and experiences of student teachers in relation to heterogeneity in school[END_REF]. One limitation of the used group discussions approach is that only collectively shared orientations can be reconstructed. What remains in the individual is hidden. But if we follow the assumption that teachers can be seen as a common group with shared experiences, collective orientations could be more important in certain circumstances.

  When you walk around during times of individual work, you could really see what the student is currently thinking about the task. As trainee that is usually very noticeable because you do not stand in front of the class but you help students in the back or walk around during individual work.

"Heterogeneity in teacher education from the start" (Heterogenität in der Lehrerbildung von Anfang an -HeLeA) is a sub-project of "TUD-SYLBER" which is part of the '"Qualitätsoffensive Lehrerbildung", a joint initiative of the Federal Government and the Länder which aims to improve the quality of teacher training. The programme is funded by the Federal Ministry of Education and Research. The authors are responsible for the content of this publication.

Following Allport (1954), stereotypes -"the pictures in our head"simplify our thinking and produce expectations about what other people are like and how they are likely to behave. In this sense, based on prior information (e.g. a student's test scores, social class, gender, ethnicity, race) a teacher develops expectations about the ability of the learner. Similar to this, beliefs are representative bits of information that a person has about an object, person or group of individuals based on certain facts or personal opinions[START_REF] Ajzen | Understanding attitudes and predicting social behavior[END_REF].

[START_REF] Helmke | Bedingungsfaktoren schulischer Leistungen [Conditional factors of academic achievement[END_REF] point out, that in addition to subject-related and teaching aspects, the individual characteristics of the pupils, e.g. language, intelligence and learning strategies are of great relevance to the students' performance[START_REF] Helmke | Bedingungsfaktoren schulischer Leistungen [Conditional factors of academic achievement[END_REF].


Paola Iannone
email: p.iannone@lboro.ac.uk

Michal Ayalon
email: mayalon@edu.haifa.ac.il

Johannes Beck
email: beck@dmuw.de

Jeremy Hodgen
email: jeremy.hodgen@ucl.ac.uk

Francesca Morselli
email: morselli@dima.unige.it

la diffusion de documents scientifiques

Keywords: Summative assessment, formative assessment, validity, feedback, assessment methods

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Introduction to the papers of TWG21: Assessment in mathematics education

Introduction to the papers of TWG21:

Assessment in mathematics education

Introduction TWG21 met for the second time in Utrecht at CERME11 and in this conference we sought to continue the work started at CERME10. The aim of the previous meeting was to ascertain where the interest of our community is when thinking about assessment, and to maintain the focus firmly on mathematics. At CERME11 we discussed 14 papers and 3 posters which helped defining such interest. We noticed again a variety of focal points: from validation of large-scale assessment instruments, to the affordances and drawbacks of online assessmentespecially in the university context -to the details of construction of individualised feedback. As in the previous meeting the papers also presented a variety of methodologies: from large quantitative studies to more nuanced qualitative investigations. Among the submissions we also received papers related to students' perspectives and teachers' perspectives on assessment. These themes were not prominent in the past meeting of the group and we welcomed the new perspectives they brought. Finally, we decided to group papers together that indicated the role that mathematics has in the assessment: this is to say papers that focus on the specifics of mathematics, such as assessing proof.

Thematic clusters

We identified 5 overarching themes that will serve to organise the papers submitted to TWG21. Below, we describe each of these themes in turn.

Large scale standardised assessment:

There is strong interest in our community for the use and validation of large-scale standardized assessment, both for benchmarking national students' attainment and for mapping common misconceptions across educational stages. Garuti and Martignone present the introduction of the INVALSI standardised tests in Italy and then investigate the possibility to assess argumentation through the multiple-choice questions included in the INVALSI tests. By using Toulmin's model (1958) to analyse the structure of arguments included in the tests the authors find that although standardised multiple-choice arguments can identify only certain aspects of argumentation, this is not to say that the aspects that these tests do identify have no educational value. They conclude by proposing that the Toulmin's model (1958) could be used to guide the construction of such tests. Klegseth, Kaspersen and Solstad use Rash Analysis to determine whether non-standardised test included in Norwegian textbooks are a valid and reliable indicator of mathematical competence. In order to do so the authors focused on the concept of function. The items tested were divided into 5 competencies categories and the authors found that, with a careful selection of such test items, the set they identified can be a reliable and valid measure of mathematical competences for these students. Drüke-Noe and Siller present a newly developed test with easy tasks which assesses basic mathematical competencies that are essential for vocational training. The test is used in years 8 and 9 in Germany. From both a content-oriented and a process-oriented perspective, preliminary test results reveal a substantial lack of these competencies in all academic tracks, and differential effects were found with respect to year and class that need to be examined beyond curricular analyses. Finally, Lasorsa, Garuti, Desimoni, Papa, Costanzo and Ceravolo also worked from the database generated by the INVALSI test for mathematical proficiency in Italy and investigated the psychometric and qualitative-educational properties of a pool of these questions. They seek to find specific cognitive obstacles that the students need to overcome to reach the desired competence levels.

Assessment with technology and Computer Aided Assessment (CAA):

As in CERME10 we received papers on the use of technology for assessment, both papers at university level. Sangwin reports the development and evaluation of an online linear algebra examination administered using the system STACK [START_REF] Sangwin | Computer aided assessment of mathematics[END_REF]. The aim of this study is to investigate whether common questions in linear algebra could be assessed automatically, which was the case for most of the questions tested. The implication for the assessment of mathematics are that tools need to be created and tested to assess explanation, justification and reasoning, and that the current questions of paperpencil test should be reconsidered since they often focus on mechanical processes. Barana and Marchisio report the implementation of automatic assessment methods both in classroom activities and in online homework tasks to enact formative assessment strategies. The aim of such implementation was to foster self-regulation in the students. Their findings illustrate how formative assessment strategies can be implemented into such a blended-learning setting and how students perceived the effectiveness of the experiment.

Assessment of mathematics: the teachers' perspective:

We received four contributions on this topic, all reporting research in school settings. Horoks, Pilet, Coppé, De Simone and Grugeon-Allys present the design of a large-scale questionnaire aimed at analysing teachers' assessment practices and how teachers use the information they gather to promote students' learning, in the context of algebra. The authors discuss the design of suitable questions to gather information on the diversity of teachers' assessing practices. Kaplan, Kan and Haser present a study that investigated preservice middle school mathematics teachers' (PST) professional development related to formative assessment practices and focus on how preservice teachers' practices changed during the teacher education program. The authors explored PSTs' performance on a task where they were asked to analyse a lesson plan without any formative assessment opportunities built in and found PSTs did not always realise the lesson plan lacked any formative assessment, especially if they had not yet attended to measurement and assessment courses and methods of teaching courses. Moreover, PSTs were keener to suggest assessing strategies based on their role as teachers, and less to insert peer assessment into the plan. Sangari, van den Heuvel-Panhuizen, Veldhuis and Gooya report preliminary results of a survey of Iranian teachers aimed at investigating to what extent teachers engaged with Descriptive Assessment, an approach for evaluating students' achievement by collecting and documenting evidence regarding their learning and performance. The survey showed that teachers engage with some of the formative assessment tasks, although not as frequently as it would be desirable. Finally, Prendergast, Treacy and O'Meara presented a study concerning teachers' perceptions of the Bonus Point Initiative (BPI) in Ireland. This initiative consists in awarding an extra 25 points in final examination results to those students who choose to study mathematics at higher level. The study adopted a mixed-method design. Findings show that, although teachers recognized the value of the BPI initiative in promoting the study of mathematics and recognizing the effort paid by students in studying maths at higher level, many of them feared that the BPI may decrease the standard of students attending to higher level courses.

Assessment of mathematics: the students' perspective: These papers focused on the impact of assessment on students' experiences and attitudes both at school and university levels. Demosthenous, Christou and Deetra Pitta-Pantazi propose a framework for designing different types of assessment tasks to elicit evidence about how students respond to the mathematical ideas presented in learning trajectories. The authors illustrate the application of the framework by drawing on a learning trajectory of fraction division for sixth grade students and discuss how the enactment of the trajectory in classroom could be adjusted to students' current understanding. Häsä, Rämö and Viii Virtanen discuss a study part of an ongoing larger project concerning student selfassessment skills in university courses. The authors developed a method enabling large cohorts of students to assess their own learning outcomes and to give their own course grades with the help of an automatic verification system. The paper explores whether the self-assessed grades correspond to the students' actual skills, and how well the automatic system can detect issues in the selfassessment. Based on an expert's evaluation of the skills of two students, the study concludes that although for large part the model works as intended, there are some cases where neither the selfassessment nor the computer verification seem to be accurate. Bóra and Juhász discuss pair-work testing. Their pilot study consisted of four tests: twice students were tested in pairs and twice students were tested individually. The researchers recorded the marks and overall performance of the two-two sets of each type of tests. Online questionnaires revealed that in general students had a positive attitude towards paired testing: they rated positively the concept and they rated their testing experience positively. In addition, the overall level of paired work was high. The authors point to the need to continue the study with a larger number of students to learn more about how paired testing improve collaboration and mastery of skills.

Aspects specific to the assessment of mathematics: Starting from the work in our previous meeting we wanted to emphasise the role of mathematics in the assessment, e.g. the specifics of assessing mathematics. We received four contribution to this topic. Bolondi, Ferretti and Santi use the framework of didactical situations [START_REF] Duval | intellectuels[END_REF] to investigate the persistence of certain mathematical misconception across educational stages (e.g. from school to university mathematics). By analysing the data collected in the database generated by the INVALSI test for mathematical proficiency in Italy the authors found that certain misconceptions concerning the operations between exponentials persist across the school to university transition, for low as well as medium attaining students. Watkins, Lamm, Kohli and Kimani investigate the relation between characteristics, beliefs, background and perception of students on student attainment in the context of US Community Colleges, a much under-researched area. Using linear regression on a large database collected for the study they are able to individuate some predictors of student achievement and conclude that rich professional development opportunities for teachers have the potential to enhance teachers' effectiveness in this context. Cusi and Telloni describe the design of individualized teaching/learning paths in the context of formative assessment at university level.

The paths are designed to be tailored to the students' responses to online test and provide tailored feedback at each stage of the test. Preliminary results show how students interact with the feedback received and highlight that few students are aware both of the difficulties they have in solving the problems and how to overcome them. The individualized teaching/learning paths have the potential to develop the awareness of strategies to overcome difficulties. Finally, Turiano, Boero and Morselli explore the emergence of a 'culture of theorems' [START_REF] Boero | Theorems in school[END_REF] through assessment involving narratives in a 10 th grade classroom. After the students were taught some Euclidian geometry they were given as part of their assessment the task to write to a friend in another school and explain what a theorem is. Preliminary analysis of the narratives shows instances of students' meta-mathematical knowledge regarding the culture of theorems.

Conclusions

The papers that were presented in the TWG21 indicated once more the variety of interests in the assessment of mathematics community, both in terms of focus and of methodologies. Again we received papers adopting quantitative methods (such as Rash analysis) and papers focusing on small in depth case studies. This reflects the variety of approaches in our field and it is normal that this variety emerges on the work of our group. However, while in the past meeting we found that authors were concerned by the lack of a shared language to discuss assessment, this time we welcomed more agreement on basic definitions. The emphasis that we detected on students' and teachers' perspectives also indicate the need for investigating how assessment affects those participating in it, following the observation we made at CERME10 of a lack of studies on this topic. Again, we sought to maintain the focus on mathematicse.g. we highlighted the impact of the discipline characteristics on the way in which we think about assessment, both summative and formative. Finally, we plan to meet again at CERME12 in Bolzano (Italy) to push forward the work of the group and out thinking on the assessment of mathematics.


Eszter Bóra
email: eszterbora@gmail.com

Péter Juhász
email: juhasz.peter@renyi.mta.hu

Students' attitudes and responses to pair-work testing in mathematics

Keywords:

Introduction

In Hungary, mathematics teachers usually teach every topic separately, one after the other. At the end of each topic the students write a test, these results are the main factors contributing to the final evaluation of a student work during the year. Other popular forms of assessments used daily are short oral individual assessments typically at the beginning of the class, short written tests (between 5-20 minutes) and checking the homework problems (OECD, 2015). There are several problems with these traditional tests taking practices.

First, the problem with the longer, end-of-topic tests are, that they can also contribute to the development of bad studying habits (e.g. last-minute studying, bad time-management), being these tests are often infrequent and their dates are known beforehand.

Second, in general, writing a test is not a perfect option for assess one's understanding of a concept or level of mastery. Since a worrying number of students develop text anxiety over the years, and so, it is hard to tell how well they mastered the given topics. In Hungary there is a surprisingly high level of test anxiety among children (Bodas & Ollandick, 2005). For these reasons, we started to experiment with routines of testing with a novel approach. We want to encourage students to perceive testing situations more like an interesting challenge, as a part of the learning process. With a group of students, we tried pair-work tests. Our reasoning behind this idea is the following: if one of them run out of ideas, the other can help. If none of them has an idea, they can realize the problems are indeed hard and they can try to develop new ideas together. If they fail to solve a problem hopefully they will not attribute it to their own failure, lack of skills, because they see their pairs struggling with it as well. Thus it helps to develop a healthy attribution theory, or mindset to mathematics, and in general, to learning.

An additional factor encouraging us to experiment with this type of test is that collaborative skills are well sought after in the workplaces. Unsurprisingly, according to many researchers the necessary 21th century skills [START_REF] Voogt | A comparative analysis of international frameworks for 21st century competences: Implications for national curriculum policies[END_REF] are: collaboration experience, critical thinking, problem-solving skills. Unfortunately, in Hungary the mathematical educational culture is only started to develop and practice more group working, cooperative techniques in a wider context [START_REF] Andrews | Opportunities to learn in the Budapest mathematics classroom[END_REF]. So, we think it is important to develop, adopt these cooperative techniques in our local contexts as well.

The novelty of our work is to examine collaboration in a testing scenario in Hungarian mathematics education.

Our study

In our current pilot study there were four tests: twice students were tested in pairs and twice students were tested individually. We recorded the marks and overall performance of two-two sets of each type of tests. The tests had the same format and length, similar difficulties and recurring topics.

The pairs were randomly selected for the pair-work tests. They did not know beforehand who their partner will be for the test. If the number of students was odd, then the first step was to know which students is willing to work alone.

After the pair-work tests we conducted an online questionnaire. Where we asked the following five questions. What percentage of the test was your contribution? How do you like the concept of writing tests in pair? How would you rate your experience during the test? How did you like your pair before the test? How did you like your pair after the test?

We made an unstructured interview with the teacher about his observations during the test, and about his opinion about the possible long-term impacts of the paired testings.

What are the students' attitudes to paired testing? We can assume from the result of the questionnaire that most student perceived the test as a fair one. We can also assume more confidently, that students have a positive attitude towards paired testing: they rated positively the concept, and they also rated their testing experience positively, despite, as in some cases, their initial not so positive attitude to their partners.

How did students respond to paired testing? Their attitudes towards each other improved. The overall level of paired work was high, the style of writing had more clarity than solo work. According to the teacher's interview, in general, students collaborated intensely but quietly, not disturbing each other.

Given the small sample size and time-frame, further research is needed to determine how paired testing improve collaboration and mastery of skills.


Christina Drüke-Noe
email: druekenoe@ph-weingarten.de

Hans-Stefan Siller
email: hans-stefan.siller@mathematik.uni-wuerzburg.de

Diagnosis of basic mathematical competencies in years 8 and 9

Keywords: mathematics tasks, diagnostic test, assessment, basic competencies, vocational training

In this paper we present a newly developed test with easy tasks which assesses basic mathematical competencies that are essential for vocational training. This test is used in years 8 and 9 in Germany. From both a content-oriented and a process-oriented perspective, preliminary test results reveal a substantial lack of these competencies in all academic tracks, and differential effects can be found with respect to year and class that need to be examined beyond curricular analyses.

The role of tasks in Mathematics

Especially in Mathematics tasks are essential and thus they are being used from various perspectives in both research and teaching. In any case they require engagement with a certain mathematical content (e.g. [START_REF] Christiansen | Task and Activity[END_REF], and they are understood here as orally or written set assignments to students to carry out a subject-oriented activity (Drüke-Noe et al., 2017).

In research, tasks can serve as documents of mathematics instruction. This applies e.g. to the project COACTIV in which tasks are used not only to assess teachers' professional knowledge but also to get insight into the level of cognitive activation potentially realized in teaching or assessing mathematics [START_REF] Kunter | Cognitive Activation in the Mathematics Classroom and Professional Competence of Teachers[END_REF]. Tasks also form the linking element between curricular standards and the professional knowledge of teachers on the one hand and the aimed-at knowledge, skills and competencies of students on the other hand [START_REF] Neubrand | Task Analysis in COACTIV: Examining the Potential for Cognitive Activation in German Mathematics Classrooms[END_REF]. Furthermore, tasks are essential in tests to assess the achievement of learning goals. For this purpose, oral and especially written tests are regularly set in Germany. These tests, which are either set internally or externally, comprise a range of tasks of a certain cognitive demand to examine competencies either formatively or summatively. Due to their backwash effect on teaching as well as their impact on the students' further educational career, centrally set external examinations at the end of various stages of education are quite prominent examples of summative written tests. Interestingly though, in several European countries as well as in different federal states of Germany such centrally set exams are often of a relatively low level of cognitive demand and cover only a limited range of competencies [START_REF] Kühn | Qualität und Vergleichbarkeit durch Bildungsstandards und zentrale Prüfungen? -Ein bundesweiter Vergleich von Prüfungsanforderungen im Fach Mathematik zum Erwerb des Mittleren Schulabschlusses[END_REF][START_REF] Drüke-Noe | Cognitive Demand of Mathematics Tasks set in European Statewide Exit Exams -Are some Competences more demanding than others[END_REF]. This also applies to internally written mathematics tests in Germany, so called class tests [START_REF] Drüke-Noe | Aufgabenkultur in Klassenarbeiten im Fach Mathematik[END_REF]. From a normative perspective, however, adequate compilations of tasks in tests are crucial as they "summarize the core components of an instructional unit and ultimately specify the level of mathematical achievement that teachers require" (Neubrand et al. 2013, p. 128) and thus make social expectations transparent for both teachers and students.

It can be concluded that tasks are "actively configurable content-related and didactic elements that serve to structure mathematics instruction" (Neubrand et al., 2013, p. 126; see also Zaslavsky, 2007, p. 434) that "provide the basis for students' cognitive activities" (Neubrand et al., 2013, p. 127) and therefore play an essential role in teaching, learning and assessing mathematics.

Tests and Basic Competencies

In Germany, as a consequence from only mediocre results in international comparative studies like e.g. TIMSS and PISA, educational standards have been implemented in almost all subjects in the last fifteen years. In the course of this, the focus of attention shifted towards regular formative examinations of learning goals and competencies, and several centrally set formative tests were introduced nationwide in Germany that serve as diagnostic instruments. These tests are designed on the basis of the German Educational Standards and are in accordance with the standards' conception. Therefore, their tasks cover all five content strands (quantity, measuring, space and shape, change and relationships, uncertainty and statistics), all six competencies (argumentation, problem solving, modeling, use of representations, working technically, communication) and all three levels of cognitive demand (low, medium, high). Two different kinds of tests are set: A first kind is compulsory comparative tests which are set in the years 3 and 8 (9-year-and 14-year-olds) which are administered as well as corrected within schools. The test results are to provide diagnostic information in which areas both teaching and learning should be improved to prepare students for a successful completion of compulsory education one or two years later at ISCEDlevel 2. A second kind of tests is centrally set every six years in the years 4 and 9. Its conception is standard-based, too, but this nationwide test is only taken by a representative sample of students and its results are only used by the education administration to monitor the school system with respect to the successful achievement of the educational standards. For both years 4 and 9 and especially for the latter one results reveal regularly that on average one quarter of all German students miss minimal educational standards [START_REF] Pant | Mathematische und naturwissenschaftliche Kompetenzen am Ende der Sekundarstufe I[END_REF]. Very similar findings come up in a German longitudinal study on Bavarian secondary students of different school tracks [START_REF] Vom Hofe | Zum Problem von Mindeststandards und der sogenannten Risikogruppe[END_REF] as well as in regular international studies like PISA (e.g. [START_REF] Neubrand | Innere Strukturen mathematischer Leistung im PISA-2000-Test [Internal structures of mathematics achievement in the PISA 2000 test[END_REF]).

To summarize, there is much empirical evidence for the mathematical achievement of year 9students that firstly implies that 16-year-olds, who leave school at the end of lower secondary education and are about to enter a phase of vocational training, lack substantial basic skills such as a reflective use of knowledge and abilities. Secondly, these students presumably lack not only the basis for an understanding of routines beyond their mechanical application but also knowledge of problem solving activities. Consequently, it seems not only necessary to diagnose such basic competencies well beforehand but alsowhich is even more importantto support these lowachievers in time both on a class-level and individually, too. More efforts need to be taken so that in the future less students fail minimal standards since a successful achievement of at least minimal standards is a precondition to successfully begin (and complete) a vocational training after the end of year 9 (or year 10).

These necessities form the basis of our task-based project "Diagnostic scaffolding of basic mathematical competencies for vocational training". In this project we assess basic mathematical competencies well before students leave school. We test entire classes and can provide teachers with quantitative and qualitative feedback on the test results. Teachers can use this feedback to support their students to better achieve basic competencies by the end of compulsory education.

Basic Competencies and the Start of Vocational Training

Based on the German Educational Standards we designed a test that only assesses basic mathematical competencies relevant for vocational training. As an adequate selection of tasks for such a test substantially relies on both an understanding of basic competencies as well as of what is necessary to successfully start a vocational training, we briefly outline essential elements of both here.

In Germany, there are different empirical and normative approaches to defining basic competencies for the end of compulsory education, which is usually the end of lower secondary education irrespective of successfully obtaining a formal school leaving certificate. However, there is no consensus as to whether basic competencies characterize what is really necessary to be prepared for a life as a citizen, for a successful start and/ or completion of a vocational training. It is equally diverse who defines basic competencies; is it teachers or researchers or those who continue working with these students after a certain stage of education? Some other task-based approaches to defining basic competencies which are relevant for our one and only focus on the end of lower secondary education are briefly outlined in the following: One primarily normative concept by [START_REF] Sill | Leistungserhebungen im Mathematikunterricht[END_REF] considers empirical results from national comparative studies and defines basic mathematical competencies needed for further education, everyday life and society. It comprises three so-called competency levels that distinguish whether certain knowledge and abilities should be immediately and automatically applicable or be easily re-activated or be only exemplary and episodal. A second concept is essential for our approach as it is explicitly based on the German Standards. It combines a normative with an empirical approach and defines basic mathematical competencies as those which all students of all educational levels must at least and permanently have at the end of compulsory education. These competencies are a prerequisite for sovereignly mastering everyday life and actively participating as politically mature citizens in social and cultural life. These competencies are equally a prerequisite for a successful and promising start into a vocational training and a pursuance of one's profession. Those who do not have these basic competencies will presumably not sufficiently be able to get along in these situations without help. These students must timely and particularly intensively be supported (Drüke-Noe et al., 2011, p. 8).

A third German and predominantly empirical approach is the "Model of Competency Levels for the Educational Standards for a qualified school leaving certificate at the end of year 9 and at the end of year 10" which was passed by the Standing Conference of the Ministers of Education and Cultural Affairs. Based on results from several large scale studies and applying probabilistic test theories which relate task difficulties to students' abilities, this model is based on a competency scale which distinguishes six competency levels. These levels describe students' abilities on the basis of task properties. The lowest two levels, which we only focus on here, define the minimal standard as a set of competencies that all students must have achieved by the end of a certain educational stage. Students at or below this minimum standard can only comprehend or apply the most simple standard argumentations or problem solving activities, use simple models or well-known visualizations, apply one-step routines or extract single pieces of information from simple texts (Kultusministerkonferenz, 2011, p. 29ff).

Several other international approaches to basic mathematical competencies also relate structural errors to procedural processes: [START_REF] Brown | Diagnostic Models for Procedural Bugs in Basic Mathematical Skills[END_REF] already do so in their "Diagnostic Models for Procedural Bugs in Basic Mathematical Skills" and stress the close relation between basic skills, procedural and conceptual knowledge. In his analyses of structural errors with respect to procedural knowledge Wu (1999, p. 1) stresses that "in mathematics, skills and understanding are completely intertwined". Ritter-Johnson, Siegler and Alibali (2001, p. 346) equally claim that "conceptual and procedural knowledge influence one another […] and develop iteratively, with increases in one type of knowledge leading to increases in the other type of knowledge".

Consequently, a traditionally rather content-focused perspective on teaching (and testing) ought to be widened to a more comprehensive one which considers both content and processes not only when putting curricular regulations into practice but also when teaching and testing basic competencies whichamongst othersare necessary to work successfully on mathematics tasks. With respect to vocational training, Weißeno and colleagues (2016, p. 3) additionally point out that "mathematical skills play an important role in developing job specific competencies in a number of occupations that require intermediate qualifications." Thus, learners do not have to be experts in the field of mathematics, but they must be able to implement the necessary basic skills, as well as be able to critically reflect upon and to apply processes in the course of solving tasks to correspond with the concept of "Higher General Education" as developed by [START_REF] Fischer | Höhere Allgemeinbildung[END_REF].

In our project, we designed a task-based test to assess students' basic mathematical competencies relevant for vocational training. Due to our test conception we can derive information on these competencies from both a content-and a process-oriented perspective. Furthermore, this information and deeper test analyses allow us to deduce a specific feedback to support teachers and students.

Test Conception and Task Features

To both evaluate basic competencies and consider a prospective ability to take up vocational training, our test addresses students who aim at getting a qualified school leaving certificate at the end of year 9 or at the end of year 10 (ISCED-level 2). To allow for sufficient time for compensation before leaving school, both versions of this test (see below) can be taken well before, i. e. in year 8 (14-year-olds) and/ or in year 9 (15-year-olds) and can be set in all school tracks.

Since the test's conception is based on the Educational Standards, it covers all five content strands, all six competencies and all three levels of cognitive demand (see Table 1). All tasks were selected correspondingly and thus allow analyses from a content-and a process-oriented perspective. Based on cognitive analyses we can describe the content addressed and the competencies needed when solving a task, which we briefly outline below by means of three examples. As only basic competencies are to be assessed, only relevant content areas of the curriculum of the preceding years 7 and 8 are tested. This curricular validity has been approved by experts. Almost all tasks are presumed to be empirically easy (i. e. at least 80 % correct answers are expected) as they were developed in accordance with characteristics of the two lowest levels of the competency levelmodel (KMK, 2011). However, from our perspective relevant basic competencies not only comprise the ability to apply certain routines and procedures to show how something works but also to use them critically and/ or to explain why something works: Therefore, we intentionally included some presumably empirically more difficult tasks in the test and only for these we anticipated less than 80 % correct answers.

Calculate the area of the triangle.

Use the illustration to explain why the area of a parallelogram is determined by applying the rule Area = Length of Base • Height.

Heike's savings book shows 180 €. The interest rate is 2 % per annum.

Calculate the amount of interest after 4 months. Three tasks (see Figure 1) are to illustrate our approach: The task "Triangle" is cognitively easy and it tests how something works. The representation immediately leads to a procedure to calculate the area by means of the given lengths of the triangle's base and height. This task ought to be empirically easy as calculating the area of this elementary figure requires just one step with small numbers. The task "Parallelogram" represents those that test why something works. The rule to determine the area of this quadrilateral is explicitly named. Using the representation students need to develop a strategy to explain why the area can be calculated by means of this rule. According to the competency level-model (KMK, 2011) it must be assumed, though, that this is no empirically easy task as it requires an argumentation. In the task "Savings book" students must initially extract relevant information from a simple text (communication) to then design a strategy and apply a basic mathematical model to calculate the amount of interest (problem solving, modeling, working technically). The two test versions (B: basic version, E: extended version) comprise 15 tasks each with three tasks per content strand that all require open answers. Eight tasks are identical in both versions and almost all the remaining seven ones are parallel versions of each other which differ either only as regards their presumed task difficulty or as regards the track-specific curriculum. The students have 45 minutes to work on the test, and they may only use pens, a ruler and a device to measure angles. Before the test was finally used, it was tried and tested beforehand and improved further.

By means of a coding scheme and applying a dichotomous coding with double digits, different types of correct, partly correct or wrong answers are identified. Further analyses of the test results allow us to provide teachers with quantitative feedback (charts on types of answers with respect to content strands) and with qualitative feedback on missing basic competencies. Thus, this diagnostic test with its combined content-and process-oriented design and the subsequent detailed feedback may not only close a gap PISA leaves but also provides teachers with class-specific information on (missing) basic competencies of prospective school leavers. Teachers can then use this detailed information to develop suitable interventions to better support their students to acquire basic competencies.

Sample and preliminary results

The test was set to a non-representative sample of students (N=367) in 16 academic and nonacademic classes (see In accordance with the standard-based test conception we present preliminary findings firstly from a content-oriented perspective and then from a process-based one. Irrespective of track and year, hardly one quarter of all students (23 %) correctly solved both tasks on the area of elementary figures (see Figure 1), and almost equally many students (26 %) solved none of these two correctly.

It is interesting to note that the task "Triangle", which rather focuses on procedures (knowing how), was solved considerably better by students from the nonacademic track (nonacademic track: 76 %, academic track: 64 %). However, considering both this task's features and the predominance of routines in Germany, a much higher percentage of correct answers could have been expected. Less surprising, far more students from the academic track were able to solve the task "Parallelogram" (nonacademic track: 6 %, academic track: 42 %) which rather focusses on understanding (knowing why). As expected, in both tracks students of year 9 solved both tasks better than those in year 8 but from a content-based perspective the overall results of both tasks are rather astounding as the area of such basic figures is an integral part of every curriculum. Error analyses reveal that students typically fail when distinguishing formulae (here: area of a rectangle and that of a parallelogram) or when explaining a rule (see "Parallelogram") although it is explicitly given here.

For analyses from a process-oriented perspective we focus on modeling here. Detailed analyses of the students' work rely on steps of the modeling cycle by [START_REF] Blum | Modellieren im Unterricht mit der "Tanken"-Aufgabe. mathematik lehren[END_REF]. As expected and irrespective of track and year, in all six modeling tasks students primarily find it difficult to understand the real situation and then to constitute a situation model. Furthermore, validating a result proves to be another difficult step even for students of the academic track. Surprisingly, in year 9 only 39,2 % of the students in the academic track solved "Savings book" correctly (nonacademic track: 20,7 %) and many typically failed when validating an even obviously wrong result like e. g. an annual amount of interest of 9000 €. Based on detailed analyses of this kind we provide teachers with both quantitative and qualitative feedback on their students' basic competencies. The qualitative feedback is based on cognitive analyses of all tasks which serve as a basis for classspecific descriptions of the most frequent mistakes per task. All these descriptions help to characterize existing and lacking basic mathematical competencies within each class and can support teachers in their successive work.

Summary and Implications

The preliminary results of our test correspond with results from representative national and international studies. Beyond this, our test which is curricular valid and addresses basic mathematical competencies relevant for vocational training once more highlights a discrepancy: Though these competencies are broadly considered to be essential, considerably high proportions of students can neither use nor apply them and thus lack crucial preconditions to successfully start a vocational training and succeed in their later professional careers. This leaves many things to do: From both a scientific and a diagnostic perspective, further analyses beyond curricular ones are necessary to explain unexpected differential effects as regards tracks, years and classes. Secondly, far greater efforts are needed to turn all students into citizens of age (cf. [START_REF] Fischer | Höhere Allgemeinbildung[END_REF] and to provide them with basic mathematical competencies. We are currently developing adaptive lesson material to be implemented (and evaluated) in teaching. This material specifically addresses students' typical difficulties as regards acquiring and keeping basic mathematical competencies.

The material aims at supporting students to better solve tasks on how something works as well as those on why something works, which means that all students should be able to write down argumentations or to validate results to name just a few examples. Last but not least it will be necessary to improve not only the tasks but also the kinds of feedback we provide the teachers with.

Figure 1 :

 1 Figure 1: The tasks "Triangle" (left), "Parallelogram" (middle) and "Savings book" (right)

Table 1 : Numbers of tasks that require one or more of the six competencies and numbers of tasks of the given level of cognitive demand

 1

				Competence			Level of cognitive demand
	Test ver-sion	Argumen-tation	Problem solving	Modeling	Use of Represen-tations	Working technically	Communi-cation	low	medium	high
	B	4	2	4	11	9	7	3	11	1

  Table 2) from two German states (Baden-Wurttemberg, Rhineland-Palatinate). The test took place at the beginning of the second semester in the school year 2016/17.

	Test version	Academic track	Noncademic track
	B	---	Baden-Wurttemberg: 2 classes in year 8 (N=52), 2 classes in year 9 (N=45)
	E	Rhineland-Palatinate: 2 classes in year 8 (N=52), 8 classes in year 9 (N=189)	Baden-Wurttemberg: 2 classes in year 9 (N=29)

Table 2 : Sample of classes and students that took the diagnostic test

 2