text:
Ralf Benölken
email: rben@wwu.de

Developing student teachers' professional knowledge of low attainments' support by "learning-teaching-laboratories"

Keywords: Professional development, learning-teaching-laboratories, low attaining students

This paper presents a study that is part of a project named MaKosi ("Mathematische Kompetenzen sichern"). It aims at the conception and evaluation of a program in which primary student teachers and children who are low attaining in arithmetic work together. The organization refers to a specific form of project seminars called "learning-teaching-laboratories". The study investigates how knowledge of identifying and supporting low attaining children develops by participating in such a program. Qualitative data were generated by learning maps in a pre-post-design and analyzed by a reconstructive pedagogic-iconological image interpretation. The results indicate a sustainable positive development of student teachers' knowledge.

Introduction

Concepts of teachers' professionalization are an important focus of current research in mathematics education (e.g., [START_REF] Dzlm | Theoretischer Rahmen des Deutschen Zentrums für Lehrerbildung Mathematik[END_REF]. Regarding a specific professional knowledge of teachers, the development of pedagogical content knowledge (PCK) including, in particular, the development of abilities to analyze children's thinking and learning is reputed to be one of the main goals [START_REF] Sowder | The mathematical education and development of teachers[END_REF]. Moreover, beyond cognitive aspects, some recent approaches on teachers' knowledge consider affective components like beliefs [START_REF] Kuntze | Pedagogical content beliefs: global, content domain-related and situation-specific components[END_REF]. Questions that arise from this are, e.g., how student teachers' education on analyzing children's learning trajectories can be realized, and as a result how cognitive and co-cognitive components of student teachers' knowledge develop. In this paper, the attention will be given to the aspects mentioned above by a synthesis of different approaches within a qualitative study: With regard to analyses of children's thinking and learning, the development of primary student teachers' knowledge about the identification and support of low attainments (ISLA) is focused on, since analyzing mistakes is assumed to be a valuable resource in this context (e.g., [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF]. As to a suitable organization of a professional development program, the approach of "learning-teaching-laboratories" (LTL) is applied, which reflects an important part of current discussions on student teachers' education in Germany [START_REF] Roth | Lehr-Lern-Labore Mathematik. Gründung eines neuen GDM-Arbeitskreises[END_REF]. Summarized, LTL provide project seminars intertwining student teachers' theoretical and practical education by working with children, i.e., via learning by teaching. The following questions will be investigated: How can a development of student teachers' knowledge about analyses of children's thinking and learning in mathematics be organized by a connection of ISLA-and LTL-concepts? How does their knowledge by taking part in an ISLA-LTL develop? First, brief overviews of the theoretical frameworks will be given. On these bases, a LTL-concept will be outlined. Finally, the study's design and results will be subsumed and discussed.

Thematic Working Group 20

Proceedings of CERME10

Theoretical frameworksbrief overviews

As to teachers' professional knowledge, the classical concept distinguishes between subject matter knowledge, PCK and curricular knowledge. Summarized, PCK refers to knowledge of possibilities regarding teaching subject matters [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF]. Independent of certain approaches, there seems to be a consensus on the fact that PCK bridges subject matter knowledge and teaching, and it designates a specific-distinctive manner of teachers' professional knowledge [START_REF] Brown | Becoming a mathematics teacher[END_REF]. According to [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF], PCK covers knowledge of content and students, knowledge of content and teaching, and knowledge of curriculum. In particular, the first mentioned aspects provide facets that are connected to analyzing children's thinking and learning as well as to providing an adequate support [START_REF] Sztajn | Learning trajectory based instruction: Toward a theory of teaching[END_REF]. Because drawing an exact distinction between cognitive aspects regarded by knowledge and affective aspects like beliefs is felt to be difficult, some current research combines both aspects to describe teachers' professional knowledge; inter alia, pedagogical content beliefs (PCB) are described as an equivalent to PCK: Convictions about handling specific instructional situations [START_REF] Kuntze | Pedagogical content beliefs: global, content domain-related and situation-specific components[END_REF] like ISLA. Recent competence frameworks of professional development programs are in line with such approaches. Beyond mathematics-related beliefs, self-oriented ones are considered including components like self-efficacy (e.g., [START_REF] Dzlm | Theoretischer Rahmen des Deutschen Zentrums für Lehrerbildung Mathematik[END_REF], which produces a more holistic view. The study's framework refers to [START_REF] Kuntze | Pedagogical content beliefs: global, content domain-related and situation-specific components[END_REF]. Thus, a combination of cognitive (PCK) and affective aspects (PCB) is assumed. Additionally, self-oriented beliefs are considered in the context of PCB, since in this way a holistic base to describe changes of knowledge by taking part in a LTL is given.

Research on individual problems in learning mathematics covers a large range: Beyond approaches that describe such problems as a social construct, or approaches focusing on learning difficulties or disabilities in a narrower sense (for a survey: [START_REF] Scherer | Assistance of students with mathematical learning difficulties: how can research support practice?[END_REF], different approaches focus on previously low achievements (e.g., [START_REF] Watson | Learning coherent mathematics through sequences of microtasks: making a difference for secondary learners[END_REF]. The perspective mentioned last mostly concentrates on arithmetic and in this context on typical phenomena such as rigidified counting (and a unilateral ordinal understanding of numbers) and an insufficient understanding of mathematical operations or the place value system (for surveys: [START_REF] Benölken | MaKosi -Ein Förder-, Lehr-und Forschungsprojekt im Themenkomplex "Rechenprobleme[END_REF][START_REF] Denvir | Understanding of number concepts in low attaining 7-9 year olds: part I[END_REF]. Mostly, a group of children is addressed which can be supported within a school's infrastructure, i.e., which does not show learning difficulties in the outlined narrower sense. As to identification or support, recent research independent of certain approaches demands a holistic view considering both cognitive and co-cognitive parameters (e.g., [START_REF] Nolte | Rechenschwäche und Fördermöglichkeiten[END_REF]. Against the background of student teachers' education, the theoretical framework of both the LTL and, thus, the study corresponds to different aspects of the above outlined approaches: As to problems in learning mathematics, low attainments in arithmetic are focused on considering both typical phenomena and a holistic view in the identification and support procedures.

As to the development of PCK, practical situations that demand, e.g., scaffolding skills are assumed to be adequate opportunities of extending knowledge [START_REF] Prediger | How to develop mathematics for teaching and for understanding[END_REF]. Existing findings indicate that one-to-one-interactions of a student teacher and a child might be a promising organizational form (e.g., [START_REF] Kilic | Pre-service mathematics teachers' scaffolding practices[END_REF]. Against this background, "learning-teaching-laboratories" aim at a mutual growth and practical application of knowledge by a specific form of academic studies combining three dimensions: First, the support of children regarding a certain topic; then, the education of student teachers in this context, e.g., as to diagnostics and support; third, research aims like theory building in the content focused on [START_REF] Roth | Lehr-Lern-Labore Mathematik. Gründung eines neuen GDM-Arbeitskreises[END_REF]. Recent research mainly concentrates on a clarification of LTL-types and on an interdisciplinary consensus about defining the term of LTL. An example is given by the following definition: LTL define a specific form of organization as to student teachers' academic studies combining children's learning with student teachers' professional development in a holistic way. In contrast to, e.g., standard lectures, seminars or practice lessons, LTL offer student teachers opportunities to develop, to enhance and to apply iteratively various skills of diagnostics, support and, thus, both teachers' professional acting and knowledge with regard to specific focuses in authentic, but complexity-reduced learning situations. [START_REF] Brüning | Untersuchungen zur Profilbildung und Evaluation von Lehr-Lern-Laboren im Entwicklungsverbund "Schülerlabore als Lehr-Lern-Labore" der DTS[END_REF](Brüning, , p. 1274; translated by the author) Hence, LTL include aspects and influences that are considered as most important by approaches on teachers' professional growth like the individuality of their learning in mutual reflection and enactment processes (e.g., [START_REF] Clarke | Elaborating a model of teacher professional growth[END_REF]. Even if LTL are conducted at more and more German universities in different "STEM"-disciplines, ongoing studies still focus on their evaluation. First impressions indicate that LTL are highly valued by student teachers and they are suited to ensure a sustainable growth of their knowledge about the respective topic (e.g., [START_REF] Brüning | Untersuchungen zur Profilbildung und Evaluation von Lehr-Lern-Laboren im Entwicklungsverbund "Schülerlabore als Lehr-Lern-Labore" der DTS[END_REF]. The study's framework refers to Brüning's definition. Its cornerstones are transferred to the context of ISLA. The demanded complexity-reduction is realized by one-to-one-interactions.

Survey of a LTL-concept in the context of ISLA

The presented study is part of the long-term project "MaKosi" that focuses on the conception and evaluation of a professional development program connecting ISLA-and LTL-approaches (for details [START_REF] Benölken | MaKosi -Ein Förder-, Lehr-und Forschungsprojekt im Themenkomplex "Rechenprobleme[END_REF]. Summarized, the aims are the support of children low attaining in arithmetic and the development of student teachers' knowledge of ISLA. The student teachers' education is organized as a combination of a theoretical course and a project seminar with children. The theoretical course covers information about approaches in the field of problems in learning mathematics as well as concepts of diagnostics and support. While the theoretical course is a regular seminar at university, the project seminar takes place at a primary school once a week about 15 times per semester. Against the outlined framework of ISLA, diagnostics triangulate different tools: In a first step, teachers are given information about the framework and they elect children providing a justification in written form. Then, parents have to fill in a declaration of consent. In a second step, children, student teachers and scientists come together to get to know each other in a playful first session. In a third step, process-diagnostics follow considering both cognitive and co-cognitive parameters; mostly, non-standardized tools such as observations on children's task solving using rating sheets or guided interviews with children, teachers or parents are applied. Every project seminar session is divided into three parts: First, a preparing workshop where student teachers and scientists come together for 15 minutes in order to highlight specific aspects of observation or other determining factors; second, a 90-minute-children-session; finally, a reflecting 75-minuteworkshop, in which each child's problems and possibilities as to an appropriate support are discussed. Within this schedule, the children's session is divided into three stages: In the beginning, a playful problem task is offered avoiding arithmetic contents to provide an adequate imagination of mathematics or to support both a positive self-perception of mathematical abilities and joy of problem solving (for example, the problem of "a ferryman, a wolf, a sheep and a head of cabbage").

At this stage, children can organize themselves considering ideas of a natural differentiation. Subsequently, one student teacher and one child turn into one-to-one-interactions of diagnostics and support in established teams for 60 minutes. Thus, the student teachers can develop, e.g., scaffolding abilities in a complexity-reduced situation. Tasks and activities applied in this context are taken from well-proven examples of literature (for examples see: [START_REF] Benölken | MaKosi -Ein Förder-, Lehr-und Forschungsprojekt im Themenkomplex "Rechenprobleme[END_REF], which the student teachers got to know in the theoretical course. They develop suggestions on both their compilation and detailed planning which are discussed during the reflecting workshop with all participating student teachers and the supervising scientists in order to ensure sustainable conducts. Each session closes with a game to support the children's joy of participating in the LTL.

The study

The study focuses on the question how student teachers' knowledge of ISLA develops by taking part in an ISLA-LTL. The participants were 25 primary student teachers; 11 (10 females, 1 male) took part in the winter semester 2015/2016, and 14 (only females) in the summer semester 2016. Mostly, they were in their third year of undergraduate studies. The study's character is explorative, i.e., generalizations were not intended, but existential propositions [START_REF] Lamnek | Qualitative Sozialforschung[END_REF] about possible developments of knowledge by participating in the LTL. Thus, a qualitative design was advisable. As to the method, qualitative data were generated according to Rott (2017) by applying learning maps in a pre-post-comparison which were anonymized by codes to ensure unbiased interpretations. In the head, the student teachers were given the impulse to craft their way between their current status and their future work at schools: "Dealing with low attainments will be a challenge as to your work as a teacher, especially due to the knowledge of identification and support: What does this mean to you personally? Which way have you covered or which way will you have to cover in the future? Please lay out your way." (translated from German) All participants designed the maps for the first time, and they had to do it before taking part in the LTL at the beginning of a semester, and, again, at its end. As to the analysis, the pre-and post-maps were compared by a reconstructive pedagogic-iconological image interpretation, which becomes more and more accepted in different scientific disciplines. Its characteristic steps were observed: (1) Discussion of previous history and selection of key images, (2) image description and analysis (with regard to the factual, expressive and form-related sense), (3) context analysis, and (4) comparative analysis [START_REF] Schulze | Bildinterpretation in der Erziehungswissenschaft[END_REF]. Data were interpreted within two meetings at the end of the summer semester 2016: The 14 participants analyzed in groups of two or three at least one, but for the most part two pairs of maps. Afterwards, the results were presented, and major observations were discussed in a plenary session.

Results

As to key images, their description and analysis, three types were identified: (1) An interrupted path, (2) a continuous path, and (3) a system of paths. The first type was found only within the pre-, the third one only within the post-, but the second type within both the pre-and post-drawings. Subsequently, we focus on the reconstructions of the examples shown by the Figures 1 and2, which were conducted in the group meetings mentioned above and which reflect typical main features. As to the factual sense of the first example's pre-map (Figure 1, left), a lack of details is obvious, which might reflect that the creator is unfamiliar with the context. The expressive sense is characterized by monotony as to, e.g., colors, which might indicate the creator's uncertainty. Regarding the sense of form, the interrupted way seems to reflect that the creator cannot (yet) imagine how to achieve the purpose. As to the factual sense, the post-map (Figure 1, right) contains more details: Different remarks are phrased; thus, the creator seems to connect many thoughts to the path. Merged stars seem to reflect interdependent experiences that influenced the creator positively, but a question mark seems to indicate obscurity about future requirements. The expressive sense is characterized by a use of different colors highlighting the significance of the experiences' connection, for instance. As to the sense of form, the continuous path obviously reflects that the creator now perceives a way to achieve the purpose, even if it is flanked by the question mark. The path precedes the current status; thus, the creator seems to have developed a more holistic view on the way he or she passed. As to the factual sense of the second example's pre-map (Figure 2, left), a main feature is a wide range of remarks, which seems to reflect that the creator already connects several aspects to the path. An important detail is the remarks' phrasing in the form of questions in most of the cases; thus, the creator rather seems to ascribe uncertainty or just a small level of recognition to him-or herself. Moreover, clouds seem to emphasize particular past and future experiences. The expressive sense is rather monotonous, e.g., as to the coloring, which might indicate that the creator refers to a matter-of-fact way. Regarding the sense of form, the continuous path is drawn as a stairway; thus, the creator obviously distinguishes different steps of his or her knowledge's complexity. Arrows emphasize secondary objects, which might reflect that the creator at least considers different complex patterns. As to the factual sense of the post-map (Figure 2, right), a wide range of remarks still can be observed, but now they are put forward in the form of declarative sentences. Signposts seem to describe possible intentions and their connections. Clouds and boxes seem to highlight important (mostly past) theoretical and (especially future) practical experiences. A computer seems to indicate an intertwining of the regarded focus with other domains. Finally, sections which the creator already had passed are characterized by continuous, but future sections by dotted lines. The expressive sense is characterized by different colors which underline the significance of main experiences; overall, a great confidence seems to be reflected. Aspects related to the sense of form confirm this impression: In contrast to the circle drawn in the first section of the map, the path continues afterwards directly, but winding to school, flanked by some concrete imaginations. The final part of the path system is drawn slightly broader compared to previous sections which might reflect that the creator will feel well prepared to enter school. In the creator's view, practical work seems to bridge impressions of running in a circle within theoretical studies, which are not connected directly to work at school, and achieving the objective, which is supposed to be a complex, but positive and manageable challenge.

As to a context analysis of the first example, the interrupted way of the pre-map (Figure 1, left) seems to reflect that knowledge on ISLA is assessed to be nonexistent or at least superficial by the creator (PCK), and he or she cannot imagine how to cope with ISLA at school (PCB). Comparing the post-map (Figure 1, right) indicates that the creator now reflects to have knowledge on ISLA (PCK), and that he or she perceives ways of handling ISLA at school (PCB). This impression is confirmed by concrete aspects and intentions given by the detailed remarks (e.g., top right, "selfresponsibility", "perspective of hope to face the topic in the future"), which were missing in the premap, and by elements like stars or colors, which underline the significance of taking part in the LTL as to developing knowledge of ISLA and, thus, of bridging theoretical knowledge and practical work. Regarding the second example, the continuous path shown by the pre-map (Figure 2, left) indicates that the creator already knows about some aspects of ISLA, even if this knowledge seems to be rather superficial (PCK) and he or she seems to be rather uncertain (PCB). Comparing this with the post-map (Figure 2, right) suggests that the complexity of the system of paths reflects an increase of knowledge (PCK), and the creator can well imagine to cope with ISLA at school (PCB). Questions posed in the pre-map (e.g., on the first step, "What has to be done?") turned to concrete intentions and planning steps (e.g., top left, an intention as to the practical semester of academic studies "Enriching scientific knowledge by specific focuses of observation."); particular attention is given to emphasize the significance of practical experiences, like taking part in the LTL, as to developing knowledge of ISLA and as to intertwining theoretical knowledge and practice.

Based on these examples, a comparative analysis of all pre-and post-maps suggests that both the student teachers' ways and their location on the ways changed. Their PCK of ISLA developed to more profound patterns, and their PCB to more confident characteristics. Put more precisely, the comparisons indicate mainly the following typical changes: As to cognitive aspects, before taking part in the LTL the student teachers' knowledge about ISLA seemed to be rather fragmentary and superficial for the most part. Moreover, most of them seemed to equate problems in learning mathematics unilaterally with learning difficulties in a narrower sense. In contrast, after participating in the LTL, the student teachers developed a complex knowledge of the entire field: They distinguished different approaches and considered phenomenology-related issues. Regarding affective aspects, before taking part in the LTL, the student teachers rather seemed to express uncertainty as to dealing with ISLA, which seems to reflect disadvantageous characteristics of selfefficacy. In contrast, after participating in the LTL, the maps indicate more proactive views: The student teachers declared more complex perceptions of problems in learning mathematics, in particular as to a child's individuality, and they proposed precise plans to develop their knowledge. Moreover, they seemed to connect ISLA closer to teachers' responsibilities, and emphasized the significance of practice as to the development of aspects such as self-efficacy. Finally, as indicated by the discussed examples, there seem to be different types: A first one representing an "optimistic novice" (Figure 1), and a second one representing an "expectant expert" (Figure 2).

Discussion

The results indicate that participating in an ISLA-LTL and, therefore, an intertwining of theory and practice contributes sustainably to a positive development of both student teacher's knowledge about ISLA and their abilities to analyze children's thinking and learning in mathematics. This observation is in line with reports emphasizing the benefits of practical work with children as to the development of knowledge, and one main reason for this might be seen in the student teachers' constructivist learning [START_REF] Sowder | The mathematical education and development of teachers[END_REF]. Thus, beyond the context of ISLA, the results suggest the hypothesis that LTL provide an appropriate professional development program for student teachers to develop their abilities in analyses of children's thinking and learning. Of course, the study's character is explorative, and it has obvious limitations; for instance, the reconstructions as to interpreting the use of colors (or, e.g., their absence) were conducted within a group, but it remains uncertain, if a consensus view is the right one [START_REF] Lamnek | Qualitative Sozialforschung[END_REF]. Subsequent research might focus on a deeper clarification as to evaluations of ISLA-LTL, and as to the benefits of LTL in general; in particular, different LTL-focuses like support of mathematics interest should be taken into account.

Figure 1 :Figure 2 :

 12 Figure 1: First example of a pre-(left) and post-map (right)

Proceedings of CERME10


domain: math shs

text:
Kristin Krogh Arnesen

Ole Enge

Yvonne Grimeland
email: yvonne.grimeland@ntnu.no

Torkel Haugan Hansen

How do prospective teachers imagine mathematical discussions on fraction comparison?

Keywords: Mathematics knowledge for teaching, representations of teaching, classroom discussion, lesson planning, rational number sense

Prospective teachers plan a short mathematical discussion on comparing fractions by writing lesson plays. We analyse how their mathematical knowledge for teaching surfaces in their written scripts, using three dimensions of the Knowledge Quartet: foundation, transformation and connection. Our findings give insight into the prospective teachers' knowledge of fractions and comparison strategies, their perspectives on mathematics and mathematics teaching, and insight into how they transform their knowledge to make it accessible to middle school students.

Introduction

Several recent studies call for a practice-based approach to research on teacher education (e.g. [START_REF] Ball | Developing practice, developing practitioners: Toward a practice-based theory of professional education[END_REF][START_REF] Grossman | Back to the future: Directions for research in teaching and teacher education[END_REF]). An integral part of teachers' work is planning for teaching. Recently, several researchers have advocated writing lesson plays as a means of learning how to plan for instruction (see [START_REF] Zazkis | Lesson plays: Planning teaching vs. teaching planning[END_REF]. Lesson plays are imagined mathematical discussions written verbatim. [START_REF] Zazkis | Lesson plays: Planning teaching vs. teaching planning[END_REF] argue that lesson plays can give a "window" for researchers to investigate mathematical knowledge for teaching. We have collected and analysed prospective teachers' (PTs') planning documents for a practice assignment in their school placement. The assignment was to write a lesson play on fractions. Several studies show that many PTs struggle to understand fractions [START_REF] Newton | An extensive analysis of preservice elementary teachers' knowledge of fractions[END_REF][START_REF] Ma | Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States[END_REF]. Siegler et al. (2010) recommend, "[p]rofessional development programs should place a high priority on improving teachers' understanding of fractions and of how to teach them". With this in mind, our research focuses on the following question: How does PTs' mathematical knowledge for teaching surface in their lesson plays on fraction comparison?

Lesson plays

This study reports our efforts to develop practice assignments in our teacher education courses with respect to the notion of high-leverage practices. [START_REF] Ball | Combining the development of practice and the practice of development in teacher education[END_REF] define "highleverage practices" to be those that, when done well, are likely to lead to improved student learning. High-leverage practices are practices which novice teachers need to learn to do, and from which they will learn more about teaching [START_REF] Lampert | Learning teaching in, from, and for practice: What do we mean?[END_REF]. Further, a high-leverage practice is such that novices can begin to master it [START_REF] Grossman | Redefining teaching, re-imagining teacher education[END_REF]. [START_REF] Lampert | Using design instructional activities to enable novices to manage ambitious mathematical teaching[END_REF] give several instructional activities that could be part of high-leverage practices in mathematics, which can be realised in a relatively short time, typically 10-15 minutes, through classroom discussions. The PTs used coursework literature to analyse and discuss a series of videos and transcripts of classroom discussions concerning computational strategies. We asked the PTs to plan a similar short mathematical discussion with a group of middle school students, age 9 to 13. This was part of a practice assignment, intended to be implemented during their school placement towards the end of the term. The PTs were asked to plan for the mathematical discussion by writing a lesson play.

Lesson plays, introduced by [START_REF] Zazkis | Lesson plays: Planning teaching vs. teaching planning[END_REF], are proposed as a way to plan teaching by writing a script for (part of) a lesson. An envisaged interaction between a teacher and a group of students is given verbatim, as an alternative for the traditional lesson plan. [START_REF] Zazkis | Lesson plays: Planning teaching vs. teaching planning[END_REF] argue that lesson plays can give an opportunity for in-depth discussions of crucial aspects of mathematics teaching before the lesson, while such discussions can only take place after the lesson if the lesson is planned using a traditional lesson plan. As such, lesson plays are not affected by [START_REF] John | Lesson planning and the student teacher: re-thinking the dominant model[END_REF] claim that traditional lesson plans do not give insight into "the substance of the particular activity" (p. 487). [START_REF] Zazkis | Lesson plays: Planning teaching vs. teaching planning[END_REF] typically give their PTs a prompt representing a mathematical error or misinterpretation, and ask the PTs to write the script of a discussion, which resolves the prompt. We asked our PTs to plan all aspects of the discussion, including formulating a mathematical aim for the discussion and a task, or a sequence of tasks, to achieve this aim. Our requirements for the discussion were that the mathematical topic was fractions, and that the aim should be to discuss some calculation or reasoning strategy on fractions. Furthermore, the script should include argumentation and some type of generalisation of concepts and/or strategies. Generalisation, argumentation and reasoning was a major focus in the coursework. The duration of the discussion should be 10-15 minutes.

Mathematical knowledge for teaching

We are interested in how PTs' mathematical knowledge for teaching surfaces in their lesson plays, and we use the Knowledge Quartet (KQ) [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: the Knowledge Quartet and the case of Naomi[END_REF] as a framework for our analysis. The KQ consists of four dimensions, three of them resting on the first, named Foundation. Foundation concerns the teacher's or PT's knowledge of mathematics and mathematics teaching as acquired in their education. It underpins a teacher's ability to make rational, reasoned choices and decisions about instruction based on knowledge of mathematics and mathematics pedagogy. The second dimension of the KQ, Transformation, is about how the teacher transforms her own subject matter knowledge of mathematics into forms which enable others to learn it. Such transformation is informed by the teacher's choice of examples and representations and how these support learning of the intended mathematical topic. Connection is about the choices a teacher makes in order to ensure the consistency of planning and teaching a topic or concept through a lesson or lessons. As such it concerns anticipation of what students will find problematic, and decisions about sequencing. Crucial is the teacher's understanding of connections between mathematical concepts and between concepts and procedures as well as anticipation of complexity when planning and teaching a topic. Contingency concerns a teacher's responses to events that were not anticipated or planned for. Since we are considering planning for teaching this dimension will not be relevant to our analysis.

As the lesson plays analysed concern comparing fractions, it is useful to clarify the implications this has for the foundation dimension of the KQ. When describing number sense, researchers state that it manifests in flexible mental computation, understanding number magnitude, making judgements about calculations, using benchmarks, and having an inclination to use and develop understanding of numbers and operations [START_REF] Mcintosh | A proposed framework for examining basic number sense[END_REF]Sowder, 1992). Researchers have identified several strategies for comparing fractions based on number sense (see [START_REF] Yang | Investigating the strategies used by pre-service teachers in Taiwan when responding to number sense questions[END_REF]. The parts strategy can be used when comparing fractions with the same numerator or denominator. The benchmark strategy refers to comparing two fractions to some well-known third fraction, typically or . When using residual thinking one builds up the fractions to 1. There are also "standard" ways of comparing fractions, which do not overtly depend on number sense, such as finding a common denominator or converting to decimals. Since the task for the PTs was to write a lesson play about reasoning with fractions, we expected the planned discussions to contain more than performing an algorithm. Using visual models of fractions can be a legitimate strategy, but researchers also warn about the limitations of relying on a visual strategy alone (Petit, Laird, Mardsen, & Ebby, 2015;[START_REF] Lamon | Teaching fractions and ratios for understanding[END_REF]. Thus, we expected strategies beyond visual strategies in the planned discussions.

Method

The participants in the study were PTs following a 4-year teacher education programme, for age 6-13, at a university in Norway. The data were collected from their responses to a coursework assignment given during their first mathematics education module, in their first year of study. Of the 178 PTs in this cohort, 32 had chosen tasks on comparing fractions for their written classroom discussion. These 32 scripts are the data analysed in this paper, in particular they were chosen for analysis because fraction comparison has good potential for reasoning based on number sense [START_REF] Yang | Investigating the strategies used by pre-service teachers in Taiwan when responding to number sense questions[END_REF]. The excerpts presented in this paper are chosen to exemplify general trends identified in the 32 lesson plays.

All four authors conducted the analysis. Together, we first analysed in detail two lesson plays, using the descriptions of the dimensions from the KQ in our analysis of the two scripts. We then individually analysed the rest of the lesson plays looking for occurrences of similar and contrasting forms of mathematical teacher knowledge related to the KQ. After this independent analysis, we compared similarities and differences in our analyses, and agreed on an interpretation of different aspects of the lesson plays, using notions included in the KQ framework.

Analysis

Of the 32 scripts, 6 used no strategies based on number sense, instead relying on visual "parts of shapes" strategies, or on algorithms such as finding a common denominator or converting to decimal numbers. In the remaining 26 scripts, the PTs used a number sense-based strategy at least once. 9 PTs used benchmarking, 13 used parts and 17 used residual thinking. In the following, we analyse some examples from the scripts in light of the mathematical content.

Anne was one of the PTs who based most of the imagined discussion on number sense-based strategies. Her stated goal of the discussion is to build understanding of the strategies of benchmarking and residual thinking. She gives two tasks designed to encourage the students to utilise these strategies, and we quote here1 two excerpts from the imagined discussion between Anne's teacher and her students while discussing the sorting of and by magnitude:

Ola: Yes, at least you see that is the same as the half of something, that was where I started.

Teacher: OK, so you believe that it is one half. But how can that help us?

Ola: Well, since it is one half, we also see that is less than one half.

Teacher: Per, can you try to elaborate Ola's thinking?

Per: is sort of lacking 3 parts to become one half. Because 5 parts is half of 10 parts.

So then is less than Teacher: Right, do the rest of you agree? Yes. OK, what do we do next?

We see that Anne, in Ola's words, uses one half as a benchmark when comparing and . We note that Ola's explanation is incomplete; it does not state why is less than one half. The PT seems to be aware of this, as she asks Per to elaborate Ola's thinking, from whom she receives the completed reasoning. In the next sample from Anne's script, a residual argument is pursued:

Teacher: All right, so now we know that is the smallest, and then comes , but which is the biggest of and ?

Per: Since is smaller than then is the most Mia: But why is that when is a lot bigger than ?

Teacher: Good question Mia, does anyone want to explain?

Ola:

When we looked at and we were looking at how much was missing to fill one whole. The one that miss the biggest part is then missing the most, and therefore that fraction is the smallest. Because is only missing a small to become one whole.

Mia:

Oh yes, now I understand, because we are looking at what is missing.

Anne does not acknowledge any students' claim without a justification. Through the whole of Anne's script, the teacher is encouraging the students to utilise their number sense when reasoning about the tasks. Similar approaches are also apparent in the rest of the 26 scripts, but not always as comprehensive as in Anne's case. One typical feature is that even though a mathematically correct conclusion is reached, there is no valid argument given by the students, and the PTs tend to accept this without comment. This is evident in Alice's script, when the students are comparing and .

Fredrik:

Yes, me and my group decided that is the biggest. I think that 3 is closer to 4, than 4 is to 8.

Teacher:

That was good thinking. [Proceeds with a different task]

Our analysis shows a general tendency in the scripts that strategies based on number sense have some kind of justification, while strategies based on algorithms and rules are more likely accepted without justification. The above excerpt from Alice's script is one of few exceptions, where she gives a correct conclusion with an attempted justification that is not valid as an argument. Judging by the teacher's response, it seems that Alice regards Fredrik's argument as valid.

The following excerpt from Christine's script shows another problem.

Sindre: must be the biggest, because that fraction is only missing 3 parts to become one whole, while is missing 5 parts to become one whole. Nina: Yes, you can also say that misses one half to be whole, while lacks more than half to be whole, since it is lacking 5, and half of is .

Teacher: That was a good explanation, Nina. Did the rest of you also understand what Nina meant? (The class agrees.)

In this excerpt, we notice that Sindre's argument is wrong even though the conclusion is correct. Christine (in the role of the teacher in the discussion) does not comment upon this, instead simply accepting Sindre's argument. Interestingly, Nina subsequently gives a valid residual argument, but Christine does not draw attention to the difference in the two arguments in her script. Our analysis shows that similar arguments that "work" on the fractions in question, but where a counterexample would prove the argument not generally valid, are typical for many of the lesson plays.

Another aspect of our findings was the PTs' choice of tasks used in the discussions. Returning to Anne's script, she uses only two tasks. They both underpin the strategy in focus, and have a natural progression in complexity: The fractions involved seem to be carefully chosen to make her target strategy suitable, and residual thinking is further highlighted by Anne asking the question "Which of the fractions are missing the most to become one whole?" at the start of the discussion. In contrast to this, Molly's choice of tasks and the sequencing chosen, seems less appropriate: Compare and ; and ; and ; and ; and and . Molly does not state explicitly a mathematical goal for the planned discussion, and her imagined discussion covers several ideas in a brief way. Moreover, the progression of difficulty in the sequence of tasks does not seem to be well thought through: in the lesson script on the first tasks, Molly's fictive students use benchmarking with one half, indicating that one half is a well-known concept for them. To then proceed with the final three tasks focusing on equivalent fractions to , seems exaggerated.

Discussion

We now relate the findings presented in the analysis to the dimensions of the Knowledge Quartet.

Foundation

For the foundation dimension, the most visible aspects are the PTs' mathematical knowledge of fractions and comparison strategies, as well as their beliefs about mathematics itself, and about mathematics teaching. In general, the PTs try to use strategies relying on number sense to compare fractions. This could indicate that the PTs value developing understanding rather than focusing on an algorithmic approach. The strategies attempted in the scripts are not always followed through in a mathematically valid argument, and some of the PTs fail to recognise the difference between valid and invalid arguments. Christine's script is an example of this.

Another finding is that very few scripts contain any attempt at discussing the generality of the strategies used. This indicates that the PTs' beliefs about mathematics might not include this as an important aspect of doing mathematics. Instead, the PTs seem to be satisfied as soon as the problem at hand is solved, as in Christine's script when neither Sindre's or Nina's arguments are investigated further from a general point of view. Recall that the task given to the PTs particularly required them to emphasise the development of their students' understanding and reasoning.

Transformation

The PTs' scripts afford good insight into their choice of examples to elicit an idea. With very few exceptions, the tasks chosen by the PTs are suitable comparison tasks where it is clear that there is at least one number sense-based strategy that could be applied.

We proceed to consider the PTs' use of questions. In the context of PTs writing an imagined discussion, we regard this as a form of teacher demonstration, and thus consider it a part of transformation. We find in most scripts a use of certain techniques and types of questions known from their coursework literature on orchestrating mathematical discussions. For example, in Anne's script, the teacher's questions structure what her students have discovered and then seek to develop their ideas further. When Anne's teacher asks Per to elaborate Ola's thinking, she succeeds in bringing to light an argument. In other scripts, the PTs seem to emphasise the use of discussion techniques in itself to such an extent that it suppresses the attention on connecting the mathematical ideas. This can be seen e.g. in the excerpt from Christine's script above, where the teacher asks a student to repeat another student's reasoning without connecting the different explanations. Sometimes the PTs fail to notice when a clarifying question is needed. This can be seen in Alice's script above, where Fredrik's attempted justification is an invalid argument in general, and yet the teacher accepts it and proceeds without further enquiry.

Connection

The sequencing of tasks, how one task should connect to the previous task, and the anticipation of what students will find problematic, is part of the connection dimension. We find that in most scripts, the sequencing of tasks is appropriate. However, we find examples of situations where the PTs do not seem to anticipate the complexity of the sequence of tasks. An example is Molly's script as discussed above. Other scripts seem to have too many tasks, given the time allotted. In these scripts the discussion moves forward smoothly with students giving the desired response quickly and effortlessly. This may indicate that these PTs do not anticipate complexity in the discussion and that the conceptual challenge for the students is underestimated. Thus, these discussions take more the form of numerous repetitions of the same procedure, which relates to the foundation dimension of the KQ and perspectives on how mathematics is learned: These PTs seem to emphasize procedural repetition as an important aspect of learning mathematics, perhaps on behalf of unpacking the mathematics of the procedures. However, some scripts include deliberate mistakes and misconceptions made by the students, which are then discussed. We see this as an anticipation of complexity.

Conclusions

Following the discussion above, we claim that lesson plays encourage the PTs to use and develop several aspects of their mathematical knowledge for teaching. For the foundation dimension, we claim that the insight we get from the scripts, is more than what we would get from simply assigning the PTs fraction comparison problems for them to solve. We note that several PTs write discussions including both valid and invalid arguments and both are accepted without further probing. For instance, Christine knows what a valid argument for comparing fractions looks like, but at the same time she accepts an invalid one. Such inconsistency in the PTs' thinking might become more visible when they plan teaching by imagining a detailed mathematical discussion.

We also claim that our findings show the importance of emphasis in mathematics teacher education on generalisation and argumentation, and how classroom discussions concerning generalisations could play out. Our PTs were asked to have those aspects in mind when writing their lesson plays, and yet it is rarely found in the scripts. How to develop the PTs' ability to emphasise this aspect more needs to be studied further. Managing classroom discussions is a complicated task for novice teachers. However, due to its high leverage on students' development of mathematical understanding it is a critical factor in mathematics teaching, and thus in teacher education.

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The original scripts were written in Norwegian, with translation to English by the authors of this paper. In the translation we have retained linguistic inaccuracies and imprecise use of terms, as in the original Norwegian.Thematic Working Group

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Acknowledgment

The authors thank Tim Rowland for many helpful comments and discussions, as well as for organising for us to come to Cambridge to work on our data in May/June 2016.


text:
Miguel Ribeiro

Fatma Aslan-Tutak

Sebastian Kuntze

Francesca Martignone

Kirsti Rø

Amanjot Toor

Keywords:

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.

Introduction to the papers of TWG20 : Mathematics teacher knowledge, beliefs, and identity

Introduction to the papers of TWG20:

Mathematics teacher knowledge, beliefs, and identity

Miguel Ribeiro 1 , Fatma Aslan-Tutak 2 , Sebastian Kuntze 3 , Francesca Martignone 4 , Kirsti Rø 5 and Amanjot Toor

Rationale

The variety of papers presented in TWG 20 at CERME 2017 connected to the growing field of teacher education. There were total of 27 papers and 4 posters that had been presented. The number and heterogeneity of the research foci, contexts, methodological and theoretical approaches, provided opportunity for in-depth discussions and reflections around the presented papers. Although the core topic proposals were expected to embrace three intertwined domains, the focus of the presented papers at TWG20 was mainly on teachers' knowledge, while the topic of teachers' beliefs and teachers' identity would appear implicitly or in the background of some of the research.

Main topics

The priority given to teachers' knowledge can be seen throughout the unbalanced number of proposals which included the 27 papers presented (30 were submitted). For the development of the work three thematic strands have been considered: (i) knowledge in mathematics education (3 papers); (ii) lesson study context and beliefs (5 papers) and (iii) teachers' knowledge" (19 papers). It is interesting to note that the differences between (i) and (iii) concerns the aims of the research (even if not perceived explicitly), and not necessarily the theoretical perspectives considered.

(i) Knowledge in mathematics education. This thematic strand included three papers, focusing on aspects different from the specificities of teachers' knowledge, even when the context was teachers' education. Although the theoretical dimensions of the presented researches addressed teachers' knowledge conceptualizations the research focused on the knowledge of prospective teachers' that is revealed at the very beginning of teachers' training.

(ii) Lesson study. The second thematic strand included five papers and one poster, dealing with mathematics teachers' professional development (PD), in the context of the development of Lesson Studies (LS), and (prospective) teachers' learning with regards their own teaching practice and students learning, as well as on how teachers perceive themselves as participants in such LS context. Although there is evidence that PD contexts can lead to improvements in teaching practices and students' learning, less is known about what and how teachers learn from PD and about its further impact on students' learning outcomes [START_REF] Borko | Professional development and teacher learning: Mapping the terrain[END_REF], and also some other intertwined variables.

Common to the presented research were the perspective of considering teachers as learners, where the research is looking at the relationships between these two elements. In this thematic strand, two particular issues arise; how can theories/approaches/perspectives on teachers' knowledge be used to analyse the impact of teachers' participation and involvement in LS for teaching practices and professional development; concerning questions around the incorporation of theory in a methodological approach for LS in order to analyse the different phases and cycles.

Further, the research developed in the context of the implementation of a LS (or a research on the LS process), needs to take into account the particularities and specificities of the cultural contexts in which it is implementedin order to acknowledge, the differences of those cultural contexts and the one where it is originated. In, and for doing so, one needs to take into account different aspects, such as the type, nature and impact of the affordances and constraints that takes into consideration; the influence of the researchers' background in the implementation, development and design of the research in the LS context; to what extent, in the context of a mathematics education research, the mathematics features of teaching is effectively the focus of attention.

(ii) Teachers' knowledge. Similar to previous CERME's, the explicit focus on teachers' knowledge has been given a major importance in the context of most of the presented proposals.

The papers were grounded on teachers' knowledge conceptualizations that have already been discussed extensively in previous CERME conferences (Mathematical Knowledge for Teaching [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]; Mathematics Teachers' Specialized Knowledge (Carrillo, Climent, Contreras & Munoz-Catalan, 2013); Knowledge Quartet [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF]; Ontosemiotic Approach [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF]). We observe that, within a period of four years (from CERME 8 to CERME 10) a certain shift related to the focus of attention occurred, namely a shift from discussing the need for different conceptualizations towards an effort to deepening on the nature of teachers' knowledge when assuming a certain conceptualization.

In CERME9, Ribeiro, Aslan-Tutak, Charalambous and Meinke (2015) suggested that the use and development of diverse conceptualizations could be perceived as both a richness of the research field and as a constraint. The richness concerns possibilities for gaining a better insight into factors that influence the development of teacher knowledge. However, there are challenges of finding a common ground for discussing the core aspects of the research field. At the current conference, the issue of a diverse conceptualization was addressed in discussions on how to investigate mathematics teachers' knowledge when assuming it to be in interplay with students' learning. In other words, the need to pay "close" attention to how we, as researchers, take into consideration the aspects of mathematics teaching and learning, being connected to teachers' intertwined knowledge as well as to the role and impact of teachers' knowledge in practice concerning the use of resources (where the focus was the teachers' knowledge involved in/for preparing and using such resourcesin a broader sensein practice and not the resources itself).

The research, and the associated discussions and reflections in the group, also bought forward one of the recurrent items in the group discussions: the need for the research on teacher education to move from a prevalent focus on what teachers do not have (the deficit perspectivein term of knowledge) to a focus on what teachers actually know, how they know it, and possible different hows that can contribute to the development of teacher's knowledge, specifically related with teacher's work of teaching. Along the discussions, and aligned with some of the presented papers, a possible direction for future research emerged on area of teachers' knowledge sustaining teachers' noticing and "earing" ability. For example, future research focusing on how mathematic teachers' pay attention to and make sense of what happens in the complexity of instructional situations (see e.g. [START_REF] Sherin | Mathematics teacher noticing: Seeing through teachers' eyes[END_REF]. Also on what aspects of one's knowledge do teachers' ground their decision makingin order to develop mathematically demanding practices, aiming at developing critical mathematical thinking for deeper mathematical understanding. One other possible focus concerns on how and why (the impact) the teachers' and researchers' knowledge influence their foci of attention and awareness.

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Emerged themes and future perspectives

We have considered three thematic strands for an operational reason, but one need to have in mind the intertwined nature of such strands. Thus, in our case, the connecting element was teachers' knowledge. Some of the discussions were grounded in ideas already discussed in previous CERME's, aiming at deepening the understanding on those aspects/dimensions while other discussions seek for an alternative and complementary path for getting such a broader understanding. A particular sensitive aspect was the need for a deeper understanding on the relationships between teachers' knowledge and practice, and for gaining such a deeper understanding some new approaches to research on teachers' knowledge were discussed, in particular studies that investigate how teachers use their knowledge to give meaning to others' solutions or to anticipate students' answers. Moreover, the role of task design in and for assessing, accessing and developing teachers' knowledge and improving practices was emphasized in the discussions. We have grouped the main research trends emerged in three groups:

-Deepening research into teachers' knowledge, beliefs, identity, and noticing  Taking into account that in some contexts mathematics teachers knowledge specificities are perceived mainly in the domain of PCK, how is the "weight" of PCK perceived in the field of research in mathematics education and how it intertwines with the specificities of the teachers' content knowledge, beliefs and identity?

 How to take into account teachers' noticing?

-Research on interactions with fields of practice  How can the focus of research be intertwined with practice and education in a more explicit manner, perceiving practice in a broader sense?

 How can we investigate whether and how teachers' knowledge affects students' learning and transitions throughout student's education? How to design and develop research aimed at approaching "simultaneously" teachers' knowledge and students learning?

 How to move from frameworks for analyzing, describing, understanding and/or evaluating teachers' knowledge, to the use of frameworks by teachers (for analyzing teaching practice)?

 How the teachers' knowledge conceptualizations take into account the notion of assessment, and how does knowledge on assessment contribute to students' learning in mathematics?

 What are the roles and knowledge (e.g., features, nature, content) of mathematic educators in teacher education (e.g. facilitator in LS; teacher trainer)?

-Research on methodological (and theoretical) challenges:

 How can we deal with (and what are the implications for) similarities and differences of aims and challenges in mathematics teacher education in different cultural contexts?

 How to clarify the findings we have, when using a particular theoretical lens for analyzing teachers' answers, comments and/or practices?


text:
Xenia-Rosemarie Reit
email: reit@math.uni-frankfurt.de

Towards an empirical validation of mathematics teachers' intuitive assessment practice exemplified by modelling tasks

Keywords: Assessment, cognitive structure, modelling tasks, mathematics teaching

The identification of intermediate steps in student solutions as a basis for assessment is a common procedure in mathematics teaching. Modelling tasks, providing more than one solution approach, are considered hard to assess. This is not least a reason for the unsatisfactory proportion of modelling in school. Assuming that task difficulty is strongly connected with assessment, the implications of a study about the difficulty of modelling tasks are discussed. Starting point for the discussion is the question whether intuitive assessment practices like the identification and scoring of intermediate steps, can be supported by empirical findings. The focus is on the influence of cognitive aspects regarding structural characteristics of solution approaches. Results indicate that a pure sequential consideration of thought structures in a solution approach lead to reasonable results and might justify its application in school due to its straightforward implementation.

Introduction

Within the mathematics community it is mostly agreed upon the positive impact of modelling tasks on the learning of students. Mathematical modelling is promoted and there are many votes for its broader implementation in school mathematics. However, several studies provide evidence that modelling is far away from playing an integral role in everyday school teaching, in Germany and also elsewhere (Blum, 2007, p. 5). [START_REF] Jordan | Klassifikationsschema für Mathematikaufgaben. Dokumentation der Aufgabenkategorisierung im COACTIV-Projekt[END_REF] confirm that the proportion of modelling in daily school routine is low. Research focusing on the teachers' point of view reveals several difficulties that teachers are confronted with. In a study of [START_REF] Schmidt | Modellieren in der Schulpraxis: Beweggründe und Hindernisse aus Lehrersicht[END_REF] it has been found out that 67% of the interviewed teachers indicate assessment as being the major challenge in the implementation of modelling tasks. [START_REF] Blum | Modellierungsaufgaben im Mathematikunterricht -Herausforderung für Schüler und Lehrer[END_REF] also speaks of an increased difficulty in the context of modelling tasks. These findings are comprehensible in view of the multiple solution approaches of modelling tasks.

A common opinion is that modelling tasks cannot be assessed as objectively as traditional task formats [START_REF] Spandaw | Modelling in mathematics teachers' professional development[END_REF]. However, if we want modelling to be part of mathematics teaching, it must part of the grading [START_REF] Hall | The assessment of modelling projects[END_REF]. With the striking quotation "What you assess is what you get" [START_REF] Niss | Assessment of mathematical applications and modelling in mathematics teaching[END_REF] also argues in favor of a provision for modelling activities in the grading. The aim of the present paper is not to discuss and contrast formative and summative assessment since the advantages of formative assessment as assessment for learning movement could be confirmed in different settings (e.g. [START_REF] Black | Assessment and classroom learning[END_REF] and are not denied. However, in view of the fact that summative assessment aspects hinder the implementation of modelling tasks in everyday school live, it is necessary to provide tools or possibilities in that direction.

Besides a number of assessment methods which aim at assessing modelling competence (e.g. [START_REF] Berry | Open university students do it by themselves[END_REF][START_REF] Haines | Mathematical modelling skills: a research instrument[END_REF], there is hardly any assessment instrument which can be used for assessing modelling tasks in everyday school live. In this context so far only Maaß suggests an assessment scheme which can be adapted to different modelling tasks by a variable weighting of several categories (Maaß, 2007, p. 40). However, an empirical validation is lacking such that the assessment scheme might serve as orientation but it cannot give detailed instruction.

On the way to an assessment scheme for a mathematics task, a common procedure of mathematics teachers (in the following referred to as "intuitive assessment practice") is to identify reasonable intermediate steps in a solution which are worthwhile to be scored. On that basis an assessment scheme is set up which determines the conditions to be fulfilled for a differentiated scoring of those intermediate steps. Hence, there is a procedural difference between the phase of identifying scoreable aspects in a solution and an assessment scheme. The former is the requirement for the latter. At this point the present paper ties on by discussing the use of so called thought structures to identify reasonable intermediate steps in solution approaches of modelling tasks in connection with its difficulty. The question of identifying intermediate steps and the influence of their structure within a solution approach to its difficulty has been analysed by [START_REF] Reit | Denkstruturanalyse als Instrument zur Bestimmung der Schwierigkeit von Modellierunsgaufgaben[END_REF]. In this study different models are developed and evaluated to determine the difficulty of solution approaches. Assuming that assessment of a mathematics task is strongly determined by its difficulty, interesting conclusions can be drawn concerning common assessment practice in school. Results of the study of [START_REF] Reit | Denkstruturanalyse als Instrument zur Bestimmung der Schwierigkeit von Modellierunsgaufgaben[END_REF] indicate that there is quantifiable influence of structural characteristics of a solution approach on its difficulty. However, it is also stated that a sequential model which is based upon a sequential arrangement of thought operations, similar to the intuitive assessment practice of mathematics teachers, can also be confirmed.

Theoretical framework

The core of the study of Reit ( 2016) is a structural analysis of students' solution approaches of modelling tasks. These thought structures of solution approaches indicate the chronology of thought operations to be done to arrive at a solution. Assuming that parallel thought operations complicate a solution approach, a non-weighting difficulty model (addition model) is contrasted with four models varying in their weighting of parallel thought operations.

Thought structure analysis

Recalling structures is a wide-spread procedure in mathematics (Bourbaki, 1961, pp. 163). In this context [START_REF] Breidenbach | Rechnen in der Volksschule: eine Methodik[END_REF] looks at the structural-substantial complexity of a word problem to decide amongst others about its difficulty. He formulates that tasks with one operation deal in the simplest case, with one issue in which three factors play a role and every factor is uniquely determined by the two others (Breidenbach, 1963, p. 200). Breidenbach named such tasks Simplex. A linking of several Simplex is called Komplex. Further developments of [START_REF] Winter | Neue Mathematik: Lehrerheft[END_REF] lead to the arithmetic tree which is still used in mathematics textbooks (Figure 1). An obvious but so far empirically not validated conclusion is that a larger number of Simplex and a more complicated nesting of them, has an effect on the difficulty of the tasks' solution [START_REF] Graumann | Mathematikunterricht in der Grundschule[END_REF]. The study of [START_REF] Reit | Denkstruturanalyse als Instrument zur Bestimmung der Schwierigkeit von Modellierunsgaufgaben[END_REF] investigates the cognitive complexity of a solution approach on the basis of its structural complexity represented by its arithmetic tree-like structure. At that point the coherence of structural considerations and cognitive psychological theories play an important role. In a study of [START_REF] Fletcher | Causal reasoning in the comprehension of simple narrative texts[END_REF] it is assumed that text comprehension is a kind of problem solving process, where the reader must find a causal chain which links start and end of a text. Furthermore they assume that information must be kept simultaneously in the working memory to be able to form such a causal chain. Results of their study show that readers must keep that information available that is the direct predecessor in the causal chain. It can be concluded that the task of the working memory is to keep information available which is necessary to link old and new information [START_REF] Baumann | Die Funktion des Arbeitsgedächtnisses beim abduktiven Schließen: Experimente zur Verfügbarkeit der mentalen Repräsentation erklärter und nicht erklärter Beobachtungen[END_REF].

By relating these findings to structural considerations of a solution approach represented as an arithmetic tree, statements can be made about its cognitive complexity. On the one hand the arithmetic tree-like structure (Figure 1) can be interpreted as causal chain since the start (given information in the task text) and end (solution of the task) is linked by chain links (intermediate steps in the solution process). On the other hand direct predecessors can be identified as relevant intermediate steps. Thus, it can be concluded that the previous intermediate step must be kept active in the working memory to master the following. The assumption that the mental processing capacity is limited [START_REF] Sweller | Cognitive load during problem solving: Effects on learning[END_REF] leads to the statement that several information which has to be kept active at the same time, complicate the solution process. Thus, it can be deduced that the load of the working memory is dependent on the number of intermediate steps necessary to master the current intermediate step. That means that the load of the working memory increases with increasing number of information needed at the respective point in the solution process.

Based on these considerations a study has been performed where theoretical difficulty of a solution approach is characterized as its cognitive complexity [START_REF] Reit | Denkstruturanalyse als Instrument zur Bestimmung der Schwierigkeit von Modellierunsgaufgaben[END_REF]. Starting point is the so called thought structure of a solution approach which can be interpreted as kind of arithmetic tree (Figure 1). To formulate a thought structure all student solutions of a modelling task have been clustered into several solution approaches according to the mathematical model or solution process used. An aim of the study was to investigate whether the number of sequential and parallel thought operations has an effect on the cognitive complexity and thus, the theoretical difficulty of a solution approach.

Study design

Approximately 1800 grade 9 students (15 years of age) from German grammar schools took part and completed a booklet consisting of three out of five modelling tasks (see modelling task "potato" in Figure 2) under seatwork conditions. The total processing time for a booklet lay within one teaching unit.

Method

In the following the method in the study of [START_REF] Reit | Denkstruturanalyse als Instrument zur Bestimmung der Schwierigkeit von Modellierunsgaufgaben[END_REF] will be briefly explained. For a detailed description of the methodical implementation it is referred to [START_REF] Reit | Denkstruturanalyse als Instrument zur Bestimmung der Schwierigkeit von Modellierunsgaufgaben[END_REF].

All student solutions of a modelling task were analyzed and different solution approaches could be identified (two to four solution approaches per modelling task). These solution approaches within one modelling task differed in their underlying mathematical model used or, if similar to this, in their solution process. Every student solution was finally assigned to a solution approach (Figure 3). A structural analysis of these solution approaches then lead to individual thought structures indicating the chronology of thought operations. Based on thought structures of solution approaches different difficulty models have been developed to translate the respective structure into a scalar difficulty value. In a first step a thought structure was mapped onto a so called thought structure vector (Figure 3). These thought structure vectors represent the tabular-compact form of a thought structure. Each vector component indicates the number of parallel thought operations on the respective level of the thought structure.

Due to the fact that it was not clear yet if parallel thought operations lead to a higher difficulty than sequential thought operations, different operationalization of a thought structure vector into a scalar value were imaginable. Therefore different difficulty models have been set up (four accounting for parallelism of thought operations by weighting them and one non-weighting model (addition model)) which lead to solution approach specific difficulties.

Industrial manufactured French fries are supposed to be equal in size and the single sticks are cut out lengthwise. Therefore not the whole potato can be used. The potato tubers look similar to the picture above, are regularly formed and approximately 10 cm in length.

How many of these potato sticks can be obtained from one potato? Reason mathematically.

Results

Whether and to what extent parallel and sequential thought operations have an influence on the difficulty was evaluated by comparison with the corresponding empirical difficulty, as a measure of the average score of a solution approach. To determine the empirical difficulty all student solutions have been assessed by two independent raters on the basis of a predefined assessment scheme set up by experts. The question was whether the theoretical difficulty reflected the associated empirical difficulty of a solution approach. In this case structural characteristics of a solution approach can be taken as a basis for assessment of modelling tasks. Of special interest are the results of the nonweighting difficulty model as analogy to mathematics teachers' intuitive assessment practice. The non-weighting difficulty model (addition model) adds up all thought operations to arrive at a theoretical difficulty as it is done, more or less intuitively, by mathematics teachers when identifying scoreable intermediate steps.

The results (Figure 4) indicate that addition and factorial model (pseudo-R²=0.83) map the coherence of theoretical and empirical difficulty best. The factorial model weights parallel thought operations. In regard of the focus of the paper the results in Figure 4 clearly show that the addition model lead to significantly better results than the most weighting models.

Discussion of results

Particular reference is made to established but so far not researched assessment practices in mathematics teaching. The focus is on whether intuitive assessment practices of mathematics teachers can be empirically confirmed and transferred to modelling tasks. Intuitive assessment practice means the common procedure of mathematics teachers of scoring intermediate steps in student solutions without accounting for structural-cognitive particularities. These intermediate steps usually then serve as a basis for an assessment scheme. The portrayed procedure is commonly used when assessing performance tasks in mathematics (summative assessment). Modelling tasks with its multiple solution approaches are considered to be hard to assess. This problem not least leads to the fact that modelling tasks are sparsely used in mathematics class. Results of a study investigating the difficulty of modelling tasks support the intuitive assessment practice of mathematics teachers and thus, legitimate transferring this assessment practice to modelling tasks.

In detail the results show that the addition model which treats sequential and parallel thought operations equally lead to reasonable results. This indicates that difficulty of a solution approach can be described well by the number of thought operations needed to arrive at a solution. By applying the addition model it is assumed that parallelism of thought operations has no influence on the complexity of a solution approach. This assumption is also made, more or less intuitively, in common assessment practice in mathematics teaching. Intermediate steps are identified and scored. Thus, on the one hand the results support the everyday procedure in mathematics teaching where the difficulty of a mathematics task is often interpreted as the number of intermediate steps to complete a solution. On the other hand the results might justify a similar assessment procedure when assessing modelling tasks. In view of the widespread problems concerning assessing modelling tasks in everyday school live as part of the grading, the results clearly highlight a possible and furthermore practicable way.

In summary it can be concluded that the so far intuitive assessment practice in school of identifying intermediate steps in a solution, can be supported by empirical findings. It can be a worthwhile procedure to identify thought structures as a basis for assessment especially when assessing modelling tasks. By assuming that assessment is connected with difficulty of the respective solution approach, parallelism of solution approaches has an influence (see the results of the factorial model) but might be neglected in favour of a straightforward applicability in everyday school practice. Thus the results of [START_REF] Reit | Denkstruturanalyse als Instrument zur Bestimmung der Schwierigkeit von Modellierunsgaufgaben[END_REF] can serve as a basis for the development of a manageable assessment scheme for modelling tasks and might promote their implementation in mathematics teaching.

Figure 1 :

 1 Figure 1: Arithmetic tree of an exemplary task following Winter and Ziegler (1969)

Figure 2 :

 2 Figure 2: Modelling task "potato" (Reit, 2016)

Figure 3 :

 3 Figure 3: Identification of solution approaches, setting up thought structures (together with its thought structure vector) and applying difficulty models which lead to theoretical difficulties

Figure 4 :

 4 Figure 4: Comparison of theoretical and empirical difficulty of solution approaches taking account for different difficulty models

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