text:
Siún Nic Mhuirí
email: siun.nicmhuiri@dcu.ie

Considering research frameworks as a tool for reflection on practices: Grain-size and levels of change

Keywords: Reflection, research framework, classroom practice(s), grain size. Thematic Working Group 19

Reflection is widely recognized as essential to teachers' learning but questions remain about what exactly teachers should reflect on and how effective reflection might be facilitated. This paper considers how research frameworks might be used as a tool to facilitate reflection on mathematics classroom practices. It will be argued that frameworks which facilitate planning and analysis of classroom practices at different levels of specificity may also target reflection at different levels. This will be illustrated with reference to frameworks which relate to talk in the mathematics classroom. It will also be argued that for research frameworks to be effective in facilitating reflection on mathematics classroom practices, they must target different levels of reflection.

Introduction

A growing body of literature recognizes the importance and complexity of the practices involved in effective mathematics teaching (e.g., [START_REF] Ball | Combining the development of practice and the practice of development in teacher education[END_REF][START_REF] Potari | Introduction to the papers of TWG19: Mathematics teacher and classroom practices[END_REF]. Classroom practices have been described as "the repeated actions in which students and teachers engage as they learn" (Boaler, 2002, p. 114). [START_REF] Dooley | Mathematics in early childhood and primary education (3-8 years)[END_REF] identify five overarching practices as essential in promoting mathematical thinking and understanding. These meta-practices are the development of a productive disposition; emphasis on mathematical modelling; the use of cognitively challenging tasks, formative assessment, and the promotion of mathematical talk. In each of these areas, research frameworks exist which may hold potential for teacher development. By 'research framework', I mean either a theoretical framework or methodological tool which 'frames' or structures a coherent set of understandings about a particular theme. For example, consider how the Math Talk Learning Community (MTLC) framework [START_REF] Hufferd-Ackles | Describing levels and components of a math-talk learning community[END_REF]) might offer a coherent set of understandings about the nature of talk in classrooms or how the Mathematical Tasks Framework [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF] offers a similar set of understandings for the use of cognitively demanding tasks. Such frameworks illustrate teacher actions that contribute to desirable metapractices for mathematics teaching and learning.

The growing emphasis on mathematics classroom practices is occurring at a time when reflection has been widely established as a guiding principle within teacher education [START_REF] Hatton | Reflection in teacher education: Towards definition and implementation[END_REF][START_REF] Zeichner | A critical analysis of reflection as a goal for teacher education[END_REF]. Mathematics education research frameworks have been used to facilitate teachers' reflection (e.g., [START_REF] Stein | Mathematical tasks as a framework for reflection: From research to practice[END_REF]. However, there is little literature to offer guidance on how frameworks might be used as a tool for reflection on mathematics classroom practices, or to guide researchers investigating this area. This paper offers a consideration of this theme. First, I will outline some key issues from the literature on reflection and will discuss the theoretical perspective from which teachers' use of research frameworks is understood. Then I will outline [START_REF] Korthagen | In search of the essence of a good teacher: Towards a more holistic approach in teacher education[END_REF] model of levels of reflection and change. Finally, I will argue that grain-size, or level of specificity, is of key importance when research frameworks are used to facilitate teachers' reflection on practices. I will illustrate this with reference to two frameworks relevant to the practices involved the promotion of mathematical talk. I will argue that for frameworks to be effective in facilitating reflection on classroom practices, they must target different levels of reflection.

Reflection and research frameworks

It is generally accepted that experience alone may not result in learning and reflection is considered to be integral to learning from practice [START_REF] Loughran | Effective reflective practice in search of meaning in learning about teaching[END_REF][START_REF] Schön | The reflective practitioner: How professionals think in action[END_REF][START_REF] Zeichner | A critical analysis of reflection as a goal for teacher education[END_REF]. The widespread adoption of reflection in teacher education programs has resulted in conflicting conceptualizations and many efforts to develop reflective practitioners are often underpinned by different ideological stances [START_REF] Hatton | Reflection in teacher education: Towards definition and implementation[END_REF][START_REF] Zeichner | A critical analysis of reflection as a goal for teacher education[END_REF]. Despite the multiplicity of interpretations, there is some agreement that reflection should involve finding solutions to real problems of practice [START_REF] Hatton | Reflection in teacher education: Towards definition and implementation[END_REF] with some authors suggesting that the initial stimulus should be a problem arising from practice (e.g., Jaworski, 1998). [START_REF] Loughran | Effective reflective practice in search of meaning in learning about teaching[END_REF] maintains that learning arises from framing and reframing the initial problem. Research frameworks may be helpful in the reframing process and towards the goal of problematizing teaching, a key component of reflective practice (Jaworski, 1998). This problematizing of teaching may arise from considering a situation from another point of view [START_REF] Loughran | Effective reflective practice in search of meaning in learning about teaching[END_REF]. Research frameworks can facilitate consideration of alternate or multiple viewpoints. For example, the MTLC framework [START_REF] Hufferd-Ackles | Describing levels and components of a math-talk learning community[END_REF] addresses both the student and teacher experience.

The reflective practice literature includes much attention to the process of reflection. [START_REF] Eraut | Schön shock: A case for reframing reflection-in-action? Teachers and Teaching: Theory and Practice[END_REF] critiques and extends [START_REF] Schön | The reflective practitioner: How professionals think in action[END_REF][START_REF] Schön | Educating the reflective practitioner[END_REF] stages of reflection-in-action and reflection-on-action to also include reflection-for-action. The extent to which research frameworks might inform teachers' reflections-in-action is limited due to the many pressures of real-time teaching but frameworks can be used to support reflection at both the planning and post-teaching phase either as a tool for formal analysis, or as a means of troubling accepted understandings of practice. The literature also highlights links between reflective practice and action research. Jaworski (1998) describes a mathematics teacher research project where the theory of the teachers' research activity aligned with a view of action research being connected to critical reflective practice. Critical reflective practice has also been conceptualized in different ways [START_REF] Hatton | Reflection in teacher education: Towards definition and implementation[END_REF]. [START_REF] Larrivee | Transforming teaching practice: Becoming the critically reflective teacher[END_REF] maintains that critical reflection is an essential part of becoming a reflective practitioner and defines it as encompassing critical inquiry and self-reflection. Critical inquiry is described as "the conscious consideration of the ethical implications and consequences of teaching practice" (p. 293). [START_REF] Zeichner | A critical analysis of reflection as a goal for teacher education[END_REF] suggests that often the obligatory content of reflection is around how well practice conforms to what is expected. This issue might arise when using research frameworks for the purposes of teacher reflection. If a research framework is considered to be an example of what has been sanctioned as 'acceptable,' then teachers' interactions with frameworks might actually serve to undermine their agency. This is not the stance that I adopt. Instead, I understand reflective practice as "the making explicit of teaching approaches and processes so that they can become the objects of critical scrutiny" (Jaworski, 1998, p. 7). I view the teacher as knower and agent for educational and social change [START_REF] Cochran-Smith | Inquiry as stance: Practitioner research for the next generation[END_REF]. The research framework is understood as a tool for teacher inquiry and reflection rather than a prescription for action. Inquiry is taken to be "a critical habit of mind that informs professional work in all its aspects" where the data arising from practice is continually interrogated (Cochran-Smith, Lytle 2009, p.121). It is from this perspective that a teacher's use of research frameworks is understood.

Theoretical perspective

Levels of reflection

Models of reflection often emphasize process or chronological phases, e.g., Gibbs (1988). In contrast, [START_REF] Korthagen | In search of the essence of a good teacher: Towards a more holistic approach in teacher education[END_REF] model (Figure 1) emphasizes the teacher as person. Korthagen describes the levels as "different perspectives from which we can look at how teachers function" (p. 80). This model should be of interest to researchers seeking to investigate the nature of teacher reflection and teacher-educators seeking to support the same process. For teacher-researchers, it may offer an overview of the landscape of reflection and serve to frame and contextualize the inquiry process. The umbrella-nature of the model is powerful because the individual levels commonly exist as distinct research domains within mathematics education. This scope is challenging to address and only key issues and contradictions are highlighted here. Korthagen explicates his model largely with reference to the literature of psychology. Some concepts may not align with sociocultural approaches present in much current mathematics education literature. Korthagen maintains that the inner levels influence the outer levels just as the outer levels can influence the inner levels, e.g., behavior can be influenced by external environmental factors as well as personal competencies, beliefs, identity and mission. A teacher's competencies will determine the behavior he/she is able to show but Korthagen suggests that competencies contain also the potential for behavior though this may not be enacted. He maintains that competencies are determined by beliefs. The mathematics education literature has also explored links and discontinuities between teacher beliefs, competencies and classroom practices (e.g., [START_REF] Stipek | Teachers' beliefs and practices related to mathematics instruction[END_REF]. Korthagen suggests that compete alignment between the levels may take a lifetime to achieve and is unlikely to occur without careful reflection on practice and self. Recently, the mathematics education literature has questioned the value of research on beliefs without due attention to teachers' participation in social practices [START_REF] Skott | Understanding the role of the teacher in emerging classroom practices: searching for patterns of participation[END_REF]. Though Korthagen's model does not address this directly, he does situate teachers' beliefs within a complex framework, with the levels of environment and mission in particular premised on the individual's engagement in a social world.

Recent research emphasizes the importance of understanding mathematics teacher competencies as personally, situationally and socially determined [START_REF] Blӧmeke | Understanding mathematics teachers' competencies as personally, situationally and socially determined[END_REF]. Where 'situational' might be connected with the environment, 'personal' and 'social' suggest links with the identity and mission levels respectively. The identity level is concerned with the personal singularity of the individual, and the mission level (or spirituality level in earlier versions of the model) is intended to acknowledge the individual's participation in both local and global communities. [START_REF] Korthagen | In search of the essence of a good teacher: Towards a more holistic approach in teacher education[END_REF] describes mission as being "about becoming aware of the meaning of one's own existence within a larger whole, and the role we see for ourselves in relation to our fellow man" (p. 85). Reflection at this level necessarily encompasses consideration of the short and long term influence of teaching on students and the larger goals of mathematics education or education more generally. This level must be considered to be important in relation to critical reflection (Larrivee, 2001). Some commonalities also exist with critical mathematics education or values education (e.g., [START_REF] Bishop | Mathematics teaching and values education-an intersection in need of research[END_REF].

While noting the importance of identity, Korthagen admits to a certain 'vagueness' around the definition of professional identity as the concept has been informed by many different research traditions. The concept has also been understood in different ways in the mathematics education literature though [START_REF] Boaler | The development of disciplinary relationships: Knowledge, practice and identity in mathematics classrooms[END_REF] and others contend that the identities students develop are strongly related to the classroom practices they have opportunities to participate in. In relation to teacher identity, it is likely that a more complete understanding of teachers' engagement with research frameworks might arise from a perspective which foregrounds the social and situated nature of identity, and acknowledges that teachers engaged in such work may be working at, or across, the boundaries of various communities of practice [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF].

Korthagen positions his model as particularly useful to the teacher-educator. He suggests that a teacher's behavior may imply reflection is needed on a particular level and the teacher-educator can orchestrate his/her interactions with the teacher accordingly. It is likely that relevant research frameworks could be introduced to focus a teacher's reflection on a particular level. It has long been accepted that research frameworks may provide an impetus to question taken-for-granted assumptions and a language to describe, analyze, and interpret practice (e.g., Erikson, 1986). The novel element here is that the grain size, or level of specificity of a research framework, becomes important as this may determine which of Korthagen's levels the framework will relate to. Existing mathematics education research frameworks range from broad general theories about (mathematics) learning to very finely grained, highly structured frameworks concerned with the teaching and learning of particular mathematical content. Korthagen's model provides a structure for considering how such differing frameworks might be used to facilitate effective reflection.

The model is particularly useful in the specific case where research frameworks are being used to facilitate reflection on and development of mathematics classroom practices. Despite classroom practices being enacted at the outer level of behavior, they are connected to both inner and outer levels and arise from a complicated interaction between mission, identity, beliefs, competencies and environment. Larrivee contends that a "deliberative code of conduct" (2000, p. 293), or conscious adoption of particular practices, results from the infusion of personal beliefs and values into a professional identity. If research frameworks, or teachers' interactions with research frameworks, confine reflection to the outer levels of Korthagen's model, there is a danger that rationalization of practice [START_REF] Loughran | Effective reflective practice in search of meaning in learning about teaching[END_REF] may occur rather than inquiry and development. While an appropriate research framework may provoke reflection across and between levels, frameworks which focus attention only on the outer levels are likely to have limited effectiveness. Teachers should have opportunities to question the relationship between inner and outer levels and reflect on how their practices align (or not) with inner levels such as beliefs, identity and mission.

A researcher interested in investigating teacher reflection who adopts the perspective outlined by Korthagen is likely to be interested in the relationships between the different levels of reflection and change. For example, if reflection appears to be occurring as a result of a perplexing situation arising from practice, then the researcher may be interested in attempting to track this to a particular level or a possible conflict between levels (e.g., a teacher is perplexed because she is struggling to implement in practice (behavior) what she believes (beliefs) to be true about 'good' mathematics teaching or in line with her mission. Similarly, a researcher investigating the extent to which research frameworks facilitate reflection may be interested in considering the extent to which a teacher's interaction with the framework(s) aligns with the various levels of Korthagen's model.

An illustration

Two research frameworks that might be used to develop the practices involved in the promotion of math talk are discussed below. This topic has been chosen as classroom interaction emerged as a strong theme in TWG19 at CERME9 [START_REF] Potari | Introduction to the papers of TWG19: Mathematics teacher and classroom practices[END_REF]. The MTLC framework arises from research which tracked a classroom community transitioning from a traditional model to one in which students helped each other learn by engaging in meaningful talk about mathematics [START_REF] Hufferd-Ackles | Describing levels and components of a math-talk learning community[END_REF]. The framework describes four levels of mathematical talk and the overall community trajectory is described as growing "to support students acting in central or leading roles and shifts from a focus on answers to a focus on mathematical thinking" (p. 88). Associated with the levels are developmental trajectories for teacher and student actions across the areas of questioning, explaining mathematical thinking, source of mathematical ideas, and responsibility for learning. Table 1 shows teacher actions at the highest level of the framework.

The MTLC framework may facilitate reflection at all levels but it raises particular challenges at the levels of beliefs, identity and mission because it presents an alternative to traditional teaching. It may challenge the idea of teacher as sole-mathematical authority and it disrupts traditional conceptualizations of teacher and student roles by emphasizing student agency. In doing so, it may provoke reflection on personal beliefs about mathematics and the teaching of mathematics. It may also foreground issues of teacher identity and mission. I used this framework in previous research and while I never formally stated my 'mission', it did help me clarify what I wished to achieve as a teacher of mathematics: students who could think mathematically, and who valued their own thinking and their responsibilities within the classroom community. The framework details teacher actions and gives some direction as to how this mission might be achieved. However, used on its own, it is unlikely to provide sufficient support for developing the complex network of teacher competencies and behavior necessary for the promotion of productive math talk. The second framework I will discuss is [START_REF] Boaler | The importance nature and impact of teacher questions[END_REF] teacher question categories (Table 2). This framework arises from analyses of practice and was developed to allow researchers investigate multiple lessons at a relatively fine grain-size. The framework directs attention to the level of behavior, specifically the questioning practices of the teacher, with scope for consideration of the learning opportunities that arise for the student. It may allow for consideration of the emphasis on relational (type 3 questions) and instrumental understanding (type 1 questions) or the opportunities created for student discussion (type 5 questions). This framework might be used as a tool for reflection, either to support planning of effective questions or to support analysis or reflection on practice. However, it is unlikely that this framework alone will hold meaning for the teacher unless its use is mediated by consideration of the 'bigger picture'. Teacher questions can only be considered effective or ineffective with reference to the overall goal for learning. This refers both to the specific mathematical goals for a lesson or unit of work, as well as goals that might be considered part of mission, such as the development of student agency and mathematical authority.

Description of Teacher Actions

Questioning: Teacher expects students to ask one another questions about their work. The teacher's questions may still guide the discourse.

Explaining Mathematical Thinking: Teacher follows along closely to student descriptions of their thinking, encouraging students to make their answers more complete. Teacher stimulates students to think more clearly about strategies.

Source of Mathematical Ideas: Teacher lets students explain and "own" new strategies. (Teacher is still engaged and deciding what is important to continue exploring.) Teacher uses student ideas as the basis for lessons or mini-extensions Responsibility for Learning: Teacher expects students to be responsible for co-evaluation of everyone's thinking. She supports students as they help one another sort out misconceptions. She helps and/or follows up when needed Teacher Question Categories 1. Gathering information, leading students through a method 4. Probing, getting students to explain their thinking 7. Extending thinking: 2. Inserting terminology 5. Generating discussion 8. Orientating and focusing:

3. Exploring mathematical meanings and/or relationships 6. Linking and applying 9. Establishing context:

The frameworks described above are of different levels of specificity. The MTLC framework attempts to describe teacher and student interactions at a broad level while Boaler and Brodie's framework facilitates fine-grained analysis of teacher questions. For this reason, they offer different affordances and constraints when considered in relation to Korthagen's levels of reflection. It is argued, that for the purposes of facilitating teacher reflection, the use of either of the frameworks alone has limitations. Potential for reflection is maximized when such frameworks are combined. It is in the interplay between reflection at the inner levels and reflection at the outer levels that more profound development may occur.

Conclusion

Many national curricula have begun to emphasize the practices, as well as the content, of mathematics education (e.g., [START_REF] Dooley | Mathematics in early childhood and primary education (3-8 years)[END_REF]. In this context, it is worth considering to what extent research frameworks might be used to support and develop teachers' reflection on practices. I have argued that any such work must strive to take into account the level(s) of reflection that a research framework might target. I also contend that the perspective outlined is of relevance to both teacher-educators and researchers. Further consideration of the methodological implications of a researcher adopting such a stance on teacher reflection is necessary. Theoretical and empirical work is also needed on teachers' use of broad and finely grained research frameworks. Such work should seek to identify the characteristics of research frameworks, and teachers' interactions with frameworks, that enable their use as tools for effective reflection.

Figure 1 :

 1 Figure 1: Korthagen's (2004) model of levels of change

Table 1 : Teacher Actions at level 3 of the MTLC framework (Hufferd-Ackles et al., 2004, p. 88 -90) Table 2: Boaler and Brodie's (2004) Teacher Question Categories

 1

Proceedings of CERME10


domain: math shs

text:
Bodil Kleve
email: bodil.kleve@hioa.no

Gerd Ånestad

The interplay between sociomathematical norms and students' use of informal mental strategies or standard algorithms

Keywords: Subtraction, place value system, learning-partners, mathematical knowledge for teaching, «Ah, mental

In a year 4 classroom, we studied students' presentations of their solutions of problem-solving tasks. Sitting in pairs (learning-partners), they solved the tasks before presenting their solutions orally in class. Based on transcripts from video recordings of the lesson, students' written notes and post interview with the teacher, the role of sociomathematical norms related to students' use of informal mental strategies and standard algorithm for subtraction is discussed. For students flexibly to carry out arithmetic operations, we suggest to develop switching between informal mental strategies and standard algorithm as a sociomathematical norm. In that respect, attention is put on mathematical knowledge for teaching (MKfT) and emphasis on place value system is suggested as amalgam between different strategies.

Background and introduction

This paper is based on a study which purpose was to identify situations in a classroom where development of existing sociomathematical norms or establishing new norms may create potential for students' learning. A video research study was carried out in a year 4 classroom (9-10 years). An earlier publication reported situations in the classroom related to argumentations, and development of existing sociomathematical norms and establishing new norms were suggested in order to increase the potential for students' learning. [START_REF] Kleve | Laeringspartner og sosiomatematiske normer som potensial for elevers laering[END_REF].

Based on students' (learning partners') written and oral presentations of a problem-solving task, where a three digits subtraction had to be carried out, sociomathematical norms [START_REF] Yackel | Sociomathematical Norms, Argumentation and Autonomy in Mathematics[END_REF] are identified and we discuss development of sociomathematical norms in light of mathematical knowledge for teaching [START_REF] Ball | Content Knowledge for Teaching? What makes it special[END_REF][START_REF] Rowland | Mental computation or standard algorithm? Children's strategy choices on mulit-digit subtractions[END_REF]. Our research question is: What role does mathematical knowledge for teaching play in order to develop sociomathematical norms, which can bridge the gap between informal mental strategies and standard algorithm for subtraction so the children can flexibly switch between different strategies? [START_REF] Raveh | Four (algorithms) in one (bag): and integrative framework of knowledge for teaching the standard algoritmes of the basic arithmetic operations[END_REF] presented an integrative framework of knowledge for teaching the standard algorithm of four arithmetic operations. Studying connections between the four basic algorithms for arithmetic operations, they encouraged teaching the standard algorithm with emphasis on conceptual understanding, putting weight on connections between the four algorithms. In our study, the focus is on the relation between informal mental strategies and the standard algorithm for subtraction.

Thematic Working Group 19

Proceedings of CERME10 Torbyns and Verschaffel (2016) analyzed students' use of mental strategies and standard algorithm on subtraction. They found that when having been introduced for the standard algorithm:

Children presumably became gradually more efficient in this algorithm, while their mastery of mental computation in general, and compensation in particular, may have stagnated or even declined (p. 112, italics in original).

Even when numbers involved were suitable for and "strongly invited to mental computation strategies", (ibid.) they found that students, when first having been introduced to the standard algorithm for subtraction, used this instead of mental strategies.

Torbyns and Verschaffel (2016) suggested that when the standard algorithm was introduced, the students would think that this newly introduced method was "the superior way to subtract larger number" (p.112). Referring to [START_REF] Yackel | Sociomathematical Norms, Argumentation and Autonomy in Mathematics[END_REF] they linked this to socio-cultural classroom norms in the classroom. Furthermore they referred to international efforts to reform elementary mathematics education which emphasizes children's abilities to flexibly apply informal mental strategies before they are introduced to the standard algorithm, and the claim that children then will continue to efficiently apply informal strategies. Torbyns and Verschaffel therefore encouraged research in more reform-oriented classrooms, comparing children from these with children taught in traditional oriented classrooms.

The classroom, in which our study took place, was reform oriented. There was a strong focus on children's use of own informal strategies and an extensive use of learning partners. The focus in this paper is on sociomathematical norms related to their use of informal mental strategies and/or use of standard algorithm for subtraction. Related to sociomathematical norms we also discuss what role mathematical knowledge for teaching play in the development of children's flexibility in using different mental strategies and the standard algorithm.

Theoretical perspectives

"Sociomathematical norms are normative aspects of mathematical discussions that are specific to students' mathematical activity" (Yackel & Cobb, 1996, p. 458). Yackel and Cobb focused on sociomathematical norms when studying "how students develop specific mathematical beliefs and values and consequently become intellectually autonomous in mathematics [ ] how they come to develop mathematical dispositions" (p. 458). They distinguished sociomathematical norms from general classroom social norms in that they are specific to the mathematical activity carried out in a classroom. In our study, we focus on an episode where use of informal mental strategies and/or standard algorithm for subtraction as mathematical activity is discussed. Sociomathematical norms can be what counts as an efficient mathematical solution, different mathematical solution and a sophisticated mathematical solution [START_REF] Cobb | Participating in Classroom Mathematical Practices[END_REF]. Cobb et al. emphasized that both social norms and sociomathematical norms are dealing with what is «Taken as shared» in the classroom.

Classroom norms are developed in collaboration between the teacher and students or between students. However, the teacher is the key person when norms are changing or new norms are established (McClain & Cobb, 2001). In our study, we consider that the teacher has great influence on sociomathematical norms, whether the norms are already established, in the process to be weakened or under development in the classroom. We therefore want to link development of sociomathematical norms to mathematical knowledge for teaching. [START_REF] Ball | Content Knowledge for Teaching? What makes it special[END_REF] developed a framework for mathematical knowledge for teaching, MKfT. They distinguished between Common content knowledge, which is mathematical knowledge possessed not necessarily for teaching, and Specialized content knowledge for teaching, which is about the teacher's way of 'unpacking' mathematics, neither necessary nor desirable for others to do. They also included a third category, Horizon content knowledge, which is about making connections between different areas and topics. "Horizon knowledge is an awareness of how mathematical topics are related over the span of mathematics included in the curriculum [ ]. It also includes the vision useful in seeing connections to much later mathematics". (Ball et al. 2008, p. 403).

In order to investigate how aspects of the teacher's mathematical knowledge surfaced in the lesson observed, the Knowledge Quartet (KQ) developed by [START_REF] Rowland | Mental computation or standard algorithm? Children's strategy choices on mulit-digit subtractions[END_REF] has been a valuable tool. The KQ has four dimensions: Foundation, Transformation, Connection, and Contingency. Foundation is informing the other three dimensions, and Connection is the dimension, which we see as linked to "Horizon content knowledge". [START_REF] Plunkett | Decomposition and all that rot[END_REF] discussed pros and cons with regard to use of informal mental strategies and standard algorithm in school, and questioned whether standard algorithms necessarily have to be taught and learned. He claimed that unlike standard algorithms, which only deal with separated digits, informal mental strategies are holistic in that they work with complete numbers and thus requires understanding. Liping [START_REF] Ma | Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States[END_REF] emphasized regrouping rather than technical use of standard algorithm. When regrouping, the subtraction algorithm will be understood on a holistic number level, rather than as separate digits. False mathematical statements as "we can't subtract a bigger number from a smaller" will be avoided. Such false statements are related to teachers' horizon content knowledge [START_REF] Ball | Content Knowledge for Teaching? What makes it special[END_REF]. [START_REF] Anghileri | From informal strategies to structured procedures: Mind the gap[END_REF] compared the Dutch approach to written division calculations in school, which extensively built on children's own informal strategies, to the English approach which was schematic and focused on separate digits. Based on the results from their study [START_REF] Anghileri | From informal strategies to structured procedures: Mind the gap[END_REF] warned against replacing informal strategies with standard algorithms. Rather one should give support to structuring informal approaches in a written record.

Referring to [START_REF] Plunkett | Decomposition and all that rot[END_REF] and that calculations are carried out in real life, [START_REF] Anghileri | Teaching number sense[END_REF] emphasized children's mental strategies as a starting point for developing more formal methods.

Based on the above, one can suggest that when introducing a standard algorithm, teachers should focus on regrouping numbers and take children's informal mental strategies as a starting point. This puts demands on the teacher's mathematical knowledge for teaching, which again will constrain the sociomathematical norms in the classroom.

Referring to among others [START_REF] Ball | Content Knowledge for Teaching? What makes it special[END_REF], [START_REF] Raveh | Four (algorithms) in one (bag): and integrative framework of knowledge for teaching the standard algoritmes of the basic arithmetic operations[END_REF] proposed a framework consisting of four components: Procedural Knowledge (PK), Knowledge of Underlying Concepts (KC), Knowledge of Similarity between the algorithms (KS) and Knowledge of Representations (KR). In our analysis, we will use components from this framework, mainly PK and KC, and some of KR. PK is about carrying out the steps correctly in the (subtraction) algorithm, while KC refers to knowledge about mathematical concepts underlying different algorithms as place-value and number regrouping. Rather than analyzing different representations of the subtraction algorithm as KR refers to in Raveh et al's framework, we emphasize the relationship between informal mental strategies, and the standard algorithm for subtraction.

In our study, we will not argue for not introducing the standard algorithm. However, we consider bridging between informal mental strategies and standard algorithm as valuable features of mathematics, which may influence students' mathematical beliefs. Development of flexibility and children's ability to switch between informal strategies and standard algorithm are linked to sociomathematical norms and to the aspects pf teachers' mathematical knowledge for teaching.

Methods

We observed two mathematics lessons in a 4th grade (mixed ability, 9-10 years old) classroom. Prior to the classroom observations, we had come to know the teacher. Her educational background was a pre-school teacher. She described her teaching as being reform oriented and that her students performed well on "transition tests". She had established an extensive use of learning partners in her classroom. She put her students together in pairs at random, same partner in all subjects, and changing partners every week. According to the teacher, the students never complained or protested against whom they received as their partners. This was established as a social norm in the class. With regard to Eli's view on mathematics teaching and learning, she emphasized the process rather than the product, saying: "For me it is not so important if the answer is correct. I am more interested in the strategy they use, that they have understood the principle behind solving such tasks". She also told us that she encouraged students to develop their own strategies in solving arithmetic problems and to discuss their strategies with their learning partner. Against this background, we wanted to study sociomathematical norms in Eli's class.

During our first visit in the class, we observed and wrote field notes. The second time we video recorded a 90 mins mathematics lesson. We used two cameras. The students were sitting in pairs and should solve different tasks, which were written on the board. After having solved the tasks, and written down their solutions, they presented their solutions orally. Towards the end of the lesson, they had some ('warm ups') whole class activities where they "worked with concepts" (teacher's expression). One of the activities was linked to the place value system.

Our analysis here is based on transcriptions of video recordings of their oral presentations and on their written work, which we collected. We also studied the video recordings together with the teacher several weeks afterwards. We interviewed the teacher and asked her to comment the different pairs' written sketches and oral presentations.

The task, on which we base our analysis, was written on the board: Tobias has two 200 NOK notes and six 10 NOK coins. He spends half on a gift, 142 NOK on a book and then he buys "pig ears" to Doffen. One ear costs 11 NOK. How many ears can he buy?

Analysis

The subtraction 230-142 had to be carried out to solve the task. Eleven out of twelve pairs of students had used the standard algorithm, however with different degree of detail in their oral presentations. The standard algorithm had become the "taken as shared" method (Cobb et al., Pair 9: 230 -142 is zero minus two, doesn't work. We have to exchange from the three, and ten minus two is eight and then we have two left there. So then we take, however four minus two doesn't work either, so then we will have to exchange from the two. Ten minus four is six plus two is eight and then we have only one left, which makes one minus one is zero. So then it is 88".

Pair 8:

We did 230 minus 142 using the standard algorithm. We got 88

Pair 7: 230 minus 142 is 88

Studying Pair 5's calculation, both their written work and oral presentation, we see that they did not use a standard algorithm for subtraction. The students regrouped the subtrahend 142 into 130 and 12. This way of calculating is flexible and requires understanding and a holistic way of thinking. They treated the complete numbers rather than separated digits [START_REF] Plunkett | Decomposition and all that rot[END_REF].

The interview with the teacher Eli, with regard to Pair 5, revealed that she did not see the way of solving and presenting this as a potential for further development. The teacher expressed her acknowledgement of different ways of doing subtractions, but that this was a cumbersome and much lengthier way. She considered one of the students in the pair as a "funny one", and that "you need to be much sharpened to follow his thoughts. However, I keep telling him that he ought to start using another strategy in order to make things go faster. So after a while you' will have to do that". This is in line with [START_REF] Plunkett | Decomposition and all that rot[END_REF] characterization of informal mental strategies: "often difficult to catch hold of "(p. 3). This comment revealed that the teacher did not see the potential in her students' mental calculations for further development. She now looked upon the standard algorithm as the most efficient and sophisticated way of carrying out subtraction, and using the standard algorithm was in the process of being established as a new sociomathematical norm.

As we can see from figure 1, the three other pairs used the standard algorithm for subtraction. All correctly written out, displaying decomposition (exchange or borrowing). This can also be interpreted as regrouping of the minuend based on the place value system. However, the students did not express any regrouping. Studying their oral presentations reveals that the students were on different levels in using the algorithm. "Standard algorithms are not easily internalized. They do not correspond to ways in which people tend to think about numbers" (Plunkett, 1979, p. 3, italics in original). The pairs only referred to digits between 0 and 10. We suggest that this is Pair 9 presented a detailed explanation of the algorithm as such. Their claims: "zero minus two, doesn't work" and "four minus two doesn't work reveal either", reveal a misconception or a "false mathematical statement" (Ma, 2010, p. 3). These students have not been presented for negative numbers yet, and false mathematical statements like these may lead to later misconceptions about negative numbers. The students in this class (except Pair 5) used the term "exchange" when regrouping the minuend 230 into 220 + 10, and when they later regrouped 220 into 120+100.

A question here is whether the students know what they are doing. According to Plunkett, use of standard algorithms encourage suspended understanding. Pair 9's explanation reflects procedural knowledge in carrying out an algorithm rather than conceptual understanding. [START_REF] Ma | Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States[END_REF] encourages regrouping rather than exchanging or borrowing (decomposing) when introducing the standard algorithm. Because then they will be working on a holistic number level rather than with separate digits.

With regard to Pair 8, they only said they had used the standard algorithm, whereas Pair 7 only presented the answer. They can be considered as having internalized the algorithm, and as [START_REF] Plunkett | Decomposition and all that rot[END_REF] puts it: "While the calculation is being carried out, one does not think much about why one does it in that way" (p. 3).

In the interview with the teacher, Eli said that when starting a new arithmetic operation, she encouraged everyone to do it his or her own way, and that she used to present all students' different informal strategies on the board. She expressed a great concern about these differences when a new arithmetic operation was being introduced. Thus, we consider that use of mental strategies for subtraction was earlier established as a sociomathematical norm. However, after having introduced the standard algorithm, this sociomathematical norm was in the process of disappearing, or at least fading, and a new sociomathematical norm was about to be developed. About the introduction of the standard algorithm, Eli said, "We practiced memory numbers and exchange in detail". Consequently, we consider such detailed explanation as a new sociomathematical norm. This sociomathematical norm is also in the process of disappearing. Everybody was now expected to use the algorithm without further explanations or comments. As we see from our data, some students used the standard algorithm naturally without further explanation, while others still explained the procedure in detail. Only one pair (5) explained the subtraction as use of mental strategies without mentioning the standard algorithm. Hence, eleven out of twelve pairs looked upon the newly learnt algorithm "as a superior way" (Torbeyns & Verschaffel, 2016, p. 112).

Discussion

Our findings suggest that although Pair 5's way of doing subtraction was not acknowledged ("he ought to start using another strategy in order to make things go faster"), the students displayed both number sense and a well-developed subtraction concept. Of those who had used the standard algorithm, some explained the procedure in detail, Pair 9, while others just referred to the algorithm. Although they might have had conceptual understanding, they did not display it. Their focus was on the skill carrying out the subtraction procedure. According to Eli, the students who used the standard algorithm had developed a more mature number sense than those still using mental strategies. Although being influenced by reform-oriented working methods, Eli expressed the necessity of learning the standard algorithm, both as a tool, an assurance to always have a method to use, and as an efficient way of doing subtraction. She looked upon standard algorithms as a supplement to informal mental strategies. However, she was not aware what research has shown; when first have been introduced to the standard algorithm for subtraction and exposed to instruction emphasizing mastery of the standard algorithm, children will gradually become more efficient in using the standard algorithm, while their use of informal and mental strategies will fade (Torbeyns & Verschaffel, 2016). The challenge is to bridge or close this gap. A goal must be to develop sociomathematical norms where students are able to switch between different strategies dependent on the nature of the numbers involved. This puts demands on the teacher's MKfT, and especially the Horizon knowledge.

There was no indication in what the teacher said in the interview that the informal mental strategies the children earlier had used had been taken as a starting point when introducing the standard algorithm. The teacher's mathematical knowledge for teaching seemed too fragile to give her courage to rely on students' mental strategies, and thus to bridge the gap. The mathematics presented for her students seemed to be fragmented. During the place value activity towards the end of the lesson students should answer questions as "what value does 1 have in 5129?" If this had been linked to the standard algorithm for subtraction, regrouping, based on place value system, could serve as an amalgam between informal strategies and the formal algorithm. This refers to Raveh et al.'s (2016) KR, which we see as knowledge about connections between informal strategies and mental strategies. Attention could here be brought to the foundation and connection aspects of the teacher's knowledge [START_REF] Rowland | Mental computation or standard algorithm? Children's strategy choices on mulit-digit subtractions[END_REF]. Knowledge of the mathematical concepts underlying the algorithm, KC [START_REF] Raveh | Four (algorithms) in one (bag): and integrative framework of knowledge for teaching the standard algoritmes of the basic arithmetic operations[END_REF] did not surface in what she said. However, she demonstrated procedural knowledge (PK) related to correct computations and the steps in the algorithm.

Our findings suggest that the teacher did not see the potential in taking earlier established sociomathematical norms about students' use of mental strategies as a starting point. We claim that linking informal mental strategies to the place value system in introducing the standard algorithm for subtraction would enhance students' ability to switch between informal strategies and the standard algorithm, dependent on the numbers involved. Regrouping based on the place value system could then serve as an amalgam between informal mental strategies and the standard algorithm.

Based on our data, we cannot say anything about the sociomathematical norms related to other areas of the mathematics in this classroom. However, the students had not yet been introduced to the standard algorithm for division. We saw that in carrying out the necessary division operation to solve the task (how many ears can he buy?) 88:11, the students used either repeated subtraction or repeated addition as (informal) mental strategies.

Encouraging teachers to rely on and see the potential in earlier established sociomathematical norms where students use informal mental strategies, are important issues for further research. In that respect, attention must be directed towards to teachers' mathematical knowledge for teaching, with special focus on Horizon knowledge and the connection dimension of the teacher's mathematical knowledge [START_REF] Ball | Content Knowledge for Teaching? What makes it special[END_REF][START_REF] Ma | Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States[END_REF][START_REF] Raveh | Four (algorithms) in one (bag): and integrative framework of knowledge for teaching the standard algoritmes of the basic arithmetic operations[END_REF][START_REF] Rowland | Mental computation or standard algorithm? Children's strategy choices on mulit-digit subtractions[END_REF].

Figure 1 :

 1 Figure 1: Four pairs' written calculations

Proceedings of CERME10


text:
Armin Jentsch
email: armin.jentsch@uni-hamburg.de

Lena Schlesinger

Measuring instructional quality in mathematics education

Keywords: Mathematics instruction, instructional quality, classroom observation techniques, instructional improvement

Instructional research has recently become more important within the framework of teaching effectiveness research. Various instruments have been developed within this research discipline in order to gain a better insight of what is really happening in the classroom. Most of these instruments mainly focus on generic aspects of instructional quality. In this paper we first describe a subject-specific dimension of instructional quality. Second, we show how these subject-specific aspects could be measured empirically with a standardized observational instrument. The results point out both good interrater agreement and satisfying reliability measures. The presented observational instrument has been developed within the study TEDS-instruct in which relations to teachers' competencies and students' achievement are analyzed.

Introduction

In German instructional research, subject-specific aspects of instructional quality have been investigated rarely until now. Three generic dimensions have been introduced which are classroom management, personal learning support and cognitive activation (e.g. [START_REF] Lipowsky | Quality of geometry instruction and its short-term impact on students' understanding of the Pythagorean Theorem[END_REF]. Although these basic dimensions were developed for mathematics instruction in the first place, they are now deemed relevant for every subject at school [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF][START_REF] Helmke | Unterrichtsqualität und Lehrerprofessionalität: Diagnose, Evaluation und Verbesserung des Unterrichts[END_REF]. Other important aspects of mathematics instruction were apparently disregarded then (e.g. representations, examples, modelling, proof). Blum and others (2006) therefore ask for a high mathematical quality of the lesson beyond the three basic dimensions.

At the same time, several instruments for measuring subject-specific aspects of instructional quality have been developed within the American debate (e.g. Learning Mathematics for Teaching Project, 2011). However, some of these instruments do not contain generic aspects even though the prognostic validity of the three basic dimensions of instructional quality has been shown empirically more than once (see [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF]Kunter et al., 2013;[START_REF] Lipowsky | Quality of geometry instruction and its short-term impact on students' understanding of the Pythagorean Theorem[END_REF]. In conclusion: to our knowledge there is no standardized instrument in existence that is based on a sound theoretical framework (e.g. the three basic dimensions) covering also subject-specific aspects regardless of the mathematical content discussed in class. Starting from these three basic dimensions, we would like to point out subject-specific characteristics of instructional quality that are deemed relevant in the literature and show how these characteristics can be measured with a standardized instrument. The purpose of this paper then is to describe the development of an observational protocol that is used to assess instructional quality in secondary mathematics classes and to present first empirical results.

Theoretical framework

In recent years educational research has shown a great scientific interest in teacher knowledge and instruction (see [START_REF] Hattie | Visible learning for teachers: Maximizing impact on learning[END_REF]. At the same time, the relation of teacher competencies and students' achievement has been analyzed [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF][START_REF] Hill | Effects of teachers' mathematical knowledge for teaching on student achievement[END_REF][START_REF] Scheerens | Review of school and instructional effectiveness. Background paper prepared for the Education for All Global Monitoring Report[END_REF]. The so-called process-mediation-product-paradigm is regarded as a theoretical framework in which these research questions are grounded. This framework describes a relation between students' learning and instruction which has to be offered by teachers and used by students [START_REF] Brophy | Teaching[END_REF][START_REF] Brophy | Observational research on generic aspects of classroom teaching[END_REF][START_REF] Helmke | Unterrichtsqualität und Lehrerprofessionalität: Diagnose, Evaluation und Verbesserung des Unterrichts[END_REF][START_REF] Oser | Effective and responsible teaching: the new synthesis[END_REF].

Three basic dimensions of instructional quality

As mentioned before, three generic dimensions of teaching quality have been developed in recent instructional research which are classroom management, personal learning support and cognitive activation. These dimensions were shown to have a positive impact on both students' learning and their motivation in class [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF][START_REF] Lipowsky | Quality of geometry instruction and its short-term impact on students' understanding of the Pythagorean Theorem[END_REF].

Classroom management focuses on quality-oriented learning time provided for students and on how effectively the teacher deals with disciplinary conflicts [START_REF] Brophy | Teaching[END_REF]. Effective classroom management is also characterized by a lesson that is organized well and has clear routines [START_REF] Lipowsky | Quality of geometry instruction and its short-term impact on students' understanding of the Pythagorean Theorem[END_REF]. The second dimension, personal learning support, includes the individual support provided by the teacher, the relationship between students and teacher as well as constructive feedback (e.g. [START_REF] Rakoczy | Motivationsunterstützung im Mathematikunterricht[END_REF]. Finally, cognitive activation refers to how problemsolving tasks are used to activate learning processes [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF][START_REF] Brophy | Teaching[END_REF]. This dimension includes the activation of previous knowledge and whether challenging tasks and questions are presented that foster students in high-level thinking activities [START_REF] Lipowsky | Quality of geometry instruction and its short-term impact on students' understanding of the Pythagorean Theorem[END_REF][START_REF] Praetorius | One lesson is all you need? Stability of instructional quality across lessons[END_REF].

Although the three basic dimensions focus on generic aspects of instructional quality, the question remains whether they could be operationalized in subject-specific way. This holds specifically for cognitive activation [START_REF] Drollinger-Vetter | Verstehenselemente und strukturelle Klarheit: Fachdidaktische Qualität der Anleitung von mathematischen Verstehensprozessen im Unterricht[END_REF]Schlesinger & Jentsch, 2016). Moreover, it is not clear which subject-specific aspects are missing in this framework and how its dimensionality changes when generic and subject-specific aspects are measured simultaneously [START_REF] Drollinger-Vetter | Verstehenselemente und strukturelle Klarheit: Fachdidaktische Qualität der Anleitung von mathematischen Verstehensprozessen im Unterricht[END_REF]. Due to these concerns, most instruments that have been developed for the analysis of instruction are only suitable for a very small number of situations (e.g. regarding the mathematical content, see [START_REF] Steinweg | Einschätzung der Qualität von Lehr-Lernsituationen im mathematischen Anfangsunterricht -ein Vorschlag[END_REF][START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF].

Measuring instructional quality

We will now focus on the question how instructional quality can be measured both reliably and validly. Praetorius and colleagues (2012) see classroom observations as a straight-forward way to do so, especially compared to the analyses of material or minutes conducted during the lesson. Helmke ( 2012) even claims that one can only speak of instructional research in a narrower sense when classroom observations are performed. The reliability of classroom observations is always an issue since observer ratings are often heavily biased and the stability of the measurement is sometimes questioned (for an overview see [START_REF] Praetorius | Messung von Unterrichtsqualität durch Ratings[END_REF]. Most authors suggest analyzing variance components beyond measures of interrater agreement to understand better which sources of error have added to the variance of the observed score (e.g. [START_REF] Praetorius | Observer ratings of instructional quality: Do they fulfill what they promise?[END_REF][START_REF] Praetorius | One lesson is all you need? Stability of instructional quality across lessons[END_REF].

Observational instruments may contain both low and high inference items [START_REF] Praetorius | Messung von Unterrichtsqualität durch Ratings[END_REF]. Codings with low inference are operationalized in a way that is strictly observable. High inference items, in comparison, need the observer to interpret what he or she sees which makes the observation much more complex [START_REF] Hugener | Dokumentation der Erhebungs-und Auswertungsinstrumente zur schweizerischdeutschen Videostudie "Unterrichtsqualität, Lernverhalten und mathematisches Verständnis[END_REF]. However, at the same time one gains a higher validity because instructional research has shown that low inference items explain only little variance when students' achievements are measured as outcome variables (e.g. [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF]. This is because the surface structure of instruction (e.g. which method is used by the teacher) and its quality may sometimes vary independently from each other (Kunter & Voss, 2013).

Developing an instrument for measuring instructional quality

The observational instrument that is presented in this paper was developed within the study of TEDS-Instruct which is a Follow-Up Study of TEDS-M (Teacher Education and Development Study in Mathematics). The main goal is to empirically investigate teachers' competencies in mathematics education at the secondary level and their influence on students achievement mediated by instructional quality. As a matter of fact, students' achievements will be collected to describe the prognostic validity of both teachers' competency tests and the observational protocol that is presented here.

Subject-specific aspects of instructional quality

For developing a subject-specific dimension of instructional quality we first analyzed which subjectspecific aspects of instruction are assumed to have an impact on students' learning which has to be examined empirically. The main goal for developing a fourth dimension with subject-specific aspects was hence to extend the existing generic theoretical framework of instructional quality. Such an extension with subject-specific aspects is not established until now (e.g. [START_REF] Steinweg | Einschätzung der Qualität von Lehr-Lernsituationen im mathematischen Anfangsunterricht -ein Vorschlag[END_REF]. During the development of the fourth dimension it became apparent that it is necessary to discuss the subject-specifity of some aspects that are included within the former three-dimensional generic framework. This discussion leads to the assumption that the three basic dimensions of instructional quality are not completely generic. However, in the fourth dimension there were included only such subject-specific aspects that were not already used to operationalize the other three dimensions.

For conceptualizing this dimension, a systematic literature survey within the databases of Web of Science, ERA and ERIC was conducted (see Schlesinger & Jentsch, 2016 for more detail). At the same time, the national debate on mathematics education and the German common core standards was reviewed. Based on the described approach, the following aspects were operationalized for the observational instrument:

 the teacher's mathematical correctness  the use of representations  mathematical competencies (modelling, problem-solving, the use of mathematical language, argumentation and proof, training mathematical tools and operations)  a constructive approach to students' mathematical errors  the quality of exercises and tasks  sense-making  teachers' mathematical explanations  appropriate examples  mathematical depth (e.g. generalizations)

Method

Based on the three basic dimensions of instructional quality and the subject-specific aspects that were condensed into a fourth dimension, we developed an observational instrument that can be used for in vivo ratings without needing videos of the lesson and that can be utilised independently from both the specific mathematical content and the academic year. Instructional quality was rated by assessing the items on a four-point scale (1="not at all true"; 4="completely true"). The instrument consists of four dimensions which are classroom management (five items), personal learning support (seven items), cognitive activation (five items) and mathematics educational quality of instruction (nine items). The data for TEDS-Instruct was collected in Hamburg from a sample of 38 teachers at the secondary level. The teachers participated on a voluntary basis. Therefore it can be assumed that they were greatly motivated to have their lessons observed. Each teacher was observed for two lessons (90 min each). Within one lesson, the instructional quality was assessed four times (every 22.5 min).

Example items Indicators

Mathematical depth

 The teacher provides generalizations  The teacher provides mathematical connections  The teacher deepens and structures mathematical knowledge

Representations

 The teacher provides various representations for mathematical objects  The teacher illustrates the linking between different representations Table 1: Two example items for the subject-specific dimension Altogether, there were six observers involved in the classroom study, all of which held at least a university degree from a mathematics teacher program. The observers were trained for the classroom observations in advance which took around 20 hours. The training had three main goals: 1) a joint understanding of the theoretical underpinnings of each rating dimension, 2) familiarizing with the observational protocol, 3) ensuring a satisfactory amount of interrater reliability. By doing so, all items and indicators were discussed thoroughly with the help of a rating manual. The goal of a joint theoretical understanding also involved the object of measurement, i.e. instructional quality.

Based on the process-mediation-product-paradigm, instruction is regarded as a learning opportunity that is individually adapted to students' skills and dispositions. Even though the focus of the observations lies mainly on the teachers' behaviour, the latter is dependent of the students' behaviour and student-teacher-interactions. Students' reactions to the learning opportunities are crucial to understand and assess the quality of instruction and are hence part of the observation, too. Nonetheless, due to pragmatic reasons no student self-reports or cognitive tests have been collected.

Before stepping into real classrooms, the observers trained their skills on videotaped lessons until they reached a certain amount of interrater agreement. Finally, a pilot study was conducted with 13 teachers in three German federal states. After each observation, the ratings were discussed intensively between the two raters. For the data collection the lessons were observed directly without using videotapes, i.e. the raters assessed all items within the lesson in vivo. Two raters were chosen randomly and rated the lesson independently from each other. For this reason it was possible to avoid systematical agreement between certain raters. In addition to these ratings, there were also produced minutes for every lesson. These minutes included teaching methods, teacher-student-interactions, students' behaviour and reactions, the mathematical content and provided materials and tasks from the lessons for a detailed description of the learning opportunities.

Results

As a first step we calculated descriptive statistics for the data that was collected by external observers. The following table contains the results of all items from the three basic dimensions and the newly developed subject-specific dimension. For ensuring interrater reliability Spearmans ρ was calculated. This is a common measure since in educational research one is more often interested in relative than in absolute decisions [START_REF] Praetorius | Observer ratings of instructional quality: Do they fulfill what they promise?[END_REF]; see also [START_REF] Shavelson | Generalizability theory: A primer[END_REF]. In the present study we reached satisfying results of .75 ≤ ρinter < .97 which can be interpreted as high or very high correlations between both observer ratings. In order to calculate the descriptive statistics the data was aggregated to a single datum per person (N = 38). By doing so, we first took the average rating of both observers and then calculated the mean of all eight measurement points per teacher.

Items

M SD rit

Classroom management (α = .83) When looking at the measures in table 2, we see that correctness has reached the highest values in the subject-specific dimension. Even though this could be seen as a ceiling effect, the statistical discrimination is quite high. The same holds for the items in the first dimension. Nonetheless, these ceiling effects are not surprising as the sample consisted of professional teachers only [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF]Blömeke et al., 2010). On the other hand, the average individual support that was observed in the lessons is quite low which is also supported by the low measures of the items "Selfregulated learning" and "Differentiation". Finally, the standard deviation of most items is high enough to conclude that a decent amount of variance was measured.

Altogether we can conclude that an acceptable internal consistency could be reached for all four dimensions. When rit = .25 is regarded as a threshold for acceptable measures of statistical discrimination, the item "Student's feedback" did not reach acceptable values and was thus excluded from further analyses which is also due to a floor effect. All other items show at least mediocre correlations to the corresponding scale which supports the claim of three generic dimensions. This is, however, supposed to be confirmed by factor analyses. Recent both exploratory and also confirmatory approaches once again support the hypothesis of three generic dimensions but suggest dividing the subject-specific dimension into two sub-dimensions which will be discussed in more detail in the presentation (Blömeke et al., submitted).

To sum up, this present study has mainly an explorative character concerning the mentioned subject-specific aspects. However, from a more content-related standpoint one can conclude that fostering specific mathematical competencies like modelling or proof has often been disregarded during lessons. Precise analyses of the used material might then be fruitful to understand better what has happened in the classroom.

Discussion and outlook

The presented instrument for measuring instructional quality shall finally be discussed concerning advantages and disadvantages compared to other instruments in the field. Since this instrument has been developed in order to be used in classrooms without analyzing video there is a chance that it could possibly be used in a broader way than instruments from video studies. Second, measuring instructional quality more than once in a given lesson may describe the learning process in more detail and can lead to more reliable data because certain aspects of instructional quality may change a lot during the lesson. The ratings then tend to be biased heavily since the observer has to give a single rating for the whole lesson [START_REF] Praetorius | Observer ratings of instructional quality: Do they fulfill what they promise?[END_REF]. Third, the instrument is suitable for most mathematics classes, academic years and mathematical contents. Finally, in this instrument generic and subject-specific aspects are combined which, in addition, can then be analyzed on their relation.

The question remains whether the present instruments' prognostic validity can be shown by analyzing the relation of instructional quality and students' achievements. It should be tested whether instructional quality can be seen as a mediator variable between teachers' competencies and students' learning, too. This might especially be interesting for mathematics educational scholars since the impact of generic aspects of instructional quality has already been shown in some studies [START_REF] Baumert | Teachers' Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress[END_REF][START_REF] Helmke | Unterrichtsqualität und Lehrerprofessionalität: Diagnose, Evaluation und Verbesserung des Unterrichts[END_REF][START_REF] Lipowsky | Quality of geometry instruction and its short-term impact on students' understanding of the Pythagorean Theorem[END_REF]. The important mathematical or mathematics educational aspects of instructional quality and their impact on both learning and other outcome variables as motivation or metacognition have still to be found. Here, our study could help to gain a little more insight.

Table 2 :

 2 Descriptive statistics for all items

	Effective use of lesson time	3.58	.33	.59
	Clear rules and routines	2.97	.19	.66
	Preventing disruptions	3.39	.45	.83
	Advance organization	2.89	.49	.55
	Working atmosphere	3.23	.51	.77
	Personal learning support (α = .714)
	Students' individual support	2.05	.45	.37
	Approach to heterogeneity/differentiation	1.26	.38	.64
	Self-regulated learning	1.48	.35	.58
	Teacher's feedback	3.07	.37	.49
	Teacher approval	3.10	.38	.49
	Students' feedback	1.05	.11	.22
	Fostering cooperative learning	1.75	.46	.27
	Cognitive activation (α = .821)
	Challenging tasks and questions	2.54	.47	.79
	Supporting metacognition	1.25	.29	.42
	Activating prior knowledge and co-construction	2.66	.37	.76
	Quality of teaching methods	2.81	.41	.66
	Securing knowledge	2.43	.48	.50
	Mathematics educational characteristics (α = .820)
	Constructive approach to students' errors	2.79	.56	.69
	Teacher's mathematical correctness	3.64	.37	.54
	Representations	2.29	.65	.39
	Exercises and tasks	2.37	.52	.63
	Examples	2.99	.42	.54

Proceedings of CERME10


text:
Helena Grundén
email: annahelena.grunden@lnu.se

Practice of planning in mathematics teaching -meaning and relations

Keywords: Planning, teaching, meaning, practice, interview

Understanding the complexity of teaching also means understanding issues outside classrooms, including planning in mathematics. Although planning is part of a mathematics teacher's everyday life, there is no shared understanding of it, and little is known about how teachers' planning is related to other practices. In response, to explore what planning means to mathematics teachers and planning's relations to other practices, interviews were conducted with teachers and their contents analyzed in several steps to generate a story of each teacher's experiences with planning. For one teacher, Fia, planning meant decisions and considerations about mathematical content and teaching situations, as well as navigating the decisions and opinions of other actors. Fia's planning is related to practices of management, mathematics teaching, and mathematics teachers, all of which influenced her planning and how her students encountered mathematics in the classroom.

Introduction

Teaching in mathematics is complex and cannot be isolated from students' learning or context. To acknowledge that complexity, "deeper explanations of teacher's decisions and actions" [START_REF] Potari | Introduction to the papers of TWG19: Mathematics teacher and classroom practices[END_REF], p. 2972) are necessary. The decisions that mathematics teachers make and the considerations they take into account before entering teaching situations influence what happens in the classroom and thereby students' opportunities for learning. To explore those considerations and decisions, the planning in mathematics teaching is vital in order to understand the complexity of teaching and to improve students' possibilities to learn. However, research has demonstrated a lack of consensus about the concept of planning in mathematics. Former studies focus mainly on the planning itself and not how planning in mathematics is a part of a complex whole and related to other practices. The lack of shared understanding, as well as of studies addressing the complexity of teaching and planning, are thus inevitable stumbling blocks when studying planning in mathematics. Still, knowledge about what teachers actually do when they plan and their constructed meaning about planning might be important information that contributes to an understanding of teachers' work. However, examining only individual aspects of teachers' work is not enough to overcome gaps in research. Teaching is complex not only because it consists of many parts, but also because those parts are framed by "contextual, epistemological, and social issues" [START_REF] Potari | Introduction to the papers of TWG19: Mathematics teacher and classroom practices[END_REF](Potari et al., , p. 2972)). To elucidate those issues, it is helpful to conceive planning in mathematics as a practice in which individual teachers act in a specific time and place, and in which their habitual ways of acting are related to other practices within "a network of practices" (Chouliaraki & Fairclough, 1999, p. 23). By acknowledging planning in mathematics as a practice, the aim of this study was to explore that practice as a part of a network of practices. To that end, it was necessary to know how teachers construct meaning while planning and reflecting upon their planning in mathematics, to what practices they relate planning and reflection, and how they express them in stories about their planning. An interview study with six teachers was designed to answer those questions. This paper recounts the story of one teacher, Fia, that includes several of the aspects and relations described by the other teachers interviewed.

Background

Planning in mathematics teaching

Although planning done by mathematics teachers bears consequences for students' learning, studies in the topic remain scarce. Research about planning in mathematics that does exist address three chief aspects: design research including learning and lesson studies, teachers' mathematical knowledge in relation to their planning, and models and templates for planning. Few studies, none of them conducted in Sweden, have focused on what teachers do in their everyday lives as teachers. National documents based on the Swedish curriculum advocate that planning should be done in a systematic way [START_REF] Skolverket | Planering och genomförande av undervisningen: för grundskolan, grundsärskolan specialskolan[END_REF] similar to that in models emphasized in research (e.g. Gomez, 2002;[START_REF] Superfine | Planning for mathematics instruction: A model of experienced teachers' planning process in the context of a reform mathematics curriculum[END_REF]. However, conversations with Swedish teachers and student teachers have indicated that planning means deciding what to do with a focus on activities, not goals or mathematical content. Such a focus on activities also appears in international research on the topic [START_REF] Akyuz | Improving the quality of mathematics teaching with effective planning practices[END_REF]National Research Council, 2001). In sum, when people talk about planning in mathematics teaching, they demonstrate no shared understanding of what it means.

Practice

One way of describing the complexity of teaching is to use the term practice, which captures both individual people's actions and more habitual, common ways of acting within a practice [START_REF] Chouliaraki | Discourse in late modernity: rethinking critical discourse analysis[END_REF]. Accordingly, conceiving planning in mathematics as a practice can afford a way to conceive teachers' actions both as individual actions and actions shared by other mathematics teachers, as well as a way to conceive the relationship between those actions and abstract structures, that is how social structures govern people's possibilities to act [START_REF] Lund | Pedagogik och diskursanalys[END_REF]. Since each practice is determined from others within a network of practices and since power relations always are present [START_REF] Chouliaraki | Discourse in late modernity: rethinking critical discourse analysis[END_REF], knowing more about teachers' planning practice becomes a way of knowing more about how the process of planning is related to other practices and how power relations are working within the network of practices. In that sense, using practice as a concept to explore planning in mathematics is a way of considering "contextual, epistemological, and social issues that frame mathematics teaching" [START_REF] Potari | Introduction to the papers of TWG19: Mathematics teacher and classroom practices[END_REF](Potari et al., , p. 2972)).

Meaning

In this study, meaning referred to "a (collectivity of) subjects' way of relating tomaking sense of, interpreting, valuing, thinking, and feeling about -a specific issue" [START_REF] Alvesson | Varieties of discourse: On the study of organizations through discourse analysis[END_REF], p. 1147). How teachers relate to planning, makes sense of planning, interprets planning, values planning, and thinks and feels about planning were thus of interest in interviews and their analysis. The meaning that teachers expressed was both transientthat is, constructed and emergent in interactions in interview situationsand durablethat is, connected to cultural and individual ideas. By conceiving meaning as partly durable, it was possible to explain how previous experiences and more habitual ways of acting formed part of the meaning that teachers expressed in interviews.

The study

An important starting point when designing the study was an interest in considerations that teachers' have and the decisions that they make that precede and influence what happens in the mathematics classroom, here called planning in mathematics teaching. With the notion that planning is both a focused, time-bound activity and what emerges from reflections and thoughts that can occur at any time, as well as given the aim to explore a concept about which there is no shared understanding, it was necessary to approach the phenomenon as unprejudiced as possible. Since all teachers have heard about and applied the concept of planning it was important to listen to the voices of teachers, hence the decision to use interviews as a method. In the larger study, from which Fia's story was taken, teachers' reflections on planning were explored. Each of six participants was asked to keep a notebook for a period of two weeks before the interview in which they were asked to record actions, reflections, and thoughts that for them were related to planning in mathematics. In the interviews, teachers referred to their notebooks and topics to talk about. The interviewer's role was to listen affirmatively by uttering encouragement and nodding, asking for clarification when something was unclear, and asking follow-up questions. By not using predetermined questions, teachers were afforded freedom in the discussion about planning, which made it possible to see beyond preunderstandings and the normative speech of planning that dominates mathematics education research. The use of notebooks provided a possibility for each teacher to return to the notebook on several occasions, and the interview situation where the notebook was used as stimuli, was a way to experience meaning as durable. In the interview situation, transient meaning was constructed and emerged in interactions both with the interviewer and with the notebook.

Analysis

Conducting interviews with notebooks as stimuli was a way of foregrounding teachers' experiences and meaning. To continue in that spirit, analysis needed to be based on the material, not predetermined categories. Along with reviewing the stories of each teacher separately, aspects hidden in stories as a whole were also sought. Those somewhat contradictory motivations required staying close to the material and keeping a distance from it. Consequently, analysis proceeded in several steps, the first of which involved reading each utterance per se, and noting what discursive action was performed by making the utterance. During that initial coding memos were written to record spontaneous reflections and ideas, as inspired by [START_REF] Charmaz | Constructing grounded theory[END_REF]. In another version of the transcripts, meaning units (that is, units considered relevant to considerations and decisions that preceded and influenced what happens in the mathematics classroom) were marked. Each unit was paired with the activity belonging to the unit in the first transcript, and by interpreting the meaning unit and the activity together an aspect of planning, considering, or decision-making emerged (Table 1).

Meaning unit (from step 2) Activity (from step 1) Aspect

We have a template that we should stick to. To see aspects hidden in the stories as a group, distance from the material was necessary. Inspired by [START_REF] Szklarski | Fenomenologi[END_REF], meaning units were therefore transformed from the first-to the third-person perspective (Table 2).

Expresses

Meaning unit Transformed meaning unit

When will I be able to plan with my colleagues? The work turns into working alone although I don't want it to. It is a lot… We make these big, long-term plans, but we never have time to see each other once we've started [implementing the plans].

When will she be able to plan with her colleagues? The work turns into working alone although she doesn´t wants it to. It is a lot… They make these big, long-term plans, but they never have time to see each other once they've started.

Do I dare consider it [special education approach] from the beginning? Do I have the energy? Do I have the time?

Does she dare consider it from the beginning? Does she have the energy? Does she have the time? The transformed meaning units were organized so that units dealing with the same aspect were grouped and read as a whole. As a result, meaning was identified and could be expressed as a product of synthesis (Table 3).

Sorted and transformed meaning units (per aspect)

Synthesis of transformed meaning units

Content

In her planning Fia sees that there are several parts to decide upon: what mathematical area the planning should cover, how to connect that to the everyday lives of students, and how to work with and make assessment in relation to students. Good activities can be reused with different foci. Fia thinks about how she will be able to apply an overall special education approach to her long-term plans.

Area [mathematical]

Planning in detail and thinking… how Relation to everyday life: How to get it [mathematics] related to the students.

At examination, it [thoughts about students] comes

How does Fia apply a special educational approach in her long-term plans?

They watched a movie again. Fia has done that several times… The story of 1[name of the movie], but that one focuses on different things. Finding… find activities that you can say, that you can get the most out of… and maybe dare to weed out those that really do not give anything. The synthesis of the transformed meaning units were organized and assembled into stories, one of each teacher. This paper presents the story of Fia, since it emphasizes several aspects visible in the other stories.

Fia's story

Fia is a mathematics-, science-, and technology teacher in compulsory school grades 7-9. She used the notebook for reflections on her planning, teaching, and decisions. In the interview, she referred to her notebook and chose topics to discuss. At several times, she also reflected upon her reflections and reached new insightsfor example, when she referred to "pedagogical plans" in her notebook and discussed how constrained she feels when she has to do her planning with a template.

Fia: I have had exactly that [referring to the template] content before, but it has not been so formal… That formality… everything has to look the same. It makes me constrained, or I don't feel free to think, or… 1,2,3,4: that must come first, then that, then that… But actually… It is also up to me! I can start to think about paragraph 4 if it's about how we should work.

In her story Fia referred to the template several times. She discussed how her school management has decided that a specific template has to be used when planning, largely to be able to collect the plans and thereby "see what is happening". Time is another constraining factor for Fia, regarding both individual and collaborative planning. Colleagues are resources in Fia's planning, and she would like more cooperation with them, also in her short-term planning. She also highly values spontaneous exchanges of ideas and experiences.

Fia's work with planning varies throughout the year. At the beginning of the school year she generally has more energy, but in the final weeks, particularly for the for the ninth graders, Fia tends to perform what she calls "spontaneous planning"that is, decides immediately before lessons what she will do. Fia argues that "spontaneous planning" can be good; the creativity that she sacrifices with templates can bloom in "spontaneous planning", and this also affords greater opportunities for student participation.

Having two groups of the same grade level at different times of the day has made Fia aware of how much the schedule influence her planning. She has also experienced how other activities planned for the students (including field days, theater visits, project periods) steal time from her mathematics teaching and thereby affect her planning. If Fia were allowed to decide upon the schedule, then she would plan for more teaching situations with one or two students. One year she had opportunities for such occasions in her schedule and felt that they helped the students very much.

The availability of materials is another factor that influences her planning. Fia discussed an occasion when she was sitting with small group of students in a room beside the main classroom while the rest of the class was supposed to work in pairs with problem solving and show their solutions on small white boards. However, since there were enough white boards for all students, each student took one and worked individually instead. For Fia, that occasion exemplified how a teaching situation is a meeting between planning and reality and how the outcome can differ from what was intended.

When planning, Fia makes several decisions, including what mathematical content to cover, how to relate that content to students' everyday lives, and how to work with the content. Referring to a film that she has shown several times, she expressed how good activities can be used several times with different focuses. Besides decisions directly related to the concrete teaching situation, Fia has considered how to adopt an overall special education perspective in her long-term planning, which she thinks can benefit all students' learning. Planning can help to "play it cool" and break norms about mathematics teaching. Fia gave examples when she has planned for working with a couple of students at a time although she had a lesson for the whole class, or when she has used the same film on several occasions for the same students. For Fia, planning is always about prioritizing and is related to feelings. She expressed how care for the students and their learning is critical when making decisions and how her own fears and energy level influence how the planning is done. Fia is constantly reflecting on previous experiences and how those experiences can be applied in future plans. She believes that even more reflection for example, after spontaneously planned lessons can be a good way to take advantage of good experiences instead of letting them go to waste.

Fia describes a practice in which the way of talking about planning in mathematics is frustrating. Some ways of planning are more valuable than others, and Fia almost excuses herself for sometimes doing what she calls spontaneous planning. Knowing that other actors might have comments influence her way of thinking, and, according to Fia, changes in teaching can lead to questioning from for example parents and students. In her notebook, she had written about how she wanted to include a special education approach in her long-term planning. She had also written: "Do I dare? Do I have the energy?" Fia said that those considerations represented fears that not all students will be challenged and that she will have to argue for her choices. Fia thinks that she has the authority to make decisions about the teaching, but that exercising that authority takes energy.

Analysis and discussion

Reading Fia's story in the light of meaning defined as "a (collectivity of) subjects' way of relating to making sense of, interpreting, valuing, thinking, and feeling about -a specific issue" [START_REF] Alvesson | Varieties of discourse: On the study of organizations through discourse analysis[END_REF](Alvesson et al., , p. 1147) makes it possible to conceive how she constructs meaning and makes sense of planning in mathematics by choosing topics to discuss, by interpreting the practice of planning, and describing what she is doing and what is important for her. She discussed how choosing mathematical content and ways to present and work with that content always are part of her planning, as well as how good activities can be re-used with a different focus. Also related to the practice of planning is how she uses, and wants to use, reflections to benefit from past experiences, and how planning can help her break norms about mathematics teaching.

Besides describing the practice of planning itself, Fia constructed meaning by, for example, discussing how decisions by school management regarding schedule, availability of materials, and templates influence her planning. From her story it is clear that she perceives models often emphasized as a support for planning (Goméz, 2002;[START_REF] Superfine | Planning for mathematics instruction: A model of experienced teachers' planning process in the context of a reform mathematics curriculum[END_REF] as constraints. This is possible to interpret as she is referring to a practice of management.

Visible in Fia's story are also norms about how mathematics teaching "should be done." Those norms influence her considerations and decisions and emerged in her story as an invisible idea of what counts as teaching in mathematics, but also as concrete examples related to opinions of students and parents. Interestingly, Fia has ideas that she thinks would benefit students' learning, but she contemplates to abandon them because she worries about parents' and students' reactions. When referring to thoughts of mathematics teaching, Fia relates the practice of planning to a practice of mathematics teaching.

Closely connected to practice of mathematics teaching are the colleagues that Fia referred to several times. She talked about a desire to make more collaborative planning, and how colleagues can be resources also in spontaneous exchange of ideas and experiences. Those parts of the story can be interpreted as her referring to a practice of mathematics teachers, in which she sees herself as a part.

In reality, the practice of management and the decisions made there to some extent determines how she can participate in the practice of mathematics teachers. The degree to which co-planning is valued in the practice of management clearly affects Fia's schedule and how much time she has with her colleagues.

Fia's story makes it clear that her actions and reflections within the practice of planning relate to other practices in different ways. Some of those relations are constraining and hinders Fia from planning the way that she wants, whereas others contribute positively to her planning in mathematics. Since those other practices influence her planning they also implicitly influence her students' possibilities to learn. Although the teacher is ultimately responsible for the teaching, results show that there are other aspects that influence the planning and, in turn, what happens in the classroom. That dynamic needs to be taken into account when discussing mathematics teaching and forming development initiatives. Viewing teachers and situations in the classroom as isolated entities poses the undesired consequence that other important aspects that also influence students' possibilities to learn mathematics are neglected.

Table 1 : Examples from analysis step 1

 1

			Formal
	requirements	from	requirements
	school administration

Table 2 : Examples from analysis step 3

 2

Table 3 : Examples from analysis step 4

 3

Proceedings of CERME10