text:
Fay Baldry
email: fb128@le.ac.uk

A teacher's orchestration of mathematics in a 'typical' classroom

Keywords: Observation, teaching methods, sociomathematical norms, variation theory

. This paper focusses on the analysis of two lessons in order to discuss the potential of this framework to foreground the mathematics made available to students and the pedagogical moves made by the teacher to bring this about. In England, class composition for secondary mathematics is usually decided by measures of prior attainment, with students of similar 'ability' grouped together. A wider study is exploring shifts in teachers' pedagogical approaches when they teach classes with different attainment profiles. Consideration here is given to whether observation of a few lessons using this framework can identify stable mathematical characteristics, so that in future the framework would allow comparisons between classes to be made.

Introduction

In England, ongoing attention is paid by politicians and educators to the mathematical attainment of students, and this includes analysing the progress of different groups [START_REF] Easby | GCSE and equivalent attainment by pupil characteristics in England: 2012/13[END_REF]. At secondary level, class composition in mathematics is often determined by measures of prior attainment, with students grouped with others of similar 'ability' and is referred to as setting [START_REF] Ireson | What are the effects of ability grouping on GCSE attainment[END_REF]. Whilst findings from research into the impact of setting do vary, the predominant view is that setting does not improve overall attainment and may indeed act as a suppressant (e.g. [START_REF] Hattie | Classroom composition and peer effects[END_REF]. Moreover, setting does increase the spread of attainment, resulting in students placed in lower attaining sets being disproportionally affected. Issues of equity are raised further as students from lower socioeconomic groups are over-represented in low sets, even when prior attainment is taken into account [START_REF] Muijs | Setting by ability-or is it? A quantitative study of determinants of set placement in English secondary schools[END_REF]. In spite of these concerns, setting in secondary mathematics classrooms appears to be firmly entrenched, with many teachers perceiving setting as the only practical way to teach a range of 'abilities' [START_REF] Wiliam | It's Not Which School but Which Set You're in That Matters: The Influence of Ability Grouping Practices on Student Progress in Mathematics[END_REF].

There is a tendency for the teaching of sets with different attainment profiles to have distinctive pedagogical characteristics; for example, contextualised language is more common in low attaining sets, shifting to more formalized mathematical language in higher attaining sets [START_REF] Dunne | The teaching and learning of pupils in low-attainment sets[END_REF]. The aim of a wider ongoing study is to explore how individual teachers shift their pedagogy when they teach classes with different attainment profiles. It is anticipated that this will offer insights into how they tailor practice for different sets and the impact this has on the mathematics made available to students. In that larger study, a conceptual framework has been developed to interpret teachers' actions in relation to their orchestration of mathematics (figure 1); in this study the framework is employed to capture practice that the teacher considers typical for them with a particular class (set).

Theoretical framework

Classrooms are widely acknowledged as dynamic environments, where the complexities cannot be captured in a simple model (e.g. [START_REF] Potari | Tackling complexity in mathematics teaching development: Using the teaching triad as a tool for reflection and analysis[END_REF]. [START_REF] Goodchild | Revisiting the didactic triangle: from the particular to the general[END_REF] argue that the didactic triangle, where vertices represent teachers, students and mathematical content, "serves as a starting point to theorise the dynamics of teaching-learning" (p. 581). For researchers, this raises a question as to how that practice could be understood without taking into account the actions of individual students. This study utilises the notion of classroom norms as a mechanism for taking into account student activity whilst maintaining a focus on the teacher.

Underpinning much of the recent research relating to teachers' practice appears to be the notion that good practice is related to an inquiry orientation [START_REF] Boesen | Developing mathematical competence: From the intended to the enacted curriculum[END_REF][START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF]. Termed the 'reform agenda' in the US, an inquiry orientation is associated with the development of conceptual understanding through the use of rich mathematical tasks, discussions and problem solving approaches [START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF]. This is often contrasted with a 'traditional' approach, characterised as students working on individual tasks that focus on the efficient application of algorithms, and delivered through transmission style teaching. Many of the existing analytical frameworks are linked to inquiry-oriented goals for the professional development of teachers (e.g. [START_REF] Boesen | Developing mathematical competence: From the intended to the enacted curriculum[END_REF][START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF]. Others focus on particular aspects of classroom practice that are considered important, such as the design of rich tasks and the management of discussion (M. [START_REF] Simon | A developing approach to studying students' learning through their mathematical activity[END_REF][START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF]. However, with evidence that traditional approaches to teaching are still common in England (S. [START_REF] Watson | Observing changes in teachers' practice as a consequence of taking part in professional development: developing a protocol for the observation of lessons[END_REF], the applicability of affordances of inquiry-oriented frameworks to the analysis of 'typical' lessons has to be questioned. For example, inquiry-orientated contexts place more value on discussion, explanation and justification, with the potential to make students' meaning making more visible, than do more traditional approaches.

The Orchestration of Mathematics Framework (OMF) was developed as a tool to build a picture of teachers' classroom practice (figure 1). Whilst there is insufficient space here for a detailed discussion, an iterative process of development was undertaken, where concepts with traction in interpreting classrooms were considered from the teachers' perspective, and their relationship to each other. For example, [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF] tracked the cognitive demand of tasks as lessons unfold; a notion that has subsequently been drawn on by many researchers (e.g. [START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF]. Here, the focus was on linking the teachers' activities with other theoretical perspectives, such as relating problems with multiple solution strategies with variation theory [START_REF] Marton | On Some Necessary Conditions of Learning[END_REF] or the press for explanations with patterns of discourse [START_REF] Imm | Talking mathematically: An analysis of discourse communities[END_REF]. In addition to demonstrating that a wide range of classroom features can have a critical effect on the learning of mathematics, research has indicated that it is not the presence or absence or particular features per se that influences the mathematics experienced by students, but rather the nuances of implementation and interdependency [START_REF] Hiebert | Understanding and improving mathematics teaching: Highlights from the TIMSS 1999 video study[END_REF]. The OMF has been designed to offer a range of lenses that can be brought into play as classroom activity unfolds. As part of this development process, the OMF was used to analyse three publically available video lessons from the TIMSS studies. Whilst not reported on here, the OMF orientated the data analysis and distinct lesson profiles were identified. The central dimensions relate to in-class activity, and in particular the teacher's orchestration of mathematics: that is, the mathematics made available in the shared space of the classroom and the actions taken by the teacher to bring this about. Two significant elements of teachers' practice are the selection of tasks and the management of classroom discourse (e.g. [START_REF] Ainley | Connecting engagement and focus in pedagogic task design[END_REF][START_REF] Stein | Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell[END_REF]. Within each dimension, there is a range of significant elements, such as the role of multiple representations and management of student responses. Variation theory draws on the idea that learning requires variation set against a backdrop of invariance [START_REF] Marton | On Some Necessary Conditions of Learning[END_REF]; A. [START_REF] Watson | Seeing an exercise as a single mathematical object: Using variation to structure sense-making[END_REF], amongst others, have drawn on this theory to explicate how the sequencing of questions or activities can make visible critical features of a concept and hence support generalisation. Moreover, variation theory also offers a mechanism as to how other previously identified beneficial features could support learning. For example, multiple representations and multiple solution strategies could be seen as holding the concept constant whilst varying representations and processes.

The notion of cognitive demand offers a way to categorize the "the level and type of thinking that a task has the potential to elicit" (Boston & Smith, 2009, p. 122). [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF] introduced a rubric where 'memorization' and 'procedures without connections to concepts' are classified as low demand, whereas 'procedures with connections to concepts' and 'doing mathematics' are classified as high demand. As such, this can be viewed as potential of the teacher's orchestration of mathematics to influence student learning. [START_REF] Cobb | Participating in Classroom Mathematical Practices[END_REF] offer an interpretative framework that coordinates social and psychological perspectives, where the social aspect is framed in terms of norms. All lessons and all interactions are unique, but norms offer a way to interpret interactions as typical or atypical, which offers the possibility of generalising beyond particular incidents. In particular, their notions of sociomathematical norms and mathematical practices can identify what is considered legitimised mathematical activity in that particular context. The lesson image, activity and interpretation cycle draws on [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] work on hypothetical learning trajectories, but extends the focus to include performance and engagement goals, as evidence indicates that not all teachers focus on learning [START_REF] Amador | Learning Trajectories, Lesson Planning, Affordances, and Constraints in the Design and Enactment of Mathematics Teaching[END_REF]. This captures the iterative planning, activity and interpretation that starts before the lesson and continues as the lesson unfolds. Teachers' interpretations of mathematical activity are predicated on what they notice; the conceptualisation of professional noticing by Jacobs, Lamb, and Philipp ( 2010) is one construct drawn on here. Taken together, the dimensions offer a way to build a picture of the classroom based on the features as they occur and interact in the course of a teacher's normal practice.

Methodology

This paper reports on a section of a pilot study, undertaken is part of a larger ongoing qualitative case study, following the interpretative tradition. The research question explored in this paper is how viable the OMF is as an analytical tool for charting a teacher's pedagogical approaches and the mathematics made available to students.

For a pilot study, two secondary mathematics teachers known to the author have been recruited. Decisions regarding selection of classes and the timing of observations resides with the teachers, in order to minimize the imposition on them and their schools. So far, data have been gathered from one teacher and two lessons with a year 8 class (12 to 13 years old), taught ten days apart. The author acted as a non-participant observer, with the OMF used as an observation pro forma. The lessons were recorded by two static video cameras and lesson artifacts, such as students' work and teaching resources, were collected. Pre-and post-lesson semi-structured interviews were conducted with the teacher, focusing on how and why the lesson was planned in the manner chosen, and on moments the teacher thought important for learning, including whether these were as anticipated or unexpected.

The audio data from lessons and interviews was transcribed. Lesson activities were coded as being mathematically relevant, organisational or not mathematically relevant. The mathematically relevant sections of the videos were then reviewed; interactions were mapped to the OMF and cross-referenced with the observation notes. The interviews were analysed for evidence of the teacher's interpretation of classroom activities and their lesson image. The analyses of the two lessons were compared; consistencies and contradictions in the dimensions of the OMF were sought, to see whether there was evidence that observation of a few lessons could build a sufficiently representative picture of a class to allow comparison with others [START_REF] Staub | Mathematics classroom cultures: Methodological and theoretical issues[END_REF]. All the analysis in this pilot study has been conducted by the author; it is anticipated that another researcher will review the dimensional analysis before the next stage of the wider study.

Findings and discussion

In this section, two linked extracts from the first lesson will be used to illustrate how the dimensions of the OMF were populated (figure 3). Then the comparison of the two lessons will be discussed.

Lesson 1

The students individually attempted to calculate the areas of six different shapes. The numerical answers were shown, in a mixed order, alongside the questions at the start. After nine minutes solutions were discussed as a whole class.

Analysis of Lesson 1

Line 78 is an example of the teacher indicating that a procedural explanation was expected as part of a response. When answering subsequent questions, most students provided a procedural explanation without prompting but other types of explanations were not offered, indicating an established sociomathematical norm. Line 78 was also the start of an IRE sequence (teacher initiation, student response, teacher evaluation), which was the predominant form of teacher-student interaction in the whole class context. In lines 80 and 113, the teacher repeated correct answers, and in line 119 reworded a more complex student explanation, claiming understanding of student reasoning. In terms of social norms, this contributed to accountability residing with the teacher. Discourse: Teacher led; Teacher requested (procedural) explanation (line 78), followed up when not provided (line 113) IRE with teacher evaluation (lines 78-80, 80-81, 113-119) Teacher re-voiced contributions; repeating correct answer (lines 80, 113), rewording more complex explanations (119) When the two lessons were compared the overall profiles were very similar, and the differences in use of context and mathematical practices could be explained by the different lesson topics. More important is the fact that contradictions between the two lesson analyses were not apparent. From these profiles a summary OMF was formed (figure 4) specific to this class. The central core, consisting of tasks, sequencing, organisation and discourse, located specific instances of the teacher's activities, with their impact interpreted in the wider framework. For example, questions were posed that could have been solved in multiple ways or with the integration of multiple representations. However, the IRE pattern of interaction, focusing on a single procedural calculation, was mirrored by the students in their work and led to the categorization of low cognitive demand.

Links between dimensions did emerge. For example, there were two occasions where pseudocontexts were used: that is real-world objects such as cars or apples were introduced, but in contrived and unrealistic ways. Students made errors in the whole class discussion that were not explored; the teacher focused on explaining the abstracted mathematical procedure whilst the students involved focused on interpreting the context. Follow-up questions were asked by the teacher, but when students' responses did not conform to the abstracted mathematical solution the teacher moved on by offering a direct demonstration of the 'correct' procedural answer. In the post lesson interview, the teacher expressed surprise that errors were made on those questions and was unclear as to why this had occurred. The activities related to the use of pseudo-context can be traced through the dimensions, contributing to the conclusion that the teacher did not explore student thinking and that mathematical competence equates to efficient production of standard solutions.

Conclusion

When the dimensions of the OMF are considered, distinctive patterns of discourse consistent across both lessons were identified, and the classroom norms indicated that these were regular patterns of interaction. Moreover, the same restricted range of task features were utilised throughout both lessons and the sequencing of questions was classified as unsystematic variation. This provides some evidence that the framework was effective in characterising a teacher's pedagogical approaches. However, whilst the teacher indicated that these lessons were 'typical', the timing of the data collection meant that they were focused on reviewing previously met material. As such, there may be features that are part of the teacher's usual repertoire but are not captured here. For the larger study, when comparisons between classes is sought, collecting data at the same time of year and when the classes are being taught similar material could ameliorate some of these issues.

All elements of the lesson that were classified as being mathematically relevant were mapped to the OMF. It may be tempting, therefore, to say that the model is sufficient to capture mathematically significant classroom events. However, the dimensions of the framework have been populated by features identified as significant in the literature and these have orientated the data collection and analysis, therefore a complete mapping could be anticipated. Instead, the question is whether the orientation this framework offers provides insights into the characteristics of teachers' pedagogical moves more powerful than a list of features and with sufficient validity to allow comparison. The evidence presented here indicates that this may indeed be the case, but further research is needed.

Figure 1 :

 1 Figure 1: Orchestration of Mathematics Framework

Figure 2 :

 2 Figure 2: Questions projected onto classroom whiteboard 78 Teacher: OK number one, what did you do and what answer did you get? 79 Azariah: I got five times two.. ten meters.. centimeters meters squared 80 Teacher: Ten centimeters squared, perfect, units are really important ok. Finley second one then. 81 Finley: I did four add eight, twelve divided by two, six.. times five makes thirty centimeters squared. ⁞ [the last question] 113 Teacher: Sixteen centimeters squared, what did you do to get it? 114 Sam: It was just the last… 115 Teacher: Just the last answer, did anyone manage to do it with the maths? Raj 116 Raj: Three times.. three times four so it's the bottom rectangle 117 Teacher: Yep 118 Raj: and that's 12 meters squared, but then you triangle the top bit, which is three times four meters and divide by two 119 Teacher: So what Raj did, and I'm guessing what most people did who managed to get that was to draw a little line there to do our 3 times 4 which is 12 meters squared and work out the triangle on top which was 4 meters squared ok good.

Figure 3 :

 3 Figure 3: Extract from lesson 1 OMF summary

Figure 4 :

 4 Figure 4: Summary OMF for lesson 1 and 2

Lesson 1: No context; Lesson 2: Pseudo-context 2/6

	OMF	Lesson Image
	Interpretation of	Goal: Performance	Plan: Exam style questions Hypotheses: Familiarisation
	classroom activity:	Tasks: Multiple solution strategies were possible, but	Classroom Norms:
	Professional noticing -	rarely explored. (4 occasions students offered	Social Norms: Agency and
	no evidence of	alternative calculation-not evaluated or compared)	accountability: resides
	exploration of student			predominantly with the
	thinking when it was	Sequencing: Questions sets unsystematic variation. All	teacher.
	not directly relatable to	questions dealt with in isolation; links between	Sociomathematical norms:
	a standard solution.	questions were not explored [Dimensions of variation	procedural explanation
		and range of permissible change not made explicit]	counts as explanation.
	Cognitive demand:	Organisation: Individual working-tables in groups of	Mathematical competence
	Potential -high	four; peer to peer discussions were had.	equates to obtaining correct
	As enacted -low	Discourse: Registers: teacher used colloquial language	answers efficiently (errors
		with no evidence of inducting students into a 'vertical	to be avoided).
		discourse'. Patterns: IRE dominant form of interaction.	Mathematical practices:
		Correct answers acknowledged, often repeated or	Lesson 1: Area equates to
		extended. Errors often ignored;	multiplication. Lesson 2:
				Proportional reasoning
				equates to multiplication

when acknowledged focused on moving to standard solution, reverting to direct explanation if initial follow-up failed.

  Extended student explanations taken over by the teacher.

Proceedings of CERME10


domain: math shs

text:
Charalampos Sakonidis
email: xsakonid@eled.duth.gr

Ove Gunnar Drageset
email: ove.drageset@uit.no

Reidar Mosvold
email: reidar.mosvold@uis.no

Jeppe Skott
email: jeppe.skott@lnu.se

Rukiye Didem

Introduction to the papers of TWG19: Mathematics teachers and classroom practices

Keywords: Mathematics teachers, classroom practice(s), teaching

Introduction

TWG19 is one of three TWGs at CERME10 dealing with issues related to teaching and teacher education (the others being TWG18 and TWG20). The group is particularly interested in studies that aim to understand the development of classroom practices and teachers' contributions to them. This includes the roles of other factors such as the available teaching-learning materials, modes of teacher collaboration at and beyond the school(s) in question, and school cultures as they relate to, for instance, teacher-student relationships, teachers' individual and communal responsibilities, and the role of curricular materials and testing. Finally, studies concerned with how micro-level interactions are informed by macro-level structures (society, culture and the political) are also of TWG19's interest.

A total of 27 contributions were initially submitted (26 papers and 1 poster), involving authors from at least 14 different countries, primarily in southern or northern Europe. The papers were grouped in five thematic areas, each assigned to one of the five team leaders for overseeing the reviewing process. Each proposal was reviewed by a team leader and two authors, and 24 contributions (22 papers and 2 posters) were accepted for presentation at the conference. Participants were expected to read the papers before the conference sessions. In the time slot allocated to each group of papers, the authors each gave a short presentation (5-7 minutes), sketching the key ideas of the work reported. This was followed by a reflection by the group leader on common themes and crosscutting issues (15 minutes), which were subsequently discussed by the participants in small groups. Ultimately, 22 papers and 1 poster are included in the conference proceedings.

The five thematic areas according to which the TWG19 contributions were grouped are: (a) problem solving and general issues related to teaching practice, (b) lesson planning, lesson study and curriculum, (c) instructional quality and assessment, (d) in-the-moment teaching actions and decision making and reflection and (e) instructional practices. In the next two sections, we discuss the contributions first within and then across these thematic areas.

Substantive issues

In this section we present research considerations and concerns in the papers in each group, including the range of theoretical perspectives and methodologies employed.

(a) Problem solving and general issues related to teaching practice

Dominant perspectives on the teachers' knowledge have changed and now focus on content knowledge closely connected to the profession. This is fueled by current reform initiatives that emphasise processes of mathematics, for instance in terms of problem solving.

Three of the four papers in this section deal with problem solving. The paper by Kleve and Ånestad concerns a Norwegian teacher, who seems inspired by a process view of mathematics. The class works with subtraction and initially uses informal mental strategies. The authors argue that there is a need to link such methods to a standard algorithm, that the flexible use of both methods is needed, and that it should become socio-mathematical norm that both are acceptable. However, the teacher's mathematical knowledge for teaching seems too weak for her to support the students in the transition from informal strategies to standard procedures. This makes it impossible for her to support socio-mathematical norms in line with the reform.

Odindo's study from Kenya is based on the expectation that problem-solving approaches may alleviate secondary students' difficulties with their final exams. He uses Learning Study (LS) to support teachers in using problem solving when teaching algebra. The paper to some extent focuses on the students, but mainly asks what the learning opportunities are for the teachers. Odindo argues that the LS format allows teachers to consider general issues of, for instance, time management, but also to focus on patterns of task variation closely related to the contents. At least implicitly, then, the paper is concerned with how LS may support mathematical knowledge for teaching.

The study by Villalonga and Andrews is also on problem solving, but less on mathematical knowledge for teaching. In fact, the teacher is conspicuously absent, as the paper deals with how Catalan students may self-scaffold when engaged in problem solving. The students use a resource, an Orientation Base (OB), which is to help them monitor their problem solving. The teacher is almost obsolete, and the OB may be read as a response to the problem that many teachers find it difficult to support their students' problem solving. OB may be seen as an attempt to circumvent this difficulty and transfer responsibility to the students.

The last paper in this group, by Mosvold and Hoover, report on a literature review of studies on mathematical knowledge for teaching. The 12 studies in the review address questions of what, how and why such knowledge plays a part for the quality of instruction. The studies argue that mathematical knowledge for teaching is important for instance for teachers' selection and adaptation of tasks, for their planning of instruction, and for how they listen to students and pursue student thinking. Based on the review, however, Mosvold and Hoover argue that the results are mixed and that there is a need to shift the emphasis towards more dynamic understandings of the relationship between mathematical knowledge for teaching and teaching.

The discussions of these papers focus on what it may mean to adopt a dynamic perspective on the knowledge-teaching relationship. How, then, we may change the emphasis from teacher characteristics (e.g., their knowledge or beliefs) to the acts of teaching? The latter perspective requires greater attention to issues of context, to what we mean by practice, and possibly to alternative frameworks that allow us to reconsider what we mean by knowledge.

(b) Lesson planning, development and curriculum

This group of papers focuses on planning for teaching, utilizing lessons learned in other contexts, and dealing with curricula and textbooks. Four papers approached these issues.

In her case study of a Swedish mathematics teacher, Grundén targets the practice of planning. Reflecting on her own planning, the teacher conceptualizes planning as making informed decisions regarding the teaching of the mathematical content in different contexts, and she relates planning to other practices of the teacher as well as practices of other teachers. Relating to the practice of other teachers is also an issue in the study of Runesson Kempe, Lövström and Hellqvist, who investigate how experiences from a Learning study can be shared and used by teachers in other contexts. Applying the results from a previous Learning study in new classrooms, the authors investigate some necessary conditions for learning about negative numbers and indicate possibilities for cultivating more effective professional practice in mathematics classrooms.

Although development and change in mathematics teaching might be teacher-driven, it is sometimes prescribed by curriculum reforms. Klothou and Sakonidis investigate the implementation of a new curriculum reform among primary mathematics teachers in Greece. They argue that contradictions in teachers' own discourses can be explained by recontextualization procedures that appear when teachers attempt to implement the reform, and inconsistencies may abide in the very discourses that teachers draw upon. Whereas some countries have official textbooks that everyone must use, French teachers are free to decide if they want to use a textbook and howif they adhere to the national curriculum. In their study on how two experienced French teachers use and adapt the content of mathematics textbooks and teacher manuals, Priolet and Mounier analyze how the teachers use the same textbook when teaching the same mathematical content. None of them follow the recommendations from the textbook completely in their lesson, and the adjustments they make tend to reduce the difficulty of textbook tasks.

These four papers provide compelling glimpses into the complex work of teaching mathematics. They discuss how authorities, schools and other teachers may provide resources that are intended to support the work of teaching, and how adapting and using such resources introduces mathematical, didactical and social demands on teachers' work. Mathematical knowledge for teaching can be described as knowledge required to deal with such demands [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF], and these four papers thus indirectly contribute to investigating more dynamic relationships between knowledge and teaching.

(c) Instructional quality and assessment

The five papers in this thematic group describe qualities of teaching. Three papers concern instrument development for assessing qualities. One such instrument is the Realization Tree Assessment tool (Weingarden, Heyd-Metzuyanim & Nachlieli). This tool is particularly interesting in the way it reduces the complexity of the lesson into a picture of the mathematical concepts discussed in order to describe qualities of the lessons. This picture organizes mathematical ideas related to what [START_REF] Sfard | Thinking as communicating[END_REF] calls saming and objectification. A second instrument presented is an innovative tool to observe, describe and evaluate metacognitive practices in mathematics (Nowinska & Praetorius). Six out of the seven dimensions developed had highly reliable ratings. A third instrument developed by Jentsch and Schlesinger starts from three established dimensions (classroom management, personal learning support, cognitive activation) and aims at adding a subject-specific dimension. This dimension includes nine characteristics -such as teachers' mathematical correctness, explanations and mathematical depthand produces results with good interrater agreement and satisfying reliability measures.

In addition, two papers study qualities of teaching using observation and interviews. Tuset investigates pre-service teachers trying ambitious teaching, allowing students to exercise authority while staying accountable to the discipline. The study finds that the pre-service teachers are able to create opportunities to engage in explorative discourses, but that their talk moves seem to be ritualized and therefore constrain students' participation. Kaldrimidou, Sakonidis and Tzekaki attempt to identify crucial elements shaping classroom mathematical meaning construction. To achieve this, they study three highly motivated and professionally active teachers' instructional practices and reflections. Findings reveal that the teachers' choices restrict the mathematical meaning because they desire to provide an easy, safe and pleasant learning environment.

These five papers illustrate two main issues for further research. The first issue regards the challenges of low and high inference observations. Low inference observations, like talk moves, explain little in themselves. On the other hand, high inference observations require extensive rater training that might result in simplification and even ritualization of the rating. What could we lose then? The second issue is that these articles illustrate the need for an instrument review. Which instruments are available for assessing qualities of teaching, what do they intend to measure, what theories do they build on, how reliable are they, and how much data and extent of rater training is needed to make them reliable?

(d) In-the-moment teaching actions and decision making and reflection

The studies in this group address mathematics instruction in a variety of ways: as teachers' management of actions and moments determining students' learning (Ferreira & da Ponte); as an activity shaping and being shaped by teachers' professional enactment in intervention or reform settings (Stouraitis; Sterner); as a practice being intentionally problematized to support teachers to develop (Potari & Psycharis; NicMhuirí). A different approach is to distinguish between papers that look at mathematics instruction as a learning-to-teach site through scrutinizing yourself or others acting it (Ferreira & da Ponte; Potari & Psycharis), as a professional activity developed through collaborative action (Sterner; Stouraitis), or through individual reflection via literature (NicMhuirí).

A range of theoretical or conceptual frameworksmostly of sociocultural originare at work in the studies reported, and reflection (on teaching practice) and collaboration are at heart of these frameworks. In particular, NicMhuirí employs a reflective practitioner's perspective in combination with a model allowing for levels of teachers' reflection to be identified. Reflection is also of concern in Potari and Psycharis' work operationalized through the construct of 'critical incidents' within a community of inquiry framework. The theoretical underpinnings of the community of practice approach are adopted by Steiner, with reflection being seen this time as a professional learning enterprise developing collaboratively. Drawing on a CHAT perspective, Stouraitis views reflection as an aspect of teachers' decision making which interacts with teaching activity. Finally, the study by Ferreira and Ponte employs features related to tasks assigned to the students and the communication established in the classroom to evaluate teaching actions.

Most papers in the group report on small, qualitative studies. Empirical data include observations of teaching, meetings and/or interviews audio-taped and transcribed. These data are predominately analyzed based on categories indicated by the literature (content analysis) or by the data themselves (grounded theory like analysis). One study uses no data, but analyzes two empirically tested constructs to exemplify the tool indicated (NicMhuirí). The results of the studies highlight various levels of mathematics teaching interacting with teachers' professional activity.

Overall, the studies in the group seek to understand how mathematics classroom teaching feeds teachers' professional practice, focusing on teachers inquiring into specific aspects of it. The relevant discussions carried out during the conference sessions raised concerns about the clarity of the terms and constructs used, the appropriateness and the functionality of the theoretical frameworks employed, and the boundaries between teaching action/practice and teacher practice.

(e) Instructional practices

The four papers in this thematic group approach instructional practices from different sociocultural perspectives. Two papers investigate teacher change during professional development programs. In particular, Venkat and Askew employ variation theory and example spaces to understand how teachers mediate primary mathematics, mainly how they generate and validate solutions as well as build mathematical connections. Şeker and Ader, on the other hand, focus on maintaining the cognitive demand of mathematical tasks, teachers' attention to students' mathematical ideas and intellectual authority in the classroom. Using the aforementioned frameworks makes it possible to illuminate different aspects of teacher practices that seem to improve based on research recommendations. Future research concerning both papers may entail a close look into the nature of professional development that influences instructional practices.

The paper by Baldry focuses on the development and viability of an analytical framework aiming to understand a 'typical' mathematical classroom in the United Kingdom. The analytical framework Orchestration of Mathematics portrays the quality of mathematics orchestration combining several theoretical constructs including cognitive demand of mathematical tasks, sociomathematical norms, hypothetical learning trajectories and professional noticing. The framework thus seems closely related to the scheme of analysis adopted by Şeker and Ader. A common theme of the two studies is teachers' difficulty in noticing and building on student thinking as well as maintaining cognitive demand of the tasks.

The importance of discourse in understanding instructional practices is evident across all papers. Drageset and Allern use drama as an innovative tool, allowing students to gain mathematical authority and engage in productive discourse patterns in making sense of mathematics. Instructional practices shaped and were shaped by student participation and responses. Future consideration for this work is to explore how teachers might implement such drama-based interaction patterns in their classrooms with the help of teacher educators.

A general trend identified in this group is using interviews together with observational data to understand teacher instructional practices, including teacher decisions and professional noticing abilities, closely connected with student participation and sociocultural norms. It would be productive to define boundaries of instructional practices and how to incorporate pre-and postobservation interviews with teachers in analysing relevant data.

Trends and developments

The studies in TWG19 address a wide range of features and factors that regulate the quality and development of classroom mathematics teaching as well as its relation to teachers' professional growth. A systematic attempt to understand, assess and trace contributions to teacher and classroom practices can be identified across the papers. Issues pursued along each of these three directions are discussed below. The section concludes with some critical considerations related to the studies hosted by TWG19.

In trying to understand teachers and classroom practices, some of the studies look at mathematics teaching in challenging circumstances (e.g., Kaldrimidou et al.; Kleve & Ånestad; Priolet & Mounier). Their results suggest that adapting teaching to effectively respond to such occasions is a difficult endeavor often leading to poor and even contradictory teaching practices. Teaching is also seen in relation to teachers' professional knowledge and practices/perspectives, with the relevant studies indicating a complex but dynamic and fertile relationship (e.g., Mosvold & Hoover; Grundén; Stouraitis). Finally, some studies consider the influence of contextual factors upon teaching practices (e.g., Baldry; Venkat & Askew). The findings reveal teachers' difficulties in administering the mediational role of these factors in order to develop effective teaching practices.

The qualities of teachers and classroom practices are assessed by focusing quantitatively or qualitatively on subject-specific rather than generic features. Studies adopting quantitative instruments highlight the value of such approaches when able to assess high inference valuations with the necessary inter-rater agreement (e.g., Jentsch & Schlesinger; Nowinska & Praetorius). The qualitative instruments, like interviews and observations (e.g., Weingarden, Heyd-Metzuyanim & Nachlieli; Tuset), found to face the same challenges of assessing high inference valuations. To do this with trustworthiness, the qualitative research typically focuses on depth of scrutiny rather than inter-rater agreement.

Certain ways of contributing to teachers and classroom practices are identified in the papers, mainly in some form of collaboration, reflection or intervention. In particular, opportunities to collaborate with other teachers to explore various aspects of teaching mathematics seem to offer possibilities for teachers to develop their professional practice (Sterner; Runesson Kempe et al.; Odindo). Teachers' reflection on classroom practice is a central constituent of this collaboration facilitated by inquiry tools. When exercised on varied levels and at specific aspects of classroom practice, affordances and constraints of this practice become evident (NicMhuirí; Psycharis & Potari). The latter appears to be also the outcome of intervention studies supporting teachers to transfer more learning responsibilities to students (Drageset & Allern; Şeker & Ader; Villalonga & Andrews).

The research activity on teachers and classroom practices included in TWG19 reflects some interesting steps forward, but it also reveals at least five sets of issues in need of critical consideration. Firstly, issues related to the methods adopted, for instance, the issue of generalizing across contexts, the role of using multiple methods, and the (dis-)advantages of different teacherresearcher relationships. Secondly, concerns about the theoretical frameworks employed, for example, selection criteria, levels of generality targeted, issues of compatibility, questions (not) addressed, 'own' frameworks. Thirdly, there is a need to carefully consider concepts and terms used. For instance, terms like 'practice', 'context' and even 'teaching' need clarification, whereas constructs like 'stability of knowledge' require further consideration. Fourthly, it is important to adopt clearly defined criteria for assessing the quality of teaching with reference to the learning of mathematics achieved as well as the wider educational goals targeted. Finally, it is necessary to consider quality criteria for research adopted, contribution to theory or practice, coherence, and sufficiency of evidence to warrant an empirical claim.

Concluding remarks

The work reported in TWG19 is part of the research on mathematics teachers classroom practices developed in recent years employing a predominately sociocultural perspective. Within this perspective, teaching is seen as a social practice in which teachers are practitioners [START_REF] Jaworski | Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching[END_REF]. Classroom practices are viewed as regular activities and norms continually developed by teachers' involvement in multiple simultaneous activities, "taking into consideration working contexts, meanings and intentions (…) the social structure of the context and its many layersclassroom, school, community, professional structure and educational and social system" (Ponte & Chapman, 2006, p. 483). These activities mutually structure and frame each other to constitute the practice of the classroom [START_REF] Skott | Understanding the role of the teacher in emerging classroom practices: searching for patterns of participation[END_REF].

There is a range of issues addressed by the studies reported that concern teachers' contribution to classroom practice in various contexts, mostly related to critical aspects of instruction-in-action and teachers either inquiring into their own teaching or working towards developing it. Collectively these contributions appear to suggest that it is valuable to shift the emphasis in this line of research from teachers to teaching. Several theoretical and analytical frameworks are used often in combination (rather than in coordination) providing multiple lenses through which certain constructs (rather than structures) are examined within particular contexts. Along the same line, different methodological approaches are pursued, mainly qualitative, seeking to capture the complexity and richness of the practices unfolding within mathematics classroom life shaping students' learning of mathematics and teachers' learning to teach mathematics alike. The findings of the studies offer some notable insights into this shaping, highlighting the importance of focusing on the micro-level of classroom practice, on the resources teachers draw on as they engage in it, and their (intentional or unintentional) professional activity.

The plurality of theoretical perspectives, constructs and analytical tools employed in the studies of teacher and classroom practice reported in TWG19 underline the dynamics of the research activity aiming at 'unpacking' teaching practice. It might be the time for the research community working in this area to consider what is already known, what is to be further examined and to develop on the basis of this evaluation a research agenda to fill the gaps. How different tools may be used considering different theoretical perspectives, decisions of whether to use an existing tool or to develop a new one and how to report the added value of using different tools require special attention in moving forward. To this end, the emphasis should be on teaching rather than on teachers, as suggested by the work presented and discussed in the context of the conference sessions.

Proceedings of CERME10


text:
Amanjot Toor

Joyce Mgombelo
email: jmgombelo@brocku.ca

Teachable moments at elementary school mathematics level

Keywords: Mathematics teaching, teachable moments, phenomenology

come

Introduction and purpose of the study

Teaching is a complex phenomenon that encompasses ever-changing situations, where teachers are challenged to positively deal with unanticipated circumstances such as teachable moments. This poster draws from an ongoing doctoral research study, a hermeneutic phenomenological study, which seeks to identify and explore teachable moments corresponding to intermediate mathematics teachers' lived experiences. The research is guided by the following questions: What does it mean to teach intermediate mathematics? What does it mean to recognize and utilize teachable moments at an intermediate mathematics level? At this stage in the research, the term teachable moment is generally defined as an opportunity that arises when connections have been made to advance learning by a learner and/or an educator. More specifically, the term moment is distinctly defined as an expected or unexpected occurrence that allows learners and/or educators to deepen their understanding. The poster focuses on the background and context of the study, by highlighting the rationale, framework and methodology of the study.

Background and context

Teachers often question the underlying purpose of teaching and its subjective meaning across three general categories: teaching subject content; teaching based on certain pedagogical tools; or teaching the child. Researchers in mathematics education have asked: What does it means to teach mathematics? How can one become an effective mathematics teacher? On one hand, researchers tend to focus on teachers' content knowledge and/or pedagogical knowledge, or an integration of the two. However, having in-depth context knowledge in mathematics does not necessarily equate to teaching it effectively [START_REF] Muri | Zero is not a number": Teachable moments and their role in effective teaching of numeracy[END_REF]. Also other researchers question how the two categories of knowledge-mathematics (subject matter) and teaching (pedagogy)-can be integrated. On the other hand, mathematics researchers such as [START_REF] Doxiadis | Embedding mathematics in the soul: Narrative as a force in mathematics education[END_REF] suggest that teaching mathematics effectively means humanizing mathematics for learners, and that "education is-should be, at its best-a process involving the complete human being" (p. 2). In this sense, humanizing mathematics means teaching it in a way that involves students as participants in mathematics, which is beyond mere content delivery or teaching certain skill sets. This poster draws from a study that aligns with this second view.

Theoretical framework

The epistemology of phenomenology centers on didactic meaning as opposed to arguing or developing abstract theory. In their discussion of the theoretical and conceptual framework for a phenomenological study, Savin-Baden and Howell Major (2012) stated that "the essence experience is so central and is to be uncovered before it is categorized, researchers do not tend to use a theoretical or conceptual framework… [because] doing so could impose presuppositions on the meaning of the experiences" (p. 221). The objective of this study is not to make broad generalizations about experiences of all mathematics teachers, but instead to examine individual teachers' personal experiences associated with a very specific phenomenon of teachable moments.

Methodology

In order to understand what teachable moments means to mathematics teachers, and how teachers use them in their day-to-day, moment-to-moment teaching, it is necessary to first gain insight into teachers' lived experiences by exploring their reasoning, beliefs, and intentions for teachable moments. As a consequence, to this underpinning, the study employs e a hermeneutic phenomenological research design founded on an epistemology of interpretivism. The study uses qualitative methods to collect data. These include semi-structured interviews, field notes and researcher journal. Participants comprise a purposeful sample of intermediate teachers with experience of teaching mathematics in Ontario, who have had lived experiences related to the phenomenon of teachable moments.

Conclusion

The "teachable moment" is viewed as a somewhat intangible pedagogical prize; a teacher might know what it feels like, yet may not identify its characteristics. Educators such as math teachers seek moments of openness and creativity with their students so they can personally experience "the psychic rewards" of teaching [START_REF] Lortie | Schoolteacher: A sociological study[END_REF]. Too often, such moments happen suddenly and slip away just as quickly, leading some teachers to conclude that such occurrences are a matter of chance. This study therefore seeks to shed some light on the characteristics of teachable moments in mathematics education to deepen teachers' understanding and use of such teachable moments in their respective practices.

Proceedings of CERME10


text:
Vanessa Crecci
email: vancrecci@gmail.com

Miguel Ribeiro

Dario Fiorentini

Mathematics teachers' knowledge and professional development in a Lesson Study context

Keywords: Teachers' professional development, interpretative knowledge, mathematics teachers' specialized knowledge, Lesson Study

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Some preliminary words

The Saturday Group [Grupo de Sábado] (GdS), which emerged in 1999, is a collaborative group that brings together teachers from schools and academics (prospective teachers, master and PhD students, teacher educators) interested in researching the teaching of mathematics in a collaborative environment. Although gatherings are held on the university landscape, there are no formal academic regulations controlling participation. Recently the participants have focused their interest in improving their practices by deepening their mathematical and pedagogical knowledge. In that context, in order to frame the work to be developed, a Lesson Study (LS) has been devised, involving teachers from primary, lower and upper secondary, prospective teachers and researchers. For the implementation of LS, a structure with three subgroups has been established (one for each school levelprimary, lower secondary and upper secondary). Complementary to the subgroups meetings (discussion and reflection upon tasks conceptualization and implementation), there will be meetings involving all the GdS elements. In those large group meetings (following the work already being done) the work goes around discussing and reflecting upon the situations emerged from teachers' practice the participants consider problematic. In particular, such discussions are aimed at contributing for developing teachers' knowledge, professional competency of noticing (in Mason's 2002 sense) and professional awareness. For the development of this project we consider what we call a hybrid methodology, where goals of two different natures are pursued. On the one hand, there is an interest in studying the development of a Lesson Study as teacher education strategy in the Brazilian context. On the other hand, the research goal concerns the teachers' professional development process, considering such development addressed through the lens of the Mathematics Teachers' Specialized Knowledge -MTSK [START_REF] Carrillo | Determining Specialized Knowledge for Mathematics Teaching[END_REF] conceptualization, intertwined with what we term of interpretative knowledge [START_REF] Jakobsen | Norwegian prospective teachers' MKT when interpreting pupils' productions on a fraction task[END_REF]. When discussing teachers' knowledge, besides the analysis of teachers' classroom practices and interviews, also a focus on the dynamics emerged in the subgroups and on the GdS will be analysed, as well as teachers' narratives grounded on their own experience (Connelly & Clandinin, 1988).

Lesson Study as a teacher education strategy

Considering the GdS context and the goals of the project in which this hybrid Lesson Study will be developed, six steps are considered. We have to recall that these steps have been emerged from the work with participantsby the nature of the GdS (a collaborative working group). The environment in which these steps are going to be developed are perceived as the context in which data for the research dimension will be gathered.

1)

Teachers identify a critical aspect from their own mathematical practice which they aim to discuss, reflect upon and improve. Such critical feature will be firstly discussed in each of the subgroups, taking into consideration the specificities of the contexts, and afterwards socialized and discussed in the large group (GdS) with all the participants;

2)

After the identification of the critical features, all the subgroup elements will discuss their own previous experiences, also grounded on some readings on documents (e.g., papers, books) where such problematic (or similar) is discussed. Such discussion aims at allowing deepening the participants MTSK, understanding and awareness on the problem at hand;

3)

Grounded on the discussions and reflections focusing on the problematic, teachers will conceptualize a task or sequence of tasks aimed at contributing for minimize the identified problem. Such tasks will be firstly prepared and discussed in each of the subgroups;

4)

The following considered stage concerns the implementation of each task prepared on the subgroups in the GdS, in order to deepen the levels of discussion both on the nature of the tasks, its mathematical goals and ways of implementation. We have to recall that in such discussion are involved teachers from different school levels which is perceived as an opportunity to intertwine different aspects of the MTSK and teachers' awareness;

5)

The next stage concerns the participants analysis of the implementation (all the participants will analyse the task, the implementation process and the teachers knowledge involved and required). Such analysis will occur firstly on the subgroups and afterwards some episodes will be discussed in the GdS;

6)

The last stage (of each cycle), concerns the writing and discussion of narratives focusing on the lived experiences.

The development of LS as a research project

Understanding LS as a proficient way to engage mathematics teachers in a professional development landscape, we focus on discussing the MTSK mobilized, involved and recognized by teachers when preparing, implementing and discussing mathematical tasks. In an intertwined manner the dimensions of the interpretative knowledge will be focus of attention at all the previously considered moments. Data collection will be taken from the video recordings of the subgroups meetings and the GdS meetings; classroom practices; interviews to teachers; interviews to students after the implementation of the tasks and teachers narratives. With the analysis we intend to contribute for a broader understanding of teachers' professional development and on the MTSK and interpretative knowledge developmental processes.

Proceedings of CERME10


text:
Niamh O'meara
email: niamh.omeara@ul.ie

Mark Prendergast
email: mark.prendergast@tcd.ie

Lorraine Harbison
email: lorraine.harbison@dcu.ie

Clare O'hara
email: clare.ohara@cso.ie

Is teacher knowledge affecting students' transition from primary to second-level mathematics education?

Keywords: Primary school mathematics education, second-level mathematics education, transition, teacher knowledge, continuity

Similar to counties such as the U.K. and the U.S.A, the Irish education system is divided into four key stages; pre-school education, primary level education, second level education and tertiary education. Transition between each of these phases has its own set of challenges but many believe the most challenging of all is the transition from primary to second level education. This quantitative, national study investigates the transition from primary to second level mathematics education from the perspective of teachers. It investigates sixth class teachers' knowledge of the mathematics curriculum and teaching strategies employed at second level and first year mathematics teachers' knowledge of the mathematics curriculum and teaching strategies favored in primary school. The results of the study highlight low levels of knowledge in these domains amongst both sixth class and first year mathematics teachers. The ramifications of this gap in teacher knowledge are also discussed in detail.

Background to the study

As is the case in Australia, the United States and the United Kingdom, Ireland's education system is divided into four phases; pre-school education, primary level education, second level education and tertiary education. The transition from primary mathematics education to second level mathematics education is one of the greatest challenges that young people experience during their school years. According to [START_REF] Bicknell | Explorations of Year 8 to Year 9 Transition in Mathematics[END_REF] the challenge presented by this transition is multifaceted and involves challenges from social, academic and systematic perspectives. As such, this is a pertinent research area and one which has been looked at in depth in recent years.

The overarching finding to emerge from the research carried out to date was that the transition from primary to second level mathematics education resulted in a decline in students' attitudes, academic performance and confidence [START_REF] Attard | Students Experiences of Mathematics During the Transition from Primary to Secondary School[END_REF]Economic and Social Research Institute [ESRI], 2007). Furthermore, [START_REF] Bicknell | Explorations of Year 8 to Year 9 Transition in Mathematics[END_REF] found that the gap between high achieving and low achieving students widened significantly during this transition period. Due to the serious nature of these consequences, researchers, such as [START_REF] Green | Moving from the World of the Known to the Unknown: The Transition from Primary to Secondary School[END_REF] and [START_REF] Attard | Students Experiences of Mathematics During the Transition from Primary to Secondary School[END_REF], have sought to investigate what constitutes effective transition and what are the main factors that contribute to an educationally poor transition for students.

In her study on students' experiences of the transition from primary to second level mathematics education in Australia, [START_REF] Attard | Students Experiences of Mathematics During the Transition from Primary to Secondary School[END_REF] listed curriculum, pedagogy, assessment strategies, social interactions and students' relationships with others, as key factors that dictate the success of transition. Likewise, [START_REF] Barber | Bridges to Assist a Difficult Crossing[END_REF] describes the transition as a set of five hurdles all of which must be overcome at once. The hurdles to be overcome to ensure a smooth transition, as listed in this study, are bureaucratic, social and emotional, curriculum, pedagogy, and management of learning. In addition to this, [START_REF] Evangelou | What Makes a Successful Transition from Primary to Secondary School: Findings from Effective Pre -School, Primary and Secondary Education 3 -14 (EPPSE) Project[END_REF] 2) stated that a successful transition for children entailed: …developing new friendships and improving their self-esteem and confidence; having settled so well in school life that they caused no concerns to their parents; showing an increasing interest in school and school work; getting used to their new routines and school organisation with great ease [and] experiencing curriculum continuity.

All research conducted into what constitutes effective transition make some reference to curriculum and pedagogical continuity. Likewise, research conducted in the area of problematic transitions all point to a lack of continuity in this regard. For example, [START_REF] Elkins | Literacy and the Transition to Secondary School[END_REF], [START_REF] Green | Moving from the World of the Known to the Unknown: The Transition from Primary to Secondary School[END_REF] and Tilleczek (2007) all found that the attainment and motivational losses that students often experience when moving from primary to second level mathematics education can, in no small way, be attributed to a lack of continuity in terms of both curriculum and pedagogical approaches.

However, in order to ensure continuity between both curriculum and pedagogical approaches it is critical that teachers who are teaching students that are about to enter or have just completed the transition process have an in-depth Mathematical Knowledge for Teaching (MKT). Such knowledge encompasses knowledge of the mathematical content previously studied and that which they will study in subsequent years [START_REF] Ball | Content Knowledge for Teaching: What makes it Special[END_REF]. [START_REF] Ernest | The Knowledge, Beliefs and Attitudes of the Mathematics Teacher: A Model[END_REF] reiterates that a teacher's MKT is not limited to knowledge of curriculum, but also knowledge of students, in order to enable them to teach mathematics effectively. The authors further ascertain that teachers, especially those involved in the transition process, must have a comprehensive MKT comprising of the curricula, students and teaching methodologies utilised before and after the transition process. Teachers who do not possess such knowledge have yet to develop the full range of knowledge domains proposed by both [START_REF] Ball | Content Knowledge for Teaching: What makes it Special[END_REF] and [START_REF] Ernest | The Knowledge, Beliefs and Attitudes of the Mathematics Teacher: A Model[END_REF] and as such, their knowledge could be considered inadequate for teaching. It is this belief, in conjunction with existing research, which led the authors to investigate the following research questions.

1. How familiar are sixth class primary school teachers with the second level mathematics syllabus and the teaching methodologies being promoted at second level and vice versa? 2. What are the consequences of these levels of MKT in terms of (a) the fluidity of the transition between primary and secondary mathematics education and (b) the teaching approach adopted by second level teachers when teaching mathematics to first year students?

Methodology

The research design for this quantitative study involved the distribution of questionnaires to a representative sample of two groups of stakeholders involved in the transition process; namely sixth class teachers in primary schools and first year mathematics teachers in second level schools 1 . For the purpose of the study two advisory groups, one involving primary teachers and another involving second level mathematics teachers, were established. Their role was to help with the development and piloting of the questionnaires and to help the authors in relation to sampling issues. To allow for comparison of responses from primary and second level teachers the questionnaires were of a similar nature and both were based on the framework for transition developed by the authors from the work of [START_REF] Anderson | School Transitions: Beginning of the End or a New Beginning[END_REF] and the models of knowledge proposed for primary teachers by [START_REF] Ball | Content Knowledge for Teaching: What makes it Special[END_REF] and for second level teachers by [START_REF] Ernest | The Knowledge, Beliefs and Attitudes of the Mathematics Teacher: A Model[END_REF]. This theoretical framework is outlined in Figure 1.

Figure 1: Theoretical framework

This study was unique in that it looked solely at the issue from the perspective of teachers. As such, only some dimensions of this model were relevant to this study namely the discontinuity pillar, the support pillar and the teacher knowledge pillar. This particular paper has an even narrower focus and looks solely at the pillar of teacher knowledge.

The sampling frame for this study was a list of all 3,300 primary schools and 723 second level schools in Ireland (DES website February 2016). The targeted sample was 700 sixth class teachers and 400 first year mathematics teachers. By consulting the primary school advisory groups, the authors established that on average, there is one sixth class teacher in each primary school in Ireland. As a result, a simple random sample of 700 primary schools was selected. Overall, the sample included 21.2% of all primary schools. Having consulted with the second level advisory group it was established that on average, there are two mathematics teachers teaching first year mathematics in each school in Ireland. Hence using this estimate, a stratified random sample of 200 second level schools around Ireland was selected. This sampling technique ensured that an accurate representation of each type of school (secondary, vocational, community and comprehensive) in Ireland was included in the sample. Overall, the sample included 27.7% of all second level schools in Ireland.

The questionnaires were distributed to the 700 primary schools and 200 second level schools in April 2015. The primary school questionnaires were sent to the principal of each school and they were asked to distribute these questionnaires to the sixth class teacher. The pack sent to each of the 700 principals included an information sheet for the principal, a teacher information sheet along with the questionnaire and a stamped address enveloped for the questionnaire to be returned in. The second level questionnaires were sent to the Head of Mathematics in each of the 200 second level schools and they were asked to distribute the questionnaires to the first year mathematics teachers in their school. The pack sent to each department head included an information sheet for their perusal, an information sheet for first year mathematics teachers along with two questionnaires and two stamped address envelopes in which the questionnaires could be returned. At both primary and second level, each stamped addressed envelope included was given a number corresponding to the school selected so the researchers could identify the schools that had not returned the completed questionnaires. Two weeks after sending the questionnaires, follow-up telephone calls to each of these schools were undertaken with the aim to increase the response rate.

Upon receipt of the completed questionnaires the quantitative data was inputted and saved into the computer programme SPSS. Descriptive analysis examined primary teachers' knowledge of the mathematics curriculum and teaching strategies employed in secondary school and second level mathematics teachers' knowledge of the mathematics curriculum and teaching strategies employed at primary level. Descriptive analysis also allowed the authors to determine how these levels of knowledge affected the approach adopted by second level teachers when teaching first year students and also to determine if the transition from primary to secondary was educationally successful from the teachers' perspective. The authors will now present the results of this analysis in an attempt to address the aforementioned research questions.

Findings

Based on the population size it was determined that, to allow for a 5% margin of error, the study would require 263 responses from sixth class teachers and 133 responses from first year mathematics teachers. The actual response rate was 296 primary school teachers (approx. 42%) and 171 second level teachers (approx. 43%). The primary teachers who responded were distributed across 271 schools (38.7% of schools surveyed) while the second level teachers who responded were distributed across 101 schools (50.5% of schools surveyed).

The first research question sought to ascertain sixth class teachers' knowledge of the first year mathematics curricula and the teaching strategies employed by first year mathematics teachers as well as first year teachers' knowledge of the sixth class curriculum and the teaching strategies adopted by sixth class teachers. The findings related to this research question are presented in Figure 2 and Figure 3. These findings demonstrate that teachers, at both levels, have a deficient understanding of the syllabus and teaching strategies that their students were/will be exposed to in their previous/next year of schooling. Over half of sixth class teachers (56%) reported that the first year mathematics syllabus was either highly unfamiliar or slightly unfamiliar to them. The corresponding figure for second level teachers was 49%. The responses in relation to knowledge of teaching strategies were even more pronounced. Almost three-quarters of sixth class teachers (73%) stated that they were highly unfamiliar or slightly unfamiliar with the teaching approaches used in mathematics classrooms at second level. Likewise, 77% of first year mathematics teachers stated that they were highly unfamiliar or slightly unfamiliar with the pedagogical approaches employed by sixth class teachers. Furthermore, only 13% of sixth class teachers claimed to be in any way familiar with the teaching approaches used by first year mathematics teachers while the corresponding figure for the first year teachers who responded was 15%.

The second research question was two-folded and sought to analyse the knock on effect of the gaps in teacher knowledge discussed previously. In order to address this research question both groups of teachers were first asked to rate their agreement with the statement "There is a fluid transition Thematic Working Group 20 between primary and secondary mathematics". The second level teachers were then further probed on these knock on effects when they were asked to describe the approach they adopt when teaching first year mathematics students upon their entry to second level. For this, they were asked to pick from a pre-determined list of four options, which included "Other". The responses received are provided in Figure 4 and Figure 5. Figure 4 shows that a large proportion of both groups of teachers believe that the transition from primary school mathematics to second level mathematics is not smooth. For example, 44.6% of sixth class teachers believe this to be the case compared with 44.4% of first year mathematics teachers. Only one teacher in both groups strongly agreed that there was a fluid transition between primary school mathematics and second level mathematics with a further 34 in each group agreeing with the sentiment. The lack of fluidity or continuity is elaborated upon further when secondary teachers were

Primary Teacher Responses

Strongly Agree Somewhat Agree Neither Agree nor Disagree Somewhat Disagree Strongly Disagree

Secondary Teacher Responses

Strongly Agree Somewhat Agree Neither Agree nor Disagree Somewhat Disagree Strongly Disagree 0 50 100 150

Number of Teachers

I see it as an opportunity for a fresh start and initially assume as little as possible about student knowledge or ability I use the Education Passport (or similar information from primary schools) to get prior information on students knowledge and ability) I use the secondary school entrance test results or my own evaluation to get prior information on student knowledge and ability Other asked to describe the approach that they adopt when teaching first year mathematics students. Of the 168 teachers who responded to this question 67.9% stated that they "See it as an opportunity for a fresh start and initially assume as little as possible about student knowledge or ability". Despite mechanisms, such as the Education Passport 2 being introduced in recent years, this study shows that teachers, most probably due to their own lack of knowledge of the primary school curriculum, continue to adopt a "fresh start" approach. This will undoubtedly lead to a disjointed and fractured transition from primary to secondary mathematics education.

Discussion and conclusion

"If a teacher is largely ignorant or uninformed he can do much harm" (Conant, 1963: 93) This research study has demonstrated that sixth class teachers have gaps in their knowledge in relation to the syllabus and pedagogical approaches being adopted at second level while the same can also be said about second level mathematics teachers in relation to the primary school mathematics syllabus and favoured pedagogical approaches. Teachers in both of these sectors do not have the full repertoire of knowledge prescribed by [START_REF] Ball | Content Knowledge for Teaching: What makes it Special[END_REF] and [START_REF] Ernest | The Knowledge, Beliefs and Attitudes of the Mathematics Teacher: A Model[END_REF]. As [START_REF] Conant | The Education of American Teachers[END_REF] points out, such gaps can be detrimental to students' progress and prove a hindrance in their academic progression. This gap in teacher knowledge can prevent teachers from adequately preparing students for the transition process or providing them with a sense of continuity when they make the transition.

For example, in a study carried out by [START_REF] Bicknell | Explorations of Year 8 to Year 9 Transition in Mathematics[END_REF] teachers expressed concerns that gaps in their own knowledge meant that they were not equipped to prepare students for the mathematics they would face at second level. Likewise, students in a study carried out by [START_REF] Green | Moving from the World of the Known to the Unknown: The Transition from Primary to Secondary School[END_REF] reported that the lack of continuity between primary and second level mathematics education, which stemmed from the lack of understanding of the mathematics syllabi and teaching strategies being employed in the years either side of the transition on the part of teachers, meant that they did not face new challenges on entry to second level and as a result their motivation and attitudes declined. Hence, internationally it has been shown that these gaps in teacher knowledge can play a role in the declining attainment levels and attitudes of students during the transition. As a result, it is critical that steps are taken to improve teachers' knowledge in this regard in order to improve students' experience of transition.

In addition to the ramifications already discussed in international literature, this study found that the knowledge levels reported by teachers had other consequences, namely the lack of fluidity in transition and the approach adopted by teachers when students enter first year. The lack of fluidity in transition reported by teachers in this study is unsurprising, as without an in-depth understanding of the previous or subsequent syllabi and teaching approaches it is difficult for teachers to ensure curriculum or pedagogical continuity. Such continuity is critical in order to allow for a educationally successful or fluid transition from primary to second level mathematics education [START_REF] Evangelou | What Makes a Successful Transition from Primary to Secondary School: Findings from Effective Pre -School, Primary and Secondary Education 3 -14 (EPPSE) Project[END_REF]. Teachers who do not possess the knowledge domains outlined in the work of [START_REF] Ball | Content Knowledge for Teaching: What makes it Special[END_REF] and [START_REF] Ernest | The Knowledge, Beliefs and Attitudes of the Mathematics Teacher: A Model[END_REF] struggle to provide curriculum and pedagogical continuity and are 2 The Education Passport was an initiative introduced in 2014. It requires primary schools to pass documentation onto second level schools which details a rounded picture of the child's progress and achievement at primary school as well as signalling to second level schools what support a child may need. The overall purpose of the Education Passport is to help the child progress and experience continuity as they move from primary education to second level education.

Thematic Working Group 20

forced to adopt a "fresh approach" with their first year students. This is the only option available to teachers who are not informed about the syllabus and/or pedagogical practices that students were exposed to in their previous year of schooling. It is not surprising that this deficiency in the area of MKT among teachers has resulted in pedagogical approaches that are not well received by students, and thus lead to boredom, lack of motivation and a consequential decline in students' attainment levels [START_REF] Bicknell | Explorations of Year 8 to Year 9 Transition in Mathematics[END_REF]. Due to such concerns, the authors believe it is of paramount importance that teachers are given the opportunity to develop knowledge of the sixth class and first year mathematics syllabi; of students in both these years; and of the teaching strategies in place across both levels. Only when such opportunities are available will teachers be in a position to develop the range of knowledge domains required for teaching and the hurdle of discontinuity will be overcome.

Figure 2 :Figure 3 :

 23 Figure 2: Primary teachers' responses when asked (a) How familiar are you with the first year mathematics syllabus? and (b) How familiar are you with the recommended teaching methods for first year mathematics?

Figure 4 :Figure 5 :

 45 Figure 4: Teachers' responses when asked their level of agreement with the statement "There is a fluid transition between the primary and secondary school mathematics curricula."

Proceedings of CERME10

In Ireland 6 th class is the final year of primary education which 1 st year is the name given to the first year of second level education.Thematic Working Group

Proceedings of CERME10