domain: math shs

text:
Claire Guille-Biel Winder

Frédérick Tempier
email: frederick.tempier@u-cergy.fr

A theoretical framework for analyzing training situations in mathematics teacher education

Keywords: Teacher education, professional development, primary education, knowledge for teaching, analysis framework

The thoughts on the primary schoolteacher training have led to the production of many resources for primary schoolteachers. Faces of the abundance of such documents, teacher educators need some tools to identify the knowledge potentially at stake in training situations and to allow them to implement such situations according to their own objectives and context. We present a five-level analysis framework that characterizes the training tasks, taking account of the activities induced by the task, according to the expected posture of the prospective teacher, to the type of the knowledge at stake and to possible degrees of decontextualization. We illustrate this analysis framework by presenting an example of a training scenario based on the principle of role-play.

Introduction

The research about primary schoolteacher education in mathematics and professional development has led to the production of many resources for educators. In France, the COPIRELEM1 group produced many of them. These resources provide "training situations" based on various training strategies [START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF], and are generally accompanied with information about their implementation (phases, steps, instructions, elements of institutionalisation) with regard to the stakes of the training. But their quality does not guarantee an accurate appropriation by teacher educators. Our questioning is: how is it possible to help teacher educators to exploit any "training situations" in a relevant way, according to their objectives?

The research literature usually provides studies about knowledge for teaching, teacher conceptions, and their evolution [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF][START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF][START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF]. Other studies present one training situation, and generally focus on its effect on the prospective teachers. For example [START_REF] Horoks | Training primary school teachers through research in mathematics' didactics[END_REF] "analyse the contents and methods of an initiation course in research in mathematics education, and […] how it can influence the beginner teachers' practises" (p.2811). To our knowledge, no study focuses on the characteristics of training situations nor provides specific framework in order to analyse any training situation. This led us to develop an analysis framework for training situations. The paper presents this COPIRELEM's work in progress.

Presentation of the analysis framework

We define a training situation as a situation that involves prospective teachers (students, pre-service or in-service teachers) and educators within an institution of teacher education. It is composed of a set of tasks that could be conducted by a teacher educator. We take into account all the tasks proposed by the teacher educator. From a training situation, the educator may elaborate a training scenario that is to say a set of chronologically organized tasks chosen among all the tasks that constitute the training situation. We voluntarily distinguish situations from "scenarios" because we intend to underline the dynamic aspect of the scenario.

In response to each task of a training situation, prospective teachers develop an activity that corresponds to "what [they engage] in during the completion of the task" (Rogalski, 2013, p.4). We distinguish five different types of activity: "mathematical activity" (doing maths during the completion of a mathematical task), "mathematical analysis activity" (analysing the maths at stake in a mathematical task), "didactical and/or pedagogical activity" (highlighting didactical and/or pedagogical choices related to the mathematical task), "didactical and/or pedagogical analysis activity" (analysing didactical and/or pedagogical choices related to the mathematical task), "problematisation activity" (identifying and investigating professional issues by mobilizing mathematical, didactical and pedagogical concepts). For each type of activity we take into account three dimensions (Fig. 1): the type of knowledge at stake; the degree of decontextualization of this knowledge; the posture of the prospective teachers expected by the teacher educator. These dimensions are specified in the next sections.

Three types of knowledge

We rely on the three types of knowledge for teaching mathematics identified by [START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF]: mathematical knowledge, pedagogical knowledge and didactical knowledge. "Mathematical knowledge corresponds to mathematics that a teacher needs to know in order to prepare, regulate and evaluate his lesson and his students" (Houdement, 2013, p.12). It "includes and specifies the content knowledge" identified by [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF]. It roughly can be related to Subject Matter Knowledge [START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF], and the specific didactical nature of mathematical knowledge can be identified to the Specialized Content Knowledge (SCK). According to [START_REF] Houdement | Au milieu du gué: entre formation des enseignants et recherche en didactique des mathématiques[END_REF], didactical knowledge is linked to the mathematical content and fed by research in the field of mathematics didactics. It corresponds to analysis of teaching and learning phenomenon and to propositions of engineering. Therefore it can be associated with at least two categories [START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF]: Knowledge of Content and Students (KCS) and Knowledge of Content and Teaching (KCT).

Pedagogical knowledge2 is characterised as "knowledge of experience" [START_REF] Portugais | Didactique des mathématiques et formation des enseignants[END_REF]. It is related to teaching and learning conceptions and to the organisation and management of the class. It is less dependent of the mathematical content than other types of knowledge. It is important to take this knowledge into account because schoolteachers deal with various school subjects. [START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques 1970-1990[END_REF] and [START_REF] Douady | The interplay between different settings. Tool-Object dialectic in the extension of mathematical ability[END_REF] identify three degrees of decontextualization of a mathematical knowledge: implicitly mobilized, explicitly mobilized in context or decontextualized (to become available in other contexts). We extend this notion to didactical and pedagogical knowledge. A mathematical knowledge is (implicitly) mobilized in context (in act) if it is used as tool [START_REF] Douady | The interplay between different settings. Tool-Object dialectic in the extension of mathematical ability[END_REF] in a mathematical task. This task can be carried out: what is asked is effectively achieved (manipulation, elaboration and writing a solution for example). But the task can only be evocated: it is mentally achieved. A mathematical knowledge is explicit in context if its use (as tool) is identified and formulated. At least, a mathematical knowledge is decontextualized if it is identified as an object of learning: a status of object is given (by the educator) to the concept used previously as tool, usually during an institutionalisation phase3 [START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques 1970-1990[END_REF]. The didactical/pedagogical knowledge is mobilized in context when the didactical/pedagogical choices are made for the considered mathematical task. It is explicit in context during the analysis about the consequences of these choices. At least, it is decontextualized when the underlying didactical/pedagogical concepts are highlighted.

Three degrees of decontextualization

Four postures of the prospective teachers

In conjunction with the teacher trainer's relationship to the prospective teachers identified by [START_REF] Sayac | Study of the Practices of educators in Mathematics, on Pre-service Teachers Training[END_REF], we define four specific postures of prospective teachers, which are expected by the educator during a training situation4 . Prospective teachers are in a posture of student relatively to the mathematical knowledge when they have to perform mathematical activity or when they are concerned with the mathematical knowledge of this activity. They are in a student/teacher posture when they investigate mathematical tasks for students or students' works, or when they analyse the conditions of implementation of a task in the classroom. They are in a teacher posture when entering in a broader questioning on classroom practices and issues of mathematical learning. Finally, they are in a practitioner/researcher posture when they problematize a professional issue related to mathematical learning or teaching.

Five study levels

In order to analyse a training situation, we define five study levels. To each level corresponds a type of activity, that induces (implicitly or explicitly) a posture of the prospective teacher (expected by the educator), and that involves different types of knowledge in a certain degree of decontextualization (see Fig. 2). Level 1. A task may induce a mathematical analysis activity related to the activity of level 0 when it highlights decontextualized mathematical knowledge and the prospective teachers are in a posture of (learning mathematics) student. In this task, the didactical and/or pedagogical knowledge can be implicitly mobilized in context and then initiates the change toward a student/teacher posture of prospective teachers.

Level 2. A task may induce a didactical and/or pedagogical activity related to the activity of level 0 when it corresponds to the analysis of implementation conditions -actual or anticipated only of the mathematical task. The didactical and pedagogical knowledge is explicit in context. The prospective teachers are in a student/teacher posture.

Level 3. A task may induce a didactical and/or pedagogical analysis activity related to the activity of level 2 when it is for example a questioning on classroom practice (specific learning tasks, professional actions...) or on issues of mathematical learning for one or several contents (curriculum, progressions...), or even a highlighting of didactical analysis concepts (didactic situation phases, types of tasks...). This analysis leads to the decontextualization of didactical and/or pedagogical knowledge. The prospective teachers are in a posture of teacher.

Level 4. A task may induce a problematisation activity when it corresponds to the problematisation of professional issues related to classroom practices, learning issues and/or didactical analysis tools.

The prospective teachers are in a posture of practitioner/researcher, especially when it comes to developing an analysis methodology of this issue and to infer results.

Each study level is based on the study of the activity of previous levels and involves some mathematical, didactical and/or pedagogical knowledge. The change from study level n to study level n + 1 is linked either to a change of the prospective teachers' posture or to a change of degree of decontextualization for at least one type of knowledge (from implicitly mobilized in context to explicit in context, from explicit in context to decontextualized). But the different activities induced by a training situation don't usually appear in a chronological order (from level 0 to level 4). For examples, see the analysis of various training situations developed in French context by the COPIRELEM group (Guille-Biel [START_REF] Winder | Proposition d'un cadre d'analyse de situations de formation des professeurs des écoles[END_REF][START_REF] Bueno-Ravel | Usage d'un cadre d'analyse pour s'approprier, concevoir et enrichir des situations de formation[END_REF]. We think that the analysis could be extended to situations based on different training strategies. That is why we present here the analysis of a training scenario based on the principle of role-play developed in an international context [START_REF] Lajoie | Role-play by pre-service elementary teachers as a means to develop professional competencies in teaching mathematics[END_REF][START_REF] Lajoie | Richness and complexity of teaching division: prospective elementary teachers' roleplaying on a division with remainder[END_REF]Lajoie, accepted).

An example of use of the analysis framework

Definition of role-play

As [START_REF] Lajoie | Role-play by pre-service elementary teachers as a means to develop professional competencies in teaching mathematics[END_REF] state "role-play involves staging a problematic situation with characters taking roles". It is used over many years in mathematics education course in UQAM (University of Québec in Montréal) and is organized as follows:

First, the 'theme' on which students will need to role-play is introduced (introduction time). Second, students then have about 30 minutes to prepare in small groups (preparation time). Third comes the play itself (play time), where students chosen by the educator come in front of the classroom and improvise a teacher-student(s) interaction (sometimes, like in the case reported here, involving the whole class). Finally, we have a whole classroom discussion on the play (discussion time). (Lajoie, accepted) We designed a role-play on the teaching of proportions based on a problem from a textbook. We use the analysis framework to illustrate an example of analysis aimed at highlight the potential of this situation.

An example of role-play

The role-play presented below is intended for pre-service schoolteacher education. We describe the different phases. Introduction time. The educator distributes to prospective teachers an excerpt from a fifth grade (10-11 year old pupils) handbook presenting a problem of proportions (Fig. 3), and various productions of pupils. The teaching issue announced by the educator is the following: to manage a class discussion about the pupils' strategies and about their ideas and solutions, in order to share them in the class community and to determine their validity and efficiency. Preparation time. The prospective teachers have to prepare the discussion class about the pupils' strategies.

Play time. At the end of the preparation time, the educator chooses prospective teachers to play the game: some of them play pupils, one of them plays the teacher, while the others are watching the discussion class and taking notes. Discussion time. The debate intends to highlight and to analyse the choices of the « teacher » during the play game: what worked well during the implementation of the discussion class? What was difficult? What seemed to be important? What alternative implementations could be realized? Institutionalisation time5 . The educator institutionalizes the knowledge at stake: he generalizes some elements about how to manage a discussion class or about proportion problems solving.

Analysis of this role-play

The initiating task is a professional situation and corresponds to a level 2 activity: the prospective teachers are initially in a student/teacher posture. But they will need « to go down » to a student posture and « to go up » to a teacher posture during the phases of the scenario. The preparation time of the discussion class leads the activity of the prospective teachers "to go back and forth" to the study levels 0, 1 and 2. The problem solving corresponds to a level 0 activity and the mathematical analysis of the problem solving to a level 1 activity. Moreover there are various strategies to solve this proportion problem. Preparing the discussion class of the pupils' strategies (level 2 activity) hence needs to analyse and rank them (from the least to the most elaborate). This analysis corresponds to a level 2 activity. The prospective teachers don't have the same activity during the playtime. The study level is different according to the role to play: mostly levels 0 and 1 for the students' roles and level 2 for the teacher's role. The discussion time corresponds to a level 2 activity when the prospective teachers analyse how the discussion class has been managed. But it can also correspond to lower levels activities, when they discuss about pupils' strategies, difficulties, mistakes and their exploitation during the discussion class. Various institutionalisations can be considered, according to the knowledge that was developed at different study levels. The institutionalized elements will be more or less developed according to the teacher educator's objectives and progression, the prospective teachers' knowledge, etc. Here are some propositions organized in ascending order of study levels. The teacher educator can institutionalize some mathematical knowledge at stake (level 1) and related to the proportionality field: various methods to solve a proportion problem, mathematical justifications and mathematical theories they are relied on. He can situate the proportion problems in the more general category of multiplicative problems, or he can explicit some didactical variables usually at stake in proportion problems (level 3). He also can identify some difficulties or mistakes revealed by the pupils' productions as « usual » and highlight mistaken conceptions: identification of quantities, choice of an adapted strategy, persistence of an « additive model », etc. At least, in regard of the announced objective of the roleplay, the teacher educator also can institutionalize some didactical knowledge, relatively to the organization of a discussion class (level 3): formulation and validation in mathematics; teacher's tasks before, during and after the discussion class…

Conclusion

The example of role-play situation shows how the analysis framework can be a tool for an a priori analysis. Moreover this example shows that the organization of the study levels is not a chronological but a hierarchical one: the initiating task can induce an activity of level 0, 1, 2, 3 or 4. But the transition to lower levels activity is often necessary. The conceptual maps of the knowledge for teaching developed by [START_REF] Houdement | Autour des stratégies utilisées pour former les maîtres du premier degré en mathématiques[END_REF] or by [START_REF] Ball | Content Knowledge for teaching, what makes it special?[END_REF] have a descriptive, predictive and prescriptive dimension (Ball & al., 2008, p.405). But beyond their interest, (Houdement, 2013, p. 21) stressed the importance of the knowledge reconfiguration in connection with the mathematical content. The analysis framework reports how, during a training situation, the types of knowledge for mathematics' teaching are dynamically hinged to one another in connection with the mathematical content. The analysis framework allows teacher educators to identify the potentialities of a full range of training situations. We intend to extend its use to study other types of training situations (for example e-learning situations). By clarifying the stakes of the various phases of the implementation, the analysis framework reveals various possible strategies for the teacher educator. Thereafter it could be a useful tool for elaborating different training scenarios. Hence, the teacher educator should be able to implement situations in a specific context according to his objectives and constraints (time and period of training, place in a progression which take into account the mathematical, didactical and pedagogical knowledge ever studied…). Besides it is possible to consider a sequence of successive scenarios. The analysis framework could also highlight various possible "training paths", which should reveal the educator's training strategy at a more global scale. A perspective is now to study how teacher educators appropriate this framework and how it supports their teaching practises.

Figure 1 :

 1 Figure 1: Three dimensions for characterizing a type of activity

Figure 2 :

 2 Figure 2: Characteristics of the five study levels Level 0. A task may induce a mathematical activity. This activity can be performed or evocated (mentally performed). The mathematical knowledge is mobilized (implicitly or explicitly) in context. The prospective teachers are in a posture of student (relatively to the mathematical knowledge).

Figure

  Figure 3: A proportion problem

The COPIRELEM is a commission dedicated to the education to the primary school. It is stemming from the network of IREM (French institute of research on mathematical education).Thematic Working Group 18Proceedings of CERME10

Proceedings of CERME10

According to[START_REF] Houdement | Au milieu du gué: entre formation des enseignants et recherche en didactique des mathématiques[END_REF]), Ball's, Phelps' and Thames' typology doesn't seem to take into account this type of knowledge.

In institutionalisation phase[START_REF] Brousseau | Theory of Didactical Situations in Mathematics: Didactique des Mathématiques 1970-1990[END_REF], the teacher gives a cultural (mathematical) status to some knowledge emerging from students' actions during the situation.

We notice that prospective teachers are not always aware of these postures.Thematic Working Group 18Proceedings of CERME10

We add this new time to the four ones proposed by[START_REF] Lajoie | Role-play by pre-service elementary teachers as a means to develop professional competencies in teaching mathematics[END_REF].


text:
C Fernandez
email: ceneida.fernandez@ua.es

S Llinares
email: sllinares@ua.es

M L Callejo
email: luz.callejo@ua.es

M Moreno
email: mmoreno@ua.es

Gloria Sánchez-Matamoros
email: gsanchezmatamoros@us.es

C Fernández

Characteristics of a learning environment to support prospective secondary mathematics teachers' noticing of students' thinking related to the limit concept

Keywords: Noticing, prospective teachers' learning, learning environment

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

Introduction and theoretical background

Research has shown that noticing is an important component of teaching expertise [START_REF] Mason | Researching your own practice: The discipline of noticing[END_REF]. Teachers need to attend to students' mathematical reasoning and make sense of it in order to teach in ways that build on students' thinking [START_REF] Choy | Snapshots of mathematics teacher noticing during task design[END_REF]Sherin, Jacobs, & Philipp, 2011). Noticing has been conceptualised from different perspectives. One of them consists of two main processes: attending to particular teaching events and making sense of these events (Sherin et al., 2011). [START_REF] Jacobs | Professional noticing of children's mathematical thinking[END_REF] particularise the notion of noticing to children's mathematical thinking, conceptualising this notion as a set of three interrelated skills: attending to children's strategies, interpreting children's mathematical thinking, and deciding how to respond on the basis of children's mathematical thinking.

Previous research has focused on pre-service teachers' ability to interpret students' mathematical thinking [START_REF] Bartell | Prospective teacher learning: recognizing evidence of conceptual understanding[END_REF][START_REF] Callejo | Prospective primary teachers' noticing of students' understanding of pattern generalization[END_REF][START_REF] Fernández | Learning to notice students' mathematical thinking through on-line discussions[END_REF][START_REF] Llinares | Changes in how prospective teachers anticipate secondary students' answers[END_REF][START_REF] Magiera | An exploratory study of preservice middle school teachers' knowledge of algebraic thinking[END_REF][START_REF] Sánchez-Matamoros | Developing pre-service teachers' noticing of students' understanding of derivative concept[END_REF] showing that the identification of the mathematical elements involved in the problem (mathematical content knowledge) plays a significant role in interpreting students' mathematical reasoning. Furthermore, previous research has shown that some contexts can help pre-service or prospective teachers develop the noticing skill: watching video clips [START_REF] Coles | Using video for professional development: the role of the discussion facilitator[END_REF][START_REF] Van Es | Learning to notice: scaffolding new teachers' interpretations of classroom interactions[END_REF], participating in online debates [START_REF] Fernández | Learning to notice students' mathematical thinking through on-line discussions[END_REF] or participating in learning environments (interventions) designed considering specific mathematical topics. For example, [START_REF] Schack | Prospective elementary school teachers' professional noticing of children's early numeracy[END_REF] in the area of early numeracy; [START_REF] Magiera | An exploratory study of preservice middle school teachers' knowledge of algebraic thinking[END_REF] in algebra; [START_REF] Callejo | Prospective primary teachers' noticing of students' understanding of pattern generalization[END_REF] in pattern generalization; [START_REF] Llinares | Changes in how prospective teachers anticipate secondary students' answers[END_REF] in classification of quadrilaterals; [START_REF] Sánchez-Matamoros | Developing pre-service teachers' noticing of students' understanding of derivative concept[END_REF] in the derivative concept; and [START_REF] Son | How preservice teachers interpret and respond to student errors: ratio and proportion in similar rectangles[END_REF] in the concepts of ratio and proportion. These previous studies underline that the skill of deciding how to respond on the basis of children's mathematical thinking is the most difficult one to develop in teacher education programs. As [START_REF] Choy | Productive Mathematical Noticing: What it is and why it matters[END_REF] pointed "the specificity of what teachers notice while necessary, is not sufficient for improved practices" (p. 187). In other words, teachers can be very specific about what they notice without having a teaching decision in mind. So, the relation between how prospective teachers develop the skills of interpreting students' mathematical thinking and deciding how to respond on the basis of students' mathematical thinking deserves further research.

On the other hand, the concept of limit of a function is a difficult notion for high school students (16-18 years old) and is a key concept in the Spanish curriculum [START_REF] Contreras | Significados pretendidos y personales en un proceso de estudio con el límite funcional[END_REF]. Cottrill and colleagues (1996) indicated that the difficulty of students' understanding of the limit concept could be the result of a limited understanding of the dynamic conception. A way of overcoming this difficulty is by coordinating the processes of approaching in the domain and in the range in different modes of representation. Knowing these characteristics of students' understanding could provide prospective teachers with information to interpret students' mathematical thinking and to make instructional decisions based on students' reasoning.

Therefore, our study analyses changes in the way that prospective teachers notice students' mathematical thinking (attending to, interpreting and deciding) in relation to the limit concept when they participate in a learning environment designed ad hoc. The learning environment designed progressively nests the skills of attending to, interpreting and deciding and its relations. We hypothesise that the structure of the learning environment help prospective teachers to decide how to respond taking into account their previous interpretations of students' mathematical thinking.

Method Participants and the learning environment

The participants were 25 prospective secondary school teachers (mathematics, physics and engineering) who were enrolled in an initial secondary mathematics teacher training program. One of the subjects of this program is focused on developing the skill of noticing students' mathematical thinking in different mathematical topics and on planning the instruction attending to students' mathematical thinking. One of the mathematical topics considered was the limit concept.

The learning environment consisted of 5 sessions of two hours each and was designed taking into account the nested nature of the skills of attending to, interpreting and deciding [START_REF] Jacobs | Professional noticing of children's mathematical thinking[END_REF]. Prospective teachers were divided into five groups of 5 persons to perform the tasks of the learning environment. Firstly, prospective teachers solved three problems related to the limit concept selected from high school textbooks (Figure 1) in order to unpack the important mathematical elements of the limit concept (session 1). Then, prospective teachers had to anticipate hypothetical students' answers to these problems reflecting different characteristics of conceptual development (session 2). That is, they had to anticipate what students are likely to do. Prospective teachers had a document with the definition of the dynamic conception of limit and its mathematical elements [START_REF] Pons | Análisis de la comprensión en estudiantes de bachillerato del concepto de límite de una función en un punto[END_REF]: (i) approaches from the right and from the left (in the domain and in the range), and (ii) coordination of the processes of approaching in the domain and in the range considering different modes of representation (graphical, algebraic and numerical). The aim of the tasks of identifying the mathematical elements in the resolution of the problems and anticipating hypothetical students' answers was to help prospective teachers focus their attention on the relationship between the specific mathematical content and students' mathematical thinking. We conjecture that focusing on this relationship is needed to develop the skill of noticing in a first step. Next, prospective teachers analysed a set of four high school students' answers (Pablo, Rebecca, Luiggi and Jorge) to the same problems. Prospective teachers had to attend to students' strategies, interpret students' mathematical thinking and propose new activities (or modify them) to help students progress in their conceptual reasoning (according to their previous interpretations of students' mathematical thinking) (session 3 and 4). The high school students' answers, provided to prospective teachers, reflected different levels of high school students' reasoning of the limit concept (Table 1; [START_REF] Pons | Análisis de la comprensión en estudiantes de bachillerato del concepto de límite de una función en un punto[END_REF]. We also provided prospective teachers with theoretical information that summarise the characteristics of high school students' reasoning of the limit concept from previous research to solve the task [START_REF] Cornu | Limits[END_REF][START_REF] Cottrill | Understanding the limit concept: Beginning whit a coordinated process scheme[END_REF][START_REF] Swinyard | What does it mean to understand the formal definition of limit?: Insights gained from engaging students in reinvention[END_REF]. In figure 2, the answers of Pablo to the three problems are given.

Prospective teachers had to answer the next three questions: (i) which mathematical elements has the student used in each problem? Indicate if he/she has had difficulties with them; (ii) identify some characteristics of how the student understands the limit of a function. Explain your answer using the mathematical elements identified before; (iii) considering the student reasoning, propose an activity that helps the student progress in their conceptual reasoning of the limit concept. Therefore, the objective of sessions 3 and 4 was that prospective teachers focus their attention on the relation between identifying-interpreting and between interpreting-deciding. We conjecture that these relationships are necessary to develop the skill of noticing. Finally, in the session 5, prospective teachers had to answer a similar task individually.

High school students

Level

Levels of students' reasoning about the dynamical conception of the limit concept

Pablo and Luiggi

High Pablo and Luiggi coordinate the processes of approaching in the domain and in the range in the three modes of representation Rebecca Low Rebecca coordinates the processes of approaching in the domain and in the range in the graphical mode of representation when limits coincide Jorge Intermediate Jorge coordinates the processes of approaching in the domain and in the range in the algebraic and graphical mode of representation (when limits coincide in this last mode of representation) Answer to Problem 1 Answer to Problem 2 a1) x1 is approaching to 1 from the left and from the right and x2 is approaching to 1 from the left and from the right a2) The images of f( x1) are approaching to 2 from the left and from the right a3) The images of g( x2) are approaching to 2 from the right and is approaching to -1 from the left b1) when x1 is approaching to 1, images of f(x1) tend to 2 b2) when x2 is approaching to 1, the images of g(x2) tend to -1 from the left and tend to 2 from the right.

Answer to Problem 3 a) Graph 3 because the limit of the function in x=2 from the right and from the left is 2. b) Graph 2 because the limit of the function in x=2 from the left and from the right is 5. c) Graph 1 because the limit of the function in x=2 is not the same from the right and from the left.

Data and analysis

Data of this study are prospective teachers' answers to the tasks of session 2 (anticipation) and sessions 3 and 4 (interpretation). Through an inductive analysis [START_REF] Strauss | Grounded theory methodology: An overview[END_REF], we generated similarities and differences about how prospective teachers conceived high school students' reasoning of the limit concept and the type of activities they provided to help students progress in their conceptual reasoning. To carry out this analysis, five researchers analysed individually prospective teachers' answers to the anticipation and interpretation tasks and then, the agreements and disagreements were discussed to reach a consensus on these issues. This analysis let us identify two ways of how prospective teachers conceived high school students' reasoning: as dichotomous (right or wrong) and as a progression (identifying different levels of students' reasoning). The type of activities that prospective teachers provided were categorised in three categories: general decisions, decision based on curricula contents and decisions based on cognitive processes. Examples of these categories are presented in the results section.

Finally, we compared categories obtained in the anticipation task with the categories obtained in the interpretation task to identify changes in the way of how prospective teachers conceived high school students' reasoning and proposed activities to help students progress in their conceptual reasoning.

Results

Our results show that prospective teachers changed the way that they conceived students' reasoning from a dichotomous to a progression way and this shift influenced the type of activities that they proposed to help students progress in their conceptual reasoning.

Changes in the way that prospective teachers conceived students' reasoning: From a dichotomous to a progression

In the anticipation task, three out of five groups conceived students' reasoning as dichotomous (right or wrong). For example, the group of prospective teachers G2 anticipated that a high school student with high level of reasoning of the limit concept (Maria) would coordinate in all modes of representation. For example, this group of prospective teachers anticipated the next answer for the algebraic representation: Furthermore, these prospective teachers (G2) anticipated that a high school student with a not suitable level of reasoning of the limit concept (Pedro) would not coordinate in any mode of representation pointing out: "Pedro only approximates (from the left or from the right) when the function is defined".

Then (in the interpretation task), four out of five groups of prospective teachers were able to interpret students' mathematical reasoning. They linked students' reasoning with the mathematical elements of the dynamic conception of limit: the approaches from the right and from the left (in the domain and in the range), and the coordination of the approaches in the domain and in the range considering different modes of representation (graphical, algebraic and numerical). For instance, the group of prospective teachers (G2) interpreted the student's answer of problem 1 (Figure 2) as:

The resolution of the student is correct (Pablo). We can notice that the student has identified the kind of function (piecewise function) since he (the student) has approximated in the range (he has calculated the approximation to x=1 from the left and from the right and the approximation to x=2 from the left and from the right) and in the domain (taking the correct definition of function in each interval). Furthermore, he has coordinated the processes of approaching in the domain and in the range since he has written, for example, that when x tends to 1 from the left, the image of the function tends to 3 (using the function 2x+1).

Maria understands the limit concept. The idea of approximation in the domain corresponds to the fact that she properly selects the branch of the function and uses the notion of approximation in the range adequately. It is demonstrated when she replaces on the limit the approach of the independent variable. This student also coordinates the approximations to establish the value of the limit according to the branch.

This group gave similar comments for the student's answers to the other two problems linking students' reasoning with the important mathematical elements in the other two modes of representations (problems 2 and 3). Afterwards, they wrote a summary about this student level of reasoning:

This student understands the limit concept since he approaches from the right and from the left (in the domain and in the range), and coordinates the processes of approaching in the domain and in the range in the three modes of representation (graphical, algebraic and numerical). This student would be in the high level of reasoning.

These prospective teachers were able to identify different levels of students' reasoning. Therefore, they conceived students' reasoning as gradual.

Changes in the type of activities they proposed to help students progress in their conceptual reasoning

Prospective teachers who conceived students' reasoning as dichotomous did not propose specific activities to help students progress in their reasoning. These prospective teachers gave general comments about teaching as instructional actions. For instance, the group of prospective teachers G2 proposed to Maria (in the anticipation task) the representation of the graph of the function of problem 1. This decision was not based on the conceptual progression of the student.

When prospective teachers interpreted students' mathematical thinking identifying different levels of students' reasoning (linking students' mathematical reasoning with the important mathematical elements), they were able to provide specific activities to help students progress in their conceptual reasoning. For the students who only coordinate the approaches in the domain and in the range in one mode of representation, they proposed new activities to integrate these mathematical elements gradually in the different modes of representation. The proposed activities required a coordination of approaches in the domain and in the range in the different modes of representation. For the students who coordinate the approximations in the domain and in the rage in all modes of representation (such as the student of Figure 2), they also provided activities to help students progress in their reasoning.

We have identified two ways in which they justified their new activities: some justifications were based on the mathematical elements and others on the cognitive processes involved. In the first case, prospective teachers focused their attention on introducing a new mathematic content. In the following example, they introduced a new type of discontinuityan avoidable discontinuity. The justification of this type of activities was based on the use of new mathematical elements (in this case, introducing other type of functions).

The activity: We would modify the function of problem 1 and we would use:

Our justification: The student (Figure 2) seems to understand the limit concept in the three modes of representation, so we would provide him a more difficult function with an avoidable discontinuity.

In the second case, prospective teachers focused their attention on the cognitive processes involved to understand the limit concept. In the next example, prospective teachers focused on the reversal as a cognitive mechanism that leads students to a new reasoning level. That is to say, prospective teachers justified the proposed activity by the need of generating learning opportunities to develop the reverse mechanism that allows the construction of cognitive objects.

The activity: Represent a graph of a function which limit in x=-1 is 4 and that there is not limit in x=1.

Our justification: with this activity students need to do the inverse process that is, they need to use all the important mathematical elements to build that function.

Discussion and conclusions

Results show that after the participation in a learning environment that progressively nests the three interrelated skills of professional noticing (attending to, interpreting and deciding), prospective secondary school teachers gained expertise in noticing. Four out of five groups of prospective teachers were able to interpret students' mathematical thinking linking students' reasoning with the important mathematical elements of the dynamic conception of limit. These findings support the claim that some characteristics of the learning environment such as considering the nested nature of the skills help prospective teachers develop the skill of noticing [START_REF] Sánchez-Matamoros | Developing pre-service teachers' noticing of students' understanding of derivative concept[END_REF][START_REF] Schack | Prospective elementary school teachers' professional noticing of children's early numeracy[END_REF]. Furthermore, prospective teachers were able to provide specific activities to help students progress in their conceptual reasoning. Therefore, the characteristics of the learning environment in which prospective teachers were engaged in the analysis of mathematical elements of limit problems, in the analysis of students' reasoning and in proposing new activities to support students' conceptual development enable them to gain more accurate understanding of the relation between the mathematical content and students' mathematical thinking. This new understanding provides prospective teachers with the needed knowledge to give their teaching decisions based on the progression of students' reasoning: from a mathematical point of view or considering the cognitive processes involved.

Figure 1 :

 1 Figure 1: The three problems related to the limit concept

Figure 2 :

 2 Figure 2: Pablo's answers to the three problems

Table 1 : Characteristics of high school students' answers

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Acknowledgment

The research reported here has been financed in part by the Project GV/2015/115 of the Conselleria de Educación, Cultura y Deporte de la Generalitat Valenciana and in part by the project EDU2014-54526-R of the Ministerio de Educación y Ciencia (Spain).


text:
Stéphane Clivaz
email: stephane.clivaz@hepl.ch

Aoibhinn Ni Shuilleabhain
email: aoibhinn.nishuilleabhain@ucd.ie

Analysing mathematics teacher learning in Lesson Study -a proposed theoretical framework

Keywords: Teacher collaboration, Lesson Study, professional development, mathematical knowledge for teaching, levels of teacher activity

. The proposed framework is situated as a tool to detail and analyse the use of mathematics teacher knowledge in planning, conducting, and reflecting on research lessons in a lesson study cycle in a primary-school case study in Switzerland.

Lesson Study and models of knowledge for teaching

Lesson study is a collaborative model of professional development which supports teacher learning (Huang & Shimizu (Eds.), 2016). Originating in Japan, this model has grown in international popularity over the past two decades, particularly in the field of mathematics education, and much research has detailed evidence of mathematics teacher learning through lesson study (e.g. [START_REF] Lewis | Improving mathematics instruction through lesson study: a theoretical model and North American case[END_REF][START_REF] Murata | Making Connections Among Student Learning, Content, and Teaching: Teacher Talk Paths in Elementary Mathematics Lesson Study[END_REF][START_REF] Ni Shuilleabhain | Developing mathematics teachers' pedagogical content knowledge in lesson study: case study findings[END_REF].

Lesson study provides teachers with opportunity to contextualize representations of their classroom activities, while also making their implicit knowledge and practices explicit through their conversations within the group [START_REF] Fujii | Designing and adapting tasks in lesson planning: a critical process of Lesson Study[END_REF]. Each lesson study cycle consists of a number of steps where teachers begin by studying the curriculum and deciding on a research theme, planning a research lesson according to that theme, conducting and observing the live research lesson, and reflecting on student learning within the lesson (see Fig. 1) [START_REF] Lewis | How does lesson study improve mathematics instruction?[END_REF][START_REF] Lewis | Improving mathematics instruction through lesson study: a theoretical model and North American case[END_REF] With increased international educational research on lesson study, there have been calls to deepen the knowledge base of the development of teacher knowledge within this model in order to provide a solid theoretical foundation for its use in teacher education [START_REF] Clivaz | French Didactique des Mathématiques and Lesson Study: a profitable dialogue[END_REF]Miyakawa & Winsløw, 2009). In this paper, we hope to contribute to the literature on professional development for mathematics teachers by analysing the mathematical knowledge utilized by teachers in their participation in lesson study, utilizing our proposed theoretical framework.

The two authors of this paper, in their analysis of teacher knowledge and learning in lesson study, seek to deliberately build on previous existing frameworks of teacher knowledge: Mathematical Knowledge for Teaching [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF] and the Levels of Teacher Activity [START_REF] Margolinas | What can the teacher learn in the classroom[END_REF]. Through analysis utilizing a combination of these frameworks [START_REF] Prediger | Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework[END_REF], we will detail features of the mathematical knowledge for teaching utilized by teachers in their participation in lesson study and will also track the movement of this knowledge.

Thematic Working Group 18

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MKT/ levels in LS: Towards a coordinated model

In this paper, we propose a framework which was developed based on data generated in two case study sites -with eight participating primary (grade 3-4) teachers in Switzerland and five lower secondary (middle school, grade 7) teachers in the Republic of Ireland. Analysis began by utilizing the Mathematical Knowledge for Teaching framework [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF] to investigate the contributions by teachers in a lesson study cycle. However, we found that this model did not fully incorporate all the elements of teacher knowledge included in the lesson study cycle, particularly in capturing the educational values and conceptions of teaching and delineating between the layers of planning sequenced content of instruction, while also attending to students' thinking during the lesson. At this point in the analysis, Margolinas et al.'s (2005) Levels of Teacher Activity was identified as a framework which could encapsulate these elements of teachers' knowledge. Building on qualitative data generated through audio/video recordings of teacher conversations during lesson study meetings, teacher notes from lesson study meetings, researcher field notes, and selected samples of student work from research lessons, we present an extended model of the categorization of knowledge required for the teaching of mathematics. Our first example of analysis presented is from the Swiss case study and future work will present further analysis from the Irish case study data. In analysing and comparing these sets of local data, we attempt to demonstrate a more global sense of this proposed framework of mathematics teacher knowledge in lesson study.

Mathematical Knowledge for Teaching

In their ground-breaking work in 2008, Ball, Thames and Phelps addressed the concepts of content and pedagogical content knowledge in their model of Mathematical Knowledge for Teaching (MKT -see upper part of Figure 1). In this paper, they identified domains of Subject Matter Knowledge (SMK) and Pedagogical Content Knowledge (PCK) used in teaching, which further defined the knowledge and skills required of mathematics teachers in relation to student learning and to mathematics content.

Research on these different categories of MKT has demonstrated direct links between teacher knowledge and high-level teaching practices [START_REF] Clivaz | Des mathématiques pour enseigner? Quelle influence les connaissances mathématiques des enseignants ont-elles sur leur enseignement à l'école primaire? Grenoble[END_REF][START_REF] Hill | Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers' Topic-Specific Knowledge of Students[END_REF] and with subsequent student learning outcomes [START_REF] Hill | The Nature and Predictors of Elementary Teachers' Mathematical Knowledge for Teaching[END_REF].

Incorporating this model of MKT with teacher learning in lesson study, research has shown that Knowledge of Content and Students and Knowledge of Content and Teaching (features of PCK as defined by [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF]) are important elements of teacher knowledge utilized in lesson study cycles [START_REF] Leavy | Looking at practice: revealing the knowledge demands of teaching data handling in the primary classroom[END_REF][START_REF] Ni Shuilleabhain | Developing Mathematics Teachers' Pedagogical Content Knowledge Through Iterative Cycles of Lesson Study[END_REF][START_REF] Tepylo | Examining Change in Teacher Mathematical Knowledge Through Lesson Study[END_REF]. However, considering the multitude of teacher knowledge and practices incorporated within each lesson study cycle in planning, conducting, and reflecting on a mathematics lesson, this model may not capture all the decisions, actions, practices, and skills required of mathematics teachers participating in lesson study.

Levels of teacher activity and MKT

To describe teacher activity, both in and outside of the classroom, Margolinas developed a model of the mathematics teacher's milieu based on [START_REF] Brousseau | Theory of didactical situations in mathematics 1970-1990[END_REF]. This model was designed to take into account the complexity of teachers' actions and to capture the broad range of activities contained in teaching and learning (Margolinas et al., 2005, p. 207). At every level of environment (or milieu) the teacher must consider all that is occurring at the current level as well as those levels that are directly above and below. These multidimensional tensions relate to a non-linear and non-hierarchical interpretation of teacher's work (Margolinas et al., 2005, p. 208). In addition to commonplace professional opportunities where teachers speak about their beliefs and experiences on general educational concepts or about teaching and learning mathematics (level +3), about teaching and learning of a particular mathematical subject (level +2), or about the lesson they are preparing (level +1), during the phases of planning and reflection in LS teachers also have opportunity to discuss their classroom activities (level 0) or observations of student activity from a lesson (level -1).

+3 Values and conceptions about learning and teaching +2 The global didactic project +1 The local didactic project 0 Didactic action -1 Observation of pupils' activity

This activity model was used by [START_REF] Clivaz | Des mathématiques pour enseigner? Quelle influence les connaissances mathématiques des enseignants ont-elles sur leur enseignement à l'école primaire? Grenoble[END_REF] and aligned with the MKT model in order to capture the movement of didactical situations, beyond the possible static characterisation which may be interpreted in the MKT model (Ball et al., 2008, p. 403). The combination of these frameworks allowed teacher knowledge to not only be analysed in terms of mathematical knowledge for teaching, but also mathematical knowledge in teaching [START_REF] Rowland | Mathematical Knowledge in Teaching[END_REF]. Similarly, Ni Shuilleabhain (2015a) used the MKT model to analyse teacher learning in lesson study, but combined this with the idea of the 'student lens' (as suggested by Fernandez, Cannon, & Chokshi, 2003, p. 180), in proposing an additional layer of the model put forward by [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF]. This concept of a 'student lens' incorporated the PCK a teacher utilises in seeing mathematics "through the eyes of their students" (Fernandez et al., 2003, p. 179).

When aligned with Margolinas et al.'s (2005) model, this layer of teacher knowledge relates partly to the -1 'Observation of pupils' activity' which can be anticipated and interpreted, but extends this observation to thinking of the mathematical content from the students' perspective. In our proposed framework, we therefore see this view of the mathematics through the eyes of the student as a layer below the observation of a students' work and include a new level of -2 level relevant to teacher knowledge titled the "student lens" (see Figure 1).

Proposed theoretical framework

Explicitly combining these two approaches to analyse the knowledge utilized by mathematics teachers during lesson study, the authors here present a new theoretical framework (see Figure 1). This framework attempts to capture the knowledge required of mathematics teachers, in the broad and complex range of teaching and learning activities, and represents teacher knowledge and activities incorporated during each phase of a lesson study cycle (see Lewis & Hurd, 2006, p. 4).

Figure 1: MKT and levels of teacher activity at lesson study phases

We first utilize the model to categorize the knowledge (MKT and levels of activity) appearing during the lesson study cycle. The knowledge about a particular mathematical topic will then be tracked over each phase of lesson study and the relations between the occurrence of this knowledge examined. At this stage, 'knowledge' is considered as collective (e.g. Ni Shuilleabhain, 2016).

Analysis

In this paper, data generated though video recordings of the Swiss case study are analysed utilising the proposed framework. Eight primary generalist teachers, new to lesson study, and two facilitators (one specialist in teaching and learning and the other a specialist in mathematics didactic (first author of this paper)) participated in the research which occurred over two academic years. Four cycles of lesson study were undertaken in this time, with a meeting held on average every two weeks during the school year [START_REF] Clivaz | Lesson Study: from professional development to research in mathematics education[END_REF]. Each of these 37 meetings (about 90' each) were videotaped and transcribed and form the base of the analysis utilizing the framework outlined above (Figure 1) and incorporating defined features of KCS and KCT as utilized in lesson study [START_REF] Ni Shuilleabhain | Developing Mathematics Teachers' Pedagogical Content Knowledge Through Iterative Cycles of Lesson Study[END_REF].

We present analysis of the first lesson study cycle where teachers chose to focus on the topic of integers and place value. The main reason for choosing this subject was the difficulty students had with whole numbers. In the first session, teachers discussed a particular difficulty their students had with counting through to new groups in base 10: Océane: The counting through to the next ten. Caroline: But each time they have to count through to (tens, hundreds, …) Stéphane (facilitator):

What's happening with counting through to the next ten? Caroline: It's… that we have no more to write here! We have to use the digits which already exist. So, we count through to come back to one… In fact… Yes, it is the abacus, in fact, we need to move by one each time we arrive at a nine at the end. We need to move by one. Océane: We exchange one packet of ten.

In this passage, during the study curriculum phase, teachers are at level of the global didactic project (+3) and this unpacking of mathematical knowledge is a Specialised Content Knowledge (SCK) i.e. the mathematical knowledge needed to perform the recurrent tasks of teaching mathematics to students (Ball et al., 2008, p. 399). At this stage, the place aspect of number system was predominant in teachers' discourse and, when the value aspect appeared, it was linked with the value. To further address this knowledge, the facilitators suggested working on students' actual mistakes. Teachers and facilitators proposed mistakes like: 5 hundreds + 12 tens + 3 units = 515 This work prompted teachers to do the task as if they themselves were students. At some moments during the activity teachers even spoke like students -placing them at the level of student lens (-2). This allowed the teachers to go deeper into potential difficulties for students and by further studying curriculum materials (referred to as kyozai kenkyu by [START_REF] Takahashi | Collaborative lesson research: maximizing the impact of lesson study[END_REF], teachers had opportunity to clarify this aspect for the research lesson. This passage is situated at the same phase, level and type of MKT as the previous excerpt above.

Anne (facilitator): […]

It's a particular type of exchange since it's in the place value system. So, we can distinguish the two dimensions: the dimension of the place and the dimension of the decimal value which is revealed in the exchanges.

Stéphane:

In fact, I prefer to talk about grouping/ungrouping instead of exchanging.

Océane: Oh, I see! Following these two excerpts, we will briefly summarize the work undertaken by these teachers planning the second research lesson and focus on this phase for analysis. The group chose a task in the form of a board game involving the exchange of "1 hundred", "1 ten" and "1 unit" cards. Following a planning exploration of the task, this research lesson was taught by one of the group and, during the post lesson discussion, teachers agreed that the task should be modified to allow students practice the exchange of values and relate these to aspects of the number system. This revised lesson was taught by another member of the group to a different group of students.

At the beginning of the game a student, Julie, arrived on the square "give 35". She had three cards of "1 unit", three cards of "1 ten" and four cards of "1 hundred". In order to get three cards of "1 ten" and two cards of "1 unit", Julie wanted to exchange two "1 hundred" cards. The teacher, Edith, wanted to explain to Julie that two "1 hundred" cards were worth more than these three cards of "1 ten" and two cards of "1 unit".

Edith: So, two hundreds -that's how many? Julie:

Two hundred. Edith:

That's two hundreds. If you tell me: "I want three tens and two units." Three tens, how many is that? Julie:

Thirty. Edith:

You told me three tens makes thirty. And what about two units? Julie:

Two. Edith:

If you put the thirty and the two together? How many is that? Julie:

Thirty-two.

Edith:

So you swap two-hundred for thirty-two! You're very generous!

In this passage situated during the conduct lesson phase, at level 0 (didactic action), the teacher converted all cards into numbers to compare them, instead of doing direct exchanges. Julie followed the teacher without expressing her own way of reasoning (which can be observed in another passage and demonstrates a 'direct exchange' way of thinking). In this case, we categorize the MKT in two ways. First as a KCS, where Edith did not notice or interpret Julie's mathematical thinking or strategies. Second as a SCK, related to the unpacking of mathematical knowledge, as detailed in the following excerpt.

During her dialogue with the class, Edith had to explain that one hundred is the same as ten tens.

Here, again, her argument is to convert to units -which requires students to already understand place value. This argument can be summarized as follows:

1 hundred = 100 units and 10 tens = 100 units therefore, 1 hundred = 10 tens

In the final lesson plan, the group of teachers reflected on this strategy and argued against it:

"Often exchanges are not really carried out and we go through the number. For example, when asked to exchange 12 hundreds into tens, many students (and adults) will go through the number 1200, namely 1200 units, to say that that 1200 is 120 tens, without being able to make a direct exchange from hundred to tens. Teachers also often explain this exchange in this way. In this case, we are in a type of vicious circle, since it means that it is necessary to have understood number system to understand the grouping/ungrouping in the place value system."

This episode appears in our data in the research lesson (conduct and observe lesson phase, level 0), in the notes of the observing teachers (conduct and observe lesson phase, level 0), in the post lesson discussion (reflect on lesson phase, level 0) and, in the above extract, in the lesson plan (reflect on lesson phase, level +2) where observations and analysis of the group were generalized and decontextualized from the particular lesson to the level of a global didactic project. In each case the knowledge represents a typical SCK.

The final example of this knowledge was found at the end of the reflect on the lesson phase. After discussing the lesson and the mathematical difficulty of directly converting hundreds into tens, Valentine (a teacher with over 30 years of teaching experience) realized she had observed a similar difficulty her own students in this topic, outside of the lesson study group. As a result of their collaborative reflection conversations, she began to realize that her students' errors were likely due to her use of only one strategy in teaching this topic:

Valentine: But, I've got a question. For example, in nine-hundred-sixty-three -how many tens are there? Ninety-six. But my students, they learned a trick -they write the number 963 and just go to the tens digit and write what is left: 96. I'm convinced they just use this trick. I probably didn't know how to explain that to them! Myself… I always convert in money! You will have nine hundred and sixty three one-franc coins. If you need to only have ten-francs notes… then you will have ninety-six tenfrancs notes.

Although this observation was not directly related to observations during the research lesson, we still categorize it as level -1 since Valentine put herself in the position of observing her students converting 963 into tens. This conversation incorporates teacher KCS in interpreting students' responses and is situated at level -1 (observation of pupils' activity).

Utilising our proposed framework and building on our analysis of teachers' collective conversations, we can detail the types and levels of knowledge incorporated by mathematic teachers in their participation of lesson study. Utilising this framework provides us with opportunity to track the knowledge included in the planning and reflection of mathematics research lesson over various phases of lesson study.

Conclusion

This paper proposes an extended theoretical framework of mathematics teacher learning in lesson study combining the existing frameworks of Mathematical Knowledge for Teaching [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF] with Levels of Teacher Activity [START_REF] Margolinas | What can the teacher learn in the classroom[END_REF]. In this paper the proposed framework is situated as a tool used to detail and analyse the use and movement of mathematics teacher knowledge in planning, conducting, and reflecting on research lessons. Based on case study data generated through mathematics teachers' participation in lesson study, we have analysed teachers' qualitative conversations and considered the potential evolution of mathematics teacher knowledge over a cycle of lesson study. Analysis to date has demonstrated that in planning and reflecting on research lessons, teacher knowledge of various forms (e.g. SCK and KCS [START_REF] Ball | Content Knowledge for Teaching: What Makes It Special[END_REF]) and across varying levels of activity [START_REF] Margolinas | What can the teacher learn in the classroom[END_REF] are incorporated in these separate phases of lesson study.

We hope this model will contribute to the literature on professional development of mathematics teachers and may serve to underpin further evidence of teacher learning in lesson study.

Table 1 : Levels of a teacher's activity (Margolinas et al., 2005, p. 207)

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