Janne Fauskanger
email: janne.fauskanger@uis.no

Kjersti Waege
email: kjersti.wage@ntnu.no

Learning to represent students' mathematical ideas through teacher time-outs in rehearsals

Keywords: Mathematics teaching, rehearsals, teacher time-outs, professional development

This study examines rehearsals embedded in cycles of enactment and investigation, and more specifically, the discussions the participants engage in when instruction is paused -Teacher Time-Outs (TTOs). The paper studies the ambitious teaching practices that are revealed in the discussions, as well as the aspects of these practices that the participants have the opportunity to learn. Approximately 60% of the rehearsal time was spent on teaching and 40% on TTOs. The use of representations was a key aspect in 28% of these discussions. It is particularly interesting that the discussions in the rehearsals make it possible for the participants to learn how to represent students' mathematical ideas in writing on the board while at the same time paying close attention to connections between student talk and mathematical correctness. Implications from these findings are also discussed.

Introduction

The aim of professional development (research) is to support in-service teachers (ISTs) in developing ambitious teaching practices (e.g. [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF][START_REF] Mcdonald | Core practices and pedagogies of teacher education: A call for a common language and collective activity[END_REF]. Ambitious teaching entails mathematical meaning making, identity building and creating equitable access to learning experiences for all students. It requires that teachers know their students as learners and engage deeply with their thinking. Ambitious teaching practices enhance student learning of complex ideas and performances [START_REF] Lampert | Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching[END_REF], Ghousseini et al., 2015). Examples of ambitious practices are aiming toward goals, eliciting and responding to students' mathematical ideas and using representations [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF] (see more examples in Table 1).

Recently, teacher education and professional development (PD) have had more focus on teaching practices and pedagogies of enactment (cf. [START_REF] Grossman | Teaching practice: A cross-professional perspective[END_REF]Lambert et al., 2013;[START_REF] Gibbons | Developing collective capacity to improve mathematics instruction: Coaching as a lever for school-wide improvement[END_REF]. The focus of this paper is on rehearsals. These researchers emerging research indicates that the use of rehearsals contributes positively to the development of ISTs' ambitious teaching practices [START_REF] Gibbons | Teacher Time Out: Educators learning together in and through practice[END_REF][START_REF] Kazemi | Getting inside rehearsals: Insights from teacher educators to support work on complex practice[END_REF][START_REF] Lampert | Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching[END_REF][START_REF] Valenta | Rehearsals in work with in-service mathematics teachers[END_REF], as well as to the development of mathematical knowledge for teaching [START_REF] Ghousseini | Rehearsals of teaching and opportunities to learn mathematical knowledge for teaching[END_REF]. Previous research highlights the need for more research on rehearsals [START_REF] Kazemi | Getting inside rehearsals: Insights from teacher educators to support work on complex practice[END_REF][START_REF] Lampert | Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching[END_REF].

We report on findings from the PD project Mastering ambitious mathematics teaching (MAM). In the MAM project, ISTs are invited to collaborate in learning cycles of enactment and investigation where the overarching aim is to learn to enact the practices that constitute ambitious mathematics teaching. Rehearsals are an important element of these cycles, and we especially examine the use of teacher time-outs (TTOs) during them. The ISTs work on instructional activities (IAs) in the rehearsals. IAs are developed to highlight and support ISTs' learning of specific ambitious teaching practices and to focus on specific mathematical ideas. The IAs in the MAM project include choral counting, quick images, number strings, problem solving and games (cf. [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF]. The ISTs learn to teach IAs through cycles of enactment and investigation. [START_REF] Lampert | Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching[END_REF] explore what teacher educators and novice teachers do together during rehearsals so they can "learn to enact the principles, practices, and knowledge entailed in ambitious teaching" (Lampert et al., 2013, p. 226). They suggest that rehearsals provide room for novice teachers to open up their instructional decisions to one another. Kazemi et al. (2016, p. 16) suggest that rehearsals, as nested in learning cycles, provide repeated opportunities for novice teachers "to investigate, reflect on, and enact teaching through coached feedback". Furthermore, rehearsals bring the student teachers "into interaction with one another around instructional decision making that is responsive to students' contributions", enabling them to "come in close contact with how their colleagues make sense of and take up the practices that are being learned" (Kazemi et al., 2016, p. 28). In the context of PD, [START_REF] Valenta | Rehearsals in work with in-service mathematics teachers[END_REF] found that the interactions between the ISTs and the teacher educator during rehearsals were mainly discussions on using mathematical representations, using a mathematical goal, paying attention to student thinking and eliciting and responding to students' mathematical ideasall important principles and practices of ambitions mathematics teaching. They suggest that rehearsals offer ISTs "the environment and opportunity to work simultaneously on a variety of aspects of practice" (Valenta & Waege, 2017, p. 3387).

The organisational TTO procedure has been highlighted as a promising way for ISTs to learn together in and through practice [START_REF] Gibbons | Teacher Time Out: Educators learning together in and through practice[END_REF]. The participants can pause the instruction to think out loud together and share decision-making with one another before continuing with the enactment. For example, Fauskanger and Bjuland's (in press) analyses suggest that TTOs enable ISTs to develop a better understanding of how to elicit students' mathematical ideas by, for example, practising questions to ask. Research indicates that TTOs provide opportunities for ISTs to make changes and to shift the focus in the interactions "from one of judgment and evaluation to one of collective consideration and opportunistic experimentation in the midst of teaching mathematics" (Gibbons, Kazemi, Hintz et al., 2017, p. 29). Gibbons, Kazemi, Hintz et al. (2017, p. 48) hypothesize that TTOs support the ISTs' "ability to be adept at momentto-moment decision-making to engage students in rich discussions" and "to cultivate learning environments where everyone [is] positioned as capable of doing substantive mathematics" at the same time as ISTs become good at "at drawing on students' multiple knowledge bases." These researchers point out, for instance, that future research should attempt to understand more about how TTOs might support ISTs in developing their instructional practices. In this paper, we address the following research questions: (a) Which (if any) of the practices entailed in ambitious mathematics teaching are the focus of attention in the TTOs taken during rehearsals? and (b)

Bearing the TTOs in mind, which aspects of the practice of using representations do the ISTs have opportunities to learn to enact? The second research question is the main focus of this paper.

Methodology Design, data material and participants

The ISTs learn to teach instructional activities (IAs) through cycles of enactment and investigation. Each cycle involves the following six steps:

1. The ISTs read an article and watch a video of a teacher's enactment of a particular IA. 2. The supervisor leads a discussion/analysis of the article and the observed video. 3. Groups of ISTs prepare to teach the IA to a group of students, guided by the supervisor. 4. One IST (the rehearsing IST) enacts the IA in a rehearsal, where the supervisor and the other ISTs act as students. During this rehearsal, all participants can ask for TTOs. 5. The IST enacts the activity with a group of students. All participants can ask for TTOs. 6. Each group of ISTs analyse the enactment, guided by the supervisor. This is followed by a guided analysis of the enactment with all the ISTs.

Thirty ISTs from 10 elementary schools participated in the project. The ISTs were divided into four groups. One group consisted of those teachers who did not want to take part in the research, whereas two out of the three remaining groups (groups 2 and 3) were randomly chosen to be part of the research reported on in this paper. The ISTs' teaching experience varies from one to 30 years. The ISTs participated in 12 sessions, which included nine cycles of enactment and investigation, over the course of two years. All the sessions were videotaped where one researcher observed each group and took notes. The data material from groups 2 and 3 consists of video recordings of the rehearsals (step 4 above) from the nine cycles of enactment, resulting in 18 recorded rehearsals.

Analytical approach

A sociocultural view of ISTs' learning frames this research as the ISTs work together and learn ambitious teaching practices in a community of practice. We analysed the video recordings of the rehearsals, looking specifically for opportunities for ISTs' learning. First, all the TTOs in the rehearsals were identified according to the following definition of a TTO: The instances where the rehearsal is explicitly paused (often with a TTO hand signal) so the participants can better understand and/or act in relation to their colleagues' thinking, pedagogical choices and/or mathematical content (cf. [START_REF] Gibbons | Teacher Time Out: Educators learning together in and through practice[END_REF]. Second, we undertook a conventional content analysis [START_REF] Hsieh | Three approaches to qualitative content analysis[END_REF] of each of the TTO episodes, i.e. we used an inductive coding process with the aim of identifying aspects of ambitious teaching practices in the TTOs and which of these the ISTs have opportunities to discuss.

Findings

In a previous study, we found that approximately 60% of the rehearsal time was spent on ISTs teaching the IAs and 40% on TTOs, showing the back-and-forth pattern between facilitating deliberate practice and posing questions (Waege & Fauskanger, submitted). Moreover, we identified 175 TTOs and found five common dimensions of ambitious teaching practice that were addressed during them (Waege & Fauskanger, submitted). An overview of these practices is given in Table 1.

Note that many of the TTOs involved simultaneous work on multiple aspects of practice, thus many of the TTOs involved more than one category. With Waege and Fauskanger's (in review) study as our point of departure, in this paper we focus on one dimension, namely the "use of representations". Table 1 shows that this dimension is an important feature in many of the TTOs (shaded in Table 1).

Use of representations

Using representations [START_REF] Lampert | Using designed instructional activities to enable novices to manage ambitious mathematics teaching[END_REF] is one of the key practices in ambitious teaching.

Representing students' mathematical ideas in writing on the board, making connections between student talk and written representations, and making connections between different kinds of representations, such as the open number line, arrays and tables are important features in 28% of the TTOs. In many of these TTOs the participants discussed how to write student ideas or strategies on the board so they could represent the students' thinking and at the same time keep mathematical correctness in focus. For instance, they discussed the use of parentheses when representing students' ideas, the use of arrows instead of the equal sign and the degree to which they should focus on the convention related to the order of factors (in Norway, i.e. 3 × 4 normally represents three groups of four). The importance of recording exactly what the students said was emphasised in several of these TTOs, however, this sometimes led to mathematical incorrectness and thus TTOs were requested.

To illustrate the TTOs in which the participants discussed how to represent the students' ideas or written representations, we provide an example from a rehearsal of a game where number sentences1 presented by the students were written on the board. In this game, the students worked on finding number sentences using given numbers. Just prior to the coded exchange, the rehearsing teacher represented what one student said by writing the following two number sentences (all supposed to equal the given number 8): 5 + 3 = 8 and 9 -5 = 4 × 3 = 12 : 6 = 2 × 4 = We join the rehearsal as one of the observing teachers intervenes:

1 OT12 : Then the question is/ (interrupted by OT2) 2 OT2:

Then the question about [correct] notation is relevant.

S:

Yes, and it can't be [written] like that (pointing to 9 -5 = 4 × 3 = 12 : 6 = 2 × 4 = 8) 4 RT:

Ok, it can't be written like this? (…)3 (session nine, game, group 3)

The use of the equal sign in the number sentence is not mathematically correct, and the observing ISTs suggested taking a closer look at the use of notation in the number sentences (Lines 1, 2). The supervisor supported their suggestions, pointing out that they needed to find another way to write the number sentences (Line 3). The use of arrows was introduced in the MAM project, but these ISTs do not appear to be used to writing an arrow instead of the equal sign when recording students' thinking. As the discussion continued, some of the ISTs suggested that the number sentence could be written the way it was because that was the way the student presented it. However, at the end of this TTO the participants concluded that they would replace the equal sign with arrows so the number sentences would be mathematically correct: 9 -5 → 4 × 3 → 12 : 6 → 2 × 4 = 8.

A second representative example is taken from another group rehearsing the same IA, but now all the number sentences were to equal 6. The following was written on the board: 5 + 1, 8 -5 + 2 + 1, 10 -8 + 2 + 5 + 1 when one of the observing ISTs asked for a TTO (referring to 10 -8 + 2 + 5 + 1):

1 OT1: Should we ask them [the students] what they are thinking as they do this? 2 RT:

Yes.

OT1:

Ask them to explain their thinking. (session nine, game, group 2)

In what followed, the participants discussed what to do when they wrote exactly what the students were saying and it turned out to be mathematically incorrect. After arguing back and forth whether to write exactly what the students were saying or to represent their ideas in a mathematically correct way, they agreed to use "circles" (e.g. 10 -8 + 2 + 5 + 1). The rehearsal continued, but similar examples were presented by ISTs acting as students (e.g. 20 -8 = 12 + 3 = 15) when the supervisor requested a TTO to ask the ISTs what they would do if these examples were presented by students in the enactment phase. In the ensuing discussion, some of the ISTs pointed out the importance of writing exactly what the students were saying, whereas others pointed to the importance of writing what was mathematically correct. This group did not reach a conclusion on this matter before continuing the rehearsal.

The next example, taken from a rehearsal of a string, shows a discussion on how to represent students' ideas in writing. The first problem in the string was to focus on a run where four students were going to run equally long where the total distance was one hundred metres. One of the observing ISTs (acting as student) said that he knew that "they have to run 25 metres each" because "four times 25 equals one hundred". When representing this on an open number line on the board, the rehearsing IST started by dividing a line segment into two and then by dividing each of these parts into two again thus representing (100 : 2) : 2 instead of 4 × 25. The supervisor asked for a TTO:

1 Supervisor: The supervisor wanted the rehearsing IST to represent exactly what the student was saying (4 × 25, Lines 1, 5) and she invited the observing ISTs to recognise this (Line 1). One of the observing ISTs recognised that the rehearsing IST had represented (100 : 2) : 2 instead of 4 × 25 (Line 2). As the discussion continued, the rehearsing IST asked if she should present the strategy (100 : 2) : 2. This was followed by a short discussion on the pros and cons relating to ISTs presenting their own ideas.

The supervisor advised the rehearsing IST to represent 4 × 25 on the number line as "25 and 25 and 25 and 25" (showing the four "jumps" by pointing to the number line). They agreed that the rehearsing IST in the enactment should illustrate 4 × 25 as four jumps of the length 25 and in addition writing 25 above each arch as illustrated in Figure 1. These three examples illustrate that the TTOs in the rehearsals could support the ISTs' learning of the practice of representing students' ideas or strategies and their paying close attention to connections between student talk and written representations. The ISTs could also learn more about the balance between representing students' mathematical ideas or strategies as presented by the students and representing these ideas or strategies in a way that is mathematically correct.

Concluding discussion

In this study we have identified and analysed ambitious teaching practices that are revealed in TTOs in rehearsals, exploring which practices the ISTs have opportunities to learn to enact during the TTOs. The findings show that ISTs have opportunities to work on and learn to enact the practices of setting up a task, using goals to work towards, supporting mathematical discussions, organising the board and using representations. Opportunities for learning these practices have also been found in previous studies (cf. [START_REF] Kazemi | Getting inside rehearsals: Insights from teacher educators to support work on complex practice[END_REF][START_REF] Lampert | Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching[END_REF][START_REF] Valenta | Rehearsals in work with in-service mathematics teachers[END_REF], which have also found opportunities in TTOs in the enactment phase of cycles of enactment and investigation (cf. [START_REF] Fauskanger | Ambisiøse undervisningspraksiser i Teacher Time Out. [Ambitious teaching practices in Teacher Time Outs[END_REF][START_REF] Gibbons | Teacher Time Out: Educators learning together in and through practice[END_REF]. In this paper, we have focused on the practice of using representations. The findings show that TTOs especially have the potential to create opportunities for ISTs to learn how to present students' mathematical ideas in writing and also show the importance of representing student ideas as accurately as possible while also focusing on the mathematical correctness of the representation. Balancing the dilemma between student talk and mathematical correctness has not previously been found in research on TTOs in rehearsals and is thus a unique finding from this study. Our findings also indicate that the ISTs might need more mathematical knowledge for teaching (MKT) and that the TTOs can be a point of departure for strengthening ISTs' MKT (cf. [START_REF] Ghousseini | Rehearsals of teaching and opportunities to learn mathematical knowledge for teaching[END_REF].

Previous research on TTOs in enactment [START_REF] Gibbons | Teacher Time Out: Educators learning together in and through practice[END_REF], as well as in rehearsals [START_REF] Lampert | Keeping it complex: Using rehearsals to support novice teacher learning of ambitious teaching[END_REF], challenges future research to elaborate on how TTOs as a procedure in PD can be sustained and developed. By using the data material from the MAM project, it is possible to study the development of TTOs in each group for a two-year period. Further analyses of the data material will also make it possible to learn more about whether and how the supervisors develop their TTO expertise (cf. [START_REF] Gibbons | Teacher Time Out: Educators learning together in and through practice[END_REF]. Similar projects in new contexts are also welcome in order to study the potential of using TTOs in new contexts. In the Norwegian context the new guidelines for initial teacher education highlight ambitious teaching practices (cf. [START_REF] Mosvold | Fra undervisningskunnskap i matematikk til kjernepraksiserendringer i grunnskolelaererutdanningens matematikkfag. [From mathematical knowledge for teaching to core practiceschanges in mathematics in initial teacher education[END_REF] and the number of days in field practice in schools has been extended. Based on the promising opportunities for ISTs to learn ambitious teaching practices through TTOs in the enactment of teaching [START_REF] Fauskanger | Ambisiøse undervisningspraksiser i Teacher Time Out. [Ambitious teaching practices in Teacher Time Outs[END_REF], studying TTOs in student teachers' field practice will be an important context for future TTO research.

Figure 1 :

 1 Figure 1: 4 × 25 represented on a number line

Table 1 : An overview of the content of the 175 analysed TTOs (Waege & Fauskanger, submitted)

 1

	Ambitious teaching practice	Example	Number of
			TTOs
	Setting up a task, i.e. the TTOs consisted of	The rehearsing IST has just posed a problem when	42
	questions and discussions on the practice of	one of the observing ISTs asks for a TTO: "There's
	launching a problem.	something I'm wondering about. Justification. Should
		she [the rehearsing IST] say something about that
		[invite the students to justify while presenting the
		task]?" (session eight, problem solving, group 3)
	Using a goal to work towards, i.e. discussions	The rehearsing IST has just written the student	35
	on the practice of directing students' attention	solution "three times ten minus one" on the board: 3 ×
	on a learning goal for the lesson.	10 -1. The supervisor intervenes by saying: "Now,
		the question about [the use of] parentheses becomes
		interesting [referring to the expression 3 × 10 -1]."
		(session three, quick image, group 2)
	Supporting mathematical discussions, e.g. using	The rehearsing IST gives the ISTs (acting as students)	23
	talk moves (cf. Kazemi & Hintz, 2014).	some time to think about the problem 400 -379, and
		then asks them to share their solutions when one of
		the observing ISTs asks: "Should we ask them [the
		students] to turn to each other and talk or should we
		let them think individually first?" (session six, strings,
		group 2)
	Using representations, i.e. representing students´	The rehearsing IST, representing a student strategy on	49
	mathematical ideas in writing (on the board),	the board, pauses and asks: "Do I write in between
	making connections between student talk and	(points on the number line)?" (session five, strings,
	representations, and making connections	group 3)
	between different kinds of representations
	(Lambert et al., 2010, Lambert et al., 2013).
	Organising the board, i.e. organisation on the	The rehearsing IST is representing a student solution	27
	smartboard and utilising space.	on the open number line when the supervisor
		intervenes by saying: "I'm thinking of another thing
		right now and that is where to put the number line and
		the tasks." (session six, strings, group 2)

  What is interesting here is that you have thought, you have explained that you were thinking four times 25 (referring to the observing IST who presented this idea). You just knew that, right. But, when you (referring to the rehearsing IST) were making the number line now, what did she do, do you recognise that [what she did represent] (looking at all the observing

		ISTs)?
	2 OT1:	She divided into two and then divided into two again.
	3 S:	Yes, you halved and then halved again.
	4 RT:	Ok.
	5 S:	It's important that we are aware of this [that this represents (100 : 2) : 2 and
		not 4 × 25]. (…) (session six, strings, group 2)

In their planning, the ISTs used the term equations. When analysing the rehearsals, however, we find that sometimes the "equations" are written as equality strings or false equations (e.g. 9 -5 = 4 × 3 = 12 : 6 =

× 4 = 8) or without an equal sign (e.g. 9 -5). For this reason we use the term "number sentence" (cf. Ghousseini, 2017).2 In the transcripts, OT represents one of the observing ISTs, RT is the rehearsing teacher and S is the supervisor.

(…) indicates that the TTO continues.

Acknowledgment

We would like to thank to all the participants in the MAM project.


Jeannette Galleguillos
email: jeannette.galleguillos@uv.cl

Miguel Ribeiro

Prospective mathematics teachers' interpretative knowledge: focus on the provided feedback

Keywords: Mathematics teachers' interpretative knowledge, error analyses, teachers' feedback

In this work, we focus on the meaning prospective mathematics teachers assign to a set of students' productions and the feedback provided to these students. Based on these findings, we discuss prospective teachers' interpretative knowledge required for providing a fruitful feedback to students, having their reasoning as the starting point. Data collection (written responses) for this investigation took place at a workshop involving 19 prospective teachers from a Chilean university. Thematic analysis of their responses revealed two distinct types of feedback, namely that grounded in the nature and focus of the interpretation provided, and that focusing on the content of prospective teachers' knowledge.

Introduction

Teachers' knowledge is one of the core factors considered when developing highly demanding mathematical practices [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. Thus, in that sense, it is essential to focus on the content of such knowledge and, in particular, on the dimensions that would allow teachers to develop such mathematical practices, having the students' own reasoning and understanding as a starting point for the mathematical discussions, as well as for determining the feedback to be provided to these students. By better understanding the provided feedback, the aim is to enrich (prospective) teachers' knowledge and experience related to teaching practice. The knowledge grounding such type of practice is perceived as specialised.

In order to deepen and broaden our understanding on the content of teachers' interpretative knowledge and promote its development, we have been developing tasks for teacher education with a specific focus of developing teachers' knowledge (e.g., [START_REF] Jakobsen | Norwegian prospective teachers' MKT when interpreting pupils' productions on a fraction task[END_REF]Policastro, Mellone, Ribeiro, & Fiorentini 2018, submitted). We have been using tasks that probe into teachers' interpretative knowledge in different teacher education contexts (initial, continuous, and complemental) as a prompt to orchestrate mathematical discussions [START_REF] Bussi | Mathematical discussion and perspective drawing in primary school[END_REF]. The aim is to develop the solvers' awareness [START_REF] Mason | Researching your own practice[END_REF] and specialised knowledge-in this case, the Mathematics Teachers' Specialized Knowledge [START_REF] Carrillo | The mathematics teacher's specialised knowledge (MTSK) model[END_REF]. Such specialised knowledge would permit teachers to give meaning to non-standard student reasoning [START_REF] Mellone | Mathematics educator transformation(s) by reflecting on students' non-standard reasoning[END_REF], as well as to better understand the causes of students' errors [START_REF] Tulis | Error management behaviour in classrooms: Teachers' responses to student mistakes[END_REF], thus treating them as real learning opportunities in [START_REF] Borasi | Capitalizing on errors as "springboards for inquiry": A teaching experiment[END_REF] sense.

In the scope of a broader project, the aim of the present study is discussing secondary prospective teachers' Interpretative Knowledge (IK) based on the feedback they provide to students when analysing and commenting on productions students provided to a specific mathematical problem. [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] emphasised the need for integration of pedagogical and disciplinary knowledge in teacher education. Other researchers contributed to the mathematics teacher development, proposing to move away from a segregated and advanced training in "pure" mathematics to focus on the development of specialised knowledge that allows the teacher to teach mathematics. Amongst several teachers' knowledge conceptualisations that have emerged, in the scope of the work we have been developing, we assume the specialised nature and content of teachers' knowledge in the sense of the Mathematics Teachers Specialized Knowledge (MTSK) conceptualisation [START_REF] Carrillo | The mathematics teacher's specialised knowledge (MTSK) model[END_REF]. Teachers' specialised knowledge grounds teachers' interpretations of the students ' comments and productions (e.g., Di Martino, Mellone, Minichini, & Ribeiro, 2016).

Theoretical Framework

Interpretative Knowledge is perceived as the knowledge that allows teachers to give sense to pupils' answers, in particular to those that contain errors, or to "non-standard" solutions, i.e., adequate answers that differ from those teachers would give or expect [START_REF] Jakobsen | Norwegian prospective teachers' MKT when interpreting pupils' productions on a fraction task[END_REF]. It sustains the teachers' ability to support the development of pupils' mathematical knowledge, starting from their own reasoning, even if students' ideas are incomplete and/or non-standard. Moreover, IK is associated with the notion of discipline of observation and, in particular, with the idea of teachers working "on becoming more sensitive to notice opportunities in the moment, to be methodical without being mechanical" (Mason, 2001, p. 61). Such knowledge thus allows teachers to consider errors and non-standard reasoning as learning opportunities [START_REF] Borasi | Capitalizing on errors as "springboards for inquiry": A teaching experiment[END_REF][START_REF] Mellone | Mathematics educator transformation(s) by reflecting on students' non-standard reasoning[END_REF].

One way to enrich prospective teachers' interpretive knowledge is to offer opportunities to face student productions and to provide and discuss feedback, as part of their teaching practice. The term feedback is defined as "information provided by an agent (e.g., teacher, peer, book, parent, self, experience) regarding aspects of one's performance or understanding" (Hattie & Timperly, 2007, p. 81). The feedback can be constructive, i.e., one that includes hints, corrections, examples or explanations. [START_REF] Kulhavy | Feedback in written instruction: The place of response certitude[END_REF] termed this type of feedback as elaborative and defined it as any method that goes beyond indicating only the answer is correct or incorrect. Still, there is no consensus that providing elaborative feedback benefits learning more than simply indicating whether the answer is correct [START_REF] Shute | Focus on formative feedback[END_REF]. Some authors even argue that complex and detailed information can counteract the effectiveness of feedback. [START_REF] Santos | The evolution of feedback practice of a mathematics teacher[END_REF] studied the evolution of written feedback provided by a middle school mathematics teacher, reporting that her feedback evolved, developing plasticity, adjusting to the specific students or to the tasks, creating reflection moments by delivering elaborate clues, and encouraging the correction of the situation.

In the present study, we examined prospective mathematics teachers' interpretative knowledge by analysing the feedback they provided to students. The quality of the feedback has been the greatest influence on student performance [START_REF] Black | Assessment and classroom learning[END_REF], thus opportunities for interpreting students' productions are perceived as chances for improving feedback and students' learning.

Methods

In the present study, we focus on the answers given by 19 prospective mathematics teachers (PTs), who worked in small groups on interpreting students' productions related to a math problem adopted from [START_REF] Cañadas | Patrones, generalización y estrategias inductivas de estudiantes de 3° y 4° de Educación Secundaria Obligatoria en el problema de las baldosas[END_REF]. Most participating PTs were in the seventh semester of a 10-semester course and had already completed courses in mathematics, pedagogy, and mathematics education, and had some school practice experience.

The tasks were conceptualised following a particular design (e.g., [START_REF] Ribeiro | Tareas para alumnos y tareas para la formación: discutiendo el conocimiento especializado del profesor y del formador de profesores de matemáticas[END_REF][START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF], whereby the first part focused on accessing and developing PTs' MTSK, while the Interpretative Knowledge -Interpretative Taskand the provided feedback was examined in the second part. Here, we focus on this last part of the task, as the goal is to examine PTs' IK when interpreting students' productions and the feedback provided. In the first step of the task implementation, PTs were required to solve a problem that would later be explored with their students (Figure 1).

Complementarily, PTs were asked to refer to and reflect upon their own difficulties when solving the problem. In the next phase of the task, PTs were provided with three student productions, and were required to give meaning to each one of these productions, as well as provide what they consider to be a fruitful feedback to each student. The productions included in the task either contained errors, were incomplete, or involved a non-standard approach to problem-solving. Here, we discuss some productions included in the task, as shown in Figure 2(a)-2(c). Imagine you have white and grey square tiles of the same size. We make a row with the white tiles and then surround the white tiles with grey tiles, as the drawing shows:

1. How many grey tiles would you need if you had 1320 white tiles and wanted to surround them in the way you did in the drawing?

2. How many grey tiles are necessary for any number of white tiles? white grey white grey

Through the proportions, you can determine the number of grey tiles needed:

, grey tiles.

(1) Grey tiles (2) According to the relationship that occurs in the rule of three (5 is to 16, as x is to y).

Figure 2(c): Javiera's productions

The data utilised in the analyses included PTs' productions when solving the Interpretative Task, focusing on PTs' perception of the problem difficulty and the feedback given to three students based on the solutions offered. The written feedback that the prospective teachers provided to the students' answer was analysed categorizing the emergent feedback.

Analyses and discussions

Data was analysed in two phases. As noted earlier, all participating PTs were first asked to solve the problem in small groups and discuss any difficulties they encountered. In the next phase, feedback the PTs provided to Camilo and Aracelli based on their productions was discussed (these students utilised a similar strategy for solving the problem).

Difficulties in solving the problem

Eight of the nine groups of PTs solved the problem correctly, while the remaining group adopted the "rule of three" which yielded incorrect answer (Figure 3). When this group examined the solutions provided by other PTs, they were able to identify their mistake (pertaining to task interpretation) and subsequently provided a correct answer.

Concerning the PTs' difficulties, four out of nine groups stated that they had difficulty understanding how to represent the figure when the number of tiles increased (e.g., "We doubt how the white tiles were placed."; "Yes, because we did not know in what way the tiles would be grouped."). These results confirm that PTs had difficulties in understanding the statement of the problem, even though the manner in which the tiles should be grouped was explained explicitly in the problem statement "we make a row of white tiles" (see Figure 1).

Categorising the feedback

Most of the PT groups pointed out that both Camilo and Aracelli (Figure 2(a) and (b)) solved the problem by incorrectly applying the "rule of three" without giving further explanation of how they should have proceeded, while some groups provided indication on the way the problem should be addressed. When analysing the feedback provided to these two students based on their productions, four categories emerged in relation to Camilo's and Aracelli's solutions:

(i) Feedback on how to solve the problem: Guide on how the students should proceed to solve the problem, particularly stating that they should think inductively. (ii) Confusing feedback: When the feedback seems to be correct, but it can be confusing for the student.

(iii) Counterexample as feedback: An example is used to refute the error exposed.

(iv) Superficial feedback: The content of such feedback was insufficient (too broad or inconsistent) to allow the solver to understand its meaning.

(i) Feedback on how to solve the problem

Two PT groups focused their feedback on explaining to the student how he/she should proceed in order to solve the problem. They specifically indicated that the problem should be approached inductively, suggesting that the student should think what happens if there is one white tile before considering, what happens if there are two, three white tiles and so on. How many grey tiles would there be in each case? Then, they prompted the student to think about what happens in general terms. We observed that these PTs could correctly interpret the students' mistakes, and thus indicate the correct approach to solving the problem.

(ii) Confusing feedback

One of the groups focused on explaining why the number of grey tiles is not proportional to that of the white tiles. They stated that "proportional thinking should not be used, since the number of grey tiles does not increase by the same amount as white tiles; thus, they are not proportional" (see Figure 4). Perform the activity by proportion, that is, for every group of 5 white tiles, there are 16 grey tiles.

. The way to approach the problem is erroneous, since proportional thinking should not be used, because the number of grey tiles does not increase by the same amount as the white tiles. Therefore, they are not proportional. So, yes, we can say that the tiles that are above and below the white tiles are in a 1:2 ratio, not counting the edges.

Then ask what happens with the number of tiles at the edges when the white tiles increases or decreases? Lead the student to make the connection between the number of upper and lower tiles plus the tiles at the edges before prompting him/her to generalise for any number of white tiles.

We posit that the last expression can lead the student to incorrectly conclude that, if the number of grey tiles increases/decreases by the same amount as that of the white tiles, those variables are proportional. Instead, feedback should focus on the fact that the number of tiles does not increase/decrease "by the same proportion."

Second, they stated, "we can affirm that the tiles that are above and below the white tiles are in proportion 1:2, without counting the edges." Thus, they explained when the figure would be proportional. Although this reasoning is true, it could be confusing to the student, because the problem is not one of proportionality. As a result, this was termed as confusing feedback. In sum, although the PTs in this group could interpret the students' mistakes, when providing an elaborated feedback, they opted for a rather complex solution, which can confuse the students.

(iii) Counterexample as feedback

Two groups used a counterexample to respond to the student (Figure 5). In this excerpt, a group provided feedback suggesting that the student should draw a figure for the case of 6 white tiles with their 18 grey tiles (if proportional), while also applying the "rule of three" that would yield 19.2 grey tiles. The group interpreted this as a contradiction, along with the impossibility of having 0.2 tiles. However, it would have been more informative to argue that there is no multiplicative relation between white and grey tiles.

This group used a counterexample in their feedback to confirm that the number of grey tiles is not proportional to that of the white tiles. The use of a counterexample requires that the teachers have a comprehension of the mathematics knowledge involved in the problem (KoT), as this would allow them to provide an explanation for student's errors. From the MTSK perspective, teachers are required to have the mathematics knowledge of the theme that allows approaching the problem in the correct way, but also ability to explain why other ways do not work. Explaining why something is incorrect requires interpreting the students' reasoning and building feedback on these mistakes. Some of the feedback on Camilo's and Aracelli's productions was superficial, that is, too vague for a clear understanding by the student. For example, PTs stated: "I would ask what is proportion, so (S)he used "simple rule of three" because (s)he assumed a direct proportion between the number of white and grey tiles. I would tell him/her to try a case where you can draw a diagram. For example, we can assume that there are 6 white tiles. So, BB = 6, BN = 18 and try the simple rule of three to see if it works.

, .

Clearly, it does not make sense to have 0.2 tiles.

that [students] can argue if it can actually be used in the exercise," or "I would ask if the rule of three is used for all cases. . . . Analyse if there is a proportion to apply the rule of three." The nature of such feedback would neither allow students to understand their mistakes nor to envisage how to proceed in order to solve the problem correctly.

Some final comments

Some prospective mathematics teachers experienced difficulties in understanding and interpreting the posed problem, which is in the space of problems (and content) they will need to work on with their students in the near future. This finding highlights the need for enhanced preparation that would ensure that prospective mathematics teachers can solve the problem independently in order to, at least, be able to perceive a correct typical answer [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF].

Data analysis further revealed two dimensions of feedback study participants provided to students: the nature and focus of the interpretation provided, and the content of prospective teachers' knowledge. In the feedback on how to solve the problem, the groups focused on explaining how students should proceed in order to obtain the correct solution, disregarding explaining why the original answer was incorrect. Feedback that was classified as confusing feedback and counterexample as feedback focused on explaining why there is a mistake. Such feedback is particularly linked with the elements of PTs' space of solutions, usually referred to as having a single approach to guide to students [START_REF] Jakobsen | Norwegian prospective teachers' MKT when interpreting pupils' productions on a fraction task[END_REF]. Further, when the PTs attempted to provide elaborate feedback, they supported their explanations with erroneous arguments, or worded it in a way that can be confusing for the students. These feedback categories reveal some critical points that need to be addressed in teacher education in order to enrich and expand the superficial or confusing feedback, empowering a meaningfully interpretation that should have students' reasoning, knowledge, and understanding as a starting point for the mathematical discussions to be developed.

Figure 1 :

 1 Figure 1: Tiles problem (Cañadas et al., 2008)

Figure

  Figure 3: Mistake in understanding the problem

Figure 4 :

 4 Figure 4: Confusing feedback

Figure 5 :

 5 Figure 5: Counterexample as feedback

Acknowledgements: This research is partially supported by the grant 2016/22557-5, São Paulo Research Foundation (FAPESP) and is a part of the activities comprising the project CONICYT PCI/Atracción de capital humano avanzado del extranjero, nº 80170101 (Chile).


Alison Godfrey
email: ag384@le.ac.uk

Using participant generated influence maps to gain insights into the influences on early career primary teachers' teaching of mathematics

Keywords: Teacher characteristics, professional development, elementary school mathematics, research tools

Set within a study looking to understand influences on early career primary teachers' teaching of mathematics through analysis of the personal perspectives of eight participants, this paper discusses the use of a 'visual data' approach. Participants were asked at the end of the two-year research period to create maps showing the relative and overlapping influences on their development over the previous two years as teachers of mathematics, describing their thinking to the interviewer as they did so. Deeper and broader responses have resulted, giving richer data than using verbal data alone, whilst this approach has proved to be a very interesting and motivating experience for the participants. New insights have emerged as to how early career teachers see the influences on their development as teachers of mathematics.

Introduction

This paper outlines how a participant generated visual technique contributed to a study that seeks to gain insights into the influences on early career primary teachers' teaching of mathematics.

Whilst there is a minimum required mathematical qualification for all those entering the teaching profession in England, early career teachers have varied mathematical backgrounds, attitudes and beliefs about the subject. They enter a complex and changeable educational landscape where schools have considerable freedom in the organisation of their mathematics teaching and offer varied opportunities for professional development (Advisory Committee on Mathematics Education, 2013). The study followed eight primary school teachers through their first two years of teaching and sought to answer the research question: How do factors related to the teacher themselves and factors related to the school context combine to influence the development of early career primary teachers' teaching of mathematics? The participants entered the teaching profession with a range of mathematical backgrounds and all studied for their teaching qualification at the University of Leicester.

The aim of the tool outlined in this paper was to support participants to articulate their views on the influences on them as early career teachers of mathematics, and the connections between these influences. Participant-generated visual data had been successfully used in previous interviews and similarly this new tool was developed to achieve greater depth and breadth in interview responses than from purely verbal techniques, enabling participants to have more control over their responses whilst providing some structure. In addition, the tool was designed to be interesting and motivating, both as an ethical end in itself and to encourage participants to give the fullest responses possible.

Theoretical backgroundinfluences on teacher development

The literature identifies a number of potential influences on teachers as they develop their teaching of mathematics, relating both to the teachers themselves and their school context. These include a teacher's beliefs, their background, including the interrelated aspects of subject knowledge, attitudes to mathematics and emotions, and their proactivity in response to reflection on their practice within their school community. A teacher's beliefs about what makes a good mathematician can influence their practice [START_REF] Ernest | Mathematics teaching[END_REF]. Ernest considers, for example, that those who believe mathematics to be about accumulating skills and rules see themselves as "instructors, aiming to develop skills mastery in their pupils, while those who see mathematics as a unified body of knowledge to be understood, see themselves as "explainers", aiming to develop conceptual understanding of this knowledge in their pupils. Recent discussions in the UK about a mastery approach in the teaching of mathematics are in line with the relational understanding described by [START_REF] Skemp | Relational understanding and instrumental understanding[END_REF] and have focused on securing "a deep, long-term, secure and adaptable understanding of the subject" (NCETM, 2015), but it cannot be assumed that all teachers seek to teach for this type of understanding.

Building on Shulman's seminal papers (1986,1987), researchers have acknowledged that a teacher's subject knowledge in various forms influences their practice. [START_REF] Askew | Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency 1995-96[END_REF] found that the connectedness of teachers' subject knowledge "in terms of the depth and multi-faceted nature" of the meanings and uses of concepts in mathematics (p. 69) is particularly influential. [START_REF] Ball | Content knowledge for teaching what makes it special[END_REF], for example, emphasise the importance of Specialised Content Knowledge (SCK), knowledge that is specific to teachers of mathematics, enabling them to explain procedures, analyse errors and consider appropriate examples. Additional background elements that teachers bring to their classroom practice are their own emotional disposition towards mathematics and perceived competence in the subject (Di [START_REF] Martino | Me and maths': Towards a definition of attitude grounded on students' narratives[END_REF].

A teacher's practice can change over time, influenced by their personal reflection on practice and proactivity to make changes within the context of their teaching community [START_REF] Turner | Growth in teacher knowledge; individual reflection and community participation[END_REF]. Although the influence of their school context is likely to have a significant impact on an individual teachers' professional development opportunities, the community within the school might have a range of different foci and agendas [START_REF] Levine | Tools for the study and design of collaborative teacher learning: The affordances of different conceptions of teacher community and activity theory[END_REF] and the current provision for Continuing Professional Development (CPD) for teachers in England is fragmented and inconsistent between schools (ACME, 2013).

My study seeks to extend the existing literature, particularly exploring the relative importance and interconnected nature of these factors through highlighting teachers' own perspectives.

Visual Data

There is agreement in the literature that participant-generated visual data, i.e., that "generated through the data collection process" (Wall, Higgins, Hall, & Woolner, 2013, p. 3), can provide additional and complementary data to verbally answered interview questions (e.g., [START_REF] Prosser | Visual methods and the visual culture of schools[END_REF][START_REF] O'kane | Research with children: Perspectives and practices[END_REF]. It can add breadth and depth, facilitate deeper thinking and assist participants to reveal additional information, views and emotions that may not have been revealed though verbal questioning alone. It is also empowering, giving participants a greater voice in the research process and allowing them more control over the content of the discussions (O' Kane, 2000), and it can be motivating and fun for the participants [START_REF] Wall | That's not quite the way we see it': The epistemological challenge of visual data[END_REF].

A range of tool designs have been used in educational research to support visual data collection. Qsorting involves the sorting of statements into a defined template alongside or in addition to a verbal interview. Data can be analysed quantitatively and thus statistical comparisons can be made between participants, offering an in-depth and systematic approach to analysing subjectivity [START_REF] Thomas | Q-sorting and MIS research: A primer[END_REF]. Diamond ranking, similarly, is a technique that can be used within interviews to explore the participant's perceptions or ideas, as they rank images or statements within a nine-section diamond-shaped structure [START_REF] Wall | That's not quite the way we see it': The epistemological challenge of visual data[END_REF]. Usually, the activity of ranking these is accompanied by the participant talking through their rationale for how they placed the items within the diamond structure and the sorting exercise acts as a stimulus for discussion.

Although the literature reviewed suggests visual techniques can be valuable to the researcher, the quality of data is dependent on the quality of the facilitation and how the data relates to other data being collected [START_REF] Wall | That's not quite the way we see it': The epistemological challenge of visual data[END_REF]. A full rationale is therefore needed for the use of my approach and how verbal questioning was used alongside the visual strategy.

Methodology

An initial interview at the end of their one year postgraduate teaching course focused on each of the eight participants' relationship with, and attitude to, mathematics and their progress in teaching the subject as a student teacher. Twice yearly interviews over the following two years, including discussion of documentation related to their progress as early career teachers, provided evidence on their ongoing development as teachers of mathematics. Interview questions were designed to probe the participants' beliefs, attitudes and subject knowledge for teaching mathematics, what they perceived to be the characteristics of effective teachers of mathematics and their perspectives on their development as teachers of the subject. Within each interview they also talked about two particular lessons: their chosen best and most challenging lessons since the previous interview. Participant-generated visual data was a particular feature of the interviews, which combined verbal questioning with a range of visual tools, including participants graphing their relationship with mathematics [START_REF] Godfrey | Factors influencing the developing effectiveness of early career teachers' mathematics teachinginitial findings[END_REF], visually organising statements about subject knowledge and discussing points about secure learning in mathematics from concept cartoons [START_REF] Samková | Using concept cartoons to investigate future teachers' knowledge[END_REF]. Through these means evidence was collected of factors influencing their development as teachers of mathematics.

For the final interviews of the project, I developed a new visual tool to assist participants in talking in greater depth and breadth about the influences on them as a teacher of mathematics and in particular to discuss the relative importance of these factors and the way the factors combined. Given the longitudinal nature of the study, a secondary purpose was to ensure that the interview was interesting and motivational since visual techniques could provide variety and interest [START_REF] Wall | That's not quite the way we see it': The epistemological challenge of visual data[END_REF].

Rather than using an open question such as, "What has influenced you as a teacher of mathematics?", I decided to provide some prompts to ensure that participants discussed the likely influences the literature suggested would have an impact on their trajectory as teachers of mathematics, even if only to discount these, thus ensuring breadth in responses. Four prompts were provided: "My own background as a learner of mathematics and my feelings about the subject" (labelled background on the photographs below and aimed at participants considering their own subject knowledge, attitude to mathematics and emotions); "My beliefs about what makes a good mathematician" (labelled beliefs); "My school context and changes within the school contexti.e. the influences that I have had from being here in this particular school" (labelled school); "My own self-imposed changes/actions through my proactivity and reflection on practice" (labelled reflection).

In order to ensure that the participants considered the relative influence of these labels, and where they overlapped, circles of translucent plastic in different sizes and colours were provided. Each participant was asked to match the largest influence to the largest circle, the second largest influence to the second largest circle, and so on. They were then asked to arrange the circles to match the impact of these influences and the relationship between them, overlapping them if appropriate, and to verbalise their thinking as they did so (see Figure 1). Thus, an interactive approach was used, with some structure given for their thinking, but without the constraints of diamond ranking or q-sorting. Participants were free to create and adapt their mapping to fit their way of thinking. The design of the tool facilitated detailed narratives from individuals, gave them some control over the content of the discussion and allowed comparison between participants.

Findings

In considering the relative influences of the factors and how these overlapped, participants reflected on how, why and to what extent these influences were connected, thus giving a depth of response that might not have been communicated purely verbally. Rama, for example, explained:

The most important […] is my proactivity and reflection on my practice. […] I always like to reflect on my lessons and try to make it better. I always try and think, was it right for my children, was it not, let's change it a little bit. Participants varied the extent to which they overlapped circles to show the significance of connections. This can be seen on Rama's map (figure 1) where there is a larger overlap between reflection and beliefs than between school and beliefs. This second overlap was created as she discussed working alongside a colleague with similar beliefs: "We actually want the same thingwe've both thought let's change it and not try to use the workbooks".

The structure of the influences map seemed to be effective in supporting participants' articulation of ideas, which they might not otherwise have discussed, thus giving additional breadth to the data collection. The use of the cards meant that participants had to mention aspects that for them had little or no influencedata which would probably not have been obtained from a straight verbal question.

Gina commented:

My own self-imposed changes are probably the least because there's been so many other things that have to be put in place that I've had to just go by the wayside with those.

Without prompting, Gina may not have mentioned this; her comments support the overwhelming influence of her school context (figure 2). For some participants, the influence map prompted their thinking beyond the structure I had proposed. Penny, for example, used the four cards as a starting point to consider additional elements in her personal situation. She was keen to stress that the impact of her school context was very small, other than being sent on courses, one specific course in particular. She described the impact that this one hour mathematics course had on her practice due to gaining both new ideas, which she implemented from the following day, and renewed enthusiasm for implementing changes in her teaching of mathematics: "The course was brilliantjust gave you that impetus to carry on, gave you ideas to do, spot on". We improvised to create a further map label for courses and also for research, referring to my project, which she considered of particular significance because it caused her to reflect, look back at her practice and consider changes she could make. Summing up how my previous interview had caused her to reflect and change her practice, she said, "I relooked at everything after your last visit." Penny's final map is shown in figure 3. Again, the overlapping here is interesting, with other influences strongly linking with the learning she gained from the course and a strong connection shown between the influence of being involved in my research and her reflection. The control that Penny and others took over the use of the map to express their opinions demonstrates that a benefit of using the tool was the ready engagement of participants who had already been interviewed several times. This aspect was further evidenced by specific comments made by Rahim, who clearly found the process stimulating:

That was a really good opportunity for me to reflect on what different things have influenced me and how I feel like I've progressed based on different elements, so that was a really helpful exercise.

As illustrated above, the analysis of the data from the influences tool is enhanced with the addition of the verbal data given alongside the visual. From this starting point, further analysis is being undertaken to explore other evidence of these influences from earlier interviews. Interview data has been summarized using a mind mapping technique to facilitate this process. Individual narratives are being developed and some comparisons made between cases, including those with different qualifications and interest in mathematics.

A brief example of how the visual and verbal data from the creation of the maps is being integrated with other data comes from one element of Gina, Penny and Rama's maps (Figure 4). These maps show a very contrasting picture of what they feel has influenced their development as teachers of primary mathematics, but for all the influence of their background as a learner of mathematics and their feelings about the subject ('BACKGROUND') was considered a relatively small influence. Gina, who has the weakest mathematical background of the three, overlapped the circles relating to her beliefs and background, giving the following commentary as she created her map: I think my own background always sticks in my mind because I know there were things that I feel like I didn't learn well enough that really impacted me later on, so I'm very hyper aware that when children don't get something that we can't just leave them there. Penny has her own mathematical background, which, in her case, includes a Master's degree in mathematics, as a separate and smaller influence than the other influences on her because, although she "loves maths", her teaching of the subject would "just happen regardless". Rama similarly stated that although her mathematical background was important, she "had grown as a person" since her days as a learner of mathematics. However, other evidence from across their series of interviews highlighted that both were keen to pass on their love of mathematics to children and that their strong subject knowledge was a foundation on which they proactively sought to develop their practice.

Conclusion

Initial analysis across all the participants in the study indicates the different perceptions early career primary teachers hold of the influences on them as teachers of mathematics. Supported by the use of the influence-mapping tool, participants were explicit about the relative impact of each of the suggested influences on their development and how these influences were inter-connected. The map supported depth and breadth of responses, whilst giving the participants some control over the content of their discussions and the motivational aspect of the tool was appreciated by the participants.

Whilst not directly comparable because of participants' different interpretations of the headings used, useful insights can be drawn from the range of influence maps. The influences of the school context and self-imposed changes through proactivity and reflection on practice were strong influences for most of the eight participants. Only two placed the four influences in the same order. Even then, their maps visually contrast with differences in the way the circles were overlapped, and this is reflected in contrasting verbal narratives. Further analysis of the broad data set is being undertaken to explore how factors related to the teacher themselves and factors related to the school context combine to influence the development of early career primary teachers' teaching of mathematics.

Figure 1 :

 1 Figure 1: Rama's influence map Various follow-up questions were planned that could be used to further probe about the impact of these influences, for example: "How do you think having a strong maths background has impacted on your teaching of the subject? Has your attitude to teaching mathematics changed? What are your beliefs about what makes a good mathematician? Have you felt well supported by your school in these first two years of teaching? How do you feel about changes imposed by your school? What motivates you to develop your own practice?"

  […] This[reflection] goes with this [beliefs] because I have really strong thoughts of what makes a good mathematician and how it should be taught in a lesson and what I want to get out of the children […] that's helped me to reflect, if that makes sense.

Figure 2 :

 2 Figure 2: Gina's influence map

Figure 3 :

 3 Figure 3: Penny's influence map

Figure 4 :

 4 Figure 4: Contrasting influence maps: Gina, Penny and Rama