text:
Einat Heyd-Metzuyanim
email: einat.metz@google.com

Shifting frames -A teacher's change towards explorative instruction

Keywords: Mathematics instruction, professional development, frames, explorative practice

The case of one middle school teacher's change in practice is examined through the lens of "shifting frames" from ritual to explorative instruction. These frames are collection of coherent mathematical as well as subjectifying (people-related) meta-rules. The teacher, who participated in a year-long PD program, started out in a ritual frame and gradually shifted to a more explorative frame. The shift was not uniform, and could be seen first in the subjectifying meta-rules, and only later (and very partially) in mathematical meta-rules of exploring objects. In addition, newly learned practices were partially distorted through the old ritual frame.

Theoretical background

I take as a starting point [START_REF] Sfard | Thinking as communicating[END_REF] view, that the gist of mathematical activity, both in the classroom and over history, is the construction and exploration of "mathematical objects." These discursive objects, like numbers, triangles, and functions are not existing objects in the physical world, yet the "metaphor of object" is used by any skilled mathematician to talk about them as though they were indeed such physical objects. Mathematical objects can have many realizations, or physical symbolic representations. A function, for instance, can be represented as a graph, a table, an algebraic expression or a verbal statement. In the process of learning, according to Sfard, students come to see all these realizations as signifying one object. Once they do that, they start talking about it as an object existing of itself, a process termed "objectification." Within this view, Sfard and Lavie (2005) theorize the process of mathematical learning as moving from a ritual, peripheral phase, where activity is first and foremost aimed at pleasing the experts of the discourse (e.g. the teacher), to an explorative phase where new mathematical narratives are produced by oneself for the sake of the activity itself. In contrast to explorative participation, the focus of ritual participation is activity itself, not the mathematical narrative produced by it. Ritual participation is often characterized by syntactic mediation, where instead of using mathematical signs as signifiers of objects, these signs are manipulated according to prescribed (often memorized) rules.

Ritual and explorative participation are governed by certain meta-rules or "patterns in the activity of the discursants" (Sfard, 2008, p. 201). These rules can be divided into mathematical meta-rules, which dictate how mathematical narratives are to be derived from each other (for example, by proof or by computation) and subjectifying meta-rules which govern the actions of people (e.g. asking questions, giving directions, talking with each other). To capture the fact that meta-rules have a certain structure and coherence, Heyd-Metzuyanim, Munter & Greeno (in review) suggested the term "frames", borrowed from socio-linguistic and socio-cognitive research to describe "a set of expectations an individual has about the situation in which she finds herself that affect what she notices and how she thinks to act" (Hammer, Elby, Scherr, & Redish, 2005, p. 9). Accordingly, they defined "frames" in mathematical classrooms as a set of meta-rules, both mathematical and social, which includes appropriate questions, answers, justifications and other discursive actions in a situation of solving a mathematical problem or performing a mathematical task. Whereas frames of explorations would be sets of meta-rules that cohere around the goal of producing mathematical narratives based on logical justifications, ritual frames would be more aligned with goals such as performing a procedure accurately according to a prescribed set of steps, and adhering to external authority.

Method

The case of Mr. M is taken from a larger study, where we followed 7 teachers and 5 teacher leaders throughout 8 months of professional development (PD) during 2014 -2015. The PD was led by Margaret Smith and Victoria Bill from the University of Pittsburgh. It centered on [START_REF] Smith | 5 Practices for Orchestrating Productive Mathematics Discussions[END_REF] "5 Practices for Orchestrating Productive Mathematics Discussions" and on Accountable Talk™ (see http://ifl.pitt.edu/index.php/educator_resources/accountable_talk). Teachers were supported via four full-day PD sessions and in-school individual coaching sessions.

Mr. M was chosen for closer inspection because he showed a steady growth in the "implementation" score of the Instructional Quality Assessment tool [START_REF] Boston | Assessing Instructional Quality in Mathematics[END_REF], used in the larger study [START_REF] Heyd-Metzuyanim | Change in teachers' practices towards explorative instruction[END_REF] to score lessons for cognitive demand. Mr. M's lesson 1 scored a 1, lesson 2: 2, lesson 3 & 4: 3 (where 4 is highest). Though his lessons never achieved the highest level of cognitive demand, in the last two lessons cognitive demand was partially preserved.

Data collection included four cycles (September, December, February and May), each containing: 1. Pre-lesson and post-lesson interviews with the teacher; 2. Lesson recording and students' worksheets. Frames of mathematical instruction were searched in four data categories:

(1) The potential of the task to engage students with different realizations of mathematical objects and with mathematical meta-rules of justification and generalizations. (2) Teacher's plans as collected from the pre-lesson interviews, as well as his post-lesson reflections. In these we looked for evidence of mathematical and subjectifying aspects of explorative frames. (3) Students' work: evidence of engagement with different realizations of objects and with connecting between them. Evidence for justifications and explanations based on mathematical logic. (4) Whole-classroom discussions: evidence for ritual vs. explorative meta-rules. Is there only talk about routines for manipulating symbols detached from their meaning, or are the symbols also related to mathematical objects?

Findings

During the first, introductory interview, Mr. M declared himself to be aligned with what may seem to be explorative instruction. Describing quality instruction, he said: … where groups get together in giving a situation in which they come up with strategies on their own, as opposed to a teacher standing up in front of the room and just saying "okay, this is how you do this, this is how you do this, now just take this problem and solve it". So … I keep trying to look for my own stuff that I'm doing, so I would look to see that in other peoples.

Notably, in remarking that such instruction was "my own stuff that I'm doing", Mr. M attested for practicing quality instruction. To this, he added: one of my philosophies (is that) there's never, ever, only one way to solve a problem, that there's many different strategies and many different pathways you can take to find different problems … Mr. M also described his challenges in encouraging students to take on agency and not rely ritually on his guidance.

I would say a majority of the students I've worked with over the last 7, 8 years pretty much when they were on a challenging problem, their first shot is to ask me to give them help and guidance along the way. I used to do that more often, …, so I tried to step back away from that for a moment, getting to know the kids a little bit better helps me understand those that are just being lazy and those that truly don't know it. Thus, in his introductory interviews, Mr. M seemed to be well aligned with "reform" mathematical pedagogy, especially with regard to letting students struggle and encouraging them to look for "different strategies" for solving problems.

Lesson 1

Despite these declarations, the first lesson revealed Mr. M's actual practices were not well aligned with the explorative frame. For the main task of the lesson, Mr. M chose the task seen in Figure 1: The task was part of a module that introduced irrational numbers through the Pythagorean Theorem. The EngageNY 1 teacher guide encourages explicitly to present the task only after students had become familiar with this Theorem. The task was taken verbatim from the module, which assumed this previous experience, thus including the "use what you know" phrase. However, Mr. M. was well aware of the fact that his students were probably not familiar with the Pythagorean Theorem. In the pre-conference of the lesson he said:

1 See https://www.engageny.org/resource/grade-8-mathematics-module-7

The triangle below is an isosceles triangle.

Use what you know about the Pythagorean Theorem to determine the approximate length of the isosceles triangle (EngageNY, Module 8.7)

Thematic Working Group 18

The first challenge that I'm expecting is that we've not introduced Pythagorean Theorem at all last year. …. Um, I'd like to hear where their conversation goes and what they're focusing on -… when they're trying to find an opposite base, … given the a 2 +b 2 =c 2 idea, do they put the numbers in the right placedo they remember the square numbers, do they remember to take the square roots of numbers to get themselves back to what thethe single variable would be? … So I'm looking to seewhat kind of challenges working with square roots and squares creates.

Though Mr. M was anticipating "challenges" in working with the problem (aligned with the PD encouraging letting students struggle), the way he talked about students coping with these challenges was through "remembering". Thus, meta-rules belonging to the explorative frame (student agency) were distorted through the ritual frame into meta-rules of fact-retrieval, and memorizing procedures.

During the lesson, the launch of the task proceeded as follows: 90. T Okay, so when we take a look at this (pointing to a right triangle on the board, the sides of which are labeled 'a', 'b,' and 'c') I'm going to give you a little bit of a formula. And it may help you today in doing some of your work (writes a 2 +b 2 =c 2 on the board) ... Who can read this out loud for me, please? 91.-95. Three students are prompted to read the formula "in different ways". One reads "a two plus b two equals c two". The second "a to the second power, b to the second power, and b to the second power", the third "a squared plus b squared equals c squared" 96. T You see, all three of those ways that they said say the exact same thing. … So when we're looking at this, can you now start to see that knowing two of the three sides will allow me to figure out the third side? … I'm gonna have you work in groups today to see if you can't use …any knowledge that you have plus anything that you've seen today. Alright?

The subjectifying meta-rules enacted in this excerpt bore resemblance to explorative frames. Students were asked to provide "different ways" of saying the "exact same thing", and were directed to "figure out" and "work in groups" to "use any information" they have to solve the problem. Yet the mathematical meta-rules, that is, the ways to derive one mathematical narrative from another, were obscure at best. a 2 +b 2 =c 2 was written on the board, detached from the right triangle, with no indication of a 2 , b 2 , and c 2 signifying the geometrical squares adjacent to the triangles sides. By that, Mr. M was treating the a 2 +b 2 =c 2 Pythagorean Theorem syntactically, as a series of signifiers, detached from their meaning as signifying geometrical objects.

Not surprisingly, since students had no access to the meaning of the Pythagorean Theorem mediated by geometrical objects, they struggled over where to place the a, b, and c labels given by the formula, on the newly presented triangle. After some leading questions from Mr. M. students concluded they should label the base of the triangle as 'C' or 'C 2 '. They subtracted 9 2 -7 2 =32, but then were unsure as to what to do with that result. Some students thought it should simply be divided between the two halves of the base, labeling the whole base as '32' and halves of it with '16'. Others simply labeled the whole base as '16' or went another step and labeled the halves with '8'. Only 3 or 4 students, out of 24, figured out that the 32 should be square-rooted and multiplied by 2. However, even they did not label the triangle's base with the resulting number 11.4 (or estimated 12). The mathematical metarules governing the classroom activity were thus primarily ritual. They could be summarized as "apply a set of symbols, somehow related to a right-triangle, to a new triangle".

It was not that Mr. M was intending students to apply the 'a','b', and 'c' symbols randomly. He did have a certain type of reasoning he was looking for. After having asked a student to come to the board and present her solution, and while the student was labeling the triangle sides, he said: 289. T Yeah, please notice -… if we looked at that formula -A squared plus B squared equals C squared, and then I asked this question of some of you: Does it make a difference where C is? …( Student answers C has to be bigger than A and B) 291. T So if it's bigger than all the other ones, then the question becomes where does the C have to go -which side? 292. Student On the longest side.

T

The longest side. Now the one misconception I saw that I didn't expect was that some of you thought that this (the base) was the longest side, the whole way across. Just be careful about that, you're focusing on just one of those two triangles, okay?

The meta-rules of the activity, as gleaned from Mr. M's words were thus "figure out where the C is according to it being bigger than a + b. Then figure out the longest side and label it as C". However, this was detached from the physical meaning of the Pythagorean Theorem. In relation to the confusion or "misconception" the students had, regarding where to place the C, Mr. M did not have much advice besides "be careful about that" [293]. There was no other indication why they should be focusing on one of the right triangles, and not the whole triangle.

In the post conference, Mr. M seemed reasonably content with the results of the activity. He explained that since students were unfamiliar with the Pythagorean Theorem, he expected them to struggle, but that that wasn't the focus of the lesson. Rather the focus was discussing imperfect squares, to which they got at the end of the discussion, when one student said 32 should be rooted and Mr. M led students to estimate the root between 5 and 6. Yet having detached the geometric, physical realization of the Pythagorean Theorem from its algebraic formula, the rationale behind the existence of imperfect squares had no way of being foregrounded. The fact that Mr. M was not concerned with students struggling considerably with something that was not "the focus of the lesson", indicates that through his frame, the whole process of working in groups and discussing solutions was not a central measure for achieving the mathematical goal of the lesson. Rather, it was a sort of a "side effect", performed for the sake of the lesson recording or for students to practice working in groups.

I now move to a similar examination of the last lesson. A description of the two middle lessons is out of the scope of this paper. However, evidence for movement from a ritual frame to a more explorative frame were starting to show during the 3rd lesson, where Mr M received, together with the rest of the teachers of the PD, a high-level task selected by the PD leaders and directed explicitly on the different solution paths that should be sought in the lesson. Lesson no. 4 seems to have reaped the benefits of this process, though as we shall see, the shift between frames was still very fragile.

Lesson 4

The first indicator that Mr. M's practice was starting to align with an explorative frame could be seen in his choice of a task. This task involved modeling of real-life processes with a quadratic equation. Following is the task as presented on the worksheet (taken from EngageNY, Algebra 1):

The baseball team pitcher was asked to participate in a demonstration for his math class. He took a baseball to the edge of the roof of the school building and threw it up into the air at a slight angle so that the ball eventually fell all the way to the ground. The class determined that the motion of the ball from the time it was thrown could be modeled closely by the function: h(t) = -16t 2 + 64t + 80, where h(t) represents the height of the ball in feet after t seconds.

Students were then asked to find the behavior of the function (maximum, minimum, vertex), to graph it and to indicate how many minutes passed until the ball fell to the ground and what was the meaning of h(0). The task thus had ample potential for using different realizations for a mathematical objectnamely the quadratic formula.

The next indication for Mr. M's movement between frames was his talk about expectations for students' work during the pre-lesson interview. This is how he presented the choice for the task:

I've been working on questions … like "how do you know that that's where your graph crosses the X axis?" So more than justwhat is that value, how do you know what it is. How did figure it out. Um, why does your graph continue past zero? In the context of the problem, what does that mean? … And try to gain some understanding of their understanding.

Instead of "remembering", Mr. M now talked about students' "understanding". He was also working hard on preparing questions that would assess students' understanding, a skill specifically taught in the PD sessions.

Examination of students' worksheets showed that most students performed the calculations involved in the problem correctly. 4 out of the 27 students wrote that h(0) indicated the height in which the ball was thrown was 80 feet, and 3 more related h(0) to the initial stage, before the ball was thrown in the air. Other students, however, either left this question blank, or stated that h(0) means "the ball is on the ground" indicating incongruence between the physical situation and the graphical realization. Thus, in contrast to the first lesson, there was evidence that a small part of the classroom was able to flexibly move between different realizations of the quadratic function.

During the whole classroom discussion, students' explanation on the board were mostly concentrated on the calculations involved in the problem. Mr. M encouraged this by calling on students to present "different ways to solve the problem" which referred to factoring by dividing the expression by 16 or by 8. He also referred to finding the value of h(0) by substituting t=0 or by just "looking" at the last term in the equation, as two different ways to find h(0). Yet these two "ways" were, in fact, both tending to the algebraic realization and could be considered as "different" only when looking at the function's expression syntactically. There was no mentioning in the whole classroom discussion of the graphical realization of the function or the physical "real life" story modeled by it. It was thus still mostly characterized by meta-rules of carrying out prescribed procedures using syntactic mediation, rather than meta-rules of exploring mathematical objects. However, some slight changes could be observed. One of the students who talked in the whole classroom discussion did, even if very briefly, mention the physical "real life" situation that the function was modelling. He talked about the ball ("the ball hasn't been thrown yet") and hinted at the height of the roof in "throw the ball at 80". Also, there was brief mentioning of the physical realization when Mr. M elicited from one student that the maximum was at "2 minutes".

In terms of social meta-rules, there were many more prompts for students to "restate" what other students have done. Mr. M made every effort to use this talk move, taught in the PD, albeit somewhat inflexibly, whenever a student made a mathematical statement he deemed as important.

Discussion

In this paper, I have used the concept of "framing" to examine subtle changes in one teacher's practice over a period of one school year, through which the teacher was receiving support both through PD sessions and through in-school coaching. These changes, including the different aspects of frames, are summarized in table 1. As can be seen from Table 1, the shift from ritual to explorative frames in Mr. M's case was not uniform. The shift occurred first in the social meta-rules and only later in mathematical meta-rules. Also, both shifts seemed to occur first at the level of declarations and only later in practice. Thus, talk about social meta-rules aligned with explorative participation was evident already in the first interview, but enacted mostly in the last lesson. Talk about expectations for explorative mathematical meta-rules was evident in the last lesson pre-conference interviews, and only very slightly evident in the enactment of the lesson. This finding corroborates earlier findings showing that teachers are quicker to adopt declarations about explorative instruction than they are to enact it (Cohen, 2001) as well as our previous findings regarding social meta-rules of teachers trying to enact "dialogic" or "reform" instruction being more aligned with explorative instruction than mathematical meta-rules (Heyd-Metzuyanim et al., in review). Most importantly, the findings reveal that teachers' learning of new practices is not a matter of acquiring new practices on a "tabula rasa" of non-existing former practices. Rather, at first, new practices are seen through the old frame. As such, they gain unpredictable "twists", such as perceiving different syntactic procedures as "different solutions" sought after, or encouraging students to restate unimportant mathematical ideas.

Figure 1 -

 1 Figure 1 -Pythagorean Task given in lesson no. 1

Table 1 -Social and Mathematical aspects of frames in two lessons

 1 Teacher seeks students "understanding", not just rule following. A small portion of students' work indicates connections between multiple realizations of the quadratic function. Still ritual meta-rules in teacher's discourse still dominate instruction.

		Social (subjectifying) meta-rules Mathematical meta-rules
	Lesson	Beginnings of explorative social	Ritual: Meta-rules only have to do with recalling
	1	meta-rules through teacher's	facts and procedures from memory. Mathematical
		declarations (students expected to	signifiers are detached from the object they are
		persevere, work together). In	representing. Students' work indicates syntactic
		practice, most of the talking is	mediation with no connection to geometrical
		done by the teacher.	realizations.
	Lesson	Explorative social meta-rules	Beginnings of explorative mathematical meta-
	4	are more dominant. Students	rules:
		are asked to restate each other's
		ideas. Two students explain their
		thinking to the whole class.
		Evidence that some student listen
		to each other.

Proceedings of CERME10

Acknowledgments

This study was supported by a Small Spencer grant no. 201500080 and by Lauren Resnick's postdoctoral fellowship, University of Pittsburgh.


domain: math shs

text:
Thomas Hahn
email: thahn@mathematik.uni-kassel.de

Andreas Eichler
email: eichler@mathematik.uni-kassel.de

Changes in beliefs of teachers on a PD course after reflecting on students' learning results

Keywords: Beliefs, belief change, professional development, reflecting students' learning results

In this paper, we discuss results of a qualitative part of a research project aiming to investigate the impact of a professional development course with a specific focus on reflecting of students' learning results on teachers' beliefs towards teaching and learning of mathematics. We refer briefly to some aspects of teachers' professional development. Afterwards, we discuss teachers' beliefs as the main theoretical construct for our research. Data for this paper about teachers' beliefs were collected by semi-structured interviews with three teachers after the professional development course. Results show that teachers report in interviews changes referring to teaching and learning of mathematics from a transmission-oriented view to a more constructivist-oriented view.

Introduction

There is a consensus that university studies and internships are not enough to prepare future teachers for all challenges in their professional career [START_REF] Mayr | Lehrer/innen als zentrale Ressource im Bildungssystem: Rekrutierung und Qualifizierung[END_REF]. For this reason, professional development (PD) is understood as being a key factor for innovating and reforming mathematics teaching in school [START_REF] Garet | What makes professional development effective? Results from a national sample of teachers[END_REF]. However, every change seems to be dependent on specific characteristics of PD: [START_REF] Desimone | Improving impact studies of teachers' professional development: Toward better conceptualizations and measures[END_REF] summed up five key features of high quality teacher PD. She described that PD will be effective, if a PD course is content-focused, enables active learning, is coherent, has a critical duration and if teachers take part in a PD course collectively. In addition, [START_REF] Franke | Understanding teachers' self-sustaining, generative change in the context of professional development[END_REF] reported that reflecting students' learning results makes teachers' learning sustainable. Reflecting students' learning results means, that teachers collect in distance phases of a PD course students' results when working in a specific learning environment or working on a specific task developed in the PD course. Afterwards these students' learning results were the basis of the next face-to-face meeting in the PD course.

Although research identified several features of PD, in most cases the effectiveness is not clear [START_REF] Yoon | Reviewing the evidence on how teacher professional development affects student achievement[END_REF]. In addition, it is not clear how they influence teachers' learning. For example, [START_REF] Franke | Understanding teachers' self-sustaining, generative change in the context of professional development[END_REF] could show that a mix of several characteristics including reflecting students' learning results is effective for teachers' learning, but they did not focus on reflecting students' learning results as a single variable. In a qualitative design, for example also [START_REF] Strahan | Promoting a collaborative professional culture in three elementary schools that have beaten the odds[END_REF] found that teachers reflecting on students' learning results increased students' achievement on elementary school level. In addition, [START_REF] Schorr | Impact at the student level: a study of the effects of a teacher development intervention on students' mathematical thinking[END_REF] showed that students' achievement increases when teachers completed a PD including the analysis of students' problem solving processes. Like a conclusion, [START_REF] Little | Looking at student work for teacher learning, teacher community, and school reform[END_REF] state that reflecting students' learning results has the potential to bring students more explicitly into deliberations of teachers. However, more research is needed to understand how teachers learn from reflecting students' learning results and whether it impacts on teachers' professional competence including particularly teachers' beliefs.

To investigate the efficiency of reflecting students' learning results in PD courses on the teachers' knowledge, beliefs or motivation, the study as a whole considers two PD courses with a quasi-experimental setting. Content, teacher trainer and learning time were mostly the same in both courses. However, we integrated reflecting students' learning results in the first PD course, but not in the second. In this paper we will not refer to quantitative results of our research that we reported elsewhere (e. g. [START_REF] Hahn | Einfluss der Reflexion von Schülerdokumenten in Lehrerfortbildungen auf fachdidaktische Aspekte der Motivation[END_REF]. We further do not refer to differences between the effects of the two PD groups, but we emphasize results of a qualitative interview study including three teachers of the PD course with reflecting students' learning results. In these interviews, we primarily refer to the teachers' beliefs concerning the teaching and learning of mathematics.

Beliefs about teaching and learning of mathematics

Beliefs can be defined as "psychologically held understandings, premises, or propositions about the world that are thought to be true. […] Beliefs might be thought of as lenses that affect one's view of some aspect of the world or as dispositions toward action." (Philipp, 2007, p. 259).

For our research a crucial question is whether beliefs could be viewed to be stable or changeable. Partly, researchers use stability as a part of their definition of beliefs [START_REF] Fives | Spring cleaning for the "messy" construct of teachers' beliefs: What are they? Which have been examined? What can they tell us[END_REF][START_REF] Liljedahl | Stability of beliefs in mathematics education: a critical analysis[END_REF]. In contradiction to that, there are studies which demonstrate belief change by a special intervention. For example, [START_REF] Decker | The relationship between quality of discourse during teacher induction classes and beginning teachers' beliefs[END_REF] reported changes referring to teaching and learning of preservice teachers and teacher trainees. [START_REF] Liljedahl | Stability of beliefs in mathematics education: a critical analysis[END_REF] analyzed related literature and conclude that there is a different meaning of stability in research studies. As a result, they propose to avoid stability in the definition of beliefs. For them, belief change is a natural process that requires a sufficient extent of time. For this reason, in this research study beliefs are described as changeable.

A further crucial question concerns the definition of the belief object that should be changed by PD. Beliefs can refer to a special subdomain of mathematics [START_REF] Eichler | Domain-specific belief systems of secondary mathematics teachers[END_REF], to mathematics itself, teaching and learning of mathematics or an overachieving orientation that is independent from a subdomain [START_REF] Staub | The nature of teachers' pedagogical content beliefs matters for students' achievement gains: Quasi-experimental evidence from elementary mathematics[END_REF]. Beliefs referring to teaching and learning of mathematics can be divided into two paradigms: the transmission-oriented and the constructivist-oriented view of learning [START_REF] Fives | Teachers' beliefs about teaching (and learning)[END_REF]. Transmission-oriented beliefs of teaching imply that knowledge is directly transmitted from teacher to the learners and learners absorb all information. In this case, learners are passive recipients. For this reason, teachers' role is to prepare all information for students to enable an effective storing and an optimal recall. In contradiction, the constructivistoriented view of learning reflects the active role of learners as constructers of their own knowledge structures. In this case, students learn new information based on their existing knowledge and beliefs to enable an integration of information in their mental networks [START_REF] Decker | The relationship between quality of discourse during teacher induction classes and beginning teachers' beliefs[END_REF][START_REF] Staub | The nature of teachers' pedagogical content beliefs matters for students' achievement gains: Quasi-experimental evidence from elementary mathematics[END_REF]. For this reason, teachers take a role as constructor of learning environments that enable students to learn self-directed. [START_REF] Voss | Mathematics teachers' beliefs[END_REF] proposed evidence that these two dimensions are not the endpoints of a continuum. Instead of this, the authors proposed to understand these two dimensions as two distinct, negative correlated dimensions. These dimensions can be assessed on different scales. For this reason, it is possible that teachers have a high extent of both views on teaching and learning of mathematics.

Based on the definition of beliefs and the two main aspects of beliefs for our research, we primarily focus on the following research question:

Which influence show the PD course with reflecting on students' learning results on teachers' beliefs referring to teaching and learning of mathematics out of teachers' perspective?

Methods

We regard three groups of teachers in our study as a whole. Two groups of teachers were enrolled in a PD course that focused on problem solving and modelling in secondary school. An ongoing task for both PD courses was to develop tasks that meet different criteria of problem solving and modelling. The first group further was asked to give these tasks to their students and to collect the results of students' work that we call results of students' learning. These results of students' learning were the basis of the next face-to-face phase of the PD course. Teachers of second group were asked to improve problem solving tasks and a third group of teachers did not get any intervention.

We conducted pre-and posttests to measure the efficiency of the mentioned specific aspect of teachers' PD, i.e. reflecting students' learning results. Further we conducted interviews with the teachers. In this paper we regard three teachers of the first group who took part in the interview session voluntarily. The interviews took place about one month after the last meeting of the PD course. In semi-structured interviews the teachers were asked to report about their changes towards beliefs of teaching and learning of mathematics. All teachers are teachers of upper secondary schools and at the age of 40 to 50 and were women. These teachers could be representative for the whole group, because it consists of 21 teachers at the mentioned age. In addition, seventeen of these teachers were women.

We analyzed the interviews with a coding method including deductive and inductive codes [START_REF] Mayring | Qualitative Inhaltsanalyse: Grundlagen und Techniken[END_REF]. The deductive codes were based on existing research referring to teaching and learning of mathematics considering the transmission-oriented and the constructivist-oriented beliefs. According to both types of beliefs, we created codes for teachers' answers. In this context, we distinguish between teachers' beliefs before and after the professional development course. The distinction is based on hints in teachers' statements which enables us to match beliefs to the appropriate point in time. In addition, we analyzed the role of "reflecting students' learning results" for teachers' learning during the PD course.

Results

The results section is structured into three parts. At first, we want to show how teachers' statements are coded according to beliefs about teaching and learning of mathematics. Second, we sum up beliefs of the three teachers. And third, the effects of "reflecting students' learning results" are considered.

In the interviews teachers were asked to report about their beliefs before and after taking part in the professional development course. In particular, they should consider changes in their statements. For example, Mrs. B states:

Mrs. B:

[…] it has changed that the tasks are different. Students should argue more and I do not have to work off stacks of tasks. I can work off all facets more determined and I do not have to say that I must work off a model and then practice, practice, practice ….

Mrs. B reported about changes in her beliefs about teaching and learning mathematics. We interpreted her statement as follows: At the end of her statement, she mentioned that she now does not have to introduce a mathematical model followed by many exercises in the lessons. Furthermore, she reported on working off stacks of tasks before the PD course. Both parts of the statement are coded as transmission-oriented beliefs, because she emphasized practicing as repetition of information or procedures which is represented in stacks of tasks. In addition, the mentioned parts of the statement were coded as beliefs at the beginning of PD course, since she reported about changes during the PD (see table 1). In the first part of her statement, she considered a change in tasks and argumentation in her classroom. She also mentioned that she can work off the mathematical ideas more determined. These parts of the statement were coded as constructivist-oriented beliefs, because Mrs. B reported a student centered teaching style where students are asked to argue about mathematical concepts and to talk about mathematical problems with each other (see student-oriented perspective in teaching in table 2). In addition, these parts of the statement were coded as beliefs after taking part in the PD course, because in the whole statement Mrs. B reported about changes. In conclusion, the statement of Mrs. B shows a belief change from a transmission-oriented view to a constructivist-oriented view of teaching.

The three teachers showed similarities in reporting aspects of a transmission orientation and a constructivists orientation. A code that seemed to be crucial for all three teachers, but was not included in the quotation of Mrs. B, was expressed by Mrs. C:

Mrs. C: Students should do more and I have to restrain myself a little bit more.

We interpret the statement as following: Mrs. C also reported changes in her beliefs. The whole statement was coded as constructivist-oriented belief, because she wanted the students to be more active in her classroom when they are learning mathematics. In addition, Mrs. C reports on restraining herself in lessons. This is in line with the constructivist view, because on this perspective, teachers are creators of learning environments and students shall learn self-directed. For this reason, it is necessary that teachers shall restrain themselves in lessons. This statement was coded as beliefs after taking part in PD course, since "more" indicates that she had other beliefs at the beginning of PD. In particular, the beliefs at the beginning compared to those reported in the statement included that Mrs. C was more in the center of the lesson and students were more passive which are in line with transmission-oriented beliefs.

The results of the analysis of the three teachers' beliefs showed that they were transmission-oriented before they took part in the PD course. However, they seem to change their beliefs towards a constructivist-orientation during the PD course. The teachers also reported existing constructivistbeliefs they had at the beginning of PD course, but they stated changes towards more constructivist beliefs while they reduced the strength of transmission-oriented beliefs. Table 1 shows the results of coding for beliefs before teachers take part in the PD course:

Mrs. A Mrs. B Mrs. C repetition of information (practicing) X X X frontal teaching X X X exact instruction X X
teacher is in center of lesson X X X Teachers reported in the interviews that they changed their beliefs referring to teaching and learning from a teacher-centered perspective to a more constructivist-oriented teaching perspective. Both tables show different beliefs of teachers before and after the PD. These tables do not imply that all teachers had only teacher-centered beliefs at the beginning of the PD. They reported also that they had constructivist beliefs. For example, the statement of Mrs. C shows that she has more constructivist beliefs about teaching. This does not imply that she has not had constructivist-oriented beliefs at the beginning of the PD course. In her statement, Mrs. C only reported about more constructivist-oriented beliefs after taking part in PD. In addition, all the coded statements show that they also have transmission-oriented beliefs after PD course, but they report to think of teaching with more studentcentered beliefs. Note, both tables only show predominant beliefs of teachers before and after PD. For this reason, peripheral beliefs about teaching and learning of mathematics are left out.

The results about beliefs show changes. For this reason, we want to know how reflection of students' learning results impact on teachers learning. In this context, Mrs. A and Mrs. C state:

Mrs. A:

[…] I have learnt a lot about my students and I have also learnt a lot about myself and for this reason I have tested some things.

Mrs. C: I think it was good. The students' learning results show how other teachers proceed in teaching, which approach they use and how they describe. Within this action, you could take new ideas that I found in students' solutions. […].

The statements of the three teachers show that "reflecting students learning results" were used to reflect the own practice of teaching. In this context, these teachers learnt about characteristics of their students and about themselves as teachers. As a consequence, Mrs. A tested some ideas contained in the PD course. In addition, the three teachers recognized other teaching styles when they reflected on students' learning results. For this reason, they also reflected about the practice of other teachers and their teaching approaches. In particular, they also reflected about their own practice compared to the one of others to get new ideas (Mrs. C). These statements are typical for all three teachers. According to this, the reflection of teaching style could be understood as one factor that result in changes of beliefs about teaching and learning.

Discussion

The data from the interviews indicate that the three teachers changed their beliefs from a more transmission orientation to a more constructivist orientation of teaching and learning of mathematics. This is shown by statements of teachers after taking part in the PD course. The phrases in both tables consider the aspects of both beliefs about teaching and beliefs about learning. Based on the analysis in the results section, statements of table 1 refer to transmission-oriented beliefs and statements of table 2 to constructivist-oriented beliefs (e.g. [START_REF] Fives | Teachers' beliefs about teaching (and learning)[END_REF]. In fact, teachers focused on constructivist-oriented beliefs, but they still expressed some transmission-oriented beliefs. This is in line with research findings of [START_REF] Voss | Mathematics teachers' beliefs[END_REF], who supposed that both types of beliefs can co-exist.

The statement of Mrs. C supports this assumption. Although she expressed transmission-oriented beliefs before the PD, she said that students should be more active, which indicates that she has had constructivist-oriented beliefs at the beginning of the PD.

Belief changes in teacher education are also shown in the study of [START_REF] Franke | Understanding teachers' self-sustaining, generative change in the context of professional development[END_REF] for teachers on primary level who examined a student-centered framework in their PD. In this context, our results are similar to those of Franke et al. on secondary level, because we also used reflecting students' learning results to emphasize student-centered teaching. Furthermore, the teachers reported that they used students' solution to reflect their own teaching and to get to know information of other teachers' approaches and also about their students. This is in line with the results of [START_REF] Little | Looking at student work for teacher learning, teacher community, and school reform[END_REF], because teachers consider the ideas of their students in their deliberations more strongly after taking part in the PD course. In addition, belief change caused by reflection is reported by [START_REF] Decker | The relationship between quality of discourse during teacher induction classes and beginning teachers' beliefs[END_REF] in the way that there is a relationship between the extent of reflection and teachers' belief change. For this reason, it is possible that all teachers of this study changed beliefs because they reflect intensively about their own practice. [START_REF] Skott | Towards a participatory approach to "beliefs" in mathematics education[END_REF] stated that substantial new experiences are necessary to change beliefs. Concerning our research, we hypothesize that the intensive reflection of the own classroom practice includes the mentioned new experiences.

There are some limitations of this research. At first the interviews took place after the whole PD course. For this reason, teacher reports about the practice before the PD course can be influenced by the experiences of the PD. For a more precise research approach it would have been necessary to interview the teachers at the beginning of the PD. Considering the external circumstances in this research project, it was not possible to interview teachers, because they took part in PD voluntarily and the first meetings last about the whole day. In addition, these qualitative data provide no proof for "reflecting students learning results" as a feature of effective PD, but there are hints that beliefs can change and teacher reflect their own practice during this part of PD. Furthermore, the teachers of the second group were not interviewed. For this reason, it is not possible to indicate whether the teachers of the second group also changed their beliefs. In addition, we do not know whether second group teachers reflect on their own practice as deep as teachers of the first group.

Future research

This part of the whole research project considers qualitative data of the first PD course. Future research should use also quantitative data (measured by items of [START_REF] Staub | The nature of teachers' pedagogical content beliefs matters for students' achievement gains: Quasi-experimental evidence from elementary mathematics[END_REF]) to support the results of teachers' belief change during PD. This could also show whether reflecting students learning results is one element that is connected with teachers' belief change empirically. In addition, it is necessary to link the qualitative data analyzed in this paper and the quantitative data which will we analyzed in the future. Combination of both resources can show the positive impact of reflecting students' learning results on teachers' knowledge, beliefs and motivation.

Table 1 : Predominant beliefs before taking part in PD course (mentioned by teachers)

 1

	Thematic Working Group 18

Table 2 : Predominant beliefs after taking part in PD course (mentioned by teachers)

 2 The following table shows statements of teachers referring to beliefs of teaching and learning of mathematics after taking part in the PD course.

	Mrs. A	Mrs. B	Mrs. C

Proceedings of CERME10


text:
Alison Godfrey
email: ag384@le.ac.uk

Factors influencing the developing effectiveness of early career teachers' mathematics teaching -initial findings

Keywords: Teacher characteristics, professional development, elementary school mathematics

Early career primary school teachers in the UK enter a varied and complex situation in terms of professional development opportunities and support for their ongoing career progression as teachers of mathematics. Literature suggests that the professional development a teacher receives impacts on both their subject knowledge for teaching and their beliefs and attitudes to the subject. This multiple case study looks to fill gaps in the research undertaken in this area through detailed analysis of the personal perspectives of the participants over a two year period. This paper reports on the initial findings of a comparative analysis of the trajectory of two teachers to date. These teachers were in seemingly similar contexts for their first year of teaching, yet had very different experiences and held very different perspectives on their development.

Introduction

Students start a primary teacher education course in England with a range of qualifications in mathematics, a range of experiences in school mathematics and a range of attitudes and beliefs about the subject. During the course student teachers develop their subject and pedagogical knowledge and gain experience in mathematics teaching. They then enter a complex and changeable situation in schools in terms of provision for their ongoing development (Advisory Committee on Mathematics Education, 2013). Using a multiple case study approach, the aim of this study is to gain a deeper understanding of how the effectiveness of early career primary school teachers' mathematics teaching develops and what impacts on this, with a particular focus on each teacher's own perspective.

Theoretical background

The notion of effectiveness as applied to mathematics teaching is complex and ideas vary about what this looks like and even what its impact should be [START_REF] Cai | What is effective mathematics teaching? A study of teachers from Australia, Mainland China, Hong Kong SAR, and the United States[END_REF]. Although generally agreed that teaching can only be considered effective if there is an impact on those being taught, i.e. effective teaching leads to effective learning and gains in understanding [START_REF] Bryan | Comparison and contrast: Similarities and differences of teachers' views of effective mathematics teaching and learning from four regions[END_REF], the notion of 'understanding' is complex. [START_REF] Skemp | Relational understanding and instrumental understanding[END_REF] made a clear distinction between instrumental understanding, simply knowing rules and procedures at a shallow level, and relational understanding which enables pupils to build conceptual schemas. Most mathematics educators agree that this second type of understanding is the most desirable and teachers internationally agree that an indicator of mathematical understanding is that pupils can use this understanding to problem solve flexibly in a range of situations (Byran et al., 2007). The significance of understanding the connected nature of mathematics is very apparent (e.g. [START_REF] Askew | Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency 1995-96[END_REF] and is consistent with the notion of relational understanding. The concept of effective learning in mathematics being based on this type of understanding within the context of appropriate intellectual challenge is endorsed by current policy makers and teacher inspection systems in the UK (Department for Education (DfE), 2013;Ofsted, 2012) and is the definition used in this study.

Thematic Working Group 18

Proceedings of CERME10

The literature suggests that teachers' effectiveness is impacted by their subject knowledge and by affective dispositions including beliefs, attitudes, motivations and emotions. Definition of these constructs are debated in the literature and indeed there are a range of ideas as to how they overlap and directions of influence [START_REF] Lewis | Disaffection with School Mathematics[END_REF]. In agreement with Hannula (2011) I consider beliefs are both cognitively and affectively situated. As a starting point for discussions of attitude and emotion, I am adopting the three dimensional model of attitude from Di Martio and Zan (2010): emotional disposition towards mathematics, vision of mathematics (relational/instrumental) and perceived competence in mathematics.

Many studies, e.g. [START_REF] Ernest | Mathematics teaching[END_REF] and [START_REF] Askew | Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency 1995-96[END_REF], have looked at teachers' beliefs about the nature of mathematics and the impact these beliefs have on their teaching. [START_REF] Ernest | Mathematics teaching[END_REF], for example, argues the most effective teachers see mathematics as a dynamic, creative, problem solving subject and adopt a 'Facilitator' teaching approach.

Teachers' attitudes to the subject, including their emotional disposition, also impact on effectiveness. Particularly in primary teaching, teachers and student teachers find themselves teaching subjects that they have not necessarily enjoyed or been particularly successful at learning themselves. Many feel insecure with their subject knowledge in mathematics and recount their experience of mathematics at school as a subject that caused difficulties and even "real emotional turbulence" (Brown, 2005 p. 21). For many teachers therefore, mathematics is linked with negative emotions, particularly anxiety, and this can impact on their wider attitudes to it. It seems that teachers even protect pupils from mathematics and, in seeking to simplify it, emphasise the step by step procedures that are likely to lead pupils into developing instrumental understanding and potentially negative attitudes [START_REF] Hodgen | Emotion, identity and teacher learning: Becoming a primary mathematics teacher[END_REF], reducing their effectiveness.

A further influence on the effectiveness of teaching of mathematics is subject knowledge. Indeed, evidence suggests that the impact of beliefs and attitudes and subject knowledge are interdependent [START_REF] Askew | Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency 1995-96[END_REF]. Most recent research on teachers' subject knowledge for mathematics uses [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF][START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF] seminal papers as a starting point. Others have applied his ideas to mathematics teaching, debated which aspects are most essential and relevant, and sought to measure or evaluate them. [START_REF] Ball | Content knowledge for teaching what makes it special[END_REF] argue that there is knowledge that is specific to teachers of mathematics and that might have an identifiable impact on the effectiveness of their teaching; Specialised Content Knowledge. This includes, for example, the understanding needed to be able to explain procedures, to analyse errors and strategies, and to consider appropriate examples. [START_REF] Baumert | Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress[END_REF] conclude that pedagogical content knowledge (PCK) makes the greatest contribution to pupil progress, but weaknesses in mathematical content knowledge are not offset by greater PCK. [START_REF] Askew | Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency 1995-96[END_REF] found that it was not the formal qualifications or the amount of subject knowledge that the teachers had which was significant in the effectiveness of their teaching, but rather the connectedness of their subject knowledge "in terms of the depth and multi-faceted nature" of the meanings and uses of concepts in mathematics (p. 69). Other authors highlight the importance of pupil voice and the ability of the teacher to choreograph classroom discourse as key characteristics of effective teaching in mathematics (e.g. [START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF], and [START_REF] Barwell | Discursive psychology as an alternative perspective on mathematics teacher knowledge[END_REF] discursive psychology perspective emphasises that knowledge is contextual and can be changed or reconstructed accordingly. There is much critiquing of Shulman's and Ball's ideas but there seems to be general agreement that the subject knowledge needed for effective teaching goes further than just having a strong conceptual knowledge of the subject being taught.

There is considerable evidence that a teacher's depth of reflection and their beliefs and attitudes towards mathematics are a crucial influence over their trajectory of development [START_REF] Martino | Me and maths': Towards a definition of attitude grounded on students' narratives[END_REF][START_REF] Turner | Growth in teacher knowledge; individual reflection and community participation[END_REF]. [START_REF] Hodgen | Emotion, identity and teacher learning: Becoming a primary mathematics teacher[END_REF] and [START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF] suggest that the development of some teachers is hindered by their linking of mathematics with emotion and that, for some teachers, professional development activities and goals should be much more about changes in their beliefs and attitudes than about improving their subject knowledge.

Teachers' work is within a social context, [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 this can have a significant impact on the nature of a teacher's professional development. The community within the school might have a range of different foci and agendas and there seems to be a wide variation in practice between schools. The very structured collaborative approach to teacher development in China seems high effective although dependent on the considerable time given to teachers to discuss, prepare and analyse their work [START_REF] Paine | Teachers working together: A dialogue on organizational and cultural perspectives of Chinese teachers[END_REF]. In contrast Ball, Ben-Peretz &Cohen (2014) consider that in the US most teachers work in isolation and the potential benefits of sharing good practice are lost. In England, although the need for quality on-going professional development opportunities for primary teachers is recognised and highlighted by Ofsted (2012), the current context is of variable provision in formal professional development (ACME, 2013).

In summary, the literature suggests that a teacher's trajectory as a teacher of mathematics is influenced by the interaction of their beliefs and attitudes, their subject knowledge and the professional development they receive, through both formal education opportunities and personal reflection, and these factors therefore influence the effectiveness of their teaching and the effectiveness of their pupils' learning. My study sits within this theoretical framework and seeks to extend the existing literature particularly though highlighting teachers' own perspectives on this process.

Methodology

A multiple case study approach is being employed to follow the trajectories of a small sample of teachers as they progress into their first two years as a qualified teacher. Four participants were chosen for the pilot study, two with mathematics qualifications beyond the minimum required for primary school teachers in England. An initial interview at the end of their one year postgraduate teaching course focused on their relationship with, and attitude to, mathematics and their progress in teaching the subject as a student teacher. To facilitate discussion, [START_REF] Lewis | Disaffection with School Mathematics[END_REF] idea of a graphic display was adopted; participants were asked to draw and explain their relationship with mathematics over time. This gave insights into participants' attitudes to mathematics and their perspectives as learners of the subject as well as in their student teacher role. Participants were able to reflect on how their relationship with the subject influenced their current teaching approaches. Twice yearly interviews, including further graphing at the end of each year, and discussion of documentation related to their progress as early career teachers, such as formal observation feedback, provide evidence of their ongoing development as teachers of mathematics. Interview questions have been designed to probe about the participants' beliefs, attitudes and subject knowledge for teaching mathematics, what they perceive 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 describe two particular lessons: their chosen 'best' and 'most challenging' lessons since the previous interview, providing insights into what they consider to be effective and ineffective teaching and their subject knowledge. In addition, they keep records of their professional development in teaching mathematics.

To begin to analyse the data, a mind map has been drawn of each data collection point for each participant. The mind map contains evidence of the participants' changing context and development as a teacher of mathematics, including both factual and non-factual information, and also the participant's interpretation and my interpretation of this evidence. The mind maps allow for a thematic approach, enabling identification of concepts and themes such as 'awareness of self as a mathematician' and 'priorities in teaching mathematics'. In regard to their relationship with mathematics and their development as teachers of the subject, the participants have stories to tell and thus narrative analysis techniques are being used to support analysis of these stories. Whilst these techniques have enabled useful analysis of the early findings, the next stage is to further research and develop these analysis methods so that data can be analysed increasingly effectively and efficiently.

Results and discussion

In this paper the focus is on the early findings of a comparative analysis of the trajectory in the first year of the study of two of the teachers, Gina and Rama (pseudonyms), who in their first year as qualified teachers were in seemingly similar contexts: they both taught children aged 5-6 years in schools with a two class entry, working alongside more experienced colleagues and teaching mathematics to their own, mixed attainment, classes. Rama has a stronger mathematics background and also chose to study a mathematics specialism as part of the teaching course, but both students achieved the highest teaching course grade in all areas of the Teachers' Standards (DfE, 2011). In their interviews, both were able to identify many ways in which they had evolved as teachers of mathematics to date, but they answered questions in ways that revealed very different perspectives on their development. In addition, their professional development records indicated that they received very different opportunities and approaches to their professional development. This raises questions about why these are so different and the extent to which this is school dependent or dependent on the approach and philosophy of the teachers themselves.

Descriptions of each participant's relationship with mathematics and how this had evolved over time revealed interesting and complex relationships (Figure 1) and it was clear that past experiences impacted their current thinking. Whilst the graphs drawn are not directly comparable as the interpretation of the vertical scale was left open, the trajectories illustrated and discussion of these provide some scope for comparison. The two peaks in Gina's relationship with mathematics indicate two different perceived aspects of success in mathematics. Firstly she recalls being successful in the subject in Year 9 (aged 13-14) when she felt she responded well to the high expectations her teachers had of her. The second period of success, during the teaching course, was due to her own development of the conceptual understanding that the literature suggests is essential for learners to gain for long term, secure understanding of mathematics. This enabled her to reflect on her previous period of 'success' which she then realised was based on superficial learning: Gina: I could do the methods, but I didn't understand them.

Thematic Working Group 18 Proceedings of CERME10 Gina's personal experience seems to have given her a particular focus in her own teaching of mathematics, particularly how strategies she is aiming to use in her teaching could prevent the children taking the path of superficial learning that she had followed. Discussing how she developed her teaching in her first few months as a qualified teacher, for example, she identified the use of representations as significant and something that she perceived as missing in her own learning: Gina: And the pictorial things, definitely. I don't think I've ever done that as a maths learner myself and I think maybe if I had it would have been much more easy.

Rama's mathematical background also seems to have impacted on her priorities in teaching mathematics. Although she finished her studies with a degree where she used mathematics, she finished primary school not enjoying the subject and lacking in confidence. She seems to have been a shy and hesitant learner, fearful of getting the wrong answer in a subject she saw as right or wrong. At secondary school she was initially placed in lower sets but talked with great enthusiasm about her Year 11 class (aged 15/16), a top set, where the environment of the classroom was such that she felt she could make mistakes without fear. She described too a change at this point from memorising how to do certain methods to taking ownership of her learning and finding her own ways of solving problems. Priorities identified by Rama in her teaching include the importance of children enjoying mathematics alongside gaining conceptual understanding, both aspects that for her were missing until the later stages of her schooling.

It seems that Gina and Rama entered two very different teaching communities with different agendas which also impacted on the nature of their professional development [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 graphs drawn at the end of their first year of teaching revealed very different perspectives on their year (see Figure 2). Rama's saw her journey as a teacher of mathematics as a smooth upward trend, with occasional dips when she taught poorer lessons. Whilst having few formal professional development opportunities, she indicated that she is confident in her own subject knowledge and in independently planning and teaching the subject. Gina described a much more structured and intensive programme with regards to the support she received. However, her graph illustrates her perspective that she has had a sometimes difficult year and narrative analysis revealed, interestingly, that the intense involvement of her school in her professional development seems to have led to a crisis in her confidence and a turbulent second half of her first year of teaching. This analysis highlights the emotional impact on Gina of the series of intervention events that followed on from a formal observation of her teaching that she labelled several times as a 'disaster', and then her subsequent recovery. In her story particular emphasis is evident of the impact on both the perceived competence and emotional disposition aspects of her attitude (Di Martino and Zan, 2010), with a consequent impact on her confidence in her ability to teach mathematics:

Gina: It seemed no matter what I was doing to change, I was still getting negative feedback and it was disheartening. There was a big period of time when I literally dreaded every single lesson, because you just think "What could happen? I don't know what I'm doing"

The records of professional development kept by Gina and Rama over the year confirm that Rama's in school support has been much more informal and she places a high priority on targets she generally sets for herself, seeking to improve the quality of her teaching through a reflective approach [START_REF] Turner | Growth in teacher knowledge; individual reflection and community participation[END_REF]. In contrast to Gina's record of events, her record is a series of targets. Contrasts in their priorities in planning and teaching of mathematics, and indications of their subject knowledge have been evident through different ideas about the characteristics of effective teachers and descriptions of their best and most challenging lessons. It seems that both teachers are seeking to teach with a relational approach, with Rama at this stage in a stronger position regarding subject knowledge. It is interesting that end of year assessment information indicates that children in both Rama and Gina's classes progressed well in their mathematical understanding over the year; this seems to have contributed to Gina finishing the year in a more confident frame of mind.

Conclusion

The findings of these case studies to date confirm my analysis of the literature in that there are two related and interwoven, but distinct, categories of factors that might influence an early career teacher's trajectory in relation to the effectiveness of their mathematics teaching -those that are related to the teacher themselves, in terms of beliefs, attitudes and subject knowledge and those that are related to their teaching context. Gina's experience during her first year of teaching provides a particularly interesting example of professional development that impacts in complex ways on beliefs and attitudes to the subject. Although it is too early to draw more than very tentative conclusions, it is interesting to ponder to what extent Gina and Rama's trajectories would have been different had they been in one another's school, and this leads to potential implications for initial teacher education.

These case studies align with current professional documentation (ACME, 2013) to suggest that there is a lack of uniformity in teachers' professional development experiences despite there being systems in place in the UK promoting uniformity. The existence of these issues has implications for the preparation of student teachers in Initial Teacher Education (ITE) and suggest that ITE providers could seek to more explicitly discuss differences and similarities in primary school approaches to teacher development and further strengthen their policies of students gaining experience in a range of contrasting placements, within which they can not only participate in whole school professional development events, but also talk with teachers about their development experiences.

Figure 1 :

 1 Figure 1: Gina and Rama's relationship with mathematics, drawn at end of teaching course

Figure 2 :

 2 Figure 2: Gina and Rama's relationship with mathematics during their first year of teaching

Proceedings of CERME10


text:
Elisabeta Eriksen
email: elisabeta.eriksen@hioa.no

Bjørn Smestad
email: bjorn.smestad@hioa.no

Annette Hessen Bjerke
email: annette.hessen@hioa.no

Camilla Rodal
email: camilla.rodal@hioa.no

Yvette Solomon
email: y.solomon@mmu.ac.uk

From emergency sirens to birdsong -Narratives of becoming a mathematics teacher

Keywords: Preservice teacher education, mathematics education, mentors

As part of a larger research project, we asked third-year PSTs to reflect on what they had learned about being mathematics teachers of the teacher education programme. The reflections were intended as presentations for first-year PSTs. In this article, we analyse the films by the third-year PSTs to understand the messages more experienced PSTs choose to communicate to novices. Concepts from Gert Biesta are the framework for the content analysis, and we find a complex picture of how qualification, socialization and subjectification interact in the narratives.

Introduction

This article presents partial results from a research project examining pre-service teachers' (PSTs) developing identities as mathematics teachers, and, in particular, their experiences of mathematics in school placement. Previously published results from the project report how first-year PSTs value what they learn from mentors in practice more than the 'theoretical' input of the university based courses, not seeing the theoretical knowledge as transferable into teaching practices [START_REF] Bjerke | A Tripartite Cooperation? The Challenges of School-University Collaboration in Mathematics Teacher Education in Norway[END_REF]. In one intervention addressing the challenge, thirdyear PSTs presented to the first-year PSTs films describing experiences of becoming mathematics teachers in the course of the first three years of the programme. In this article, we analyze the thirdyear PSTs' presentations to understand how they view their own emerging professional identities.

Our research question is: What domains of their educational experiences do PSTs highlight in their presentations of their first three years in mathematics teacher education?

Research background and theoretical underpinnings

While learning to teach is about acquiring professional knowledge and skills, it is also about developing a teacher identity [START_REF] Haniford | Tracing one teacher candidate's discursive identity work[END_REF]. Adding to the identity work of experienced teachers, PSTs have the responsibility of successfully positioning themselves in relation to their teacher education programmes and cooperating teachers [START_REF] Haniford | Tracing one teacher candidate's discursive identity work[END_REF]. Identity formation is driven by the individual's goal state of what he/she wants to become [START_REF] Smeby | Connecting to professional knowledge[END_REF]. [START_REF] Biesta | Becoming educationally wise: Towards a virtue-based approach to teaching and teacher education[END_REF] discusses these processes under the headings of qualification, socialisation and subjectification, and we choose his concepts as the starting point of our analyses of PSTs' narratives.

For [START_REF] Biesta | Becoming educationally wise: Towards a virtue-based approach to teaching and teacher education[END_REF], all education (including teacher education) is a question of judgement, because educators' decisions about the purpose of what they do occur within domains that may be in synergy with each other, but may also be in conflict. He notes three such domains, which interact and overlap: the domains of qualification, socialisation and subjectification. Qualification is about knowledge, skills and dispositions; socialisation and subjectification can be seen as opposites: while Thematic Working Group 18 Proceedings of CERME10 socialisation applies to the induction of novices into existing practices, subjectification denotes how education contributes to a process of individuation, of becoming an independent subject. Educational judgements are underpinned by an understanding of the interdependencies between the three domains. In teacher education, the situation is more complex still: the purpose should not just be PSTs own qualification, socialisation and subjectification, but also to enable PSTs to become 'educationally wise', aspiring towards virtuosity in making educational judgements themselves. Such judgements are situated: they are made during the practice of teaching, and cannot be set out in advance, or in general -they are rooted in concrete situations and relate to the need to handle tensions and see possible synergies. To become 'educationally wise', one needs experience, together with opportunities to see more experienced others in action, and to discuss those actions in terms of the virtuosity of judgement which underpins them.

Method

The 32 PSTs in this study were enrolled in a four-year programme for primary school teachers (grades 1-7, ages 6-13) in Norway. They had chosen to continue with mathematics beyond the compulsory course spanning the two first years in teacher education, and were asked to look back on their mathematics education course and four school placements and to prepare group presentations describing their development as mathematics teachers in grades 1-7. They were asked to reflect on what they know now which would have been good to know in their first year; how they plan their mathematics teaching now compared to in their first year; what they have learned along the way; and what are the pitfalls and experiences to bear in mind for future mathematics teachers.

Six presentations were made -all short films (F1-F6) incorporating line drawings and sometimes photographs, with voice-over commentary and music. Five of the six films were organised as developmental stories from their novice anticipation and preparation of their first placement, to their reflective stance as third-year PSTs. The remaining film focused on four pitfalls for novice PSTs.

To analyse the data, we operationalised [START_REF] Biesta | Becoming educationally wise: Towards a virtue-based approach to teaching and teacher education[END_REF] concepts in terms of: descriptions of their knowledge, skills and dispositions; processes and accounts of making educational judgements and justification of judgement; and accounts of being and becoming a mathematics teacher, as exemplified below:

Qualification: References to knowledge, skills and dispositions, processes and practices from the teaching profession.

Socialisation: References to learning and to expectations in the school context. Subjectification: References to inner feelings, identity and becoming a teacher and to perception of self (as a teacher).

The presentations were transcribed and analysed in several steps. First, they were coded according to Biesta's terms qualification, socialisation and subjectification, synergies and conflicts between these domains, and professional judgement. Then disagreements among the researchers were resolved, and all presentations were re-read and coded by another member of the group. A decision was made to organise the analysis in two parts -one around PSTs' novice anticipation, and one around their reflective stance in third year. Finally, we re-read each presentation to make sure that important longitudinal messages were not lost in our attempt to organise the analysis in two parts.

Looking back at early experiences in school placement -Emergency sirens

Qualification

In the films, there are a few examples of what PSTs learned in the first year, for instance pieces of 'theory', such as the importance of using multiple representations, giving feedback, and using the "didaktisk relasjonsmodell" (F1) (a hexagon connecting elements relevant for lesson planning: topic, learning objectives, the pupils in that class, etc.). The first-year PSTs are, however, uncertain about how to put the knowledge into practice. The uncertainty about effective use of manipulatives is most usual (F1, F4, F5), but other dilemmas are also identified:

I also thought about all the theory I learned at university: Piaget's theory of stages of development, Bruner's representation theory and theory of scaffolding, Vygotsky's focus on cooperation, Bandura and his theories on motivation and self-efficacy. How should I use these theories to plan a lesson on fractions? [...] Which pupils should work together: the ones that are on the same level in terms of subject knowledge, or should the strong ones help the weak? (F1) Several of the films show students meeting concepts as an overwhelming mass of words (Figure 1). At the same time, there are many statements about elements of qualification the PSTs perceive as lacking, both in terms of knowledge ("Do I know enough about this topic?" (F5)) and of processes: However, the anxiety of seeing the mentor write "like there was no tomorrow" is followed by "I had so many questions" -suggesting that the mentor is regarded as a person to ask for advice, as well.

Subjectification

A basis for subjectification is the development of a certain degree of self-confidence. In the description of the first year, we see little self-confidence -uncertainty and fear dominates:

What if I don't succeed? (F5) I went from being one of the best in mathematics to being perplexed when the pupils asked me questions about the subject. (F1)

Conflicts and synergies between domains

The three domains of education overlap. These non-empty intersections are implicitly present in the films. There are clear examples that a perceived lack of qualification (being overwhelmed by new concepts and by making sense of these in practice) leads to a lack of self-confidence -"I felt unsure and very, very small" (F1) -which we regard as part of subjectification. This can also work the other way: lack of confidence leads to lack in qualification:

In the first school placement, I struggled a lot with getting the class to settle down. Later on I came to think it was because I did not feel like a confident and clear classroom manager. (F1) There can also be a conflict between qualification and subjectification, in the sense that learning more makes you aware of your shortcomings:

The more I learnt, the more I discovered what I didn't know. Inside the domain of qualifications there are interactions between elements. In one case, the confidence in mathematics is shaken by the practice of teaching: I went from being one of the best in mathematics to being perplexed when pupils asked me questions. (F1) At the same time, during the first year the process of lesson planning is weighed down by the awareness that there are many considerations to be taken. This is visible in form of the time that goes into writing a lesson plan (shown with clocks in the films), and the number of books that fill the desk in the process (Figure 2).

There can be a tension between socialisation and subjectification in meeting the mentor: in one example, the role model (supposed to provide socialisation) is so impressive that the PST's selfconfidence suffers:

The meeting with the mentor was scary. I saw him as a Superman who really knew his work. He was confident, clear and, not the least, had strong subject knowledge. (F1) With an emerging sense of agency, the PST questions the mentor's view and asks herself: "Should we always stick to the textbook?" (F3).

To conclude, there are synergies between (a lack of) qualification and (a lack of) subjectification, but also a conflict between qualification and subjectification, as well as between socialization and subjectification.

Practicing educational judgement

Judgement is difficult. A lack of self-confidence leads to a very detailed plan with little room for judgement on-the-fly.

As a first-year PST the plan for the lesson was a long script. We had written down word for word what to say during the lesson. We were dependent on this script and could not improvise along the way. We even planned how to explain simple mathematical things that we actually knew well. This is also about lack of experience and confidence as a teacher. (F6) At the same time, a lack of qualification translates into constraints on opportunities for judgement in the process of lesson planning:

It's natural to continue from where we left off, it's not like I have other suggestions. (F2)

Looking at their recent experiences in school placement -Birdsong

Qualification

Changes from first to third year are visible in all aspects of qualification, from subject knowledge and knowledge of students and teaching, to the practices of teaching. In terms of knowledge, some films refer to knowing more mathematics, but, in terms of mathematics pedagogy, the films stress that the understanding is deeper, the knowledge can be operationalised to a greater extent.

The process of lesson planning during the first year involved long hours dedicated to the task (F1, F4, F6), and resulted in long scripts produced for each lesson (F4, F6). The films highlight, in comparison, how much quicker lesson planning goes (F1, F6), and how much shorter the scripts become (F1, F4) by third year, but the films give different suggestions on how to take advantage of the reduced burden, from watching TV and playing with the dog (F6) to investing time and energy on the 'frills' of differentiation and using a variety of teaching methods (F1).

Teaching practices out of reach during the first year are now on the agenda (F1): motivating pupils, providing them with opportunities to feel both confident and challenged, seeing the individuals as well as the class as a whole, giving more room to children's contributions, and encouraging enquiry.

Socialisation

The main presence that embodies the socialisation component is the teacher mentor, although some PSTs also mention peers and other colleagues playing a role. At this stage the mentor has transitioned from a feared judge to a colleague (F1), in some cases a role model (F2, F4), although disagreements between the views of PSTs and their mentor may occur, for example regarding the role of textbooks (F3). However, adopting established practices of the teaching community, such as body language (F1) or ways of saying or doing things in the classroom, seems to be perceived by PSTs as a sign of having become teacher-like: I've even put together extra handouts [for those who might need another type of challenge]. (F2)

Subjectification

Through the journey from first to third year, the PSTs have grown into teachers who are aware that teaching is not just about what you know, it is about making choices about complex situations. As there are no deterministic answers to these dilemmas, neither objectively speaking nor in terms of what is the established way of the teaching community, these choices come down to the individual, they are drawing on the domain of subjectification: "We're more aware that there should be a reason behind our choices" (F4), "I understand my own thoughts" (F5). In their third year, we hear the PSTs stress the importance of trusting their own choices (F1, F2, F4), and being yourself (F2).

Planning lessons is now an altogether more positive experience, described with attributes such as joy, and belief in oneself. Importantly, some of the PSTs realize that becoming a teacher is a continuous process, and experimenting is a part of it: Don't be afraid to try out new things. (F2)

A lesson plan can never be too good. It's like a piece of silverware that you take out and polish from time to time. (F4)

Conflicts and synergies between domains

As PSTs become more comfortable as teachers (subjectification), some find reassurance in their theoretical knowledge (qualification) as well as their awareness of what is acceptable among teachers (socialisation):

Not everything has to be perfect [...]. The theory I used to think about while planning lessons in my first year is now under my skin. (F1)

The routines of teachers (socialisation) also contribute to being more successful in the practices of teaching, such as lesson planning (qualification): "You don't have to reinvent the wheel (F1)". There is an aspect of growing confidence (subjectification) when the PSTs reuse lesson plans they have had positive experiences with (F1).

Unlike in their first year, lesson planning in the third year takes less time (F1, F6) and the scripts for the lesson are shorter (F1, F4) or even disappear altogether ("We've thrown out the script", F6). The that, looking back on their first-year school placement, PSTs remember a combination of a lack of qualifications, unclear expectations from mentors and low self-confidence. The fear many PSTs report on, seems rooted in their low self-confidence and the unclear expectations. Although firstyear PSTs are allowed to try different approaches and to fail, the same combination of a lack of repertoire, uncertainty of the mentor's role and lack of self-confidence holds them back. By the third year their qualifications have increased, their role as PSTs is clearer and their self-confidence has grown. Because of this, they also find themselves practicing educational judgement more often.

Such narratives, perhaps in combination with logs from early placements, could be part of educational experiences, supporting PSTs' identity work. First-years might also benefit from watching the films, as not all challenges discussed can be dealt with by the teacher educators. While teacher education can and should make explicit what is expected of PSTs in their school placement, it cannot rush becoming educationally wise. However, we hypothesise that creating spaces where first-and third-year PSTs can discuss their experiences would contribute to the domain of socialisation and subjectification for both groups. Analyzing the students' contributions in terms of Biesta's concepts reveal the complex relationships between qualification, socialization and subjectification in teacher education. The three domains are interdependent, with conflicts and synergies which influences PSTs overall experience. More insight into these conflicts and synergies may contribute to better understanding of PSTs' experiences of their school placements.

  Figure 1: Knowledge in overwhelming amounts in the first year (F1, F4, F5) Socialisation The mentor is, naturally, the main role model for students in their first year of teacher education, and the mentor features in most of the (few) examples we find of socialisation when describing the first year. The PSTs are uncertain about what the mentor expects of them, other than using the "didaktisk relasjonsmodell" (mentioned above) which is common in Norway: As first-year students we used it [the didaktisk relasjonsmodell] slavishly (F6) ...I have to carry on from here [in the textbook], that must be what [the mentor] expects (F2) The mentor can be viewed as an evaluator: Shit! This took the entire hour. The mentor glares at me. It didn't go as planned. (F3)

  [...] Based on Piaget's theory I knew most of the pupils were at the concrete-operational stage. But which of Bruner's representations should I use? [...] Or should I use the strange Cuisenaire rods that I still haven't really gotten to grips with? (F1) Drowned in the curriculum he feels puzzled. What is most important? (F4)

Figure 2 :

 2 Figure 2: Lesson planning during first-year school placements (F1, F4, F6)In another example, a PST's attempt at making a choice outside of the textbook is struck down by the mentor:Hmmm….I think maybe we should stick to the textbook. (F3)

Proceedings of CERME10

change is attributed in general to an increase in confidence (subjectification) but in some cases also to an aggregated influence of all three domains: … more confident in myself and the mathematics, I know more about the pupils' level in mathematics, I have become a clear leader, I dare to make mistakes, I am better at dealing with things as they happen. (F1)

In another film, the three domains come together in synergy to express the PSTs' development:

By contrast with first year, when we used the the syllabus for the course a lot, we now have more knowledge of the subject and of pedagogy. We've become better at making use of our own knowledge, we cooperate more closely with colleagues. (F6)

The way these sentences are linked, makes it possible to interpret it as meaning that better qualification leads to better self-confidence (subjectification) which again leads to better cooperation with others (socialization).

Practicing educational judgement

Increased self-confidence by the third year is not synonymous with knowing just what to do: How can I connect algorithms and conceptual understanding? I need to be able to show them different strategies, to be sure as many as possible understand. How many strategies for division are there? Maybe they come with some I haven't thought of? Maybe some misconceptions will surface during the lesson? How can I then, in the best possible way, deal with this? (F4)

The difference from the first year is being able to deal with dilemmas, to practice professional judgement, guided by what they see as the goals of teaching:

There's still a lot to think about, but I understand my own thoughts now, I know where I'm heading (F5)

In the third year practicing educational judgement features as a defining factor of the PST-mentor relationship at this stage: the detailed scripts for lesson plans that were in the first year in part written for the sake of the mentor (and in part to boost one's confidence, to feel prepared) are now shorter. A mentor's voice sets expectations: Just show me that you are aware of the choices you make, and that you can argue for them (F2) During the third year, educational judgement is visible in reflections on one's own teaching: Teaching analysis draws on and at the same time feeds into the domain of qualification and perhaps also socialization. This way of assessing oneself -and the knowledge that you do it well -can be regarded as an engine for development also after graduation, it feeds into subjectification.

Concluding remarks

The titles of two subsections of the analysis reflect the soundtrack of a film where the experiences of first-and third-year school placements are introduced with emergency sirens and birdsong respectively. In terms of [START_REF] Biesta | Becoming educationally wise: Towards a virtue-based approach to teaching and teacher education[END_REF] framework, the overall picture the presentations paint is