Prospective teachers development of adaptive expertise

Teaching and Teacher Education 49 (2015) 108e117

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Prospective teachers development of adaptive expertise* Glenda Anthony*, Jodie Hunter 1, Roberta Hunter 1 Institute of Education, Massey University, PO Box 11222, Palmerston North, 4442, New Zealand

h i g h l i g h t s
Developing adaptive expertise evident in shift of focus from self to students.
Adaptive expertise evident in developing understandings of complexity of teaching.
Building formal theories of practice by engaging in everyday theories.
Practice-based pedagogies supported development of adaptive expertise.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 August 2014 Received in revised form 10 February 2015 Accepted 18 March 2015 Available online

This article responds to calls for graduating teacher standards that reflect a vision of teachers as adaptive experts. Drawing on prospective teachers' reflections of their learning within a mathematics classroom inquiry course, we examine the development of expertise as characterized by shifts in teacher focus from self to student and from simple to increasingly complex understandings about teaching and learning. We argue that the instructional dynamics linked to practice-based pedagogies within our teacher education program, inclusive of opportunities to experiment, risk-take, and engage directly with learner outcomes, supported the development of prospective teachers' professional stance aligned to adaptive expertise. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Adaptive expertise Teacher education Prospective teachers Practice-based pedagogies

1. Introduction Seen by policy makers as both the cause of and a solution for education problems, teacher education is frequently criticized for not producing teachers of sufficient quality while simultaneously being viewed as “an ideal site for increasing teacher quality, providing it is subject to reform” (Ell & Grudnoff, 2012, p. 79). In many parts of the world, the desire to ensure the preparation of ‘quality’ teachers has prompted “unprecedented and politicized attention to teacher preparation/certification and the policies and accountability systems that govern them and measure their effectiveness” (Cochran-Smith & Villegas, 2015, p. 10). In reference to graduating standards, we, like others (e.g., Fairbanks et al., 2010; Griffiths, 2013), ask what is necessary and possible beyond knowledge and skill sets? Discussions in the New Zealand context

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Research conducted at Massey University, New Zealand. * Corresponding author. Tel.: þ64 63569099. E-mail addresses: [email protected] (G. Anthony), [email protected]. nz (J. Hunter), [email protected] (R. Hunter). 1 Tel.: þ64 63569099. http://dx.doi.org/10.1016/j.tate.2015.03.010 0742-051X/© 2015 Elsevier Ltd. All rights reserved.

are currently informed by Aitken, Sinnema, and Meyer's (2013) proposed Teaching for Better Learning model that details a set of graduating standards structured around “a series of inquiries designed to establish learning priorities and teaching strategies, examine the enactment of strategies and their impact, determine professional learning priorities, and critique the education system” (p. 4). Envisaging graduating teachers as “inquiring professionals who are focused on better learning for themselves and their students” (p. 30), Aitken et al. (2013) argue that standards must “emphasise the context-dependent nature of effective teaching and, therefore, adaptive expertise as the hallmark of a professional teacher” (p. 4). For mathematics education particularly, where reforms demand significant shifts towards inquiry-based mathematics learning communities (Hunter & Anthony, 2011), adaptive expertise is viewed as essential in order to minimize the possibility of beginning teachers' socialisation to the more familiar transmission modes of mathematics teaching. However, despite agreement that adaptive teaching expertise is a worthy goal of teacher education (Hammerness, DarlingHammond, & Bransford, 2005) little is currently known about expectations of adaptive expertise capabilities for beginning teachers,

G. Anthony et al. / Teaching and Teacher Education 49 (2015) 108e117

nor about ways to develop adaptive expertise within initial teacher education contexts (Soslau, 2012). In seeking to add to current understandings this paper explores two prospective teachers' (PTs') professional learning associated with expertise as marked by shifts in beliefs and values concerning learners/learning and teachers/ teaching. Responding to Thames and Zoest's (2013) call for more research that “deliberately presses into the instructional dynamic” (p. 585) associated with teacher learning, we examine the mediating influences of our instructional design and teacher educator pedagogy. To that end, we build on the work of others (Grossman, Hammerness, & McDonald, 2009; Lampert et al., 2013) who advocate teacher education reforms that feature “teaching as a central element to learning to teach” (McDonald et al., 2014, p. 500). We begin by reviewing the literature on expertise with a view to understanding the nature of adaptive expertise that we might reasonably expect of graduating teachers. With reference to the extant literature on practice-based teacher education, we discuss key components of our mathematics education programdwith a focus on a Classroom Inquiry (CI) courseddesigned to support PTs' development of expertise. Utilizing Timperley's (2013) distinction between shifts towards routine and adaptive expertise (see Section 2), we provide exemplars of what counts as evidence of the development of expertise. We conclude with a review of PTs' perceptions of instructional design features of the Classroom Inquiry (CI) and the implication for supporting PTs' development of adaptive expertise.

2. Teacher expertisedroutine and adaptive Defining expertise in teaching is a longstanding challenge. As noted by Ainley and Luntley (2007), attempts to describe the knowledge base of teachers may “offer tools for analysing particular aspects of practice, but fails to provide an adequate account of what is required to function effectively minute by minute in the classroom” (p. 4). In today's classrooms, teaching expertise demands skilful balancing of varied content and pedagogical knowledge alongside “consideration of the contingency of pedagogical relations connected to the embodiment of both teachers and students, and of the sociocultural context of a classroom” (Griffiths, 2013, p. 223). Importantly, expertise in this sense is not directly related to teaching experiencedthe traditional novice versus expert divisiondbut rather, considered as a component of professionalism. A useful distinction when applying descriptions of expertise is the contrast between ‘routine’ and ‘adaptive’ expertise (Hatano & Inagaki, 1986). The focus for the routine expert is on applying a core set of skills and routines with improved fluency and efficiency. Routines capture the certainties within teaching, and as such can be anticipated and can become part of a knowledge base for learning how to teach. For example, PTs' learning can include knowledge of typical misconceptions around learning mathematical concepts and patterns in students' responses to tasks that embody these concepts. Such knowledge ensures that performance is “highly competent, as long as the issues the individual deals with fall within the realm of the familiar” (Schoenfeld, 2011, p. 332). However, when learning the work of teaching, graduates need more than competency that involves being fluent with routines; they need competency that enables them to “innovate when necessary, rethinking key ideas, practices, and values in order to respond to nonroutine inputs” (Lampert, 2010, p. 24). Signifying adaptive expertise, they pursue the knowledge of why and under which conditions certain approaches have to be used or new approaches have to be devised.

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Characterized by flexible, innovative, and creative competencies, adaptive expertise can be viewed as a psycho-social construct that includes dimensions of concern, control, curiosity, and confidence (Savickas, 2005). Koh, Hong, and Seah (2014) added the competency of commitmentdthe propensity to experiment with new and different activities so that new possibilities can be generated. These constructs are captured in Timperley's (2013) depiction of adaptive teachers as driven by a “moral imperative to promote the engagement, learning and well-being of each of their students” and who “engage in ongoing inquiry with the aim of building the knowledge that is the core of professionalism” (p. 5). To illustrate the trajectories of developing routine and adaptive expertise, Timperley (2013) proposes a framework that highlights shifts that PTs might make on their learning journey. The first shiftda focus from self to studentsdconcerns the interrelated issues of identity, efficacy/agency, and normality (see Table 1). For example, Timperley contends that indicators of PTs' shift in focus from self as a learner towards the enactment of effective learning environments is an example of professionalism associated with routine expertise. Moreover, PTs who shift in focus towards the teacher as one who promotes valued outcomes for each learner exhibit aspects of professionalism associated with adaptive expertise. The second shift concerns PTs' understandings of teaching, including ideas about knowledge, interactions and responsibilities, and the location of learning. As summarized in Table 2, coming to appreciate the complexity of teaching routine expertise involves the recognition that “what students learn is filtered through their personal frames of reference, and they take account of this when constructing classroom environments” (Timperley, 2013, p. 8). In developing adaptive expertise, PTs come to view teaching as the coconstruction of knowledge that involves responsive, reciprocal power-sharing relationship with their learners and their learning communities. In defining markers of developing expertise, Timperley (2013) takes care to note that these shifts are not mutually exclusive. While it is important that PTs master routines, what distinguishes adaptive teachers is their constant attention to the impact of teaching and learning routines on students' engagement, learning, and wellbeing. In mathematics education, teaching approaches associated with adaptive expertise have been variously described as “ambitious,” “dialogic,” “reform-oriented,” “responsive” (Stylianides & Stylianides, 2014), and “responsible” (Ball & Forzani, 2011). In the next section we attend to how teacher education can support the development of such expertise, with an overview of the current turn toward practice-based teacher education (Zeichner, 2012) and the contextual background of our study. 3. Supporting adaptive expertise in practice-based teacher education As far back as the 1980s, Hatano and Inagaki (1986) argued that to avoid the ‘halt’ in expertise growth, the development of professional expertise required a balance between the development of effective routines and the development of conceptual understanding. However, while there is agreement that adaptive expertise entails the basic components of routine expertise (Stylianides & Stylianides, 2014), recent studies challenge the necessity of a developmental sequence from a routine expert to adaptive expert, arguing that “adaptive expertise should be understood as a fundamentally different conception of professionalism” (Timperley, € nings, Segers, and van 2013, p. 9). Bohle Carbonell, Stelmeijer, Ko €nboer's (2014) review of adaptive expertise studies across Merrie workplace settings noted that training activities that stimulate

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Table 1 Shifting focus from self to students: markers of routine (unshaded cell) and adaptive expertise (shaded cells). Self

Student

Identity

Self as person learning how to teach.

Efficacy/agency

Self-preservation: surviving the reality shock.

Normality

Self as the norm: All learners are like them and learn as they do.

Job is to construct an effective learning environment. Professional identity is focused on promoting valued outcomes for each learner Strong sense of self-efficacy that provides the confidence necessary for teaching. Agency depends on developing relationships with learners that promotes their learningdparticularly priority learners. Realizes that some learners are different from them Learn to identify and use cultural and linguistic resources of diverse learners

Adapted from Timperley, 2013.

learners to experiment, to make errors, and to try out different solutions methods benefit the creation of a flexible knowledge base associated with adaptive expertise. These experiential learning activities, Ericsson (2014) argues, must provide individuals with: challenges that go beyond their current level of reliable performancedideally in a safe learning context that allows immediate feedback and gradual refinement by repetition. These learning environments can be viewed as scaffolds that facilitate attainment of a higher level of performance. Later the scaffolds can be gradually eliminated so performance can be embedded and elicited in the natural environments in the domain of expertise. (p. 192) Within teacher education, alternatives to the ‘practice makes perfect’ foundations associated with routine expertise (Aitken et al., 2013; Darling-Hammond, 2014) extend Britzman's (1991) ‘practice makes practice’ argument to “view teaching not only as a resource for learning to teach but as a central element of learning to teach” (McDonald et al., 2014, p. 500). In linking the ‘how’ and ‘what’ of teacher education, McDonald et al. contend that conceptual coherence of practice-based reforms requires simultaneous innovation of “organizational structures and policies, content and curriculum, and teacher education pedagogy” (p. 501). Curriculum reforms are typically organized around a set of core teaching practices (Forzani, 2014; Reid, 2011), a set of normative principles to guide teachers' judgement in the use of these practices, and the knowledge needed to teach discipline content (Lampert et al., 2013). In mathematics education these core instructional practices include, for example, eliciting and responding to students' thinking, representing mathematical thinking, orchestrating group work and mathematical argumentation, and positioning students as competent learners. The Learning In, From, and For Teaching Practice study (see Lampert et al., 2013) provides a seminal model of teacher education practice-based pedagogy reform. Based on utilization of “approximations of practice” (Grossman et al., 2009, p. 238), PTs learn the work of teaching through enactments of teaching in the form of

public rehearsals with teacher educator coaching. Lampert et al. contend that pressing the rehearsing PTs to consider what they are doing in relation to aspects of practice or the underlying principles of ambitious mathematics teaching supports engagement in theory building and development of shared conceptual framework aligned to adaptive expertise. Likewise, Timperley (2013), in advocating for practice-based approaches, suggests that opportunities to reflect on practice-based tasks in carefully constructed learning communities promote metacognition and self-regulated learningdcharacteristics of adaptive expertise. Over the last 3 years we have engaged in a project, Learning the Work of Ambitious Mathematics Teaching (see Anthony & Hunter, 2013), involving rehearsals of instructional activities as part of PTs' mathematics methods courses and enactment of approximations of practice within a Classroom Inquiry (CI) course. The findings reported in this paper draw on the data generated from the CI completed by 23 PTs in the final semester of their 4-year teacher education program. Working in groups of four, the PTs were required to plan, teach, and review their teaching of a group of students aged 9e11 years over an eight lesson sequence. Teaching sessions comprised an introductory activity (e.g., quick image), small group work on a rich task, and a plenary session. Building on PTs' earlier learning from practice-based cycles of rehearsal, planning and school-based enactments of core practices within their mathematics methods courses, the CI afforded extended opportunities for PTs to experience the relational demands associated with launching a problem, eliciting and responding to students' mathematical thinking, utilizing a range of representations, connecting the big ideas in mathematics, and positioning students as competent. The program of learning utilized a cyclical ‘teaching-as-inquiry’ process (Timperley, Wilson, Barrar, & Fung, 2007) designed to support the development of PTs' professional knowledge through a deliberate mix of theory, practice, and attending to how their students respond and learn (see Fig. 1). The focusing phase required PTs to establish baseline information about their students' learning needs and experiences. Data needed to include an understanding of what the students already

Table 2 Shifting understandings about teaching and learning: markers of routine (unshaded cells) and adaptive expertise (shaded cells). Simplicity

Complexity

Complexity and knowledge

Teaching is about transmitting knowledge.

Interactions, relationships and responsibilities

If taught in ways that are familiar students should learn.

Location of learning

Learning happens primarily in the classroom.

Teaching needs to flexibly provide opportunities to learn accepted knowledge. Teaching is the co-construction of knowledge. Students' responsibility to learn, teacher to present as well as possible. Teaching and learning is a function of complex relationships between teachers and students, whanau [families] and communities. Learning is a complex interaction between home, community and school. The school needs to build on this learning. Learning draws on resources from multiple environments. Teaching develops powerful educational connections between these environments.

Adapted from Timperley, 2013.

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Fig. 1. Inquiry and knowledge-building cycle. Aadapted from Timperley et al., 2007.

knew about the mathematics content and mathematical practices associated with argumentation and current experiences with collaborative group work. In preparing for the teaching phase, the PTs needed to attend explicitly to their own learning needs. They were encouraged to connect the evidence from research and prior learning from rehearsals (e.g., the use of talk moves) to support their planning of the school tasks and teaching/learning experiences. Drawing on conceptual frameworks of participation and communication within group work (see Hunter, 2008; Smith & Stein, 2011), the third phase challenged PTs to consider what it means to enact group work effectively; to look critically at student learning participatory experiences and outcomes with a view to assessing the effectiveness of their individual and group learning and actions to promote student learning. To develop teacher agency and dampen the effects of enculturation into existing teaching modes, it was important that we challenged and supported PTs to build theories of practice that bridged formal and everyday knowledge (Lampert, 2010). Back at the university, group and whole-class reflections of videos of their weekly teaching sessions helped PTs “figure out what they do and do not yet understand about how their students are performing and what to do about it” (Hammerness et al., 2005, p. 377). Reflections were also encouraged through completion of journal entries related to each phase of the CI as part of the formative assessment requirements. Informed by PTs' reflections and our own observations of each cycle of school teaching, additional university-based rehearsals opportunities were scheduled to support the learning of key instructional routines (e.g., launching a problem). Our vision was that the practice-based opportunities to learn in the university and school setting would occasion PTs' learning in ways that supported them to continue to learn in and from their practice. In doing so, we hoped that they would develop those attributes of professionalism associated with inquiry, collective responsibility and knowledge co-construction that signify adaptive expertisedexpertise that is crucial for mathematics teachers to “do teaching that is more socially and intellectually ambitious than the current norm” (Lampert et al., 2013, p. 241).

4. A way into the data In looking to explore the nature of PTs' developing expertise and to further understand the impact of our instructional design we retrospectively examined the data associated with the CIda subset of data generated from our larger formative design research (Anderson & Shattuck, 2012) study on pedagogies of practice within teacher education. The CI data set comprised assessment artefacts (Journal entries coded J#1, J#2, J#3) plus our field notes of the school teaching sessions and university group feedback sessions. In addition, we interviewed 14 PTs at the beginning and end of the CI (coded I#1, I#2) about their perceptions of group work in mathematics lessons, self-assessment of their current and future use of group work and mathematical discussion, and their perceptions of the learning process in the CI. In this paper we utilize a case study approach based on data from two PTs selected as ‘telling’ cases (Mitchell, 1984). Our interest in selecting these cases concerned the feasibility, or otherwise, for diverse learners within teacher education programs to develop adaptive expertise. Pip and Troy were chosen to represent PTs with contrasting personal histories with regard to learning outcomes assessed by prior university-based mathematics education courses. In contrast to Troy who had achieved ‘A’ rankings for prior mathematics methods courses, Pip had recently repeated the final mathematics methods course to obtain a ‘C’ grade. She self-selected into the mathematics (as opposed to say reading or drama) CI option in order to “build her self-confidence and knowledge about maths teaching” (PI#1). She had no previous experience of observing collaborative mixed-ability group workd“it's always been ability-based small group work and I've seen no whole-class math discussions” (PI#1). Troy noted similar experiences of group work, both personally and in school-based field experiences. However, possibly aligned with his high-achievement levels in all three mathematics methods, Troy spontaneously provided an elaborated critique on current classroom practices: I haven't really seen this kind of teaching … what I've seen is just a lot of grouping for grouping sake without the things behind it

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that make it make sense. So I've seen lots of teachers good at questioning, at eliciting individual students' thinking and responses but haven't seen that more student supporting student and working together yet. [TI#1] Our examination of these two data sets involved both inductive and deductive analysis of the interview and journal transcripts (Strauss & Corbin, 2008). We remained open to emerging themes (e.g., views on ability grouping and student identity) while at the same time attending to evidence relating to markers of expertise as identified by the descriptors in Timperley's (2013) framework. Patterns of shifts were identified through multiple analytic passes to track changes across successive journal entries and in conjunction with initial and final interviews. In some instances respondents' reflections and responses detailed their perceived changes in practices, beliefs, and understandings with reference to the CI in particular, and occasionally with reference to previous mathematics methods courses and experiences. The coding also confirmed that the development of routine and adaptive expertise were not mutually exclusive. This was particularly the case for Pip, where responses in interviews 1 and 2 represented a mix of routine and adaptive markers. In contrast, as signposted in the initial elaborated response to experiences with group work in schools as noted above, Troy's interviews were coded almost exclusively as evidence of adaptive expertise. However, we note that this data relates to PTs' experiences in their fourth year of study, and thus in terms of mapping a developmental trajectory it is unknown whether Troy would have previously provided responses more aligned to routine expertise. In Section 5 we provide a narrative of Pip and Troy's developing expertise during their final semester of the 4-year teacher education program through the lens of shifts in focus from self to students as related to identity, efficacy/agency and normality (see Table 1) and shifts to increasingly complex understanding about the teaching and learning nexus (see Table 2).

5. Pip's developing adaptive expertise 5.1. Shifts in focus from self to student Early in the CI it was evident that Pip's shifting of focus from self to students was strongly aligned to her concerns around mathematics identity and confidence. Initial interview responses reflected her conscious attention to developing relevant knowledge and skills associated with constructing an effective learning environment, similar to that promoted in rehearsals during the mathematics methods courses. For example, Pip expressed a desire to master skills in scaffolding, questioning, and facilitating her students to: make connections without me having to say do it this way and also for them to come to a consensus that this is the answer and that doesn't always have to be me, the teacher, saying this is right, this is the answer. [PI#1] Practising these new skills was changing the way she thought about herself as a mathematics teacher: My own self confidence has been a challenge, and I've found just in the small things that we've done, working out the problem and anticipating what they might do, has been really good to allay those fears that I have about whether I have enough knowledge. [PI#1]

In Pip's second journal, efforts to develop effective learning environments expanded to include a focus on children developing positive and productive relationships with mathematics: Getting children to know that maths is hard and thinking is hard, but if we work together we are all learning. So those are key. … And also another thing was trying to draw out our more reticent students and some of those who had attitudes, who sat back. [PJ#2] We conjecture that these experiences were challenging Pip to move away from believing that her own experiences of learning mathematics were necessarily the norm; she was developing an understanding that individuals and groups are diverse in their learning needs. As noted in her second journal: One girl was very confident and competent and held a lot of mathematical status. … There were two boys who were very quiet. One of these boys you could see him working out in his head as he would use his hands and be looking up, like visualizing in his mind, but he found it very hard to explain his thinking. Then there was E, we had to position her within a group who would spend the time to explain their thinking.

5.2. Understandings about teaching and learning As the CI progressed, Pip's increased confidence about how to facilitate group work and discussions mirrored her increasingly complex understandings of teaching and learning. Excited by the challenge of “letting children use what they know” in contrast to her previous experiences of “standing up the front and teaching”, Pip's PT group was keen to explore ways to support learner agency within the group task activity. As Pip noted: All the children can hear everybody's thinking, so it caters for all levels, wherever they are at, and it can lead on to higher learning. … For the learner they have to know what's going on, they have to know the strategy. That's part of their responsibility to know what's going on and to offer an idea and to take risks and then having to be accountable for explaining and justifying their thinking. [PI#1] In the first block of teaching, Pip's PT group decided to assign their students roles (e.g., facilitator, recorder, and reporter) within the group task. Subsequently, in an effort to support a more collaborative approach, the group task roles were removed. Moreover, rather than see participation as solely a responsibility of the individual, Pip provided evidence that she appreciated that norms and knowledge were co-constructed: To have an inclusive classroom your classroom has to be built where everyone feels safe to share their ideas, take risks and it's okay to be wrong. … I think that the discussions and the listening adds to the relationships and I think other children can also make connections with each other because they are seeing them in a new way, not just how they are perceived, they can see them in another light. [PI#1] In supporting collective sense making, Pip acknowledged that co-construction of mathematical knowledge was a new idea for her and would require her to develop new ways of being: It's those fears, getting away from being the fount of all knowledge and putting it back to the group. And letting go,

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because we do like to control. And also stepping back from the discussions, listening to the children react without the teacher having to prompt. [PI#1] As the inquiry progressed, Pip was seen to evaluate her experimentation with ways to support students to participate. She evaluated this in terms of student responsibility to themselves and their peers: Immediately it [collaborative participation] takes it away from ability-based groupings; it allows all children to participate at their level, wherever they are at. They also have a responsibility to the big group too. I was really keen that if you had one strategy everyone had to know what that strategy was so that you could all explain it so that everyone could participate. [PI#2] By the end of the CI Pip appeared more confident that she could support the learning of those who were struggling alongside those students who appeared at first glance well-advanced. In particular, her focus on learning rather than performance was noted: Flexible group and whole-class discussion allowed me and I hope the children to realize they all have something to offer. It was empowering when we stepped away from having to get the correct answer. There is this huge learning in the misconceptions, and trusting that the students can work out a solution and that everyone was learning, participating and engaged in the mathematics. [PI#2] Overall, in locating mathematics learning Pip was beginning to consider the social aspects of the learning process, and also reflect on possible impacts on other learning areas: we are changing how people interact with each other. So it's not going to be just setting up our environment but I think over time if I keep at it I can see it as a really positive move, not just maths too but all the curriculum areas, my whole classroom will change. [PI#2]

5.3. Developing adaptive-expertise In summary, with Pip we saw glimpses of theorizing, interspersed with indications of confidence building related to the fluency and efficiency of a set of skills and routines. Pip's developing agency remained tempered with concerns about routine expertise. She continued to voice her learning in terms of skill development associated with a sense of self-efficacy that provides the confidence necessary for teaching: I wasn't good at maths and knowing about the research about how teachers who are confident and have good attitudes about maths pass that on to their students, but doing maths how we've done it this way I feel more confident that I can go into the classroom. It's changed my attitude about how I feel about myself. Being able to facilitate discussion and bringing children's thinking out has been a really important part for my learning. [PI#2] Evidence of developing adaptive expertise is seen in her move towards linking teacher actions with student learning and then taking action to experiment and modify instruction as needed in her quest to support students to engage in complex mathematical practices such as giving explanations, making connections, and using representation. While conscious of Bray's (2011) contention

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that those teachers with “sound knowledge are more apt to notice and respond to critical learning moments in the lesson, and are more likely to be able to use students' thinking as springboards for inquiry in the context of class discussions” (p. 35), we wonder whether Pip's emergent adaptive expertise will be sufficient to support generative learning as she transitions into the workplace.

6. Troy's developing adaptive expertise 6.1. Shifts in focus from self to student To use practices, principles and knowledge adaptively teachers must be able to “strategically move away from planned curriculum components to better support the contextual needs of their pupils … and recognize the need to refine, change, and try out different decisions while paying close attention to the impact on their pupils” (Soslau, 2012, p. 768). From the start of the CI Troy made it clear that he was open to having his assumptions about learning and teaching mathematics tested: So what I knew was that repetition and practice was the way to learn maths, so you do one thing over and over again until you become proficient at it and then you move on to the next thing. … As far as challenging me as a teacher I just think I've got a lot to learn. [TI#1] In addition to indicating that challenging his assumptions was a long-term way of learning, Troy was explicit that challenges came not just from linking the ‘what’ he needed to learn to his past experiences but also to evidence of ‘student learning’. For example, he was clear that his assessment of the value of group work (in his classroom teaching episodes) would be assessed against learning outcomes: “the ultimate bottom line is does it help them do maths better?” In his first journal entry his assessment that group work was productive referenced a range of valued participatory and mathematical practice outcomes: It is probably more engaging than your typical ability-based group work, a rich problem; lots of kids come in with their ideas and lots of groups working well. I think they can take those ideas and use them. It's giving everyone a bit of expression; hopefully they can see themselves as a more of a mathematician than they would have otherwise. [TJ#1] As another example, when watching the video re-play of the lessons, as required in the CI process, Troy noted the need to focus on not only what “went alright” but also to think hard about “what could be done differently”, evidenced through learning outcomes for individual students: Probably the bigger goal is more for the kids. … five of them have displayed good working together practices but maybe they could be even better. Others really need to take that personal responsibility to say if they don't understand and things like that. So I think you start to set individual goals for students. And then there's that overall goal for the group itselfdat the end are they going to be working better? [TI#1] During the CI, our observations of Troy's planning and teaching combined with his reflections provided multiple points of evidence that he was particularly aware of those children who are not traditionally well-served in mathematics classrooms. Marking his developing adaptive expertise, his journal entries reflected a growing awareness that teacher agency was linked to relationships

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with students that promote learning. As his knowledge of the students increased over the CI we saw evidence related to shifts in normality from self to increased awareness of diversity. Troy noted the importance of his PT group planning for individual student outcomes: E [a student] is a very reluctant participant. We aim to encourage her participation by devising simpler problems and highlighting how her strategies/solutions relate to other more complex problems. C's [another student] change, in contrast, will be providing clear, accessible explanation of his strategies to his peers. [TJ#2] In the final interview, Troy made a point of noting how the student group work afforded productive mathematical engagement for specific learners: The group [of students] we were with I could make a guess about what they would be like in a traditional maths lesson, who would be the reluctant ones and who would be the keen ones no matter what because they already love maths and see themselves as great at it, but it [this way] seemed to me to be a lot more engaging for most of the group more of the time. You certainly had lots more opportunity for members of the group to participate more that they might otherwise. [TI#2]

6.2. Understandings about teaching and learning In looking at shifts in Troy's understandings of teaching and learning, it was apparent that Troy was initially in a stronger position than Pip. From the start of the CI, Troy confidently expressed a view that teaching was much more than transmitting mathematical knowledge. He liked that collaborative group work was “not so teacher reliant”. He wanted mathematics to become real, accessible, and relevant to his students: I think you get to the point where students are helping students and learning from students and that's probably much more related to real world problem solving than constantly being reliant on someone in a position of authority to feed you and support you to find answers to problems. [TI#1] Aligned to this position, Troy was quick to focus on students' thinking as a resource. However, he was aware that the ‘how’ to use this resource would require rethinking his teaching moves, especially those associated with eliciting and responding: I think I get the goal of strategically choosing who is going to share and when but again hopefully practice makes perfect. … just getting really good at listening, listening to students and really being able to pull out where they are at, what they're thinking or what they're not thinking as the case might be. I just have to work on all of them to be honest, one thing at a time. [TI#1] Troy also looked closely at the relational opportunities for learningdnot just in terms of surface evidence of students' participation in the discourse but in terms of the quality of the mathematical explanation: Student-to-student and student-to-students inter-relational stuff, just ways of talking, there's a difference between just being nice and being quiet when someone is talking and active listening, there's a difference between explaining your thinking and explaining it well. So there's a lot of almost quite observable,

what you could call hard skills that they can learn. Speaking with certain words, the language they can use, using certain phrases that can really help them out there. [TI#1] As the CI progressed it became evident that Troy drew on his earlier public rehearsal opportunities (in the methods classes) to evaluate his ability to implement a range of core instructional practices. Troy reflected on his use of thinking time in terms of occasioning student opportunities to learn more effectively: As far as the specific teacher strategies, I think I've got much better with what I call easing back and just listening. And not needing to redirect their thinking instead of letting them get there themselves. To an extent I came to appreciate that if you give them time they will get there. And you can tell when they are really stuck and they need more of a push or some more pointed questions to help get them going in the right direction. [TI#2]

6.3. Developing adaptive-expertise As a result of the CI, Troy appeared confident that his teaching skills would improve with more opportunities to practise. His focus on understanding how he could adapt his teaching to ensure that collaborative inquiry processes could benefit each of the diverse learners in his group was evidenced by a rich multidimensional image of student learning outcomes. Building an understanding of the deeply relational process of teaching and the ways in which he could and should continue to learn in and from practice signified encouraging moves towards adaptive expertise development. For Troy, his theorizing about teacher and student learning was enhanced with evaluative critique of past, current, and future practice. 7. Supporting the development of adaptive expertise Mediated by personal histories, beliefs and everyday practice theories (Fairbanks et al., 2010), we take heart that these two PTs both moved along developmental trajectories towards ‘baby’ adaptive expertise (Hatano & Oura, 2003). In this section we discuss elements of the CI design that were viewed as supportive or otherwise for the development of adaptive expertise. The central element connecting the phases of the cyclic inquiry model was the explicit linking of teaching to student learning outcomes. Starting with the requirement to gather baseline student data (e.g., Pip reported on the “variety of personalities with a variety of mathematical self-efficacies” [PJ#1]), the continued attention to student learning outcomes impacted on both shifts from self to student and towards more complex understandings of teaching and learning. For Pip, increased awareness of the impact on her teaching on diverse learners was implicated in her attempt to resolve tensions between her formal and everyday knowledge of ability grouping structures that dominates New Zealand classrooms. Pip noted: I can see that thinking about your groupings, not just letting the students randomly choose is a big part. I can see it being another way to change the perception that maths is only for those people with a maths brain … and making this fun for everyone, it's not just for the bright and clever, it's for everybody. [PI#2] Likewise, we see how the press to evaluate instructional practices in terms to student learning played out for Troy:

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I avoided interrupting students' discussion because they were progressing mathematically and were, at times, demonstrating the actions/behaviours we were aiming for. However, with my encouragement these discussions may have been more productive, and some students' participation more robust. … We should have checked more carefully individual students' understanding during small-group time. For example, I called on E to share her group's strategies believing she understood it when in fact she did not. [TJ#2] Repeated opportunities to engage with the group of learners over a sequence of time supported the PTs to experiment with ‘what works and what doesn't’ in terms of instructional organisation to support learning. For example, Pip's group experimentation included: Our first session we were worried about management issues, for example putting your hand up and respecting others when they were talking. … later we were concerned that explaining and justifying would be difficult so we decided to appoint one person in the group to explain the group thinking. … this proved to be too restrictive and we removed the roles. [PJ#1] Adaptive expertise as characterized by evaluating and adapting instruction was further supported by the utilization of research frameworks (see Anthony & Walshaw, 2009; Hunter, 2008; Smith & Stein, 2011) and expectations that PTs source addition research literature. These frameworks also supported the exploration of where different theories of learning fall with respect to the efficiency and innovation dimensions of teaching. Specifically, the ethic of care and positioning of students advocated within frameworks sharpened PTs' focus on the diverse learning needs within their student groups. For example, as Troy became increasingly aware of the situated and relational aspects of learning mathematics, he began to question the notion of ‘fixed ability’: I think that insight that wherever they start from is really good and maybe it demonstrated just how much their attitude plays a role their learning, how good or not they feel about their maths has just as much impact; it's really what defines their ability to do it. Not that there aren't certain kids that are innately more inclined to be better at maths than others, but those that can think at maths time, that starts to really snowball and become a self-fulfilling prophecy because for every moment that they don't engage with the lesson because they think they can't is one more moment that they miss the opportunity to learn. So this approach seems to keep them a bit more engaged more oftendmaybe it's because they don't have to work on it alone, I'm not sure. [TI#2] Individual and group reflections on the teaching episodes, inclusive of planning, supported PTs' shifts in understanding of teaching and learning. For example, Troy's everyday awareness of the practicalities of teaching was tempered by evidence of the nature and frequency of student learning opportunities as the organizer for his formal theories of practice: I think it's more manageable than the other approach [experienced on practicum] where you have four to five groups where you are doing separate planning, separate lessons for, and so when you rotate these some don't get any interaction with you, so one day they have a maths game, that's not very good learning. … I'm a new teacher so I need to know both, but this way I know leads to more teacher-student time and more

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student-to-student time and it hones in on more the maths itself. [TJ#2] Troy's reflections, in particular, appear to act as “a means of not only uncovering his preconceptions, but also analysing and reducing them to workable localised theories for teaching” (Gelfuso & Dennis, 2014, p. 3). For beginning teachers an important aspect of adaptive expertise involves the ability to learn from others (Le Fevre, 2014). Participation in the CI was deliberately organized in peer groups in order to support PTs' learning to work in teams. Both Pip and Troy remarked on the value of peer collaboration in relation to planning, teaching observations, and post-lesson reflections. Pip, in particular, noted how the collaborative planning acted to counter her lack confidence of mathematics knowledge. Citing both emotional and practical benefits, Troy noted that working with a group was a refreshing change from his cultural and university learning experiences to date: I've really come to appreciate the strength of the collective. … just being able to bounce ideas off each other. Whole class sessions, we get a few suggestions and a bit of feedback and it's interesting to hear how other groups are working, they might be differentdso I really like the group aspect. And it just makes it more manageable for the first go at teaching being responsible for teaching and not the observations or videotaping. [TI#1] Likewise Pip noted the emotional aspects of the CI process. For her, the processes of learning to work with colleagues and opening up one's practice for public scrutiny involved an element of risk taking: Even though it's a group and you're teaching and you're learning, you are getting videoed. So I feel that you are on show; that you're going to be critiqued. But as I've done one or two of the lessons you just get in and you just forget about that. My thoughts are that if you make mistakes that's good. I'm here to learn, we're here to learn. [PI#1] Overall, the act of engagement in an inquiry and knowledge building cycle (see Fig. 1) appears to have enhanced PTs' valuing of teacher inquiry as a way of learningdthe essence of adaptive expertise. Pip, in reference to feedback from peers, remarked: “You don't know you do stuff, you think you are being an effective teacher, an equitable teacher but sometimes you're not” [PI#2]. In a similar view, Troy's reflections indicated a growing metacognitive awareness of the need to figure out what he can and cannot do in terms of supporting learning: It's new, so doing it well is a challenge in itself, I think really knowing which questions to ask at the right time, the timing of the questions so that they can build on what they were doing previously. And secondly, and this may be over a longer term, but there were patterns developing, like there were explainers, kids who could do the maths really quickly and were also supporting others to understand and not a lot of what you would want to see which is others explaining the thinking and the whole group valuing that we didn't get that before, but that's probably a longer term thing and we were probably too ambitious to get it too soon. [TI#2] In summary, key elements of the CI design that supported our PTs' development of adaptive expertise included the cyclic inquiry

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process focused around core practices of ambitious mathematics teachingdmost notably the noticing and responding to students' thinking, the use of research-based literature and frameworks to support evaluation and links to student learning outcomes, the opportunity to practise in a safe learning environment, and the support of the learning community established through collaborative practice-based activities (e.g., group planning and public rehearsals).

8. Discussion and implications Aligned with our goal of preparing PTs to engage in equitable and culturally responsive mathematics teaching, the CI involved the enactment of core instructional practices rehearsed in methods classes. The teaching as inquiry model pressed PTs to link teaching actions to student learning. The individual and group reflective activities not only exposed PTs' preconceptions, but also prompted the PTs to analyse and rework them as localized theories of teaching. In particular, this CI, with its focus on teaching and learning rich mathematical collaborative group tasks, disrupted PTs' everyday understandings of ability group work organisation (and associated fixed ability mindset) provoking (re)thinking about the role of students' participation in sense-making in the mathematics lesson. In considering the multiple professional learnings for each of our PTs, shifts in understandings from self to students and towards increasing complex understandings about teaching and learning, as detailed in Timperley's (2013) framework, provided a useful way to map PTs' developing routine and adaptive expertise. For both PTs the opportunity to engage with cycles of experimentation, embedded in inquiry as a model of professional learning, supported shifts aligned to the development of adaptive expertise markers. Most notably, these PTs evidenced strong convictions concerning valued mathematical practices (e.g., mathematical argumentation) and outcomes for diverse learners. As the CI progressed, their reports reflected a shift of focus from developing efficacy towards building agency for developing the relationships and teaching practices to achieve these valued outcomes. Exploratory in nature, we acknowledge that our interpretation of evidence of expertise was confined to two PT data sets. Moreover, in reporting their experiences and learning about their teaching and responsibilities, it may well be that interview and journal responses were tainted by the PTs' ability and inclination to critique their own learning. As Troy noted in the final interview: Every lesson we did something well but I thought maybe we should have done this better or maybe we should have tried to target that kid's participation a bit better. But in the group sharings I noticed that everyone had really positive things to say, which is good, but there's no way that in a group full of brand new teachers, essentially teaching maths in a way that is completely different to the way we learned, that we didn't make a few mistakes and had quite a few things that we did poorly, or could have done better. So I just kind of wonder how, like was I effective or am I just re-storying this to fit the outcome that I was really hoping for. [TI#2] Recognizing that teacher expertise, in the widest sense, is a function of one's orientations, goals, and resources that are tightly linked and develop slowly over time (Schoenfeld, 2011), we do not claim to have ‘connected the dots’ between markers of adaptive expertise and future teacher actions and enhanced student learning. However, we hope that PTs' adaptive expertise capabilities, be they emergent, will support them to take responsibility for

ongoing learning and adaption of their practice in response to assessment of student learning outcomes. Thus, in returning to the debate on graduating teacher standards we offer our exploratory cases as a first step to illustrate how the development of adaptive expertise can realistically be supported within practice-based teacher education programs. We, along with others, are increasingly convinced that knowledge about teaching cannot be separated from its enactment; that is, teachers do not learn new things then learn how to implement them in the classroom. In reviewing teacher education reforms, we offer structured practice-based approaches, such as CI, as an alternative to calls to increase the practicum/field-based time. However, if we are to take these graduating standards seriously we as teacher educators need also to be adaptive experts with regard to our pedagogies. In moving to incorporate practice-based approaches, we need to ensure that PT reflective processes are more than description and feelings. Our experience suggests that for inquiry to generate “warranted assertabilities” (Gelfuso & Dennis, 2014, p. 3) there needs to be purposeful instructional design. For us, these included rehearsals and enactments of teaching in safe settings, utilization of research-based frameworks, carefully constructed learning communities, and inquirybased assessments. In mathematics education, an ability to take an agentic position towards improving practice is critical. For beginning teachers, the challenges involve enacting a personal vision of teaching (Fairbanks et al., 2010), coping with potential tensions of fit between individual and school perspectives (Anthony, 2010), and resisting the status quo in a way that counters existing hegemonic practices. While adaptive expertise offers a way forward, we need to be cautioned by Troy's assessment in his final interview: I think it's almost an impossible ask in the space of one course to produce in most of the people competent teachers in this sense but having them experience it leads to a definite cognitive change, it really just attacks the structure of what most people think maths teaching should be and that's a start and then really it's up to schools and the system to better support the right practice out there. [TI#2] Responding to demands for beginning teachers to be adaptive experts will require understanding much more about the nature and development of expertise within current and alternative teacher education programs. With a focus on adaptive expertise, urgent priorities include research on the effectiveness of different pedagogies within teacher education (McDonald, Kazemi, & Kavanagh, 2013), research on specific feature of programs linked to PT learning (Ghousseini & Herbst, 2014), and research related to how teacher educators can generate and use formative assessment data concerning PTs' developing adaptive expertise to inform their own instructional decisions. Our study, like others primarily in the descriptive stage, needs also to be extended to determine the role of adaptive expertise in the workplace transition and continued teacher learning trajectories.

Acknowledgements The research reported in this paper was supported by the Teaching and Learning Research Initiative fund administered by the New Zealand Council of Educational Research. This work is grounded in the collective work of the Learning the Work of Ambitious Mathematics Teaching project, whose members are Dayle Anderson, Robin Averill, Tim Burgess, Michael Drake, Roger Harvey, and Peter Rawlins and the authors.

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