The growth of reflective practice among three beginning secondary mathematics teachers

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The growth of reflective practiceamong three beginning secondarymathematics teachersMichael Cavanagh a & Anne Prescott ba Department of Education , Macquarie Universityb Faculty of Arts and Social Sciences – Education, The Universityof Technology , Sydney, AustraliaPublished online: 15 Apr 2010.

To cite this article: Michael Cavanagh & Anne Prescott (2010) The growth of reflective practiceamong three beginning secondary mathematics teachers, Asia-Pacific Journal of Teacher Education,38:2, 147-159, DOI: 10.1080/13598661003678968

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Asia-Pacific Journal of Teacher EducationVol. 38, No. 2, May 2010, 147–159

ISSN 1359-866X print/ISSN 1469-2945 online© 2010 Australian Teacher Education AssociationDOI: 10.1080/13598661003678968http://www.informaworld.com

CAPJ1359-866X1469-2945Asia-Pacific Journal of Teacher Education, Vol. 38, No. 2, Mar 2010: pp. 0–0Asia-Pacific Journal of Teacher EducationThe growth of reflective practice among three beginning secondary mathematics teachersAsia-Pacific Journal of Teacher EducationM. Cavanagh and A. PrescottMichael Cavanagha* and Anne Prescottb

aDepartment of Education, Macquarie University; bFaculty of Arts and SocialSciences – Education, The University of Technology, Sydney, Australia

(Received 30 April 2009; final version received 2 February 2010)

This paper reports a study of three beginning secondary mathematics teachers and howtheir reflective practice developed during a one-year university teacher educationprogram and concurrent professional fieldwork experience or practicum. The participantswere interviewed three times during the practicum and once more in their first year ofteaching, to investigate the nature and depth of their self-reflections about the practicumand their relationship with their supervising teachers. A three-stage, hierarchical modelof reflective practice (Lee, 2005) was used to interpret the interview responses. Resultsshow some improvement in the participants’ ability to reflect on their teaching duringthe practicum, while also highlighting the importance of the practicum school contextin their professional formation and professional development. The beginning teachersdeveloped a greater capacity for reflection in their first year of teaching, but even thentheir responses were generally descriptive in nature rather than demonstrating criticalreflection. We suggest some reasons for this.

Keywords: practicum; reflective practice; secondary mathematics

Introduction

Becoming a teacher involves participation in university studies and school experience,known in Australia as the practicum. In both of these contexts, beginning teachers areencouraged to adopt a critical attitude toward their classroom practice by engaging inongoing and focused reflection. This reflection involves constantly framing questions inresponse to classroom observations and experiences, seeking answers and re-framing newquestions as they arise. Reflective practice is therefore widely regarded as one of thehallmarks of quality teaching. But learning to become a reflective practitioner is no easytask, especially when ideas about what constitutes good teaching that come from theuniversity appear, in the eyes of the beginning teacher, in disagreement with the teachingpractices of the practicum classroom.

The critical importance of the practicum in teacher preparation has been noted in arecent report prepared for the Australian Parliament (House of Representatives StandingCommittee on Education and Vocational Training, 2007). The report outlined some keyfeatures of a high-quality practicum, the first of which is ensuring that it ‘integrates theoreticalknowledge and professional practice across the three domains of a teacher education

*Corresponding author. Email: [email protected]

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148 M. Cavanagh and A. Prescott

program: “content” knowledge gained through a liberal education, professional knowledge,pedagogical skills and insights’ (p. 74). In this paper we examine the reflective insights ofthree beginning secondary mathematics teachers as they enter the teaching profession. Wefocus on their reflections on their practicum experiences both during the practicum and asfirst-year teachers.

Background

Reflective practice among pre-service teachers

The ability to reflect critically on one’s classroom practice is generally regarded as anessential part of any teacher’s professional growth (Jaworski, 2006) and is especiallyimportant for beginning teachers (Artzt, 1999; Dinkelman, 2000; Kaminski, 2003). Butpre-service teachers often either do not reflect on their practice (Alger, 2006; Shoffner,2008) or do so in a superficial way (Bean & Stevens, 2002; Collier, 1999). Instead, theyfocus mainly on the technical skills of teaching (Le Cornu & Ewing, 2008) and practicalconcerns, such as planning and classroom management (Moore, 2003; Nyaumwe, 2004),rather than listening to, and becoming more aware of, the students they teach (Jaworski &Gellert, 2003).

The school-based practicum is the environment in which the nexus between theoryand practice has the potential to become most acutely apparent to pre-service teachers(Moore, 2003). However, practicum classrooms may not reflect the reformist vision of theuniversity course (Goos, 1999) and many pre-service teachers have practicum experienceswhere they are discouraged from trying out new teaching approaches, thus reinforcingmore traditional pedagogies first seen in their own school days (Grootenboer, 2005/2006).

The institutional environment of the school has a profound influence on novice teachers(Hodkinson & Hodkinson, 1999) because they have a strong desire to win the approvaland acceptance of their supervising teachers (Roberts & Graham, 2008). As a result,pre-service teachers are prone to internalise and reproduce the routines and norms of theirplacement schools (Jaworski & Gellert, 2003) and copy the teaching techniques theyobserve, leading to a simplistic and technically-based view of teaching (Putnam & Borko,2000). Such passive imitation of supervising teachers does little to encourage a stance ofcritical reflection or promote the beginning teachers’ capacity to learn from their fieldexperiences in any meaningful way (Zeichner, 1992).

Yost, Senter and Forlenza-Bailey (2000) recommend that research focus on the role ofteacher education experiences, including the school-based practicum, in helping pre-serviceteachers become more adept at critical thinking. The purpose of this paper is, therefore, todocument the reflective practices of a small number of students in their teacher educationprogram.

Theoretical perspectives on reflective thinking

According to Dewey (1933), reflective thinking has two fundamental characteristics: (1)A state of doubt, hesitation, perplexity or mental difficulty, in which thinking originates;and (2) an act of searching, hunting and inquiring, to find material that will resolve thedoubt, and settle and dispose of the perplexity. Rodgers (2002) distils four key componentsof reflective thinking from Dewey’s work. First, she highlights reflection as a process ofmaking sense of one’s experiences – not found in routine actions undertaken withoutawareness, but grounded in educative situations that broaden the mind and open up new

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Asia-Pacific Journal of Teacher Education 149

possibilities of thought and action. Second, reflection must become systematised into away of thinking that is both disciplined and rigorous; hence, the need to provide time forreflection between thought and action, particularly for novice practitioners. Third, reflectionoccurs via interaction with others since ‘when one is accountable to a group, one feels aresponsibility toward others that is more compelling than the responsibility we feel only toourselves’ (Rodgers, 2002, p. 857). Fourth, reflection requires an attitude of mind thatprizes personal and intellectual growth through open-mindedness and a shift in focus fromself-absorption to self-awareness.

Schön (1987) views the practicum as the place where the theoretical curriculum andpractical experience are integrated, and rejects a view of the practicum as ‘technical trainingor seeking right answers’ (p. 39). He contrasts a technical practicum where professionalknowledge is seen as ‘facts, rules and procedures applied nonproblematically’ (p. 39) witha reflective practicum where we ‘make new sense of uncertain, unique or conflictedsituations of practice’ (p. 39). Schön highlights the crucial role of the senior practitioneror coach (i.e. the supervising teacher), noting that coaches in a reflective practicum do notsimply observe performance to detect errors or point out correct procedures, but rather‘emphasise indeterminate zones of practice and reflective conversations’ (p. 40).

In order to develop a model for analysing the reflective practice of the participants inthe study, we turned to what other authors have theorised about the reflective tendenciesof pre-service and practising teachers. Most authors typically identify three levels ofreflection, which progress from simple descriptions of an event, often focusing on technicalaspects of teaching, such as delivery of content or classroom management, to considerationsof the problems of teaching where alternative viewpoints are examined thoughtfully,raising new questions that may be resolved (Edwards-Groves & Gray, 2008). We nowdescribe three models that are representative of the research. Each demonstrates commonelements of reflective practice – an initial preoccupation with practical concerns of teachingleading to a search for causes to explain actions and, finally, to imagining alternativecourses of action with a greater focus on student learning.

Lee (2005) reported on the content and depth of pre-service teachers’ reflective thinking.She identified recall as the first stage of reflective practice, in which a beginning teacherdescribes experiences without looking for alternative explanations and teaches in waysthat they have observed or were taught themselves without questioning them. Lee’s nextstage is that of rationalisation, which involves a search for causes to help explain experi-ences and guide the development of general principles for future actions. Lee’s third stage,reflectivity, occurs when a novice teacher begins to analyse experiences from variousperspectives and identify the influence of their supervising teacher with the intention offuture changes in action.

Van Manen (1977) (as cited in Power, Clarke & Hine (2002)) developed a differentframework to describe student teachers’ understanding of their reflections. The first levelis that of technical rationality, which is simply attending to strategies that work or fail inclassroom settings. Next is practical action, where the student teacher focuses on thelearning experiences of pupils and starts to recognise teaching as problematic. Finally,there is critical reflection, in which the student teacher deliberates on the moral and socialimplications of classroom practices.

Muir and Beswick (2007) looked at more experienced teachers and also identified athree-level model. This model starts with the technical description of classroom events,where teachers provide general accounts of classroom experiences, often focusing on thetechnical aspects of teaching (such as delivery of content or maintaining students’ attention)without any consideration of the value or importance of those experiences. Level 2 is

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deliberate reflection, in which critical incidents are identified and explained. And level 3is critical reflection, where the teacher moves beyond identifying and explaining criticalincidents to consider the perspectives of others and contemplate alternative actions.

To summarise, Muir and Beswick are concerned with the reflective practice of experi-enced teachers and Van Manen, while studying pre-service teachers, emphasises the moraland social implications of teaching. We have adopted Lee’s framework because her focuson pre-service teachers’ reflective practice and their relationship with their supervisingteachers is a primary concern of our own research. The following examples, taken fromour own experience, illustrate Lee’s three levels of reflective practice:

• Recall – General descriptions of classroom practice; evaluating the success or failureof actions; focusing on the technical aspects of teaching.

• I ran out of material at the end of the lesson.• The class liked working on the computers.

• Rationalisation – Accounts of critical incidents; searching for causes to guide futureactions.

• The small group activity did not go well because the class has not done anythinglike that before; they are used to a more structured approach in lessons.

• I tried to teach them multiplication of fractions but they didn’t know their tables.A lot of students got confused and I had class management problems. I shouldhave revised some [multiplication] tables questions in the quiz and maybe kept thenumbers smaller.

• Reflectivity – Analysis of experiences; considering various perspectives; identifyingthe influence of supervising teachers.

• I want to do more group work but my supervising teacher says I’m not giving theclass enough practice exercises. I want her to write me a good report.

• During prac I was always buffered by my supervising teacher, who prepared mea lot for the lessons. He said, ‘Look this is what we’re doing’; he gave me guide-lines and because of his presence things were a bit different. It was good, but itwas not the full reality. I wish he’d let me try things on my own more and learnfrom my mistakes.

Lee’s three levels of reflective practice are closely aligned to Schön’s (1987) notion oftechnical and reflective practica. At Lee’s recall level, the focus is on the technical aspectsof teaching, where the pre-service teacher’s classroom actions occur without analysis andteaching is viewed non-problematically. This is equivalent to Schön’s technical practi-cum. Lee’s second and third levels indicate a shift towards what Schön calls a reflectivepracticum, where the pre-service teacher begins to identify and explore the uncertaintiesof teaching and the unique characteristics of the practicum context. It is the extent towhich the pre-service teacher can make sense of these uncertainties that distinguishesLee’s levels of rationalisation (a search for causes) and reflectivity (a deeper analysis ofexperience).

Schön also identifies the important role played by the supervising teacher in assistingthe pre-service teacher progress to a more reflective practicum. In a technical practicum

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(recall level) the supervising teacher simply observes the lesson and offers practicaladvice about the technical aspects of teaching, such as classroom management. Bycontrast, in a reflective practicum (rationalisation and reflectivity levels) the supervisorengages the pre-service teacher in professional dialogue that seeks to analyse the ambigu-ous nature of classroom interactions to promote reflective practice.

In the present study, we sought to: (1) Document the reflective practices of three pre-service teachers during the practicum; and (2) identify factors that facilitated orconstrained their progress towards reflectivity. In particular, we investigated how thesupervising teacher and the practicum school context influenced the professional formationof each pre-service teacher in our study.

The present study

Setting

The present paper reports part of a study conducted at two universities in Sydney. Ethicalclearance has been given for the research and informed consent was given by all partici-pants. The data came from three students enrolled in a Graduate Diploma in Education[Grad Dip], a one-year full-time equivalent, professional qualification for secondaryteaching, comprising units in education, curriculum and methodology. The program isavailable to graduates with academic qualifications in mathematics or a related field;they are typically mature-aged and decide to become mathematics teachers after previ-ous work experience. A supervised 50-day practicum is completed in a single schoolunder the direction of one supervising teacher, predominantly in one teaching day perweek over the course of an entire school year. The extended timeframe of the practicumand its concurrence with methodology units are intended to integrate theory and practicein the Grad Dip program.

The mathematics methodology units in the Grad Dip are designed to allow studentsopportunities for reflection and analysis of their practicum experiences informed by theoryand research perspectives. The classes incorporate many features recommended in theliterature as likely catalysts for reflection among pre-service teachers. These includeensuring that practicum supervision is carried out by the methodology lecturer (Borko &Mayfield, 1995), course assignments based on reflective tasks completed in conjunctionwith the supervising teacher (Le Cornu & Ewing, 2008) and action research projects onstudent learning (Ebby, 2000). In addition, video analysis of selected classroom situationsallows the methodology class to say what they might do in the circumstances. By compar-ing different courses of action, the pre-service teachers identify ‘the surprising possibilityof theories of action alternative to their own’ (Schön, 1987, p. 323). There are also regular‘school experience discussions’, similar to the episodes and issues framework suggestedby Jaworski and Watson (2001) (as cited in Jaworski & Gellert (2003)). In these discussions,pre-service teachers ‘bring significant episodes (or anecdotes) from their teaching to share. . . [and] are encouraged by tutors to take a positively critical stance towards such issues,relating the questions that arise to teaching situations they have personally experienced’(p. 847). There are also ‘joint lesson planning’ tasks based on the lesson study model(Stigler & Hiebert, 1999), in which pre-service teachers work in small groups over anumber of weeks to prepare a comprehensive lesson plan and associated resources. Thegroups revise their initial ideas in response to detailed written evaluations from designatedpeer reactors and the tutor, present the revised lesson plan to the methodology class forfurther discussion and analysis, and make final revisions in response to that feedback.

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The present study reports on part of a larger research project that occurred during thefinal implementation of a new mathematics syllabus for Years 7 to 10 (Board of StudiesNew South Wales [BOS NSW], 2002) that emphasised Working Mathematically, anapproach to teaching mathematics that provides ‘opportunities for students to engage ingenuine mathematical activity and to develop the skills to become flexible and creativeusers of mathematics’ (BOS NSW, 2002, p. 45). Working Mathematically is a majortheme of the mathematics methodology units in the Grad Dip and a focus of our research.

Participants

In the larger study, a random sample of 16 Grad Dip students, eight from each of two par-ticipating universities, were interviewed individually prior to, during and after completionof their practicum (Cavanagh & Prescott, 2007). Four of the students obtained full-timeteaching positions in metropolitan Sydney and agreed to be interviewed once more inTerm 2 of their first year after graduation (Prescott & Cavanagh, 2008). Here we presentcase studies of three of these students, to be referred to by pseudonyms throughout thispaper. We chose these three students because they exemplify important aspects of how webelieve beginning teachers develop their reflective practice in the context of the practicum.

Data collection and analysis

The second author conducted all of the interviews during the Grad Dip year and the firstauthor conducted the final round of interviews in June of the following year, when theparticipants had completed their studies.

Semi-structured interviews of approximately 20 minutes duration took place with eachpre-service teacher in February, June and November of the Grad Dip year. There werebroad themes that we wanted to cover in the interviews: the development of each pre-serviceteacher’s reflective practice, their relationship with their supervising teacher, the practicumcontext, and how they were implementing Working Mathematically. In particular, WorkingMathematically provided a focus for talking about participants’ classroom practice, andtheir comments about Working Mathematically gave us an insight into their growth inreflectivity.

In the first round of interviews, the interviewer asked participants to recall their ownschool days, describe their favourite mathematics teachers, discuss their motivation forchoosing a mathematics teaching career, and share their concerns about the impendingpracticum. In the second and third interviews, the pre-service teachers described their ownand their supervisors’ mathematics lessons, and gave an impromptu outline for a lesson ona specified topic. In the final interview, the participants were asked to review their practi-cum experiences from the stance of a first-year teacher. For all of the interviews, we chosefocus questions that might allow participants to discuss their teaching practices, the con-text of their practicum schools and the roles played by their supervising teachers. In thisway we hoped to gain some insight into how they developed their reflective thinking.

After each round of interviews, the second author transcribed all the recordings, usingpseudonyms for each participant. We had already developed a provisional list of codesbased on our research focus areas, and we used this list to undertake a preliminary analysisof each interview. Other issues, specific to each participant, emerged from the data andwere used as the basis for questions in subsequent interviews. This grounded approach(Glasser & Strauss, 1967) led to a refining of the provisional codes and the developmentof new codes. Finally, we used Lee’s framework (2005) to revisit the data, looking for

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illustrative examples of the themes we had developed, to form the three narrative casestudies (Miles & Huberman, 1994).

The case studies

Sam

Sam, in his late twenties, worked as an IT trainer prior to the Grad Dip. He was quietlyspoken and reserved, often emphasising the affective traits of teaching. In the first interview,he described his best teacher as ‘very caring’ and his own motivations for teaching as ‘agreat need to help people and I guess I’m caring, so teaching and helping students issomething I enjoy. I care a lot about students’. His dominant memory of mathematicslessons at school was one where the teacher ‘came in, wrote on the board, we wrote notesand we did exercises in class and we did exercises for homework’. In contemplating hisapproaching practicum, Sam expressed a desire to ‘engage students in practical activities;to try to link the concepts to the real world’. However, outlining how he might teach alesson on decimals, he said, ‘On the board, I’d write up a heading and I’d write the theorybehind it. You need to keep the [decimal] point above the point. And then run throughsome examples.’

Sam’s reflective thinking

Sam completed his practicum in a school located in a low socio-economic area whereclassroom management was a challenge for most teachers, and he quickly became preoc-cupied with keeping students ‘under control’. In the second interview, he described hissupervisor as a good teacher because she ‘was extremely stern and used a lot of bodylanguage and a lot of screaming to keep the kids under control. The amount of learningwasn’t that great, but she kept them under control.’ Early in his practicum Sam worriedthat he was becoming ‘too focussed on trying to implement the lesson as I planned. Hopefully,as I progress I’ll be able to focus less on what I’m trying to teach and more on the disci-pline.’ Later, in the third interview, Sam described one of the best lessons he had taught asone where ‘I didn’t get much discipline issues’. In the same interview, Sam continued toreport difficulties with classroom management and complained that his practicum schooldid not supply textbooks for students; he pleaded, ‘I need a textbook’.

Sam commented in the second interview that his supervising teacher suggested that hetry hands-on activities in lessons as a means of improving student behaviour, and sheprovided him with many resources he could use. However, Sam reported in the third inter-view that the supervising teacher did not model this approach in her own lessons and so hewas unsure about how to implement it effectively. Preparing so many hands-on lessonswas ‘taking up a lot of time’ and when he tried using concrete materials in the classroom itseemed only to exacerbate his management problems and he commented that the practi-cum was ‘wearing me out and I get de-motivated’. But Sam was not able to explain whyhis classroom management problems continued nor could he offer any alternative strate-gies for dealing with them other than persisting with the hands-on activities suggested byhis supervising teacher.

It was only after the practicum, when Sam had been in a school with similar manage-ment issues to his practicum placement, that he began to recognise some other sources ofthe difficulties he experienced in trying to manage his classes. In the final interview, hecommented on students’ poor behaviour saying, ‘I think it could stem from students not

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understanding or having learning difficulties.’ He also spoke about some of the unrulyclasses he was currently teaching, ‘I don’t like teaching something that’s simply just got tobe taught for the sake of it; and that confuses them so then I have class managementproblems. It’s sort of a vicious circle.’ Sam went on to say that he would like to ‘throw outthe [mathematics] program’ because he realised the futility of teaching some topics, likefractions, when students ‘don’t know their tables and can’t work with whole numbers’.Instead, Sam said he would prefer to devote more time in lessons to basic numeracy but hehad not done so because of ‘pressure to cover the program’. He commented, ‘As a first-yearteacher you’re afraid to tread on toes. You don’t want to do anything that’s too out therebecause you’re on probation so you don’t want to be taking too many risks.’

John

John worked previously in research and development for a telecommunications companybut a redundancy had prompted a career change into teaching. Married with two childrenand in his late forties, John decided to undertake the Grad Dip because his wife, a teacherherself, recognised his ability to explain concepts clearly to his own children. Not surprisingly,John described mathematics teaching in the first interview as ‘clarity in explanationswhere everyone understands’, since this was what he remembered from his own schooling.John commented that he expected his practicum classes would be ‘well behaved,motivated, like I was when I was in secondary school’.

John’s reflective thinking

John’s practicum occurred in a middle-income suburb at a school that he described in thesecond interview as ‘good, with minor class management issues’. In the same interview,he characterised a successful lesson from his supervisor as ‘chalk and talk’ where ‘content[was] delivered quickly and clearly’ and ‘key principles [were] well explained’. However,as his practicum continued and John heard more about alternative pedagogies at university,he began to question some of the practices of his supervising teacher and to recognise thelimitations of his own schooling.

In the third interview, he described constructivism as ‘a breath of fresh air thatchanged my attitude to teaching dramatically’ but he was disillusioned that ‘the methodswe discuss at uni [are] not practised at school’. For John, this disconnect became moreapparent when he tried unsuccessfully to implement a Working Mathematically approachin his lessons. Also in the third interview, he recognised that his difficulties arose becausestudents were not used to Working Mathematically and the ‘school culture doesn’tpromote this’. Even though John was ‘allowed to do lessons my way’, he resolved tofollow his supervisor’s more traditional style and continue to ‘actively discuss [with thesupervisor] what we learn at uni’. John described how his supervisor was ‘lending asympathetic ear’ and had challenged John with ‘if you can prove to me that it [WorkingMathematically] can work, I will try to adopt it more’. However, the challenge proved toogreat for John and, as his subsequent attempts at Working Mathematically did not succeedas he had hoped, he began to question the efficacy of the approach because he did notthink he could sustain it and complete all of the syllabus content. He concluded the thirdinterview by explaining that it was better to ‘proceed with caution’, adopt a transmissivestyle and build his confidence.

As a first-year teacher, John repeated his claim that students were not prepared toengage in higher-order thinking because they want ‘memory work, a lot of drill’. Hence,

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‘it’s easier to develop your teaching skills in an instrumental teaching scenario and onceyou’ve got the basic skills, then you can start to try new things’. John also reconfirmed hisideas about the need for clear explanations in teaching, noting ‘your ability to explainthings must be flawless’. He stated, ‘I’ll be focussing on my ability to explain things, tryand keep at the right level, use the right words’.

Peter

In his mid thirties and with a doctorate in chemistry, Peter had long been interested inbecoming a teacher. He was gregarious and spoke at length in the first interview about hispassion for teaching, noting that from high school and all through university he had ‘spenta lot of time teaching other students’, an experience that had convinced him that he was‘good at it’. He described his best mathematics teacher as one whose lessons were ‘quiteunstructured and we could explore things we wanted to explore. So lessons were a lotmore discovery-oriented and we did the exercises for homework’. Peter described his aimsin teaching ‘to inspire students’ interest in maths’ and show students who may not likemaths how it can be ‘applicable and important in their lives’. He said he wanted to‘stretch’ the students.

Peter’s reflective thinking

Peter’s practicum took place in a comprehensive government high school whereachievement levels in mathematics had historically not been high. In the second inter-view, Peter described his supervisor as a ‘traditional’ teacher whose lessons werealways ‘very structured with a quick quiz, review of homework, examples andwell-graded practice exercises’. He concluded that ‘students need structure so lessonsare predictable’ and described one of his own lessons on angles as successful because‘they [students] remembered the rules next lesson’. Later, in the third interview, Peterreported that he had tried activities designed to engage students more in their learningwithout much success because the ‘class wasn’t used to doing group work’. He alsocommented that his supervising teacher had very strong views about how mathematicsshould be taught and had made it equally clear that he expected Peter to follow a trans-missive style. Peter said he became frustrated when his supervisor did not support aWorking Mathematically approach and when he criticised Peter for the lack of practiceexercises he provided for students. Peter reported in the third interview that he hadeventually decided that a ‘good [practicum] report necessitated following his way’, buthe looked for opportunities to try out new ideas when his supervisor was absent fromthe class.

After biding his time for much of the practicum, Peter’s responses in the fourth interviewwere markedly different. He described his first few weeks as a full-time teacher as ‘a realhigh’ because he was ‘finally able to do what I wanted to do’ and had ‘more freedom to trythings, to experiment’ rather than having to ‘please my supervising teacher’. He now tried‘to begin [lessons] with concrete examples and generalise them to formulate the rules’because that was his own preferred learning style. However, Peter noted that he could nottake an investigative approach in every topic just yet due to work pressures, such asincreased preparation time and the need to keep up with parallel classes for exams. Herecognised that it was ‘easier just to pick up the textbook’ sometimes because ‘students[are] not interested in taking up the challenge of Working Mathematically’. Peter alsocommented that he was ‘still learning the content’ as he had not taught most topics before,

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so he was relying on textbooks more than he would like but would adopt his preferredWorking Mathematically style as he felt ‘more confident’.

Discussion

The initial reflections of all three participants were at the level of descriptive recall, focusingon the technical aspects of teaching, such as Sam’s anxiety about managing behaviour,John’s insistence on providing clear explanations and Peter’s observation that studentsprefer more structured lessons. This finding is consistent with previous studies that havehighlighted the dominance of procedural matters during the practicum (Moore, 2003) andindicates the pre-service teachers’ focus on practical concerns as an essential first step indeveloping their reflective practice (Shoffner, 2008).

There was some development, admittedly at different rates and to varying degrees, inthe reflective stances taken by the participants. Sam and John both advanced to the level ofpractical rationalisation, as evidenced by Sam’s recognition in the final interview thatlearning difficulties were the likely source of students’ poor behaviour and John’s realisationin the third interview that the practicum school culture and his supervisor’s pedagogymade adopting Working Mathematically more difficult. In his third interview, Peterappeared to exhibit critical reflection, albeit in an incomplete way, by recognising that hecould wait until the practicum was over to incorporate more student-centred activities inhis own classes. It is also worth noting that all of the participants showed a greater capacityfor critical reflection on their practicum experiences in the fourth interview, perhapsconfirming Schön’s (1987) dictum that important kinds of learning reveal themselves‘only when a student moves out of the practicum into another setting’ (p. 299).

A number of factors appeared to assist the development of the participants’ reflectivepractice. The extended time of the practicum allowed them to ‘complete reflectiveprocesses concerning a range of pedagogical issues rather than (perhaps) to recogniseproblems but not have sufficient time to work towards resolving them’ (Lee & Loughran,2000, p. 85). Peter and John both reported that they were able to maintain their desire forWorking Mathematically despite the traditional style of their supervisors – Peter by waitinguntil his supervisor was absent and John by deciding to concentrate first on developing hisconfidence as a teacher in the classroom.

The concurrence of the practicum with university studies and the dichotomy betweenthe reform approach in methods units and the traditional pedagogies of the practicumclassrooms also served to problematise the fieldwork experience for John and Peter. Eachacknowledged the disconnect between the two contexts and eventually chose to adopt thestyle of their supervisors for pragmatic reasons. Peter also used tactical compliance(Roberts & Graham, 2008), maintaining a productive working relationship with his super-vising teacher while moving beyond the limitations his supervisor imposed. Later, Petertook the initiative in ‘seeking out opportunities for control and experiment according tothe relationships and circumstances of . . . [his] particular placement’ (Roberts & Graham,2008, p. 1406) by inclusion of Working Mathematically activities during his supervisor’sabsences.

The case studies also show how the conditions that student teachers find at their practi-cum school can influence their professional formation and development. These conditionsare mediated by the individuality of each student teacher and influenced by the dominantconcerns expressed by their supervising teacher. For example, Sam’s preoccupation withclassroom management is perhaps not surprising given the nature of his practicum schooland his supervising teacher’s emphasis on student behaviour as the key indicator of his

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professional progress. John’s emphasis on clear teacher explanations as the main form ofinstruction resulted from his personal philosophy of teaching and was reinforced by theteaching he observed during his practicum, which discouraged him from thinking aboutalternatives.

Our study indicates that practicum classrooms do not necessarily need to be idealconstructivist models to promote reflection among novice teachers, but they do need to‘provide enough room for student teachers to envisage alternatives, interact with children,and try out new approaches’ (Ebby, 2000, p. 95). The reports of all three participantsindicate that there was little space made available for considering various perspectives oroffering alternative strategies during the practicum, so it is perhaps not surprising thatevidence of critical reflection was so sparse. Sam’s supervisor simply provided him withresources he could use, John’s lent him a sympathetic ear but did not encourage his class-room experimentation, and Peter’s was hostile to Working Mathematically, despite thesyllabus requirements. All three case studies therefore demonstrate the crucial role of thesupervising teacher, while at the same time confirming the supervisors’ tendency to giveadvice rather than help student teachers ‘problematise what they observe, and to reflect onwhat might be done to improve things’ (Jaworski & Gellert, 2003, p. 836).

Conclusion

The results of the present study show that reflective practice can be a daunting process,particularly for beginning teachers who may lack the wisdom that can come with greaterclassroom experience (Da Ponte & Brunheira, 2001). Despite numerous opportunities forreflection in the Grad Dip program aligned to the pre-service teachers’ concurrent practi-cum, the participants made little progress towards developing their own reflective practice.

Perhaps the practicum experience, where situations of stress are commonplace, makesself-reflection too difficult for novice teachers (Lee & Loughran, 2000), particularly ifthey attach little importance to reflective practice or perceive it to be of little benefit.Instead, the work of reflection may be more likely to occur when beginning teachers makethe transition to their own classrooms where they can shape their own professional identi-ties more freely (Bean & Stevens, 2002).

Muir and Beswick (2007) note that critical reflection is unlikely among teachers in theabsence of an external voice that can serve to challenge current ideas and practices. It islikely, therefore, that the opportunities we provided in the interviews allowed the partici-pants to reflect in ways that might not have otherwise occurred to them. Moreover, eventhough the novice teachers in our study may have gained some skill in reflection, it ispossible that they did not possess the disposition to become reflective practitioners ontheir own. For this reason, it was encouraging to hear that two participants, Sam and Peter,both have mentors in their new schools who they regard highly and use regularly assounding boards to discuss their classroom practice.

Perhaps the most important external voice of the practicum, that of the supervisingteacher, was absent from our study. We plan to extend our research by investigating howsupervising teachers envisage their role in working with student teachers and the ways inwhich they promote reflective practice among them.

Notes on contributorsDr Michael Cavanagh is a senior lecturer in mathematics education in the Department of Educationat Macquarie University.

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Dr Anne Prescott is a senior lecturer in mathematics education in the Faculty of Arts and SocialSciences – Education at the University of Technology, Sydney.

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Documents

The growth of reflective practice among three beginning secondary mathematics teachers