THE NATURE OF MATHEMATICS TEACHERS’ REFLECTIVE PRACTICE

THE NATURE OF MATHEMATICS TEACHERS’ REFLECTIVE PRACTICE

THE NATURE OF MATHEMATICS TEACHERS’ REFLECTIVE

PRACTICE

by

Anna Barbara Posthuma

A thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy (PhD)

Department of Science, Mathematics and Technology Education

Faculty of Education

University of Pretoria

Supervisor: Prof JG Maree

Co-Supervisor: Dr GH Stols

2011 i

© U n i i v e r r s s i i t y o f f P r r e t t o r r i i a

Acknowledgements

There are a number of people who graciously helped me to complete this research. I wish to personally and publicly thank all who supported me during the course of this research.

Thank you Prof Kobus Maree, my supervisor, for the excellent professional and personal support that you provided in the completion of this research. Dr Gerrit Stols my cosupervisor, I am grateful for your unwavering belief in me and encouragement to stay positive while engaged in this research. Mr A.K. Welman, thank you for editing the language of this thesis in such a professional way.

My family encouraged me for the duration of the research project and I want to thank Arend

(my husband), for his faith in me; my daughter Jacomien Muller, for always being willing to listen to my ideas; and my children (Arend and Drinie Posthuma, Louise and Eben Karsten and Dave Muller) for your love and inspiration.

Without the five mathematics teachers who agreed to share their teaching lives with me this research would not have been possible. Thank you for giving up your free afternoons for group reflections and the positive feedback I received from you towards the end of the research project. An empowered teacher is a reflective decision-maker, who finds joy in

learning and in investigating the teaching and learning process—one who views learning as construction and teaching as a facilitating process to enhance and enrich development

(Fosnot, 1989, p. xi).

Lastly, I dedicate this research study to my grandchildren, Ciska and Anoux. ii

Declaration

I hereby declare that this thesis submitted for the degree Doctor of Philosophy at the

University of Pretoria, South Africa, is my own original work and has not been submitted for any degree or examination at any other institution of higher education. I further declare that all sources cited or quoted are indicated and acknowledged by means of a comprehensive list of references.

Anna Barbara Posthuma

28 June 2011.

iii

Summary

Thoughts about reflection and reflective practice have evolved over many decades, through carefully constructed theory and research applications, mainly based on the work of Dewey

(1933) and Schӧn (1983). Evidence also exists in the literature that the ability to reflect on practice is considered a necessity for effective instruction (Sowder, 2007). By reflecting critically teachers become more positive in the search for a new understanding of their teaching practice and design more ways to deal with the challenges that confront them daily.

When teachers act reflectively, they consider carefully the problems in their own teaching and think about how those problems are related to their educational or social context. They are aware of the consequences of their teaching and how their own assumptions or beliefs can influence their teaching.

This main purpose of my research study was to explore the nature of mathematics teachers’ reflective practice in the context of lesson study. To achieve this aim, an in-depth exploration of five mathematics teachers’ reflection before, during and after teaching a lesson was conducted. The possible relationship between these teachers’ reflection and their classroom practice was also examined. The research also aimed to explore whether and how mathematics teachers’ reflections differ from the conceptualisations of reflection in classroom practice as found in the literature. Contextual factors that might influence the nature of mathematics teachers’ reflective practice were also investigated.

My findings indicate that the mathematics teachers in my sample have a limited understanding of the concept of reflection. Furthermore, based on lesson plan analysis, there was no evidence that these teachers reflect-for-action. However, they all reflected on-action verbally and in writing, and three of the five teachers reflected-in-action while teaching. They all reflected on Level R1 (recall level of reflection) and Level R2 (rationalisation level of reflection) and three teachers reflected critically on their learners’ understanding of mathematics and their own teaching of concepts towards the end of the research project

(Lee, 2005). Language and the lesson study group experience emerged as contextual factors that seemed to influence the teachers’ reflection.

Although the research study’s results cannot be generalised due to the small sample, I believe that through engaging in the lesson study experience the five teachers of this study iv

improved their reflective practice, reporting an increase in self-knowledge and finding new ways of teaching mathematics to learners.

Key words: Reflection; Reflective practice; Reflective thinking; Mathematics; Reflectionfor-action; Reflection-in-action; Reflection-on-action. v

Table of Contents

Chapter 1 .............................................................................................................................. 1

1.1 Orientation and background ....................................................................................... 1

1.1.1

1.1.2

1.1.3

Earliest views on teachers as reflective practitioners: Dewey and Schön .............1

International research on teachers’ reflective practice .........................................2

Research on teachers’ reflective practice reported in Southern Africa .................3

1.2. The context of this research study .......................................................................4

1.3 Rationale for the study ............................................................................................... 5

1.3.1

1.3.2

Personal interest in teachers’ classroom practice ................................................5

Concern about mathematics teachers’ ability to reflect on their practice ..............5

1.3.3 Evidence for the need to reflect critically on teaching practice from research .......6

1.4 Statement of purpose................................................................................................. 6

1.5 Research questions ................................................................................................... 6

1.6 Methodological considerations ................................................................................... 7

Chapter 2 .............................................................................................................................. 9

2.1 Introduction ................................................................................................................ 9

2.2 Theoretical perspectives on reflective practice in education ....................................... 9

2.2.1

2.2.1.1

Dewey’s approach to reflective practice ...............................................................9

Critical evaluation of Dewey’s work 10

2.2.2 Schön’s approach to reflective practice .............................................................. 12

2.2.2.1 Critical evaluation of Schӧn’s work 12

2.3 Categories of reflection in education ........................................................................ 14

2.3.1

2.3.2

2.3.3

Van Manen’s levels of reflection ........................................................................ 14

Valli’s levels of reflection .................................................................................... 16

Jay and Johnson’s levels of reflection ................................................................ 17 vi

2.3.4 Hatton and Smith’s levels of reflection ............................................................... 17

2.3.5 Lee’s levels of reflection ..................................................................................... 18

2.4 Research studies in education investigating teacher reflection ................................ 19

2.4.1 Overview of research studies dealing with preservice teachers’ reflections........ 20

2.4.1.1 Concerns about the definition of reflection as revealed in studies dealing with preservice teachers’ reflections 21

2.4.1.2

2.4.1.3

2.4.1.4

2.4.1.5

2.4.1.6

Research studies reporting on the nature of preservice teachers’ reflections 23

Research studies reporting on the content of preservice teachers’ reflections 25

Research studies reporting on the moment of student teachers’ reflection 26

Research studies reporting on the benefits of or barriers to reflection

Research studies reporting on contextual factors that influence reflection

27

28

2.4.2 Summary ........................................................................................................... 28

2.5 The reflective practice of practising teachers ........................................................... 33

2.5.1

2.5.2

2.5.3

2.5.4

2.5.5

2.5.5.1

2.5.5.2

2.5.5.3

Introduction ........................................................................................................ 33

Teacher reflection through self-study ................................................................. 34

Teacher reflection on feedback from learners .................................................... 35

Teacher reflection in communities ..................................................................... 35

Opportunities that potentially allow teachers to become reflective practitioners . 37

Professional development as an opportunity for teacher reflection

Professional learning through lesson study

Professional learning through action research

37

38

39

2.5.6

2.5.7

2.5.7.1

2.5.7.2

Characteristics of a reflective practitioner........................................................... 39

Benefits of reflective practice ............................................................................. 41

Gaining professional and personal knowledge

Changing/improving practice

41

42 vii

2.6 Mathematics teachers’ reflective practice ................................................................. 43

2.6.1

2.6.2

2.6.3

2.6.3.1

Introduction ........................................................................................................ 43

Research studies linking reflection and beliefs about action .............................. 45

Research studies dealing with teachers’ reflection-on-action ............................. 47

Lesson study contexts that enhance reflection-on-action 48

2.6.4 Research studies dealing with teachers’ reflection-for-action ............................. 51

2.6.5 Research studies dealing with teachers’ reflection-in-action .............................. 53

2.7 The conceptual framework for this research study ................................................... 58

2.7.1 Definition of reflection and reflective practice to guide this research study ......... 60

2.8 Conclusion ............................................................................................................... 61

Chapter 3 62

3.1 Introduction .............................................................................................................. 62

3.2 Paradigmatic assumptions and perspectives ........................................................... 62

3.2.1

3.2.2

Ontological assumptions .................................................................................... 62

Epistemological assumptions ............................................................................. 63

3.2.3 Assumptions about human nature ..................................................................... 64

3.3 Research site and sampling ..................................................................................... 64

3.4 Data-gathering procedures ...................................................................................... 64

3.5 Data-gathering instruments ...................................................................................... 66

3.5.1

3.5.2

3.5.2.1

3.5.2.2

Interviews .......................................................................................................... 67

Observations ..................................................................................................... 67

Individual teacher observation

Lesson study group observation

68

68

3.5.3 Document collection .......................................................................................... 68 viii

3.5.3.1

3.5.3.2

Lesson plans

Researcher diary

68

69

3.6 Strategies for data analysis ...................................................................................... 71

3.7 Quality assurance: data verification ......................................................................... 72

3.8 Summary of the layout of the research design ......................................................... 74

3.9 Ethical considerations .............................................................................................. 76

3.10 Limitations ............................................................................................................... 76

3.11 Conclusion ............................................................................................................... 77

Chapter 4 ............................................................................................................................ 78

4.1 Introduction .............................................................................................................. 78

4.2 Coding the data ....................................................................................................... 78

4.2.1

4.2.1.1

Coding the transcripts of the interviews with the participants ............................. 78

Inclusion criteria that determined the coding of the data 80

4.2.2 Deductive style of indentifying themes ............................................................... 84

4.2.2.1

4.2.2.2

4.2.2.3

4.2.2.4

Theme 1: Understanding of reflection

Theme 2: Reflection-on-action

Theme 3: Reflection-for-action

Theme 4: Reflection-in-action

85

86

87

87

4.2.2.5

4.2.2.6

Theme 5: Content of reflection

Theme 6: Contextual factors

88

88

4.3 Personal ethnographies of the participants .............................................................. 89

4.3.1

4.3.2

4.3.3

4.3.4

Dianne ............................................................................................................... 90

Mary .................................................................................................................. 90

Morgan .............................................................................................................. 91

Sipho ................................................................................................................. 91

4.3.5 Vicky .................................................................................................................. 91

4.4 Participants’ reflections in relation to the themes of the study .................................. 92 ix

4.4.1

4.4.2

4.4.2.1

4.4.2.2

4.4.2.3

4.4.2.4

4.4.2.5

Theme 1: Participants’ understanding of reflection ............................................. 92

Theme 2: Reflection-on-action ........................................................................... 94

Dianne

Mary

Morgan

Sipho

Vicky

94

95

96

98

100

4.4.3

4.4.3.1

4.4.3.2

4.4.3.3

4.4.3.4

4.4.3.5

Theme 3: Reflection-for-action ......................................................................... 103

Dianne’s lesson planning

Mary’s lesson planning

Morgan’s lesson planning

Sipho’s lesson planning

Vicky’s lesson planning

103

104

105

106

106

4.4.4

4.4.5

4.4.6

Theme 4: Reflection-in-action .......................................................................... 107

Theme 5: Content of reflection ......................................................................... 108

Theme 6: Contextual factors that influence reflection ....................................... 110

4.5 Results of lesson study group reflections ............................................................... 112

4.5.1

4.5.1.1

4.5.1.2

4.5.1.3

4.5.1.4

4.5.1.5

Discussion of lesson study group reflections on lessons observed .................. 112

Lesson study group reflection on Mary’s lesson

Lesson study group reflection on Morgan’s lesson

Lesson study group reflection on Vicky’s lesson

Lesson study group reflection on Sipho’s lesson

Lesson study group reflection on Dianne’s lesson

112

113

113

114

114

4.5.2

4.5.2.1

4.5.2.2

4.5.2.3

Final group reflection on being part of the lesson study group ......................... 116

Lesson planning

Teaching mathematics

Self-observation

116

117

117 x

4.5.2.4

4.5.2.5

Observation of colleagues

Group discussions

117

118

4.6 The reflective journey of each individual participant ............................................... 119

4.6.1

4.6.2

4.6.3

Dianne ............................................................................................................. 120

Mary ................................................................................................................ 121

Morgan ............................................................................................................ 122

4.6.4

4.6.5

Sipho ............................................................................................................... 123

Vicky ................................................................................................................ 124

4.7 Conclusion ............................................................................................................. 124

Chapter 5 .......................................................................................................................... 127

5.1 Introduction ............................................................................................................ 127

5.2 Chapter summary .................................................................................................. 127

5.3 Verification of research questions .......................................................................... 128

5.3.1

5.3.2

5.3.3

5.3.4

Research question 1 ........................................................................................ 128

Research question 2 ........................................................................................ 131

Research question 3 ........................................................................................ 132

Summary of verification of research questions ................................................. 132

5.4 What would I have done differently? ...................................................................... 135

5.5 Providing for errors in my conclusions ................................................................... 136

5.6 Conclusions ........................................................................................................... 136

5.7 Recommendations ................................................................................................. 137

5.8 Limitations of the study .......................................................................................... 138

5.9 Last reflections....................................................................................................... 138

References ........................................................................................................................ 140 xi

Figure 2.2

Figure 2.3

Figure 2.4

Figure 3.1

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

List of figures

Figure 2.1

Lenses used to assess studies dealing with preservice teachers’ reflections

Four dimensions of teachers’ professional practice

Framework for reflective teaching

Lee’s reflection levels (2005)

Illustration of the lesson study cycles for this research study

Theme: Understanding of reflection

Theme: Reflection-on-action

Theme: Reflection-for-action

Theme: Reflection-in-action

Theme: Content of reflection

Theme: Contextual factors

Benefits of lesson study

The reflective journey of individual participants xii

List of Tables

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 2.6

Table 3.1:

Table 3.2

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

Table 4.7

Table 4.8

Table 4.9

Table 4.10

Table 4.11

Table 5.1

Summary of levels of reflection proposed by researchers

Summary of studies on preservice teachers’ reflective practice

Summary of characteristics of a reflective practitioner from different theorists

Interpretive framework delineating studies on teacher reflection and action

Summary of different modes of reflection

Summary of research on mathematics teachers’ reflective practice

Relationship between research questions and data collection techniques

Summary and layout of research design

Timeline of the data gathering process

Inclusion criteria for coding the data

Exclusion criteria for coding data

List of codes created for the two interviews

Biographical detail of participants

Participants’ understanding of reflection

Summary of participants’ reflection-on-action

Participants’ level (content) of reflection

Contextual factors that influence participants’ reflections

Summary of content and level of reflection revealed during the lesson study group reflections after each lesson

Participants’ reflections in relation to the themes of this study

Summary of verification of research questions xiii

List of appendices

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Lesson plan template

Letter of consent

Letter of permission to the department

Interview and group reflection schedules

Ethical clearance certificate xiv

List of abbreviations

ACE

CPTD

BMI

Advanced Certificate in Education

Continuing Professional Teacher Development

Body Mass Index xv

Chapter 1

Introduction and contextualisation

1.1

Orientation and background

This research study seeks to explore mathematics teachers’ reflective practice in the context of lesson study

1

. In the orientation to this research study I discuss some views on teachers’ reflective practice that have been expressed internationally and nationally to situate my research within this broader framework.

1.1.1 Earliest views on teachers as reflective practitioners: Dewey and Schön

One of the first American educational theorists to view teachers as reflective practitioners was Dewey. According to Dewey (1933) true reflective practice takes place only when the individual is faced with a real problem that he or she needs to resolve in a rational manner.

In his book (first published in 1910), How we think, Dewey (1933) links the process of reflection to attributes of the ideal teacher. According to Dewey (1933) ideal teachers acquire the habit of on-going thoughtfulness and examination of the beliefs and theories they use to inform their instruction of learners. This process of reflection helps teachers develop specific orientations, such as open-mindedness, responsibility, and wholeheartedness.

Although Dewey’s reflective thinking was popularly endorsed, it was not widely practised in teacher education (Lee & Tan, 2004). The reason for this neglect was, according to Adler

(1990), the strong influence of behaviourist ideas in teacher education. Lee and Tan (2004) report that during the 1980s teacher education reform and research shifted to issues surrounding teachers’ thinking and their professional knowledge, and this shift sparked an interest in the teacher as a decision-maker.

After Schön’s (1983) publication of The reflective practitioner, the slogans of ‘reflective teaching’, ‘action research’, ‘research-based’ and ‘inquiry-oriented’ teacher education have been embraced by both teacher educators and educational researchers throughout the world (Zeichner, 1994). Schön (1983) introduced the terms reflection-in-action and reflection-

on-action to describe teachers’ thinking in their classroom practice. The term reflection-in-

action (Schön, 1987) is used to describe teachers’ reflection on certain matters while they

1

Lesson Study (or kenkyu jugyo) refers to a process in which teachers progressively strive to improve their teaching methods by working with other teachers to examine and comment on one another’s teaching techniques (Baba, 2007).

1

are teaching (e.g. Are the group of learners engaged in the task at hand? Are they bored?

Should I move on to a new topic?). Schön (1987) uses the term reflection-on-action for retrospective thinking, or thinking ‘after the event’. Schön’s ideas were attractive to teacher educators because he closely described daily classroom situations that teachers encounter and the kinds of thinking processes that accompany teachers’ work (Lee & Tan, 2004).

According to a number of researchers (e.g. Gimenez, 1999; Lee & Tan, 2004; Zeichner,

1994) the range of interpretations of teacher reflection is extremely wide and there is no single definition of the concept, which leaves me with a dilemma. How can I best study a phenomenon that has been fairly vaguely defined and is so widely interpreted? I address this dilemma by synthesising insights from the vast amount of literature to develop a definition of reflection and reflective practice in Chapter 2.

A brief summary of some of the international and national research on teachers’ reflective practice follows.

1.1.2 International research on teachers’ reflective practice

International research on teachers’ reflective practice has focused on different research lines. Whereas some researchers have attempted to document and describe the

processes

of teachers’ reflections and associated actions, and the relationship between these processes and teacher development (e.g. LaBoskey, 1994; Russell & Munby, 1991), others

(cited in Zeichner, 1994) have focused on studying the

social and individual conditions

which influence the reflections of teachers (e.g. Ashcroft & Griffiths, 1989; Erickson &

Mackinnon, 1991; Grimmett & Crehan, 1990; Richert, 1992; Wubbels & Korthagen, 1990). In addition Sparks-Langer and Colton (1991) studied and identified three

elements of

teachers’ reflective practice: the cognitive element, which is concerned with the knowledge that teachers need to make good decisions in and on the classroom situation; the

critical element, which relates to social justice and ethics in education; and the narrative

element, which has to do with teacher accounts of their own experiences in the classroom.

Some researchers focused on yet another facet of the phenomenon under discussion, viz. the benefits of reflective practice. York-Barr, Sommers, Ghere and Montie (2006) for instance claim that by engaging in reflective practices, educators increase their learning and improve their practice. This view is supported by Sowder (2007) who argues that reflective teachers plan more effectively because they anticipate students' difficulties. They know what prior knowledge must be present to understand something new and they know how to scaffold knowledge to assist students in developing understanding. Hill, Sleep, Lewis and

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Ball (2007, p. 145) conclude that teachers who can describe, explain and reflect on their work are potentially better teachers, because the ability to articulate one’s practice is an

indicator of deliberateness, and the ability to write cogent reflections is an indicator of

analytic capacity, both of which may predict student achievement.

According to Hill et al. (2007)

the relationship between articulate reflection and effective

instruction has not yet been established clearly. Sowder (2007) however believes that the ability to reflect on practice is a necessity for effective instruction and cites Darling-

Hammond (p. 198) as follows:

Teachers need to be able to analyze and reflect on their practice, to assess the effects of their teaching and to refine and improve their instruction. They must continuously evaluate what students are thinking and understanding and reshape

their plans to take account of what they have discovered.

1.1.3 Research on teachers’ reflective practice reported in Southern Africa

In the Southern African context Nyaumwe (2007) documents four chronological phases that

Zimbabwean preservice mathematics teachers' conceptions of reflections went through: conceptions of classroom management, personal survival, teaching situations and individual learner needs. Case studies of four preservice teachers doing twelve weeks of teaching practice in two different schools provided data for the study through narratives of post-lesson reflective texts and interviews. The findings provide insight into the phases that preservice teachers go through to become reflective practitioners.

Polaki and Morobe (2007) were interested in the issues that teachers in Lesotho focused on as they attempted to reflect on their lessons. When challenged to reflect critically upon their lessons, these teachers either focused on organizational factors or on what the learners were unable to do during the lesson. They never made remarks about how their own actions could have been modified to better support learners' development of mathematical concepts.

Nyanjom (2009) investigated the relationship between mentoring and teachers’ reflective practice at a technical college in Botswana and reports that reflective practice enhances the learning and development of educators. She concludes that reflective practice will assist educators to obtain clarity on issues that pose challenges to their practice (Nyanjom, 2009).

According to Hill (2003) reflective practice is one of the themes in current education discourse which impacts on teacher education in South Africa. She researched the

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relationship between globalisation, reflective practice and assessment and concludes that reflexivity is adaptive discursive behaviour which connects researchers in South African contexts with multiple layers of disembedded relations in global space that impinge on how we function in our situation (Hill, 2003).

1.2. The context of this research study

The context of this research study is lesson study, which could be considered as a special type of case study. In lesson study the focus is on the concrete examination of practice and the testing of new ideas in actual classrooms. This examination of practice is a collaborative exercise in which a group of teachers design, reflect on, and deliver mathematics lessons to enhance learner achievement. Research has shown that lesson study impacts on teachers’ understanding of learner thinking; it enhances teachers’ content knowledge and awareness of new approaches to teaching; it helps teachers to connect their practices to school goals and broader goals; and it creates a demand for improved instruction and allows competing views to be heard during the reflection stage of the lesson study cycle (Lewis, cited in

Sowder, 2007). According to Friedman (2005) the habits of personal reflections on one's teaching that occur during the lesson study process are habits that remain with teachers long after the research lesson is over. This is one of the reasons why I decided to situate my research in this context.

1.2.1 Origin of lesson study

Lesson study has played an important role in professional development in Japan since the beginning of the public education system more than a hundred years ago. One of the reasons for this popularity might be that lesson study provides Japanese teachers with opportunities to do the following: 1) make sense of educational ideas within their practices;

2) change their perspectives on teaching and learning; 3) learn to see their practices from a child’s perspective; and 4) enjoy collaborative support among colleagues (Takahashi,

Watanabe & Yoshida, 2006).

1.2.2 Definition

Lesson study is defined as a form of action research that allows teachers to work with each

other collaboratively as reflective practitioners (Yoshida, cited in Jita, Maree & Ndlalane,

2007, p. 461). The lesson study process consists of a cyclical process and has the following basic components: 1) collaborative planning; 2) lesson observation; 3) reflection on the

4

lesson; and 4) implementation of changes. In Chapter 2 I will discuss lesson study in more detail.

Against this background the rationale for this research study follows.

1.3 Rationale for the study

I would like to believe that the vast majority of teachers have chosen the teaching profession in the hope of making a positive difference in the lives and development of young people.

The rationale for the current research study is embedded in this personal interest in mathematics teachers’ classroom practice on the one hand and a broader interest in making a contribution to the science and art of mathematics instruction on the other.

1.3.1 Personal interest in teachers’ classroom practice

In the past fifteen years I have often observed teachers teaching mathematics lessons. I have also been involved with a mathematics Advanced Certificate in Education (ACE) programme in which students (who are experienced teachers) submit a portfolio with lessons they planned and taught as part of their formative assessment. In most cases they report that the lesson went well and if they ever had to teach the lesson again they would teach it in exactly the same manner. The examples of the mathematical activities they include indicate that they very seldom address higher-order thinking levels in their learners. They also report that the learners were “very happy with the lesson” and “gained a lot of knowledge”.

1.3.2 Concern about mathematics teachers’ ability to reflect on their practice

The reflections of the ACE students on their lessons appear to be very superficial and can be positioned on level 1 of Van Manen’s hierarchical model of levels of reflectivity. Van

Manen (1977) distinguished between three distinct levels of reflective practice. The first level is concerned with the effective application of skills and technical knowledge in the classroom setting. Nyaumwe (2007) found that preservice teachers at this level narrate pedagogy, information on learners, content mastery, and availability of instructional resources or make superficial conclusions and recommendations on their instructional practice. The second level, according to Van Manen (1977), involves reflection on the assumptions underlying specific classroom practices, as well as on the consequences of particular strategies, curricula, etc. Critical reflection occurs on the third level which entails the questioning of moral, ethical, and other types of normative criteria related directly and indirectly to the classroom (Van Manen, 1977).

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1.3.3 Evidence for the need to reflect critically on teaching practice from research

A further incentive to undertake this study is drawn from the research on the importance of reflective practice for effective instruction. In my review of the literature I found that there is little evidence of research on mathematics teachers’ reflective practice in the South African context, which strengthens my rationale to undertake this study. Yet the ability to reflect on practice is considered a necessity for effective instruction (Sowder, 2007). According to

Hillier (2005) there are two reasons to reflect on practice: 1) to change existing practices that will in the long term not actually help learners learn effectively and 2) by reflecting critically teachers become more positive in the search for a new understanding of their teaching practice and find more ways to deal with the challenges that confront them daily. When teachers act reflectively, they consider carefully the problems in their own teaching and think about how those problems are related to their educational or social context. They are aware of the consequences of their teaching and how their own assumptions or beliefs can influence their teaching.

Having provided the rationale for conducting the study the statement of purpose will now discussed.

1.4 Statement of purpose

The main purpose of this research study is to explore mathematics teachers’ reflective practice in a lesson study context. To achieve this aim an in-depth exploration of mathematics teachers’ reflection before, during and after teaching a lesson will be conducted. The possible relationship between mathematics teachers’ reflection and their classroom practice will also be examined. The research will aim to explore whether and how mathematics teachers’ reflection differs from the conceptualisations of reflection in classroom practice as found in the literature. The study will also seek to examine how contextual factors influence the nature of mathematics teachers’ reflective practice.

The research questions that will guide this inquiry will subsequently be discussed.

1.5 Research questions

Given these purposes and objectives and against the background of my working assumptions, the study will seek to address the following main questions:

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Question 1: What is the nature of mathematics teachers’ reflective practice?

To address this main question, the following subquestions will guide the enquiry: a) How do mathematics teachers understand the concept of reflection? b) How do mathematics teachers reflect before, during and after teaching? c) What is the possible relationship between mathematics teachers’ reflection and their classroom practice?

Question 2: How do contextual factors influence mathematics teachers’ reflective practice?

Question 3: What is the potential significance of mathematics teachers’ reflective practice for the science and art of mathematics teaching?

1.6 Methodological considerations

My principal concern is to understand the way in which the teachers in the lesson study group create, modify and interpret the social context in which they function as they plan, teach and reflect on the lesson. Therefore a qualitative inquiry with an epistemological perspective of the interpretive paradigm will underpin this study as I seek to explore the nature of these mathematics teachers’ reflective practice. Within a qualitative approach, I propose a case study design.

1.7

Possible contribution of this research study

The importance of this research rests on its unique connection of reflective practice relating to teaching, specifically in the field of mathematics. Typically educators who are committed to excellent teaching continually seek growth and improvement, as the art of teaching is never a finished product. A changing community of learners requires teachers to grow professionally to be able to justify their pedagogy and educational philosophies. The rationale for this study stems from the premise that mathematics teachers need to find a vehicle for growth and improvement. The development of a reflective process can serve as an important technique in increasing self-knowledge and seeking new ways of educating learners in mathematics. The study can add to research findings concerning reflective practice and contribute to the discussion on the usefulness of including teacher reflection in professional learning programmes.

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1.8 Structure of the thesis

Chapter 2 focuses on a review of the literature relating to teachers’ reflective practice and lesson study, in order to situate this research study. In Chapter 2 a conceptual framework for investigating mathematics teachers’ reflective practice is described. Chapter 3 describes the methodologies, data collection methods and data analyses for this study. Validity issues and ethical considerations are also discussed. In Chapter 4 the data obtained from the participants are presented and discussed. Chapter 5 provides a discussion on the findings related to the research questions, as well as a final summary, conclusions and recommendations for further research on teachers’ reflective practice.

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Chapter 2

Literature review

2.1 Introduction

In this chapter I review the theoretical underpinnings of teacher reflection and reflective practice as found in the literature. Research studies dealing with the reflective practice of preservice teachers as well as practicing teachers will be investigated. I focus on teacher reflection in general and mathematics teachers’ reflective practice in particular. The different meanings of reflection found in the literature will be explored and a definition of teacher reflection for the purpose of this study will be developed. The conceptual framework for this study is based on this review and exploration, and the focus of my study is highlighted in this chapter.

2.2 Theoretical perspectives

on reflective practice in education

Thoughts on reflection and reflective practice have evolved over many decades, if not centuries, through carefully constructed theory and research applications (York-Barr,

Sommers, Ghere & Montie, 2006). John Dewey is frequently recognised as the eminent

20th-century influence on reflection in education (Ottesen, 2007; Pollard, 2002; Rodgers,

2002; York-Barr, et al., 2006; Zeichner & Liston, 1996). The seminal work of Donald Schön

(1983, 1987) has also inspired a renewed interest in reflective practice in the field of education (Lee & Tan, 2004; Valli, 1997). The contributions of Dewey and Schön are discussed in the following section.

2.2.1 Dewey’s approach to reflective practice

Dewey (1933) views the purpose of education as promoting intellectual, social, and moral growth of the individual in order to create a strong democratic society. His interest is in how people think when faced with real and relevant problems. Dewey (1933, p. 17) states that it is reflection that

emancipates us from merely impulsive and routine activity ... enables us to direct our activities with foresight and to plan according to ends-in-view, or purposes of which we are aware. It enables us to act in deliberate and intentional fashion ... to know

what we are about when we act.

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According to Dewey (1933, p. 6) reflection is the active, persistent and careful consideration

of any belief or supposed form of knowledge in the light of the grounds that support it and

the further conclusions to which it tends. Rodgers (2002) distilled four criteria from Dewey’s writing that characterise Dewey’s concept of reflection and the purposes they serve:

1) Reflection is a meaning-making process that moves a learner from one experience into the next with deeper understanding of its relationship with and connections to

2) other experiences and ideas.

Reflection is a systematic, rigorous, disciplined way of thinking, with its roots in scientific enquiry.

3)

4)

Reflection needs to happen in communities, in interaction with others.

Reflection requires attitudes that value the personal and intellectual growth of oneself and of others.

Rodgers (2002) concludes that Dewey was precise in what it means to think reflectively. For

Dewey (1933, p. 4) reflection involves not simply a sequence of ideas but a consequence –

a consecutive ordering in such a way that each determines the next as its proper outcome,

while each outcome in turn leans back on, or refers to its predecessors.

As far as reflective practice is concerned Dewey (1933) claims that true reflective practice takes place only when the individual is faced with a real problem that he or she needs to resolve and seeks to resolve in a rational manner. He identified five general phases of reflective thinking: a problem situation, a tentative interpretation of the elements of the problem situation, careful survey of all attainable considerations which will define and clarify the problem, elaboration of a tentative hypothesis, and testing of the hypothesis. This reflective process of inquiry could be applied to perplexed, troubled, or confused situations in order to bring about a “cleared-up, unified, resolved situation” (Dewey, 1933, p. 106).

According to Lee and Tan (2004) these ideas bring to mind an image of teaching as an inquiry into problematic situations where the reflective teacher is engaged in deliberative inquiry as he/she tries to resolve each problematic situation.

2.2.1.1 Critical evaluation of Dewey’s work

Hillier (2005, p. 15) highlights two points in Dewey’s approach to reflective practice. The first is that his suggestion of the use of a careful survey and elaboration of a tentative hypothesis constitutes a reflective approach to action, rather than a simple trial and error approach to action. Second, Dewey suggests that hypotheses are formulated and then tested through taking action. He draws a distinction between routine action (where external circumstances, habit and tradition, and externally perceived authority, are dominant, and where reasons for practices have not been considered actively) and reflective action (where actions are

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persistently and carefully considered and justifications developed for them) (Dewey, 1933).

In the context of teaching, teachers who act routinely accept the circumstances within which they teach, and will not question the curriculum or the social conditions of their schools, while teachers who act reflectively learn from their experiences, and are proactive in trying out new ideas and solutions to existing problems. These teachers are aware that any action they take leads to new challenges (Hillier, 2005).

According to Pollard and Tann (1993), Dewey’s notion of reflective action, when developed and applied to teaching, is both challenging and exciting. They identified six key characteristics of its implications for teaching (Pollard & Tann, 1993, p. 9):

1)

2)

3)

4)

5)

6)

7)

Reflective teaching implies an active concern with aims and consequences, as well as means and technical efficiency. The reflective teacher should consider not only the immediate aims and consequences of classroom work, but also acknowledge the political process and be willing to contribute to it both as a citizen and as a professional.

Reflective teaching is applied in a cyclical or spiralling process, in which teachers not only plan but also monitor, evaluate and revise their own practice continuously.

Reflective teaching requires competence in methods of classroom enquiry, to support the development of teaching competence.

Reflective teaching requires attitudes of open-mindedness, responsibility and wholeheartedness.

Reflective teaching is based on teacher judgement, which is informed partly by selfreflection and partly by insights from educational disciplines.

Reflective teaching, professional learning and personal fulfilment are enhanced through collaboration and dialogue with colleagues.

A seventh characteristic is added by Pollard (2002): Reflective teaching enables teachers to creatively mediate externally developed frameworks for teaching and learning. Pollard (2002, p. 23) concludes that the aim of reflective practice is to

support a shift from routine actions rooted in common-sense thinking to reflective

action stemming from professional thinking.

Valli (1997) observes that Dewey’s theory separates teachers’ thinking from their action, which means that theory and practice are kept apart. Whereas Dewey highlights a link between intentional reflection and intelligent action (a theme also found in the work of

Schӧn), Schӧn further notes that skilful practice may reveal a kind of knowing that does not stem from a prior intellectual operation (Schӧn, 1992).

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Schön’s work, by contrast, emphasises that professionals continuously face unique situations that they frame and reframe in the light of previous experience, and he recognises the embedded reflection in practice (1983). The common thread between Dewey and Schön rests in the idea of inquiry and experiment in practice as the basis for the development of professional knowledge (Butke, 2003). However, Butke (2003) argues that reflecting on practice through a critical lens is typically related to Dewey’s approach, whereas Schön’s reflection-in-action is based on the notion of the intelligent knowing-in-action that teachers do as they act and interact within the immediacy of problematic situations.

2.2.2 Schön’s approach to reflective practice

Schön (1983) developed his ideas about the reflective practitioner in response to three criticisms of the prevailing positivist epistemology of practice. In the positivistic view, good knowledge had to be scientific and systematic, which Schön called technical rationality

(1987, p. 3). His first criticism relates to the fact that in many professions the product is more important than the process of getting there, a criticism directed at outcome-based projects in education and training. The second criticism is that researchers are not working with practitioners and practitioners are not finding out about recent research. Thirdly, Schön

(1987) argues that there is a separation of knowing from doing and subsequently developed his epistemology of practice to argue the importance of practical knowledge. He argues that in a professional practice there are problems that can be solved through the application of research-based theory and technique, but there are also problems that can only be solved by a form of professional knowing, a form of artistry he called reflection-in-action (Schön,

1983, 1987). Reflection-in-action is a process that is prompted by experience and over which we have limited control (Russell & Munby, 1991). According to Russell and Munby (1991, p.

164), the essence of reflection-in-action is hearing differently or seeing differently, a process

that Schön calls reframing.

Schön (1983, 1987) uses the term reflection-on-action for retrospective thinking, or thinking after the event. The sort of thinking characterised by reflection-on-action involves careful considerations of familiar data when one thinks critically about what has taken place

(Russell & Munby, 1991). A teacher’s reflection-on-action will involve all the different thoughts and feelings he/she has about the teaching of the lesson.

2.2.2.1 Critical evaluation of Schӧn’s work

According to Kinsella (2007, p. 102) the popularity of Schön’s theory is tied in part to his critique of technical rationality, and to his acknowledgement of the significance of practitioner

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experience and indeterminate zones of practice in the development of expertise. Schön tapped into a growing disillusion with technical rationality that coincided with a crisis of knowledge across a range of disciplines (Kinsella, 2007).

A number of researchers (e.g. Court, 1988; Van Manen, 1995) however question the possibility of reflection-in-action and maintain that only limited true reflection is possible while teaching. For example Court (1988) argues that Schön’s examples of “reflection-in-action” appear to involve removing oneself from the action in order to reflect, and thus the term may not be appropriate. Van Manen (1995) maintains that the classroom teacher must constantly act on the spot and cannot step back and postpone acting in order to first reflect on the various alternatives to this action and the consequences of the various alternatives.

However, Russell and Munby (1991) argue that it is only from a researcher’s perspective that reflection-in-action is difficult to detect and challenging to document.

According to Hughes (2009) Schön’s work has been criticised because it does not allow for the complexity of ways in which people reflect on and consider their actions. She argues that

while sometimes reflection is immediate, at other times it is deferred with a need for distance

from the event (Hughes, 2009, p. 453). She cites Brockbank and McGill (1998) who believe that the action which follows a reflection can be instantaneous or postponed (Hughes, 2009).

Newman (1999) maintains that there are fundamental difficulties with Schön’s theories which call into question the often uncritical use made of his ideas. Newman (1999) attributes the popularity of Schön’s ideas in the area of teacher education to Schön’s claim that we need to close the alleged gaps between means and ends, between research and practice and between knowing and doing. His argument is based particularly on criticisms of just how critically reflective Schön’s case studies are and suggest that a better alternative to describe reflective practice would be “critical practice” or “practical philosophy” (Newman, 1999, p.

159). According to Newman (1999, p. 160) both terms suggest an approach which

practitioners can adopt in the different social contexts in which they find themselves.

Zeichner and Liston (1996) argue that although Schön’s conception of reflection-in-action and reflection-on-action and the accompanying spiral of appreciation, action, and reappreciation add both texture and substance to Dewey’s understanding, two features need to be added. First, although reflection can at times be a solitary and highly individualistic affair, it can also be enhanced by communication and dialogue with others. Second, reflection needs to focus not only within the classroom but on the contexts in which teaching and schooling are embedded (Zeichner & Liston, 1996). Hughes (2009) supports this argument and believes that an individual’s reflection on practice must draw on the norms

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and agreed behaviours of the professional community in which s/he is participating. Thus, a

teacher’s reflections will be influenced by the practice of others in the local school, university or college environments as well as by current thinking on what is good educational practice

and peer support for reflection will be invaluable (Hughes, 2009, pp. 453-454).

According to Farrell (2004) the focus of reflective practice has dulled somewhat in the late

1990s, with some individuals in education believing it was just one more bandwagon that administrators and university researchers had jumped upon. He believes that it has become unclear just what reflective practice really means to the practising teacher. Hillier (2005) reasons that reflective practice has become a byword for a range of practices and meanings which do little to challenge our tacit assumptions and implicit, informal knowledge.

Nonetheless, there continues to be tremendous conceptual and practical confusion surrounding what reflective practice is and in what ways it is distinct from other modes of reflective theorising (Clark, 2001; Fenstermacher, 1988; Procee, 2006). The debate over different definitions and approaches on reflection and reflective practice is addressed at a later stage in this review of the literature.

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Before I review the research studies in education based on reflection, it is necessary to explain how theorists in the past three decades have assembled working models to define and categorise the reflective process (e.g. Valli, 1997; Van Manen, 1977).

2.3 Categories of reflection in education

The depth of teachers’ reflection can be measured at different levels as identified by a number of researchers (Dewey, 1933; Hatton & Smith, 1995; Jay & Johnson, 2002; Lee,

2005; Valli, 1992; Van Manen, 1977). For example Dewey (1933) has identified the following phases of reflection: interpretation of experience (recognition of possible solutions to the problem); description of experience (problematising or intellectualising the situation); analysis (generating hypotheses that might lead to possible solutions) and overt action on the part of the thinker (experimenting, testing hypothesis) (Lloyd, 2005; Mewborn, 1999).

2.3.1 Van Manen’s levels of reflection

In my review of the literature on reflective practice I realised that Van Manen’s levels of reflection (1977) are still used extensively to determine the depth of reflection during action. I will therefore start the discussion on the categories of reflection with Van Manen’s levels of reflection.

2

See paragraph 2.4.1.1

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Van Manen (1977) distinguished between three distinct levels of reflective practice. The first level is concerned with the effective application of skills and technical knowledge in the classroom setting. Level one, technical rationality, consists of responses that deal with the technical application of educational knowledge and basic curriculum principles, such as are the students doing what the teacher asked them to do? At this level reflection entails only the appropriate selection and use of instructional strategies in the classroom (Van Manen,

1977). The contexts of the classroom, school, community, and/or society are not taken into consideration. This is the most basic level of reflection and is concerned with the efficiency and effectiveness of the means used to attain ends which are accepted as given (Zeichner,

1994). In essence, the first level is not reflective in the sense that it will result in changing behaviour. It is merely a reaction to an observation that a problem exists, and in that sense it links with behaviouristic ideas. Zeichner (1994, p. 13) argues that the reason why teachers reflect on this very superficial level is because most schools are hostile to critical enquiry.

However, many researchers agree that this technical level of reflection is also important because it relates to the everyday world of the teachers (Griffiths & Tann, 1992; Zeichner,

1994).

At the second level, practical action, the teacher becomes more concerned with clarifying assumptions and predispositions while assessing the educational consequences towards which a teaching action leads. The teacher analyses student and teacher behaviour to see if and how goals are met. In other words, at the second level of reflectivity teachers would begin applying educational criteria to teaching practice to make individual and independent decisions on pedagogical matters (Van Manen, 1977). The outcomes for students are also investigated on this level (York-Barr et al., 2006).

The third level is critical reflection. At this level, educators are concerned with worth of knowledge and the social circumstances useful to students without personal bias. The teacher asks her/himself several questions such as what were the strong points in the lesson, what should be changed, and was the content that was covered important to the students. At this level there will be a concern for justice, equity and the satisfaction of important human purposes within the larger social context. Critical reflection entails the questioning of moral, ethical, and other types of normative criteria related directly and indirectly to the classroom (Van Manen, 1977). A number of researchers question the essentiality of the role of critical reflection in education (Valli, 1997; Zeichner & Liston, 1987), emphasising that educators must critically examine how instructional and other school practices contribute to social equity and to the establishment of a just and humane society.

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According to Van Manen (1995) reflection is central to the life of the educator. He notes that it is in the very nature of the pedagogical relation that the teacher reflectively deals with children, rather than doing so unthinkingly, dogmatically, or prejudicially. However, he argues that the concept of reflection is challenging and may refer to a complex array of cognitively and philosophically distinct methods and attitudes (Van Manen, 1995).

2.3.2 Valli’s levels of reflection

Valli (1992) identified six different types of reflection. The lowest level is behavioural, which she admits is prescribed, not reflective content (Valli, 1992, p. 220). According to Valli (1992, p. 217), technical reflection is the second level of reflection, focusing on general instruction and management practices based on research. The focus of this type of reflection is on the narrow domain of teaching techniques or skills. The quality of reflection is to match one’s own performance to external guidelines. Other levels of reflection that she proposes are:

reflection-in and on-action, which focus on one’s own teaching performance and making decisions based on one’s own unique situation; deliberative reflection, which can focus on a wide array of teaching related practices and concerns but involves intentional consideration of assumptions, different perspectives, and research findings; personalistic reflection, which focuses on one’s own growth and relationships with students and involves learning to listen to one’s own inner voice, as well as the voices of others; and critical reflection, which focuses on social, moral and political dimensions of education and involves making judgements based on ethical criteria (Valli, 1992, pp. 217-219).

Zeichner and Liston (1987) acknowledge the importance of reflection at all the levels suggested by Van Manen and Valli, but encourage teachers to critically reflect also on curriculum goals, educational ends as well as school and societal structures. The focus here seems to be on the teacher and not on the teaching and learning situation. However, several other researchers have identified teacher reflection on their practice and student learning as critical to the success of reform (e.g. Artzt, Armour-Thomas & Curcio, 2008; Darling-

Hammond, 1998). According to Darling-Hammond (1998, p. 8)

teachers need to be able to analyze and reflect on their practice, to assess the effects of their teaching, and to refine and improve their instruction. They must continuously evaluate what students are thinking and understanding and reshape

their plans to take account of what they have discovered.

According to Hatton and Smith (1995) there are several fundamental flaws in Valli’s conception of reflection levels, especially with regard to the placement of Schӧn’s reflection-

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in-action at level 3. From his own description this would appear to be the most complex and demanding kind of reflection which needs considerable experience (Schӧn, 1983).

2.3.3 Jay and Johnson’s levels of reflection

Jay and Johnson (2002) examined the various facets of reflection with respect to teaching and subsequently outlined a systematic classification of reflective thought on three dimensions.

The first level of reflection is descriptive reflection which involves describing a situation or a problem. Such problems may be specific and explicit, as when teachers know that the curriculum is not working for their students and find they need to make a change, or vague and implicit, as when teachers sense a resistant tone from a class but do not know why (Jay

& Johnson, 2002).

The second level of reflection is comparative reflection which involves thinking about the situation from different perspectives. As opposed to a technical approach to teaching, in which a teacher accepts a problem immediately and sets about trying to solve it, a reflective practitioner looks for distinct ways to pose a problem and attempts to get a different

purchase on the students and the issues involved (Zeichner & Liston, 1996, pp. 4-5). On this level teachers try to solve a problem while also questioning their values and beliefs.

The third level of reflection is critical reflection and on this level teachers consider all the different perspectives of a situation or problem and all the players involved: teachers, students, the school, and the community (Jay & Johnson, 2002).

Akbari (2007, p. 195) considers these levels of reflection useful, especially the last level, critical reflection, which she calls the decision-making stage resulting from careful analysis of

the situation and deliberation. This last stage forms the basis for the formulation of alternative ways of teaching or approaching the problem on the part of the teacher.

2.3.4 Hatton and Smith’s levels of reflection

Hatton and Smith (1995) contend that the reflective process was more developmental than hierarchical in nature. They defined three distinct levels of reflection. On the most basic level

(in agreement with Van Manen (1970) and Valli (2002)), they place technical rationality.

The nature of reflection on this level is a technical decision making about immediate behaviours and skills. On this level one begins to examine one’s use of essential skills.

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Hatton and Smith’s (1995) second level of reflection is reflection-on-action. On this level they distinguish between descriptive, dialogic, and critical reflection (Hatton & Smith, 1995, p. 45). The nature of reflection-on-action is descriptive when one is analysing one’s performance, giving reasons for actions; dialogic when one is weighing competing claims and viewpoints, exploring alternative ways of solving problems; and critical when one is thinking about the effect of one’s actions upon others, taking into account social, political and cultural forces.

On the highest level they define reflection-in-action, where one is dealing with on-the-spot professional problems as they arise (Hatton & Smith, 1995).

2.3.5 Lee’s levels of reflection

Lee (2005, p. 703) proposes the following levels to assess the content and depth of reflective thinking:

On a recall level (R1) one describes what they experienced, interprets the situation based on recalling their experiences without looking for alternative explanations, and attempts to imitate ways that they have observed or were taught.

On a rationalization level (R2) one looks for relationships between parts of their experiences, interprets the situation with rationale, searches for ‘‘why it was,’’ and generalizes their experiences or comes up with guiding principles.

On a reflectivity level (R3) one approaches experiences with the intention of changing/ improving in the future, analyses experiences from various perspectives, and is able to see the influence of cooperating teachers on students’ values/behaviour/achievement.

The following table summarises the proposed levels of reflection by the researchers discussed in this section. From the table it appears that although there is little agreement among researchers on the labels used to describe the various levels of reflection, the levels generally appear not to overlap each other. However, Hatton and Smith (1995) believe that

Schӧn’s reflection-on-action and reflection-in-action incorporate all levels of reflection, even critical reflection.

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Table 2.1

Summary of levels of reflection proposed by researchers

Researcher

Dewey (1933)

Hatton and Smith (1995)

Jay and Johnson (2002)

Lee (2005)

Valli (1992)

Van Manen (1977)

Reflective levels

Interpretation of experience

Description of experience

Analysis

Action

Technical rationality

Reflection-on-action

Reflection-in-action

Descriptive reflection

Comparative reflection

Critical reflection

Recall level

Rationalisation level

Reflectivity level

Behavioural

Technical reflection

Reflection-in and on-action

Deliberative reflection

Personalistic reflection

Technical rationality

Practical action

Critical reflection

As can be seen from the table these theorists/researchers use different terms to identify the levels of reflective thinking. According to Lee (2005) reflection on Level 1 is mainly concerned with mastery and/or application of technical means for achieving given educational ends, and includes a simple description of observation or a focus on behaviours or skills from past experience. Reflection on Level 2 is directed at an interpretive understanding of the meanings of educational experiences and choices of action within a particular social and institutional context (Lee, 2005). Reflection on Level 3 links classroom practice to the broader arena of political, moral, and ethical forces (Lee, 2005). Reflection on the first level directs a teacher’s practice, while reflection on the second level informs a teacher’s practice through examining his/her beliefs that guide actions in light of context.

Reflection on the third level transforms practice, for it reconstructs experience in light of a

life characterized by justice and equality (Lee, 2005, p. 703).

2.4

Research studies in education investigating teacher reflection

In this section I will first report on research studies in teacher education investigating preservice teachers’ reflection, and then explore research studies dealing with practising teachers’ reflections. Lastly I will focus on research studies dealing with mathematics

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teachers’ reflective practice. I will use these studies to explore the different meanings of reflection found in the literature, the content of teacher reflection (what do teachers reflect on?), the nature of student teachers’ and teachers’ reflection (how do they reflect?), the moment of reflection (when do they reflect?), the benefits of and barriers to reflection reported in the literature, and the contextual factors that might influence teacher reflection.

2.4.1 Overview of research studies dealing with preservice teachers’ reflections

3)

4)

5)

6)

From my literature review I have found that researchers focus on different research lines when investigating preservice teachers’ reflections. Whereas many researchers are concerned about the definition of reflection (Hatton & Smith, 1995; LaBoskey, 1994; Lee &

Tan, 2004), some focus on the content of student teachers’ reflections (LaBoskey, 1994;

Lee, 2005; Liou, 2000; Mewborn, 1999; Pedro, 2001), while others explore the nature of these reflections (Loughran, 2002; McKeny, 2006; Ottesen, 2007). Some researchers address the moment of reflection (depending on the methods they use to collect their data)

(Pedro, 2001), a few mention benefits of reflection (LaBoskey, 1999) and some consider

contextual factors that play a role in student teachers’ reflection on practice (Lee & Tan,

2004). It seems that research on reflection in teacher education can be explored using the following lenses (as illustrated in Figure 2.1):

1)

2)

definition of reflection or reflective thinking; the content of the preservice teachers’ reflections; the nature of their reflection; the moment of reflection;

benefits of reflection or barriers to reflection; and

contextual factors that influence preservice teachers’ reflections.

Definition

Contextual factors

Content

Lenses

Benefits Nature

Moment

Figure 2.1 Lenses used to explore studies dealing with preservice teachers’ reflections

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A brief discussion of studies on reflection in teacher education follows. As will become clear, the studies focus on different facets of reflection, as depicted in Figure 2.1.

2.4.1.1 Concerns about the definition of reflection as revealed in studies dealing with preservice teachers’ reflections

According to Hatton and Smith (1995) reflection is claimed as a goal in many teacher preparation programs, but its

definition and how it might be fostered in student teachers are problematic issues. They cite Dewey (1933) who considered reflection as a special form of problem solving and debate whether reflection is limited to thought processes about action, or whether it is more inextricably bound up in action (Hatton & Smith, 1995).

Although Schӧn (1983; 1987) clearly relates reflection to action, using his terms “reflectionin-action” and “reflection-on-action”, other researchers seem to view reflection as a special form of thought, (Artzt, Armour-Thomas & Curcio, 2004; Sparks-Langer & Colton, 1991).

According to Rodgers (2002) thinking, particularly reflective thinking or inquiry, is essential to both teachers’ and students’ learning. However, she claims that although the cry for

accomplishment in systematic, reflective thinking is clear, it is more difficult to distinguish

what systematic, reflective thinking is (Rodgers, 2002, p. 842). She mentions four problems associated with this lack of definition that make achievement of such a standard difficult (p.

843):

1) It is unclear how systematic reflection is different from other types of thought.

2) It is difficult to assess a skill that is vaguely defined.

3) Without a clear picture of what reflection looks like, it has lost its ability to be seen and therefore has begun to lose its value.

4) Without a clear definition, it is difficult to research the effects of reflective teacher education and professional development on teachers’ practice and students’ learning.

According to LaBoskey (1994) one problem in using the term “reflection” in teacher education is that it is not made clear which particular meaning one has in mind. A second problem is that the definitions are not used consistently by the theorists, researchers, or teacher educators who employ them (ibid.). These views are supported by a number of researchers (e.g. Rodgers, 2002; Sparks-Langer & Colton, 1991; York-Barr et al., 2006).

However, she agrees that there are many well-constructed meanings of reflection and mentions Dewey’s notion of reflection as the active, persistent and careful consideration of

any belief or supposed form of knowledge in the light of the grounds that support it and the

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further conclusions to which it tends (LaBoskey, 1994, pp. 3-4). Dewey (1933, pp. 100-101) characterises reflection as a specialised form of thinking and describes its function as

transform[ing] a situation in which there is experienced obscurity, doubt, conflict, disturbance

of some sort, into a situation that is clear, coherent, settled, harmonious.

LaBoskey’s definition of reflective thinking is based on what Dewey (1933, p. 8) called

grounded belief: Reflection thus implies that something is believed in (or disbelieved in), not

on its own direct account, but through something else which stands as witness, evidence,

proof, voucher, warrant; that is, as ground of belief. LaBoskey (1994, p. 9) contends that good teaching requires thoughtful, caring decision making and that good teachers are constantly making decisions and formulating ideas about educational goals, practices and outcomes, and these definitions and ideas are subjected to careful reconsideration in the

light of information from current theory and practice, from feedback from the particular

context, and from speculation as to the moral and ethical consequences of their results.

However, she claims that reflective thinking is a necessary, though not sufficient, component

of the teaching process (LaBoskey, 1994, p. 122).

Pedro’s (2001) qualitative study explores how five preservice teachers construct meanings of reflection, and how these meanings inform their practice. He (2001) found that the preservice teachers who were participants in his study possessed varying notions of reflection in teaching as interpreted in their definitions of reflection. Two of these participants stated that they looked back on their actions to think about what they could have changed; while three other participants said that they wanted to see what they could do to change or improve their performance in the future.

Lee and Tan (2004) propose four critical attributes as criteria to define and distinguish reflection from ordinary forms of thinking. The first criterion is an examination of practice, where reflection is not merely recalling a teaching incident in a general manner, but reflective thinking is seen as focused and directed at particular issues or concerns about practice (Lee

& Tan, 2004). Reflective thinking is triggered by the need to examine one’s practice because there is an awareness of some problematic aspects (ibid., 2004). The second attribute of reflection which they mention is reflexivity or reflective awareness, which unites the external world (the classroom situation) with the internal world (personal beliefs, assumptions and values) of the practitioner (ibid., 2004). Thirdly they regard reflection as a constructive process on which the reflective teacher is not merely practising his/her craft passively or without questions, but responds actively to potentially problematic situations (ibid., 2004).

The fourth attribute that Lee and Tan (2004) regard as a criterion to define reflection is a

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process of transformation, whereby in the teacher education context, reflection aims at fostering professional growth which will leave the teacher with transformed understandings of the situation and a heightened sense of the self-as-teacher.

McKeny (2006, p. 18) agrees with other researchers (e.g. LaBoskey, 1994; Lee & Tan, 2004;

Pedro, 2001) that there is little consensus as to what constitutes reflection and what teacher

educators might do to foster its development. He defines reflective teaching as the active

use of reflection on the part of the participants in both oral and written forms as a way of coming to understand the complex task of teaching and coming to understand their efforts in

the act of teaching (p. 15). Reflection by preservice teachers is operationally defined as an

interactive process of reporting, reviewing and rethinking previously held beliefs and dispositions about teaching and learning as a means of developing personally useful and

practical knowledge into action within the classroom (McKeny, 2006, p. 19).

2.4.1.2 Research studies reporting on the nature of preservice teachers’ reflections

It seems as if different interpretations are attached to what the “nature” of a teacher’s reflection constitutes. LaBoskey (1994) investigated the nature and stability of reflection in preservice teacher education and a possible means for its measurement. Her research includes an exploration of the

interactions between individual beliefs, attitudes,

emotions and inquiry skills and a particular reflective strategy. Twelve student teachers enrolled in the same teacher education programme were selected as participants. Six of these students were rated as most reflective at the time of enrolment and called the alert

novices, whereas the other six were rated as least reflective and called the commonsense

thinkers (LaBoskey, 1994, p. xii). She found that the alert novices were concerned with “why” they were doing what they were doing, and with the meanings and implications of their values and philosophies (LaBoskey, 1994). An example of a “why” question is: “Why am I teaching what I am teaching in the way that I am teaching it?” (p. 108). Only two of the commonsense thinkers asked “why” questions, which means that four students were not concerned with reflecting on what they were doing. This finding is confirmed by Russell et al.

(1988, p. 88) who cite examples of both beginning and experienced teachers who seem

unable to reflect on their practices, unable to reframe their problems and unable to interpret

their practices in more than one way.

The nature of the preservice teachers’ reflections in Pedro’s study (which he calls the

context in which reflection takes place) was revealed through their self-reflections, verbal reflections and written reflections and journals (2001). All the students reflected on a technical level, and some engaged in reflections at the interpretive level when they moved

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beyond thinking about their skills and knowledge, to think about the consequences of their actions, and the goals of teaching (Pedro, 2001). He reports that the preservice teachers in

this study demonstrated their ability to reflect with their peers and others about technical, as well as interpretive, and to some extent critical issues that they had dealt with in their

practice (p. 158).

Lee (2005) investigated three student teachers’ reflective thinking in field experiences in

Korea. The data collection involved interviews, observations, written journal entries and questionnaires. Lee (2005) found that these student teachers’ reflective thinking levels were affected by the mode of communication. Some showed strength in written reflections while others reflected more deeply in the verbal format. It seems as if it is important to create various opportunities for reflective thinking, rather than to limit students/teachers to a particular approach. The students in this study all reflected on the three levels proposed by

Lee (2005): Recall level, Rationalisation level and Reflectivity level. However, Lee (2005) found that the development of student teachers’ reflectivity is influenced by conditions such as the cooperating teacher’s characteristics, teaching opportunities and the teaching context.

Ottesen (2007) also investigated how student teachers reflect on their practice and what they accomplish through these reflections and found that although reflection is evident in nearly every session, it is commonly neither systematic, nor extended in time. She found that the objects of reflection emerge from puzzling or disturbing aspects of teaching experiences, or some problem they experience when planning their lessons (Ottesen, 2007).

According to McKeny (2006) the nature of preservice teachers’ reflection is revealed only when they experience cognitive dissonance. As long as classroom activities and processes

run smoothly, there is no true call for learning and reflection (ibid., 2006, p. 421). When a problem arises the level of reflection is technical and practical in the sense that the problem situation is considered, potential action is considered and concerns for the possible implications for student learning are considered (McKeny, 2006).

Husu, Toom and Patrikainen (2008) also studied the quality (depth) of eight student teachers’ reflection using the procedure of guided reflection and report that contrary to many previous studies (they cite Dinkelman, 2000; Francis, 1995; Harrington, Quinn-Leering, &

Hodson, 1996) student teachers are capable of using various kinds of reflection when analysing their teaching practices. Furthermore, Husu et al. (2008) share the common assumption that it is hard for student teachers to move beyond immediate concerns of their teaching practice (habituation) to addressing long-term inquiries in their profession.

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Ward and McCotter (2004) analysed exemplars of student teachers’ reflection to study their

levels of reflection by using a reflection rubric that consists of four levels of reflection:

Routine (low level reflection lacking questioning and a sense of responsibility for change);

Technical (reflection is used as a means to solve specific problems); Dialogic (discussion and consideration of the views of others) and Transformative (questions fundamental assumptions and purpose more deeply). They found that the reflections of beginning teachers reinforce the fact that reaching levels of transformative reflection is unusual and difficult (Ward & McCotter, 2004). Watts and Lawson (2009), who used Ward and McCotter’s rubric (2004) in a meta-analysis activity where students identified the quality of critical reflection in their lesson evaluations, found that the activity can result in a qualitative improvement in the nature of critical reflection (Watts & Lawson, 2009).

2.4.1.3 Research studies reporting on the content of preservice teachers’ reflections

The content of reflection of the participants in LaBoskey’s (1994) study consists of reflections on the student, the teacher and the lesson. Only a few of these preservice teachers reflect on the personal enjoyment and degree of enlightenment gained from the teaching experience. In this study the data were collected using case investigation write-

ups, freewrites, questionnaires, supervisor summaries and interviews (LaBoskey, 1994, pp.

32-34). The preservice students were not observed in the classroom.

Lee (2005) assessed student teachers’ reflections and found that there are variations in the content, and that the pace at which reflective thinking deepens depends on the student teachers’ personal background, field experience contexts, and the mode of communication.

The three student teachers in this study reflected on pedagogical issues (curriculum/content, instructional skills, lesson preparation, and teaching styles), learner behaviour, and the gap between the ideal and the reality of education (Lee, 2005). The study provides insights into how to measure the quality of reflective thinking and how to enhance reflective thinking and cultivate reflective practitioners, including the kinds of experiences that could be incorporated in a teacher education program (Lee, 2005).

Liou (2000) studied 20 student teachers’ written reports on their teaching practice and found that student teachers reflect on topics related to teaching and that the level of their reflections is more descriptive than critical. The student teachers wrote about theories of teaching and teaching approaches, classroom management and evaluating own and other teachers’ teaching in their reports.

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In a study of four preservice teachers during a field experience connected to a mathematics methods course, Pedro (2001) found that curriculum matters seemed to interest four of the preservice teachers in his study. The preservice teachers thought about their lesson planning, and how they could adapt the lessons to suit the diverse needs of the students.

Assessment was another area that interested these preservice teachers. They reflected on the various forms of assessment they used in their student teaching practice. The issue of the diversity of students with special needs (and also diversity of race) seemed important to the preservice teachers who reflected on their concern for students with special needs in the classroom.

Mewborn (1999) investigated elements of mathematics teaching and learning that four preservice elementary teachers’ found problematic during a field-based mathematical methods course. She identified four areas of concern that these preservice teachers reflected on (listed in the order they were addressed): 1) matters of classroom organization

(physical arrangement of classroom) and management (behaviour of individual children); 2) mathematics pedagogy; 3) children’s mathematical thinking; and 4) mathematics content.

2.4.1.4 Research studies reporting on the moment of student teachers’ reflection

Student teachers reflect on their actions when encouraged to do so through reflective writings and journal entries, in interviews or during conversations with their mentors (Griffin,

2003). The moment of reflection will therefore usually be after-action. During classroom observations researchers might be able to see reflection-in-action. Hatton and Smith (1995) argue that it is therefore important to employ different methods to investigate teachers’ reflections.

Griffin (2003) used critical incidents in a supervised field experience to increase the capacity of preservice teachers to develop reflective and critical thinking skills. These critical incidents provide a deeper and more profound level of reflection-on-action because it goes beyond a detailed description of an event that attracted attention, to analysis of a reflection on the meaning of the event (Griffin, 2003). The incidents were collected after the participants’ field experience and reflections were analysed by a panel. The majority of the reflections were placed on Van Manen’s (1977) Levels 1 and 2. The critical incidents appeared to assist concrete thinkers to look beyond themselves and the immediate situation to larger, contextual issues (Griffin, 2005).

The moment of reflection for the four preservice teachers in Pedro’s study (2001) was after teaching a lesson, in other words they reflected on-action. They thought about actions that

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did not go well in the classroom, and they questioned what could be done to change those actions (Pedro, 2001). Only one of the student teachers in this study reported how he reflected in-action (Pedro, 2001, p. 148): I can reflect instantly, and I can tell by the class

atmosphere if students understand things and you can instantly change your mind. Three of the preservice teachers also thought about how they would change their actions in future

(reflection-for-action) (ibid., p. 114). The data was obtained through individual interviews, reflective journals and observation.

As far as the moment of reflection is concerned, Lee and Tan (2004) found that the participants’ in their study only reflected when they encountered problems in their lessons.

This over-emphasis on teaching problems prevented student teachers from deliberating on other aspects of teaching with their supervisors (Lee & Tan, 2004). The data for this study was collected using observations, post-conferences, interviews and artefacts (Lee & Tan,

2004).

2.4.1.5 Research studies reporting on the benefits of or barriers to reflection

It has been suggested by a number of researchers that there are certain benefits of reflective practice (Brubacher, Case & Reagan, 1994; Craft & Paige-Smith, 2008; Farrell,

2004; Loughran, 2002; Sowder, 2007; Valli, 1992). For example, Zeichner and Liston (1996, p. xvii) believe that it is through reflection on our teaching that we become more skilled, more capable, and in general better teachers. It seems that when teachers reflect on their practice they identify problems they experience while teaching and are able to make sense of their learners’ understanding of concepts. García, Sánchez, Escudero and Llinares (2006) report that through reflection they have developed their own identity as teacher educators and as a consequence of reflection, their practice as teacher educators started to be modified and led to new understandings about how their student teachers learn.

However, LaBoskey (1994) found that some of the preservice teachers who were less likely to reflect were very good teachers who were skilful, well organised, and productive in the classroom. From her research it seems that reflection is not a prerequisite for being an effective teacher, although she claims that

we cannot afford to have teachers who are unwilling or unable to analyze the sources, meanings, and implications of their beliefs about their students and the learning process; who do not attempt to examine the nature of problems and their

underlying causes or to explore alternative solutions (LaBoskey; 1994, p. 123).

Loughran (2002) examined the value of reflection as a meaningful way of approaching learning about teaching so that a better understanding of teaching, and teaching about

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teaching might develop. He concludes that many teacher education programs have

incorporated views of reflection into their course structures, but the effectiveness and forms

of adoption may well be limited by the largely traditional nature of the programs to begin with

(Loughran, 2002, p. 42). Lee (2004) reports that reflective teacher education programmes lack clear conceptual focus because the concept of reflection is used in a generic sense. He believes that without supportive contexts, reflection fails to support student teachers’ professional development, especially when the supervisors lack clear a understanding of reflection, which can be seen as a barrier to promoting reflective practice (Lee, 2004).

2.4.1.6 Research studies reporting on contextual factors that influence reflection

Lee and Tan (2004) investigated how reflection was implemented in the Malaysian teacher education context. Their findings indicate that student teachers reflect not only publicly through existing mechanisms (e.g. post-conference discussions, post-lesson analyses and weekly journals), but also privately by examining their own teaching, their pupils, and their beliefs or values on teaching. These private reflections were obtained through informal interviews. The public reflections were very weak in contrast with the private reflections that were rich and varied (Lee & Tan, 2004). Some reflections were focused mainly on technical skills such as how to implement activities, give instructions, or manage the pupils. On the private level Lee and Tan (2004, p. 126) found that students thought about complex issues

that were not directly related to day-to-day teaching events, for example the heavy

responsibilities of teaching, their values, and their inadequacies as teachers. Two significant findings were: 1) student teachers’ reflective practices lack an element of enquiry, and 2) reflective practices were carried out individualistically.

Lee and Tan (2004) identified the following main

contextual factors that influence

reflection: 1) Interpersonal contexts play a crucial role on student teachers’ understandings and practice of reflections. They report (Lee & Tan, 2004) that the mentor lecturers did not guide the students’ reflections and the student teachers were left to learn to teach purely from their own experience. 2) Personal dispositions play a role in student teachers’ practice of reflection. Lee and Tan (2004, p 137) report that not all individual student teachers are

equally predisposed to be reflective on their practices. Competence and confidence have emerged as important factors.

2.4.2 Summary

Table 2.2 provides a summary of the studies discussed above dealing with preservice teachers’ reflection.

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Table 2.2

Summary of studies on preservice teachers’ reflective practice

LaBoskey

(1994)

Mewborn

(1999)

Liou

(2000)

Pedro

(2001)

Definition of reflection

Constant reconsideration

Reflection is qualitatively different from recollection or rationalisation.

Action is an integral part of the reflective process

Critical reflection is examining teaching experiences as a basis for evaluation and decision making and as a source for change

Thinking about an action to make some change

Content of reflection

Student, teacher and the lesson

Nature of reflection

(Reflection levels)

Revealed by “why” questions, directed at the roots of problems and the meanings of ideas and actions

(Van Manen’s Levels

1, 2 and 3)

Used Dewey’s phases of reflective thinking

Classroom context;

Pedagogy of teaching mathematics;

Children’s mathematical thinking;

Mathematical content

Practical teaching issues and evaluating other teachers’ and own teaching

Mostly descriptive reflection and less critical reflection

Moment of reflection

After action

After action

No systematic procedures for reflection-inand on-action evident in the study

Curriculum, assessment and diversity of students

Van Manen’s levels:

Level 1: (technical reflection) and Level

2: (practical or interpretive reflection)

Reflection-onaction and reflection-foraction

Obtained by using …

The written word

Individual interviews, group discussions, individual journals and teaching episodes

When thinking reflectively about the multiple aspects of their teaching, field experiences can have a positive effect on student teachers’ learning about teaching mathematics

Written reports by the students

Benefits/Barriers of reflection

Not evident in the study

Interviews, reflective journals and observation

Reflection as a conceptual device to help think about knowledge, and better teaching skills

Contextual factors that influence reflection

Not evident in the study

Teaching environment that promotes investigation and inquiry into the problems of mathematics teaching

Critical reflection raises teachers’ awareness about teaching, enables deeper understanding of themselves as teachers and triggers positive changes in practice

Support needed from the educational system and the sociocultural context

Not evident in the study

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Lee and

Tan (2004)

Ward and

McCotter

(2004)

Examination of practice in response to problem situation to obtain professional growth

Reflection is situated in practice, cyclic in nature and makes use of multiple perspectives

Professional and own growth

Prior knowledge, instructional strategy, struggling students

Observations, postconferences, interviews and artefacts

Publicly on a technical level in problem situations

(Level 1) and privately on practical and even critical levels (Levels 2 and

3)

Hatton and Smith’s levels

Reflection-inaction and onaction

After the action with reflective notes

Reflective text, lesson plans, samples of student work

When teachers act reflectively, they consider carefully the problems in their own teaching and think about how those problems are related to their educational and social contexts

Not discussing the qualities of good reflection with student teachers is a barrier to their reflective practice

Interpersonal contexts

(mentor teachers) and personal disposition

(competence and confidence) of teacher

Scaffolding needed to reach higher levels of reflection

Lee (2005)

McKeny

(2006)

Ottesen

(2007)

Any form of thinking

Rethinking previously held beliefs about teaching in order to develop as a teacher

Reflection is seen as embedded in and emerging from activity.

Thus, reflection is a social activity.

Discipline, instructional skills, relationship with students

Mathematics content, teaching styles, own competence as a teacher

Puzzling or disturbing aspects of teaching experiences

Recall level (R1);

Rationalisation level

(R2); Reflectivity level

(R3)

Only in a problem situation and then on a technical and practical level (Levels

1 and 2)

Reflections focus on what they had observed, done and were unable to do, and hoped to do in future

Reflection-onaction

Reflection as induction; reflection aimed at conceptual development; reflection on experience

When confronted with puzzling situation

Interviews, observations and written documents

Questionnaires, reflective writing, observations, interviews, focus groups, supervisor feedback forms

Analyses of recorded conversations between student teachers and mentors

The lack of a clear definition of reflection and vague criteria to assess the quality of reflective thinking create problems in implementing reflective activities

Cooperating teachers’ characteristics; teaching context

Teacher growth, personal growth, professional growth and building a supportive community of learning

Mentor teachers, honesty and integrity of feedback, and personal freedom to explore own thinking

The motivation for reflective action must be sought in the object to be transformed

Interpersonal contexts

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Husu,

Toom and

Patrikainen

(2008)

Reflection is believed to be a genuine way of fostering change

Critical incidents selected by the student teachers

Habitual reflection and introspective reflection

After critical incidents

Analysing critical incidents

Students reported professional growth

Implementing the procedure of guided reflection in students’ teaching practicum

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From Table 2.2 it seems that most researchers agree that student teachers reflect on their actions when a problem situation arises in class. However, the nature of their reflection differs. Most preservice teachers reflect only on a technical level (Level 1), concerned with the effective application of skills and technical knowledge in the classroom setting

(Brubacher, Case & Reagan, 1994). Some of these preservice teachers reflect on a practical level (Level 2), concerned with the assumptions underlying specific classroom practices as well as the consequences of particular strategies, curricula, and so on (ibid., 1994). A few preservice teachers do reflect on a critical level (Level 3), which entails the questioning of moral, ethical, and other types of normative criteria related directly and indirectly to the classroom (ibid., 1994)

According to LaBoskey (1994) reflection is incorporated in the goals and practices of the teacher education programmes in her study, and students must engage in acts of reflection in order to learn during the programme and beyond. Most of the preservice teachers reflect on-action after the teaching experience; although a few reflect for-action (also after the teaching experience but considering future actions should they experience the same problem situation again). They reflect on the student, the teacher and the lesson, but also on curricula, assessment and student diversity. The preservice mathematics teachers also reflect on the content they have to teach.

One of the benefits of reflection is to grow as a teacher, both professionally and personally

(Sowder, 2007). However, LaBoskey (1994) found that some of those students whom she regarded as less reflective than others also turned out to be effective teachers. This is in contrast with the findings of a number of theorists and researchers (e.g. Dewey, 1933;

Schӧn, 1983, 1987). According to Brubacher, Case and Reagan (1994, p. 18) good teaching

requires reflective, rational, and conscious decision making.

One of the major contextual factors which seem to have an impact on preservice student’s reflections is the role of mentor teachers. However, personal dispositions (like the competence and confidence of the preservice teacher), together with the quality of feedback from mentor teachers and university supervisors also appear to play a significant role.

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2.5 The reflective practice of practising teachers

2.5.1 Introduction

It is generally acknowledged that reflection is an important part of the professional behaviour of teachers and essential for the stimulation of their professional development (Luttenberg &

Bergen, 2008; Schön, 1983, 1987). Krainer (2001) regards action, reflection, autonomy and networking as four dimensions of teachers’ professional practice (illustrated in the figure below). Krainer (2001) argues that most teachers are placed in the first quadrant, where there is much action and autonomy but less reflection and networking, in the sense of critical dialogue about one’s teaching with colleagues, mathematics educators, the school authority, the public, and so forth. The author regards the promotion of reflection and networking as a powerful intervention strategy in the professional development of teachers (Krainer, 2001).

Autonomy

Reflection

Four

Dimensions

Action

Networking

Figure 2.2 Four dimensions of teachers’ professional practice (Adapted from Source:

Krainer, 2001, p. 288)

In this section I will review the literature on practising teachers’ reflective practices in more detail. I will also focus on evidence in the literature on the conditions that allow teachers to become reflective practitioners, the characteristics of a reflective teacher, why it is necessary to develop teacher reflection and the benefits of reflective practice. I will also investigate possible barriers to teacher reflection.

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Brookfield (1995) proposes four lenses for teachers to become

critical reflective

practitioners: 1) the autobiographical lens (or self-reflection), 2) the students' eyes (student feedback), 3) colleagues' experiences (peer advice, mentoring and feedback) and 4) theoretical literature (teachers who research, present, or publish scholarly literature). In the following sections I will review the literature on reflective practice using these lenses.

2.5.2 Teacher reflection through self-study

Self-study involves inquiring into one’s thinking, learning and instructional practices

(Chapman, 2008). According to Moss (2008, p. xiii) reflection is generally regarded as a dialogic process, in which the dialogue may be with the inner self (interior listening). Ghaye and Ghaye (1998) refer to this process as a reflective conversation, which allows one to consider and question the values that one is committed to. Although these conversations may initially be private conversations with the self, at some point they are articulated with others (McIntosh, 2010). According to McIntosh (2010, p. 47) questions such as What is my

practice like? Why is it like this? How has it come to be this way? What are the effects of my

practice? and How can I improve what I do? enable a critical distance from reflective practice and the context in which it takes place. York-Barr et al. (2006) suggest a very similar process, which they call the 4-step process for guiding individual reflection, consisting of the following questions: 1) What happened (description) 2) Why? (analysis, interpretation) 3) So what? (overall meaning and application) and 4) Now what? (implications for action).

A number of researchers have investigated their own reflections while teaching as a result of

self-study (Attard, 2008; Bartlett & Burton, 2006; LaBoskey, 2004; Loughran, 2007).

Loughran (2007, p. 12) reports that

a central purpose in self-study is uncovering deeper understandings of the relationship between teaching about teaching and learning about

teaching. According to Dinkelman (2003) self-study is the intentional and systematic inquiry into one’s own practice.

Hillier (2005) suggests that one way to start this process is to focus on a critical incident that occurred while teaching. According to Griffin (2003, p. 208) a

critical incident provides a deeper and more profound level of reflection because it goes beyond a detailed description

of an event that attracted attention, to analysis of and reflection on the meaning of the event.

An example of a critical incident that forced him to critically reflect on his practice is mentioned by Kwok (2005). In his class, dealing with spirituality, two students raised the issue of race, and he was unprepared to deal with this sensitive issue (being of Asian origin himself). This incident caused him to reflect deeply on his own teaching of spirituality with a diverse group of students.

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However, Convery (1998) argues that, for many teachers, the central impediment to fundamentally improving their practice is their self-protective individualism. He questions the possibility of improving practice through individual self-study. In his experience collaboration was crucial in helping him to develop beyond a reflective practice which focused on techniques for improving classroom experiences, to a reflexive appreciation of his actions

(Convery, 1998). Moss (2008) agrees with this argument and claims that much of the time reflection involves relationships with others, listening to others and being listened to, thus contributing to a community of practice. In the next sections I will briefly review literature on teachers’ reflection on feedback they receive from their learners and from their colleagues.

2.5.3 Teacher reflection on feedback from learners

Loughran, (2002, p. 33) states that for reflection to genuinely be

a lens into the world of practice, it is important that the nature of reflection be identified in such a way as to offer ways of questioning taken-for-granted assumptions and encouraging one to see his or her

practice through others’ eyes. In my review of the literature I have found a paucity of evidence of teachers allowing learners to provide feedback on their teaching. One exception is Lighthall (2000, p. 154), who describes how he investigated his own teaching practice by reflecting after class with his students in, what he calls, a pedagogical laboratory. During this reflection the students revealed details about his teaching that were important to them.

Another researcher who allowed his learners (student teachers) to comment on his practice is Russell (2007) who, i

n the last 3 or 4 minutes of each class, gave each student an index card or small piece of paper and asked for responses to questions such as “What is the main idea you are taking from today’s class?” and “What further questions do you have about something we did or discussed?” As the year proceeds, there are times when comments are entered anonymously on an electronic bulletin board, where all members of a class may read them. Russell (2007) reports that he was impressed by the value of this practice as a way of fostering clear communication between teacher

(himself) and learners (students) and also among his learners (students).

2.5.4 Teacher reflection in communities

In order to achieve critical reflection, Day (1999) argues that other teachers are needed in the process. Systematic investigation of practice with the help of a critical colleague can enhance the reflective process.

Teachers may for instance find it beneficial to come together in groups or teams to discuss their teaching in a supportive atmosphere (Farrell, 2004).

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According to Pollard (2002) the value of engaging in reflective activity is almost always enhanced if it can be carried out in association with other colleagues. York-Barr et al. (2006) agree with Pollard and maintain that reflecting on practice with another person has the potential to greatly enrich understanding and support improvements in practice. They believe that reflecting with a partner can assist in gaining awareness of fixed assumptions and help a teacher to view events from another perspective (York-Barr et al., 2006). They distinguish between reflecting with partners (two or three people) and group or team reflection (teacher communities). When reflecting with partners listening, thinking and coaching are central to

fostering reflective practice (York-Barr et al., 2006, p. 114). They contend that reflective practice in the context of horizontal relationships is sometimes more powerful than in hierarchical relationships (for example where a novice teacher is coached by an expert teacher).

Farrell (2004) suggests the following activities to enhance teacher reflection:

1) Group discussions, in which teachers talk to their colleagues and build on one another’s insights to analyse and interpret classroom data and their experiences in the school. Discussion and collaboration within a group facilitate the sharing of

2)

3)

4) different knowledge, skills, expertise and viewpoints.

Classroom observations, where teachers observe their colleagues’ classrooms.

However, Farrell (2004) argues that these observations should be descriptions of classroom events and not judgments ont what should or should not occur in class.

Teaching journals, which Farrell (2004) believes are excellent tools to aid reflection.

Teachers can write in their journals at any time to record criticism, doubts, frustrations, questions, the joys of teaching, and the results of experiments.

Teaching portfolios foster reflection because to compile them teachers must examine their professional strengths and weaknesses (Farrell, 2004).

Another way that teachers can reflect critically is by describing

a significant event or practice (Hillier, 2005). The reflective cycle begins when the teacher describes a significant event he/she experienced in their teaching. It then enters a phase of interpretation, where the teacher looks for significance of the event being reflected on. In the next phase colleagues assist the teacher to compare their theories in practice with their espoused theories, and in the final phase the cycle moves to reconstructing, through which the teacher devises new ways of proceeding (Hillier, 2005).

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2.5.5 Opportunities that potentially allow teachers to become reflective practitioners

Reflective teaching requires that teachers examine their values and beliefs concerning teaching and learning so that they can take more responsibility for their actions in the classroom (Korthagen, 1993). Most theorists (Farrell, 2004; Hillier, 2005; York-Barr et al.,

2006) agree that teachers need time and opportunity to reflect on their practice. Richert

(1992) studied the conditions that influence the reflective capabilities of novice teachers through journal writing, portfolio-inspired reflection essays, conversations with peers, and conversations with more experienced teachers. The results of this study indicate that a structured opportunity to reflect, time, and safety, all emerged as important elements

(Richert, 1992).

In the next sections I review the literature to attempt to identify situations that potentially allow teachers to become reflective practitioners (conditions that promote teacher reflection), for example professional development opportunities, lesson study and action research.

2.5.5.1 Professional development as an opportunity for teacher reflection

Professional development provides an opportunity for teachers to reflect critically on their practice. The ultimate goal of any professional development program supporting school mathematics reform should be to develop among teachers the mindset that they are lifelong inquirers (Borasi & Fonzi, 2002). This means both developing the appropriate expectations and mindset, and providing teachers with strategies and skills to inquire effectively.

According to Barnett (1998) teacher inquiry plays a central role in many of the prevailing conceptions of teacher learning including critical reflection, reflection-in and on-action, personal and pedagogical theorizing, narrative inquiry, action research and teacher research.

Professional development helps teachers to develop a sense of self as a teacher of mathematics (Sowder, 2007). Knowledge of self develops when teachers regularly engage in reflection, in, on, and about their values, purposes, emotions and relationships (Day &

Sachs, 2004, p. 9). Shedding anxiety about the teaching of mathematics can lead to a sense of empowerment (Sowder, 2007) for teachers engaged in a professional development programme. These teachers work collaboratively rather than individually and have more opportunities for reflection.

Borko and Putman (1995, 1996) reviewed the literature on professional development programmes, much of it in mathematics education, and concluded that there was substantial evidence showing that teachers in these programmes did experience significant changes in

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their instructional practices, depending on opportunities for teachers to construct knowledge of subject matter and pedagogy in an environment that supports and encourages risk taking and reflection (Borko & Putman, 1995, p. 59).

2.5.5.2 Professional learning through lesson study

Darling-Hammond et al. (2009) report that in comparison to Asian and European countries, not enough time for professional learning is structured into the work lives of teachers’ of the United States. In a professional learning environment t eachers meet on a regular schedule in learning teams organized by grade-level or content-area assignments and share responsibility for their students’ success (Darling-Hammond et al., 2009). These authors

(2009) propose that teachers should devote non-classroom time to collaborative planning,

lesson study, peer observations and action research.

Lesson study refers to a process in which teachers progressively strive to improve their teaching methods by working with other teachers to examine and criticise one another’s teaching techniques (Baba, 2007). According to Rutledge and Benedicto (2007) it is actually a form of action research that allows teachers to work with each other collaboratively as reflective practitioners. The lesson study process is cyclic and has the following basic components: 1) collaborative planning; 2) lesson observation; 3) reflection on the lesson; and 4) implementations of changes.

Lesson study has played an important role in professional development in Japan since the beginning of the public education system more than a hundred years ago. One of the reasons for this popularity might be that lesson study provides Japanese teachers with opportunities to do the following: a) make sense of educational ideas within their practices; b) change their perspectives on teaching and learning; c) learn to see their practices from a child’s perspective; and d) enjoy collaborative support among colleagues (Takahashi,

Watanabe & Yoshida, 2006).

In South Africa a school-based in-service education intervention programme, modelled along the lines of the Japanese lesson study, was launched in 2000 in Mpumalanga (Jita,

Maree & Ndlalane, 2007). The conceptual framework for this initiative is based on the social constructivist theories of learning which assert the importance of learning in collaboration

(Jita, Maree & Ndlalane, 2007). According to Jita, Maree and Ndlalane (2007) the lesson study approach has managed to establish a system in which teachers have grown accustomed to relying on each other, coaching, leading discussions and exploring alternative solutions to problems experienced in their teaching of mathematics.

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Another study by Coe, Carl and Frick (2010) in a rural primary school in the Western Cape province sought to determine the value that a group of teachers placed on the process of lesson study as a model for their own learning and instructional improvement. The findings

(Coe, Carl & Frick, 2010) highlight the following benefits of lesson study:

1) Lesson study offers an effective strategy to bring teachers our of isolation, allowing them to experience meaningful collaboration with fellow teachers.

2) The process of lesson study is embedded within the classroom context by setting goals and then planning instruction with the purpose of moving the learners closer to the goals. A connection between the content of the research lesson and the

3)

4) remainder of the curriculum is established. Furthermore lesson study provides an opportunity to observe the learners during the research lesson. The post-lesson discussion is also valuable to validate and develop the perceptions of learners in relation to the prescribed goal.

Lesson study has been experienced as the catalyst for transforming new instructional strategies into routine classroom practice.

Continuous support is embedded within the model of lesson study.

2.5.5.3 Professional learning through action research

Action research is an ongoing process of systematic study in which teachers examine their own teaching and learners’ learning through descriptive reporting, purposeful conversation, collegial sharing, and reflection for the purpose of improving classroom practice (Sowder,

2007, p. 191). According to Sowder (2007) action research has multiple forms: teachers might work alone to pursue a research interest, they might work together in inquiry teams, or they might work with university researchers. Some concerns have been raised about the validity of action research by Jaworski (1998) who notes that the results reported by teachers should be seen in the context of these teachers’ classrooms. Action research is a cyclical process of planning, acting and reflecting which teachers pursue to work on an element of their own practice (Goodchild, 2008).

In the next section I discuss the characteristics of a reflective teacher, as revealed in the literature. The benefits of teacher reflection and possible barriers to reflection will also be discussed.

2.5.6 Characteristics of a reflective practitioner

Dewey (1933) has suggested that teachers who want to be reflective practitioners must possess three characteristics. They must be open-minded, responsible and

wholehearted. A teacher who is open-minded is willing to listen to more than one side of an

39

issue and give attention to alternative views (Farrell, 2004; Zeichner & Liston, 1996). A responsible teacher will carefully consider the consequences of his/her actions, especially as they impact on the students. When teachers act reflectively, they consider carefully the problems in their own teaching and think about how those problems are related to their educational and social contexts (Lee & Tan, 2004; Zeichner & Liston, 1996). Reflective teachers are aware of the consequences of their teaching and how their own assumptions or beliefs can influence their teaching. To be wholehearted implies a willingness to take risks and work through fears and uncertainties (Farrell, 2004). According to Zeichner and Liston

(1996) teachers who are wholehearted regularly examine their own assumptions and beliefs and the results of their actions. Such teachers approach all situations with the attitude that they can learn something new (Zeichner & Liston, 1996).

Zeichner and Liston (1996, p.6) mention five key features that are central to reflective teachers. Teachers who are reflective 1) examine, frame and attempt to solve the dilemmas of classroom practice; 2) are aware of and question the assumptions and values they bring to teaching; 3) are attentive to the institutional and cultural contexts in which they teach; 4) take part in curriculum development and school change efforts, and 5) take responsibility for their own professional development.

Procee (2006) believes that reflective practitioners think about their experiences in practice and view them as opportunities to learn. They are concerned about the contexts of their practices and the implications for action, and they reflect on their assumptions and their theories of practice, and take action grounded in self-awareness. Finally, reflective practitioners recognise and seek to act from a place of praxis, a balanced coming together of action and reflection (Procee, 2006).

A summary of the discussion in Section 2.5.6 is given in Table 2.3.

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Table 2.3

theorists

Summary of characteristics of a reflective practitioner according to different

Theorist

Dewey (1933)

Zeichner and Liston (1996)

Procee (2006)

Characteristics of a reflective practitioner

Open-minded

Responsible

Wholehearted

Eager to solve dilemmas of classroom practice

Awareness of own assumptions

Attentive to institutional and cultural contexts

Participation in curriculum development and school changes

Take responsibility for own professional development

Consider experiences and learning opportunities

Concerned about context of practice and implication for action

Reflect on own assumptions

Action grounded in self-awareness

Act from a place of praxis

2.5.7 Benefits of reflective practice

The benefits of adopting a reflective practice are central to the purpose of this research study and can be divided in two broad categories: 1) gaining professional and personal knowledge and 2) changing/improving practice.

2.5.7.1 Gaining professional and personal knowledge

Korthagen (2001) believes that reflection broadens and deepens the professional development of teachers and thus their competence (Korthagen, 2001). He echoes Schӧn’s

(1983) argument that reflection helps teachers to find solutions in their own practice to problems which experts cannot solve with theories. Schӧn, (1987) maintains that w hen teachers are encouraged to develop a habit of reflection, they are more able to conceptualize and explain their classroom practices, thus gaining personal knowledge.

Darling-Hammond et al. (2009) mention that in some Swiss states, the new teachers in each district meet in reflective practice groups twice a month with an experienced teacher who is trained to facilitate their discussions of common problems for new teachers. In Singapore a

Teacher’s Network was established in 1998 to produce life-long learners by making schools a learning environment for everyone from teachers to policymakers and having knowledge spiral up and down the system. The network’s mission serves as a catalyst and support for

41

teacher-initiated development through sharing, collaboration, and reflection. It has six main interrelated components: 1) learning circles, 2) teacher-led workshops, 3) conferences, 4) well-being program, 5) a Web site, and 6) publications (Darling-Hammond et al., 2009).

2.5.7.2 Changing/improving practice

The primary benefit of reflective practice for teachers is a deeper understanding of their own teaching style and ultimately greater effectiveness as a teacher. According to Butke (2003) reflective teachers have the opportunity to think about their teaching behaviours and the context in which they occur, and through the cycle of looking back on events, making judgements about them, and then altering their practice based on craft, research, and ethical knowledge, teachers can effectively change their practice.

Rodgers (2002, p. 863) mirrors Dewey’s view of reflection as a vehicle used in the

transformation of raw experience into meaningful theory that is grounded in experience.

According to him the process of reflection and steps of observation and description require the teacher to confront the complexity of learners and their learning, of themselves and their teaching, the content they teach and the contexts in which they operate (Rodgers, 2002).

When teachers reflect after a lesson, they think back on their work, relating events that took place in the classroom to their understanding, taking into account their knowledge and expectations that influenced their plans for the lesson. These reflections might lead to further actions, changing current practices and seeking further resources that influence their planning of follow-up lessons.

2.5.8 Barriers to reflection

Butke (2003) divides obstacles that may be encountered in the process of reflection into the following categories: cultural barriers, issues of time, personal risk, and motivation.

Brookfield (1995) mentions three cultural barriers: the culture of silence (teachers do not discuss their teaching practice with colleagues); the culture of individualism (teachers work in isolation) and the culture of secrecy (teachers are reluctant to reveal weaknesses, uncertainties and frustrations).

The perceived lack of

time is, according to Butke (2003) a major constraint to reflection.

Darling-Hammond et al. (2009) confirms that teachers lack time and opportunities to view each other’s teaching, learn from mentors and work collaboratively.

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Personal risks associated with critical reflection that can act as barriers against becoming a reflective teacher are for example the fear of being found out as a teacher who really does not know what he/she is doing (Brookfield, 1995).

According to Butke (2003) there is little incentive for a teacher to break away from habits and routines in the hope of advancing a teaching career. Teachers lack the motivation to become reflective practitioners.

2.6

Mathematics teachers’ reflective practice

2.6.1 Introduction

In this section of the literature review I focus on research studies dealing with mathematics teachers’ reflective practice. In my review of the literature I realised that research on mathematics teaching covers a vast range of themes, for example mathematical discourse or teacher change. These research studies address mathematics teachers’ reflection on their practice indirectly, which presents me with a dilemma. Do I include these studies’ findings or not? I have decided to include some of these studies to strengthen my argument that it is necessary for mathematics teachers to reflect on their practice in order to become more effective teachers, which will impact on their learners’ understanding of mathematics.

A substantial body of research on teacher reflection and action has been conducted over the past four decades (e.g. Adler, 1990; Artzt, Armour-Thomas & Curcio, 2008; Brookfield, 1995;

Butke, 2003; Convery, 1998; Griffin, 2003; Hughes, 2009; Korthagen, 1993; LaBoskey,

1994; Lee, 2005; Lee & Tan, 2004; Loughran, 2002; Mewborn, 1999; Nyaumwe, 2007;

Pedro, 2001; Van Manen, 1977). This research contains a wealth of information on teachers’ thinking about their daily work in classrooms. In this section I focus on studies that deal with mathematics teachers’ thinking about their actions, before they teach lessons (reflection-foraction), while teaching lessons (reflection-in-action) and after they have taught lessons

(reflection-on-action). The two domains that I focus on in this review are action and

reflection. I report on those studies that link these two domains within the context of mathematics teaching.

My interest was to find established researched-based studies on teacher reflection and action. I focus only on practicing mathematics teachers’ reflection and action and exclude any studies on mathematics teacher education, as this has been dealt with in Section 2.4.

Studies dealing with teachers’ self-study or action research are considered. I also refer to

43

studies that deal with teacher reflection in lesson study groups. In this discussion I focus on recent work (the term “recent” is defined here as the period 2000 – 2010).

Marcos and Tillema (2006) developed an analytical framework (see Table 2.4 for the adapted version) to review research on action and reflection. A brief discussion of this framework follows.

Table 2.4 Interpretive framework delineating studies on teacher reflection and action

(Source: Adapted from Marcos & Tillema, 2006, p. 115)

Process measured

Object studied

Teachers’ thinking

What teachers say

(a) Reflective thinking about teaching

Object: beliefs, prior knowledge about teaching (reported thinking)

Instrumental approach: questioning

What teachers do (c) Reflection (be)for(e) action

Object: prospective reflection about or in teaching (recorded thinking)

Instrumental approach: written documents (plans and designs)

Teachers’ action

(b) Reflection on action

Object: retrospective reflection on teaching (reported action)

Instrumental approach: written documents (narrative inquiry)

(d) Learning by being engaged in/from action

Object: action (professional practice in teaching, recorded action)

Instrumental approach:

observation

From the table it appears as if this framework can be used to position studies in four categories (Marcos & Tillema, 2006): a) b) c) d)

Descriptive studies investigating how teachers’ beliefs and prior knowledge interpret their work (studies focusing on the teachers’ voice).

Studies on reported action; investigating retrospective accounts of actions that look back to interpret what was done ( reflection-on-action).

Studies on teacher thinking; investigating teacher plans and intentions, taking their background and beliefs before taking action into account ( reflection-for-action).

Studies on observed action; investigating in-depth how action itself exemplifies teacher knowledge ( reflection-in-action).

Although these categories seem to be mutually exclusive, from my perusal of the literature I have found that researchers investigate combinations of aspects of teachers’ reflective

44

practice. I am not convinced that researchers always differentiate between e.g. reflection-inaction and reflection-on-action in a reliable and valid manner. It seems necessary to clarify these terms once again, for the purpose of this review. The concept of teaching as

reflection-in-action refers to the teacher’s thinking about the teaching-learning process or problem-solving teaching/learning situations while directly engaged in teaching. Russell and Munby (1992, p. 4) consider the essence of reflection-in-action to be hearing differently or seeing differently. For Farrell (1998, p. 12) it is reflection that gives rise to on-the-spot

experimentation. A teacher will demonstrate effective reflection-in-action when he/she changes his/her teaching approach in class after recognising that the approach is not working. The concept of reflection-on-action refers to the immediate thoughts after

teaching the lesson on a) what the teacher might have done differently to meet the needs of learners even more explicitly, b) how the lesson could have been modified to solicit other particular kinds of thinking and representations of understanding from the learners, and c) how to solve logistical issues such as optimal learner groupings, ease of distribution of materials, or pacing of the lesson (Bruce, 2009). Reflection-on-action refers to recalling, explaining and evaluating after a lesson and includes thinking about reflections-in-action that were part of the lesson (Reed, Davis & Nyabanyaba, 2003). A description of these modes of reflection is summarised below in Table 2.5. Reflection-for-action is described by Farrell

(1998, p. 12) as proactive in nature, where the teacher uses ideas from his/her reflections inaction and on-action to plan reflectively for future lessons.

Table 2.5 Summary of different modes of reflection

Source: Adapted from Boon Tiong, (2001)

Nature of thought

Reflection-for-action

Reflection-in-action

Reflection-on-action

Description

Before action – it leads to the design of actions and reactions

During action – it leads to modification of action and learning while carrying out the designed action

After action – it leads to retrospective evaluation and learning from remembered actions

From my review of the literature it seems as if most of the research studies focus on teachers’ reflection-on-action. This may be because the research design of these studies allow for teachers to reflect on their practice during interviews.

2.6.2 Research studies linking reflection and beliefs about action

Cross (2009) conducted a collective case study to investigate the relationship between mathematics teachers’ beliefs and their classroom practices. The study was part of a larger project focusing on the effects of mathematical argumentation (discourse) and writing on the

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learners’ understanding of Algebra. Five teachers of two high schools in a suburban county in the south-eastern United States agreed to be participants in this study. After orientation to the project the researcher observed each teacher twice while teaching. This was followed by a semistructured interview with each teacher to establish his/her views on mathematics as a discipline, mathematics pedagogy and learners’ learning of mathematics. Over the next ten weeks each teacher was observed twice again, while the researcher took detailed field notes. After each observation the researcher and teacher had an informal discussion to elicit thoughts related to specific actions and decisions made during the lesson. Notes were taken during the discussions and copies of lesson plans as well as samples of work by learners were collected.

The teachers’ narratives were examined and both similarities and differences regarding their views were observed. Three themes emerged: 1) A view of mathematics as computation versus a way of thinking, 2) Using demonstration rather than guidance as a teaching strategy and 3) Learners learn mathematics through practice rather than understanding.

Three teachers described mathematics as formulas, procedures and calculations and two teachers considered thought processes and mental actions of the individual as fundamental aspects of mathematics (Cross, 2009). These views were translated into the teachers’ classroom practice in two ways, the kinds of activities they designed and how they interacted with their learners.

As far as teacher reflection (gathered through reflective conversations with the researcher) is concerned, Cross (2009) reports that only by the end of the project, three teachers were beginning to question the effectiveness of their current practices. These teachers reported that although they had learnt alternative methods of designing and orchestrating instruction, they were not confident they could adopt these practices holistically, given the curricular and institutional constraints (Cross, 2009). This reflection on the part of the teachers was prompted by the fact that their learners performed poorly in conceptually rich tasks in relation to the learners in the larger project (Cross, 2009). She claims that it is clear that although the teachers welcomed the new practices they were filtered through the old belief system, resulting in minimal overall change (Cross, 2009). According to Cross (2009) belief change must be an ongoing process of awareness, confrontation and reflection. Her findings sustain

Phillipp’s (2007) argument that reflection is the critical factor for supporting teachers’ changing beliefs and practices.

Another study which deals with the relationship between teachers’ reflection and their beliefs was conducted by Warfield, Wood and Lehman (2005). Their sample consisted of seven

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novice elementary mathematics teachers in a district that includes the portion of a county surrounding a mid-sized Midwestern city. The importance of reflection in learning, as well as the role of reflection in helping teachers connect their beliefs and practice, led them to question whether there were relationships among teachers’ beliefs, their reflection, and their learning (Warfield, Wood & Lehman, 2005).

The seven participants engaged in private reflection on their teaching and the learning of their learners. Videotape was used to help the teachers reflect on their practice. In the first year of the project the teachers videotaped their mathematics lessons once a month. They developed a Personal Plan of Action (PPA) based on a dilemma they had encountered in their teaching, worked on that dilemma throughout the month, and used a structured procedure to analyse and reflect on their teaching (Warfield, Wood & Lehman, 2005). This procedure consisted of writing expectations related to the PPA prior to teaching the lesson; teaching and videotaping the lesson; watching the videotape and making detailed records of discourse that had occurred in the class discussion portion of the lesson; and comparing and contrasting their expectations with events that actually occurred. The expectations, records of discourse, and comparisons of expectations and what they observed on their tapes were written in reflective journals.

Warfield, Wood and Lehman (2005) found that four of the seven teachers did not learn to teach in ways that encouraged children to become autonomous learners. They often did not understand their learner's thinking and did not encourage the learners to clearly explain and justify their reasoning. These teachers also frequently interfered with their learner's thinking.

They based instructional decisions on the expectations of external voices rather than on their children's thinking (Warfield, Wood & Lehman, 2005). As a result they did not reflect deeply about either their learner's mathematics or about their own teaching. Instead their thinking about teaching focused on classroom management and procedures. However, the remaining three teachers allowed their learners to solve problems in their own ways and expected them to both explain and justify their reasoning and to listen to and question the reasoning of other students. They also learned to reflect about their children's mathematics and about their own roles in developing learner's thinking (Warfield, Wood & Lehman, 2005).

2.6.3 Research studies dealing with teachers’ reflection-on-action

In this section I report on studies that deal with reflection-on-action. According to García,

Sánchez and Esquadero (2006) reflection-on-action can be 1) generated spontaneously with the help of researchers 2) included in mathematics teacher education programmes and professional development through the use of narratives and 3) generated in research

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projects in which teachers and researchers collaborate. It seems as if teachers reflect on their actions when instigated to do so.

Although there are numerous studies in the literature on teachers’ reflective practice I am going to focus on those studies that investigated teachers’ reflection-on-action within the contexts of lesson study and action research.

2.6.3.1 Lesson study contexts that enhance reflection-on-action

The context of my research is lesson study, and in this section I focus on research dealing with mathematics teachers’ reflection on their classroom practice in lesson study groups.

Lesson study involves the planning of a research lesson (designed to focus on a predetermined goal) by a group of teachers (Ono & Ferreira, 2010). The lesson is observed by the other teachers, recorded and reflected upon and discussed by the group. During the process of lesson study teachers of various levels of experience interact to examine their practice through the implementation of, and reflection on a research lesson (Ono & Ferreira,

2010).

A study by McDonald (2009), conducted in a semi-rural area at a P-12 College west of

Brisbane, Australia, investigated the relationship between teacher professional development, teacher growth and any changes to learner outcomes in the context of a lesson study professional development model. This study was conducted over a one-year period with five teacher participants. Qualitative data were collected through interviews with teachers and learners, participant observation, teacher and learner questionnaires, field notes by the researcher and quantitative data were collected through pre- and post-tests for learners. The teacher questionnaires required participants to reflect and comment on current beliefs and values concerning mathematics learning and the place of problem-solving in mathematics instruction. The interviews provided opportunities to reflect on changes to beliefs or values as a result of participating in the lesson study model of professional development. The teachers in this study reflected on their own content knowledge, their pedagogical content knowledge and their professional confidence. They also reflected on their learners’ achievement.

Although the data analysis needs to be treated with caution due to the small sample size,

McDonald (2009) reports an increase in content and pedagogical-content knowledge of teachers (resulting from the collaborative planning and feedback during the lesson study process), and changes to their belief that problem-solving is an activity for the more able learners. Reflective practice in the context of this study refers to a deliberate and planned

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process of reviewing and critical thinking about teacher practice, with the purpose of increasing learning opportunities for learners and modifying research lessons for teachers

(McDonald, 2009). However, McDonald (2009) reports that it was due to her involvement as a researcher that the teacher-participants in her study focused on the learning of learners rather than on their own teaching in their planning of the research lesson.

Bruce and Ladky (2009) conducted a study on what happens between the stages of the lesson study cycle. During focus group interviews which occurred on three occasions, these researchers asked twelve mathematics teachers to describe the informal activities that took place between the formal stages of the lesson study cycle (Bruce & Ladky, 2009).

The first two stages involve identifying the lesson study goal and planning the first research lesson. Between these two stages Bruce and Ladky (2009) report that the teachers were busy with 1) searching and researching the internet, data-bases, and teacher resources on the topic in focus; 2) conceptualizing (through brainstorming, self-talk and informal conversations) valuable tangents for the lesson; 3) investigating and exploring the use of manipulatives and technological tools (such as using the white board and video) with learners to expand the teacher and learners’ repertoire; and, 4) monitoring and keeping up with details such as on-going learner assessment which provided insights into learners’ learning and assisted in the planning of lessons.

Between the second (planning stage) and the third stage (implementing the planned lesson)

Bruce and Ladky (2009) report that the participants were e-mailing each other, planning pre- and post-lessons in the sequence carefully, and considering learner groupings. The teachers were committed to documenting the full lesson sequence because one of their primary goals was to provide learners with multiple opportunities to learn and understand complex mathematical ideas (Bruce & Ladky, 2009).

Between the third stage (lesson implementation) and the fourth stage (reflection on and evaluation of the lesson) Bruce and Ladky (2009) report that the immediate reflection-onaction teachers engaged in included thoughts about 1) what the teacher might have done differently to meet the needs of learners even more explicitly, 2) how the lesson could have been modified to solicit other particular kinds of thinking and representations of understanding from the learners, and 3) how to solve logistical issues such as optimal learner groupings, ease of distribution of materials, and pacing of the lesson. The teachers immediately began planning the follow-up lessons based on the observations of the research lesson. It seems, from this study, that it is very important that teachers focus constantly on the goal of the lesson study research lesson, and maintain contact with each other.

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In Bruce and Ladky’s study (2009) the participants in the lesson study group were committed to document the full lesson sequence in the planning stage because one of their primary goals was to provide learners with multiple opportunities to learn and understand complex mathematical ideas. However, in their reflection-on-action they focus on what the teacher might have done differently to meet the needs of learners even more explicitly; how the lesson could have been modified to solicit other particular kinds of thinking and representations of understanding from the learners; and how to solve logistical issues such as optimal learner groupings, ease of distribution of materials, and pacing of the lesson

(Bruce and Ladky, 2009).

In another study Taylor et al. (2005) followed an action research approach to document a systematic inquiry into improving the classroom practice of four teachers, using the

Japanese lesson study model of professional development. The study was conducted for 15 months in rural Carlinville, Illinois. Data was provided by carefully recorded field notes, meeting summaries, video recordings and interviews. The four participants confessed to similar teaching styles reflecting the way they plan lessons and their expectations of their learners during the interviews.

In the planning stage of the lesson study cycle the participants decided on a goal for the research lesson. According to Taylor et al. (2005) the lesson study group decided 1) to allow learners to do their own thinking and design their own way of solving a two-step word problem, 2) to give learners time to share their mathematical thinking with their classmates, and 3) to listen to their learners’ mathematical thinking and become more flexible in their approaches to teaching two-step word problems. In the planning of the lesson they focused on what to teach, selecting a problem, thinking about logistics (where and when to present the lesson, how to display the problem for the whole group, the classroom management, etc.), materials (to be displayed on the board or overhead, hand out individual copies, provide rough paper, etc.), teacher script (what should be said, how much help should be given, etc.) and time management. The next phase of the lesson study cycle consisted of the teaching and collaborate reflection on the lesson. The lesson was videorecorded and the participants reflected afterwards on the different solutions that the learners produced to the problem. The lesson was revised to increase learner understanding and the lesson study cycle was repeated.

Taylor et al. (2005) report on the following benefits of the lesson study professional development model: an effective detailed lesson plan achieves the goal of more effective learning by learners; the lesson study model provides a highly motivated structure for

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planning and teaching a lesson; reflecting and thinking in the company of other teachers allow for sharing, interacting, questioning assumptions, and reassessing common practices; observing a lesson enables a shift in thinking from a teaching focus to a learning focus; focusing on learner thinking provides opportunities for feedback to support changes in teaching mathematics; and lesson study transforms working relationships and conversations between teachers.

2.6.3.2 Action research contexts that enhance teacher reflection-on-action

Lesson study is not the only professional development context that involves teachers’ reflection-on-action. According to Aldridge, Fraser and Sebela (2004) teacher action research may also promote teachers to become reflective practitioners. They investigated the success of (among others) using journals as a means of encouraging teachers to reflect on teaching strategies and improve their learning environments with a group of South African teachers (Aldridge, Fraser & Sebela, 2004). The second phase of their study focused on action research, involving two teachers and one mathematics class. The two teachers identified constructivist aspects of the learning environment that they would like to improve: using spiralling cycles of questioning, planning, implementing, collecting data and reflecting

(Aldridge, Fraser & Sebela, 2009). The teachers were required to keep a teaching journal to use as a means of reflection.

Aldridge, Fraser and Sebela (2009) report that during the 12-week intervention phase, weekly observations of the classes of the two teachers were used to determine whether they were using their reflections in their classroom practice and to provide encouragement and feedback during the process. The results of the study indicate that the use of journals help teachers to keep on track and to think about possible solutions to problems, as well as encourage them to reflect and plan future activities (Aldridge, Fraser & Sebela, 2009).

2.6.4 Research studies dealing with teachers’ reflection-for-action

Studies dealing with teachers’ planning (reflection-for-action) intend to explore how teachers’ thoughts are put in practice. Such studies should appraise planning before the action takes place and offer a comparison between the plans and performance to reveal their fit or alignment with the intended outcome (Marcos & Tillema, 2006). Some of these studies

(Scherer and Steinbring, 2006; Taylor et al., 2005) mention the collaborative planning of lessons by their participants but do not elaborate on the analysis of the lesson plan and how it aligns with what actually happens in class. Marcos and Tillema (2006) suggest that lesson plans and intentions should be appraised in their natural temporal order (before the action

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commences, through written plans and designs), and that one should look for discrepancies, that is, compare differences between plans and practices. I did not find evidence of such comparisons in any of the studies that I reviewed.

According to Scherer and Steinbring (2006) one could focus on many different aspects of improving the quality of mathematics teaching. They argue that for a better understanding of learners’ mathematical learning processes or teaching and learning in general, reflection on, and analysis of concrete classroom situations are of major importance (Scherer &

Steinbring, 2006). Their research focuses on the joint reflection between teachers and researchers on the participating teacher’s own classroom interaction by means of concrete examples (Scherer & Steinbring, 2006). In a three-year project in Germany two researchers and an assistant worked with three mathematics teachers, teaching Grade 3 and Grade 4 in an elementary school, to improve their classroom practice. The study focused on the professional teaching activity of the participating teachers and the systematic reflection that followed each teaching activity. The researchers collaborated with the teachers in intensive discussion and development of didactical ideas on the mathematics topic, and the transition of informal strategies to standard algorithms. The researchers however, were not involved in the actual planning of the lessons or designing of the worksheets.

Data were collected through informal observations by researchers (taking field notes) in initial mathematics lessons taught by the teachers. They also collected learners’ work for analysis. Observations of three teaching experiments taught by the teachers followed, each containing 3 – 6 lessons (about fifty lessons were recorded). A mutual guest observation (to which the researchers were invited by the teachers) and subsequent reflection provided more data. There were also group meetings where the researchers and teachers reflected on the learners’ work and teaching episodes. Final reflections took place during in-service courses.

According to Scherer and Steinbring (2006) the actual project focused mainly on reflectionfor-action in the beginning through discussions or reflections on didactical orientations or on the planning of teaching experiments. However, Scherer and Steinbring (2006) report that in the course of the project, the focus shifted to reflection-on-action while teachers and researchers reflected jointly on video documents. The study does not explicitly explain the link between planning and action. There is no evidence of any document analysis reported in the study. The study also does not report on the planning of the lesson and possible adaptations to the plan while teaching the lesson as a result of unexpected events happening in class. The focus on the teachers’ reflection-on-action is more clearly described

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and a definite link established between their actions and their reflection on their actions in class.

2.6.5 Research studies dealing with teachers’ reflection-in-action

Eraut (cited in Jaworski, 2004) suggests that teaching is too complex for reflecting-in-action to be a serious option for most teachers. According to Jaworski (2004) Eraut emphasises that a teacher has to be constantly assessing the situation, responding to incidents, deciding whether to change the activity, and be alert for opportunities to tackle difficult issues.

However, Jaworski (2004) reports that from her own experience and research with teachers

(Jaworski, 1994) reflection-in-action does happen with consequences for immediate teaching action.

Recent research by Ross and Bruce (2005, p. 4) on teachers’ self-assessment confirm that reflection-in-action is possible and occur as self-assessments in the moment. Artzt, Armour-

Thomas and Curcio (2008, p. 138) refer to reflection-in-action as thinking on your feet. They describe a case study of one teacher’s reflection-in-action to illustrate this interactive aspect of teaching and in their final commentary on the case, Artzt, Armour-Thomas and Curcio

(2008, p. 140) conclude that during a lesson teachers must continually assess the

understanding of their learners to regulate their instruction in ways that will meet the

learners’ needs.

Reflection-in-action is also addressed in a study by Leikin and Dinur (2003) who conducted research on one teacher’s flexibility in the course of a whole-class mathematics discussion in

Israel. Leikin and Dinur (2003, p. 1) consider a teacher being flexible at a particular point of the discussion if s/he adjusts the planned learning trajectory in accordance with students’ contributions that differ from those that s/he expects of them. In this study Leikin and Dinur

(2003) focus on a teacher’s flexibility associated with situations in which learners’ replies are unexpected by the teacher. To describe the teacher’s flexibility they compare the teacher’s plans regarding the lesson with the actual events and procedures that occur in the classroom (Leikin & Dinur, 2003).

The first purpose of Leikin and Dinur’s (2003) study was to zoom in on the teacher-learner interactions in the context of a whole-class mathematical discussion in order to describe patterns of flexibility. The second purpose was to analyse how different types of teacher knowledge influence teacher flexibility. The data collection and analysis were on-going, using a qualitative approach. The data were collected in triads of planning, teaching in the classroom, and stimulated recall. The three elements of each triad were connected by a

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particular lesson. All the lessons chosen for the investigation included a whole-class discussion. The data was video-recorded and transcribed. Additionally the researcher took written field notes while collecting the data. When analysing the data they performed multiple observations of the videotapes and careful reading of the transcripts (Leikin & Dinur, 2003).

Leikin and Dinur’s (2003) analysis focuses on the teacher’s behaviour in the cases where learners’ replies differed from those that the teacher had expected. At the stage of stimulated recall, based on chosen episodes, the teacher was asked to discuss the lesson and to analyse how and why her plans coincided or did not coincide with the real management of the lesson (Leikin & Dinur, 2003). She explained to the researcher her reasons for the decisions taken in the course of the whole-class discussion.

In their findings Leikin and Dinur (2003, p. 8) state that they applied the idea of the

mathematics teaching cycle to the micro-situations in a junior-high school classroom in order to develop a theory of teacher flexibility-in-action that corresponds to teacher reflection-in

action. They identified four patterns of teacher flexibility that differ concerning: 1) outcomes

(representing teacher-learners interactions associated with an unexpected learner’s reply which lead to a learning trajectory that ends differently from the one planned by the teacher);

2) strategies (representing teacher-learners interactions associated with an unexpected solution strategy/explanation suggested by a learner); 3) sequencing (representing teacherlearners interactions associated with a connection between equivalent properties of mathematical objects, whose direction is opposite to the one expected by the teacher) and

4) scopes (representing teacher-learners interactions associated with questions/conjectures that are “bigger” than those which the teacher deems possible for discussion in the particular classroom) (Leikin & Dinur, 2003).

A summary of the research reviewed follows in Table 2.6.

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Table 2.6 Summary of research on mathematics teachers’ reflective practice

Study

Cross (2009)

McDonald (2009 )

Bruce & Ladky (2009) (b), (c)

Aldridge, Fraser &

Sebela (2004)

(b)

Scherer & Steinbring

(2006)

Taylor et al. (2005)

(b), (c)

(b), (c)

Leikin & Dinur (2003) (c), (d)

Object of study

(Categories of

Marcos & Tillema,

2006)

(a), (b), (c), (d)

(a)

(a), (b), (c)

Instrument(s)

I, ID, WD, O, V,

FN, Q, T;

I; O; WD; FN

I; O; Q; FN; T

I

WD

O; V; WD; FN; ID

FN, V, I

V; FN

Sample

NvT, ExT

ExT (5)

ExT (12)

ExT (2)

ExT (3)

ExT (4)

ExT (1)

Limitations mentioned

ExT, NvT (5) Small sample

Unable to use consistent and paired data collection methods; researcher acts as the professional development facilitator

None

Small sample

Small sample

Small sample

Small sample

Link between reflection and action

Reflection on beliefs prompted by interview questions and informal discussions with researcher

Reflection-for and -on action in lesson study cycle; reflection about beliefs in teacher questionnaire; interviews provided opportunities for reflection-on practice and about beliefs

Reflection about actions between stages of lesson study cycle

Reflection-on-action through the use of journal writing and action research

Joint reflection between teachers and researchers on their practice

Reflection-for-action and on-action in lesson study cycle

Reflection prompted by researcher probes of behaviour in-action

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(a): beliefs about teaching; (b): reflection-on-action; (c): reflection-for-action; (d): reflection-in-action;

I: interviews; O: observation; V: video recordings; WD: written documents; FN: field notes; ID: informal discussions: T: tests; Q: questionnaires;

ExT: experienced teachers; NvT: novice teachers

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The table portrays the research I reviewed on the link between teacher reflection and action.

Two of these studies involve teachers’ beliefs about their teaching (Cross, 2009; McDonald,

2009). Five of the studies I reviewed deal with mathematics teachers’ reflection on their practice (Aldridge, Fraser & Sebela, 2004; Bruce & Ladky, 2009; McDonald, 2009; Scherer &

Steinbring, 2006; Taylor et al., 2005). The recognition of teachers as reflective practitioners, who as professionals learn from experience and construct knowledge for their practice, lies at the heart of this domain of studies (Marcos & Tillema, 2006). Three of these studies probe teachers’ reflection-on-action in the context of lesson study (Bruce & Ladky, 2009;

McDonald, 2009; Taylor et al., 2005), while Aldridge, Fraser and Sebela (2004) used action research to determine their participants’ reflection-on-action. Reflection-on-action is part of action research and the lesson study cycle, and the design of these studies therefore allows teachers’ to reflect on their teaching practice. Two of these studies used video recordings to stimulate teacher reflection-on-action (Bruce & Ladky, 2009; Taylor et al., 2005). According to Marcos and Tillema (2006) the stimulated recall technique is appropriate to assess reflective processes because it is tied to a specific context (lesson) and has memorable references to reflected practices. Moreover they claim that participants’ freedom to reflect remains intact: they are not subject to researcher guidance (framed), but may report reflections whenever they want (by stopping the video and commenting on it) (Marcos &

Tillema, 2006).

None of the studies I reviewed deal exclusively with teachers’ reflection-for-action. In addition, I reviewed only one study that deals with mathematics teachers’ reflection-in-action

(Leikin & Dinur, 2003), mainly because there was a lack of evidence of such research in the literature. According to Marcos and Tillema (2006) these studies are most difficult to conduct because they require a wealth of data collection methods and a careful analysis of different data sources. They also maintain that they have not been able to identify many studies in this domain, nor have they been able to determine a number of strategies that can analyse teacher activity in action.

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2.7 The conceptual framework for this research study

My investigation is influenced by the conceptual framework for my research study, as visualised below, which I call the Framework for reflective teaching of mathematics (FRTM).

Contextual factors

Teacher reflection

Nature of reflection

(Quality)

Moment

(When)

of reflection

Level 1: Technical/

Descriptive/Recall

Level 2:

Deliberate/Comparative/Rati onalisation

Level 3: Critical reflection/Reflective level

Reflectionfor-action

(before)

Reflectionin-action

(during)

Reflectionon-action

(after)

Teacher’s classroom practice

Figure 2.3

Framework for reflective teaching of mathematics

In this visual representation it is acknowledged that contextual factors may influence mathematics teachers’ reflection and ultimately also their reflective practice. Lee and Tan

(2004) identified personal dispositions, interpersonal relationships, instructional and curricular practices and institutional values, norms and practices as contextual factors that influence mathematics student teachers’ understanding of reflection. I believe that these are not the only contextual factors that may influence mathematics teachers’ reflective practice in the South African context, and this study will explore the possibility that other factors such as language, culture and socio-economic circumstances may play a role. The context of this research study is lesson study, and this context may also influence mathematics teachers’ reflective practice.

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This research study seeks to examine teacher reflection by inter alia focusing on the

nature of reflection and the moment of reflection. The nature of mathematics teachers’ reflections will be explored by investigating whether they reflect on their classroom practice,

how they reflect and the content of their reflections (what they reflect on while teaching mathematics). The level of teacher reflection will be determined by using Lee’s (2005) levels of reflection as visualised below.

Level 1: Recall level of reflection (R1)

The participant describes what (s)he experienced, interprets the situation based on recalling the experience without looking for alternative explanations, and attempts to imitate ways of dealing with the situation using what (s)he has observed or was taught

Level 2: Rationalisation level of reflection (R2)

The participant looks for relationships between parts of their experiences, interprets the situation with rationale, searches for ‘‘why it was,’’ and generalises the experiences or comes up with guiding principles

Level 3: Reflective level (R3)

The participant approaches experiences with the intention of changing/ improving in the future, analyses experiences from various perspectives, and is able to see the influence of cooperating teachers on learners’ values/behaviour/achievement

Figure 2.4

Lee’s reflection levels (2005)

The moment of reflection will be explored against the theoretical background of Schön’s terms reflection-in-action and reflection-on-action (1983). The term reflection-for-action is different from the previous types of reflection in that it is proactive. Killon and Todnew (cited in Farrell, 2004, p. 31) argue that reflection-for-action is the desired outcome of both previous types of reflection. They believe that we undertake reflection, not so much to revisit

the past or to become aware of the metacognitive process one is experiencing, (both noble

reasons in themselves) but to guide future action (the more practical purpose).

As indicated in the visual representation of the conceptual framework, teacher reflection, which includes the nature of reflection and the moment of reflection, impacts on a teacher’s

classroom practice and vice versa. Classroom practice involves what the teacher does

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before entering the classroom, in terms of his or her planning and preparation; while in the classroom, both while functioning as an educator and in all the other roles expected of the teacher; and retrospectively after she/he has left the classroom (Brubacher, Case & Reagan,

1994). The arrows in the conceptual framework indicate a possible reciprocated relationship between the concepts or influence of one concept on another.

Finally, I present the definition of reflection that guides this research study.

2.7.1 Definition of reflection and reflective practice to guide this research study

The criteria that Lee and Tan (2004) propose to define teacher reflection

3

and reflective practice accommodate the interpretations of Dewey (1933) and Schön (1983) in the following way. Dewey (1933) maintains that teachers should acquire a habit of on-going thoughtfulness and examination of the beliefs and theories they use to inform their instruction of students (which relates to the first two criteria that Lee and Tan (2004) mention). Schön (1983) regards teaching as so complicated that teachers cannot merely apply what they have learned in an inflexible manner. They have to reflect-in-action when the practitioner is suddenly confronted with a problematic situation and must resolve it and reflect-on-action after a teaching episode to determine whether matters were resolved in a satisfactory manner (Schön, 1983).

The tentative operational definition of teacher reflection and reflective practice for this study, based on these criteria, follows.

Teacher reflection is an interrogation of practice before, during and after the act of teaching (reflection-for-practice, reflection-in-practice and reflection-on-practice), asking questions about the effectiveness of the teaching and learning experience and how these might be refined to meet the needs of the learner. The teacher is reflectively aware of the context in which he/she teaches as well as his/her own beliefs, knowledge and values regarding not only mathematics, but also the learners in the class. Reflection on practice happens actively in response to potentially problematic situations and allows for professional growth and change.

3

See Section 2.4.1.1

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2.8 Conclusion

In this chapter I have reviewed theoretical perspectives on reflective practice in the literature and have given an overview of a number of research studies dealing with teacher reflection, focusing on mathematics teachers’ reflective practice. I have also provided the conceptual framework for this study. In Chapter 3 I will discuss the research design that will guide this study.

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Chapter 3

Research Design

3.1 Introduction

A research design is a plan or strategy that moves from the underlying philosophical assumptions to specifying the selection of respondents, the data-gathering techniques to be used and the data analysis to be done (Nieuwenhuis, 2010).

Within the qualitative approach, the design chosen for this research study is a case study design. According to Cohen, Manion and Morrison (2005) a case study provides a unique example of real people in real situations. Nieuwenhuis (2010) argues that case studies offer a multiperspective analysis in which the researcher considers not just the voice and perspective of one or two participants in a situation, but also the views of other relevant groups and the interaction between them. A key strength of the case study method is the use of multiple sources and techniques in the data-gathering process, which includes interviews, observation and field notes.

In this chapter I firstly discuss the paradigmatic assumptions and perspectives underlying my research. I then reveal the research site and sampling of this research study, followed by a discussion of the data-gathering procedures. Subsequently the strategies for data analysis, quality assurance and ethical considerations that guide this study are discussed. Lastly I mention perceived limitations of this research study.

3.2 Paradigmatic assumptions and perspectives

A paradigm is a set of assumptions or beliefs about fundamental aspects of reality which give rise to a particular world-view (Nieuwenhuis, 2010).

In defining my paradigmatic perspective as a qualitative researcher, I am aware that I approach my research with certain basic assumptions about the world and how it should be studied. In this section I address my fundamental assumptions about the nature of reality (ontology), the relationship between knower and known (epistemology) and my assumptions about human nature.

3.2.1 Ontological assumptions

This research study seeks to explore the nature of mathematics teachers’ reflection in their classrooms and in the lesson study group. I believe that teachers’ construction of reality lies in their sense-making and negotiation of the external world (the context of classroom, school

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and community) and their interpretation of this world. This research study therefore follows a qualitative research approach that focuses on teachers’ reflections on their classroom practice. Qualitative research is based on a philosophy that views reality and truth as subjective, multifaceted and a shared social experience (McMillan & Schumacher, 2006). Its goal is to understand the situation from the participants’ perspective. I am undertaking this research in the belief that human life can only be understood from within and not as a form of external reality, and that the human mind is the purposive source of meaning

(Nieuwenhuis, 2010). The focus of this study is therefore on teachers’ subjective experiences and how they share these experiences with their fellow teachers.

3.2.2 Epistemological assumptions

In terms of epistemology an underlying assumption I bring into the inquiry is that people create reality by learning from others, teaching others and reflecting with others on their own knowledge. In my research study I allow for a rich understanding of social reality by using the context of lesson study, which allows teachers to share their ideas about their classroom practice with one another. Lesson study provides teachers with the opportunity to learn from one another’s experiences, thus building up a shared body of knowledge. I do not believe that precise, systematic and theoretical answers to complex human behaviour are possible, and that is one of the reasons why this research study will be qualitative. According to

Nieuwenhuis (2010) knowledge should emerge from the local context and should privilege the voice of the insiders (p. 56), taking into account what people say, do and feel, and how they make meaning of the phenomena under investigation. In the context of the lesson study group my role as researcher is to understand the teachers’ reflections from their point of view, and not from my own.

Because this research study seeks to understand the nature of teachers' reflection and their reflective practice it is situated within an interpretive paradigm. The central endeavour in the context of the interpretive paradigm is to understand the subjective world of human experience (Cohen, Manion & Morrison, 2005). It is characterized by a concern for the individual and efforts are made to get inside the person and to understand from within (ibid.,

2005).

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3.2.3 Assumptions about human nature

As a researcher I share my participants’ frame of reference and try to understand how their views shape the action which they take within that reality (Beck, cited in Cohen, Manion &

Morrison, 2005). The experiences of teachers regarding their reflective practice are investigated against the social context of interaction between fellow teachers as well as in the social context of mathematics classrooms. I acknowledge that an interactive relationship between me as researcher and the participants exists. The experiences and narratives of the teachers are the medium through which this research study explores their reflective practice.

These assumptions impact on my methodological choices and require consideration of different research methods. Because this study addresses teachers’ reflective practice, I wish to adopt an approach that has been described by various authors as “reflective”

(Evans, 2002). According to Alvesson and Sköldberg (2002) reflective research has two basic characteristics: careful interpretation and reflection. Reflection means interpreting one’s own interpretations, looking at one’s own perspectives from other perspectives, and turning a self-critical eye onto one’s own authority as interpreter and author (Alvesson &

Sköldberg, 2002).

3.3 Research site and sampling

The research site for this study is in the Thabo Mofutsanyana district in the Free State.

Permission to access a school in this district was obtained from the Free State Education

Department. A meeting with the principal of the school was arranged and permission from him was obtained to meet with potential participants. The participants in this research study are five mathematics teachers, teaching Grades 8 - 11. One participant teaches

Mathematical Literacy. The criteria for selection as a participant include the factors of convenience, access, and willingness to participate. The expectations of the study were presented to potential participants verbally and in writing. Meetings with these teachers took place in the teachers’ school environment.

3.4 Data-gathering procedures

This research study takes place in the context of lesson study. Underlying the practice of lesson study is the idea that teachers can best learn from and improve their practice by seeing other teachers teach (Stephens & Isoda, 2007). There is an expectation that teachers who have developed deep understanding of and skills in subject matter pedagogy should be encouraged to share their knowledge and experience with colleagues (ibid., 2007). Whereas

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the focus appears to be on the teacher, the final focus is on the cultivation of learners’ interest and on the quality of their learning (ibid., 2007).

The lesson study cycle involves a planning phase, a teaching phase and a feedback

(reflection) phase. In this research study the participants decided to cooperatively plan a lesson on equations, which is one of the topics that is covered by Grades 8 – 12. The planning phase will last two weeks. The lesson study group will plan a lesson on linear equations for Grade 8. Each teacher will then adapt the Grade 8 lesson to the grade that he/she teaches, taking into account that the content and context of each grade should show progression from simple to complex. During the teaching phase of the lesson study cycle the planned lesson will be taught by the Grade 8 teacher. This lesson will be observed by the researcher together with an assistant who will manage the video recorder. During the feedback (reflection) phase of the lesson study cycle the participants of the lesson study group will view the video-recorded lesson the same afternoon in a post-conference to improve on the lesson plan. The focus will not only be on the teachers’ presentation but also on the learners’ understanding of the concepts that were taught. Figure 3.1 illustrates the lesson study cycles for this research study.

Lesson study group: planning phase

Cycle 1: Grade 8 lesson on linear equations

Cycle 2: Grade 9 and 10 lesson plans improved

Cycle 3: Grade 11 lesson plan improved

Cycle 4: Mathematical Literacy lesson plan improved

Lesson study group: feedback phase

Cycle 1:Group reflection

Cycle 2: Group reflection

Cycle 3: Group reflection

Cycle 4: Group reflection

Teaching phase (Individual teacher)

Cycle 1: Grade 8 teacher teaches lesson

Cycle 2: Grade 9 and 10 teachers teach lesson

Cycle 3: Grade 11 teacher teaches lesson

Cycle 4: Mathematical Literacy teacher teaches lesson

Figure 3.1

Illustration of the lesson study cycles for this research study

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The figure illustrates the continuous cycles of lesson study: planning, teaching and reflection.

This research study will have four lesson study cycles. The first cycle will involve the planning of a Grade 8 lesson on linear equations, the teaching of the lesson and the postconference during which the group will reflect on the learners’ understanding of the equation concepts. The second lesson study cycle will involve the adaptation of the Grade 8 lesson plan to Grades 9 and 10. The teaching of this revised lesson will be followed by a reflection session during which the participants will try once more to improve the lesson plan. This cycle is repeated again by the Grade 11 teacher and lastly by the Mathematical Literacy teacher.

My principal concern is with understanding the way in which the teachers in the lesson study group create, modify and interpret the social context in which they function as they plan, teach and reflect on the lesson. Therefore a qualitative inquiry with an epistemological perspective of the interpretive paradigm will underpin this study as I seek to explore the nature of these mathematics teachers’ reflective practice.

In this research study my role as researcher will be that of participant observer. Participant observation is useful when the focus of interest is how activities and interactions within a setting give meaning to beliefs or behaviours. This fits in with the assumption that everyone in a group or organization is influenced by assumptions and beliefs that they take for granted. It is therefore considered the qualitative method of choice when the situation or issue of interest is obscured or hidden from public knowledge and there are differences between what people say and what they do.

My role as researcher will also take on a reflective stance: interpreting my own interpretations, looking at my own perspectives from other perspectives, and turning a selfcritical eye onto my own authority as interpreter and author.

3.5 Data-gathering instruments

A discussion of the methods of data collection follows. The link between the research questions posed by this study and the method of data collection is provided after the discussion.

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3.5.1 Interviews

The aim of qualitative interviews is to see the world through the eyes of the participant in order to obtain rich descriptive data that will help to understand the participant’s construction of knowledge and social reality (Nieuwenhuis, 2010). According to Bernard and Ryan (2009) probing is the key to successful in-depth interviewing and they mention the following probing techniques (which I will keep in mind while interviewing the participants in my study):

The silent probe (waiting for a response

The echo probe (repeating the last thing someone has said and asking them to continue)

The uh-huh probe (making affirmative statements)

The tell-me-more probe (asking questions like “Could you tell me more about that?” or “Why do you say that?”)

I will conduct a semistructured interview

4

with each of the five teachers individually before the research study commences. A list of prepared questions will be used as a guide to explore these teachers’ understanding of reflection. During this interview I will also use a lesson plan from each teacher as a discussion document to probe whether, how, when and on what they reflect when teaching their lessons.

A second individual interview

5

with each teacher will be conducted after each lesson observation to probe their experiences while teaching the lesson, as well as to understand any deviations from their lesson plan (reflection-in-action and on-action).

A final group interview

6

with all the participants will be conducted after the last lesson study cycle to establish how the reflexive processes of lesson study affect their classroom practice.

This group interview will take place during the final phase of the research study.

3.5.2 Observations

Observation is an essential data-gathering technique which allows the researcher to hear, see and begin to experience reality as participants in the research group do (Nieuwenhuis,

2010). Bernard and Ryan (2009) argue that observation behaviour should be recorded as accurately as possible, in order to produce unique valuable qualitative data.

4

See Appendix D

5

See Appendix D

6

See Appendix D

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3.5.2.1 Individual teacher observation

Each teacher will be observed while teaching the planned lesson. I will observe this lesson which will be video-recorded by an assistant. The video-recording will be viewed afterwards by the lesson study group to determine whether the goal of the lesson (improving learners’ understanding of equations) was met. It might be necessary to view the video-recording more than once in order to focus on the teacher’s reflection-for, on- and in-action. I will take field-notes during this group reflection.

3.5.2.2 Lesson study group observation

Teachers in the lesson study group will be observed during the reflection-on-action stage

(while reflecting on the lesson in the post-conference phase). These observations will reveal the nature of the teachers’ reflective practice. I believe that the lesson study process will foster the participant teachers’ reflective awareness and hope to see evidence of this in their classroom practice. The lesson study group will be video-recorded and audio-taped while they reflect on the lessons. I will take fieldnotes while observing the participants.

Throughout all the observations, I will try to remain sensitive to the ethnographic data emerging from the participants’ professional lives.

3.5.3 Document collection

Simply stated, document collection is about learning from things (Lehman, 2003). According to Hodder (cited in Lehman, 2003) d

ocument collection is important in qualitative inquiry for the following reasons: it provides easy and low cost access to information; the data may differ from what is interpreted from direct observation and interviewing, allowing the qualitative researcher to explore multiple voices and conflicting interpretations; and material culture is more permanent than the spoken word and can provide historical insight.

In this research study I will collect data from the participants’ lesson plans.

3.5.3.1 Lesson plans

Lesson plans of teachers provide striking evidence of the whole nature of teaching and classroom life (Burton & Bartlett, 2005). A set of lesson plans from each teacher before participation in the research study will be analysed to establish levels and moments of reflection before participating in the lesson study group. The lesson plans might also reveal

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their assumptions about their learners’ mathematics knowledge and their own views on mathematics teaching. The lesson plans will provide additional data regarding the main research question.

The lesson plan of the lesson study group will also be analysed to reveal the quality of their collective reflection. I believe that reflection is better carried out in collaboration, but there might be limitations involved that I am unaware of at this stage.

3.5.3.2 Researcher diary

I will enter my own reflections in a researcher diary. In this diary I will reflect on my role as researcher during the research process, and record possible dilemmas or unanticipated incidents that might occur. Such reflections might help in framing my own dilemmas, and serve to clarify issues and keep the focus on the study. New understandings might emerge as previous views regarding the research process are re-assessed. I intend to reflect not only on issues concerning the participants involved but also on the process of the research in my own personal journal. Entries will be made during each lesson study session, and on the days that I meet with the teachers. It is important for me to reflect on: the progress of the study, my communication with the teachers, the reactions of the teachers to the study, and my observations of the congruencies in what I see in their teaching versus their reflections in the interviews. I will share some of my reflections with the teachers when they are beneficial to the study or if they are helpful to the individual teacher. Throughout the writing process I might become aware of what my biases and interests are, and that those biases might play a part in the reflections that the participants reveal to me.

The knowledge gained by this exercise might, together with the knowledge gained by analysing the teachers’ reflective diaries, contribute to the existing body of knowledge that exists in the literature on teacher reflection.

Table 3.1 summarises the relationship between the research questions and the data collection techniques described in this section.

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Table 3.1:

Relationship between research questions and data collection techniques

Research question Data collection technique Purpose

Question 1: What is the nature of mathematics teachers’ reflective practice?

How do mathematics teachers reflect before, during and after

Interview

To investigate whether teachers reflect on their practice teaching?

Document collection

(lesson plan)

Observation

Field notes

To explore how teachers reflect on their classroom practice

To determine the moment of reflection ( when do teachers reflect on their practice: before, during and/or after the lesson?)

To examine the content of mathematics teachers’ reflection

(what they reflect on)

To determine the level of teachers’ reflection using Lee’s levels of reflection (2005):

R1: Recall

R2: Rationalisation

R3: Reflective

What is the possible relationship between mathematics teachers’ reflection and their classroom practice?

Question 2: How do contextual factors influence mathematics teachers’ reflective practice?

Question 3: What is the potential significance of mathematics teachers’ reflective practice for theory building in mathematics teaching?

Interview

Observation

Field notes

Interview

Observation

Field notes

Researcher diary

Field notes

To explore possible benefits of being a reflective practitioner when teaching mathematics

To gain a sense of how the context of lesson study influences teachers’ classroom practice.

To explore other possible contextual factors that influence teachers’ reflective practice

To contribute to the body of knowledge on mathematics teachers’ reflective practice

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3.6 Strategies for data analysis

The data for this research study will be obtained through interviews with the participants, observations of both participants teaching a lesson and the lesson study group reflecting on the lessons, and document analysis. I intend to analyse the qualitative data using both an inductive and deductive approach. According to Nieuwenhuis (2010) the main purpose of an inductive analysis is to allow research findings to emerge from the frequent, dominant or significant themes inherent in raw data, without the restraints imposed by a more structured theoretical orientation.

Nieuwenhuis (2010) claims that the data analysis in a qualitative study tends to be an ongoing and iterative process, implying that data collection, processing, analysis, and reporting are intertwined. I will therefore continuously consult my fieldnotes to verify conclusions, as well as solicit feedback from the participants to clarify gaps which have been noticed. My goal is to summarise what I see and hear in terms of words, phrases, themes or patterns, to help my understanding and interpretation of what emerges from the data.

Throughout this process I will keep my research questions in mind.

The data gathered during the lesson study cycles will be analysed during and after the datagathering process, based on Creswell’s (2003) approach. I plan to inductively analyse the gathered data using content analysis and conversation analysis to develop themes, patterns and categories that identify and describe the participants’ reflectivity whilst planning and teaching lessons. I will use the computer software program Atlas.ti 6 to assist me with the data management, the coding, categorisation, abstracting and conceptualising stages of the analysis. Atlas.ti 6 allows for the analysis of textual, graphical and audio data.

1)

2)

3)

Data will be obtained through interviews (individual and group), observations (videorecorded), field notes and document analyses (lesson plans).

I will organise and prepare the data for analyses. Interviews (individual and group) will be transcribed verbatim. My goal is to summarise what I see and hear in terms of common words, phrases, themes or patterns that would aid my understanding and interpretation of that which is emerging (Nieuwenhuis, 2010).

The transcripts of the interviews will be read and re-read and I will watch and rewatch the video recordings to familiarise myself with general patterns that emerge and to gain information about the depth, reliability and usability of the information.

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4)

5)

6)

I will then assign codes to meaningful segments of text in a transcript, using the computer programme Atlas.ti 6. Coding is the process of reading carefully during the transcribed data, and dividing it into meaningful analytical units (Nieuwenhuis, 2010).

The next phase of the data analysis process involves the organisation or combination of related codes into themes or categories (known as “families” in Atlas.ti 6).

Interpretation and explanation of the data follow. All conclusions reached will be based on verifiable data.

According to Nieuwenhuis (2010) computer-aided data analysis can, on face value, appear deceptively easy, where the coding, clustering and searching functions make data analysis quick and satisfying. He argues that there are no short cuts to the demanding process of reading and rereading the data, and searching to unfold the meanings constructed by the participants of the study (Nieuwenhuis, 2010). I will take this argument into account when using the computer-aided data analysis programme.

3.7 Quality assurance: data verification

It is important in any study to ensure that the research is valid and reliable. The analogous criterion in naturalistic inquiry to establish validity and reliability is trustworthiness (Lehman,

2003).

3.7.1 Trustworthiness

According to Nieuwenhuis (2010) trustworthiness is of the utmost importance in qualitative research. The qualitative data being collected from this research study is in the form of observations, interviews and document analysis. The observations and interviews will be electronically recorded and transcribed. Participants will have the opportunity to review these transcriptions at the end of the entire data collection period to ensure accuracy and provide additional research data.

4)

5)

6)

To enhance the trustworthiness of qualitative research studies Nieuwenhuis (2010, pp.

113-115) suggests that the following steps be taken:

1)

2)

3)

Using multiple data sources

Verifying raw data

Keeping notes on research decisions taken

Greater trustworthiness in coding data

Stakeholder checks

Verifying and validating your findings

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7)

8)

Controlling for bias

Avoiding generalisation

9) Choosing your quotes carefully

10) Maintaining confidentiality and anonymity

11) Stating the limitations of your study upfront.

To ensure the trustworthiness of this research study, data from multiple sources will be used to help me verify my findings. For example data collected through interviews will be verified with information gathered from the observations and the document analysis. In addition, the transcripts and fieldnotes will be submitted to the participants to correct factual errors.

During the second interview the participants will be asked to verify whether my interpretation of what they have shared with me during the course of the study is correct. I will write down my thoughts and decisions during the research process and document the category labels I create. Any revisions I make to categories and any observations I note concerning the data will be written in my researcher diary.

3.7.2 Triangulation

Qualitative inquirers use several major procedures to ensure that their research produces highly credible findings. Triangulation is a general term in naturalistic inquiry, incorporating the use of multiple methods, various investigators, diverse theories, and different resources to establish credibility in a qualitative study (Cohen, Manion, & Morrison, 2005). According to

Terre Blanche and Durrheim (cited in Maree & Van der Westhuizen, 2009, p. 34)

triangulation is essential to ensure interpretive validity and establish data

trustworthiness. In this study I will use observation, interviewing, and document collection as multiple sources to ensure that the trustworthiness of the data. I will also compare patterns that emerge from the data with other theories found in the literature review.

3.7.3 Crystallisation

Crystallisation refers to the practice of “validating” results by using multiple methods of data collection and analysis (Maree & Van der Westhuizen, 2009). Different perspectives that all reflect the unique reality and identity of participants are necessary to provide for a complex and deeper understanding of the phenomenon (Nieuwenhuis, 2010). Richardson (2005) proposes the use of the term “crystallisation” rather than “triangulation” in qualitative research, asserting that the central image for qualitative inquiry should be the crystal, not the triangle, because crystallization provides us with a deepened, complex and thoroughly

partial understanding of the topic (p. 963).

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To establish the trustworthiness of this research study, various techniques will be employed to gather data and peer debriefing will be used to provide feedback on my notes and to verify my evolving interpretations of the study.

3.8 Summary of the layout of the research design

Table 3.2 summarises the layout of the research design for this research study.

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Table 3.2

Summary and layout of research design

Purpose

Qualitative investigation

Data gathering

29-07-2010 and 30-07-2010:

• Initial semi-structured individual interview with each teacher

24-02-2011 until 05-05-2011:

• Lesson plan analysis

• Classroom observations, video recorded

• Post-lesson individual interview with each teacher

• Lesson study group feedback sessions on each observed lesson

• Final group interview

Data analysis

Analysis of data during and after the data-gathering process using Atlas.ti 6

Follow the seven steps for data analysis proposed by

Creswell (2003):

• Gathering data

• Organising data

• Overview of data

• Coding

• Creating categories

• Report writing

• Interpretation and crystallisation

• Final report

Participants

Five practising mathematics teachers teaching

Grade 8 – 11 of whom one teaches

Mathematical

Literacy

Trustworthiness

Key criteria of trustworthiness (Lincoln &

Guba, 1985, cited in Nieuwenhuis, 2010) are: credibility, applicability, dependability and confirmability.

The following steps are necessary:

• Engaging in multiple methods of data collection (observation, interviews and document analyses);

• Describing those findings which crystallise from the data will add to the trustworthiness of the research (Nieuwenhuis, 2010).

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3.9 Ethical considerations

According to Bogdan and Biklen (2003) two issues dominate traditional official guidelines of ethics in research with human participants: informed consent and the protection of participants from harm. These guidelines ensure that participants enter research studies voluntarily, understand the nature of the study and the dangers and obligations that are involved, and are not exposed to risks that are greater than the gains they might derive.

I verbally briefed each participant and presented the following information in writing using

Butke’s guidelines (2003): 1) the purpose of the study; 2) risks involved in the study, which may include the discomfort of analysing a teaching practice and the loss of time for other facets of life; 3) general procedures of the study; 4) demands upon participants’ time in the study; 5) timeline of the study; 6) confidentiality concerning anonymity of participants in the study, which include the use of pseudonyms (however the five participants would know each other and would be intertwined in the reflective process via the lesson study meetings); 7) rights of participants in the study which include one that determines that the participant is acting in a voluntary role and may withdraw at any time without penalty; 8) the phone numbers of the researcher; and 9) benefits of the study to the participant and the profession.

I have asked each teacher to sign a permission contract indicating consent to participate in the study. In addition, the following principles will guide my process of ensuring ethical research (University of Pretoria, 2010): the principles of respect for personal autonomy, benevolence and justice.

3.10 Limitations

One limitation of this research study relates to the lack of generalisability of case studies. It is, however, not my intention to generalise these results of individual cases but to add to the body of knowledge on the nature of mathematics teachers’ reflective practice as well as to generate new research questions and hypothesis.

A second limitation of this research study relates to fact that the five participants are of the same cultural and language group, teaching at the same school. I would have preferred a more diverse sample, excluding ethnic and geographical biases.

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3.11 Conclusion

In this chapter I have described the research design and methodology that will guide this research study. In Chapter 4 I will discuss the research results obtained using this qualitative research design.

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Chapter 5

Conclusions and implications

5.1

Introduction

In Chapter 4 the results of the data analysis of this research study were discussed. In this chapter I provide the conclusions and implications of the study. After summarising the chapters the research questions are verified, followed by a discussion of my own reflections on the procedure I followed to conduct the study as revealed in my researcher diary. I also make provision in this chapter for the fact that I may have been wrong in my interpretation of the participants’ reflective practice. This is followed by conclusions, recommendations and final reflections on the research study.

5.2

Chapter summary

In Chapter 1 I introduced and contextualised the research study. This study aimed to explore mathematics teachers’ reflections before, during and after teaching a lesson. I situated my investigation in the context of lesson study, mainly because I believed that teachers would reveal their reflections more openly in a setting where they jointly discuss their teaching experiences. In this chapter I also formulated three research questions to guide the inquiry. I briefly discussed methodological considerations, as well as possible contributions and limitations of the study.

In Chapter 2 I reviewed the theoretical underpinnings of teacher reflection and reflective practice as found in the literature. Research studies dealing with the reflective practice of pre-service and practising teachers were investigated. I focused on teacher reflection in general and mathematics teachers’ reflective practice in particular. The content and depth of teacher reflection are measured at different levels and in this chapter I discussed various categories of reflection as depicted by different researchers and theorists. I also explored the different meanings of reflection found in the literature and developed a tentative definition of reflection and conceptual framework for this study, based on this review and exploration.

Chapter 3 described the research design that guided this case study. An in-depth exploration of five mathematics teachers’ reflection before, during and after teaching a lesson was qualitatively conducted through interviews, classroom observations and lesson study group reflections on the lesson observed. The teachers’ lesson plans and reflective writings were

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also analysed. The strategies for data analysis were discussed, and measures of quality assurance were provided. Ethical considerations were presented and the possible limitations of the study were indicated.

In Chapter 4 I discussed the results of the analysis of this research study. The data were analysed qualitatively using the computer programme Atlas.ti 6. The interview transcripts were coded and six themes, related to the conceptual framework of the study and the research questions that guided the study, were created. The voices of the five mathematics teachers were heard, explaining their understanding of reflection, their reflection-for-action, reflection-in-action and reflection-on-action, as well as the contextual factors perceived to influence their reflections. Each teacher’s level of reflection was established by using Lee’s

(2005) and Jay and Johnson’s (2002) levels of reflection.

5.3

Verification of research questions

In this section I discuss the interpretations of the participants’ reflective practice as they relate to the research questions.

The research questions that guided the study were the following:

Question 1: What is the nature of mathematics teachers’ reflective practice?

To address this main question, the following subquestions guided the enquiry: a) How do mathematics teachers understand the concept of reflection? b) How do mathematics teachers reflect before, during and after teaching? c) What is the possible relationship between mathematics teachers’ reflection and their classroom practice?

Question 2: How do contextual factors influence mathematics teachers’ reflective practice?

Question 3: What is the potential significance of mathematics teachers’ reflective practice for the science and art of mathematics teaching?

5.3.1 Research question 1

To understand the nature of mathematics teachers’ reflective practice the study attempted to answer three subquestions. The first subquestion deals with mathematics teachers’

understanding of reflection.

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The findings of the study indicate that only two of the five mathematics teachers understood the concept of reflection, as measured by the working definition of reflection defined in

Chapter 2.7. Morgan stated that he looked back on his actions to think about what he could have done differently to help the learners gain a better understanding of the concepts he taught. Vicky said that because of her reflection on her learners’ circumstances she was able to change her way of teaching. For her reflection meant looking forward to transform her

actions. This finding is very much in line with findings in the literature that suggest that there is no single definition of reflective practice (Osterman & Kottkamp, 1993; Sparks-Langer,

1992; Zeichner & Liston, 1987).

The second subquestion asked

how mathematics teachers reflect before, during and

after teaching a lesson. This subquestion relates to teachers’ reflection-for-action, reflection-on-action and reflection-in-action. The findings of the study indicate that none of the teachers involved in the study completed the sections in the lesson plan that relate to reflection on their expectations of their learners. They did not tailor their planning with the idiosyncratic needs of their learners. According to Butke (2003) a teacher’s lesson plan formulates the concrete, tangible product of reflection-for-action. In this research study the planning, teaching and reflection on the lesson observed formed a cycle in which the reflection-for-action dimension was an extension of the reflection-on-action where certain understandings had been reached concerning the previous teaching episode, and how the next teaching episode would be affected. However, the five teachers’ lesson plans did show an improvement from the original lesson plans presented at the initial interview to the final lesson plans submitted for the observation lesson.

The five teachers all reflected on their actions during the initial interviews and the postobservation interviews ( reflection-on-action). They reflected on various aspects of their teaching, their time management, their classroom arrangement, their learners’ lack of understanding of basic concepts and understanding of mathematics amongst other things.

One teacher, Mary, reflected on the curriculum that does not allow question papers to be set in such a way that most of the learners will pass the papers. She also reflected on the learners’ use of calculators and obsession with cell phones and blamed these for the learners’ lack of interest in mathematics. Only one teacher, Sipho, reflected on his learners’ thinking while they were busy doing mathematics. The five teachers’ reflection-on-action are in line with the understandings of Dewey (1933) and Schӧn (1983, 1987) who focused on reflection as a method of thinking about experience that leads to inquiry and problem solving

13

. The finding also confirms Hatton and Smith’s (2006) argument that as

13

Discussed in Chapter 2, Sections 2.2.1 and 2.2.2

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professionals, teachers should frame and reframe complex and ambiguous problems that they face, test out various interpretations and modify their actions as a result.

Three teachers displayed moments of reflection-in-action while teaching their observation lessons. All three explained the deviations from their lesson plans in terms of reflection while teaching the lesson. By reflection-in-action they solved problems that had emerged from their planning. Morgan realised while teaching that he should expose his learners to a more difficult example, Sipho said he realised that he should have revised the distributive property in more detail and Vicky wanted to prepare her class for more challenging binomial products because they were to write a general Grade 9 exam. By reflection-in-action they found a way to re-appreciate a problematic situation. Schӧn (1987) described reflection-in-action as that moment when a teacher becomes surprised, interprets something in the teaching-learning situation as a problem and invents procedures to solve the problem.

To answer the third subquestion ( What is the possible relationship between

mathematics teachers’ reflection and their classroom practice?) the study considered each participant’s reflection in relation to his/her observation lesson. Mary, who reflected mostly on Level R1 (recall) (Lee, 2005) did not involve her learners actively in her lesson. My impression was that she did not understand her learners' thinking and did not encourage the learners to clearly explain and justify their reasoning. She also frequently interfered with her learners’ thinking. According to Warfield, Wood and Lehman (2003) such teachers base instructional decisions on the expectations of external voices rather than on their learners’ thinking. They do not reflect deeply about either their learners’ mathematics or about their own teaching (Warfield, Wood & Lehman, 2003).

Morgan however, reflected critically (Level R3) (Lee, 2005) on his learners’ understanding of mathematics concepts and how to possibly transform his own practice to improve their understanding. In his observation lesson he allowed learners to solve problems in their own ways and expected them to both explain and justify their reasoning and to listen to and question the reasoning of other learners, actively involving all learners in class.

To sum up, the first question deals with the nature of mathematics teachers’ reflective practice. From my investigation I found that only two of the five mathematics teachers understood the concept of reflection (based on the working definition of reflection for this research study) before the onset of the research. Although the lesson plan template made provision for reflection-for-action (with space provided for writing about expectations of how the content will be received by the learners), none of the teachers completed these sections.

Three teachers reflected in-action while they were teaching and changed their lesson plans

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to adapt to unexpected happenings in the classroom; and all five teachers reflected onaction during the interviews and in their lesson study group discussions. They reflected on pedagogical matters (classroom management, time management, teaching style, learners’ understanding of mathematics), personal issues (language, shortcomings as a teacher), external factors (curriculum, interferences while teaching, class size), and critical issues

(learners’ needs, learners’ thinking). Their reflections were rated on Lee’s (2005) levels of reflection and ranged from R1 (recall level of reflection which is descriptive in nature), R2

(rationalisation level of reflection) and R3 (reflective level, thinking critically about their own teaching and the impact of their actions on their learners’ understanding of mathematics).

Furthermore, by relating each teacher’s level of reflective thinking to his/her observation lesson, it seems that a possible relationship might exist between a teacher’s reflection and his/her instructional decisions. Teachers who were reflecting on a critical level of reflection seemed to pay more attention to their learners’ thinking about mathematics and how their own instruction of mathematics might influence their learners’ understanding of mathematics.

5.3.2 Research question 2

To answer the second research question (

How do contextual factors influence

mathematics teachers’ reflective practice?), the participants’ reflections during the interviews as well as during the lesson study group reflections were analysed. Contextual factors that emerged through these reflections were 1) the opportunity created for reflection by the lesson study group experience and 2) language.

Support for the first contextual factor, the lesson study group experience is provided by Coe, Carl and Frick (2010), who mention that 1) lesson study can act as an agent of change in a culture of isolation; 2) participants become comfortable with having colleagues observe them teach; 3) an increase in content knowledge is realised by participating members; and 4) lesson study provides an approach that is continuously effective in meeting the needs of learners.

Language as a possible contextual factor is confirmed by Reed, Davis and

Nyabanyaba’s (2003) who suggest that teachers may need to be apprenticed into reflective discourses, whether in their main language or an additional one

.

To sum up, the second question deals with how contextual factors influence mathematics teachers’ reflective practice and from my investigation I found that teachers reflect on their practice when granted the opportunity to do so (through interviews and within the context of lesson study). In addition language seemed to influence the teachers’ ability to reflect on their practice.

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5.3.3 Research question 3

The third research question,

What is the potential significance of mathematics teachers’

reflective practice for the science and art of mathematics teaching? encapsulates the other two main questions. The results of this study indicate that not all teachers are reflective practitioners, but by creating an opportunity for reflection, through lesson study, the participants of this study did reflect-on and in-action. Their individual reflections were descriptive, rationalising their actions and not directed to transformation of their classroom practice. However, in the lesson study group reflections the individual participants achieved a critical reflective level, and this finding can be utilised to bring teachers out of isolation by way of meaningful collaboration. The findings highlight that a lesson study model has potential for effective continuous professional teacher development (CPTD) within the South

African context (Coe, Carl & Frick, 2010).

To sum up, the third question deals with the significance of mathematics teachers’ reflective practice and I found that through the lesson study experience the teachers in my study were able to reflect on their classroom practice, reporting that they had gained a lot from observing themselves and their colleagues’ teaching.

5.3.4 Summary of verification of research questions

In Table 5.1 a summary of the findings of this research study to verify the research questions of the study is provided.

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Table 5.1

Summary of verification of research questions

Research questions

1) What is the nature of mathematics teachers’ reflective practice?

Verification

Mathematics teachers reflect in-action and on-action, but not for-action. The content of their reflections indicate that they reflect mainly on pedagogical matters, personal issues, external factors and critical matters. The level (depth) of their reflections range from Level 1 (recall/descriptive), Level 2

(rationalisation/comparative) to Level 3 (reflective level/critical reflection). They reflect whenever they are given the opportunity to do so (during interviews and in the context of the lesson study group). Furthermore, a number of contextual factors influence their reflective practice. In the current study, the lesson study experience seemed to have a positive influence on their reflective practice. In addition, inadequate language and verbalisation skills seemed to hamper reflective abilities. a) How do mathematics teachers understand the concept of reflection?

Whereas one teacher understood the concept of reflection as looking back on action, another teacher understood the concept of reflection as looking forward to guide action. The other three teachers were unable to explain their understanding of the concept of reflection. b) How do mathematics teachers reflect before, during and after teaching?

No evidence was found in this study that mathematics teachers reflect before their teaching of a lesson, even though the lesson plan template made provision for such reflection.

Three teachers reflected during their teaching by deviating from their written lesson plans and adapting their examples and classroom arrangements to cater for unexpected events or responses from learners.

All five teachers reflected after their teaching by recalling or describing certain incidents (R1 level of reflection). Moreover, all teachers increasingly rationalised or generalised their experiences as the research study progressed (R2 level of reflection). However, only two of the teachers were able to eventually reflect critically on their action (R3 reflective level) during the lesson study group reflections (Lee, 2005).

Individually, teachers reflected verbally on their actions during the interviews and cooperative participation during the lesson study group reflections. Their post-observation written reflections on their actions in class were at a recall level only (describing what had happened in class). c) What is the possible relationship between mathematics teachers’

Evidence was found that a relationship between the teachers’ reflection and their classroom practice exist.

Teachers who reflected on a lower level (R1 level of reflection) neglected to allow learners to solve problems using their own methods and communicate their findings to their fellow learners in contrast with

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reflection and their classroom practice? teachers who reflected on a more critical level (R3 level of reflection) about their learners’ understanding of concepts and their own classroom practice.

2) How do contextual factors influence mathematics teachers’ reflective practice?

The lesson study context experience proved to be a positive influence on all the teachers’ reflective practice. All teachers reported positively on the cooperative planning of a lesson, revealing that they learned much from the experience of planning with the goal to improve learners’ understanding of concepts.

Additionally, they reported that they were teaching with more confidence as a result of watching themselves as teachers and learning from watching their colleagues. They also reported a sense of increased and deeper awareness of their learners’ needs and the importance of involving learners in their lessons. Lastly, two of the teachers regarded the lesson study experience as self-research that enabled them to compare themselves with their colleagues and observe their own actions critically while watching the postobservation videos.

Teachers’ inadequate linguistic skills and inability to verbalise basic mathematical concepts properly seemed to influence mathematics teachers’ reflective practice negatively. One teacher constantly reflected on her poor command of English and the fact that she code-switched to Sesotho to explain content to her learners. As the research project progressed, the teachers increasingly talked more openly and freely during the group reflections in English, in contrast to their first planning session when they all wanted to talk in Sesotho. A plausible explanation for this phenomenon seems to be that teachers’ lack of experience ion teaching Mathematics in English impacted their self-confidence negatively.

3) What is the potential significance of mathematics teachers’ reflective practice for the science and art of mathematics teaching?

The results of this study show that when mathematics teachers’ are made aware of their reflections on their practice in a context of working cooperatively, they are encouraged to reflect at a more critical level. This finding has some potential value for planning professional learning programmes, where teachers can be encouraged to talk about their classroom experiences, share their joys and challenges with each other and strive to build a community of reflective practitioners to enhance their learners’ understanding of mathematics.

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5.4

What would I have done differently?

As a novice researcher I learned a lot during the course of this research study. My own reflective capabilities grew and I am now more able to reflect on my own role as researcher and teacher educator. I also learned more about the reality of teaching in a rural school using a language of instruction that is not your home language, teaching mathematics without resources, and trying to deal with the pressure from the school management and district office to increase the pass rate of the learners. This caused me to expand my thinking about the development of mathematics teachers. I now realise that effective professional learning programmes need to establish supportive and interactive communities that use reflection as a means for growth.

What would I have done differently? I would have liked to extend the opportunities for participant reflection, but due to time constraints this was not possible since I could only meet the participants once a week on a Thursday afternoon.

In addition, I initially wanted the participants to complete a daily reflective diary, so that I could follow their teaching lives during the week, but they were reluctant to submit their diaries as arranged. Lee (2005) argues that it is important to create various opportunities and climates where reflective thinking about their practice can flourish rather than to limit teachers to a particular approach. With this view in mind, I then asked the participants in my study to write a one-page reflection on the observation lesson, so that I had at least some form of reflective writing from each participant. I agree with Russell (1993) who suggests that some teachers need support in learning how to reflect. If I were to repeat this study, I would provide for reflection on critical incidents and ask the teachers to write reflectively about these incidents on a weekly basis.

When I originally planned this research study, I proposed to invite teachers teaching the same grade (e.g. Grade 10) from neighbouring schools to become participants to this study.

However, the district official suggested that I use only one school, with the result that the participants of my study were teaching mathematics to learners ranging from Grade 8 – 11.

This turned out to be a beneficial arrangement as I realised that the teachers learned more about learners’ understanding and misunderstanding of concepts by observing each other’s video-recorded lessons, than they would have if the group had remained homogenous as originally planned.

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Finally, when I coded the data I realised that language emerged as a possible contextual factor that influences teachers’ reflection. There is a possibility that the participants in my study might have reflected more openly and freely in their own language if an interpreter was available from the start of the research project.

5.5

Providing for errors in my conclusions

I engaged with five mathematics teachers who opened their classrooms to me and revealed themselves as teachers and human beings in the context of my study. I have made some decisions on their reflective capabilities as teachers, and I have to provide for the fact that I may have been wrong in my conclusions. I have tried to ensure that the results of this study are trustworthy. I have used interviews, video recordings of lessons, reflective writings and group reflections to gather data. I have also asked two experienced colleagues to verify my coding of the data and the families I created from the coding of the data, in order to enhance the trustworthiness of the data analysis process. The emerging reality I described in my results chapter (Chapter 4) was a crystallised reality obtained from different perspectives that all reflected the unique reality and identity of the participants of this research study

(Nieuwenhuis, p. 81, 2010).

However, I often struggled to make sense of the abundance of data, frequently consulting with my supervisor and co-supervisor to share my feelings of disequilibrium. Although I sometimes became confused, I persevered and with time succeeded to piece together a picture of each participant’s reflective journey. I have learned that teachers, once they are aware of their reflective capabilities, realise the value that reflection can add to their classroom practice. I am convinced that reflection created opportunities for the participants in my study to grow as teachers, as professionals and personally.

5.6

Conclusions

This research study focused on mathematics teachers’ understanding of reflection, the content and level of their reflections before, during and after teaching a lesson, and the contextual factors that influenced their reflections. The findings indicate that the participants of this study experienced growth in their reflective practice as the research study progressed. Although their lesson planning improved, they still neglected to reflect on their expectations of learners in their planning and their final reflective writings were on a descriptive/recall level (Level 1) only. All the participants reflected on their actions in class during the interviews and lesson study group reflection sessions. The content of their

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reflections differed, ranging from pedagogical issues, curricular matters, personal issues and critical issues. The level of their reflection-on-action (Jay & Johnson, 2002; Lee, 2005) during the interviews ranged from Level 1 (descriptive/recall) to Level 2

(comparative/rationalisation), but during the lesson study group reflections they reflected on

Level 3 (critical/reflective) on each other’s lessons as well as on their own teaching. Three of the participants reflected in-action, as indicated by their deviations of their lesson plans. The contextual factors that seemed to influence the participants’ reflections are the opportunity the lesson study group provided for collaborative reflection and language. The participants reflected positively on the lesson study experience, and mentioned the following benefits: 1)

They felt that they improved their lesson planning; 2) They gained confidence as mathematics teachers; 3) They reported a deeper awareness of learners’ needs; 4) They learned from colleagues, watching each others’ video recordings and discussing their own and each others’ lessons after observation of the video; and 5) They felt that the lesson study experience helped them to research themselves as teachers, seeing themselves in their classrooms, becoming aware of not only their mistakes, but also the positive aspects of their teaching.

5.7

Recommendations

Additional questions have been generated by this research study. For example, it would be worthwhile to undertake a follow-up study with the participants of this study to understand the long-term effects of reflective process.

Furthermore, I believe that there might be other contextual factors that influence mathematics teachers’ reflective practice that need to be researched further, for example gender, personality characteristics and culture. The reason why I consider gender to possibly influence the reflective process is because the male participants in my study were very reluctant to write about their reflections in a reflective diary. Personality characteristics might play a role in a teachers’ reflective practice and I base this belief on Dewey’s (1933) three attitudes that he considered to be integral to reflection: openmindedness, responsibility and wholeheartedness. Reflection might also be culturally bound, for example Lee and Tan

(2004) investigated student teachers’ reflective practice in Malaysia and found that their private reflections were on a deeper level than their public reflections. An intercultural study, for example comparing South African mathematics teachers’ reflective practice to that of a different culture will provide for a more comprehensive body of knowledge on reflective practice.

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Lastly, the results of this study can be used in the planning of future CPTD programmes. The positive feedback of the participants on the lesson study process suggests that lesson study should be initiated in other settings. Further research should explore lesson study as a model in South Africa for successful CPTD programmes.

5.8

Limitations of the study

As already stated in Chapter 3 Section 3.9, the small number of participants (five) makes it impossible to generalise the data from this research study. However, within the qualitative paradigm I tried to study the reflective practice of these five participants from multiple perspectives, using various methodologies.

Another limitation of this research study pertains to the researcher’s inability to speak the home language of the participants. Language emerged as a contextual factor that possibly influences the participants’ of this study’s reflective practice, although I allowed for an interpreter during the last group reflection. I evaluated the participants’ verbal and written reflections as they struggled to express themselves in English, and I have to make provision for the fact that I may have misinterpreted their understanding of reflection.

5.9

Last reflections

As much as any qualitative study can draw any conclusions, this research study represented not only the professional and personal development of the five teachers who were so willing and eager to share their stories with me, but also the personal and professional culmination of three years of hard work. During this time I gained new insight into mathematics teachers’ classroom practice, as well as the realities which these teachers have to deal with every day.

The lesson study context of my research offered an effective strategy to bring teachers out of isolation, allowing them to experience meaningful collaboration with their fellow teachers, observing and criticising their own and each other’s lessons and planning lessons together.

All five of these teachers proved to be reflective practitioners to a greater or lesser degree.

However, these teachers need ongoing support if they are going to continue to develop as reflective practitioners and they need professional learning opportunities to help them to develop their reflective capabilities.

In Chapter 1 I mentioned that the importance of this research study rests on its unique connection of reflective practice with mathematics teaching. I am convinced that through engaging in the lesson study experience the five participants of this study improved their

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reflective practice, reporting an increase in self-knowledge and finding new ways of teaching mathematics to learners. As for myself, my own reflective journey and involvement with mathematics teachers’ professional learning will continue. In the words of Carl Friedrich

Gauss (cited in Fleming & Varberg, 1992)

Does the pursuit of truth give you as much pleasure as before? Surely it is not the knowing but the learning, not the possessing but the acquiring, not the being-there but the getting-there, that afford the greatest satisfaction. If I have clarified and exhausted something, I leave it in order to go again into the dark. Thus is that insatiable man so strange: when he has completed a structure it is not in order to dwell in it comfortably, but to start another.

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APPENDICES

Lesson plan template Appendix A:

Lesson Study: Lesson Plan

Date:

Educator:

Lesson duration:

Pre-active stage of the lesson:

Topic:

Understanding this topic is important because …..

What are your goals for this lesson?

What will learners understand as a result of this Lesson?

(what do you want learners to know and be able to do?)

What evidence will show learners understand?

(what performance tasks, products, projects, etc. will demonstrate understanding?)

What materials will you need?

Prior knowledge of learners

Future knowledge

(following the lesson)

153

Interactive stage of the lesson:

Introduction (Please provide full details of your expectations) Reflection (after the lesson)

(How will you introduce the lesson? Give full details)

Learner Activity

(provide full details of the activity, as well as your expectations of learners’ responses)

• class arrangement

• assessment

• teacher activity

(role)

• learner activity

(role)

Post-active stage of the lesson

Class discussion

(how do you plan to wrap up the lesson)

Your feedback to the learners about the activity and learning that took place

Final reflections on the lesson

(complete this part as soon as you have taught the lesson)

154

Appendix B: Letter of consent

FACULTY OF EDUCATION

13 May 2010

Mrs AB Posthuma

Groenkloof campus [email protected]

Tel: 082 293 5533

Dear Ms/Mr ………………..………………

Letter of consent to the mathematics teacher

I am conducting a research study for my doctoral degree in mathematics education at the University of Pretoria. This letter is written to invite you to be a participant in this study. The Free State

Department of Education granted permission to allow you to participate in the study.

The study seeks to investigate the nature of mathematics teachers’ reflective practice. In order to do this, the following techniques will be employed: 1) I will interview you before the onset of the research study. Please bring one of your lesson plans to the interview for discussion. The purpose of this interview is to help me to understand the nature of your reflections in your classroom practice. 2) You will form part of a lesson study group with other mathematics teachers. In this group you will collaboratively plan a mathematics lesson. One teacher in the group will teach the planned lesson to his/her class. This lesson will be video-recorded and this video recording will be shown to the group afterwards so that they can discuss the teaching of the lesson and reflect on ways to improve the lesson plan. The revised lesson will then be taught by another member of the group and the cycle will continue. 3) You will be asked to keep a reflective journal and share experiences related to your teaching. 4) You will be asked to participate in a final group interview with your fellow teachers of the lesson study group in order to reflect upon your teaching practice and the experience of planning and teaching lessons collaboratively. Audiotapes of the interviews will be made so that each teacher’s verbal reflections can be used as data. Participation is entirely voluntary and you may withdraw from the study at any time if you so choose.

Should you declare yourself willing to participate in this study, confidentiality will be guaranteed. You will be asked to read the transcripts of the interviews to ensure the trustworthiness of the data. Your decision to accept/decline involvement in this research will not influence your teaching career in any way, nor will your participation be reflected in your performance appraisal.

If you are willing to participate in this study, please sign this letter as a declaration of your consent.

Yours sincerely

………………………………………………………….

Mrs AB Posthuma

………………………………………………………….

Supervisor: Prof JG Maree

Participant’s signature ........................................................

E-mail address ……………………………………………….

Date: ………………………………

Date: ………………………………

Date:………………………………

Contact number ………………..

155

Appendix C: Letter of permission

to department

FACULTY OF EDUCATION

Mrs AB Posthuma

Aldoel Building F235

Groenkloof Campus [email protected]

Tel: 082 293 5533

13 May 2010

Free State Department of Education

Dear Sir/ Madam

Request from FDE for permission to do classroom observations and to conduct interviews

I am currently enrolled as a doctoral student at the University of Pretoria, where I am also a lecturer in the Department of Science, Mathematics and Technology Education. The title of my proposed thesis is as follows: The nature of mathematics teachers’ reflective practice. The importance of this research rests on its unique connection of reflective practice relating to teaching, specifically in the field of mathematics. The rationale for this study stems from the premise that mathematics teachers need to find a vehicle for growth and improvement. The development of a reflective process can serve as an important technique in increasing self-knowledge and seeking new ways of educating learners in mathematics. The study can add to research findings concerning reflective practice and contribute to the discussion on the usefulness of reflective practice as a reform effort in teacher education.

In order to collect data for this project, I would like to invite mathematics teachers of at least two neighbouring schools in the Thabo Mofutsanyana district to participate in forming a lesson study group. In lesson study the focus is on the concrete examination of practice and the testing of new ideas in actual classrooms. This examination of practice is a collaborative exercise in which a group of teachers design, reflect on, and deliver mathematics lessons to enhance student achievement.

I will interview the teachers individually and observe them in the lesson study group while they are planning lessons and reflecting on their classroom practice. The teacher who teaches the lesson will also be observed and video-recorded. My observations will be unobtrusive and confidentiality of both participants and the institutions will be ensured.

I hereby formally request your permission to observe and interview mathematics teachers at schools in the Thabo Mofutsanyana district in the second term of this year. I trust that my request will meet with a favourable response.

Yours faithfully

…………………………………..

Mrs AB Posthuma

156

Appendix D: Interview and group reflection schedules

INTERVIEW SCHEDULE 1

Individual semi-structured interview

Name of school

Name of researcher

Name of teacher

Male/ female

Date of interview

Teacher’s qualification

Level of Mathematics education

Barbara Posthuma

No of years teaching Mathematics

Home language

Thank you for taking part in my study. As you are aware I am working on my thesis. The topic is the nature of mathematics teachers’ reflective practice. I want to get your perspective of reflection and have some questions that I want to ask you. Please feel free to elaborate on your answers, the more you can tell me, the more I will be able to represent your views in the thesis.

1.

2.

3.

What do you understand by the term reflection?

Do you reflect on your classroom practice in mathematics?

Can you give me an example of when you reflected on your classroom practice? i. How do you reflect on your practice? ii. When do you reflect on your practice? iii. What do you reflect on after teaching a lesson? iv. Do any particular factors influence your reflections? v. Have you

gained anything by reflecting on your practice? If the answer is “yes”, how? If the answer is “no”, why not?

157

Discussion of the lesson in lesson plan provided during initial interview:

What did you like best about this lesson?

What, in your opinion, were essential strengths of the lesson?

What, in your opinion, were essential challenges experienced during the lesson?

Suppose you had to teach the lesson again, what aspects, if anything, would you change about the lesson?

Did your learners achieve the lesson outcomes? Why do you say so?

What, if any, unanticipated learning outcomes resulted from the lesson?

Can you think of another way you might have taught this lesson?

Can you think of alternative approaches to teaching this lesson that might improve the learning process?

Do you think the content covered was meaningful to the learners? Why?

Is there anything else that you would like to add to what has already been said in this regard?

INTERVIEW SCHEDULE 2

Post-observation interview

What, in your opinion, were essential strengths of the lesson?

What, in your opinion, were essential challenges experienced during the lesson?

What, if anything, would you change about the lesson?

Do you think the lesson was successful? Why?

Should you feel that the lesson was not successful, please elaborate?

What conditions were important to achieving the outcomes?

What, if any, unanticipated learning outcomes resulted from the lesson?

Can you think of other ways in which you might have taught this lesson?

Can you think of alternative approaches to teaching this lesson that might improve the learning process?

Do you think the content covered was meaningful to learners? Why?

What, if any, moral or ethical concerns occurred as a result of the lesson?

Is there anything else that you would like to add to what has already been said in this regard?

158

Final group Interview

Name of school

Name of researcher

Name of teacher

Gender

Date of interview

In which ways, if at all, did participation in the lesson study group alter your teaching practice?

What are your thoughts about the lesson study group?

What are your feelings about the lesson study group?

How did you experience working collaboratively with colleagues in the lesson study group?

Did you gain anything from this experience? If the answer is “yes”, in which ways and what did you gain?

Were there any negative experiences in the lesson study group? If the answer is “yes”, please elaborate?

As a result of your lesson study group experience, describe a specific instance(s) in which you have been reflective or specific instance(s) in which your teacher thinking was changed in some way

159

Appendix E: Ethical clearance certificate

160

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