professional growth through working together...

PROFESSIONAL GROWTH THROUGH WORKING TOGETHER:

A STUDY OF RECIPROCAL BENEFITS FOR TEACHER AND EDUCATION ADVISOR

THROUGH CLASSROOM BASED PROFESSIONAL DEVELOPMENT

Judith Elizabeth Hartnett Dip Teach (Primary), Grad Dip (Computer Ed), MEd

Submitted in fulfilment of the requirements for the degree of

Doctor of Education

Centre for Learning Innovation

Faculty of Education

Queensland University of Technology

2011

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Professional growth through working together: A study of reciprocal benefits for teacher and education advisor through classroom-based professional development.

Keywords

Change sequence Education advisor Growth network Interconnected Model of Professional Growth Mathematics Mental computation Practising classroom teacher Professional growth Reflection Teacher professional development

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Abstract

Teacher professional development provided by education advisors as one-off,

centrally offered sessions does not always result in change in teacher knowledge,

beliefs, attitudes or practice in the classroom. As the mathematics education advisor in

this study, I set out to investigate a particular method of professional development so as

to influence change in a practising classroom teacher’s knowledge and practices. The

particular method of professional development utilised in this study was based on

several principles of effective teacher professional development and saw me working

regularly in a classroom with the classroom teacher as well as providing ongoing

support for her for a full school year. The intention was to document the effects of this

particular method of professional development in terms of the classroom teacher’s and

my professional growth to provide insights for others working as education advisors.

The professional development for the classroom teacher consisted of two

components. The first was the co-operative development and implementation of a

mental computation instructional program for the Year 3 class. The second component

was the provision of ongoing support for the classroom teacher by the education

advisor. The design of the professional development and the mental computation

instructional program were progressively refined throughout the year. The education

advisor fulfilled multiple roles in the study as teacher in the classroom, teacher educator

working with the classroom teacher and researcher.

Examples of the professional growth of the classroom teacher and the education

advisor which occurred as sequences of changes (growth networks, Hollingsworth,

1999) in the domains of the professional world of the classroom teacher and education

advisor were drawn from the large body of data collected through regular face-to-face

and email communications between the classroom teacher and the education advisor as

well as from transcripts of a structured interview. The Interconnected Model of

Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999) was used to

summarise and represent each example of the classroom teacher’s professional growth.

A modified version of this model was used to summarise and represent the professional

growth of the education advisor.

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This study confirmed that the method of professional development utilised could

lead to significant teacher professional growth related directly to her work in the

classroom. Using the Interconnected Model of Professional Growth to summarise and

represent the classroom teacher’s professional growth and the modified version for my

professional growth assisted with the recognition of examples of how we both changed.

This model has potential to be used more widely by education advisors when preparing,

implementing, evaluating and following-up on planned teacher professional

development activities. The mental computation instructional program developed and

trialled in the study was shown to be a successful way of sequencing and managing the

teaching of mental computation strategies and related number sense understandings to

Year 3 students.

This study was conducted in one classroom, with one teacher in one school. The

strength of this study was the depth of teacher support provided made possible by the

particular method of the professional development, and the depth of analysis of the

process. In another school, or with another teacher, this might not have been as

successful. While I set out to change my practice as an education advisor I did not

expect the depth of learning I experienced in terms of my knowledge, beliefs, attitudes

and practices as an educator of teachers. This study has changed the way in which I plan

to work as an education advisor in the future.

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Table of Contents

Keywords ................................................................................................................................................2 Abstract ...................................................................................................................................................3 Table of Contents ....................................................................................................................................5 List of Figures .........................................................................................................................................8 List of Tables ........................................................................................................................................10 Statement of Original Authorship .........................................................................................................11 Acknowledgments.................................................................................................................................12 1. INTRODUCTION ...................................................................................................................13 1.1 Background................................................................................................................................14 1.2 Stimulus for the study ................................................................................................................16 1.3 Purposes.....................................................................................................................................17 1.4 Context.......................................................................................................................................18 1.5 Significance ...............................................................................................................................18 1.6 Thesis Outline ............................................................................................................................19 2. LITERATURE REVIEW .......................................................................................................20 2.1 Teacher change ..........................................................................................................................20

2.1.1 Teacher knowledge.........................................................................................................23 2.1.2 Models of teaching and learning ....................................................................................25 2.1.3 Models of teacher change ...............................................................................................27 2.1.4 Teacher professional growth ..........................................................................................35 2.1.5 Modelling teacher professional growth ..........................................................................38

2.2 Effective teacher professional development ..............................................................................42 2.2.1 Principles of effective teacher professional development ..............................................44 2.2.2 Methods of teacher professional development ...............................................................45 2.2.3 The importance of reflection ..........................................................................................50 2.2.4 The change environment ................................................................................................54

2.3 Education advisor change ..........................................................................................................56 2.3.1 Modelling education advisor change..............................................................................58 2.3.2 Professional development of education advisors............................................................63

2.4 Roles of an education advisor ....................................................................................................66 2.5 Mental computation ...................................................................................................................68

2.5.1 Components of mental computation ...............................................................................69 2.5.2 Categorisation of mental computation strategies............................................................71 2.5.3 Teaching mental computation.........................................................................................72 2.5.4 Assessing mental computation .......................................................................................78 2.5.5 Making thinking visible..................................................................................................79

2.6 Implications for this study .........................................................................................................80 2.6.1 Implications for professional growth..............................................................................80 2.6.2 Implications for the professional development...............................................................81 2.6.3 Implications for the mental computation instructional program.....................................81

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3. RESEARCH DESIGN.............................................................................................................83 3.1 Overview....................................................................................................................................83 3.2 Methodology..............................................................................................................................84 3.3 The change environment............................................................................................................86

3.3.1 The school context..........................................................................................................87 3.3.2 The education advisor’s work context............................................................................88 3.3.3 Subjects...........................................................................................................................88 3.3.4 The new mathematics syllabus .......................................................................................89

3.4 Data collection ...........................................................................................................................90 3.4.1 Class teacher and education advisor reflections .............................................................90 3.4.2 End of year interview .....................................................................................................92 3.4.3 Monitoring student learning ...........................................................................................92

3.5 Data analysis ..............................................................................................................................93 3.5.1 Representing the classroom teacher’s professional growth............................................95 3.5.2 Representing the education advisor’s professional growth ............................................97

3.6 The professional development ...................................................................................................99 3.6.1 Mental computation instructional program...................................................................101 3.6.2 Ongoing support by the education advisor ...................................................................104

3.7 Quality criteria .........................................................................................................................106 3.8 Ethics considerations ...............................................................................................................107 4. THE CLASSROOM TEACHER’S PROFESSIONAL GROWTH ..................................109 4.1 Knowledge of mental computation strategies ..........................................................................111

4.1.1 Vignette 4.1: From teacher to personal user of mental computation strategies ............111 4.1.2 Discussion of Vignette 4.1............................................................................................112

4.2 Belief in students making thinking visible...............................................................................114 4.2.1 Vignette 4.2: From in the students’ heads to on paper .................................................114 4.2.2 Discussion of Vignette 4.2............................................................................................115

4.3 Inclusion of number sense activities ........................................................................................118 4.3.1 Vignette 4.3: From ‘doing’ a strategy to understanding a strategy...............................120 4.3.2 Discussion of Vignette 4.3............................................................................................121

4.4 Resources to support learning..................................................................................................122 4.4.1 Vignette 4.4: From standard number board to alternative number board.....................124 4.4.2 Discussion of Vignette 4.4............................................................................................125

4.5 Meeting the varying needs of students.....................................................................................127 4.5.1 Vignette 4.5: From directed use of levelled activities to student selection of levelled

activities........................................................................................................................128 4.5.2 Discussion of Vignette 4.5............................................................................................130

4.6 Summary..................................................................................................................................131 5. EDUCATION ADVISOR PROFESSIONAL GROWTH..................................................133 5.1 Co-operative teaching ..............................................................................................................134

5.1.1 Vignette 5.1: From presenter to collaborator...............................................................134 5.1.2 Discussion of Vignette 5.1............................................................................................135

5.2 Co-operative planning..............................................................................................................139 5.2.1 Vignette 5.2: From planning for to planning with........................................................140 5.2.2 Discussion of Vignette 5.2............................................................................................140

5.3 Buliding relationships ..............................................................................................................143 5.3.1 Vignette 5.3: From outsider to insider..........................................................................143 5.3.2 Discussion of Vignette 5.3............................................................................................143

5.4 Working in classrooms ............................................................................................................146

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5.4.1 Vignette 4: From central professional development to classroom-based professional development. ................................................................................................................146

5.5 Education advisor as private consultant...................................................................................149 5.5.1 Vignette 5.5: From education advisor to many to education advisor to a few ............149 5.5.2 Discussion of vignette 5.5 ............................................................................................150

5.6 Summary..................................................................................................................................151 6. CONCLUSIONS ....................................................................................................................153 6.1 Professional growth of the classroom teacher .........................................................................156

6.1.1 Representing the classroom teacher’s professional growth..........................................157 6.2 Professional growth of the education advisor..........................................................................159

6.2.1 Representing the professional growth of the education advisor ...................................160 6.3 Factors influencing the professional growth of the education advisor ....................................164 6.4 Implications for education advisor practice .............................................................................167 6.5 Limitations of the study ...........................................................................................................171 6.6 Implications for further research..............................................................................................173 6.7 Professional Growth: A personal perspective..........................................................................175 REFERENCES..................................................................................................................................176 APPENDICES ...................................................................................................................................193

Appendix 1: Strategy Categorisation framework.....................................................................193 Appendix 2: Strategy Categorisation Framework strategy category abbreviation codes ........194 Appendix 3: Examples of Beishuizen’s mental computation strategies linked to the SCF .....195 Appendix 4: End of year interview questions..........................................................................196 Appendix 5: Tests to monitor student learning.......................................................................198 Appendix 5a: List of potentially efficient strategies for each addition item on the assessment

instruments. ..................................................................................................................201 Appendix 5b: List of potentially efficient strategies for each subtraction item on the assessment

instruments. ..................................................................................................................202 Appendix 6: Overview of Mental compuation instructional program.....................................203 Appendix 7: Sample lesson and communication between the classroom teacher and education

advisor ..........................................................................................................................205 Appendix 8: Student learning data...........................................................................................207

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List of Figures

Figure 2.1 A linear model of the teaching-learning process (described by Steinbring, 1998) .............25 Figure 2.2 Teaching and learning mathematics as autonomous systems (Steinbring, 1998)................26 Figure 2.3 Teaching triad (Jaworski 1992, 1994).................................................................................27 Figure 2.4. Psychotherapeutic model of educational change (e.g., Lewin, 1935). ...............................27 Figure 2.5 An alternative model of educational change (Guskey, 1986) .............................................28 Figure 2.6 Cyclic model of teacher professional development Clarke (1988) .....................................29 Figure 2.7 A dynamic model of professional growth (Clarke & Peter, 1993).....................................30 Figure 2.8 Role of classroom experimentation in teacher change (Clarke & Peter, 1993).................33 Figure 2.9 Role of teacher knowledge and beliefs in teacher change (Clarke & Peter, 1993) .............34 Figure 2.10 Interconnected Model of Teacher Professional Growth (Teacher Professional Growth

Consortium, 1994)........................................................................................................................35 Figure 2.11 Example of a change sequence (Hollingsworth, 1999) .....................................................36 Figure 2.12 Example of a growth network (Hollingsworth, 1999) .....................................................37 Figure 2.13 The Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002;

Hollingsworth, 1999) ...................................................................................................................40 Figure 2.14 Reflection as a mediating process in the Interconnected Model of Professional Growth

(Clarke & Hollingsworth, 2002, Hollingsworth, 1999) ...............................................................51 Figure 2.15 A growth network where change is mediated particularly through reflection .................53 Figure 2.16 Influences of the change environment on teacher professional growth ...........................54 Figure 2.17 Modified teaching triad of teacher educators (Zaslavsky & Leikin, 2004).......................60 Figure 2.18 Further modification of Jaworski’s (2009) teaching triad (Zaslavsky, 2009) ...................60 Figure 2.19 Zaslavsky & Leikin (2004)’s extension of Steinbring (1998)’s model of mathematics

educator change............................................................................................................................61 Figure 2.20 Generalisation of Steinbring’s (1998) model of teaching and learning mathematics

(Zaslavsky, 2009) .........................................................................................................................62 Figure 2.21 Four foci model of teacher education (Tzur, 2001)..........................................................64 Figure 2.22 Number board showing jumps on and off the decade .......................................................75 Figure 2.23 Numbered Line, Un-Numbered Line and Empty Number Line (showing 14+8) ...........76 Figure 3.2 The Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002;

Hollingsworth, 1999) ...................................................................................................................95 Figure 3.3 Modified version of Interconnected Model of Professional Growth

(based on Clarke & Hollingsworth, 2002; Hollingsworth, 1999) ................................................98 Figure 4.1 Classroom teacher’s professional growth in relation to personal knowledge of mental

computation strategies................................................................................................................113 Figure 4.2 Recording of two students’ thinking showing incorrect application of a strategy ..........115 Figure 4.3 Classroom teacher’s professional growth in relation to students making their thinking

visible .........................................................................................................................................117 Figure 4.4 Number board showing two strategies for 26+19 ............................................................120

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Figure 4.5 Classroom teacher’s professional growth in relation to the inclusion of number sense activities to support mental computation strategy development.................................................121

Figure 4.6 A standard number board showing Counting in Tens and Ones to solve 34+23 .............123 Figure 4.7 Using a standard number board for addition requiring bridging of a ten (36+26) ...........124 Figure 4.8. An alternative number board for computations requiring bridging of a ten.....................124 Figure 4.9 Using an alternative number board for addition requiring bridging of ten (36+26) .......125 Figure 4.10 Classroom teacher professional growth in relation to the different use of a known

resource - number boards ...........................................................................................................126 Figure 4.11 Classroom teacher’s professional growth in relation to catering for the range of

student abilities...........................................................................................................................131 Figure 5.1 Education advisor professional growth in relation to co-operative teaching. ..................138 Figure 5.2 Education advisor professional growth in relation to co-operative teaching ...................142 Figure 5.3 Professional growth of the education advisor in relation to developing relationships.....145 Figure 5.4 Education advisor professional growth in relation to working in classrooms..................148 Figure 3.1 Format of Pre, Post, Mid-year and Short tests ................................................................199 Figure A1 Percentage of response types on the pre and post tests.....................................................208 Figure A2. Use of recording matching methods or tools demonstrated or discussed during lessons

(i.e. empty number line and number boards)..............................................................................208 Figure A3. Student use of Strategy Categorisation Framework labels to describe strategies used.....209 Figure A4 Students’ own description of the thinking and strategy used...........................................209 Figure A5 State test data for all content strands and overall numeracy scores. ................................210 Figure A6 Year 3 state-wide test percentage correct for each of the mental computation items. .....211 Figure A7 Student use of the same strategy for the same item on Short-test 1 and 2 ........................219 Figure A8 Student use of the different strategy for the same item on Short-test 1 and 2 ..................219

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List of Tables

Table 2.1 Domains and components of the knowledge base of teaching (Borko & Putnam, 1995) .....24 Table 2.2 Alignment of teacher change models and teacher knowledges.............................................31 Table 3.2 Descriptions of the relationships between domains in the IMPG model for this study ........96 Table 3.3 Overview of the mental computation instructional program ..............................................103 Table 4.1 Percentage of items responses where strategies were identifiable per instrument .............116 Table A1 Categorisation of Mental Computation Strategies (Hartnett 2007)....................................193 Table A2 Strategy Categorisation Framework codes.........................................................................194 Table A3 Beishuizen strategy categories with examples (in Klein, Beishuizen & Treffers, 1998)

and links to Strategy Categorisation Framework (Hartnett, 2007) for addition examples........195 Table A4 Beishuizen strategy categories with examples (in Klein, Beishuizen & Treffers, 1998)

and links to Strategy Categorisation Framework (Hartnett, 2007) for subtraction examples...195 Table A5 Items on each of the researcher designed tests ...................................................................198 Table A6 State-wide test mental computation questions. ...................................................................200 Table A7 Percentage of items answered correctly on all instruments...............................................210 Table A8 Efficient strategies chosen of the identifiable strategies .....................................................212 Table A9 Accuracy of efficient and inefficient strategies for addition items ......................................213 Table A10 Accuracy of efficient and inefficient strategies for subtraction items ...............................213 Table A11 Different strategies evident in addition items for each instrument and in total ...............214 Table A12 Different strategies evident in subtraction items for each instrument and in total ...........215 Table A13 Individual student flexibility for addition items per instrument .......................................217 Table A14 Individual student flexibility for subtraction items per instrument ..................................218 Table A15 Strategy use on Short-test 1 and Short-test 2 instruments ...............................................219

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Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

Signature: _________________________

Date: 28th July 2011

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Acknowledgments

This thesis is the result of many years of work. I have been supported by many

people along the journey that set out to document and share some of the successes I was

having working as an education advisor supporting teachers in their classrooms to

improve the learning and teaching of mathematics. This thesis has taken many forms

over the years of its development. My initial supervisor Associate Professor Annette

Baturo helped me to shape my original ideas about supporting the teaching of mental

computation strategies. Professor Tom Cooper helped me to reshape the thesis to focus

on my role as an education advisor. Professor Stephen Ritchie helped me to refine the

thesis as a professional study of the reciprocal benefits for the classroom teacher and

myself as the education advisor. Doctor Mal Shield provided ongoing support and a

valued overall perspective for the project as this document were finalised. To these

colleagues I extend my sincere thanks. Without your help this thesis would not have

emerged from my work.

I would like to thank Mary Hare, the teacher who welcomed me into her

classroom for a year and allowed me to trial some of the ideas and the framework for

organising computation instruction and activities I had developed. Mary was a

successful teacher with many years of experience when we conducted this study. Her

willingness to learn and her honesty in reflecting on her learning and on the processes

we used during the study allowed this thesis to emerge from the data gathered over the

year we spent together. I would also like to thank the administration team and staff of

Mater Dei Catholic Primary School for their support during the year of the study and

after.

I also would like to acknowledge and thank my husband Russell and daughter

Kelsey for their ongoing support and tolerance of the hours I spent working on this

project. I wasn’t going to give up and you were there right through the set backs and the

successes. Thank you. ☺

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1. Introduction

Teaching is a profession undergoing continual change. As educational research

identifies innovations in content knowledge and pedagogical practices, schools and

teachers are implored to change to meet their overall aim of improving student learning

outcomes. Historically teacher change has been linked to planned teacher professional

development events that provided a stimulus to change. Guskey (1986) noted that the

provision of professional development was often in response to a perceived deficit in

teacher skills and knowledge. Other professional development activities focus on

changing existing teacher knowledge and skills. Justi and van Driel (2006) observed

that “there is a general agreement in the educational research community about the

importance of teachers’ professional development as one of the ways to improve

education. However, there is no consensus about how such a process occurs.” (p. 437).

Professional development for teachers needs to be facilitated by educators who

are knowledgeable and capable in relation to the innovations in content and pedagogy

that are the focus of change. Education advisors are educators who facilitate teacher

professional development. The educational literature refers to teacher educators,

although often this is in the context of pre-service and post-graduate teacher education

in universities. In this thesis the term education advisor is used to describe educators

who work with practising classroom teachers to support their professional growth.

Clarke and Peter (1993) introduced the term “teacher professional growth” (p. 167) to

characterise the learning aspects of teacher change as a process. Clarke and

Hollingsworth (2002) stated that “if we are to facilitate the professional development of

teachers we must understand the process by which teachers grow professionally and the

conditions that support and promote that growth” (p. 947). They further described

teacher professional growth as an inevitable and continuing learning process and

proposed a model of teacher professional growth to “offer a powerful framework to

support the analyses of those studying teacher change (or growth) and the planning of

those responsible for teacher professional development” (p. 947).

Education advisors need to maintain the recency of their own knowledge and

practices to best support the professional growth of teachers. While literature exists to

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describe the teaching and learning process (e.g., Jaworski, 1992; Lewin, 1935;

Steinbring, 1998), teacher change (e.g., Clarke, 1988; Clarke & Peter, 1993; Guskey,

1986) and the professional growth of teachers (Clarke & Peter, 1993; Clarke &

Hollingsworth, 2002; Hollingsworth, 1999), there is less literature that focuses on the

learning and professional development of education advisors.

The study reported in this thesis investigated a particular method of mathematics

professional development for one practising classroom teacher. The method of

professional development was designed to be more than a single session or single

session with follow-up. These “one-shot workshops” (Clarke & Hollingsworth, 2002,

p. 967) have been criticised in the literature as not providing effective or sustained

teacher change (Clarke, 1988; Fullan, Hill & Crevola, 2006; Loucks-Horsley, 1998).

The professional development method adopted in this study utilised many principles of

effective teacher professional development (Guskey, 2002; Mewborn, 2003; Nisbet,

Warren & Cooper, 2003). The professional growth experienced by the classroom

teacher and the professional growth of the education advisor were monitored and

reported. Factors which influenced the professional growth were identified so as to

provide recommendations for the professional development of other education advisors.

The education advisor in this study was also the researcher, so in this study I was

investigating my own professional growth as an education advisor as I implemented

professional development for the classroom teacher. The professional development was

designed to facilitate the classroom teacher’s professional growth in relation to the

teaching of mental computation strategies as part of the mathematics curriculum.

1.1 BACKGROUND

Many countries have completed reviews and implemented major new

mathematics curriculum policies. Some examples of these mathematics curriculum

policies and programs include National Numeracy in the United Kingdom (Department

for Education and Employment, 1999); Adding It Up in the United States of America

(Kilpatrick, Swafford, & Findell, 2001); and the Numeracy Professional Development

Project in New Zealand (Ministry of Education, 2007). In Australia, mathematics

curriculum documents have continued to be developed in each state (e.g., Board of

Studies New South Wales, 2006; Queensland Studies Authority, 2004, 2007) and more

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recently at a national level (Australian Curriculum Assessment and Reporting

Authority, 2008, 2009).

Mathematics education is exposed to a great deal of public scrutiny today with

the government reform agendas for education considering data from International

studies, for example, the Trends in International Mathematics and Science Study

(TIMSS) and the Program for International Student Assessment (PISA), as well as the

National Literacy and Numeracy Assessment Plan (NAPLAN), a high-stakes testing

program. Impetus for change in mathematics education is high and such educational

reforms assume that teachers will maintain currency in the knowledge and practices of

teaching mathematics.

The implementation of educational reforms requires classroom teachers to

participate in professional development and assumes there will be competent educators

to facilitate such professional development. This study drew upon the implementation

of a new Mathematics syllabus (Queensland Studies Authority, 2004) in the Australian

state of Queensland. I was working as a mathematics education advisor supporting

teachers to implement this new syllabus. Concerns about a lack of change in classrooms

as a result of professional development activities had been raised at a system level by

the team of education advisors I worked with. As part of my role as an education

advisor supporting schools to implement the new Mathematics syllabus and for this

study, I chose to work with one teacher in her classroom to support change in relation to

a particular aspect of the new Mathematics syllabus and to study and document my

professional growth as an education advisor as well as the professional growth of the

classroom teacher.

The aspect of the new mathematics syllabus chosen as the content focus for the

professional development in this study was mental computation. The inclusion of

mental computation in the new syllabus required a significant change in the

computation instructional focus of teachers as the previous syllabus had aimed for the

development of the traditional written algorithms. The professional development of the

classroom teacher involved the development and implementation of a comprehensive

mental computation instructional program. The professional development featured the

enhancement of the classroom teacher’s own knowledge about mental computation

strategies as well as the development of the mental computation instructional program

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that included the teaching of associated number sense understandings. Resources to

support mental computation learning and teaching to cater for the wide range of abilities

of the students in the Year 3 class, as well as tests of the students’ use of the strategies

were developed as part of the study.

1.2 STIMULUS FOR THE STUDY

Before becoming the researcher conducting this study, I had worked as a full-

time education advisor supporting teachers in relation to mathematics education for

eight years. At the time of the study, I worked as the Education Officer: Mathematics,

one of a team of curriculum advisors who provided support for approximately 130

primary schools (Years 1-7) and secondary schools (Years 8-12). Questions about the

effectiveness of the style of professional development being offered by the curriculum

team had been raised in team meetings prior to this study. In particular we had

discussed how the curriculum changes presented to teachers through our professional

development activities, which were generally offered as half or full day sessions for

large numbers of teachers, could have a greater impact at the classroom level. We had

also discussed whether such group professional learning sessions were the most

effective way of stimulating the desired changes. Goals to meet stated systemic aims

were set each year by the individual advisors and by our whole team, yet scarce data

were collected to judge the effectiveness of the work conducted. A desire to change

from large-scale centrally-offered professional development to smaller, more focussed

school-based support had been discussed

In the year prior to this study the Queensland Studies Authority (QSA), the

statutory body that developed curriculum in Queensland for all three educational

systems (government, Catholic and independent), had released a new Mathematics

syllabus. The Queensland Mathematics Years 1-10 Syllabus (QSA, 2004) which

provided “a framework for planning learning experiences and assessment opportunities

through which students demonstrate what they know and can do with what they know in

the Years 1-10 Mathematics key learning area” (p. 13). This syllabus consisted of five

content strands, broken up into eleven topics. The outcomes for computation involving

the four operations were contained in the Number strand in the topics of “Addition and

Subtraction” and “Multiplication and Division” and the core content listed computation

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methods as “mental computations, written recordings, calculators and computers” (p.

45). The related number sense components were in the “Number Concepts” topic.

The classroom teacher in this study had participated in a professional

development workshop on mental computation that I conducted in my role as education

advisor supporting the implementation of the new syllabus. After the session the

classroom teacher indicated a personal interest in learning more about mental

computation strategies and ideas for including them as a focus of the mathematics

program for her Year 3 class. I was interested in changing my ways of supporting

classroom teachers and recognised the interest of this teacher to change her knowledge

and practice as an opportunity to trial a different approach to professional development

and to document change in the teacher’s knowledge of computation strategies and her

pedagogical practices. I initiated this study as part of my role of providing professional

development to teachers, but with a specific intention of gathering data about the

changes and professional growth of myself and the classroom teacher. The study

utilised aspects of teacher change research, effective professional development and

models of learning and teaching to analyse and document the ongoing, classroom-based

approach to professional development that I facilitated as an education advisor during

this study.

1.3 PURPOSES

The overall aim of this study was to investigate a particular method of teacher

professional development through the implementation of classroom-based professional

development and to investigate the professional growth of the classroom teacher and

myself as a result of our work together. The research set out to document the

professional growth we experienced so as to inform the knowledge and practice of

education advisors who work to support practising classroom teachers.

The following research questions were used to pursue the overall aim of the

study.

1) What professional growth did the classroom teacher experience as a result

of the professional development conducted in her classroom?

2) What professional growth did I experience as an education advisor as a

result of conducting the professional development?

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3) What factors influenced the professional growth of the classroom teacher

and the education advisor?

1.4 CONTEXT

This study was conducted in a suburban Catholic school in Brisbane,

Queensland, Australia. The school was supportive of teacher professional development

and the school administration gave consent for the study to be conducted in only one of

the three Year 3 classrooms in the school as they believed the learnings of the

classroom teacher involved in the study could be shared with the other staff members

both during and after the year of the study. The classroom teacher had a total of 37

years experience, 25 of those years teaching Year 3. The class consisted of 27 students

approximately 8 years old, with a range of abilities in mathematics.

1.5 SIGNIFICANCE

While much has been written about the principles and models of teacher change

and effective teacher professional development, a newer field of research has begun to

conceptualise the role and multiple dimensions relating to the education of teachers.

Much of the literature about those educators who teach teachers focuses on those who

teach pre-service and post-graduate teacher education courses offered through

universities. There is less literature available that relates to the role of education

advisors who work with practising classroom teachers and even less literature in

relation to their professional growth or the improvement of their knowledge and

working practices.

This study provided an opportunity to investigate a particular method of

professional development and to study the professional growth of a classroom teacher

and an education advisor through the implementation of classroom-based professional

development. The professional growth of the classroom teacher and the education

advisor are documented, detailing specific examples of change and professional growth.

The results show the success of this particular method of professional development and

provide recommendations for others working as education advisors.

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1.6 THESIS OUTLINE

Chapter 1 outlines the stimulus and background for the study, the context in

which it was conducted, the purposes of the study and an overview of the thesis.

Chapter 2 reviews the literature on teacher change, professional growth and

effective teacher professional development and describes the research relating to the

role of education advisors. The literature review also outlines the mental computation

literature used to guide the development and implementation of the instructional

program.

Chapter 3 details the theoretical basis for the study as a case study. It also

outlines how the professional growth of the classroom teacher and the education advisor

were analysed and represented. This chapter also outlines the context for the study and

the participants.

Chapter 4 reports examples of change that led to the classroom teacher’s

professional growth. Each example provides some background, a vignette describing a

significant event which led to ongoing change and the classroom teacher’s professional

growth. Each vignette is discussed and the professional growth is represented using the

Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002;

Hollingsworth, 1999).

Chapter 5 describes my professional growth as an education advisor, outlining

examples of change in the same format as Chapter 4, using vignettes of significant

events and descriptions of the ongoing change leading to my professional growth. The

examples of the my professional growth are represented using a modified version of the


Hollingsworth, 1999)

Chapter 6 reports the conclusions of the study by firstly revisiting the aims of the

study. The research questions are answered with reference to the literature, and

implications are discussed particular to inform education advisor practice and further

research.

20


2. Literature Review

Education is a process by which accumulated knowledge, skills and values are

transmitted from one generation to another. The times in which we are currently living

have been described as an information or knowledge age. Any consideration of the

education of students should acknowledge that we are living in a society faced with

rapid social, economic, technological and cultural change impacting on the way we

prepare young people to be effective citizens. Students need education to equip them

with the skills required to learn, to transfer learning, and to live in this rapidly changing

world. “As societies have confronted the challenges brought about by globalisation and

new technologies, especially information technologies, the critical importance of

education has become obvious to all” (Fullan, Hill & Crevola, 2006, p.1). For teachers

to be able to best educate students in this changing world they need to maintain

currency of knowledge and practices. Education advisors, in their role as providers of

professional development for teachers, need to maintain currency of their knowledge

and practices to facilitate teacher change.

This chapter reviews the literature on models of teaching and learning and

models of teacher change, principles of effective professional development and

particularly teacher professional growth. It also outlines literature relating to the roles of

teacher educators and models of teacher educator change. These models are described

and compared so as to shape the model of teacher change and education advisor change

used in this study. This chapter also provides a review of the strategy-based mental

computation educational literature that formed the focus of the professional

development implemented by the education advisor.

2.1 TEACHER CHANGE

The world is changing and hence education is also changing, yet many teachers

and society in general expect stability and familiarity in education. Hoban (2002)

described what he saw as a paradox embedded within the teaching profession. He noted

that in this rapidly changing society, teachers are often reluctant to change their

practices for a range of social and contextual reasons. The reasons include a perceived

21


need to conform and teach as colleagues do and external pressures to cover curriculum

in short time frames not allowing time for new ideas. Duke (2004) also observed that

the influence of parents provided further factors impacting on change in education.

Parents want schools to prepare students for life in this ever-changing society, yet they

prefer the recognisable consistency of schools that operate in the same way and offer

the same types of experiences as they remembered when they were students.

The culture that exists in schools is a major influencing factor in teacher change.

This has long been recognised in the literature. For example, Lortie (1975) described

how some teachers are quite content to maintain the familiar and to continue with

traditional or conventional teaching methods rather than adopt new paradigms and

practices. This problem is still being identified more recently. Elmore (2006) reported

on the “power and resilience of the default culture of public schools” which he

described as “the deeply rooted beliefs, structures, artefacts and symbols of an

increasingly dysfunctional and obsolete set of institutions” (p. xi). Elmore continued to

state that:

the best ideas of reformers have, as yet, proven no match for the inertia of a powerful

resident culture. I am increasingly convinced that the work of reform is not about

‘changing’ the institutions and practices of schools but about deliberately displacing one

culture with another. (p. xii)

Recommended changes to mathematics education originate in educational

research. As educational research into content and improved pedagogies emerges,

schools and teachers are implored to change. For research to impact on student learning

it needs to be translated into teachers’ classroom practice. Research findings are shared

with the wider community through publication in books and journals, however as Staub

(2004) noted, research publications only have a “weak impact” on change at the school

level (p. 41), raising doubts as to whether such research is transferred into classroom

practice.

The task of implementing new reforms identified in research through the

provision of professional development is seldom completed by the researchers

themselves. Teachers need to be assisted, through provision of professional

development activities, to enact changes identified in the literature. The provision of

teacher professional development is most often undertaken by education advisors who

22


tend to be, as described by Resnick and Glennan (2002), educators seeking career

alternatives but not in administrative roles. The development of the knowledge and

practices of these education advisors to enable the transformation of factors identified in

educational research into reform in classrooms is therefore a factor influencing teacher

change.

Other pressures for schools and teachers to change include societal and

government expectations of change together with public scrutiny of the results of high-

stakes testing. Some people outside the field of education see the need for professional

development as an indication that teachers are doing an inadequate job. Howey and

Vaughan (1983) reported that the post-depression years saw a major emphasis on

teacher professional development. This need for professional development was

identified as implying a deficit in teacher knowledge and practice due to circumstances

of the time. Often the professional development conducted in response to these deficit

views of education was the offering of a single session to deliver the knowledge or

skills deemed to be missing from teacher repertoires. Researchers have provided

evidence of the limitations and ineffectiveness of this once-off style of professional

development (Fullan, Hill & Crevola, 2006; Guskey, 1986; Lovitt & Clarke, 1988;

Wood & Thompson, 1980). Other researchers (Fullan, Hill & Crevola, 2006; Fullan &

Stiegelbauer, 1991; Schön, 1983, 1987) referred to effective teacher professional

development as being a longer-term process of change to improve skills in the art of

teaching rather than a short term fix of a perceived deficit. Literature describing

alternative purposes of teacher professional development dates back many decades.

Jackson (1974) described a “professional growth approach” to teacher professional

development which did not focus on the perspective that teacher change was needed

due to deficits in knowledge and practice. He described teachers who sought greater

fulfilment through professional development as practitioners of the art of teaching

rather than those in need of repair of personal inadequacies.

The practices of a past era are unlikely to provide the knowledge and skills

needed in this newer era. Shulman (1986), in a classic work on teacher knowledge,

noted the change in the knowledge required for teachers from the late 1800s to more

recent times. In the late 1800s the knowledge expected of teachers was almost entirely

content focussed (e.g., written arithmetic, oral grammar and geography), whereas more

23


recently the focus of required teacher knowledge has shifted to the processes associated

with teaching, for example teaching methods, the recognition of individual differences

and the development of instructional plans. The focus on what teachers need to know

has changed from content alone to the processes of teaching as well as content

knowledge. The professional development of teachers needs to consider several aspects

of teacher knowledge.

2.1.1 TEACHER KNOWLEDGE

Descriptions of the types of knowledge required by teachers have been the focus

of writing in the educational literature particularly in the past twenty years. Elbaz

(1981) discussed teachers’ “practical knowledge” which she identified as what teachers

know that others who are not teachers do not, and how they consider this knowledge.

She described teachers having knowledge of self, mileu, subject matter, curriculum

development and instruction. She also stated that this knowledge is represented in

practice as rules, practical principles and images. Schön (1983) described teacher

knowledge as more than “knowing things” but also knowing how to identify and solve

professional problems and knowing how to construct knowledge in general, particularly

through reflection on practice.

Borko and Putnam (1995) stated that the central goal of teacher professional

development should be the “elaboration and expansion of a teacher’s knowledge base”

(p. 58). One important component of teachers’ ongoing learning and change is the

growth of their professional knowledge base. Borko and Putnam, in their description of

a teacher’s knowledge base, reviewed the cognitive psychology literature (e.g.,

Anderson, 1984) and stated that:

Teachers’ thinking is directly influenced by their knowledge. Their thinking,

in turn, determines their actions in the classroom. Thus, to understand

teaching we must study teachers’ knowledge systems… Similarly, to help

teachers change their practice, we must help them to expand and elaborate

their knowledge systems. (p. 37)

A paper often quoted as the definitive list of types of teacher knowledge was

published by Shulman (1986). In his paper Shulman proposed a model of the teacher

knowledge that included a blend of content knowledge and pedagogical knowledge. The

complete model outlined seven types of knowledge: knowledge of content, general

24


pedagogical knowledge, curriculum knowledge, pedagogical content knowledge,

knowledge of students, knowledge of educational contexts and knowledge of

educational ends, purposes and values. Shulman emphasised the importance of

pedagogical content knowledge and described it as a “missing paradigm” (p. 7) linking

knowledge of pedagogy with knowledge of content. Most often Shulman is quoted as

referring to three particular types of teacher knowledge: content knowledge, curriculum

knowledge and pedagogical content knowledge.

Based on the work of Shulman (1986) and colleagues (e.g., Shulman &

Grossman, 1988), Borko and Putnam (1995) proposed a model of the knowledge base

of teaching organised around three domains of knowledge that they argued were

particularly relevant to teachers’ instructional practices. The three domains were:

general pedagogical knowledge, subject-matter knowledge and pedagogical content

knowledge. Table 2.1 outlines these domains and their components as proposed by

Borko and Putnam (1995).

Table 2.1 Domains and components of the knowledge base of teaching (Borko & Putnam, 1995)

Domains Components

General pedagogical knowledge

Learning environments and instructional strategies Classroom management Knowledge of learners and learning

Subject matter knowledge

Knowledge of content and substantive structures Syntactic structures

Pedagogical content knowledge

Overarching conception of teaching a subject Knowledge of instructional strategies and representations Knowledge of students’ understandings and potential misunderstandings Knowledge of curriculum and curricular materials

Other researchers have described domains of the broader role of teaching as

more than just teaching practice. Clarke and Peter (1993) described four domains that

encompass the teacher’s world: the personal domain (teacher beliefs and attitudes); the

domain of practice (incorporating the knowledge types described by Shulman (1986)

and summarised by Borko and Putnam (1995)); the domain of consequence (outcomes

salient to the teacher’s practice including student learning outcomes, teacher control,

25


motivation, student development of new ideas); and the external domain (sources of

information, stimulus or support including professional development opportunities).

Clarke and Peter (1993) described a model of teacher professional growth that

recognised how educational change was possible in any of these domains and that when

change in one domain led to change in another, professional growth was possible. This

model of educational change is discussed in detail in Section 2.1.3.

2.1.2 MODELS OF TEACHING AND LEARNING

Steinbring (1998) noted that descriptions in the literature pertaining to subject

matter knowledge gave “no specific attention to the needs regarding the teaching and

learning of this subject matter knowledge” (p. 157) and that this knowledge, along with

pedagogical content knowledge, were often considered as two major components of

teacher professional knowledge. Steinbring saw the process of developing subject

matter knowledge as a key aspect of the teaching-learning process and stated that often

the process of developing these knowledges was seen as linear: mathematical

knowledge is transformed into school mathematical knowledge by the teacher and then

this knowledge is conveyed to the students (see Figure 2.1). This model highlighted the

role of the teacher in facilitating the transfer of the subject matter knowledge to the

student.

Figure 2.1 A linear model of the teaching-learning process (described by Steinbring, 1998)

Steinbring (1998) acknowledged that learning was not simply a process

influenced by the teacher but that the students also played an important role. He

considered that while teachers can offer learning environments and tasks for the

students they cannot directly steer the learning. This needs to be constructed by students

themselves. Teachers observe these learning activities and achievements and then offer

further learning accordingly. Steinbring saw these two systems, the students’ learning

processes and the interactive teaching process between the student and the teacher as

Scientific mathematical knowledge

School mathematical knowledge

Mathematical knowledge of the

student Teacher prepares knowledge for mathematics teaching

Teacher conveys mathematical knowledge to the students

26


autonomous but related. He proposed an alternative model that outlined interactions

between specific aspects of the professional knowledge of teachers (see Figure 2.2).

Figure 2.2 Teaching and learning mathematics as autonomous systems (Steinbring, 1998)

This model depicts effectively two independent systems. A consequence of the

relative independence of these systems was that the mathematical subject matter

knowledge in a school environment as offered by a teacher was seen as different from

the knowledge constructed by students themselves. Steinbring (1998) stated that this

difference, which he believed relied on the context and social setting in which the

students learned, placed demands on teachers to diagnose students’ constructions of

mathematical knowledge and to vary learning offerings accordingly. This observation

supported Shulman’s position that pedagogical content knowledge was a vital type of

professional knowledge for teachers.

Other models developed to describe teaching-learning processes also identified

multiple components. Some included multiple connections between the components

leading to non-linear models. Jaworski (1992, 1994) explained a model of what she

considered “the important elements of teaching” (p. 107). This model was developed

from a study which categorised student and teacher actions in a classroom. After many

observations of teachers and students interacting in mathematics classrooms, three main

elements emerged: provision of mathematical challenge for students; sensitivity to

students; and management of learning. These three elements were a synthesis of many

other categories identified during the study. This model highlighted that student

learning related to the management of the learning environment and sensitivity to

students within the learning environment. This model, describing the considerations of a

mathematics teacher, was referred to as the teaching triad (see Figure 2.3).

Scientific and curricular mathematical knowledge

School mathematical knowledge, concepts and problems

Subjective interpretation of mathematical knowledge

Teacher makes learning offers for the students

Teacher observes and varies the learning offers

Students try to work out and solve problems

Students reflect and generalise their solutions

27


Challenging content for students

Management of Student Learning

Sensitivity to students

Figure 2.3 Teaching triad (Jaworski 1992, 1994)

Jaworski’s (1992, 1994) teaching triad provides a model of teaching and

learning that is consistent with constructivist perspectives. The triad highlights the

importance of the inclusion of mathematical challenges as well as the management of

learning being sensitive to the students’ reasoning and needs. This model, in the same

way as Steinbring’s (1998) model, proposes that teachers are responsible for designing

learning tasks that provide students with challenges and opportunities to think and

construct knowledge. By observing the students and reflecting on their observations,

teachers construct understanding and develop practices which allow them to modify

learning opportunities in ways that are beneficial to their students. As a result it can be

surmised that the learning opportunities that teachers provide for their students are a

function of their own knowledge and practices.

2.1.3 MODELS OF TEACHER CHANGE

Early teacher change research, for example Lewin (1935), identified models of

educational change that focussed on changing teachers’ beliefs, attitudes and

perceptions. This perspective, based on psychotherapeutic models, held that once

teachers’ beliefs, attitudes and perceptions were changed, change would occur in their

practice and then influence student learning outcomes (see Figure 2.4).

Figure 2.4. Psychotherapeutic model of educational change (e.g., Lewin, 1935).

Professional development / learning opportunity

Change in teachers’ classroom practices

Change in student learning outcomes

Change in teacher beliefs and attitudes

28


This model implies a linear sequence of consequences. It also places the

improvement of student learning outcomes as the end or ultimate aim of professional

development activity. Clarke and Hollingsworth (2002) recognised this aim as a

“plausible and legitimate educational agenda” but stated that its placement as the aim of

teacher professional development was misleading (p. 949). Other researchers have

described how they believe that the assumptions underlying this model are misplaced

(e.g., Fullan, 1982; Guskey, 1986). These researchers argued that teacher change

models that assume such a linear sequence fail to consider the complexities of the

process of teacher change.

Later research on teacher change indicated that this linear model (Figure 2.4) did

not lead to the desired outcome of changing teacher practice to enhance student learning

outcomes. Guskey (1986) described how teachers change by taking practices that work

in their classrooms into their repertoire. Strategies or practices that do not work are

discarded. He argued that the key factor in whether a change in teacher practice is

maintained is the demonstration of learning success by the students. Built on this

understanding, Guskey proposed an alternative model of teacher change based on the

idea that teacher beliefs and practices only changed significantly after change in student

learning outcomes is evidenced, and that this typically resulted from changes teachers

had made in their classroom practices (see Figure 2.5).

Figure 2.5 An alternative model of educational change (Guskey, 1986)

Guskey acknowledged that although this model did not account for all the

variables which could be associated with the teacher change process, it was deliberately

simple. It provided an ordered framework to assist the understanding of the dynamics of

teacher change. He also stated that there were inherent interrelationships between the

components.

Clarke (1988) proposed a cyclic model of teacher change (see Figure 2.6) that

overcame a potential deficiency in the linear model proposed by Guskey (1986).

Professional development / learning opportunity

Change in teachers’ classroom practices

Change in student learning outcomes

Change in teacher beliefs and attitudes

29


Figure 2.6 Cyclic model of teacher professional development Clarke (1988)

Clarke’s model utilised the same components as Guskey’s (1986) model but

offered two additions in terms of modelling teacher change. The first addition in

Clarke’s (1998) model of teacher change was the ongoing, cyclic nature of this change.

This model saw teacher professional development as an ongoing process where

successive changes are built over time and where each stage is mediated by the

particular aspects of each teacher’s personal situation. Change in one domain is

translated to change in another domain through mediation. Change in classroom

practice as a result of staff development (indicated as 1. in Figure 2.6) is mediated

through teacher participation in and interpretation of the professional development

content. Change in student learning as a result of teacher classroom practice (indicated

as 2 in Figure 2.6) is mediated by student participation in and interpretation of the

activities. Change in teacher beliefs as a result of student learning (indicated as 3 in

Figure 2.6) is mediated by interpretation of the class experience where teacher

inferences link perceived gains to new practices. The final change represented in this

model is between teacher beliefs and staff development activities (indicated as 4 in

Figure 2.6) which are mediated by access to suitable professional development

activities.

The other additional element offered in Clarke’s (1988) model of teacher change

was the possibility of joining or exiting the cycle at any point. A teacher could

improvise a new activity as part of their own classroom practice and demonstrate the

positive effect on student learning outcomes, leading to a change in the teacher’s beliefs

Classroom practice

Staff development activity

Teacher Beliefs Student

Learning

1.

2.

3.

4.

30


that could be shared with other teachers, and the cycle continues. This demonstrates a

start of the change cycle initiated by a stimulus other than external input from some

form of teacher professional development. It would also be possible to exit the cycle at

any stage if obstacles to progress are encountered. For example if a new teaching

approach utilised after a professional development activity was not valued or did not

suit a particular school community, the teacher change could cease due to lack of

support or recognition.

A study of the evaluation of a professional development for secondary

mathematics teachers conducted by Clarke, Carlin and Peter (1992) led to the

refinement of Clarke’s (1988) model of teacher change. Analysis of the data from that

study using Clarke’s model highlighted the multiple possible pathways for teacher

change and challenged the linear or sequential models proposed previously. Other

developments from the study included a broadening of the conception of each of the

domains of teacher change. For example, organised teacher professional development

sessions were not the only form of external stimulus for change evidenced during the

study and student learning outcomes were not the only consequences of changes to

teacher practice. The authors noted that teacher change could occur in the absence of

external stimulus or support. This had been proposed by Clarke (1988) and evidence

was found to support this idea in the study by Clarke, Carlin and Peter (1992). A refined

model of teacher change was then described by Clarke and Peter (1993) as a “dynamic

model of teacher professional growth” (p. 170) (see Figure 2.7).

Figure 2.7 A dynamic model of professional growth (Clarke & Peter, 1993)

External source of information and stimulus or support

Teacher knowledge and beliefs

Valued outcomes

Classroom Experimentation

Enactive mediating processes Reflective mediating processes

Personal Domain

External Domain

Domain of Practice

Domain of Inference

31


Clarke and Peter’s (1993) model depicted two distinct categories of constructs,

the analytic domains and mediating processes. The analytic domains matched the

elements in earlier teacher change models (e.g., Guskey, 1985, 1986). The four domains

were: the Personal Domain, the Domain of Practice, the Domain of Inference and the

External Domain. These domains constituted the teacher’s professional world with the

External Domain (depicted by the square in the model) being the only domain outside

the teacher’s personal world (depicted by circles). The Personal Domain depicted

teacher knowledge and beliefs. The Domain of Practice included teacher

experimentation in their classroom. The Domain of Inference described the valued

professional outcomes that may include student learning outcomes, teacher satisfaction,

teacher effectiveness and efficiency, and increased student enjoyment. The External

Domain included the sources of information, stimulus or support accessed by a teacher.

These external sources could include professional developments, professional reading,

staff meetings or informal conversations with colleagues.

These four domains correspond closely with the four domains described in both

the early model of educational change (Lewin, 1935) and the model proposed by

Guskey (1986). They also relate to the types of teacher knowledge described o these

paradigms, and the connections.

Table 2.2 Alignment of teacher change models and teacher knowledges

Domains of growth (Clarke &

Hollingsworth, 2002)

Domains of change

(Guskey, 1986)

Teacher knowledges (Shulman, 1986)

Teacher knowledges (Borko & Putnam, 1995)

Personal domain

Knowledge and beliefs

Content knowledge Curriculum knowledge Pedagogical content knowledge

General pedagogical knowledge Subject matter knowledge Pedagogical content knowledge

Domain of practice

Classroom practice

General pedagogical knowledge Pedagogical content knowledge Contextual knowledge Knowledge of students

General pedagogical knowledge Subject matter knowledge Pedagogical content knowledge

Domain of consequence

Student learning outcomes

Knowledge of educational ends

External domain

Professional development/ learning opportunity

32


The Clarke and Peter (1993) model proposed that a change could be ‘located’ in

one of the four domains. A change in information provided by an external stimulus or

source could be located in the External Domain. A change in personal knowledge or

belief would be situated in the Personal Domain. A change in pedagogy or teaching

strategy would be situated in the domain of practice, while a change in learning

outcomes demonstrated by students would be located in the Domain of Inference. The

Domain of Inference implied that change in this domain was inferred or interpreted by

the teacher. Clarke and colleagues at the Teacher Professional Growth Consortium

(1994) changed the name of this domain to the Domain of Consequence to better

explain that changes were “of consequence” (p. 2) to the teacher seeking to explain

changes in knowledge, beliefs or practices. They described how increased student talk

could represent an observed outcome of a new teaching practice. However, this

outcome could be interpreted by one teacher as increased classroom noise, whereas

another teacher could interpret this same behaviour as increased student engagement. So

the Domain of Inference/Consequence was not simply an observable change but the

interpreted meaning or consequence of the observation.

The format of the Clarke and Peter (1993) model was deliberately non-linear and

inferred that change in one domain was able to lead to change in any of the other

domains and that there were a multitude of possible pathways for change to occur. The

change was mediated by either enactive or reflective processes. Enactment was used in

this model to describe more than just acting but rather to describe the translation of a

belief or pedagogical model into action most often resulting in change in the Domain of

Practice. The other mediating process was reflection where change in one domain is

reflected upon leading to change in another domain. This reflective consideration

resulted most often in change in the Personal Domain of a teacher’s knowledge, beliefs

and attitudes or change in learning outcomes, the Domain of Consequence. The

importance of reflection is described further in Section 2.2.3.

The Clarke and Peter (1993) model of teacher change was different from

previous models in its recognition and representation of the multiple pathways that can

lead to teacher change. While the linear pathway (external stimulus, change in teacher

practice, change in student learning outcomes and then change in teacher beliefs and

practices) as depicted by Guskey (1986) was still included, the many other pathways

33


and the potential ongoing nature of teacher change was able to be captured by this

model.

Teacher change generally results in change in teacher knowledge or teacher

practice. The Clarke and Peter model represents the way change in one domain can lead

to changes in other domains. These changes result in change to knowledge and beliefs

or classroom practice. When change in each of these domains is considered individually

the relevance of each of the mediating processes is more clearly evident. Figure 2.8

shows the mediating processes relating to change in the Domain of Practice.

Figure 2.8 Role of classroom experimentation in teacher change (Clarke & Peter, 1993)

Of the multiple possible change pathways it was noted that change in the

Domain of Practice occurred as a result of the mediating process of enactment of

stimulus from an external source or enactment of a change in personal knowledge or

beliefs (shown as red arrows in Figure 2.8) and that the only change caused by change

in the Domain of Practice was a change in the Domain of Consequence through

reflection on the valued student or personal outcomes of the classroom experimentation

(shown as a blue arrow in Figure 2.8). Clarke and Peter (1993) noted that classroom

experimentation in teacher change takes three forms: as informed mimicry of advocated

innovative practice arising from a teacher’s response to an external stimulus (arrow a in



Valued outcomes


Personal Domain

External Domain

Domain of Practice

Domain of Inference

Enactment

Reflection

a

c

b

34


Figure 2.8); as an enaction of personal knowledge and beliefs (arrow b in Figure 2.8); or

as a stimulus for reflection on outcomes of consequence (arrow c in Figure 2.8).

Similar analysis of the Personal Domain of teacher knowledge and beliefs

provides further perspectives on the complexities of teacher change. Figure 2.9 shows

that the pathways leading to change in teacher knowledge and beliefs (shown in red) are

through the mediating process of reflection on either stimulus or support from the

external domain (arrow a in Figure 2.9) or on the valued outcomes of a teaching

experiment (arrow b in Figure 2.9). A change in teacher knowledge and beliefs (shown

as blue arrows)may result from three actions: participation in or engagement with a

stimulus from an external source (arrow c in Figure 2.9); enactment of practice in the

form of classroom experimentation (arrow d in Figure 2.9); or reassessment of valued

student or personal outcomes through reflection (arrow e in Figure 2.9).

Figure 2.9 Role of teacher knowledge and beliefs in teacher change (Clarke & Peter, 1993)

Clarke and Peter (1993) noted that their model had “significant implications for

future teacher professional developments” (p. 174). They also recognised the

complexity of professional growth, noting possible growth pathways that could allow

the providers of professional development to anticipate and encourage all avenues to

professional growth.



Valued outcomes


Enactment

Reflection

Personal Domain

External Domain

Domain of Practice

Domain of Inference

a

b

c

d

e

35


A further refinement of the Clarke and Peter (1993) dynamic model of

professional growth was achieved by a consortium of education professionals (Teacher

Professional Growth Consortium, 1994) who described the modified model as the

Interconnected Model of Teacher Professional Growth. This model reflected the

multiple pathways of possible teacher change presented by Clarke and Peter (1993)

while including some additional possible enactive or reflective connections. These new

connections, shown in blue in Figure 2.10, recognised that reflection on change in the

Domain of Practice could lead to change in teacher knowledge, beliefs and attitudes and

that enactment of valued outcomes of learning could lead to change in classroom

practice.

Figure 2.10 Interconnected Model of Teacher Professional Growth (Teacher Professional Growth Consortium, 1994)

2.1.4 TEACHER PROFESSIONAL GROWTH

Teacher professional growth is related to but different from teacher change.

Clarke and Peter (1993) called their modification of the teacher change model presented

by Clarke (1988) a “model of professional growth” rather than as another model of

teacher change. The Teacher Professional Growth Consortium (1994) provided details

of how their model of professional growth (see Figure 2.10) depicted change in each of

the four domains of the teacher’s world and that there are multiple ways that change in

Knowledge and beliefs

Salient outcomes


External source of information and stimulus

Enactment

Reflection

Personal Domain

External Domain

Domain of Practice

Domain of Consequence

36


one domain can lead to change in another through the mediating processes of enactment

and reflection.

Hollingsworth (1999) described what she considered to be the difference

between change and professional growth in her study of the professional growth of

primary teachers involved in mathematics professional development. She used the

Interconnected Model of Teacher Professional Growth (Teacher Professional Growth

Consortium, 1994) to represent what she called change sequences and growth networks.

It was the differentiation between these two types of change that she used to define

teacher professional growth. Hollingsworth described change in one domain linked

through enactment or reflection to change in another domain as a change sequence.

Figure 2.11 depicts one of Hollingsworth’s examples of a change sequence that

occurred when a teacher was exposed to a new teaching strategy as part of the program

in her study. The stimulus represented change in the teacher’s External Domain (E).

The teacher decided to explore the use of this strategy in the classroom leading to

change in the Domain of Practice (P). The arrow in Figure 2.11 represents this change

sequence.

Figure 2.11 Example of a change sequence (Hollingsworth, 1999)

In this example the change in the External Domain (the introduction of the new

teaching strategy as part of the professional development) was translated through the

mediating process of enactment (represented by the solid arrow) into a change in the

Domain of Practice when the teacher experimented with the strategy in the classroom.

Hollingsworth noted that it was also possible to make a change in one domain that did

not lead to change in others.

S

P K

E

Enactment

Reflection

37


Hollingsworth (1999) stated that change sequences, like the example above

(Figure 2.11), did not represent what she considered to be professional growth. In her

study the term growth was reserved for change that was enduring or lasting. She

referred to change sequences where the change occurred over time as growth networks

and identified these as teacher professional growth. An example quoted in her thesis

describes how a teacher had been exposed to the new teaching strategy (change in the

External Domain) and then experimented in classrooms (change in the Domain of

Practice). The change sequence is depicted in Figure 2.12. The teacher may have

reflected on the consequences of the new teaching strategy and noted that the outcomes

of the experiment included improved student learning and greater student engagement,

as well as increases in her own satisfaction with regard to teaching (Domain of

Consequence). This change in salient outcomes led to a change in the teacher’s belief

about the teaching strategy (Personal Domain), and consequently the strategy was

included in the teacher’s regular classroom practice (change in the Domain of Practice).

This example, depicted in Figure 2.12, constituted what Hollingsworth described as a

growth network and what she considered to be teacher professional growth.

Figure 2.12 Example of a growth network (Hollingsworth, 1999)

In this example, the teacher enacts change after an external stimulus (arrow 1),

then reflects on the consequences in terms of observed salient student-learning

outcomes (arrow 2). The teacher then evaluates the change in relation to its significance

to their own personal knowledge and beliefs (arrow 3), leading to further modifications

to practice (arrow 4). Hollingsworth analysed the professional growth of teachers in her

study by identifying change networks and representing them using the Interconnected

Model of Teacher Professional Growth (Teacher Professional Growth Consortium,

1994).

S

P K

E

1.

2.Enactment

Reflection

3.

4.

38


In the conclusion of her study, Hollingsworth (1999) stated how she had initially

set out to study the change in a group of teachers participating in a particular

professional development program and that she had recognised that the term change

was “insufficiently precise” (p. 318). Hollingsworth described four benefits of making

clear a distinction between teacher change and growth. Firstly, she noted that many of

the teachers in the study changed aspects of their ideas or practices and many tried

similar strategies. However their growth over time was “highly individualistic” (p. 319).

Secondly, the distinction between change and growth was useful for examining

different processes involved in changing teachers’ knowledge, beliefs, attitudes and

practices, in particular the change sequences and growth networks for representing

change and growth. Thirdly, the distinction was useful for describing different aspects

of teachers’ professional activity. Finally, the distinction between change and growth

“was useful for considering the complex roles of, and relationships between the

individual teacher, the professional development program and the school context, with

respect to teacher growth” (p. 319).

In her study, Hollingsworth (1999) identified three types of professional growth

that she referred to as: adoption, where there was lasting change in teacher practice

and/or beliefs that matched the ideas presented in the program of professional

development; misrepresentation, where lasting change in teacher practice and/or beliefs

developed through a misinterpretation of the ideas from the professional development

program; and rejection, where the lasting change in teacher beliefs was directly related

to the rejection of the ideas presented in the professional development program. In her

study Hollingsworth only considered the examples associated with the adoption type of

professional growth to be professional growth as she had defined it for the study.

2.1.5 MODELLING TEACHER PROFESSIONAL GROWTH

Hollingsworth used the Interconnected Model of Teacher Professional Growth

(Teacher Professional Growth Consortium, 1994) “to interpret and represent the

empirical data collected … and consequently to facilitate understanding of the process

of growth experienced by the teachers involved in the study” (p. 127). She concluded

that the capacity of the model to represent the many different forms of change was one

of its distinguishing features and using it in her study had presented opportunities to

39


consider the effectiveness of the model. As one of the recommendations of her study

she proposed some modification of the model.

One of the changes to the model proposed by Hollingsworth (1999) was to

change the focus of the Domain of Practice from classroom experimentation to

professional experimentation. This change allowed for extension of the model into other

professional settings in addition to classrooms. It also recognised that teachers engage

in professional learning outside their classrooms by processes such as professional

collaboration and collegiality. By making this change, other professions could use the

model to assist with considerations of professional development foci such as

organisational strategies, administrative skills or other professional activities, and make

connections to professional growth.

A second change to the model was identified as a result of Hollingsworth (1999)

noticing that most of the teachers who participated in the professional development in

her study showed change in attitudes toward the teaching of mathematics. Of particular

note were increased enthusiasm for and satisfaction with teaching the subject.

Hollingsworth proposed the addition of attitudes to the description of the Personal

Domain, referring to teacher knowledge, beliefs and attitudes.

A third change to the Interconnected Model of Teacher Professional Growth

was the acknowledgement of the importance of the school context on teacher

professional growth. Hollingsworth (1999) recognised that any processes of

professional growth represented by the model occur “within the constraints and

affordances of the enveloping change environment” (p. 329). She stated that access to,

participation in, experimentation with and application of professional development

influenced individual teacher growth. Her suggestion was to enclose the model within a

border representing the change environment. The change environment is discussed in

further detail in relation to teacher professional development in Section 2.2.4.

The fourth change suggested by Hollingsworth (1999) was to the title of the

model. She retitled the model as the Interconnected Model of Professional Growth

(rather than teacher professional growth) to reflect changes made to the model and to

distinguish the changed model from the previous versions. Figure 2.13 shows the model

with modifications made by Hollingsworth (1999). The Interconnected Model of

Professional Growth was described in detail by Clarke and Hollingsworth (2002).

40


Figure 2.13 The Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999)

Clarke and Hollingsworth (2002) described teacher development as a learning

process that could be related to learning theory. They argued that teacher professional

growth concepts linked to both cognitive and situative learning theories (Cobb &

Bowers, 1999). Cognitive learning theories focus on the development of teacher

knowledge. One of the general outcomes of teacher professional development is the

facilitation of change in the various types of teacher knowledge (see Section 2.1.1). In

terms of teacher professional growth, change in a teacher’s knowledge would occur in

the Personal Domain of the Interconnected Model of Professional Growth. Situated

learning theories recognise the inevitable link between teaching and the professional

context in which it occurs. Situated theories of learning relate to the change in teacher

practices. In terms of teacher professional growth this change links to change in the

Domain of Practice in the Interconnected Model of Professional Growth.

The Interconnected Model of Professional Growth provides an effective way to

model and represent professional growth as growth networks where change in one

domain of the teacher’s world can be shown to lead to change in other domains. Other

researchers have used this model to analyse and represent professional growth of

teachers and of professional developers.

Knowledge beliefs and attitude

Salient outcomes

Professional Experimentation


Enactment

Reflection

Personal Domain

External Domain

Domain of Practice


The change environment

41


Justi and Van Driel (2006) used the Interconnected Model of Professional

Growth to represent change observed in science teachers’ content and pedagogical

content knowledge about the role of models in science education. The teachers in the

study participated in a program of professional development in The Netherlands and

data were collected from a variety of sources including interviews, questionnaires and

discussions about research projects conducted by the teachers after the professional

development. The authors used the Interconnected Model of Professional Growth to

analyse the change identified in the data collected throughout the study. Justi and Van

Driel described the connections indicating change in one domain leading to change in

another as relationships between the domains. They identified and numbered all nine

possible relationships between pairs of domains in the model and described each using

an example specific to the use of models in science education. They described the nine

relationships as “types of change” (p. 444).

The study by Justi and Van Driel was conducted over a short period of time and

so the researchers did not feel they could define a growth network as change over time.

Instead, they counted the number of relationships between domains where a relationship

was where change in one domain resulted in change in another. They identified whether

the aspects of teacher knowledge that they were studying had changed or not for each

teacher and identified whether the change was a change sequence (one or two

relationships between domains) or growth network (more than two relationships). These

researchers did not look at the progression of change between the domains but at each

change in one domain leading to change in another domain as a different type of

change.

Diezmann, Fox, deVries, Siemon and Norris (2007) used the Interconnected

Model of Professional Growth to analyse the learning of a team of five “professional

developers” (p. 94) who worked together to support a group of teachers participating in

a mathematics project to improve the learning of Years 1 to 3 students. The professional

developers in their study were working with practising classroom teachers in workshops

but were also visiting the schools, similar to my role as education advisor in this study.

Diezmann, Fox, deVries, Siemon and Norrie (2007) analysed the learnings of the

professional development team members as change in one domain of the Interconnected

42


Model of Professional Growth or as change that occurred through enactment or

reflection between pairs of domains.

2.2 EFFECTIVE TEACHER PROFESSIONAL DEVELOPMENT

Professional development is a term used to describe systematic efforts to bring

about change in the beliefs and attitudes of teachers, in their classroom practices and in

the learning outcomes of students (Guskey, 2002). In post depression times, teacher

professional development was often enacted in response to perceived deficits in teacher

knowledge or practice. The goal of such professional development was to provide a fix

for the perceived deficit. Jackson (1974) described teachers who sought greater

fulfilment in the art of teaching through professional development. Guskey (2002)

stated that the ultimate goal of professional development was the improvement of

student learning outcomes.

Teacher professional development needs to be a component of all plans for

educational improvement. Guskey and Huberman (1995) noted that the current and

ongoing emphasis on professional development comes from the “growing recognition

of education as a dynamic, professional field” (p. 1). Although it is true that teachers are

often required to participate in professional development, Guskey (1986) argued that

“most teachers engage in staff development because they want to become better

teachers” (p. 6). Teachers want to have a positive influence on the students they teach

and prepare them as best they can for life beyond school. Teachers realise that

professional development can assist them to improve and do their jobs well.

The format or style of teacher professional development needs to model what

constitutes effective teaching. There are alternate views of teaching and so there are

alternate views of teacher professional development. Hoban (2002) stated that “if

teaching is conceived to be a labour to be mastered, then it is logical to assume that

teacher learning involves attending workshops to gain additional knowledge and skills

to increase mastery” (p. 11). This view of teacher professional development was

described by Day (1999) as “additive”, because it implied the accumulation of strategies

being added to an existing repertoire. Argyris and Schön (1974) described this form of

professional development as “single-loop learning” (p. 19) where constancy is

maintained by designing actions that satisfy the immediate challenges. As a new

43


challenge emerges, a solution or suggested action is added to the repertoire to manage

that challenge and maintain constancy. This linear, action–consequence approach to

teacher professional development assumes that change occurs as a natural outcome of

having received new information or curriculum materials.

An alternate perspective of teacher professional development arises from the

belief that teaching is a complex practice with dynamic differences in the context of

each school and each classroom. Hoban (2002) stated that “teaching is more than the

delivery of prescribed knowledge using a repertoire of strategies, ..[it] is a dynamic

relationship that changes with different students and contexts” (p. 165). Fullan, Hill and

Crevola (2006) characterised what occurs in classrooms to be more than a teacher

teaching. The classroom involves “interaction between students, teachers and resources

in specific but constantly changing contexts” (p. 29). The complexity of teaching

requires that teachers themselves be life-long learners and the professional development

of teachers needs to be supported by frameworks to promote learning over time,

including reflection and rethinking. Argyris and Schön (1974) described this type of

learning as “double-loop learning” where instead of changing actions to meet

immediate variables, a change is devised to alter the bigger picture and lead to

consequential change of a whole system of theories and practices. Professional

development that aims for this type of learning makes a greater, longer-term change to

teachers than the single-loop or additive learning.

Teacher change requires the status-quo that exists in schools to be questioned.

Many teachers who teach in schools today teach as they were taught rather than

confronting the challenges of societal change, and ongoing educational change. Richert

(1997) stated that teacher education does little to challenge the systems of schools as

they are. He argued that the reward system of schools (salaries, promotion) do not tend

to stimulate teachers to examine the purpose of their work in the first place, or to

explore alternative conceptions of what is and what might be to accomplish different

ends. For better or for worse, schools are persistently stable places. Meier (1995) noted

that “the thing that is wrong with prescriptive teaching is not that it doesn’t work... it’s

that it does” (p. 604). The professional development of teachers needs to do more than

address immediate concerns. To be effective it needs to contribute to the longer-term

professional growth of teachers.

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2.2.1 PRINCIPLES OF EFFECTIVE TEACHER PROFESSIONAL DEVELOPMENT

There has been much written about effective mathematics teacher professional

development by mathematics education researchers. By highlighting practices and

principles behind successful and unsuccessful professional development, the research

provides a guide to the provision of successful teacher professional learning. Many

researchers in the past decades have presented lists of principles of effective teacher

professional development. Owen, Johnson, Clarke, Lovitt and Morony (1988) outlined

nine principles for effective professional development based on a review of literature

and their analysis of the “best” professional development practices occurring across

Australia. Clarke (1994) developed ten key principles for making professional

development more effective. Johnson (1993, 1996) also provided a list of features of

effective teacher professional development with a particular focus on characteristics of

the teacher. Mewborn (2003) completed a more recent review of research on effective

professional development for teachers of mathematics and identified three key themes.

These key themes were: the need for the provision of sufficient time; professional

development in school-based contexts; and the inclusion of discussion with supportive

colleagues. These three themes mirrored principles that had been included in other lists

of principles of effective teacher professional development.

Guskey (2003) also reviewed the literature on effective teacher professional

development. He compared 13 lists of principles of effective teacher professional

development and found 21 different attributes. He noted that there seemed to be varying

determinations of effectiveness and while many of the principles appeared in a majority

of the lists there were many principles that were mentioned in only a few. The five most

frequently occurring principles were: enhancing teachers’ content and pedagogical

knowledge; providing sufficient time and resources; promoting collegial and

collaborative exchange; establishing procedures for evaluating the professional

development experience; and conducting school-based professional development.

Guskey described how many of the statements appeared to be “yes-but” statements

where the principle was important but there was more to it than it being just a checklist

type statement. An example quoted was “yes, professional development should provide

sufficient time and resources, but such time and resources must be used wisely,

focussing on activities that positively affect learning and learners” (p. 750).

45


Another group of researchers compared a range professional development

projects. Nisbet, Warren and Cooper (2003) observed that success in these projects was

associated with addressing teacher ownership of the professional development,

continuity of the professional development over time, pertinence to classroom practice,

and the provision of opportunities for personal reflection, as well as discussion with and

support from an experienced mentor.

Another factor that needs to be considered in the preparation of professional

development activities for teachers is the context in which they work. Guskey (1995)

stated that because of the powerful and dynamic influence of context, it is impossible to

make precise statements about the elements of effective professional development.

Instead he offered procedural guidelines that he considered were critical to the

professional development process. These guidelines were: to realise that change is an

individual as well as organisational process; to approach change in small steps but have

an eye to the big picture; to work in teams providing regular opportunities for sharing

and include procedures to enable feedback; to provide follow-up and support; and to

integrate programs and show how they fit into the bigger focus of a developing

professional knowledge base. Those who provide teacher professional development

need to consider the principles of and guidelines for effective development and then

consider how adhering to such principles will impact on their work with the teachers.

2.2.2 METHODS OF TEACHER PROFESSIONAL DEVELOPMENT

The professional development of teachers takes many forms. Most often,

professional development sessions or workshops are run by education advisors and are

held in schools or in central locations. These sessions may provide opportunities to

share information with large groups of teachers, but research has questioned the impact

and effectiveness of this approach in regard to change at the classroom level (e.g.,

Fullan, 1995, Wilson & Berne, 1999). Other approaches which target smaller groups of

teachers or even individuals, while more personalised and often more successful, are

time consuming and resource expensive.

Guskey (2000) summarised the work of others including Sparks and Loucks-

Housley (1989) to provide a comprehensive list of different methods of teacher

professional development, namely training, observation, curriculum development, study

46


groups, inquiry/action research, individually-guided activities and mentoring. The

characteristics, benefits and limitations of each method are summarised below.

Training

Training methods of teacher professional development are usually large-scale

and generally offered centrally as workshops or seminars. This method of professional

development is cost-effective as large numbers of educators can attend at the same time.

Joyce and Showers (1995) stated that this method is useful for the appropriate and

consistent delivery of new strategies by what they referred to as using executive control.

Guskey (1996) stated that this method could lead to shared knowledge and common

vocabulary and that it could be used to dispel rumours that can emerge when only a

select few are able to access professional development opportunities. Disadvantages of

this method of teacher professional development include the lack of opportunity for

choice and individualisation of the program and goals of the sessions. This also means

that this method is not appropriate for varied levels of skill or expertise in those

attending.

Observation

Professional development that focuses on the learner’s observation of others or

on being observed themselves is another method. The learning is derived from the

discussion and analysis of what was seen. Peer coaching and clinical supervision also

utilise aspects of this method. Showers and Joyce (1996) noted that one benefit of this

model is that learning occurs for both the observer and the observee. Ackland (1991)

also detailed benefits of this approach in terms of it breaking down the isolation of

teaching through the sharing of goals and the utilisation of opportunities to work with

others. Learning in this method of teacher professional development can occur through

the provision of an alternative perspective on teaching, particularly when the

observation sessions are planned in advance with the focus outlined for discussion in

follow-up sessions. A version of this professional development method referred to as

Lesson Study was used in schools in Japan. Groups of teachers planned a lesson

together, observed and took notes while one teacher presented the lesson and then met

to discuss the lesson that may then be modified and taught by another teacher in the

group while their colleagues observed (Fernandez, 2002)

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Curriculum development

Some teachers are able to be involved in the development or review of

curriculum, the planning of strategies to improve instruction or the solving of some

particular problem relating to education. Activities such as these provide rich

opportunities for professional development. Curriculum development usually involves

the acquisition of new knowledge through research, discussion and observation of trials.

Opportunities provided by this method of professional development also allow for

awareness of perspectives that practising teachers do not normally meet. For example,

the perspective of school administrators might be considered by classroom teachers

working on such projects. The benefits arising from this type of professional

development include increases in specific knowledge as well as collaboration and

shared decision-making skills. The disadvantages include the small proportion of

teachers able to access such opportunities. Guskey and Peterson (1996) noted that there

is potential for some educators involved in curriculum development to possess dominant

viewpoints that require expertise and confidence to challenge. Practising teachers may

not feel they can make such challenges.

Study groups

Study groups involve a project for a number of people, often an entire school

staff, and are carried out over a long period of time, for example over a full school year.

An issue or particular goal is identified and small groups of educators focus on different

aspects of the project to plan, trial and report back on considerations and developments.

Murphy (1992, 1997) reported that this method of professional development helped to

facilitate the implementation of curricular or instructional innovations and to

collaboratively plan school improvements. The potential to reinforce the whole school

as a learning community can be encouraged through the use of this method of teacher

professional development. The project needs to be well planned and each group needs

to clearly understand their role in the overall plan. This approach to professional

development takes time and as reported by Guskey (2000), there is a tendency for group

discussions to be opinion-based rather than research-based and for some individuals to

dominate while others remain uninvolved.

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Inquiry/action research

The inquiry or action research method of teacher professional development

allows teachers to investigate options to solve problems or to answer role-related

questions. The action research cycle has been well documented as a method for research

about teaching (e.g., Kemmis & McTaggert, 2000). When teachers use the method to

investigate ways to improve their practice or to try new initiatives, the process can

provide professional development. As noted by O’Hanlon (1996), this method can be

used by individuals, small groups or whole school staffs. By participating in action

research, teachers become reflective practitioners and more systematic and thoughtful

decision makers (Sparks & Simmons, 1989). Loucks-Horsley, Harding, Arbuckle,

Murray, Dubea and Williams (1987) also stated that action research as a method of

professional development can narrow the gap between educational research and

educational practice. This method of professional development is another which

requires a significant amount of time to follow the iterations of the cycle and also

requires initiative on the part of those involved to devise the focus question and to plan

and maintain the cycle.

Individually guided activities

In this method of teacher professional development, individual educators

determine their own goals and select activities that they will complete to fulfil these

goals. This method is based on the assumption that the best person to determine the

professional development needs of a teacher is the teacher. Guskey (2000) noted that a

further assumption is that the teacher will be capable of the self-direction and have the

motivation needed to complete the activities. The activities can also be part of a

structured program where individuals complete their own tasks after exposure in a

larger group. The major advantage of this method of professional development is the

flexibility and individualisation that is possible. Teachers involved can be self-reflective

and participate in self-analysis through strategies including writing personal histories or

journals, reviewing audio or video recordings, or participating in cognitive coaching,

case study or role play (Langer & Colton, 1994). Sparks and Loucks-Horsley (1989)

noted that this method is quite isolating and can result in teachers identifying the same

learning as others before them, although the inclusion of collaborative sharing can assist

with this.

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Mentoring

Pairing an experienced and successful educator with a less-experienced

colleague to work toward agreed goals can provide professional development for all

involved. Vygotsky (1978) discussed what he called “more knowledgeable others” who

often participated in the role of mentor, assisting other teachers to change their beliefs

and practices. Guskey (2000) noted that the mentor needs to have credibility among

their colleagues and be recognised for their ability to initiate curriculum and school

change as well as be highly competent and respected. Activities completed with a

mentor can include regular discussion, sharing of ideas and strategies, reflection and

observations. This approach to professional development is highly individualised and

can foster long-term productive professional relationships (Drago-Sevenson, 1997 cited

in Guskey, 2000). However, this method is time-consuming with the participants

needing to set aside times to work and reflect together. Guskey warned that unless

mentoring was supplemented by other forms of teacher professional development, it can

limit opportunities for broader collaboration and collegial sharing.

Co-teaching

There are a number of ways in which the teaching of one group of students is

shared by more than one teacher. Sharing the teaching provides an opportunity for

professional development in similar ways to observation described above. Shared

teaching is more likely to be a more frequent occurrence than observation. Many

schools combine two classes with two teachers who together teach the whole group.

Tobin and Roth (2006) made a clear distinction between team teaching and what

they termed co-teaching. These authors described team teaching as a division of labour

where teachers share tasks, often according to their personal competencies or

inclinations. With team teaching, although there may be two teachers in a classroom,

they tend to work individually. Co-teaching, in comparison, was depicted by Tobin and

Roth as analogous to co-piloting. There are always two pilots flying a plane and they

work together. They not only fly together, they also learn from each other as they fly by

developing “shared understanding of the situation in which they are together” (p. 3).

The varied methods of teacher professional development each have advantages

and disadvantages. Methods of professional development need to be selected to best suit

the purpose and the context in which they are conducted. Guskey (1995) argued against

50


the idea that there is a best method of teacher professional development, asserting that

issues of context are crucial in determining which approaches to take.

2.2.3 THE IMPORTANCE OF REFLECTION

Reflection is identified as an important component of effective teacher

professional development. The concept of reflection and its relationship to teacher

education has long been discussed in the education literature. Dewey (1910) described

reflection as teachers taking “active, persistent and careful consideration” (p. 6). He

stated that the use of reflection helped to explain how humans expand their thinking and

therefore learn by continually noticing that particular consequences followed particular

activities. Dewey proposed that learning occurred through reflection on these

consequences. Such activity-consequence relationships allowed new concepts to be

abstracted from unanticipated, surprising consequences of known activities. Dewey

stated that teaching requires the teacher to identify the conceptual state of the child,

present a task to stimulate experiences and to then direct reflective thinking about the

activity and consequences of the activity.

Piaget (1970) also described how he believed humans formed logical thoughts.

He stated that thought was actively created rather than absorbed from the world around

us, and observed that “to know is to transform reality in order to understand how a

certain state is brought about” (p. 15). Piaget argued that learning required reflection on

relationships between actions and expected goals. Once these relationships had been

established, a learner was able to anticipate actions based on the possible outcomes of

future events. Piaget, similarly to Dewey, saw the role of the teacher as firstly, to

observe the actions of students so as to identify their mental schemes, and then to

engage them in inquiry and reflection to transform these schemes into new abstractions

or learning. He also suggested that the processes of inquiry and reflection were valuable

for those who educate teachers to enable them to be active in their support of the

development of teachers’ professional knowledge and practice.

Schön (1983) also proposed that reflection was crucial to the development of

professional knowledge. By applying a process of reflecting and acting recursively,

where the result of a reflection can be reflected upon, deeper meaning and learning can

result. Schön stated that learning specialised types of knowledge and practices, such as

those related to the profession of teaching, could be improved by participation in the

51


activities of the profession and through reflection on these activities. He also noted that

reflection could be guided by experts in the field to shape and direct the learning.

LaBoskey (1993) observed that reflective thinking is “a constant and careful

reconsideration of a teacher’s beliefs and actions in 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” (p. 9).

Educational literature in relation to teacher professional development has also

identified reflection as a valued activity when working with teachers (Clarke, 1994;

Nisbet, Warren & Cooper, 2003). The Interconnected Model of Teacher Professional

Growth (Clarke & Hollingsworth, 2002) and earlier versions of this model (Clarke,

1988; Clarke & Peter, 1993) emphasised the importance of reflection and enactment as

enablers or mediators of professional growth and teacher change. Reflection is a

significant aspect of this model. Figure 2.14 shows the model with just the reflective

connections between domains indicated.

Figure 2.14 Reflection as a mediating process in the Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002, Hollingsworth, 1999)

In this model the only pathway leading to change in the Personal Domain of

teacher knowledge, beliefs and attitude is through the mediating process of reflection.

Teachers can reflect on external information or stimuli which could lead to change in


Salient outcomes



Reflection

Personal Domain

External Domain

Domain of Practice



52


their knowledge, beliefs or attitudes (arrow a). Similarly, teachers could reflect on

experimentation with practices in their classrooms or their participation in other

professional activities (arrow b). Teachers could also reflect on the consequences of

observed changes in student or personal outcomes (arrow c). An example of reflection

mediating change in a classroom could involve a teacher who trialled a new teaching

strategy in her classroom (change in the Domain of Practice). By reflecting on the new

strategy the teacher could change her own knowledge, beliefs or attitudes about the

strategy and teaching in general based on this stimulus (change in Personal Domain). In

this example it is the mediating process of reflection that sees a change in the Domain

of Practice result in change in the Personal Domain.

The Interconnected Model of Teacher Professional Growth (Clarke &

Hollingsworth, 2002; Hollingsworth, 1999) also shows that consequences of

professional experimentation (changes in the Domain of Practice) are also only

mediated through reflection. The two domains that can be changed as a consequence of

reflection on changes to the Domain of Practice are the Personal Domain (arrow b) and

the Domain of Consequence (arrow c). An example of this change could be that after

classroom experimentation teaching mental computation strategies rather than written

algorithms (change in the Domain of Practice), a teacher could reflect on the practice,

noting changes to their knowledge, beliefs and attitudes about the teaching of

computation (change in the Personal Domain). Another consequence of experimentation

with teaching mental computation strategies could be the teacher observing and

reflecting on change in the students’ behaviour when they discuss and compare different

strategies used (change in the Domain of Consequence).

The other change mediated by reflection in the Interconnected Model of

Professional Growth model occurs when change in a teacher’s knowledge, beliefs or

attitudes (Personal Domain) is reflected on in relation to the observed and interpreted

changes to the Domain of Consequence (arrow d). For example, a teacher who has

changed her belief in the value of mental computation as a focus of computational

instruction could reflect on how they value students discussing their use of strategies

after seeing the practice in their classroom. These examples represent change

sequences, where change in one domain leads to change in another domain. Reflection

can also be the mediating process in growth networks where change in one domain

53


leads to lasting change after a sequence of changes in other domains. An example of a

growth network could occur when a teacher participates in professional development

and identifies new information about teaching mental computation strategies (change in

the External Domain). After enacting some of these strategies in the classroom (change

in the Domain of Practice) the teacher reflects on the learning behaviours of the students

and interprets this as a change (Domain of Consequence). Then through further

reflection on these observations she changes her beliefs about the teaching of

computation (change in the Personal Domain). This growth network is depicted in

Figure 2.15)

Figure 2.15 A growth network where change is mediated particularly through reflection

To support the professional growth of teachers and to expect change in

classroom practice to lead to change in either teacher knowledge and beliefs or salient

outcomes, it is clear that the inclusion of reflection as part of the process is needed to

enhance the potential of the change. Many researchers have described the benefits of

reflection as a necessary component of effective teacher professional development.

Glazer, Abbott and Harris (2004) commented on the use of reflection in

professional development. They stated that in some professional development activities

reflection is sometimes completely absent. These authors also noted how tightly

structured-reflection can restrict participants from reflecting in an open-ended manner,


Salient outcomes



Enactment

Reflection

Personal Domain

External Domain

Domain of Practice


1

2 3


54


resulting in less ownership of the professional development. DeBard and Guidera

(1999) described the benefits of reflection through discussion. They suggested that face-

to-face reflection can be a little threatening and sometimes fast-paced reflection

sessions including exchanges of ideas can end with participants not having an

opportunity to reflect in any depth. These authors suggested that the use of computer-

mediated communication can encourage more in-depth reflection and more meaningful

discussion than possible with face-to-face reflection. The asynchronous nature of

computer-mediated communication allows participants to reflect and provide comments

that have truly been reflected on over a longer period of time.

2.2.4 THE CHANGE ENVIRONMENT

It is recognised in the literature that the effectiveness of teacher professional

development programs is influenced by contextual factors. Clarke and Hollingsworth

(2002) and Hollingsworth (1999) included what they described as the change

environment as part of the Interconnected Model of Professional Growth.

Hollingsworth (1999), in her study of teacher professional growth, described the school

contexts in which teachers worked as a particular aspect of the change environment that

influenced their response to professional development. She concluded that the school

context had a substantial impact on teachers’ professional growth. She explained how

the school context influenced each stage of what she described as the professional

growth process. This growth process had four stages: access to professional

development, participation in professional development, experimentation with ideas

from professional development and the application of professional development ideas

(see Figure 2.16)

Figure 2.16 Influences of the change environment on teacher professional growth

Factors involved with access to professional development included the provision

of information about professional development opportunities and individual or whole

school professional development plans. Participation in professional development was

Change Environment (School Context)

Access Participation Experimentation Application

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influenced by the provision of financial assistance for attendance and release from

classroom duties to participate. Participation in professional development involves

additional responsibilities in the form of communicating ideas to and mentoring other

staff, leading to greater professional recognition. At the experimentation stage,

Hollingsworth (1999) identified factors that influenced whether professional

development resulted in change or professional growth. The factors included the

feasibility of the ideas from the professional development for their school context,

acceptance from colleagues of ideas being trialled, and the process of experimentation.

Finally, the application of ideas from professional development can be constrained in

some school contexts by factors including school policies relating to educational

philosophy and classroom management, funding, resource availability and the physical

characteristics of the school setting.

Particular examples of teachers who were encouraged and discouraged to

implement changes from professional development activities were provided by

Hollingsworth (1999). One teacher who worked in an environment that was conducive

to change described how he was supported by school staff, the professional

development tutors, the resources and equipment available in the school, the

mathematics ethos of the school and the professional development culture evident in the

school. For other teachers in the study the absence of particular conditions impeded

their professional growth. Factors included a lack of co-ordination in leadership with

respect to mathematics, little collegial activity and no obvious commitment to

mathematics professional development. Even when teachers were themselves

enthusiastic toward the professional development and desired change, they found it

difficult when the school context was not supportive.

Clarke and Hollingsworth (2002) described how aspects of the change

environment could either afford or constrain change in any of the domains represented

in the Interconnected Model of Professional Growth. Aspects of the change

environment that they mentioned specifically included opportunities to participate in

professional development, school subscriptions to professional journals, administrative

encouragement of teachers to experiment with innovative teaching strategies, the

encouragement of collegial discussion and the structural provision of opportunities to

share and reflect on each other’s practice.

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The Interconnected Model of Professional Growth (Clarke & Hollingsworth,

2002; Hollingsworth, 1999) places the domains of the teacher’s world within the change

environment represented by a box surrounding the model to identify how “change in

every domain and the effect of every mediating process are facilitated or retarded by the

affordances and constraints of the workplace context and each teacher (or other

professional)” (p. 965). The change environment includes contextual factors that have

potential to influence professional growth.

2.3 EDUCATION ADVISOR CHANGE

Recent calls for reform in education, supported by research and debated in

professional and political forums, need to be translated into change in the classroom.

Zaslavsky and Leikin (2004) stated that calls for reform in mathematics education make

an assumption that there are educators available in roles to support classroom teachers

to make the recommended changes. Education advisors who support practising teachers

do exist, however their availability, levels of expertise and experience vary

considerably.

There are generally two types of education advisors. The first, who have been

recognised extensively in the research literature (e.g., Goos, 2009; Tatto, Lerman, &

Novotna, 2009), are educators who work to support the development of pre-service

teachers, generally in university settings. The others are those who work in schools or

as employees of an educational system whose role it is to support practising teachers.

Ball and Even (2009) argued that working with practising teachers was different from

work associated with the education of prospective mathematics teachers. They noted

that the settings are quite different and usually the education of pre-service teachers

follows a formal program over time, whereas the education of practising teachers is less

structured and is conducted in a variety of settings. In addition, professional

development for practising teachers usually has no formal diploma or necessity to

demonstrate results of the learning at the conclusion of sessions or courses. Some

educational systems and registration bodies require a particular number of professional

development hours per year to maintain currency and there are many ways for these

hours to be completed legitimately, however, there usually is no assessment of learning.

Certificates are issued indicating the number of hours of attendance only.

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Education advisors who conduct teacher professional development with

practising teachers have varied experience and preparation for their role. Some

education advisors who support teachers in schools are full-time classroom teachers

who lead professional development sessions or provide support for their colleagues as a

part of their role as a school staff member as well as completing their teaching roles.

Others work part-time as classroom teachers and part-time in a curriculum leadership

role in schools, planning and supporting curriculum initiatives. Another variation of this

role involves education advisors who are employed by education systems to provide

support to implement particular reforms and to offer general ongoing support. Often

these teacher educators work supporting teachers from many schools. Another group of

education advisors work as independent consultants who are hired by schools to support

the local professional development of classroom teachers, often on a once-only basis or

short series of workshops.

Teachers who work to provide professional development for teachers have wide

and varying experiences as education advisors. Pope and Mewborn (2009) noted that

people who become education advisors often have been successful teachers who are

looking for a change. Pope and Mewborn also noted that teachers who became

education advisors had often been involved with the supervision and mentoring of pre-

service teachers during school placements. Whatever their experience, education

advisors are a valuable component of the process of implementing educational reforms

in schools. Education advisors are teachers, however the role has more complexities

than the role of teacher of students. Staub, West, and DiPrima (2003) stated that those

who support teachers need to be “excellent teachers” in terms of discipline knowledge

and teaching focus (p. 1). Classroom teachers need to know about how students learn as

well as and about how to teach.

A classroom teacher’s knowledge and practices focus on improving student

learning outcomes. An education advisor needs a broader consideration of teaching and

learning. Loughran (1997b) described his transition from classroom teacher to pre-

service teacher educator. He found that the knowledge of content and pedagogy he had

acquired as an experienced classroom teacher were not sufficient for the task of

teaching about teaching. By working with pre-service teachers he identified that to have

professional knowledge of teaching about teaching he needed understanding of: student

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teachers’ needs and concerns in their transition from student to teacher; appropriate

ways and time for challenging the student teachers’ beliefs about teaching and learning;

a range of school teaching situations (content, year level); and approaches and practices

in supervision, pedagogy and teaching about teaching. Tzur (2001) also described his

development from being a learner of mathematics, to a teacher of mathematics, to a

teacher of teachers of mathematics and finally to being a mentor of those who teach

teachers of mathematics, highlighting the change of focus needed for each role. While

education advisors are teachers, and the role is effectively one of teaching teachers, it is

more complex than the original teaching role they were trained for. Education advisors

require support and opportunities to improve their knowledge and practices to best fulfil

their roles, just as classroom teachers require support to improve their knowledge and

practices.

Teaching requires more than content knowledge (see Section 2.1.1), the role of

education advisor also requires more than knowledge of how to teach. To develop the

knowledge and practices of teacher educators, details of the education advisor’s role

need to be identified to enable the description and analysis of models of education

advisor change. So just as the components or domains of the teacher’s world needed to

be identified to analyse teacher professional growth, the components or domains of the

teacher educators’ world need to be identified to analyse the professional growth of

education advisors.

2.3.1 MODELLING EDUCATION ADVISOR CHANGE

In Section 2.1, ways of modelling classroom teacher change were discussed, in

particular the Interconnected Model of Professional Growth (Clarke & Hollingsworth,

2002; Hollingsworth, 1999). This model depicts how change can occur in any one of

four domains of the teacher’s world and it offers a method of representing the many

possible pathways for change leading to teacher professional growth. The model was

developed from models focussed specifically on teachers (Clarke & Peter, 1993) to be

applicable to other professionals (Clarke & Hollingsworth, 2002; Hollingsworth, 1999).

The model has also been used by researchers to describe the learning of other

educational professionals. Diezmann, Fox, DeVries, Siemon and Norris (2007) used this

model to describe the learning of education professionals they referred to as

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professional developers. These professional developers were providers of professional

development in roles that align to the role of education advisor in this study.

Zaslavsky and Leikin (2004) offered models that could be used to describe the

change or learning of professionals who teach teachers. These models were based on

models of teaching and learning that originally focussed on classroom teacher change.

Zaslavsky and Leikin modified the models to relate to those professionals who support

the change of these teachers. Jaworski (1992, 1994) and Steinbring (1998) provided

models of teaching and learning depicting the connections between subject matter and

pedagogical knowledges as necessary aspects of teachers’ professional knowledge and

practices. These models were described in Section 2.1.1 of this thesis. To examine the

growth of those educators in roles supporting the professional development of teachers,

Zaslavsky and Leikin (2004) modified both Jaworski’s and Steinbring’s models to

identify the knowledge and practice and domains of the world of teacher educators so as

to model their change. In the study reported in this thesis, teacher educators are

described as education advisors. The modified models therefore apply to the modelling

of education advisor change.

Zaslavsky and Leikin (2004) broadened the teaching triad model (see Figure 2.3)

proposed by Jaworksi (1992, 1994) for classroom teachers, to describe and analyse the

underlying processes of the professional development of mathematics education

advisors. In Jaworski’s original model, mathematics was identified as providing the

challenging content for students in classrooms. The modified version of this model

identified the complete teaching triad as providing the challenging content for those

who teach mathematics teachers (see Figure 2.17). Hence, this modified version of

Jaworski’s teaching triad considers three domains of teacher educators (education

advisors) as: challenging content for the mathematics teachers, sensitivity to

mathematics teachers and management of mathematics teachers’ learning.

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Challenging tasks

Management of learning

Sensitivity to learners

Figure 2.17 Modified teaching triad of teacher educators (Zaslavsky & Leikin, 2004)

Zaslavsky (2009) further modified the teaching triad to make it generalisable

and able to describe considerations and practices of a wider range of educators by

substituting the term “learners” for “students” and “challenging tasks” for

“mathematical challenge”. This version of the triad could then be used to describe the

role of classroom teachers, education advisors and those who educate education

advisors (see Figure 2.18). The model still highlighted the need for tasks presented to

learners to be challenging and that management of learning relates to having sensitivity

and knowledge of the learners.

Figure 2.18 Further modification of Jaworski’s (2009) teaching triad (Zaslavsky, 2009)

Zaslavsky and Leikin (2004) also adapted Steinbring’s teaching-learning process

model (see Figure 2.2) to offer ways in which teachers of mathematics, mathematics

education advisors and those who supported the education advisors could learn from

practice. In this model Zaslavsky and Leikin referred to the different roles as

Challenging content for mathematics teachers

Management of mathematics teachers’ learning

Sensitivity to mathematics teachers

Challenging content for students

Management of Student Learning

Sensitivity to students

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mathematics teachers (MTs), mathematics teacher educators (MTEs) and mathematics

teacher educators’ educators (MTEEs). Their three-layered model includes three

interrelated teacher-learner configurations, each of which includes two autonomous

systems, one describing the action teachers engage in and the other the actions of the

learner. Figure 2.19 shows this model diagrammatically.

Figure 2.19 Zaslavsky & Leikin (2004)’s extension of Steinbring (1998)’s model of mathematics educator change

This model maintains the original model as presented by Steinbring (1998) in

the centre and adds a layer for each of the other educators, namely the education

advisors (MTEs in this model) and those who support the education advisors (MTEEs

in this model). The model recognises that each of these roles are effectively a teaching

role and that the interaction needs to be between the teacher and the student through the

teacher making learning offers and the student reflecting on and generalising solutions

for the learning to occur. In each layer the roles are slightly different. In the central

layer of the model the teacher is the classroom teacher and the learner is the student. In

the next layer the teacher is the education advisor (MTE in this model) and the learner

is the mathematics teacher (MT in this model). In the third layer the teacher is the

educator supporting the education advisor (MTEE in this model) and the learner is the

education advisor (MTE).

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Zaslavsky (2009) further generalised Steinbring’s (1998) model, as she did with

that of Jaworski (1992, 1994), to make it applicable to a broader group of mathematics

educators, beyond classroom teachers. By changing the term “students” to “learners”

and “teachers” to “facilitators”, the model was able to be generalised and depicted as a

single layer. This model was then able to be applied to the learning from practice for

mathematics teachers, mathematics education advisors and teachers who supported

education advisors (See Figure 2.20). Zaslavsky (2009) stated that both of the new

broader models (Figure 2.18 and 2.20):

can be applied to various learning settings, differing in the specific facilitator-

learner identities (i.e., mathematics teacher-students, mathematics teacher

educator-mathematics teacher, mathematics teacher educator educator-

mathematics teacher educator) and in the nature and content of the learning. (p.

107)

In this model it was the use of tasks by the teacher/facilitators and the reflection

on and generalisation of solutions which led to learning.

Figure 2.20 Generalisation of Steinbring’s (1998) model of teaching and learning mathematics (Zaslavsky, 2009)

The analysis and modification of teacher change models by Zaslavsky and

Leikin (2004) and Zaslavsky (2009) to generalise models of teacher educator change

provided an opportunity to describe and analyse the development of the knowledge and

practices of teacher educators. The layering of these models of teaching and learning

provide recognition that education advisors and those who support the development of

education advisors are all teachers and that the development of the knowledge and

Facilitator constructs knowledge

Learners construct knowledge

Learners engage in tasks

Facilitator makes learning offers

Facilitator observes and varies the learning offers

Learners work on tasks and try to solve them

Learners reflect on and generalise solutions

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practices of education advisors can be investigated using teaching-learning models just

as it can for classroom teachers.

2.3.2 PROFESSIONAL DEVELOPMENT OF EDUCATION ADVISORS

There is much literature that has outlined the development of classroom teachers

and pre-service teachers. Literature on the development of education advisors or teacher

educators is much less common. Ball and Even (2009) noted that “one issue on which

the international community could focus and build a collective capacity is the

identification and development of teacher developers” (p. 256). Other than the attempts

by Zaslavsky and Leikin (2004) and Zaslavsky (2009) to provide models of the learning

of those who educate teachers, little literature exists on the provision of support for the

learning and ongoing development of educators in role of education advisors.

Even (2008) described what she saw as the challenges of educating mathematics

teacher educators who work with practising teachers. She stated one of the key

problems underlying this challenge was that there was very little literature on the

education of teacher educators. She stated that because practitioners in this role were

not identified as a distinct professional group, there had been little research on what

these teacher educators have to know and be able to do to support teachers’ learning,

and how the necessary knowledge and skills are developed. Another problem making

the education of teacher educators a challenge is the lack of a term to describe those in

the role. Ball and Even (2009) stated that “no single word or phrase exists to describe

the professionals who work with teachers” (p. 257). Some of the descriptions of these

people include teacher educators, education advisors, supervisors, professional

development providers, professional developers, teacher developers, facilitators,

teacher-leaders, lead teachers, teachers of teachers and in some specific cases, mentors

and coaches. The education of educators to work with practising mathematics teachers

is a challenge that needs to be addressed to work toward improved student opportunities

to learn mathematics. In the study reported in this thesis the term education advisor is

used to identify the teacher working with and supporting the classroom teacher. In the

context of this study this term was understood by the participants and while it has been

pointed out that the role of advisor is still a teaching role, the term teacher would be

used exclusively to refer to the classroom teacher.

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The range of support for teacher educators to develop their knowledge and skills

varies from no preparation to specific courses usually related to particular reform

agendas. In Queensland, Australia, in 1995 the implementation of the Shaping the

Future report (Wiltshire, 1994) and the Mathematics – A Curriculum Profile for

Australian Schools (Curriculum Corporation, 1994) saw the employment of

approximately 110 education advisors who participated in a one-week initial course

before working with teachers in their assigned geographical districts. Another example

of planned professional learning for teacher educators, described by Ball and Even

(2009), was a special course in Israel for providers of professional development. This

course was conducted over a number of years. Other teacher educators take on roles

supporting teachers and develop their skills with no specific support. Tzur (2001)

proposed a framework to conceptualise the “complex terrain … that is termed

mathematics teacher educator development” (p. 259). His framework consisted of four

stages of development: learning mathematics, learning to teach mathematics, learning to

teach teachers of mathematics and learning to mentor teacher educators (see Figure

2.21).

Figure 2.21 Four foci model of teacher education (Tzur, 2001)

In his conception of the framework, Tzur (2001) considered his many roles as an

educator and realised how his movement through the different roles had relied on

reflection on the activities of previous roles. For example, success as a teacher of

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mathematics required reflection on the learning of mathematics done by students.

Success as a teacher educator required reflection on the teaching of mathematics as well

as on the learning of mathematics.

Tzur (2001) described how his model utilised recursive reflection, where

reflection-in-action (Schön, 1983) can be turned into an object of the reflective process,

and then this new object can be reflected on, and so the cycle continues. The other

models used to describe the roles of teacher educator (Zaslavsky, 2009; Zaslavsky &

Leikin, 2004) also highlighted the reflective interactions between aspects of the role. In

a similar way of thinking about the progression of learning, Tzur (2001) used reflection

as the instigator of advancement to higher levels of the model. While the model was not

linear, progression to higher levels was facilitated by reflection (represented as arrows)

on activity-effect relationships at the lower levels (see Figure 2.21).

In Tzur’s (2001) model each role (learner of mathematics, teacher of

mathematics, etc) has a specific focus which the other roles within the model do not

have. Tzur described how links between the different roles in the model are made

through reflection. In Figure 2.21 the arrow marked as 1 indicates the reflection by

mathematics teachers on what it means to learn mathematics. The mathematics teacher

educators (education advisors) reflect on learning about teaching mathematics learning

(arrow 2) and learning mathematics (arrow 3). This extra layer of reflection

differentiates the role of mathematics teacher educator from that of mathematics teacher

and student. It also highlights the interconnection between knowledges required for

these varying roles. Finally, in this model the mentor of mathematics teacher educators

reflects on what it means to teach teachers (arrow 4), which also includes reflection on

the roles teaching mathematics (arrow 5) and learning mathematics (arrow 6). Tzur

(2001) made the point that a key to understanding the model was to recognise that:

development from a lower to a higher level is not a simple extension, that is,

doing more of and better of the same thing. On the contrary, development

entails a conceptual leap that results from making one’s and other’s activities

and ways of thinking at a lower level the explicit focus of reflection. (p. 275)

This model focuses on learning and situates the learning of mathematics as the

beginning point. As mentioned above, the model is not intended to be hierarchical but it

does depict a developmental sequence from learning mathematics, to learning to teach

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mathematics, to learning to teach teachers, to learning to mentor or support those who

teach the teachers. The model shows how those in each of the roles are learners. Each

role has progressively more to learn about in each of the separate but interrelated roles.

The roles in Tzur’s (2001) model relate closely to those in Zaslavsky and Leikin’s

(2004) extension of Steinbring’s model (Figure 2.19). “Learn Math” in Tzur’s model

equates with the role of a student or learner in the classroom. “Learn Math Teaching”

represents the mathematics teachers, which can be compared to Zaslavsky and Leikin’s

(2004) MT, who needs to learn about mathematics teaching as they develop and

improve their own role. Those in the mathematics teacher role will also be reflecting on

student learning of mathematics as it relates to their role. The next layer of the model,

“Learn Educating Teachers”, relates to mathematics teacher educators or education

advisors (MTE) whose learning focus is on how to best teach mathematics teachers.

Evidence of success in this role will be found through reflection on the teaching of

those they have taught and this should influence student learning. The final layer,

“Learn Mentoring Educators”, describes the learning of mentors of mathematics teacher

educators or mathematics teacher educator educators (MTEE). In this role the learning

is about all the other inner layers but in particular about what it means to educate an

educator.

2.4 ROLES OF AN EDUCATION ADVISOR

The complexity of the work of those who teach teachers is highlighted in the

range of models devised to try to capture the dimensions of these professionals. It is

clear from the models presented above, that being an effective, even excellent teacher is

not enough to ensure that someone is a good education advisor. Consideration of the

multiple roles of an education advisor as a mathematics learner, as a teacher, as a

teacher of teachers and as a mentor to others like themselves, needs to be made when

the aim is to support their professional growth.

All mathematics education advisors would have started their relationship with

school mathematics as students themselves. Richert (1997) stated that “teachers learn

what schools and teaching are first as students, and perpetuate through deed and action

these conceptions in their role as teachers” (p. 74). The concept of life-long learning

(QSA, 2004) recognises that learning does not cease at the end of schooling. If

education advisors are to change their knowledge and practices they will need to learn

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from their experiences. Tzur’s (2001) model (see Figure 2.21) identified how reflection

on being a learner of mathematics can assist with success in the role of mathematics

teacher and roles that support teachers of mathematics.

Classroom teachers learn about teaching through interactions with their students

and with peers in the school context, as well as through involvement in professional

development activities. Education advisors learn through their interaction with teachers

in a variety of professional development contexts, through engagement with their peers

and through professional development activities including professional reading. The

classroom teacher’s learning is focussed on the improvement of student learning

outcomes in the classroom. Education advisors learning is focussed on improving the

knowledge and practices of teachers they work with. The education advisor must also

understand the learning of students to enable them to advise the teachers who will

transfer their learning to the classroom and ultimately to the students.

Education advisors have almost certainly been classroom teachers themselves.

Staub, West and DiPrima (2003) argued that those who teach teachers need to be

excellent teachers to lead content and practical change in teachers. Generally education

advisors do not teach in classrooms where the focus is on the school students. Instead

their students are the teachers who teach the school students. So one of the roles of an

education advisor is clearly that of teacher but the focus is different to that of the

teachers they teach. Education advisors are teachers, therefore models of teaching and

learning like the teaching triad proposed by Jaworski (1992, 1994) are also applicable

to education advisors. Zaslavsky’s (2009) modification of the teaching triad (see Figure

2.18) highlights how the knowledge and practice required of education advisors

includes engaging classroom teachers in challenging tasks that stimulate their thinking

about teaching, as well as exhibiting sensitivity to the teachers as learners and managing

the learning of the teachers.

Learning about teaching is equally demanding. Loughran (1997a) identified that

for student teachers to learn about teaching they need to operate at two levels, as do

their teachers. One level concerns the need to learn about learning through the

experiences they are offered. The other level concerns the simultaneous learning about

teaching. These important elements of teacher knowledge become the focus of learning

where teachers are the recipients of instruction. Teaching about teaching is also no easy

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task, and the professional growth of education advisors needs to acknowledge their

multiple roles.

2.5 MENTAL COMPUTATION

The professional development provided for the classroom teacher in this study

focussed on the teaching of mathematics and specifically the use of mental computation

strategies rather than traditional written algorithms as the basis of a Year 3 computation

instructional program. This aspect of mathematics was chosen due to its significance in

the then new Mathematics syllabus.

Traditionally, computation forms a major component of school mathematics.

The dominance of computation and in particular the use of traditional written

algorithms in school mathematics programs has led many to believe that success in

school mathematics equates to an ability to complete computational algorithms

(Langrall, Mooney, Nisbet & Jones, 2008; Mingus & Grassl, 1998). Mental

computation as it relates to the development of the basic facts and the use of strategies

for computation beyond basic facts is an area of mathematics education that has gained

prominence in research literature and curriculum documents, both internationally and in

Australia, in the past twenty years.

Concerns regarding the perceived overdependence on procedural understanding

compared to conceptual understanding in mathematics (Hiebert & Lefevre, 1986) led to

questioning of the school focus on written methods of calculation. The benefits of a

focus on mental computation strategies were outlined in the literature, including: a

deeper understanding of the structure and properties of numbers (Plunkett, 1979; Reys,

1984; Thompson, 1999); the development of problem solving and thinking skills

(Callingham, 2005; Plunkett, 1979); and better alignment of school mathematics to the

mathematics used beyond the classroom (Australian Education Council, 1991;

Callingham & Watson, 2008; Northcote & McIntosh, 1999).

In Australia, A National Statement on Mathematics for Australian Schools

(Australian Education Council (AEC), 1991) stated that:

People need to carry out straightforward calculations mentally, and students

should regard mental arithmetic as a first resort in many situations where a

calculation is needed. (AEC, 1991, p. 109)

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Mental computation has been defined in basically two ways by researchers. One

view taken in the literature focuses on how the computation was completed. The

completion of a computation in the head with no recording is considered to be mental

computation and computation done on paper is written computation (McIntosh, 2005;

Reys, 1992; Sowder, 1988; Trafton, 1978). The other view of mental computation

considers the thinking processes involved. The researchers taking this approach to

describing mental computation did not concentrate on whether there was any recoding

done and acknowledged that the use of pencil and paper, including informal jottings and

student invented recordings could be a legitimate aspect of mental computation

(Anghileri, 1999; Beishuizen, 1997; Stacey, Varughese and Marston, 2003). In this

study mental computation was considered based on two major components that are

reported in the following section.

2.5.1 COMPONENTS OF MENTAL COMPUTATION

As mental computation involves more than the application of a remembered

procedure (Callingham, 2005), it is more complex than the straightforward application

of the series of steps of a traditional written algorithm. To plan a successful

computation program, the varied components of mental computation need to be

considered. Many researchers have outlined what they believe are the main components

of mental computation. Thompson (1999) formulated a model for mental computation

that comprised four components: facts, skills, understandings and attitudes. He argued

that success with mental computation required the possession of all four components.

Sowder and Wheeler (1989) proposed a similar classification that Morgan (1999)

reworked when he proposed his framework for mental computation which included:

affective components; conceptual components (number sense); related concepts and

skills; and strategies for computing mentally (pp. 64-65).

To enable the preparation of a mental computation instructional program during

this study, the components of mental computation were considered to be number sense

and mental computation strategies. The following sections describe each component in

further detail.

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Number sense

The term number sense has been defined and used by many researchers seeking

to describe a more conceptual focus for mathematics. The National Council for

Teachers of Mathematics (1989, p. 39-40) stated that children with good number sense:

“have well understood number meanings; have developed multiple relationships among

numbers; recognise the relative magnitude of numbers; know the relative effects of

operating on numbers; and have developed referents for measures of common objects

and situations within their environment”. Sowder (1992) and Resnick (1989) extended

this list and made a direct link to mental computation by including, in their definition of

number sense, an ability to: perform mental computations with non-standard strategies

and take advantage of the ability to compose and decompose numbers, use numbers

flexibly to estimate numerical answers to computations and to realise when an estimate

is appropriate, and judge the reasonableness of solutions obtained, dependent on their

belief that mathematics makes sense and that they are capable of finding sense in a

numerical situation.

Other authors have also stated direct links between number sense and success

with mental computation. McIntosh, Reys, Reys, Bana and Farrell (1997) described

number sense as having a “general understanding of number and operations along with

an ability and inclination to use this understanding in flexible ways” (p. 3). Callingham

(2005) highlighted research in mental computation that characterised it as “identifying

and describing students’ strategies for addressing particular kinds of calculations, often

within a framework of number sense” (p. 193). Students with sound number sense can

analyse computation situations using this sense and make informed decisions when

choosing how to complete the problems. Markovits (1989) noted that students with

number sense tend to analyse the whole problem first rather than immediately applying

a standard algorithm.

Mental computation strategies

The other component of mental computation is the strategies students use to

complete computations mentally, with the head as opposed to mental recall of

memorised facts (Beishuizen, 1997). Threlfall (1998) described computation strategies

as “approaches used by children when calculating involving a flexible and inferential

use of number knowledge” (p. 71). In a later paper Threlfall, (2000) stated that mental

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computation strategies required students to “construct a sequence of transformations of

a number problem to arrive at a solution as opposed to just knowing, simply counting or

making a mental representation of a paper and pencil method” (p. 30). It is this

description of a mental computation strategy as a sequence of transformations that has

been adopted for this study.

Research on students’ use of strategies for basic facts has been reported in the

mathematics education literature (e.g., Ashfield, 1989; Carpenter & Moser, 1984). Basic

facts are generally considered to be the addition, subtraction, multiplication and division

of numbers from zero to nine or zero to ten. Other research has identified strategies used

for computation with larger numbers by students, and competent adult calculators (e.g.,

Beishuizen, 1985; Cooper, Heirdsfield & Irons, 1996). For this study, the program was

planned to include computation strategies that would be applicable to basic facts as well

as to computations beyond the basic facts. For mental computation to form the basis of

an instructional program, some organisational structure was required to facilitate the

program’s depth and breadth.

2.5.2 CATEGORISATION OF MENTAL COMPUTATION STRATEGIES

A mental computation instructional program needs to consist of a

comprehensive range of strategies applicable to the year level of the students being

taught. A review of the literature identified many studies which had focussed on the

identification and categorisation of mental computation strategies (Beishuizen, 1985;

Cooper, Heirdsfield, & Irons, 1996; Fuson, Wearne, Hiebert, Murray, Human, Olivier,

Carpenter, & Fennema 1997; Klein, Beishuizen, & Treffers, 1998; McIntosh, 2005;

Morgan, 1999; Reys, Reys, Nohda, & Emori, 1995; Thompson, 1999). The study

reported in this thesis utilised a comprehensive framework compiled by the researcher

from a review of many published categorisations for both basic fact strategies and

strategies for computations beyond basic facts (Hartnett, 2007).

The Strategy Categorisation Framework (Hartnett, 2007) comprised five main

strategy categories: Count On and/or Back; Adjust and Compensate; Break Up

Numbers; Double and/or Halve and Use Place Value. Twenty-one subcategories were

identified providing details of variations in the possible use of each category. Together

the categories and subcategories of the Strategy Categorisation Framework provided a

comprehensive list of mental computation strategies that could form the basis of a

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primary school mental computation program. The organisation of mental computation

strategies in the Strategy Categorisation Framework is shown in Appendix 1. A series of

codes were developed for quick reference to each strategy (see Appendix 2). The use of

the Strategy Categorisation Framework for the development of the mental computation

instructional program is described in Section 3.6.1.

2.5.3 TEACHING MENTAL COMPUTATION

The sections presented below briefly detail considerations for the teaching of

mental computation. Reference to the literature is described in relation to the

instructional focus of the teaching and whether this involves the students inventing

strategies or being directly taught the strategies, the resources that have been found

useful to support the teaching of mental computation, and the sequencing of a mental

computation program.

Instructional focus

A question in relation to the teaching of mental computation is whether teachers

should directly teach strategies for the completion of mental computation problems or

whether the pedagogical focus should be on the students inventing the strategies

themselves. In the literature reports about the approaches used to teach mental

computation vary. Some studies involved students inventing their own mental

computation strategies (e.g., Blöte, Klein, & Beishuizen, 2000; Buzeika, 1999; Hedrén,

1999; Kamii, Lewis & Livingston, 1993; Kamii & Dominick, 1998). Other researchers,

including Ames and Ames (1989), stated that instruction based on students inventing

their own strategies was more suited to average or better students. They argued that less

capable students would benefit from more structured instruction in which the teacher

helped them to construct their strategies to solve problems. This view was supported by

Foxman and Beishuizen (1999) and Murphy (2004) who reported that when there was

an emphasis on students spontaneously inventing their own mental strategies it was the

higher attaining students who employed a range of strategies whereas below average

students often relied on inefficient counting procedures or taught algorithms.

An alternative approach to teaching mental computation involves students being

directly instructed about particular mental computation strategies. Threlfall (1998)

stated that he believed “whole heartedly” (p. 72) that children should be taught to be

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more efficient and effective in mental computation. However, he cautioned about the

structured teaching of strategies which involved showing and practising each approach

on a suitable example. Threlfall’s concern was that the students would be trained to

respond in particular ways to carefully selected examples. This position was supported

by Murphy (2004) who set out to identify the extent of use of a particular strategy,

compensation (Adjust and Compensate), in student work after a period of direct

instruction that focussed specifically on that strategy. Overall findings showed that all

students made some use of the taught strategy but that their success related to their

number sense knowledge rather than replication of the strategy. Murphy noted that

strategies can be introduced to students through whole class instruction but the use of

the strategies was reliant on the student’s personal knowledge. Sowder (1992) supported

direct instruction of mental computation strategies by stating that ‘‘there is evidence

that instruction on mental computation can lead to both increased understanding of

number and flexibility in working with numbers” (p. 14).

Another consideration in regard to whether strategies should be invented by

students or taught directly concerns whether students would invent the more efficient or

more complex strategies linked to a greater depth of number sense as noted by Sowder

(1992). In the United Kingdom, a report by the School Curriculum and Assessment

Authority (1997) recommended that the development of mental computation strategies

“should not be left to chance” (p. 29). The National Numeracy Strategy (Department for

Education and Employment, 1999) stated that some mental strategies may develop

intuitively but that others would need to be taught explicitly. Murphy (2004) agreed and

stated that there was a possible danger that if more complex, deductive mental

calculation strategies were not taught, some children may be denied access to them.

Wigley (1996) commented that mental arithmetic needs to be taught using approaches

which offer more than one method, as using only one method was considered too rigid.

He also argued that leaving pupils to find their own methods could deprive many of the

more advanced strategies.

A comprehensive approach to teaching mental computation can draw from both

of these instructional focuses. The benefits of students constructing their own

knowledge are well documented and accepted. Concerns about the repertoire of

strategies invented not including some efficient strategies are also valid.

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Resources to support learning

Currently the most commonly used representative model for supporting the

teaching and learning of computation in Queensland primary schools is Dienes multi-

base arithmetic blocks (MAB). These are used for all four operations to model

decomposition methods and are also used to represent and teach base ten place value

concepts. Miura and Okamato (2003) questioned the use of these blocks as a model for

supporting the development of number concepts. For students to successfully develop

flexible strategies for mental computation they need multiple representations of number,

that is, constructions of numbers other than tens and ones are needed. Multiple

representations of number have been described as a component of sound number sense

(see Section 2.5.1).

If students are to use a variety of mental computation strategies they require a

well-developed number sense (Callingham, 2005) which includes knowledge and

application of a range of representations of number. Researchers have stated how the

use of structured base-ten blocks were not appropriate for some of the strategies

considered to be more efficient where both numbers are often not partitioned into

standard place value parts (Cooper, Heirdsfield & Irons, 1996; Klein, Beishuizen &

Treffers, 1998; Resnick, 1982). For a comprehensive mental computation instructional

program, resources to support the learning and use of efficient strategies would need to

offer other representations of numbers than simply tens and ones representation.

Number boards are another resource discussed in the literature that have been

used to support students’ understanding of number concepts and to support computation

(Beishuizen, 1993; Bobis & Bobis, 2005). Standard number boards are generally in the

form of a ten by ten grid where the numbers 1-100 or 0-99 are represented in rows.

While this model is also related to a base of ten, it has more flexibility and allows for

further conceptual understanding than the MAB blocks. This resource allows students to

model addition and subtraction of tens as vertical jumps and addition or subtraction of

ones as horizontal jumps using the 10x10 structure of the board, a benefit identified by

Bobis & Bobis (2005) who described how the use of a hundred board assisted students

to develop number sense, for example, counting in tens on the decade (10, 20, 30..), and

off the decade (1, 11, 21..) can be visualised and acted out on number boards. Figure

2.22 shows these jumps on a 1-100 number board.

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1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Figure 2.22 Number board showing jumps on and off the decade

Beishuizen (1993) found that base-ten materials (MAB) evoked decomposition

or place value strategies for addition and subtraction whereas the hundred square

simulated a pattern of counting in tens, encouraging counting on and back strategies.

Klein, Besihuizen and Treffers (1998) noted that with growing experience during the

1980s, the hundred square with its pre-structured character, although providing a better

model for counting in ten strategies than base-ten blocks, did not allow for informal

computation methods and was an overly complicated learning aid for weaker students.

Further resources were considered by Dutch researchers to allow a more flexible

approach to computation especially the number line.

The number line is another resource to support the teaching of mental

computation. Researchers have reported limitations with numbered lines where all or

some numbers are marked. Bobis and Bobis (2005) noted how numbered lines could

lock students into using the numbers as presented and that the use of marked number

lines, containing extra unnecessarily marked numbers caused confusion for students and

resulted in the use of an inefficient strategy like counting in ones. Un-numbered lines

provide evenly spaced gradations and students can add number labels to the scale to suit

the computation they are completing. Gravemeijer (1994a) found that marked number

lines were being associated with a rigid ruler with fixed, predetermined distances and

relating particularly to measurement contexts.

An empty number line has no marked gradations and students are able to add

whatever marks they like to show the progression of their computation. Treffers and

deMoor (1990), in their revision of the Dutch primary mathematics curriculum, devised

the Empty Number Line (ENL) by removing the numbers and the evenly spaced marks.

Jumps on the decade Jumps off the decade

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The empty number line allows students to communicate their own solution procedures

as well as facilitating solution of these procedures. Another reported reason to use an

empty number line is that students were cognitively involved in their actions. The ENL

“… shows which part of the operation has been carried out and what remains to be

done” (Bobis & Bobis, 2005, p. 447). The empty number line allows students to use

whatever strategies they prefer and provides an open structure to record their thinking.

Figure 2.23 shows an empty number line compared to a numbered line, an un-numbered

line and an empty number line being used to calculate 14+8.

Numbered line Un-numbered line (students add the number labels)

Empty number line

Figure 2.23 Numbered Line, Un-Numbered Line and Empty Number Line (showing 14+8)

In the Netherlands, where the ENL was a resource included as part of the

curriculum documents, it was found that students were more cognitively involved in

what they were doing and the ENL allowed students to work with less formal strategies

for completing computations. It also provides the opportunity to raise the level of

students’ activity to give them freedom to develop more sophisticated strategies.

Sequencing mental computation instruction

Little research has been identified that suggests a sequence for the development

of mental computation with primary school-aged students. Studies have reported that a

teaching program was utilised, either as the focus or as an intervention, but few details

of the actual instructional sequence are available in the published documents. Klein and

Beishuizen (1994) suggested an order for teaching mental computation strategies based

on strategies identified in their research (1010, N10, A10 and 10s – described in

Appendix 3). This sequence focuses on the introduction of one strategy at a time before

moving onto introduce another and clearly takes a direct instructional focus where the

students are taught the strategies rather than inventing them for themselves.

+ 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

+ 2 + 2 + 2 + 2

14 20 22

+ 6 + 2

14 15 16 17 18 19 20 21 22

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One study which provided some guidance as to the instructional program used

for intervention in a classroom was reported by Heirdsfield (2005). Heirdsfield provided

a class teacher with a sequenced program to enhance the development of addition and

subtraction mental computation for students in a Year 3 classroom over a ten week

period. The suggested teaching sequence was used to introduce number combinations in

conjunction with the use of the empty number line and number board (0-100 and 0-99).

The sequence followed was:

1. jumping in tens forwards and backwards from multiples of ten;

2. jumping in tens forwards and backwards from numbers other than multiples

of ten;

3. relate the previous step (jumping forwards and backwards) to addition and

subtraction;

4. further addition and subtraction without bridging tens; and

5. further addition and subtraction, bridging tens.

This teaching sequence differs from that suggested by Klein and Beishuizen

(1994) in that the sequence focuses on the development of processes to assist with

computation strategies rather than teaching particular strategies. The students in

Heirdsfield’s study were encouraged to use number sense understandings and processes

to invent ways to add and subtract.

Another consideration for the sequencing of a mental computation program

needs to be the place, or not, of the traditional written algorithm. McIntosh (2005)

reported on studies he had completed which had assisted teachers to provide instruction

in regard to mental computation. He stated that teachers who had been involved in his

study still believed that there was a place for the written algorithms but that they would

delay the introduction until at least the latter part of the primary school. The teachers’

belief was that when the written algorithm is not the focus of instruction in the early

years of primary school, the development of strategies could support the students’

developing understandings of number.

Murphy (2004) suggested that by categorising and analysing mental

computation strategies it was possible to present a process that can be used to organise

learning in a mathematics classroom. In the study reported in this thesis, the Strategy

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Categorisation Framework (Hartnett, 2007) was used to provide the overall structure of

the instructional program based on the categories and sub-categories of the strategies in

the framework. Further details of the development of the mental computation

instructional program are presented in Section 3.6.1.

2.5.4 ASSESSING MENTAL COMPUTATION

The identification of what it means to know or have learned mathematics has

changed over time. Traditional mathematics curricula consider understanding of

mathematics to mean mastery of a body of facts and procedures. Another perspective

states that competency is not just the knowledge base but the ability to be able to use the

knowledge to solve problems effectively and efficiently (DeCorte, Greer, &

Verschaffel, 1996; Schoenfeld, 1985, 1992). Reasons for the inclusion of mental

computation in the mathematics curriculum, as described in Section 2.5, match the

contemporary view of mathematics education. The assessment of students’ successful

learning about mental computation needs to consider the effectiveness and efficiency of

the use of strategies, not just mastery or recall of a set of learned facts and procedures.

Mathematical proficiency was the term used by Kilpatrick, Swafford, and

Findell, (2001), to describe a comprehensive view of what constituted successful

learning of mathematics. Other authors have reported aspects of mathematical

proficiency in more detail including the NCTM (2000) who described computational

fluency in the Principles and Standards for School Mathematics. Computational

fluency was seen as having efficient, accurate and generalisable methods for computing

based on well-understood properties and number relationships. Other authors have

characterised computational fluency including Russell (2000) who described three

components: efficiency, accuracy, and flexibility. Russell suggested that teachers could

use these computational fluency criteria to assess students’ approaches to computation

and that by analysing the responses students make to computational problems teachers

can assess the students’ understanding of operation concepts as well as their choice of

strategies.

To provide a measure of the computational ability of students in this study, and

to reassure the classroom teacher that the students were not being disadvantaged by the

approach being taken, three aspects of computational fluency, namely accuracy,

efficiency, and flexibility, were chosen to provide a framework on which judgements

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about student learning and thus the success of the instructional program could be made.

To determine efficiency and flexibility of strategy choice by students it is necessary to

be able to identify the strategy used for a computation. By making their thinking visible

through a description of their thinking and strategy, choice is possible.

2.5.5 MAKING THINKING VISIBLE

Many researchers have gathered oral responses from their subjects about their

experiences, thought processes and strategies. Ericsson and Simon (1993) detailed the

use of think-aloud protocols as a data collection method where verbal reports were

gathered and analysed by researchers. For teachers to be able to make decisions about

students’ computational fluency the strategies they use need to be identifiable. If mental

computation is considered as computation in the head “without the use of external

recording devices” (Sowder, 1988, p. 182) then teachers would need to interview

students about their work. This practice would be time consuming for a practising

teacher. Treffers (1991) argued that the use of written work was to be encouraged. He

stated that mental computation does not exclude pencil and paper. By using mental

arithmetic along with pencil and paper work, the flexible thought processes that are

essential to mental computation can be displayed. To enable teachers to judge how

effective students are in regard to mental computation, their thinking about strategy

choice and application of the strategies needs to be captured as well as their answers.

Ideally this should be possible in the course of normal classroom routines. Written

recording of strategies used can facilitate this.

To capture their thinking and strategy use, the students in this study were

encouraged to record their thinking in lessons and on the assessment instruments.

Asking students to record responses to computation questions in writing does not

automatically elicit their thinking strategies. Asking them to show their thinking on

paper could elicit the traditional written algorithms rather than the thinking and strategy

choices used for mental computation. Scaffolding the recording of student thinking by

including the development and practice of language for describing strategies used can

assist students to more accurately record their thinking. Klein, Beishuizen and Treffers

(1998) encouraged students to use particular verbal labels to describe different

strategies. They used, for example, G for Gewoon (Dutch for Normal, referring to the

N10 strategy), SPV for Spring Verder (Dutch for Jump Further referring the N10C

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strategy) and S for Splitsen (Dutch for Splitting referring to 1010). The researchers

noted that

these verbal labels not only facilitated classroom communication about the

different solutions but also prompted students to first look thoughtfully at a

problem before solving it… to their teachers’ surprise, most second graders

rather easily learned to use these labels in an adequate way. (p. 448-449)

Once the identification of student thinking and strategy choice is achieved, even

in very informal written descriptions, teachers can assess the students in terms of their

accuracy, efficiency, and flexibility of mental computation strategy use.

2.6 IMPLICATIONS FOR THIS STUDY

This study set out to investigate a particular method of teacher professional

development which was classroom-based and involved the ongoing support of an

education advisor and the development and implementation of a mental computation

instructional program. The professional growth of both the classroom teacher and the

education advisor were studied to identify the successes in the professional

development and the mental computation program but to also provide insights for the

work of other education advisors working with practising classroom teachers.

The literature was reviewed to provide directions for the study in relation to the

professional growth of the classroom teacher and of the education advisor, the method

of professional development, and the mental computation instructional program which

provided focus for the professional development for the classroom teacher.

2.6.1 IMPLICATIONS FOR PROFESSIONAL GROWTH

The complexity of the role of teacher was identified and the range of

knowledges and practices needed to fulfil the role were highlighted. Models

representing the complexities of teaching were presented as were models of teacher

change. A distinction was made between teacher change and teacher professional

growth and the development of the Interconnected Model of Professional Growth

(Clarke and Hollingsworth, 2002; Hollingsworth, 1999) from earlier versions relating to

teacher change was described. This model was used in the study to analyse and

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represent the professional growth of the classroom teacher who participated in the

professional development provided by the education advisor.

2.6.2 IMPLICATIONS FOR THE PROFESSIONAL DEVELOPMENT

In this chapter, the principles and methods of effective teacher professional

development were reviewed in the literature and the findings of many researchers were

summarised to present a guide to successful professional development. This research

provided the basis for the actions of the education advisor in this study. The role of

education advisors who provide professional development for teachers was described

and models linking the complexities for those who teach teachers to the complexities of

teaching were identified. Descriptions of organised professional development for

education advisors were explored and the multiple roles of an education advisor as

teacher, teacher educator and researcher and the relationships between those roles were

highlighted.

The importance of education advisors in supporting teacher change and

professional growth was evident from the literature. The literature also provided

insights into the method of professional development chosen for the study, working in

the classroom with the teacher, so as to enhance its potential to successfully change the

classroom teacher’s knowledge, beliefs, attitudes and practices as well as the value of

including reflection as part of professional development activities.

2.6.3 IMPLICATIONS FOR THE MENTAL COMPUTATION INSTRUCTIONAL PROGRAM

The focus of the professional development provided for the classroom teacher in

this study was the inclusion of mental computation strategies as a major aspect of the

classroom mathematics program. The review of the literature relating to mental

computation outlined the components which needed to be considered in the

development of a comprehensive mental computation program. These components were

number sense and computation strategies. A definition of a mental computation strategy

was chosen from those in the literature, describing a strategy as “a sequence of

transformations of a number problem to arrive at a solution as opposed to just knowing,

simply counting or making a mental representation of a pencil and paper method”

(Threlfall, 2000, p. 30). This definition guided the selection of what constituted

strategies for inclusion in the mental computation instructional program. A

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comprehensive list of strategies was identified, providing a base on which to build the

mental computation instructional program which aimed to provide the students with

experience with a range of effective strategies.

Factors involved with the teaching of mental computation, including varying

views about the instructional focus and the assessment of student success, were

described. For this study it was decided to focus on direct instruction and to assess the

students according to their accuracy, efficiency and flexibility of strategy use. The

recognition of the need for students to make their thinking and strategy choices visible

to allow for the identification of strategies used was highlighted. This practice would be

vital for the identification of learning outcomes, particularly for efficiency and

flexibility, as strategy choice needed to be identifiable for judgements about these

elements to be made.

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3. Research design

3.1 OVERVIEW

This study was conducted to investigate a particular method of professional

development that I believed would be effective for instigating teacher change at the

classroom level. The method of professional development that I adopted for this study

utilised many of the principles of effective teacher professional development

highlighted in the literature (see Section 2.2.1). This was done to increase the likelihood

of the professional development being successful. I worked in the classroom along with

the classroom teacher, facilitating mental computation learning for the students and the

professional development for the classroom teacher. The intention was to document the

effects of this particular method of professional development in terms of the classroom

teacher’s and my professional growth and to provide insights for others working as

education advisors.

As the education advisor and the researcher in this study, I researched my own

practice as I worked with the classroom teacher. Part of the significance of this study

was that both the professional development and the mental computation instructional

program were planned and implemented as the study progressed in the classroom over a

full school year. The length of this professional development constituted an extended

period of time compared to common professional development opportunities for

classroom teachers. The professional development included demonstration lessons to

show the content and pedagogy included in the instructional program and provided

individual, ongoing support for the classroom teacher, both in person and via email. In

this way I was able to relate the professional development and mental computation

instructional program directly to the teacher’s classroom and the students’ learning

needs. In this study I fulfilled multiple roles: learner, classroom teacher, teacher

educator and researcher. The classroom teacher was a learner and a teacher. This

chapter outlines the details of the design and methodology of the study.

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3.2 METHODOLOGY

As an education advisor employed by a large education system it was my role to

support a large number of teachers in relation to mathematics education. By conducting

this study I had hoped to change my way of working with practising classroom teachers

by working for an extended period of time in a classroom rather than through provision

of one-off professional development sessions or school-based staff meetings. The

investigation was conducted as a case study (Stake, 1994). The case investigated in this

study involved myself as an education advisor facilitating professional development for

one teacher in her classroom. Stake described how cases of interest in education tend to

be unique and yet similar in many ways. My intention was to draw conclusions from

this case to inform the practice of other education advisors seeking similar changes by

working with this classroom teacher in her classroom and studying our professional

growth.

Case study is a form of ethnographic design which Creswell (2002) described as

“qualitative research procedures for describing, analysing, and interpreting a culture-

shaping group’s shared patterns of behaviour, beliefs, and language that develop over

time” (p. 481). In this context culture refers to the practices of a group rather than

ethnicity. Erickson (1986) described another approach to qualitative research as

interpretive. This was where individuals who knew their field were placed so as to

observe the actions of others in a particular setting and to interpret their observations in

terms of their knowledge. Davis (1995) outlined what she saw as a distinction between

interpretive qualitative studies and ethnographic studies. She stated that “the former

focus on the construction or co-construction of meaning within a particular social

setting (e.g., classroom) whereas the latter focus on the shared meaning of a particular

social group (culture) and/or on interactions among cultural groups” (p. 427). This study

took place in a classroom where the classroom teacher and I co-constructed the meaning

of our interactions with the students and each other. The social setting played an

important role in the conduct and results of the study. There were two groups

represented in this study. The classroom teacher represented teachers in general and I

represented education advisors. Observing both the classroom teacher and my

professional growth enabled conclusions to be drawn from this study that were

applicable to other teachers and education advisors. The study was interpretive.

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Case studies tend to generate large quantities of data from a number of real life

sources. Robson (1993) defined case study as “a strategy for doing research which

involves an empirical investigation of a particular contemporary phenomenon within its

real life context using multiple sources of evidence” (p. 146). Creswell (2008) described

case study as “an in-depth exploration of a bounded system based on extensive data

collection” (p. 476). The term bounded system, first used by Smith (1978), refers to the

case being investigated as being separated from other research in terms of time, place or

physical boundaries. In this study, the case being studied involved myself and one

classroom teacher participating in classroom-based professional development in relation

to teaching mental computation strategies to Year 3 students. The context for the study

comprised a Year 3 class learning mathematics designated by a new syllabus in a school

with particular characteristics. Details of the context are provided in Section 3.4. It is

from studying this particular case that advice for other education advisors was drawn.

This study was largely qualitative. Guba and Lincoln (1994) described how

much qualitative research is based on a holistic view that social phenomena, human

dilemmas, and the nature of cases are situational and influenced by happenings of many

kinds. Stake (1994) argued that a case study researcher focuses on issues regarding the

specific case. He described a process of issue development starting with a topical issue

being identified, followed by researchers posing a fore-shadowed problem. Then as data

are collected and interpreted the issue develops and reforms as assertions. Malinowski

(1984, cited in Stake, 1994) provided commentary on the difference between a fore-

shadowed problem and a preconceived idea. He stated that preconceived ideas can

hinder the work of a researcher if evidence that develops is not considered because it

does not match what was expected. However, a fore-shadowed problem can help to

shape a theory by focussing on data collected from issue-related observations.

Stake (1994, 2005) described three types of case study: intrinsic, instrumental

and collective. In an intrinsic case study the case itself is of interest. In an instrumental

case study a specific case is used to illustrate and illuminate the issue being

investigated. In a collective case study multiple cases are described and compared to

provide insight into an issue. In this study an instrumental case study design was

utilised where the data gathered through the multiple methods (see Section 3.4) allowed

for the issue being investigated, namely the provision of effective professional

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development by education advisors that leads to the professional growth of classroom

teachers, to be addressed. As with instrumental case studies, the data collected in this

study detailed the ordinary activities involving the day-to-day development and

implementation of the mental computation instructional program. The classroom

teacher and my reflections and communications were examined in depth to help pursue

the intent of providing insights for the practice of education advisors.

As described above, a case study is interpretive research. Creswell (2002)

outlined a series of steps for conducting a case study: “1) identify intent and the type of

design, and relate the intent to your research problem; 2) discuss approval and access

considerations; 3) use appropriate data collection procedures; 4) analyse and interpret

the data; and 5) write the report consistent with your design” (p. 496-497). The intent,

background, stimulus and research problem for the study were described in Chapter 1.

They type of design is described here. The access considerations are described in

relation to the context of the case in Section 3.3, the quality criteria and ethics

considerations are described in Section 3.7 and 3.8. The data collection procedures are

explained in Section 3.4 and the data analysis processes are provided in Section 3.5.

The data itself is presented in Chapter 4 in relation to the professional growth for the

classroom teacher and my professional growth is reported in Chapter 5. Chapter 6

provides the conclusions drawn from the data in relation to the literature (Chapter 2)

and the data. This thesis forms the complete report of this study.

3.3 THE CHANGE ENVIRONMENT

The change environment is a component of the Interconnected Model of

Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999),

represented as encompassing the domains of the professional’s world in which change

occurs. The change environment in the model recognises that the environment in which

change occurs has the potential to influence the success and depth of the change.

Hollingsworth (1999) added the change environment to the Interconnected Model of

Professional Growth as a recommendation of her thesis, stating that “the context in

which teachers work can have a substantial impact on their professional growth” (p.

321). The change environment in relation to teacher professional growth was described

in more detail in Section 2.2.4.

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The environment in which change occurred in this study influenced the

professional growth of both the classroom teacher and the education advisor. Factors

considered to be part of the change environment in this study included the school

context, the classroom context, the education advisor’s work context, the subjects and

the new mathematics syllabus. These factors are explained in the following sections.

3.3.1 THE SCHOOL CONTEXT

The school in which this study was conducted was a Catholic primary school

catering for students in years Prep to Year 7 (ages 5 to 12) in a mid to high socio-

economic suburb of Brisbane. The enrolment of the school was approaching 300

students in the year of this study. The school staff members were regular participants in

centrally-offered professional development programs and was an early adopter of

educational initiatives in many different areas of the curriculum including literacy,

outcomes-based education and numeracy as a particular focus in the early years. The

school had been successful in applying for several small systemic grants for school-

based projects to improve both student learning outcomes and teacher knowledge and

practices. The principal was young and had been appointed to the school the year before

the study as his second principal appointment. The previous principal had taken a

particular interest in the professional development of her staff. The assistant principal

and curriculum support teacher were both active in progressing the teachers’

professional development and supporting them to implement new initiatives and project

findings. As a result of the support from the administration and staff participation in

many professional development offerings, the school had a positive perspective on

change and a culture of support for new ideas.

The principal was not concerned that the study would take place in only one

classroom. He was confident that learning from the study would be shared with other

staff members as the study progressed and after its conclusion. The principal believed

that learning from the study would be able to influence change in the whole school’s

approach to the teaching of computation. This had been a reason why the whole school

had participated in a session with the education advisor prior to the classroom teacher

identifying her interest to learn more and participate in this study. This positive

approach to the professional development in this study as a potential learning

opportunity for all staff provided a supportive environment for the conduct of this study.

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3.3.2 THE EDUCATION ADVISOR’S WORK CONTEXT

As an education advisor, I was a member of a curriculum team whose role it was

to support schools to improve student learning outcomes, support the development of

effective and innovative learning resources and environments and to provide teacher

professional development and curriculum leadership through collaborative practices and

partnerships. This curriculum team met regularly as a whole team and in small groups to

maintain currency in relation to educational issues, system priorities and support for

schools. The members of the curriculum team had all been selected as experienced

practitioners in their particular specialist areas. In the year of this study and for eight

years prior to this study, I had worked as a member of this curriculum team as the

Education Officer Mathematics. Some of the team were discipline specialists, like me,

while others offered more general curriculum and pedagogical support.

The context in which the team worked was highly professional and stimulating

with regular consideration of current and topical educational issues. The curriculum

team was also mutually supportive and collegial. As the education advisors in the team,

we were encouraged to find what we considered to be our most effective ways of

working. Whole team meetings often focussed on effective ways to provide professional

development for classroom teachers to enact changes at the system, school and

classroom level. It was this focus on effective ways of working in the role which

stimulated me to research the provision of effective professional development and my

professional growth. The Senior Education Officers in the curriculum team were

supportive of me conducting this study as part of my normal role.

3.3.3 SUBJECTS

The subjects in this study were the classroom teacher and myself, as the

education advisor and researcher. I had been a primary school teacher and had a range

of teaching experience in terms of year levels taught and school contexts. During the

conduct of this study I was working as the Education Officer: Mathematics with

Brisbane Catholic Education, supporting the implementation of the new Years 1 to 10

Mathematics syllabus (QSA, 2004) in Brisbane Catholic Education schools. I had also

fulfilled a similar role earlier in my career supporting other mathematics curriculum

implementation initiatives in State (government) schools.

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The classroom teacher was an experienced teacher, having taught thirty-seven

years prior to the year of the study, of which twenty-five of those years had been

teaching Year 3, the focus year level for the study. I knew that the classroom teacher

was interested in the inclusion of mental strategies in the class program for the Number

strand of the new syllabus as she had approached me after a professional development

session I had conducted at the school. She perceived that there would be benefits for her

class and so was keen to participate in this study when I suggested it. The twenty-seven

Year 3 students (approximately eight years of age) also participated in the study. Year 3

was chosen for the study as traditionally formal addition and subtraction algorithms for

numbers beyond single digits were a major instructional focus in that year of schooling.

The students in this study formed part of the change environment due to their potential

to impact on the professional growth of the classroom teacher and myself as the

education advisor.

3.3.4 THE NEW MATHEMATICS SYLLABUS

At the time of this study a new state mathematics syllabus, the Years 1 to 10

Mathematics syllabus (QSA, 2004), had been developed and had been made available to

schools two years prior to the commencement of the study. This syllabus was outcomes-

based rather than year-level input specific. Content was presented as outcome

statements in six levels (for years 1 to 10) rather than individual year levels. The

syllabus had been developed following a review of local as well as worldwide research

into mathematics education and as such included changes in both content and pedagogy.

One change of note for this study was a move away from traditional written algorithms

as the focus of computation instruction to an emphasis on mental computation as the

first method of computation when solving problems. The syllabus mentioned mental

computation strategies and provided some examples. Having worked with many groups

of teachers leading up to the release of this syllabus, I believed that generally teachers

felt that they did not have the knowledge or practices to make mental strategies the

focus of computation instruction. I had a particular interest in mental computation and

mental computation strategies and was eager to assist schools to make changes to align

their practices with those in the new curriculum documents. I had developed and

conducted teacher professional development sessions on this topic at centrally-offered

venues and at school staff meetings. The new mathematics syllabus represented a

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significant aspect of the change environment in which the classroom teacher’s

professional growth was situated as well as being a factor influencing my professional

growth as an education advisor.

3.4 DATA COLLECTION

Data were collected from a variety of sources detailing the day-to-day activities

of the classroom. To enable my professional growth as an education advisor to be

analysed, the professional growth of the classroom teacher needed to be analysed as an

outcome of my actions in facilitating the professional development. Examples of

professional growth were identified for both the classroom teacher and myself from a

large body of data. Student learning data were also collected to evaluate and inform the

development of the mental computation instructional program.

Qualitative data were sourced from formal and informal communications

between the education advisor and the classroom teacher, including reflections recorded

as email communications or researcher field notes (Section 3.3.1) and interviews

(Section 3.3.2). Student learning data were collected in the form of written computation

tests administered across the school year as well as the state-wide test administered to

all Year 3 students in Queensland (Section 3.3.3). The data were analysed to identify

examples of professional growth that included significant events reported as vignettes

and then discussed in Chapter 4 for the classroom teacher and in Chapter 5 for me as the

education advisor. The analysis of the data is described in Section 3.4. Section 3.4.1

explains how the examples of the classroom teacher’s professional growth were

represented using the Interconnected Model of Professional Growth (Clarke &

Hollingsworth, 2002; Hollingsworth, 1999) and Section 3.4.2 describes how the

examples of my professional growth were represented using a modified version of this

model.

3.4.1 CLASS TEACHER AND EDUCATION ADVISOR REFLECTIONS

The main sources of data relating to the knowledge and practices of the

classroom teacher and myself were the ongoing communications and reflections

recorded in emails or as researcher field notes. The other source of data for reflections

on learning during the year was the end-of-year formal interview. The ongoing

discussions differed from formal or semi-structured interviews (e.g. Grossman, 1990;

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Nias, 1989) in that specific questions or themes were not pursued deliberately. The

discussions were open-ended (Elbaz, 1983) and occurred over a longer period of time.

Sikes, Measor and Woods (1985) used what they termed “prolonged interviews” (p. 14)

in their investigation of a group of teachers, their careers and the contexts in which they

worked. They considered that this method of data collection enables a more holistic

understanding of the teachers’ perceptions, provides a greater depth of response and

allows for a greater appreciation of the context in which the teachers worked. In this

study it was important that the communications were kept informal so the classroom

teacher felt comfortable to discuss and reflect on any aspect of the activities conducted

as part of the professional development.

The classroom teacher and I knew each other prior to the study as I had

presented professional development sessions at the school and had worked with her on a

project relating to mathematics planning prior to the study. A professional relationship

already existed between the classroom teacher and myself which meant that the

communications between us could be informal from the beginning of the study and that

the classroom teacher was comfortable communicating in a reflective manner with me.

This familiarity, together with the use of email for delayed reflection, enabled data in

the form of these communications to be collected from very early in the study. In some

emails to the classroom teacher I encouraged her to reflect on particular aspects of a

lesson or topic by asking informal questions about how the follow-up lessons went, how

the students responded and what she knew or thought about particular topics. These

questions were included more often at the beginning of the year to elicit reflections.

However, as the year progressed I needed to do less prompting in order to receive

reflective feedback or questions from the classroom teacher.

The intended use of ongoing reflection between the classroom teacher and

myself as data for the study was specified, discussed and agreed upon at the beginning

of the year. The classroom teacher was comfortable with using email and had access to

a computer in her classroom allowing for frequent and informal communication. Email

was chosen for its convenience, accessibility and its ability to preserve a record of the

reflections. Early in the study we realised that there would not be enough time to reflect

or discuss lesson outcomes in person during my visits to the school. The use of email

for the discussions and reflections seemed a logical solution. A benefit of the use of

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email for reflections of this type was outlined in Section 2.2.3, noting that the

asynchronous format of the communication allowed time for deeper reflection than is

possible in synchronous communication (DeBard & Guidera, 1999). The reflections

provided feedback that was used to refine the design of the instructional program and

the professional development (see Section 3.6) to match the needs of the students and

the classroom teacher. At times throughout the year the class teacher and I also were

able to reflect in face-to-face discussions when I visited the classroom. At the end of

each lesson, when there was time, the class teacher and I would discuss the lesson and

student learning. Often we found we would also individually reflect further on the

lesson and then share extra thoughts via email. I recorded details of the face-to-face

discussions and my own reflections as field notes.

3.4.2 END OF YEAR INTERVIEW

At the end of the study I interviewed the class teacher. The interview was semi0-

structured (Creswell, 2002) where some questions were close-ended and some were

open-ended. The open-ended questions in particular were designed to allow the

classroom teacher to provide personal experiences and reflections. The questions I

asked were provided to her prior to the interview so she could consider and plan for her

responses. The list of questions is provided in Appendix 4. The questions related to the

main aspects of the study: the computation program - the content and the pedagogy,

student learning, the professional development and her own learning. The interview was

audio-taped and transcribed directly for analysis as data for the study. The transcriptions

were scrutinised to identify evidence of classroom teacher learning. Many of the

examples of this learning were reported in the vignettes provided as data in Chapters 4

and 5. The end of year interview constituted a reflection on the whole year of classroom

activities. The format of the interview, while addressing particular questions, was kept

informal and conducted as a conversation.

3.4.3 MONITORING STUDENT LEARNING

In this study it was necessary to monitor the students’ learning so the classroom

teacher and I had information on which to base our decisions about the development of

the mental computation instructional program. It was also necessary to demonstrate the

success of the program so the classroom teacher could be reassured that the change was

not adversely affecting the students’ learning. To monitor the learning outcomes of the

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students, I developed computation tests that were used periodically throughout the year.

The tests consisted of addition and subtraction questions chosen to provide a range of

difficulty for Year 3 students and to offer a range of possible strategies. The tests

provided data in the form of accuracy scores as well as counts of clearly recorded or

described strategies that were used to identify the students’ efficiency and flexibility of

strategy choices. The computations tests were administered at the beginning and end of

the year and a shorter version was used twice during the year. The State-wide Year 3

Aspects of Numeracy test was another source of data to monitor the students’ learning

in relation to mathematics in general and mental computation more specifically in

comparison to other students. Details of these tests are included in Appendix 5.

3.5 DATA ANALYSIS

The steps for conducting a case study described by Creswell (2002) were listed

in Section 3.2.1. The fourth step of the process outlined the analysis and interpretation

of data. Creswell describes the process as “developing a description, analysing your

data for themes, and providing an interpretation of the meaning of your information (p.

498). Analytical strategies used in case study research include the coding of data and

identification of themes. Coding relies on making assertions about the frequency of

themes and identification of important events during the study and in the data.

Krippendorff (2004) identified that frequencies of themes indicate the emergence of

trends in qualitative research data.

As described by Robson (1993) and Stake (1994, 1995, 2005), case study

research generates and considers a large amount of data from multiple sources so the

analysis of the data needs to be well organised and is likely to take time. Using a

process of cycling back through the data is common practice in qualitative research

(Creswell, 2002). In this study a very large quantity of narrative data was collected

through the regular email reflections and communications between the classroom

teacher and myself. During the study data were considered as needed to guide the

development of the mental computation instructional program. Data were analysed at

the conclusion of the year working in the classroom to identify examples of the

classroom teacher’s and my professional growth.

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The data collected in this study were predominantly written text. Effectively

these texts were a written form of discussion between the classroom teacher and myself.

Creswell (2002) stated that “the primary form for representing and reporting findings in

qualitative research is a narrative discussion” (p. 274). There is no set form for these

narrative discussions. Creswell (2008) explained how narrative is a feasible way of

collecting data because it is a common device used in everyday interactions. Coffey and

Atkinson (1996) stated that:

there are no formulae or recipes for the ‘best way’ to analyse the

stories we elicit and collect. Indeed, one of the strengths of thinking

about our data as narrative is that this opens up the possibilities for a

variety of analytic strategies. (p. 158)

Data collected were analysed to identify evidence to answer the research

questions that guided this study. The first question asked: What professional growth did

the classroom teacher experience as a result of the professional development conducted

in her classroom? The data were reviewed to find significant events relating to the

classroom teacher’s professional growth. The significant events were presented as

vignettes including a brief description of the setting followed by a discussion of the

changes evidenced. Direct quotes from the written communications relating to

significant events that indicated change in the knowledge or practice of the classroom

teacher or myself were also included in the discussion. Each example of the classroom

teacher’s professional growth was then summarised and represented using the



A similar process was used to analyse evidence and identify change for the

second research question which asked: What professional growth did the education

advisor experience as a result of the professional development conducted in her

classroom? The body of data was reviewed to identify significant events that triggered

or indicated changes and which provided examples of my professional growth. The

examples of my professional growth domains were summarised and represented using a

modified version of the Interconnected Model of Professional Growth.

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3.5.1 REPRESENTING THE CLASSROOM TEACHER’S PROFESSIONAL GROWTH

In this study the Interconnected Model of Professional Growth (Clarke &

Hollingsworth, 2002; Hollingsworth, 1999) was used to summarise and represent the

classroom teacher’s professional growth. The development of this model was outlined

in Section 2.1.4. The model is re-presented in Figure 3.2.

Figure 3.2 The Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999)

This model depicts four domains of the professional’s world (External Domain,

Personal Domain, Domain of Practice and Domain of Consequence) as well as the

multiple possible connections between these domains. Each connection represents how

change in one domain can lead to change in another domain through the mediating

processes of either enactment or reflection. Change in one domain leading to change in

another was described by Hollingsworth (1999) as a change sequence. A series of

changes between domains was referred to as a growth network. It was the growth

networks indicating enduring change that represented professional growth in this study.

Hollingsworth (1999) used the Interconnected Model of Professional Growth to

represent the professional growth of teachers in her study as growth networks. In an

example growth network provided in her thesis (see Figure 2.1.2), Hollingsworth

numbered the arrows to show the sequence of changes in the growth network. She did


Salient outcomes



Enactment

Reflection

Personal Domain

External Domain

Domain of Practice



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not use the numbered arrows in her representations of the professional growth of the

teachers in her study. In this study I decided to utilise these numbered arrows to

represent the sequence of changes in each growth network depicting the classroom

teacher’s professional growth.

In a study of the role of models in science teaching, Justi and Van Driel (2006)

used the Interconnected Model of Professional Growth to represent change observed in

science teachers’ content and pedagogical content knowledge. They identified and

described examples of what they referred to as possible relationships between pairs of

domains specific to the use of models in science education. A similar analysis of the

change relationships between the domains was used in this thesis. This assisted with the

representation of changes in the classroom teacher’s knowledge and practices identified

in the data. These relationships are represented in Table 3.2.

Table 3.2 Descriptions of the relationships between domains in the IMPG model for this study

Relationship between domains

Description

E P Enact a practice based on external stimulus P K Reflect on the implementation of a practice P S Reflect on a practice in terms of salient outcomes K P Enact a practice based on knowledge K E Actively seek further external input S K Reflect on salient outcomes of a practice S P Enact a practice based on salient outcomes K S Reflect on knowledge in terms of salient outcomes E K Reflect on external stimulus in terms of knowledge

Enact Reflect

The data collected in this study in the form of written reflections, email

communications, field notes and interview transcripts were analysed to identify

instances of change in the classroom teacher. The analysis took place after the year of

working in the classroom had been completed. Vignettes describing significant events

in these examples of professional growth were then reported. The Interconnected Model

of Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999) with its

connections between domains mediated by reflection or enactment was used to identify

how a change in one domain led to a series of changes in other domains. As described

above, in this study numbered arrows were used to indicate the sequence of changes

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between domains of the model with the changes numbered to show their sequence. The

examples of the classroom teacher’s professional growth are reported in Chapter 4.

3.5.2 REPRESENTING THE EDUCATION ADVISOR’S PROFESSIONAL GROWTH

The Interconnected Model of Professional Growth, (Clarke & Hollingsworth,

2002; Hollingsworth, 1999) was developed and refined with classroom teachers as the

focus. Hollingsworth (1999) concluded that the model would be applicable for other

education professionals and other professions. Education advisors are education

professionals, therefore the model can capture the domains of their world as it does for a

teacher and represent their professional growth resulting from changes in external

stimulus, knowledge, beliefs and attitudes, practice and salient outcomes. Diezmann,

Fox, deVries, Siemon and Norris (2007) used the Interconnected Model of Professional

Growth to analyse the learning of a team of five “professional developers” (p. 94) who

worked together to support a group of teachers participating in a mathematics project to

improve the learning of Years 1 to 3 students. These professional developers were

educators working with practising classroom teachers in a role similar to that of the

education advisor in this study.

While reviewing the literature it was noted that models of learning and teaching

practice (e.g. Jaworski, 1992, 1994; Steinbring, 1998) had been modified by researchers

to represent the practices of those who educate teachers (education advisors). Zaslavsky

and Leikin (2004) modified Jaworski’s (1992, 1994) teaching triad model of teaching

practice to enable it to represent a wider range of educators. This modification identified

Jaworski’s complete original teaching triad, that described the considerations for

teachers of mathematics, as one of the three considerations for mathematics teacher

educators (education advisors). Zaslavsky and Leikin (2004) also extended another

learning and teaching model proposed by Steinbring (1998) to apply to teacher

educators as well as those who mentor education advisors. With this model they used

the structure of the model in layers to represent the learning process of students,

teachers and teacher educators. The modification of the learning and teaching model for

other educators (education advisors) was described in Section 2.3.1.

As the researcher in this study I modified the Interconnected Model of

Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999) to represent

my professional growth as an education advisor. The modification considered each of

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the domains of the model with a particular focus on education advisors rather than

classroom teachers. The External Domain, the Personal Domain and the Domain of

Practice related to the education advisor much as they did to classroom teachers. It was

the Domain of Consequence that I considered to have a different focus for the education

advisor. The Domain of Consequence represents the salient outcomes of consequence.

For the classroom teacher the outcomes of consequence most often relate to the

students’ learning. As the education advisor facilitating the professional development

for the classroom teacher, I saw her learning as outcomes of consequence for my work.

Therefore I represented my Domain of Consequence by the entire Interconnected Model

of Professional Growth as it applied to the classroom teacher. That is, the teacher’s

professional growth as a consequence of the professional development constituted the

salient learning outcomes of consequence for me as the education advisor. This

modification to the model was similar to the adaption by Zaslavsky and Leiken (2004)

of Jaworkski’s (1992, 1994) teaching triad. The modified version of the model is

presented in Figure 3.3.

Figure 3.3 Modified version of Interconnected Model of Professional Growth (based on Clarke & Hollingsworth, 2002; Hollingsworth, 1999)

The External Domain (E) or source of information and stimulus for a teacher is

most often professional development sessions or interactions with colleagues. In terms

of an External Domain for the education advisor there are very few role-related

K P

E

S

S

PK

E

Enactment

Reflection


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professional development opportunities. With a lack of specific support for education

advisors to improve their knowledge and practice, they need to seek their own external

information and stimulus to support their own development. The External Domain for

an education advisor can involve reviews of educational literature, attendance at

conferences and professional discussions and interactions with colleagues rather than

specific professional development sessions. The Personal Domain (K) of knowledge,

beliefs and attitudes and the Domain of Practice (P) apply to the role of education

advisor as they do to the role of teacher.

Examples of my professional growth were identified in the data as significant

events that were reported as vignettes as was done for the classroom teacher’s

professional growth. These significant events triggered a series of changes in my

knowledge and practices. The modified Interconnected Model of Professional Growth

was used to summarise and represent the sequence of changes including the significant

events described in the vignettes. Numbered arrows were used to identify the sequence

and pattern of changes that formed a particular growth network. Examples of my

professional growth in this study are represented and described in detail in Chapter 5 as

vignettes followed by a discussion of the significant event and a representation using

the model as described in this section.

3.6 THE PROFESSIONAL DEVELOPMENT

The overall aim of this research was to implement a particular method of

professional development and to study the professional growth of the classroom teacher

and myself in the role of education advisor facilitating the professional development.

My professional growth was analysed and represented in relation to the facilitation of

the classroom-based professional development for the classroom teacher. The classroom

teacher’s professional growth was analysed and represented as the participant in the

professional development. To enable our professional growth to be studied, the

professional development I was providing for the classroom teacher needed to be

successful. While it is recognised that my professional growth could still be studied

should the professional development be unsuccessful, it would be unlikely that the

classroom teacher would have continued with the study for the extended period of time

had the professional development, and the mental computation instructional program

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that formed a major part of the professional development, failed or had been perceived

to fail. The classroom teacher was happy to continue the study for the year.

To give the professional development the greatest chance of success I reviewed

the literature in relation to effective teacher professional development and chose to

include many of the identified principles of effective teacher professional development

in my way of working with the classroom teacher. These principles were detailed in

Section 2.2.1. The professional development in this study: focussed on the classroom

teacher feeling ownership of the process (Nisbet, Warren & Cooper, 2003); was school-

based as it was conducted in the teacher’s classroom (Guskey, 2002); was conducted

over an extended period of time (Mewborn, 2003; Nisbet, Warren & Cooper, 2003)

with regular feedback (Nisbet, Warren & Cooper, 2003); and included collegial support

(Guskey, 2002). Although it was not planned at the beginning of the study, the

classroom teacher and I ended up co-teaching (Tobin & Roth, 2006), another effective

professional development method identified in the literature. The professional growth of

the classroom teacher and the education advisor reported in Chapters 4 and 5 as well as

the student learning results provide evidence that the professional development was

successful.

The professional development method utilised in this study differed from the

types of professional development I had offered previously. The professional

development also differed from the types of professional development the classroom

teacher had experienced prior to this study. Two factors made this professional

development different. The first of these factors involved the development and

implementation of a mental computation instructional program that suited the needs of

the classroom teacher and the students. The type of professional development and the

focus of the computation instructional program were discussed and agreed on prior to

the study commencing. Further details about this aspect of the professional development

are provided in Section 3.6.1. The second factor was that the professional development

was situated in the classroom and I was able to provided ongoing, personalised support

through my regular visits and email communications. Through this ongoing support the

classroom teacher could access regular personal advice from me whenever she felt it

was needed. This factor is described in Section 3.6.2.

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3.6.1 MENTAL COMPUTATION INSTRUCTIONAL PROGRAM

The mental computation instructional program was important in this study for

two reasons. Firstly, the design and implementation of this program was the focus of the

professional development I provided for the classroom teacher. Secondly, successful

implementation of the program, in terms of student learning outcomes, would be more

likely to lead to change in the classroom teacher’s knowledge and practices, and

therefore result in her professional growth which was the purpose of the professional

development.

The mental computation instructional program aimed to develop computation

strategies and the related number sense understandings. These two components of

mental computation (strategies and number sense) were described in Section 2.5.1. The

program used in this study was designed to incorporate learning in the Addition and

Subtraction and Multiplication and Division outcomes from the Years 1 – 10

Mathematics Syllabus (QSA, 2004). It was also expected that other understandings

included in the Number Concepts outcomes from this syllabus would be incorporated to

develop the students’ number sense. The overall structure and sequence of the program

were linked to the Strategy Categorisation Framework (Hartnett, 2007) that provided a

comprehensive list of strategies applicable to primary school students, organised into

five major strategy categories. The Strategy Categorisation Framework was detailed in

Section 2.5.2.

The overall aim of the mental computation instructional program was for the

students to become accurate, efficient and flexible users of mental computation

strategies. These aspects of were described in Section 2.5.4 in relation to the assessment

of mental computation. Particular attention was paid to encouraging the students to

record their thinking and strategy use throughout the year. Recording enables teachers

to identify the students’ strategies as well as their conceptual understandings and

misconceptions as a basis for the further development of lessons and activities. Further

details about students making their thinking visible were provided in Section 2.5.5. The

recording of thinking and strategies was modelled and encouraged from my initial

classroom visit and throughout the year. The students were told that the teachers

(classroom teacher and I) wanted to see what they were thinking and how they solved

each problem. The answers would be important but it was how they worked out the

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answers that would provide data to assess their efficiency and flexibility of strategy

choices.

Some assumptions were also discussed from the outset. The classroom teacher

and I were aware that the students would possess some mental computation strategies

prior to the lessons and that they may or may not be conscious of these strategies. The

Year 2 teacher who had taught the majority of the students reported that she had used

strategies for addition and subtraction basic facts including counting on and doubles.

There was no assumption that because the students had not had formal instruction in

mental computation strategies for addition and subtraction of two-digit numbers that

they would not be able to use strategies to complete computations.

Another decision made with the support of the class teacher was that the

students would not be taught the traditional written algorithm for any of the four

operations during this school year. The classroom teacher agreed to this change to her

usual program.

Overview of the mental computation instructional program

At the beginning of the study the classroom teacher and I met to discuss initial

plans, goals and the overall structure of the mental computation instructional program.

We had a general discussion about mental computation strategies and number sense

concepts that would be applicable to Year 3 students. With reference to the syllabus

document (QSA, 2004) and the Strategy Categorisation Framework (Hartnett, 2007) the

content and sequence of the instructional program was discussed. The Strategy

Categorisation Framework provided a list of mental computation strategies organised

into five strategy categories and eleven sub-categories (see Section 2.5.2). The school

year in Queensland is divided into four terms of approximately 10 weeks each. It was

decided that one major strategy category from the Strategy Categorisation Framework

(Count On/Back, Break Up Numbers, Adjust and Compensate and Use Doubles) would

provide the focus for each term, together with activities to develop the related number

sense concepts including place value concepts and operation concepts, for example the

concept of addition.

By the end of the year it was intended that the students would have a

comprehensive repertoire of strategies for calculations involving addition and

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subtraction of two-digit whole numbers and some larger numbers, and multiplication

and division basic facts and some two-digit numbers. A logical sequence for the

development of the strategy categories was discussed and the overall program sequence

was decided. The order in which the strategies would be introduced was chosen to

provide what the class teacher and I considered to be a logical sequence, moving from

strategies the students had some previous experience with in relation to basic facts (e.g.,

Count on and Back) to the strategies more suited to multi-digit numbers (e.g., Break Up

Numbers). Table 3.3 provides the overview of the mental computation instructional

program. A more detailed overview of the mental computation instructional program for

the whole year is provided in Appendix 6.

Table 3.3 Overview of the mental computation instructional program

Term Strategy in focus 1 (Feb-April) Count On / Back for addition and subtraction 2 (April- June) Adjust and Compensate for addition and subtraction 3 (July – Sept) Break Up Numbers for addition and subtraction 4 (Oct – Dec) Use Doubles and /or Halves – multiplication and division facts

Consolidate addition and subtraction - focus on student choice of strategy

The Use Place Value strategy category was not allocated to a term on its own

but it was covered incidentally throughout all terms when multiples of ten or counting

in tens was the focus.

In the literature the instructional focus for teaching mental computation included

two different pedagogical styles. Some studies on mental computation instruction

reported the instructional focus being on the students inventing their own strategies

(Blöte, Klein & Beishuizen, 2000; Buzieka, 1999; Hedren, 1999; Kamii, Lewis &

Livingston, 1993; Kamii & Dominick, 1998). An alternative approach was the use of

direct instruction about particular mental computation strategies. Some researchers

stated that a preference for direct instruction led to an increased understanding of

number and flexibility when working with numbers (Sowder, 1992). Other researchers

were concerned that if students were not directly taught particular strategies they would

not become part of their repertoire (MacLellan, 2001; Murphy, 2004). Other effective

teaching focuses (flexibility, discussion and conceptual understanding) were discussed

in the literature (e.g., Morgan, 1999). Section 2.5.3 provided details of considerations

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for the teaching of mental computation. The classroom teacher and I discussed the

teaching approach that would be used in this study at the beginning of the year and

decided to directly teach the particular strategies to provide exposure to strategies from

each of the major strategy categories of the Strategy Categorisation Framework during

the year. The outline of the program in terms of the strategies in focus for each term and

the general teaching approach was decided prior to the start of the classroom-based part

of the study.

Development of the mental computation instructional program

The week-to-week details of the mental computation instructional program were

developed progressively throughout the year. The development of the program was

based on the needs of the students and the classroom teacher as identified in reflective

discussions following the demonstration lesson I conducted in the classroom once a

week, usually on a Tuesday morning. These discussions would guide the direction and

development of the program. The classroom teacher would lead follow-up lessons

during the rest of the week based on materials I provided when I came for the

demonstration lesson. In the first few weeks of the study, while we were establishing

the routines, I visited the classroom and presented the demonstration lesson without

providing information to the classroom teacher prior to the lesson. The classroom

teacher soon began to assist students during the lesson and by Week 5 I was emailing an

outline of my plans for the demonstration lesson to the classroom teacher prior to my

visit. As the year progressed the classroom teacher became more involved in both the

teaching and the planning of the lessons for the students. These changes to my practice

as an education advisor are reported in Chapter 5. The classroom teacher and I

developed a routine of reflecting on the program as it developed via email

communications. An example lesson plan and some of the email correspondence about

the lessons and follow-up lessons are provided in Appendix 7.

3.6.2 ONGOING SUPPORT BY THE EDUCATION ADVISOR

A feature of the professional development conducted in this study that made it

different from professional development I had previously provided was the classroom

teacher’s ongoing access to my support. This approach was also novel for the teacher.

The plan for the professional development was based on research about effective

principles of teacher professional development (see Section 2.2). Analysis of the

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literature highlighted the heightened potential for teacher learning when they have

ongoing access to expert support (Nisbet, Warren & Cooper, 2007). I was able to

support the classroom teacher utilising my knowledge and experience in mathematics

education and in particular mental computation strategies. I was more knowledgeable in

regard to these areas of the program. The classroom teacher spent more time in the

classroom with the students than I did and so she was more knowledgeable in regard to

the students’ learning needs and abilities. Working together to plan and implement the

mental computation instructional program made this professional development activity

different for both the classroom teacher and myself.

As described above I visited the classroom each week and presented a

demonstration lesson to the students for the classroom teacher to observe and participate

in. This regular visit formed part of the ongoing support. Whenever possible the

classroom teacher and I would discuss the lesson directly after it was completed. When

this was not possible we discussed the lesson at a more convenient time via email.

These email communications formed the other aspect of the ongoing support provided

for the classroom teacher and formed a large component of the data collected in this

study. The communications were deliberately informal and reflective in nature. The use

of these communications as data for the study was explained in Section 3.6.1.

There were many times during the year when the classroom teacher contacted

me via email seeking information or teaching strategies to assist with the

implementation of the mental computation instructional program. Some of these emails

asked for ideas in relation to the implementation of suggested follow-up lessons. Other

times requests related to mathematics education topics beyond mental computation and

as such beyond the focus of this study. I was happy to provide whatever support I could

for the classroom teacher and for the learning of the students in her class.

Over the course of the year spent in the classroom and due at least in part to the

ongoing support and communication, the classroom teacher and I developed a close

working relationship as well as a social relationship. By keeping the communications

unstructured and informal, the relationship that developed was able to contribute to the

success of the study in terms of both the classroom teacher’s and my professional

growth.

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3.7 QUALITY CRITERIA

There has been criticism of case study as a research methodology due to the lack

of generalisability from the particular case. Such criticism often stems from a view that

there is little or no substantial indication of the degree to which the case is

representative of other cases (Gay, 1987). Stake (1994) stated that a case study has a

broader scope of influence than other quantitative forms of research that focus on

experiments and testing hypotheses. Qualitative case study researchers explore the

complexities of the context to connect ordinary practice in “natural habitats to the

abstractions and concerns of diverse academic disciplines” (Stake, 1994, p. 239). While

case studies aim to develop a better understanding of a unique case, the researcher and

readers of the research can extrapolate the findings to provide a better understanding of

other similar cases. The case in this study was my role as an education advisor

providing professional development to one classroom teacher. The approach to

classroom professional development in this study was intended to inform the role of

other education advisors. Other professionals can draw their own conclusions from the

results presented in this thesis.

The second perceived limitation involves the potential for researcher bias (Gay,

1987). Researchers who conduct case studies are interested in determining meaning

from their results. However, their work still needs to be rigorous, accountable, reliable

and valid. Methods that can be employed to reduce researcher bias include triangulation

using multiple sources of data such as observational field notes, documents and

interviews (Creswell, 2002). In this study data were collected by several methods

including documentation in the form of email communication between the classroom

teacher and myself. Another technique to limit researcher bias is member validations

(Bloor, 1997; Lankshear & Knobel, 2004), member checking (Creswell, 2002) or

reciprocity (Lather, 1991, 2000) where participants’ thoughts are articulated, recorded

and then returned to the participants for clarification, confirmation and/or amendment.

This practice was utilised in this study as the data were analysed by providing the

classroom teacher with copies of the analysis for her to read through and validate the

trustworthiness of the report of the experience. Ultimately the study was of me as the

researcher and it portrays my observations of what occurred. The account provided is

truthful rather than biased.

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Another limitation in case study research is the effect of the researcher’s

presence on the actions of the subjects. In this study, I was a subject of the study and not

just a neutral observer or participant. The mental computation instructional program the

classroom teacher and I co-planned and co-taught would not have occurred had this

study not been conducted in this classroom. In this sense my presence was intended to

make a difference to the way this class would have operated had the study not been

conducted. Whether the students and teacher would have behaved differently had the

teacher taught the program without my participation cannot be tested.

The fourth criticism of case study as a design is that there are difficulties with

repeating a case study. This is a true limitation of many case studies as often the

research is incident-based and/or placed within a particular context. Simpson (1992)

detailed how replicating findings in qualitative research is near impossible due to the

subjective, spontaneous and interactive nature of the research. It is in response to such

claims that researchers typically describe their cases in such detail that readers can

vicariously experience the reported happenings and draw their own conclusions (Stake,

2000). In this study the change environment in which this study was conducted was

described in detail (see Section 3.3) so that other education advisors and researchers

would recognise the context in which this study took place. This was done to enable

comparison with other settings so that aspects of this study could be replicated. Detailed

descriptions and vignettes of specific events that were significant in the identification of

the classroom teacher’s and my professional growth were provided for the same reason.

In Chapter 6 of this thesis I present a list of actions I recommend for other education

advisors based on my learning in this study. Other education advisors and researchers

should be able to take these recommendations and implement some or all of them and in

this way will be replicating the positive aspects of my changed knowledge and practice.

3.8 ETHICS CONSIDERATIONS

This research was conducted in a school. This study received ethics approval

from the Queensland University of Technology Human Research Ethics Committee.

The approval number was 2965H. As per university guidelines the classroom teacher

gave her own permission to participate in the study and parental permission was sought

and received on behalf of all the students in the class. The researcher ensured the

development and maintenance of ethical and respectful relationships between the

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participants and the researcher. The establishment and maintenance of these

relationships were focused on the two concerns of ethics: informed consent and the

protection of the participants from harm.

The ethical considerations for this study followed the four principles of the

Australian Association for Research in Education (AARE) code of ethics. These

principles, edited by Bibby (2009) are outlined below:

1) The consequences of a piece of research, including the effects on the

participants and the social consequences of its publication and application

must enhance the general welfare.

2) Researchers should be aware of the variety of human goods and the variety

of views on the good life, and the complex relation of education with these.

They should recognise that educational research is an ethical matter, and

that its purpose should be the development of human good.

3) No risk of significant harm to an individual is permissible unless either that

harm is remedied or the person is of age and has given informed consent to

the risk. Public benefit, however great, is insufficient justification.

4) Respect for the dignity and worth of persons and the welfare of students,

research participants, and the public generally shall take precedence over

self-interest of researchers, or the interests of employers, clients, colleagues

or groups.

The participants in the study were assured full anonymity. The classroom

teacher nor the school were identified and the students were all assigned pseudonyms in

the reporting of results.

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4. The classroom teacher’s professional growth

The aim of this study was to implement a particular method of professional

development to better align the outcomes of professional development with change in a

classroom teacher’s knowledge and practices. As described in Section 2.1.4 enduring

sequences of change in this study are considered to depict professional growth. The

particular method of professional development that I facilitated as the education advisor

in this study was classroom based and involved the development and implementation of

a mental computation instructional program. It was important for the classroom teacher

to see that the professional development and in particular the mental computation

instructional program was successful for this study to continue for the intended whole

school year. The classroom teacher needed to see change in salient outcomes in terms of

learning outcomes of her students and learning outcomes for herself as this was the

intention of the professional development.

The mental computation instructional program, being implemented as part of the

professional development, presented a very different way of teaching computation

compared to the classroom teacher’s previous practice. While the classroom teacher

knew and trusted my knowledge and skills and believed in the ideas behind the

program, she needed to be sure that the students were not being disadvantaged through

the development and implementation of this program. To achieve this reassurance the

classroom teacher and I regularly observed and discussed the students’ learning and we

assessed the students’ learning and application of the strategies using tests administered

throughout the year. As these students were in Year 3 they also participated in the state-

wide Year 3 test of mathematics content and skills that took place in August of the year

of the study. This test enabled comparison of the class to state-wide averages, providing

another means to ensure the program being followed was not adversely affecting the

students.

One observable outcome of the professional development was that the students

started to make their thinking and mental computation strategy choices visible by

recording strategy choices by name as part of their responses to the assessment items.

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This was one of the goals of the mental computation instructional program and was a

significant outcome as it allowed the classroom teacher and myself to identify strategies

being used as well as other aspects of the students’ thinking. The classroom teacher and

I were able to discern the students’ efficiency and flexibility of strategy choices when

the strategies being used were identifiable, which along with accuracy, form what

Russell (2000) considered the assessable aspects of mental computation. In terms of the

Interconnected Model of Professional Growth, the Domain of Consequence for the

classroom teacher included the learning outcomes demonstrated by the students. The

results of the students’ learning are presented in Appendix 8.

To summarise the student learning data, it was noted by the middle of the year

that the students displayed an adequate ability to describe strategies they were using and

this allowed for the classification of their strategies used to answer computations on

assessment items (See Appendix 8). The data from the second half of the year showed

that the students made efficient choices and that they chose a variety of computation

strategies both individually and as a class, indicating flexibility of strategy choice. The

students’ accuracy increased across the year, particularly for subtraction items. The

class showed it was above average in all sections of the state-wide test and while this

was not necessarily caused by the study, it reassured the classroom teacher that the

approach being taken was not adversely affecting the learning of the students in the

class. The classroom teacher was satisfied with the students learning outcomes as the

year progressed and was happy to continue with the study for the full school year.

The evidence of the professional growth of the classroom teacher is provided in

this chapter through descriptions of significant events that signified change in the

classroom teacher’s knowledge or practice. These examples of the classroom teacher’s

professional growth have been reported through the use of vignettes as is often the

practice in interpretive research. These vignettes describe the instances demonstrating

change and to the classroom teacher’s professional growth. Each example is

summarised and represented using the Interconnected Model of Professional Growth

(Clarke & Hollingsworth, 2002; Hollingsworth, 1999). The changes in the growth

networks representing the professional growth of the classroom teacher were numbered

to indicate the sequence in which the changes occurred. The significant event described

in the vignette is represented in the Interconnected Model of Professional Growth as a

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blue arrow in the sequence of changes between domains. The following sections outline

examples of the classroom teacher’s professional growth.

4.1 KNOWLEDGE OF MENTAL COMPUTATION STRATEGIES

At the beginning of the study the classroom teacher told me that she had very

little personal knowledge of mental computation strategies. At the initial staff

professional development session she recognised the potential of using mental

computation strategies as a part of the mathematics program for students the age of

those in her class. The realisation of this potential was the major reason the classroom

teacher approached me to investigate programming instruction of such strategies in her

classroom. The classroom teacher recognised that lack of personal knowledge of mental

computation strategies would hinder her ability to make this approach a focus of her

classroom mathematics program.

The classroom teacher’s personal knowledge of mental computation strategies

emerged as an example of her professional growth after a particular sequence of

activities and discussions in the classroom in Term 2 of the school year. I had visited

the classroom to facilitate lessons on the Adjust Two Numbers strategy as part of the

series of lessons that term on the Adjust and Compensate category of strategies. I had

demonstrated the strategy to the students, starting with two single digit numbers, one of

which was a 9, prompting adjustment to make this number 10. Follow-up lessons

involved the addition of single digit numbers where one of the numbers was 8, then

addition of double-digit numbers where one of the numbers ended in 8 or 9. Number

sense activities involving the addition of 10 and single digit numbers were included in

the lessons involving this strategy along with activities working with pairs of numbers

which totalled 10. An event which occurred two weeks after the beginning of this series

of lessons showed that the classroom teacher had developed new personal knowledge of

mental computation strategies. This event is described in the following vignette.

4.1.1 VIGNETTE 4.1: FROM TEACHER TO PERSONAL USER OF MENTAL COMPUTATION STRATEGIES

After participating in the demonstration lessons and conducting follow-up

lessons on the Adjust Two Numbers strategy, the classroom teacher commented that she

had found herself considering mental computation strategies when completing some

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computations herself. Two weeks following the demonstration lesson on this new

strategy the classroom teacher described how she had challenged herself to use the

Adjust Two Numbers strategy when combining student marks for the two content

quizzes for science.

I used the Adjust Two Numbers strategy during the week myself. The students had completed two small quizzes before the holidays for science. I had to add the two scores, which were both out of 25 together for each student. To practise this strategy I made myself add all the pairs of scores together using Adjust Two Numbers. For some of them I probably didn’t need to but I did it for practice. I was quite excited by the end because I knew I was doing it quite well and that this practice would help me be a better teacher of mental computation. (Classroom teacher, via email, Week 8)

4.1.2 DISCUSSION OF VIGNETTE 4.1

The external stimulus for this change in the classroom teacher’s knowledge was

the demonstration lesson and follow-up lessons provided as part of the mental

computation instructional program. The classroom teacher commented in an informal

discussion after the demonstration lesson in Week 7 that she had not been familiar with

the Adjust Two Numbers strategy before the lesson and considered this strategy to be

potentially very useful. In the email in Week 8 reported above, she indicated that she

had been considering the strategy and that she would like to know more about the

application of this strategy beyond the lesson expectations for Year 3 students.

I have been thinking about the Adjust Two Numbers strategy we have been working on this week and last week. This is such a powerful strategy and quite simple. You really need to consider the numbers you are working with to choose the adjustments but then once the adjustment is done the computation is so much easier. Does it work for other operations? I don’t want to go there with the Year 3s but I would like to know for myself. (Classroom teacher via email, Week 8).

I was able to provide examples beyond those which the students in the class had

solved and discussed further applications with operations other than addition and

subtraction by providing examples in a series of emails. This information was reflected

on and the new strategies provided new pedagogical content knowledge for the

classroom teacher. After this consideration the classroom teacher used the strategy

when adding the science quiz scores as reported in the vignette. Reflection on the use of

this strategy and her comment about how her use of the strategy strengthened her ability

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to teach it shows she developed pedagogical content knowledge as well as content

knowledge.

This episode identifies professional growth in the classroom teacher’s personal

knowledge about mental computation strategies. This growth was able to be represented

as connections between the domains of the Interconnected Model of Professional

Growth (See Figure 4.1)

Figure 4.1 Classroom teacher’s professional growth in relation to personal knowledge of mental computation strategies

The classroom teacher watched demonstration lessons on a particular mental

computation strategy (E) and then reflected on this strategy, gaining personal

knowledge of the strategy (1). The strategy was new and of interest so she sought

further input about the strategy by asking questions about examples beyond those being

used with the students (2). Further reflection on the new information led to deeper

knowledge of the strategy (3) which then led to enactment of this knowledge as a

change in personal practice when she used it to add the science quiz marks (4).

Reflection on the use of the strategy added further knowledge and a level of confidence

with this strategy (5).

A comment from the end of year interview provided an insight into her personal

learning about the mental computation strategies in general.

My own maths understanding was limited, as a child I struggled with maths. I had a fairly narrow range of strategies available to me. I think that for me the strategies you have used have broadened my own ability to attempt computations using strategies where before I would have taken a long time over it or have gone back to the traditional way. My new knowledge of the different strategies gives me confidence to use them for calculations in my own life. (Classroom teacher, End of Year Interview, Week 36)

S

PK

E1, 3

2

Enactment

Reflection

4

5


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4.2 BELIEF IN STUDENTS MAKING THINKING VISIBLE

Another goal of the mental computation instructional program and the

professional development for the classroom teacher was the development of ways for

the students to record their thinking. In the literature, reference had been made by a

number of researchers to the benefits of including written recordings of strategies when

students complete mental computations. Treffers (1991) argued that the use of written

work and the recording of student thinking in conjunction with strategy use was to be

encouraged. Klein, Beishuizen, and Treffers (1998) encouraged the use of particular

labels to describe the strategies the students had used. McIntosh (2002) developed a

process where teachers in his study helped their students to develop and refine their

written recordings. These processes enable the identification of the strategies used by

students as well as the identification of the conceptual understandings and

misunderstandings of students.

As the mental computation instructional program was implemented during the

study, the students were encouraged to record their strategy use and to show what they

were thinking. The classroom teacher noticed the benefit of the students recording their

thinking in Week 10 when the class was learning about the Breaking Up Two Numbers

strategy. The following vignette describes the context of this change in the classroom

teacher’s belief in the value of the students recording their thinking.

4.2.1 VIGNETTE 4.2: FROM IN THE STUDENTS’ HEADS TO ON PAPER

The focus early in Term 3 was the Breaking Up Numbers category of strategies.

The students were led through activities to break up either one number of both numbers

in a two-number computation. They were encouraged to show their thinking in writing

using arrows or by rewriting the numbers showing the parts the numbers had been

broken into. Initially the examples provided were addition operations. In week 21

examples involving subtraction were included. The classroom teacher collected the

activity sheets they had been working on that week and noticed the way two different

students had recorded their thinking for one of the questions showed their

misunderstanding of the strategy. Figure 4.2 shows these work samples.

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Figure 4.2 Recording of two students’ thinking showing incorrect application of a strategy

The classroom teacher emailed me describing what she had noticed about the

students recording their thinking and strategy use.

Have a look at the attachment. I have scanned the work from yesterday of Lucas, Trevor, Eleanor and Wayne. I am amazed at what I can see. These children have all made mistakes in their computations but they are similar but different. They all broke the numbers into their place value parts as we have been doing in class. They all did the 30-10=20 the same. The way they dealt with the ones is interesting. Lucas and Trevor added them then took this away from the 20. Eleanor and Wayne subtracted the ones and then took that from the 20. What amazes me is how clearly I can see what they were thinking. (Classroom teacher via email, Week 21).

These student work samples and the classroom teacher’s analysis of them

provided evidence of her recognition of the value of the students showing their thinking

and strategy use.


The classroom teacher was aware that one of the aims of the mental computation

instructional program was that the students would be recording their thinking. They

were encouraged to do this from the start of the year. Data collected during the year

showed a marked increase in the percentage of identifiable strategies used by the mid-

year test, this percentage remaining relatively constant for the remainder of the year (see

Table 4.1).

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Table 4.1 Percentage of items responses where strategies were identifiable per instrument

Instrument Pre-test (%)

Mid-year(%)

Short 1 (%)

Short 2 (%)

Post-test (%)

Addition items 5.7 66.6 58.2 66.8 68.8 Subtraction items 1.1 55.6 62.1 59.6 58.0

This particular event, where the classroom teacher noticed the errors made by a

number of students in the class, was significant for her professional growth. The

particular error noted in Figure 4.2 had also been noted in the research literature

(Beishuizen, 1993) as being common in young students. The classroom teacher

reflected on what this meant for these particular students and analysed the pattern of

errors that she had noticed. This analysis led to her developing further knowledge about

the students but also about the value of having the students record their strategy choices

and their thinking so that as the teacher she could put in place lessons and activities to

correct their misconception.

After the classroom teacher emailed me about her realisation we discussed the

misconception and inaccurate application of this strategy. We developed a plan to help

the class understand why the error occurred and to develop strategies that were less

likely to lead to errors when breaking up numbers in computations involving

subtraction. In the following week’s demonstration lesson, a comparison of the Break

Up Two Numbers Using Place Value with the Break Up One Number Using Place

Value for computations involving subtraction was presented. The students completed

more examples in the demonstration lesson and in the follow-up lessons led by the

classroom teacher. The teacher continued to encourage the students to record their

thinking and strategy choices. The classroom teacher reflected on the difference in an

email later that week.

I think we have all learned a lot this week. I understand how valuable it is to be able to see the students’ thinking and which strategies they have chosen so we can plan activities to strengthen their good understandings and correct their misplaced understandings. I also think the students have seen the value of them recording their strategies so we can help them. Many of them had made different errors in their subtraction problems and I found some of them looking back over what they had done to find their own mistakes. This is so positive. (Classroom teacher via email, Week 22).

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In the end of year interview the classroom teacher was asked her opinion of the

practice of encouraging the students to record their thinking and strategy usage. She

responded positively, indicating her growth in appreciation of this teaching practice.

Does the use of informal recording methods, which the students are being

encouraged to use, give you as teacher, valid and reliable information to judge their

progress?

Yes, I think it [written recording of thinking] gives me more understanding of how they’re thinking. If students do most of their computation mentally it would be difficult to account for what they are learning. It is hard to get around to talk to all the students about their strategies so getting them to record them works well for most of them. I have seen more of what they understand from them recording their thinking than I have with other methods of computation in the past. I also think they can see more of what they understand and can refer back to how they completed other examples and notice the differences during the year. This is quite amazing considering they are only 8 years old. So yes I think this method is valid and reliable and gives me really valuable information about their understandings. (Classroom teacher, End of year interview, Week 36)

This example of another growth network for the classroom teacher is

represented in Figure 4.3.

Figure 4.3 Classroom teacher’s professional growth in relation to students making their thinking visible

The external stimulus (E) was provided by my highlighting the importance of

the students discussing and in particular recording their thinking. The classroom teacher

adopted this practice of encouraging the students to record their thinking and strategy

use in lessons she presented during the year as part of the way we were working with

the students (1). The classroom teacher noticed the value of this practice during Term 3

as reported in vignette 4.2 and reflected on what she noticed about the students’

1

2 3

4

S

PK

E

Enactment

Reflection

5


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learning outcomes (2). The classroom teacher reflected on the knowledge she gained

about the students’ understandings from recordings as part of the mental computation

program (3). The classroom teacher continued to encourage the students to record their

thinking and strategy use during the rest of the year and beyond (4). The classroom

teacher reflected on her belief that this was a valuable practice in the end of year

interview (5).

4.3 INCLUSION OF NUMBER SENSE ACTIVITIES

In the literature, number sense is identified as an important component of mental

computation (Morgan, 1999; Sowder & Wheeler, 1989). Just teaching students

computation strategies would be akin to teaching them procedural algorithms if the

connections to deep understanding of numbers and how they can work to simplify

computations were not included in the instruction. Ensuring that the classroom teacher

understood that the inclusion of lessons or components of lessons focussing on the

development of the number sense understandings associated with the different strategies

was an important component of the mental computation program. This was one of the

goals of the professional development (see Section 3.4.1)

The example of the classroom teacher’s professional growth in relation to the

inclusion of number sense activities along with strategy lessons relates to number sense

understandings associated with the Adjust One Number and Compensate strategy in

Term 2. There were other examples of the use of number sense in conjunction with the

teaching of other strategies. This example is reported to show a particular event when

the teacher noted the importance of number sense understandings.

The number sense understandings required for the Adjust One Number and

Compensate strategy compliment the knowledge students need about how to use the

strategy. The students need to be able to: recognise numbers which are close to a

multiple of ten; adjust one number in a computation to make a multiple of ten, complete

the computation using this adjusted number; and then understand how to complete an

appropriate compensation for the adjustment. This example of the depth of

understanding of numbers and operations required for the successful use of this strategy

was considered an example of number sense and matches definitions described in

Section 2.5.1.

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In Week 14 the demonstration lesson I facilitated focussed on the Adjust One

Number and Compensate strategy with two-digit addition examples. The demonstration

lesson introduced the concept of adjusting one number to make a multiple of ten and

how that made a computation easier to complete. The students were shown how the

answer obtained in this way was not the exact answer but was close to the exact answer.

The follow-up lessons were designed to focus on the compensation aspect of this

strategy. The intention was for the students to decide which way to compensate for their

adjustment to make the answer exact. After the follow-up lesson, the classroom teacher

reported that the students were experiencing difficulty in understanding the way to

compensate for the adjustment to make a multiple of ten, for example, 27+19 was

changed to 27+20 and the students needed to work out whether to compensate by

adding 1 or subtracting 1.

The classroom teacher contacted me to reflect on the first follow-up lessons that

week and expressed her concern that the students were still having difficulty with the

strategy.

I didn’t think this strategy would be as difficult for so many of the children. They can change one of the numbers to a multiple of ten OK and do the new computation but they just can’t work out which way to fix it. I tried to get them to think about whether they had added extra or taken away some when they had changed the number and that doing the opposite would work to fix it but many of them can’t see it. Any suggestions for a way to help them understand this strategy? (Classroom teacher, via email. Week 14).

I suggested a focus on the number sense understanding of how to compensate for

the adjustment to enable the students to find the exact answer using this strategy. To

help the students to see and understand the adjustment and especially the compensation,

I suggested the classroom teacher use a number board as we had with the Counting

On/Back strategies in Term 1. By showing the addition of 19 by adding 10 (move down

one row) and then adding 9 by nine jumps to the right (requiring them to move the next

row down) as well as showing it as adding 20 (moving two rows down) and subtracting

1, the students saw the two strategies resulted in the same answer and they would be

able to see which way to compensate for the adjustment of making 19 into 20 to make

the computation easier. Figure 4.4 shows this use of the number board to help the

students see and understand the compensation for 26+19 with the Count On strategy in

red and the Adjust and Compensate Strategy in blue.

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Figure 4.4 Number board showing two strategies for 26+19

The following vignette describes the classroom teacher’s realisation of the

importance of understanding why and not just what to do.

4.3.1 VIGNETTE 4.3: FROM ‘DOING’ A STRATEGY TO UNDERSTANDING A STRATEGY

The classroom teacher implemented the modified follow-up lesson after

communicating with me about her concern that the students were not able to use the

strategy effectively. She commenced by using the number board to show an alternative

strategy that the students were confident would solve the computation. She then showed

the new strategy for the students to see why the adjustment needed to be a particular

way. The classroom teacher and I discussed the lesson by email after it was completed.

She reported the success of the majority of the class. A day after the lesson and our

correspondence about its success, the classroom teacher sent the following email.

I have realised something about the lessons this week. We both wanted the students to be successful but there was a slight difference in our focus. I was trying to get the students to do the strategy. You were trying to get them to understand why the strategy worked. Both are important but once they could see how to decide on the fix after they had changed the numbers they were able to do the strategy every time. Small change but big difference. Next week when we do subtraction problems I will be looking for their understanding of why again. I am sure the number boards will be used again for some of the students to help them understand the strategy for subtraction. (Classroom teacher, via email Week 14).

The classroom teacher had seen the importance of the students understanding the

number sense concepts that underpin the use of computation strategies.

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The strategy in focus in the week described above required the students to

understand the number sense underpinning the strategy. As the lesson that I was

delivering and the follow-up lessons the classroom teacher led were planned as the year

progressed, we were able to make the type of adjustment to the program demonstrated

during this week of the study. The classroom teacher’s professional growth in relation

to this event is represented in Figure 4.5

Figure 4.5 Classroom teacher’s professional growth in relation to the inclusion of number sense activities to support mental computation strategy development

I facilitated a demonstration lesson and had planned the outline of the follow-up

lessons for the teacher to use later in the week (E). The classroom teacher reflected on

the strategy in terms of her knowledge and understanding of the mathematics (1) and of

her students (2). The follow-up lesson was implemented (3) and the classroom teacher

reflected on the difficulty that some students were having with identifying how to

compensate for the adjustment they had made (4). The classroom teacher considered

what she knew about the strategy and how she could support the students to use it

effectively (5). She contacted me again asking for ideas (6). I suggested the use of the

number board which she implemented in another follow-up lesson later in the week (7).

It was the reflection on this lesson in terms of the student learning outcomes (8) and her

realisation about the slight difference in our focus for the lesson (9) that led to the

classroom teacher changing her practice to include number concepts as well as the

strategies so the students understand why a strategy works and not just how to do it

(10).

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When prompted to reflect on the association of number sense understandings

related to the mental computation strategies in the end of year interview, the classroom

teacher reinforced her learning by recalling this series of lessons.

What have you learnt as part of us working together in relation to the

connection between number sense and mental computation strategies?

Remember the couple of weeks when we introduced the change one number and fix it strategy and there were some students who needed more support. They didn’t understand how to fix the computation after they had changed it. You suggested the number board activity to show them which way to adjust. I realised how a seemingly simple understanding of why a strategy works can disrupt the successful application of a perfectly good strategy. (Classroom teacher, End of year interview, Week 36)

4.4 RESOURCES TO SUPPORT LEARNING

The instructional program and therefore the professional development for the

classroom teacher included the use of particular resources to support the teaching and

learning of the mental computation strategies. The main resources utilised were number

boards and the empty number line. The literature pertaining to the resources and their

use in supporting mathematics and mental computation development was discussed in

Section 2.5.3. The inclusion of the resources in the mental computation program was

reported in Section 3.4.2. Games (e.g., board games, card games and matching games)

were also used to support learning in this study, sometimes from commercially available

resources and at other times developed by the education advisor, to support particular

concepts and strategies being taught to the students.

One example of the use of a resource (the number board) was reported in section

4.3 above. The classroom teacher’s professional growth resulting in change of practice

reported in this section is characterised by two different examples in relation to the use

of resources to support the mental computation instructional program. The first example

involved learning about an alternative use of a resource the classroom teacher was

familiar with, the number board. The second was learning about a resource that she had

not used previously, the empty number line. The Interconnected Model of Professional

Growth is used to show the different growth networks for the classroom teacher’s

learning about the use of these two resources.

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Number board

During Term 1 the standard number board, where the numbers 1-100 or 0-99 are

presented in ten rows of ten, was used to support the students’ use of the Count

On/Back strategy. The structure of the number board assists students using the counting

in tens and ones strategy to solve addition and subtraction problems. Adding ten is

shown by jumps down one row and adding one is shown by jumps to the right. With the

board the counting patterns, especially the counting in tens patterns, are visible to the

students to assist with their computation. Figure 4.6 shows how a standard 1-100

number board can be used to show 34+23 as counting on two tens and three ones. This

strategy relates closely with Breaking Up One Number Using Place Value, but in Term

1 the focus was on using counting to complete addition and subtraction computations.

In the classroom lessons in this study, each student had a copy of the number board and

a transparent counter which they moved around the board to complete the action of

counting on and back to add or subtract

Figure 4.6 A standard number board showing Counting in Tens and Ones to solve 34+23

Difficulties with using this teaching resource can arise when computations

require the bridging of a ten as the students need to count off one end of the number

board to the next row. Figure 4.7 shows an example of a computation requiring bridging

of ten (36 + 26) using a standard number board.

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Figure 4.7 Using a standard number board for addition requiring bridging of a ten (36+26)

Students could count a jump at the end and beginning of both rows or they could

move to the wrong row. Both of these actions can result in an incorrect answer. An

event occurred in Week 5 that provided an example of the classroom teacher’s learning

about another form of number board and its value in supporting students’ learning of

this strategy.

4.4.1 VIGNETTE 4.4: FROM STANDARD NUMBER BOARD TO ALTERNATIVE NUMBER BOARD

To alleviate the potential for errors when students used number boards for

computations involving the bridging of a ten I introduced an alternative number board

to the class in a demonstration lesson in Week 5. This number board maintained the

same structure as a standard number board (rows of 10 numbers) but did not start at

either 0 or 1, positioning the multiples of ten away from the left or the right-most

column. Figure 4.8 shows an alternative number board which begins at 34 and places

the multiples of ten closer to the centre of the board.

34 35 36 37 38 39 40 41 42 43

44 45 46 47 48 49 50 51 52 53

54 55 56 57 58 59 60 61 62 63

64 65 66 67 68 69 70 71 72 73

Figure 4.8. An alternative number board for computations requiring bridging of a ten.

Computations requiring bridging a ten could be completed on these number

boards without students needing to change rows. Figure 4.9 shows the use of the

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alternative number board shown above to solve the computation which requires the

bridging of a multiple of ten (36+ 26).

Figure 4.9 Using an alternative number board for addition requiring bridging of ten (36+26)

The use of the alternative number board provided an opportunity for students to

build on their experience with standard number boards and use the structure of the

board to support their counting in tens and ones, reducing potential errors when

computations needed to bridge ten. The lesson in Week 5 had the students using

alternative number boards and moving counters around the boards to solve

computations. The classroom teacher had never seen alternative number boards and so

this lesson provided significant learning for her as she saw a new use for a resource that

she already valued.


The demonstration lesson using the alternative number board was the stimulus

for this example of the classroom teacher’s professional growth. She was familiar with

number boards but had never seen the alternative version that I had found useful to

overcome the problems with computations requiring the bridging of a ten. Prior to this

lesson I had sent the classroom teacher details of the lesson I had planned for her

information and so she could participate more actively in the lesson. In the lesson plan I

mentioned my intention to focus on addition computations requiring the bridging of a

ten using number boards. I had not included details of the alternative number boards. It

didn’t occur to me that the use of the alternative number boards would be a learning

experience for the classroom teacher. I knew she had used number boards for addition

in the past as we had discussed them prior to the lessons using this strategy. Based on

my experience as a classroom teacher and mathematics education advisor I had

predicted the students could have trouble with computations requiring bridging of a ten

using standard boards. I had included this lesson using the alternative boards to

particularly deal with this potential computational problem. The classroom teacher

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reflected on the use of these boards after the demonstration lesson when she saw them

for the first time.

The number boards you used in the lesson today were different from ones that I have used before. When I saw in your lesson plan that you would be looking at examples that needed regrouping I must say I didn’t really understand how the number board could be helpful with regrouping questions. However these number boards with the tens in the middle fixed that problem. I noticed how some of the students including Michelle and Ian really seemed to understand this strategy when they had struggled previously with additions that needed regrouping. I will be interested to try this strategy with these number boards like this again this week. I will let you know how it goes. Thanks for today I have another useful resource to add to my growing collection. (Classroom teacher, via email, Week 5)

The classroom teacher used the alternative number boards again during the week

in a follow-up lesson where the students completed subtraction computations using the

Counting Back strategy. She reported her success in using this new version of number

boards.

I tried the lesson where the children used the number boards with the tens in the middle again but this time for subtraction. The same process for the lesson worked well, I just listed some problems on the board and they used the number boards to show their calculation by subtracting the tens, then the ones. The children seem to be getting the hang of this strategy using these number boards and the change to subtraction caused very little problems. These number boards are really helpful (Classroom teacher, via email, Week 5).

This example of the classroom teacher’s professional growth is represented in

Figure 4.10.

Figure 4.10 Classroom teacher professional growth in relation to the different use of a known resource - number boards

The classroom teacher observed the demonstration lesson where the education

advisor introduced the students to the use of alternative number boards for the Count On

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strategy for addition (E). She reflected on the alternative number boards (vignette 4.4)

as a new version of the number boards she was familiar with (1). Later that week the

classroom teacher enacted a follow-up lesson using these boards with the students to

solve further computations as reinforcement of concepts and skills from the

demonstration lesson (2). The classroom teacher added this new resource to her

professional repertoire (4). The classroom teacher had learned that the alternative

number boards were a valuable resource for supporting the development of mental

computation strategies (5).

At the end of the year the teacher was asked about the use of number boards in

general and reported that:

I had used number boards with the students before although you have used them more often and in different ways than I have in the past. For example, getting the students to choose the numbers to include on the number boards made them consider the numbers in the problems carefully as well as supporting the strategy where they used the tens and the ones in the numbers in separate steps of the solution. I have used these activities again since the focus on this strategy and they have worked well. I think using the number boards in these flexible ways has helped some of the slower students to understand the concepts and to have the confidence to try some more difficult problems using their number boards or designing part of a number board to suit the question. (Classroom teacher, End-of-year interview, Week 36)

4.5 MEETING THE VARYING NEEDS OF STUDENTS

The class involved in this study included students with a wide range of abilities.

The classroom teacher had a better understanding than me of the abilities of each of the

students simply because she spent the majority of time with them. Consideration of the

range of abilities was a particular emphasis during the development of the mental

computation instructional program. Despite the approach of direct instruction of a

particular strategy being the focus for most of the demonstration lessons, these lessons

were designed to be investigative to allow students to work on problems in ways that

suited them. Through the implementation of a variety of teaching approaches the

education advisor and the classroom teacher were able to consider and cater for the

range of learners. The example of the classroom teacher’s professional growth provided

in this section highlights the utilisation of her knowledge of the students’

understandings and abilities and how a particular practice I suggested allowed her to

incorporate a new teaching technique to cater for the range of abilities.

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During the first week of Term 3 (Week 18) the demonstration lesson I had

planned focussed on the introduction of a new strategy category (Break Up Numbers).

The first lesson of the term included the use of standard and alternative number boards

which the students had used in both previous terms for other computation strategies.

The strategy in focus for the first lesson that term was Break Up One Number Using

Place Value. This strategy related very closely to the Counting On in Tens and Ones

strategy from Term 1. I planned for this lesson to revise this strategy in a way that

would provide a link and introduction to the new strategy. The lesson included an

introduction to the idea of breaking numbers into place value parts (tens and ones) and

then the use of number boards to show the addition of the tens as jumps down a row and

ones by moving to the right using alternative number boards as described in detail in

Section 4.4 above.

In the demonstration lesson, the students were successful performing addition

and subtraction computations by breaking one of the numbers into tens and ones and

moving their transparent counters around different number boards. For the last activity

in the lesson, each student was given a handout depicting four number boards, one

standard and three alternative, each starting on a different number. One of the

alternative number boards started on 76 and so bridged one-hundred to allow for

computations involving one three-digit number. The students were guided to choose a

number board they thought would be appropriate for a given computation. This

culminating activity was successful with the students generally being able to make

choices of number boards based on the numbers in each computation.

4.5.1 VIGNETTE 4.5: FROM DIRECTED USE OF LEVELLED ACTIVITIES TO STUDENT SELECTION OF LEVELLED ACTIVITIES.

I had prepared a series of three worksheets for the classroom teacher to use that

week for a follow-up lesson. The worksheets varied in difficulty so as to cater for the

range of abilities in the classroom. The easiest worksheet focussed on addition of

multiples of ten only, the most difficult worksheet included a three-digit number and

computations that extended just beyond the given number board so as to challenge

students to extend or visualise the numbers beyond the given board. I explained the

aims and expected difficulties of each worksheet with the classroom teacher in the notes

accompanying the lesson.

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The classroom teacher implemented a follow-up lesson using the worksheets of

varying levels of difficulty by choosing a worksheet she believed would be at an

appropriate level for each student and asking them to complete the sheet after a short

general introduction. The classroom teacher commented on how the follow-up lesson

had gone in an email later that week.

To be honest I don’t think this lesson went all that well. I revised the strategy by giving the children some examples to complete on the number board in their scrapbooks and then gave each of them one of the worksheets you had sent. The different worksheets for different students caused quite a bit of interest and questions about who had which one to start with. Quite a few finished their worksheets with no problems. Many of those who had the most difficult one were successful. Paul, Mark, Owen and Bridget for example had no problems at all and finished quite quickly. The easiest worksheet was a little too easy and so I gave Rachel, Lisa, Michelle and Warren an extra copy of the middle sheet to work on together. There were quite a few who struggled with their assigned sheets too. I can see what you were thinking with the different worksheets as there is such a range of ability in this class but I actually found it a bit difficult to manage with so much going on. (Classroom teacher, via email, Week 18)

In a reply email to the classroom teacher I suggested an alteration to the way the

various worksheets were used to meet the needs of the variety of learners in the

classroom. My suggestion was for the students to make the choices about which

worksheet they wanted to do rather than have them assigned. I also suggested copying

the different worksheets on different coloured paper so the students could see which

was which and didn’t need to ask as many questions at the start of the lesson. I also

suggested the classroom teacher copy some extras of the middle and upper worksheets

so students were able to return for another sheet if they completed one before the end of

the lesson. The idea was to see if they could manage their own time to free the

classroom teacher to help those who needed her but also for them to make decisions

about their ability level when choosing a worksheet.

The classroom teacher implemented the lesson as I suggested using a slightly

varied set of the worksheets from the previous lesson. The computations on the

worksheets were changed and a fourth more challenging worksheet was added to the

selection. The classroom teacher emailed again after this lesson reporting the changes.

The lesson today was very interesting for me. I wasn’t sure how letting the children choose their own worksheet would go. I must say I was pleasantly surprised. Several of the choices interested me. Michelle chose a worksheet that I thought was much too hard for her. It was really but I am so pleased she decided to choose something challenging as her confidence in

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mathematics has not been high all this year. Another interesting choice was Wayne. He is a very capable chap but chose a worksheet which he probably knew he would get all right. I thought he might be influenced by Paul and Mark who chose the most difficult worksheet to start with but he stuck to his own ability rather than theirs. I was pleased to that he then went on to try the most difficult sheet. This lesson has taught me a different teaching approach and I have seen more about the students abilities than I did by a choosing the worksheet for them. (Classroom teacher via email, Week 18)

The plan was to cater for the range of abilities in the classroom by differentiating

the level of complexity of the worksheets. By acting, reflecting and re-planning the

classroom teacher and I found a way of working that better met our aim and involved

the students in making learning choices for themselves.


The classroom teacher’s learning described in this vignette related to her trying

and being successful with a teaching practice she had not used before. We had

discussed the range of abilities in the class in relation to particular mathematical

knowledge as well as in relation to the different computation strategies we were

teaching as part of the study. The demonstration lesson at the beginning of this week

focussed on a new computation strategy and because it linked closely to another

strategy had been taught earlier in the year, it served as revision of this strategy as well.

This highlighted the range of abilities in the class as well as providing a means for

scaffolding learning by returning to a strategy that was familiar for those who needed

support as well as introducing the new strategy variation for the more capable students.

The opportunity for the way of differentiating the learning during this week was

not planned ahead but developed as the classroom teacher and I corresponded after the

initial demonstration lesson. I prepared the range of worksheets as a way of showing the

connection to the previous strategy and beyond. The classroom teacher used these in a

way that made sense to her. I suggested a way that made sense to me which she tried

and found very helpful. In the end-of-year interview I asked the classroom teacher about

how the mental computation instructional program met the varying needs of the

students in her class and she recalled this lesson, showing it had a lasting impact on her

teaching practice.

This example of the classroom teacher’s professional growth is represented

using the Interconnected Model of Professional Growth in Figure 4.11.

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Figure 4.11 Classroom teacher’s professional growth in relation to catering for the range of student abilities

The demonstration lesson by the education advisor (E) introduced the new

strategy (Breaking Up Numbers) and made links to the strategy used in Term 1

(Counting On /Back). The variety of worksheets was provided as part of the materials

for the follow-up lessons. The classroom teacher reflected on the mathematics involved

in each worksheet (1) and on the learning outcomes of the students in the class after the

demonstration lesson that week (2). She implemented the follow-up lesson, assigning

one worksheet to each student (3), and reflected after the lesson on the students’

learning (4) and her knowledge of the teaching approach she had used (5). The

classroom teacher asked me for some ideas of how to challenge the more able students

as well as supporting the others (6) and then implemented the lesson where the students

made the choices about the worksheet they would complete themselves (7). The

classroom teacher reflected on the students’ learning in this lesson (8) and on her new

knowledge of the teaching approach used in the lesson (9).

4.6 SUMMARY

The classroom teacher participated in the professional development to change

her knowledge and practice in regard to mental computation. She understood that the

new Mathematics syllabus included mental computation and that this was an aspect of

mathematics education which needed to be updated personally. The focus on mental

computation and some of the teaching strategies included in the professional

development during the year provided opportunities for the classroom teacher’s


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This chapter explained the professional growth of the classroom teacher by

providing particular examples of change in the domains of the classroom teacher’s

personal and professional worlds. The examples reported in this chapter provide

evidence of how change in one domain led to ongoing change in the teacher’s world. It

was ongoing change of this nature that Hollingsworth (1999) described as professional

growth as compared to teacher change. Each of the vignettes described in this chapter

provides details of a significant event leading to a particular change in the classroom

teacher’s knowledge and practice as a result of her being involved in the professional

development.

The examples of the classroom teacher’s professional growth reported in this

chapter were chosen to highlight the range of the learning she experienced during the

year of the study. Some examples of her professional growth, like the change in her own

knowledge and use of mental computation strategies, related to growth in her personal

knowledge. Other examples of the classroom teacher’s professional growth, like the

importance of linking number sense understandings with mental computation strategies

and new ways to use teaching resources like the number boards, demonstrated growth in

her teaching practice. Professional growth resulting in change in her belief in the value

of students making their thinking visible when completing computations using mental

strategies was also described in this chapter. Each example of the classroom teacher’s

professional growth was represented using the Interconnected Model of Professional

Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999) that depicted the growth

network and how change in different domains of the classroom teacher’s world were

mediated to influence change in other domains through enactment and reflection. The

significant events described in the vignettes were able to be identified as part of the

growth networks. The classroom teacher described this professional development as the

“best professional development I have had in terms of my learning and my teaching”

(Classroom teacher, end of year interview, Week 36).

The professional growth of the education advisor is reported in a similar way,

through the identification of examples of change and growth, described using vignettes

and represented using the modified version of the Interconnected Model of Professional

Growth. These aspects of my change are reported in Chapter 5.

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5. Education Advisor Professional Growth

The aim of this study was to implement a particular method of teacher

professional development and identify the professional growth experienced by the

classroom teacher and myself as the education advisor facilitating the professional

development. As described in Section 3.4, this professional development for the

classroom teacher included two components: the development and implementation of

the mental computation instructional program and the ongoing access to and interaction

with me in my role as education advisor for support. The success of the professional

development was demonstrated by the examples of the classroom teacher’s professional

growth reported in Chapter 4.

This chapter portrays my professional growth as the education advisor. During

the year of this study I continued to work as an education advisor supporting a large

number of schools as well as working in the classroom of the teacher in this study. The

data in this chapter are presented as a series of examples of my professional growth

resulting in change in my knowledge or practices. Each example of my professional

growth portrayed in this chapter is presented with a background description of the

context and activities that led to the change followed by a vignette describing the

significant event that triggered a sequence of changes. Each vignette describes a

different event and is followed by a discussion of the change with reference to

reflections by me, the classroom teacher and other contributors to my growth as an

education advisor.

Each example of my professional growth is summarised using the modified

Interconnected Model of Professional Growth to represent the sequence of changes. The

changes in each growth network were numbered to indicate the sequence in which the

changes occurred. Five significant changes to my knowledge or practice are presented

in chronological order, indicating the progression of my development as an education

advisor over the year I worked in the classroom.

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5.1 CO-OPERATIVE TEACHING

A variety of terms have been used in the literature to describe teachers working

together in a classroom. Tobin and Roth (2006) detailed what they saw as the difference

between team teaching and co-teaching. This difference was described in Section 2.2.2.

Each week I visited the classroom and taught one lesson on a particular mental

computation strategy, or on a number sense understanding, sometimes utilising a

particular resource. In the early stages of the study my role was as a presenter. I visited

the classroom and provided demonstration lessons and advice while the classroom

teacher observed. After the first two demonstration lessons the classroom teacher started

to assist students as needed during my lesson. By Week 5 of Term 1, I was sending the

plans of my lessons to the classroom teacher prior to the lesson so she knew in advance

what the strategy, concepts and activities in the demonstration lesson would be. With

this information the classroom teacher began to participate in the lesson rather than just

observing. By the seventh week of Term 1 the classroom teacher was involving herself

more in the teaching of the lessons. We were working as team teachers (rather than co-

teachers) in the way explained by Tobin and Roth (2006).

The classroom teacher facilitated follow-up lessons based on the concepts and

strategies in the demonstration lesson each week in the days following my visit. The

follow-up lessons were planned for the teacher, at least in the first term. Early in Term 2

an event occurred that triggered a change in the way we were working together. The

vignette below describes my professional growth in relation to the conduct of

classroom-based professional development.

5.1.1 VIGNETTE 5.1: FROM PRESENTER TO COLLABORATOR

When I started this study I had a mind-set that I had comprehensive knowledge

of concepts, strategies and teaching practices in relation to mental computation and that

I was going to assist the classroom teacher to develop some of these understandings and

practices by her watching me teach and through discussions and reflection after the

demonstration lessons. I was not conscious that I had this way of thinking until in Week

10, three weeks into the second term, when the classroom teacher emailed me for

guidance. Instead of asking for ideas to try she suggested a change to the way we were

teaching so I could help her to teach the concept rather than me coming to do it for her.

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When you come in next Tuesday could we go through the Adjust Two Numbers strategy for subtraction with the class again please? I tried to explain it again today but the students are still confusing it with the same strategy for addition. You explained the difference last week but I think they need it explained again, and I think I do too. Would you mind if I started the lesson the way I did today and you step in to help clarify anything I am not explaining or doing clearly. I would like you to see what I am doing so you can help me and we can both help the children to understand the strategy. I also think it will be nice for the children to see us working together. I would like them to see that I asked for help with something I was not good at and that by working together we solved the ‘problem’ of helping them to understand. Being able to ask for help be such an important lesson for this age group and I would like to gain skill in explaining some of these strategies more clearly. (Classroom teacher via email, Week 11).

I visited the classroom in Week 12 and the classroom teacher and I co-taught the

lesson, working and learning together with the students. The classroom teacher started

the lesson and we worked together to help the students to develop an understanding of

the difference between using the Adjust Two Numbers strategy for addition and

subtraction. I recognised that the classroom teacher was an experienced teacher and

although she was the recipient of the professional development her learning did not

need to come solely from watching and reflecting on my practices. By working with me

rather than just observing my practices she was involved in constructing her own

learning. During this lesson I was very conscious of how I was teaching and explaining

the concepts, more so than I would have been had I just taught the lesson myself. I also

learned a valuable insight into the benefits of recipients of professional development

being able to shape the learning to suit their needs by asking for help. As the facilitator

of the learning I was able to be more tuned to the needs of the learner (the classroom

teacher) rather than just presenting the knowledge and practices.


Prior to this study and due to my personal interest in mental computation as a

topical issue in mathematics education I developed an extensive knowledge of mental

computation strategies and related concepts and issues obtained through reading the

literature, involving myself in discussions with colleagues and attending and presenting

at mathematics education research conferences and teacher conferences. I knew that the

classroom teacher lacked personal knowledge about mental computation strategies as

well as ideas for teaching the strategies as a replacement for the traditional algorithms

that she had been teaching for many years.

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My initial approach to assisting the classroom teacher to develop both

knowledge and teaching practices to implement a change to mental computation

strategies in her classroom was to demonstrate my knowledge of the content and

pedagogy when I visited the classroom each week. I initially believed my role to be that

of expert and that the classroom teacher would learn by observing me teach and through

our reflections after the lessons. The email from the classroom teacher suggesting a

different approach to the demonstration lessons led me to reconsider my role. The

classroom teacher recognised the benefit of co-teaching rather than just observing me

teaching. This was not a way of working I had considered at the beginning of the study.

Also her desire to teach the lesson so the students could see that she had solved the

problem experienced in teaching the Adjust Two Numbers strategy for subtraction

previously added significance to this event. This event led me to reflect on my role as

education advisor. I made the following entry in my field notes after the classroom

teacher had suggested we teach together.

The email from the classroom teacher suggesting we teach together has me considering my role in this study. [Classroom teacher’s name] is an experienced teacher who is working hard this year to understand and try to teach some very different methods of computation as well as new teaching approaches and resources. I haven’t given her enough credit as an experienced teacher. She wants to experience teaching this lesson so the students understand and I can help her with that. Also she has the best interests of the students at heart and by suggesting we re-teach the concept lesson for Adjust Two Numbers for subtraction together as she wants the students to learn not only the maths but an important life lesson about asking for help when you need it. I was concentrating on the students learning in my role as teacher and had lost some sight of the classroom teacher as learner in this professional development (Education Advisor Field Notes, Week 10).

The classroom teacher began the lesson in Week 12 and we worked together to

help the students to understand the difference when using the Adjust Two Numbers

strategy for subtraction rather than for addition. The classroom teacher had a greater

depth of knowledge of the individual students and their learning needs and I knew the

connections and concepts relating to this mental computation strategy. This particular

lesson was the first time the classroom teacher and I had taught together rather than me

doing the teaching and the classroom teacher assisting me. During the lesson we utilised

each other’s strengths, not individually as in team-teaching, but collaboratively as

needed to make the lesson successful. I took an opportunity during the lesson to revise

the concepts of addition as joining and subtraction as difference to help the students see

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why the strategy was used in a different way for each of these operations. The

classroom teacher was aware that the students could remember the previous lesson

when she had not been able to make the difference between the different applications of

the strategy clear to them. What I realised was that by working together, especially with

her leading the lesson, we showed the students that it is a good idea to seek assistance

when you realise that you need help and that someone else can provide that help. By the

end of the lesson we saw successful application of this strategy for addition and

subtraction by many of the students.

After this lesson I returned to the literature to investigate research in relation to

co-operative teaching. It was this research that led me to the significant work of Tobin

and Roth (2006). The analogy proposed by Tobin and Roth (2006) of co-pilots flying

together was appropriate for this lesson where we taught and learned together. The

power relationship between us was equally distributed, not just focussed on me as the

expert. The classroom teacher learned how to explain the difference in the application

of the strategy. I learned that co-teaching was indeed an effective way of working as an

education advisor, and the students learned how and why the Adjust Two Numbers

strategy needs to be applied differently for subtraction problems as compared to

addition problems.

The classroom teacher’s reflection after this lesson notes that she learned from

this experience.

Thanks for the lesson today I think it worked really well. I can see the difference much more clearly between the two versions of this strategy and I think the students do as well. But also I can see how we were able to teach about this difference by looking at the concepts of addition and subtraction more generally. The children saw that my difficulty in explaining the difference between these strategies last week was overcome by us working together. I will work on this strategy with the children some more later this week – for my own practice as well a for the children’s learning. (Classroom teacher, via email, Week 12)

The practice of co-teaching was not initially planned to be part of the study. The use of

this as a method of professional development emerged throughout the study, stimulated

by the event described in the vignette. This experience triggered a change in my

practice as an education advisor. From Week 3 of Term 2 I worked to include the

classroom teacher in the teaching as much as possible. In the end-of-year interview the

classroom teacher stated that:

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I have thoroughly enjoyed this year and I have learned so much both personally and professionally. I wish I could have you as a teaching partner all the time. This has been the best professional development I have ever had in terms of my own learning and my teaching. Thank-you. (Classroom teacher, end of year interview, Week 36)

This example of my professional growth as an education advisor is represented

using the Modified Interconnected Model of Professional Growth (see Figure 5.1).

Figure 5.1 Education advisor professional growth in relation to co-operative teaching.

The email from the classroom teacher (vignette 5.1) provided the external

stimulus for this example of my professional growth (E). I reflected on the email

realising that my approach to the demonstration lessons had been very much that of me

as expert presenting the lesson expecting that the classroom teacher would learn by

watching (1). The classroom teacher and I taught the lesson on the Adjust Two Numbers

strategy for subtraction together meeting the classroom teacher’s learning needs as well

as the learning needs of the students (2). I reflected particularly on the classroom

teacher’s learning in this lesson based on her email reflection and my own observations

and thoughts about it (3). The classroom teacher’s email caused me to reflect on her

learning needs in this study and to realise she was a competent experienced teacher, not

just someone needing to learn from me demonstrating my expertise (3). This reflection

led me to develop new knowledge about the added benefits of using demonstration

lessons as part of teacher professional development, especially when the classroom

teacher can be involved in co-teaching the lesson (4). I returned to the literature to

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research co-teaching (5). The result of this significant event in the study was my

changed practice in relation to co-teaching. I prepared an outline for each demonstration

lesson and sent that to the classroom teacher prior to the lesson so she was able to

contribute to the teaching of the lesson. Through my research and the differentiation

between team teaching and co-teaching, I worked to ensure we worked together rather

than just being two teachers in the same classroom (6).

5.2 CO-OPERATIVE PLANNING

My practice prior to this study had included planning lessons and units of work

on behalf of teachers and providing these unit and lesson plans for the teachers to

implement. An example of this aspect of my role was the preparation, publication and

dissemination of a series of units of work for teachers to use as part of the Consistency

of Teacher Judgement process that had been planned to focus on Mathematics. Teachers

implemented these units of work and gathered student work samples that supported

professional discussions about the different levels in the syllabus on the student-free day

in October when teachers from several schools met. The aim of this process was

consistency in these discussions so we felt that by planning these units for the teachers

they would be aligned more accurately to the concepts of the new Mathematics

syllabus. This preparation of material for teachers was a major focus of my role outside

of this study, so I found myself working this way during the study as well. My change

to co-planning with the classroom teacher in the study provides another example of my

professional growth as an education advisor.

At the beginning of the year I planned the demonstration lessons and

implemented the plan with the teacher observing and participating more as a teaching

assistant. I sent the classroom teacher a copy of my plan for her information and as a

contribution to her learning about the mental computation strategies and effective

teaching practices. As explained in Section 5.1 my role changed from presenter to co-

teacher early in Term 2. In relation to the mental computation instructional program, I

believed that the development of this program was my task as I had the in-depth

knowledge of the strategies and the connections to number sense activities that were

important for a comprehensive program. During Term 2, not long after I had realised

the benefits of co-teaching with the classroom teacher, another event occurred that

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triggered a change in my belief about the role of an education advisor in relation to co-

planning.

5.2.1 VIGNETTE 5.2: FROM PLANNING FOR TO PLANNING WITH

During this study the mental computation instructional program was written as

an initial overview for each term, listing the strategies to be focussed on but with no

week to week detail. I planned the demonstration lessons and the follow-up lessons for

the classroom teacher to implement after my lesson each week. In the eighth week of

Term 2 (Week 14 of the study) I was very busy with other aspects of my role and

needed to travel out of Brisbane to visit schools. I was not going to be able to visit the

class for the demonstration lesson and just did not have time to prepare details of what I

intended to do or details of possible follow-up lessons for that week. I sent an email to

the classroom teacher apologising for my lack of preparation and received the following

reply.

Don’t worry about it [not being able to do the demonstration lesson the following week]. I can plan an activity using what we did last week. I will try subtraction examples to follow on from the addition ones from last week and will use the number boards the same way you did. I am sure we will cope. You worry about what you have to get organised for your trip and we will see you next week. (Classroom teacher, via email, Week 14)

I realised that the learning in the classroom in relation to the mental computation

program did not rely solely on me. The classroom teacher was experienced and had

developed familiarity with the way we had been working and with the content and was

happy (and I think quite enthusiastic) to plan and implement a lesson for herself.


My ideas about the planning of the mental computation instructional program

came from my research and discussions into mental computation strategies and related

number sense understandings over several years prior to the study. This research

involved reviewing the literature, consultations with colleagues and attendance and

presentations at conferences (as described in 5.1.1). As a result of this research I

developed a Strategy Categorisation Framework which listed what I considered major

categories of mental computation strategies that were worthy of inclusion in a primary

school computation program (see Section 2.5.2). This framework was used as the

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overall guide to the development of the mental computation instructional program

where we chose to focus on one of the main strategy categories each term (see Section

3.4.2). As the study began I planned demonstration lessons and follow-up lessons to

provide a progressive development of concepts and skills for the strategy in focus for

the first term. After each lesson the classroom teacher and I discussed the learning from

that lesson – the students’ as well as the classroom teacher’s. An example of such

communication is provided below. I had reflected on one particular lesson, voicing my

concern about the students’ varying abilities to see that counting in multiples of ten

could be used as a strategy for adding multiples of ten.

I think you are right about them not making the connection between counting in tens and adding tens. Michelle, Rachel and Warren were struggling with the counting sequence but others that I thought would see the connection, like Amber, Wayne and Tim did not seem to get it either. I hadn’t realised the usefulness of counting in tens but not starting at a multiple of ten when adding multiples of ten. You are teaching the class and me at the same time. (Classroom teacher via email, Week 5).

The reflections on the lessons guided my preparation and implementation of the

next demonstration and follow-up lessons. I did not change my practice until Week 14

when Vignette 5.2 triggered my reflections, as recorded in my field notes below:

I realised today that my approach to the professional development during this study so far has been somewhat condescending. I have assumed the classroom teacher knows very little about mental computation and that she needs me to help her with the knowledge and the teaching practice to go with it. She has more years of teaching experience than I do and although I have had some particularly rich learning experiences in the past few years as an education advisor I do not know it all. I have realised that I am actually using a teaching practice which I readily discourage. The ‘student as an empty vessel needing to be filled’ perception of teaching is against my personal teaching philosophy but effectively this is what I have been doing so far in this study. Until today I didn’t realise this. I realise now that if I not only co-teach with the classroom teacher but co-plan with her as well she is likely to learn more than by just watching me and following my plans. (Education Advisor Field Notes, Week 14)

My reflection on this event led to a change in my understanding of myself as an

education advisor. I did have knowledge and skills worth sharing with classroom

teachers. I could plan well-structured and sequenced activities for them to implement.

The planning helped the lessons to be successful but did not lead to the depth of

learning for the classroom teacher that I had in mind when I set out to implement this

way of working as an education advisor. After this week I planned in less detail and

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discussed my ideas with the teacher more, usually via email for convenience, before my

demonstration lessons. I noticed that the classroom teacher’s communications with me

after my visits to the classroom started to include more suggestions about what could be

part of the next demonstration lesson. The co-planning seemed to lead to a greater depth

of understanding about the mental computation strategies when she was considering

how to teach them. The classroom teacher was also better prepared to co-teach the

lessons with me when we had both participated in the planning of the lessons.

This example of my growth as education advisor resulting in a change in my

knowledge can be represented using the modified Interconnected Model of Professional

Growth (see Figure 5.2).

Figure 5.2 Education advisor professional growth in relation to co-operative teaching

The email from the classroom teacher indicating that she was comfortable with

planning and implementing a lesson to continue the mental computation program

provided the external stimulus for this example of my professional growth (E). I

reflected on my role as education advisor and noted that I was working in opposition to

my personal teaching philosophy (1). I changed my practice to include more co-

planning of activities for the students with the classroom teacher (2). I reflected on the

classroom teacher’s learning and noticed changes in her contribution based on her

understandings (3). I continued to include the classroom teacher in the planning of the

demonstration and follow-up lessons for the remainder of the study (4).

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5.3 BULIDING RELATIONSHIPS

In my practice as an education advisor prior to this study I had rarely developed

personal relationships with teachers for whom I provided professional development. In

some schools I developed a connection with some of the curriculum leaders who I saw

more often than the classroom teachers. During this study I had the opportunity to

develop a close working relationship with the classroom teacher as I saw her every

week and we continually emailed back and forth, sharing ideas and reflecting on

activities in the classroom. I had not considered the building of relationships as a

significant part of my role prior to the study. I realised that I would be likely to develop

a closer relationship with the classroom teacher in the study by spending time and

corresponding with her, but I did not this consider this to be a significant aspect of the

study or of my role as an education advisor. An event occurred during Term 2 that

highlighted the value of building a relationship with the teacher and the students. This

event is depicted in the following vignette.

5.3.1 VIGNETTE 5.3: FROM OUTSIDER TO INSIDER

This study was conducted in a Catholic school where part of the religious

education program involved each class preparing and leading a mass or liturgy for the

whole school at least once during the year (depending on the age of the students). The

class I was working with had their turn to plan the school gathering in May. The class

invited me to attend the liturgy they prepared. I realised that I wasn’t just a teacher who

visited the class once a week to do some maths lessons but that they considered me to

be a part of the culture of the class and they wanted to include me in the liturgy they had

planned for the whole school.


Part of my personal knowledge about the profession of teaching is that it relies

on interpersonal contact with a range of people. In the classroom, relationships develop

amongst the students and between the students and the teacher. In a school teachers

develop relationships with their colleagues and the parents of their students. As an

education advisor I had developed effective working relationships with the other

members of the curriculum team but only limited relationships with teachers in schools,

mainly due to the lack of time spent with particular individuals. What does develop,

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assuming the education advisor does an effective job, is a professional relationship

where the education advisor is known and respected. This is not like a working or

personal relationship. So I began this study knowing that I would be likely to get to

know the classroom teacher and the students but I hadn’t considered the impact on the

study of developing a professional or personal relationship.

During the study I visited the classroom and worked with the students each week

and corresponded with the classroom teacher, often several times each week. It was

inevitable that I developed familiarity with the classroom teacher and the students. I

noted this growing familiarity in my field notes in Week 9

I am finding myself really looking forward to Tuesday mornings when I visit the Yr 3s. When I arrive now I am often greeted by a cheerful “Here comes Mrs H” which always makes me smile. I am enjoying preparing the lessons for the class and I am starting to consider individual students in my planning as I get to know them. (Education Advisor Field Notes, Week 9).

I also noticed as the study progressed that the emails between the classroom

teacher and myself became less formal, indicating her increasing familiarity with me.

While the relationship was not especially personal, it became more casual and familiar

as the year progressed. When the class invited me to attend the liturgy they had

prepared for the school I realised that the students and the classroom teacher accepted

me as part of their class and school life and wanted me to participate in this important

event in their class calendar. I was not just someone doing a job facilitating lessons for

the students and professional learning opportunities for the classroom teacher. I had

developed a more meaningful relationship with the members of the class.

I had been an education advisor for eight years prior to this study without my

own class to work with. The invitation to participate in the class celebration made me

realise that this was almost like having my own class again. I had not considered how

the role of education advisor lacked much of these interpersonal aspects of teaching. I

realised how I missed having ongoing input into the development of a particular group

of students and planning lessons and activities specifically for their needs. After

attending the class liturgy I noted a personal reflection as part of the field notes making

a particular connection to the importance of building professional and interpersonal

relationships in the profession of teaching.

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Today I attended the liturgy planned by the class in this study. I was quite touched to receive the invitation to attend from the classroom teacher and especially noted that she had mentioned that the students were keen for me to attend. They really seem to consider me a part of the class even though I only visit once per week. (Education Advisor Field Notes, Week 12)

After attending the liturgy I changed my approach to my visits. I was more

conscious of the interpersonal connections and felt that the students and classroom

teacher had accepted me as effectively a member of their class. I felt less like a visitor

and more that I belonged, even if I only came once a week. This experience added to

my perception of effective education advisor practice and while I recognise that this sort

of relationship develops over time, I came to believe that much more can be achieved

once the perception of my role changes from outsider to insider.

This example of my professional growth can be represented in the form of

Figure 5.3.

Figure 5.3 Professional growth of the education advisor in relation to developing relationships

The classroom teacher sent me an email inviting me to attend the liturgy planned

by the class as part of the school religious gatherings (E). I reflected on the deeper

meaning of the invitation realising the connection that had developed between myself

and the classroom teacher and the students and how I had not experienced such a

connection since working as an education advisor (1). I noticed that the

communications between the classroom teacher and myself had become less formal and

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I reflected on this in terms of the classroom teacher’s willingness to ask questions and

her learning (2). I continued to communicate with the classroom teacher noting a more

social and less formal tone to the communications (3). I reflected on my realisation that

the development of relationships was an important aspect to the role of education

advisor when time was available for them to develop.

5.4 WORKING IN CLASSROOMS

The teacher professional development literature cites many authors who believe

that conducting classroom-based activities is one of the key principles of effective

teacher professional development (Clarke, 1994; Guskey, 2003; Owen, Johnson, Clarke,

Lovitt & Morony, 1988; Mewborn, 2003). I conducted this study in one classroom to

trial the provision of classroom-based professional development as an alternative to

centrally organised “one-off” sessions. I believed that individualised professional

development activities for teachers would provide better learning opportunities,

however I also realised that the provision of centrally-offered professional development

was a more efficient use of time and resources. The team of education advisors with

whom I worked with had begun to offer school-based professional development

activities, most often presented in a staff meeting for an hour or an hour and a half after

school. The audience was smaller and the message and activities were able to be

discussed in terms of the local context. By initiating this study I saw an opportunity to

trial classroom-based professional development. I did recognise that this method was

unlikely to be transferable to the large number of schools I was supporting. During the

year of this study an opportunity to work differently presented itself and, because of my

involvement in the study, I believe I reacted differently to the request. The following

vignette describes this opportunity.

5.4.1 VIGNETTE 4: FROM CENTRAL PROFESSIONAL DEVELOPMENT TO CLASSROOM-BASED PROFESSIONAL DEVELOPMENT.

In the year of the study, much of the work for the team of education advisors I

worked with was aimed at supporting Consistency of Teacher Judgement (CTJ)

activities across all the schools we supported. The focus of this activity in the year of

the study was the strand of Patterns and Algebra in the then new Mathematics syllabus.

As part of my role I had developed a staff meeting-length professional development that

included background information and details of activities that could be used in different

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year-level classrooms. I spent much of my time when not working directly on this study

visiting many schools to ‘do the CTJ session’ in after-school staff meetings.

About halfway through the year of the study a school contacted me requesting

this session at their school. It occurred to me when I was speaking with the school

principal on the phone that I could combine some classroom-based professional

development with the school-based staff meeting. I offered to work with teachers in the

school in a way similar to how I had been working with the classroom teacher in my

study. I would implement demonstration lessons in classrooms during the day to show

some of the activities I planned to discuss at the staff meeting that afternoon. Then in

the staff meeting we would discuss the concepts and the lessons that had been

completed during the day. The principal thought this was a good idea and so I enacted

this method of professional development for that school. I was able to present the

content at the staff meeting with examples from the different lessons conducted during

the day, supported by the classroom teachers. I noted that, when compared with

previous deliveries of this same professional development session, the teachers were

more engaged in discussions relating to the content and participated by recounting the

activities from the classrooms and the students’ reactions.

5.4.2 Discussion of Vignette 5.4

The external stimulus for this change to my practice began with my experiences

facilitating demonstration lessons as part of this study. When the opportunity arose to

work with other teachers in a similar way, I decided to try the method of professional

development that I was finding effective in the context of the study.

The vignette depicted above indicates a significant event in my work as an

education advisor when I took the successful practices I had experienced in this study to

other classrooms. After this positive experience of using the method of school-based

professional development from this study, I started to offer to work in classrooms more

regularly as an aspect of my role beyond the study. Instead of visiting a school for a

staff meeting on a particular topic I offered to be at the school all day, working in

classrooms with the students and teachers on the topic of the afternoon’s staff meeting.

Teachers were generally more engaged in the professional development and the

following meeting when there was an opportunity to share and discuss the outcomes of

the lessons conducted during the day. In some cases after-school staff meetings were

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not possible. In those cases I discussed the lessons with the classroom teachers when I

could, often over lunch in the staffroom or on playground duty.

Evidence of the resultant change in my regular practice to include demonstration

lessons was noted in my Performance Review at the end of my contract in the year after

the study was concluded. A commendation from this report concerned the benefits seen

by classroom teachers and principals of the inclusion of demonstration lessons as part of

a more classroom-based focus of the professional development. This was noted as an

aspect of my practice as an education advisor that differed from the practice of other

education advisors in the curriculum team.

Of particular mention are Judy’s demonstration lessons which have been universally acknowledged and commended by all involved as not only modelling good teaching practice but as also having considerable positive impact on student learning outcomes in mathematics. (Performance Appraisal Panel Report, 21 August 2007, p. 1)

This example of my professional growth is represented in Figure 5.4 using the

modified version of the Interconnected Model of Professional Growth.

Figure 5.4 Education advisor professional growth in relation to working in classrooms.

The classroom-based method of professional development utilised in this study

formed the external stimulus for this example of my professional growth (E). I enacted

the practice of classroom-based professional development followed by discussion and

reflection in a staff meeting in another school during the year of this study (1). I

reflected on the learning outcomes demonstrated by the classroom teachers at the

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school, compared with teachers experiencing just the same staff meeting that I had

facilitated in other schools that year (2). Whenever I had the opportunity I offered and

facilitated classroom-based demonstration lessons followed by staff meetings or

discussions with the classroom teachers to reflect on the experience (3). My reflection

on the success of these practices led to me changing my beliefs about the best way to

work as an education advisor and the general value of working in classrooms (4). This

change in belief led to me changing my practice as an education advisor to include

demonstration lessons whenever possible.

5.5 EDUCATION ADVISOR AS PRIVATE CONSULTANT

After the year when I worked in the classroom to complete the data collection

stage of this study I continued to work as a systemic education advisor supporting

approximately 100 schools belonging to the same education system. As explained in

Section 5.4, I changed my way of supporting schools to include working in classrooms

wherever possible. Due to the large number of schools I was supporting I was not able

to include as much of the co-teaching and co-planning to accompany the classroom

demonstration lessons. Often a single staff meeting was included in my visit to the

school and sometimes follow-up visits to demonstrate other aspects of mathematics

content or pedagogy were included.

As this thesis is being finalised I am looking toward beginning a new career as a

private mathematics education consultant working with schools that contract my

support. This will be similar in some ways to my work as a system-wide education

advisor, yet different as I will only need to support the contracted schools rather than

attempting to support a large number across a vast geographic area. A recent

negotiation with one school as to how I could support them in the upcoming school year

provides another event to demonstrate my growth as an education advisor. The initial

plan for working with this school is described in the following vignette.

5.5.1 VIGNETTE 5.5: FROM EDUCATION ADVISOR TO MANY TO EDUCATION ADVISOR TO A FEW

I recently visited a school to plan how I could support the introduction of an

inquiry-based school mathematics program. In a meeting with the principal and the

mathematics curriculum support teacher we discussed how I could best support the

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teachers to change to an inquiry/investigation approach to the teaching of mathematics.

I mentioned my willingness and preference for working in classrooms along with

teachers and this approach was enthusiastically received by the principal and curriculum

support teacher. We also discussed the importance of co-planning, co-teaching and

reflective discussions (in person as well as electronically). We decided on a way of

working that would be repeated on my several visits to the school.

Firstly I will meet with all the classroom teachers involved in the first stage of

the professional development to discuss my intended approach and the intended

outcomes of introducing an inquiry approach. I intend to share my plans for the

demonstration lesson, including a draft criteria sheet showing the mathematics learning

intended in the lesson. I will teach the demonstration lesson with 3 or 4 teachers in the

room observing and participating as they felt comfortable. Immediately after the lesson

the participating teachers and I will discuss the lesson, highlighting the mathematics

learning observed as well as the teaching approach used. The second phase of the

professional development will see me return to the school and co-plan another

mathematics inquiry with the teachers. I will participate as a co-teacher rather than as

the leader. We will then discuss this lesson. As mathematical inquiries are planned and

taught, an electronic record of them along with student work samples, assessment

criteria and activities will be collected for other teachers to attempt. Over several visits

and through the provision of ongoing support for a year it is hoped that teachers at this

school will value inquiry and it will form a more significant aspect of their mathematics

programs than it does currently.


The professional development plan discussed in vignette 5 depicts growth in my

approach to my role as an education advisor informed by this study. In my new role as a

private consultant I do not need to focus on supporting a large number of schools and

can I choose to support a smaller number of teachers and schools in more depth. To

make this as effective as possible I looked to my learnings from this study to determine

a way of working to achieve the desired changes at the school. The way of working

described in vignette 5 is not likely to be the only way my professional growth as an

education advisor could manifest itself to support change in schools and particularly in

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classrooms, but it shows a change in approach from my way of working prior to this

study.

As this professional development model is yet to be implemented there is no

further evidence to support this as an example of my growth. The principal and the

mathematics curriculum support teacher were very enthusiastic about this plan and

indicated that they had never had the opportunity to support the teachers in such a way,

anticipating that it would be highly effective. The move toward the digital sharing of the

outcomes of the inquiries co-planned, co-taught and reflected on is an addition I am

considering for other professional development plans.

This example of my professional growth has not been summarised and

represented using the modified Interconnected Model of Professional Growth as this

change in my way of working has not yet been implemented so the results of this

approach cannot be reflected on and have not been enacted.

5.6 SUMMARY

This chapter described my professional growth as the education advisor through

the provision of examples of change in my knowledge and practices. The structure of

this chapter followed the structure of Chapter 4 where examples of my professional

growth including a significant event reported as a vignette were discussed. These

examples of sequences of changes that occurred over an extended period of time form

my professional growth as described by Hollingsworth (1999).

My professional growth was linked to the classroom teacher’s professional

growth as it was change to her knowledge and practice that was focus of the

professional development. While I may have grown professionally from the experiences

even if the teacher had not displayed significant learning, I was clearly encouraged and

stimulated by her positive reactions to the professional development. As discussed in

Chapter 4, the classroom teacher’s professional growth through the continuation of the

professional development activities also required the students in her class to

demonstrate effective learning to show that the mental computation instructional

program was not causing any detrimental effects. The examples of my professional

growth in this chapter were summarised and represented using the modified version of

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the Interconnected Model of Professional Growth starting with the significant event

reported in the vignette.

The professional growth of the classroom teacher and myself as the education

advisor provided the evidence on which to evaluate the particular method of

professional development used in this study. The following chapter draws together what

was learnt about our professional growth and the professional development through

reference to the research questions, the research literature and the data presented in

Chapters 4 and 5 to draw conclusions and make recommendations for the work of other

education advisors and researchers.

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6. Conclusions

This study set out to investigate a particular method of professional

development for practising classroom teachers. The inspiration for the study came

from my experience working for several years supporting classroom teachers to

change their knowledge and practices in regard to mathematics education. The

professional development being used generally involved sessions offered at a central

location with teachers from many schools. Concerns over the transfer of learning

from these sessions to the classroom led me, and members of the curriculum team I

worked as member of, to seek alternative methods of professional development.

When the opportunity to conduct classroom-based professional development for one

teacher arose I recognised the potential to try a different way of working. By

formalising the experience as a doctoral study I would be able to share my learnings

with other education advisors and contribute to the knowledge of my profession.

The classroom teacher in this study approached me after I had conducted an

after school workshop in relation to some of the changes in computation outlined in

the new Mathematics syllabus. She identified the shift to a greater focus on mental

computation and strategies for computation rather than algorithms as a personal

challenge, but one that she could see potential advantages in for her class of Year 3

students. I suggested we work together to implement the change to mental

computation in her classroom over the following school year. The administration of

the school and my supervisors were supportive of the idea to trial this particular

method of professional development so the classroom teacher and I began to make

plans for the study to occur in the following school year.

Mental computation was an area of mathematics education and the

mathematics syllabus that I had taken a particular interest in prior to and during the

development of the new syllabus. I had built a comprehensive knowledge of this area

through my own professional reading and experimentation, discussions with

colleagues and attendance at conferences. As part of my interest in this area I

developed a Strategy Categorisation Framework (Hartnett, 2007) that I believed

could assist teachers to include mental computation strategies in their classroom

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programs. The Strategy Categorisation Framework is described in Section 2.5.2 and

the framework is provided in Appendix 1. This study enabled me to use this

framework to provide the general structure of the mental computation instructional

program, which along with my ongoing support formed the basis of the professional

development I facilitated.

The classroom teacher and I chose to focus on one major strategy category

and the associated number sense activities for each term as the program’s overall

structure. The details of the program developed as the year progressed, depending on

the arising needs of the students and the classroom teacher. We also decided to use

direct instruction as the general teaching approach rather than having the students

invent strategies for themselves. This way we believed we ensured that the program

provided the students and the classroom teacher with an introduction to a wide

repertoire of strategies over the year. The aim of the study, therefore, was to

investigate a particular method of professional development that should result in

change at the classroom level. The classroom teacher’s and my professional growth

were investigated from data collected during the year I worked in her classroom.

Evidence of student learning was also collected to reassure the classroom teacher

that the teaching approach and program we were using was not detrimental to the

students’ learning.

A set of research questions were developed to pursue the overall aim of this

study. The classroom teacher’s professional growth was the focus of the first

research question: What professional growth did the classroom teacher experience

as a result of the professional development conducted in her classroom? Data were

collected in the form of personal reflections by both the classroom teacher and me on

lessons, mathematical concepts in focus and activities. The data was recorded as

emails and field notes as well as transcripts of interviews. These data were reviewed

after the completion of the year in the classroom to identify examples of the

classroom teacher’s professional growth. Many examples were identified and a

number of them were reported through the use of vignettes and discussion in Chapter

4. Each of the reported examples was summarised and represented using the



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The second research question focussed on my professional growth: What

professional growth did I experience as the education advisor as a result of

conducting the professional development? I knew I would be changing my practice

through the use of classroom-based rather than centrally-offered, one-off

professional development sessions. After my review of the literature on teacher

change and effective professional development I believed that working with the

teacher in her classroom and providing ongoing support over an extended period of

time would be effective in providing an opportunity for the classroom teacher’s

professional growth. I anticipated that I would experience professional growth in

terms of lasting change to my knowledge and practices. However, I was not able to

predict what professional growth I would experience during the study beyond the

major change to my practice of working in the classroom with the teacher. The data

were scrutinised to identify examples of my professional growth. These examples

were reported in Chapter 5 through the use of the discussion of vignettes, with each

example of my professional growth represented using a modified version of the

Interconnected Model of Professional Growth (see Section 3.2.2).

The classroom-based method of professional development utilised in this

study informed recommendations for education advisors in their roles supporting

classroom teachers. My professional growth was documented to identify factors

which would bring reciprocal benefits for education advisors and teachers who work

together in classrooms to improve student learning. The third research question

aimed to indentify the influences of my professional growth by asking: What factors

influenced my professional growth as an education advisor?

Each of the research questions are discussed in the following sections, with

reference to the literature and the results reported in Chapters 4 and 5, to synthesise

the findings of this study in relation to the classroom teacher’s professional growth

(Section 6.1), my professional growth as an education advisor (Section 6.2), and

factors influencing my professional growth (6.3). Implications for education advisors

supporting practising teachers (Section 6.4) and for further research (Section 6.5) are

also offered. Limitations of this study are reported in Section 6.6.

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6.1 PROFESSIONAL GROWTH OF THE CLASSROOM TEACHER

The overall purpose of choosing classroom-based professional development

for this study was to see a better alignment between the intended learning facilitated

through professional development and change in the classroom. Change at the

classroom level in this study was evidenced through the examples of the classroom

teacher’s professional growth reported in Chapter 4.

The classroom teacher displayed change in the three types of knowledge

described in Section 2.1.1: general pedagogical knowledge, subject matter

knowledge and pedagogical content knowledge (Borko & Putnam, 1995). There was

also evidence of change in her practices. The examples of the classroom teacher’s

professional growth described in Chapter 4 showed that she changed her personal

knowledge of mental computation strategies (Section 4.1). She also changed her

beliefs about the value of students making their thinking visible (Section 4.2), which

was a focus of the mental computation program implemented as part of the

professional development. Section 4.3 reported on a change in the classroom

teacher’s practice to include the teaching of number sense understandings in

conjunction with the teaching of mental computation strategies. The classroom

teacher also experienced a change in her knowledge about the use of a particular

resource, the number board, in a way she had not used before the study (Section 4.4).

The final change was in the classroom teacher’s practice of utilising a particular

pedagogical strategy to meet the diverse learning needs of the students in her class

(Section 4.5).

The vignettes reported in Chapter 4 were not isolated incidents but occurred

as part of sequences of observable changes. The examples of the classroom teacher’s

professional growth reported through the vignettes were represented as connected

sequences of change between the different domains of the classroom teacher’s world.

These growth networks were represented using the Interconnected Model of

Professional Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999).

Conclusions about the use of this model are reported in the following section.

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6.1.1 REPRESENTING THE CLASSROOM TEACHER’S PROFESSIONAL GROWTH

The Interconnected Model of Professional Growth was initially developed by

Clarke (1988) and the Teacher Professional Growth Consortium (1994) to represent

the complexities of teacher change. It was then further developed by Clarke &

Hollingsworth (2002) and Hollingsworth (1999) to represent professional growth

more generally. This model depicts what the authors saw as the four domains of the

world of a professional as well as the numerous possible connections between these

domains that represent how change in one domain can be mediated through

enactment or reflection to lead to change in other domains. The examples of the

classroom teacher’s professional growth in this study were represented as growth

networks (Hollingsworth, 1999) using this model.

Using the Interconnected Model of Professional Growth to summarise and

represent the classroom teacher’s professional growth as sequences of changes, I

found it difficult to decide where to culminate a particular growth network. Most

examples of professional growth involved all four domains. I realised that each of the

examples of the classroom teacher’s professional growth I had chosen to report

resulted in change to either her Personal Domain (knowledge, beliefs and attitudes)

or her Domain of Practice (professional experimentation). The other domains were

part of the sequence of changes but change to the External Domain or the Domain of

Consequence were not a culminating change. They always led to either change in

knowledge (Personal Domain) or change in practice (Domain of Practice). Another

decision I found I had to make was which of these two domains should represent the

final change in the sequence. Professional knowledge and practice are very closely

linked. Growth networks resulting in change in the classroom teacher’s knowledge,

for example the change in the her personal knowledge of the use of alternative

number boards reported in Section 4.4, would inevitably lead to change in practice,

for example using these number boards in lessons, that would lead to a realisation of

their value (knowledge), and further practice and so on. In each example I made the

decision based on what I thought best represented the professional growth of the

classroom teacher. As with all professional development the hope is that the learning

leads to ongoing changes. I chose enough of the sequence to represent the example

of the classroom teacher’s professional growth.

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The Interconnected Model of Professional Growth was used to represent the

sequence of changes leading to the event described in each vignette and related

changes after the event. As professional growth is by definition enduring change it is

to be expected that the significant events reported in the vignettes would involve an

extended growth network. So I decided to show what I considered to be enough of

the growth network to represent the event.

Another conclusion in relation to the use of this model was the difficulty I

had at times identifying change in the classroom teacher’s knowledge. Changes in

her practice were easier to recognise. The detailed reflections and communications

by email were revisited, looking for examples of change in any of the domains of the

classroom teacher’s professional world. Because these communications were in

writing I was able to search for words indicating a change of knowledge or practice.

I was always conscious that I was interpreting this from the data. To counter any

potential misinterpretations, I continued to communicate with the classroom teacher

after the year in the classroom with her as I was analysing the data and reporting on

the study during the preparation of this thesis The use of this model for the

representation of her professional growth meant that I specifically reviewed the data

looking for examples of changes in her knowledge relating to each of the domains.

Choosing to represent the classroom teacher’s learning and change using this model

led to a more in-depth analysis of her growth.

With each of the examples of the classroom teacher’s professional growth

represented using the Interconnected Model of Professional Growth I was able to

identify patterns in the representations between the different examples. I noticed that

in the examples reported each of the growth networks started with an external

stimulus (change in the External Domain). In all of the examples reported, except

one, that stimulus was one of my demonstration lessons. In the other example the

external stimulus was a discussion the classroom teacher and I had about the students

recording their thinking and strategy use. This evidence reinforced the importance of

my role as education advisor and that providing this method of professional

development was leading to change at the classroom level.

Use of the Interconnected Model of Professional Growth to represent the

examples of the classroom teacher’s professional growth allowed me to observe that

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there was no pattern in the placement of the significant event described in each

vignette as part of the growth network. In one example, the significant event was the

first change after the demonstration lesson (see Section 4.4). In another (Section 4.2)

it was the second change. Yet another example saw it placed in the middle of the

sequence (Section 4.1) and in the other two examples the significant event was at the

end of the growth network (Section 4.3 & 4.5). The significant changes that I

identified in the data reported as the vignettes occurred at different points in the

sequence of changes. I did not always occur straight after the stimulus and often

occurred after reflection and enactment of other changes.

A further observation about the sequence of changes representing each

example of the classroom teacher’s professional growth was that in each growth

network the significant event reported in the vignette was mediated through

reflection. This result highlighted the importance of reflection that had been

identified by several significant authors from Dewey (1910) and Piaget (1970) to

Schön (1983) in relation to teacher change and the principles of effective

professional development (see Section 2.2.3).

Using the Interconnected Model of Professional Growth to represent the

classroom teacher’s professional growth stimulated the consideration of the factors

involved with that growth. The classroom teacher’s professional growth could then

be represented as the sequence of changes that led to that growth. The creation of a

visual representation of the sequence of changes involved in the classroom teacher’s

professional growth was helpful in considering a wider set of factors leading to

change than I might have if the model had not been used. By using it to summarise

each reported example of professional growth, the model also made me carefully

consider the sequence of events that led to the resultant change.

6.2 PROFESSIONAL GROWTH OF THE EDUCATION ADVISOR

By setting out to conduct the professional development in this study in a

classroom rather than at a central venue or as a school staff meeting I was making a

change in my practice as an education advisor. Apart from this expected change there

were other examples of my professional growth that led to other changes in my

knowledge and practice. I experienced professional growth that resulted in change in

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my knowledge and practices. The examples of my professional growth reported in

Chapter 5 showed that I changed my approach to the demonstration lessons from

presenting the lesson for the classroom teacher to co-teaching it with her (Section

5.1). This change to incorporate co-teaching led me to co-plan the demonstration and

follow-up lessons with her as well (Section 5.2). Another change I experienced

during this study involved my realisation of the importance of developing

relationships as part of professional development. The invitation to participate in a

liturgy planned and facilitated by the class indicated that I had been accepted as a

member of the class rather than being considered a visitor (Section 5.3). I also

realised the importance of the close professional relationship I had developed with

the teacher to the on-going success of the program. The change reported in Section

5.4 described the change in my practice by working in classrooms. Section 5.5

detailed how I plan to conduct my new role as a private mathematics education

consultant based on change in my knowledge and practices.

Each of the examples of the classroom teacher’s professional growth were

summarised and represented using the Interconnected Model of Professional Growth

(Clarke & Hollingsworth, 2002; Hollingsworth, 1999). As discussed in Section 3.2.2,

this model could have been used to represent my professional growth as well but I

chose to modify the model slightly to recognise that the salient outcomes of change

for me (Domain of Consequence) was the entire representation of the professional

growth of the classroom teacher. Conclusions about the use of this model to

represent my professional growth are provided in the following section.

6.2.1 REPRESENTING THE PROFESSIONAL GROWTH OF THE EDUCATION ADVISOR

Using the modified version of the Interconnected Model of Professional

Growth, rather than the original model, made little difference to the way I

represented my professional growth compared to the representation of the classroom

teacher’s professional growth. The domains were the same except that the modified

version of the model made recognition of the connection between the classroom

teacher’s professional growth and mine by showing her growth as salient outcomes

for me. Representing my professional growth using the model showed that the

changes to my knowledge and practices involved a sequence of changes and were

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not isolated events. It also showed that the classroom teacher’s professional growth

was an important factor for consideration in the sequence of changes that constituted

examples of my professional growth. The events reported in the vignettes occurred

as one change in the sequence of changes that together represented my professional

growth.

Representing my professional growth using the modified Interconnected

Model of Professional Growth highlighted a difference in comparison to using the

original model to represent the classroom teacher’s professional growth. With

representations of the classroom teacher’s professional growth I was easily able to

include events and changes that occurred before the significant event reported in a

vignette. I expected that this would be the same when I represented my professional

growth using the modified model, however I found that this was not so simple.

When I tried to depict the events leading up to the significant event depicted

in the vignette appeared misplaced and seemed to completely change the growth

network. For example, with my change from presenter of the professional

development to collaborator and co-teacher reported in Section 5.1, I initially

considered that the external stimulus (E) to this event was my review of the

literature, discussions with colleagues and attendance and presentation at research

and teacher education conferences in relation to mental computation I had

experienced leading into this study. The growth network continued as I reflected on

these information sources to develop my own comprehensive knowledge of mental

computation strategies and pedagogies (E K). I then implemented this knowledge

that I had developed as demonstration lessons (K P). I continued to work in this

way until the email reported in the vignette (Section 5.1) asking me to help the

classroom teacher to reteach a lesson led me to change my practice. The event

described in this vignette did not work as an enactment or reflection to continue this

sequence as a growth network. It was interesting that using the model to represent

my professional growth did not work in the same way as it did for representing the

professional growth of the classroom teacher.

After a reconsideration of the use of the model to represent my professional

growth I tried representing the examples of my professional growth as starting from

the significant event described in the vignette rather than outlining the sequence

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leading to this event. In this way the significant event formed the external stimulus

(External Domain) from which to start the growth networks to represent my

professional growth. Using the modified Interconnected Model of Professional

Growth this way was more successful in representing the examples of my

professional growth including the significant events described in the vignettes.

In this study professional growth was depicted as an ordered sequence of

events that included a significant event reported as a vignette. For the classroom

teacher each significant event was able to be situated anywhere in the sequence of

changes that led to professional growth. For me as the education advisor the

significant event needed to be the start of the sequence of changes to make the

reflection and enactment of subsequent changes flow. While it is recognised that

there were other factors and other changes which had occurred prior to the

significant event, it did not make sense to represent them as part of the example of

my professional growth even though this worked for representing the classroom

teacher’s professional growth.

Hollingsworth (1999) numbered the sequence of changes in an example

growth network when she was describing the difference between a change sequence

and a growth network (p. 131). In her data collection processes she represented

examples of the teacher’s professional growth using the Interconnected Model of

Professional Growth but not as numbered sequences as I did although in her

descriptions of the teacher’s professional growth a sequence was able to be

understood which related to the model. The inclusion of the numbered sequence as

an important part of the representation of professional growth was a difference in the

use of this model in this study to how others that have used this model.

Almost all of the examples of professional growth presented by

Hollingsworth (1999) include the External Domain. However, as the sequence is not

provided it is unclear whether this domain was the start of change depicted in her

examples. Other studies that have used the Interconnected Model of Professional

Growth (Diezmann, Fox, DeVries, Siemon, & Norris, 2007; Justi & VanDriel, 2006)

did not depict professional growth as the extended sequence of changes as I have.

These studies identified change in one or two domains, not a connected sequence.

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Another factor relating to the use of the model to represent professional

growth was that as the researcher I had to decide where the sequence of changes in

each example of professional growth culminated. I found that each growth network

representing my professional growth resulted in change in either the Personal

Domain or the Domain of Practice, as it had for the classroom teacher’s professional

growth. For each example of my professional growth I made a decision about how

much of the change sequence best illustrated the significant changes in the growth

network and where it made most sense to culminate the sequence of changes.

In relation to the identification of change in knowledge, I had more clarity in

this decision for my professional growth than for the classroom teacher’s as it was

easier for me to identify change in my own knowledge than it was to identify change

in the someone else’s knowledge. I could analyse my own thoughts whereas I had to

make observations, draw conclusions from the data and check my interpretation with

the classroom teacher to identify change in her knowledge. The Personal Domain

(knowledge, beliefs and attitudes) and the Domain of Practice (professional

experimentation) are very closely related, meaning that examples of significant

change in my knowledge, beliefs or attitudes were likely to lead to change in my

practice and significant changes in my practice led me to reflect on my knowledge.

In terms of representing my professional growth as growth networks, I had to decide

when to stop representing the sequence and which domain the example of

professional growth had led to change in. Three of the four examples of my

professional growth represented using the modified version of the Interconnected

Model of Professional Growth saw the growth network depicted result in change to

my practice.

As with using the Interconnected Model of Professional Growth to represent

the classroom teacher’s professional growth, I found that representing my

professional growth using the modified Interconnected Model of Professional

Growth helped with the considerations of factors influencing the changes and the

sequence of changes that influenced my professional growth. Use of the model was

one factor which influenced my professional growth. The following section details

the range of factors that influenced my professional growth as an education advisor.

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6.3 FACTORS INFLUENCING THE PROFESSIONAL GROWTH OF THE EDUCATION ADVISOR

An aim of this study was to investigate my professional growth as an

education advisor through implementation of a particular method of professional

development. A number of factors influenced my professional growth. The third

research question framing this study asked: What factors influenced the professional

growth of the classroom teacher and the education advisor?

The method of professional development utilised in this study was one of the

factors that influenced my professional growth. The professional development was

classroom-based and this was a new way of working for me. In the literature Guskey

(2003) identified the use of school-based activities was a principle of effective

teacher professional development. This study took the professional development

even closer to the classroom teacher’s world by conducting it in her classroom.

Nisbet, Warren and Cooper (2003) described how pertinence to classroom practice

was another principle of effective teacher professional development. The classroom-

based nature of this professional development allowed me to put into practice many

of the teaching practices and concepts relating to mental computation that I had been

describing to teachers in previous professional development sessions not conducted

in classrooms. The classroom-based nature of the professional development utilised

in this study enabled the activities to be tailored to be pertinent to the classroom

teacher and her students.

The inclusion of reflection as a major component of the professional

development was another factor that contributed to my professional growth as well

as the classroom teacher’s professional growth. Reflection is a valuable component

of teacher change often discussed in the literature (e.g. Dewey, 1910; Piaget, 1970 &

Schön, 1983). Reflection is oneof the two processes mediating change in one domain

leading to change in another domain in the Interconnected Model of Professional

Growth (Clarke & Hollingsworth, 2002; Hollingsworth, 1999), highlighting its

importance in relation to professional growth. In this study it was noted that all the

significant events described in the vignettes in the examples of my professional

growth were mediated through reflection. In centrally-offered professional

development sessions I had conducted prior to this study I had rarely included

reflection in workshop sessions, often due to my belief that genuine consideration of

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such reflection, while it could help me refine my practices, was less likely to lead to

change in the practices of the classroom teachers attending the professional

development. Although the process of personal reflection can be beneficial, this was

unlikely to be followed up or supported further by me once the classroom teachers

returned to their classrooms. Another form of reflection that I rarely used was having

participants complete evaluation or feedback forms at the end of a workshop or

session. These sources of data had been shown to provide few worthwhile comments

when asked for at the end of a day when participants are ready to leave and

considered the professional development to be over.

The classroom teacher and I found that reflecting on the lessons and the

students’ learning and providing each other with feedback as we worked together for

the year of the study was quite a natural process that did not need to be facilitated.

The only difficulty was that often the classroom teacher and I were not able to reflect

and discuss my demonstration lessons before I had to leave the school for the day

due to time constraints. We developed a process of communicating and reflecting on

lessons and the students’ learning by email after my visits. DeBard and Guidera

(1999) described how computer-mediated asynchronous methods when used for

reflection allowed participants time to consider their responses. This method of

reflection was highly effective in this study as a way to share our thoughts and it also

generated a large body of data that I was able to take time to analyse after the year

spent in the classroom. In centrally-offered professional development the reflections

need to be completed immediately, resulting in less-considered responses. The use of

asynchronous reflection via email was another factor which influenced my


The collegial relationship that developed between the classroom teacher and

me was a further factor contributing to my professional growth. Mewborn (2003)

described a key theme in the teacher professional development literature as the

inclusion of discussion with supportive colleagues. In this study the classroom

teacher and I were colleagues working together in the same classroom to facilitate

the learning of the students. As the relationship developed we began to work as co-

teachers (Tobin & Roth, 2006) as opposed to team teachers (see Section 2.2.2). The

classroom teacher’s and my relationship developed over time and was a factor in my

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professional growth reported in Chapter 5 (see Section 5.3). The personal

relationship that developed meant that the classroom teacher felt comfortable to ask

questions and to request particular lessons or actions from me. Our interactions were

informal, allowing for frequent and honest requests, discussions and reflections.

This study took place over a full school year. This extended period of time

allowed the professional development to progress at a pace that matched the

development of the classroom teacher and her students. The length of time of the

study possibly contributed to the effectiveness of the professional development.

Conducting professional development over an extended period of time was a

principle of effective teacher professional development included in the writing of

many authors including Owen, Johnson, Clarke, Lovitt and Morony (1988), Clarke

(1994), Mewborn (2003) and Guskey (2003). The mental computation instructional

program, one of the components of the professional development, was developed

progressively throughout the study to provide a full school year of instruction based

on the needs of the students and the classroom teacher. The extended time for this

study was an important factor in the success of the professional development because

if the classroom teacher was likely to include the strategies and number sense

activities in her future planning the co-operative development complete year of

activities was a major factor in the success of this transferring beyond the study.

The classroom teacher in this study had ongoing support from me to assist her

to develop and implement the changes to the computation program in the year of the

study. As part of my own professional development prior to the study I had gathered

a comprehensive knowledge of mental computation strategies, the associated number

sense understandings and activities and resources to support the teaching of mental

computation. Nisbet, Warren and Cooper (2003) described how support from an

experienced mentor contributed to the success of teacher professional development.

The classroom teacher also accessed other aspects of my personal knowledge of

mathematics and mathematics education during the year we worked together. My

ability to provide ongoing support in this study was another factor influencing my


In summary, the factors identified in this study as influencing my professional

growth were: the classroom-based nature of the professional development; the use of

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reflection and in particular the use of email to capture our considered reflections; the

extended period of time taken for the study; and the classroom teacher’s ongoing

access to my comprehensive knowledge of mental computation strategies and

mathematics education. In the next section implications for the role of education

advisors working to support practising classroom teachers are drawn from these

factors and the study in general.

6.4 IMPLICATIONS FOR EDUCATION ADVISOR PRACTICE

This study set out to trial a particular method of professional development

and through the monitoring and reporting of the professional growth of the classroom

teacher and myself in the role of education advisor recommendations for the practice

of education advisors were identified. The intention was that implications from this

study could contribute to the work of education advisors who support practising

classroom teachers. Informed by the professional growth experienced by the

classroom teacher, I analysed my own professional growth in relation to the

successful provision of the method of professional development facilitated in this

study. My experience of success leads to several recommended practices for

education advisors, namely facilitating classroom-based professional development,

including reflection especially via email, utilising an extended period of time,

fostering relationships with the teachers who are the focus of the professional

development and if possible their students, as well as providing ongoing access to

expert knowledge and support. The remainder of this section outlines further

considerations for education advisors concluded from this study.

At the beginning of the study I approached my role as the deliverer of

knowledge and demonstrator of practices for the classroom teacher. I had developed

a comprehensive understanding of mental computation and teaching strategies for the

implementation of a strategies focussed approach to computation from my research

prior to this study. I was operating from a perspective where I had the knowledge and

practices and I just needed to show these to the classroom teacher and she would be

transformed. Early in the study (Week 9) I realised this was how I was working and

that this was against my personal philosophy as a teacher. When I worked as a

classroom teacher I encouraged my students to inquire and discover their own

understandings. The classroom teacher in this study was an experienced professional

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and after the event described in Section 4.1 I realised her potential as a colleague and

co-teacher rather than a passive recipient of the professional development. Although

I was not aware of the practice of co-teaching at the beginning of the study, further

research after the classroom teacher suggested that we work together to support the

students’ understanding of a particular mental computation strategy (Vignette 4.1)

led me to the work of Tobin and Roth (2006).

The key aspect of co-teaching is the existence of more than one teacher in a

classroom working together to teach a group of students. Tobin and Roth (2006)

described how co-teaching serves two purposes. It provides more opportunities for

the students to learn as well as providing opportunities for the teachers to grow in

their abilities to teach. Tobin and Roth also make the point that teaching is

“something that is done” (p. 12) and that reading about, talking about or watching

teaching is not teaching. While it is possible to learn about teaching through these

practices, the knowledge gained about teaching needs to be enacted. By co-teaching

the ideas gained by working with another teacher can be enacted and reflected on as

well as learned. Co-teaching as a form of professional development and as a way of

working for education advisors was confirmed in this study and is to be

recommended for others working in similar roles supporting practising teachers.

Further reading of the work of Tobin and Roth (2006) has highlighted another

practice that would be beneficial in the work of education advisors. These authors

highlighted the practice of using cogenerative dialogue in their studies focussing on

teacher change. Tobin and Roth described how co-teachers become “like each one

another in the ways in which they talk, gesture, observe the class, use space and time,

access materials and interact with one another and the students” (p.81). Teachers

may not be aware of their learnings and that by talking to others including the

students, using cogenerative dialogues, they can identify what works, what does not

work and any practices that might disadvantage participants. The perspectives of

others can then become the objects of reflection. The outcomes of these processes are

co-generated involving and earning respect and responsibilities to enact changes in

the classroom. The inclusion of cogenerative dialogues could have contributed

another level of analysis of the professional development utilised in this study had I

been aware of this aspect of the research prior to the study. This practice, however,

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can be recommended as a practice for other education advisors to consider as a

conclusion of this study.

This study was conducted over a full school year. This allowed the mental

computation instructional program developed as part of the professional

development to cover the entire year. Another benefit of the extended period of time

taken to conduct this study was that it allowed the way the classroom teacher and I

worked to evolve. The time allowed us to develop an interpersonal and collegial

working relationship which was reported as one of the factors influencing the

classroom teacher’s and my professional growth of and the success of the

professional development program. A more detailed look at the classroom teacher’s

professional growth indicated that the significant events reported as vignettes in

Chapter 4 all occurred in the first half of the year. The final one was in Week 3 of

Term 3, just after the mid-year school holidays. In terms of my professional growth

the examples were all from Term 1 and 2 of the study. It could be inferred from these

data that this study maintained momentum for the second half of the year but did not

lead to further significant professional growth for either the classroom teacher or

myself. The influence of the study had been effective before the end of the year and a

similar set of results and conclusions could be possible in a significantly shorter

period of time with the classroom teacher continuing the development of the mental

computation program based on the overview developed at the beginning of the year

and support from me from a distance rather than in the classroom. Another

implication of that can be drawn from this study is that the outcomes achieved in

relation to my professional growth and the professional growth of the classroom

teacher could have been achieved in less than a school year.

The mental computation instructional program the classroom teacher and I

developed during this study was a successful component of the professional

development. Analysis of the students’ performance on the researcher designed tests

and the state-wide test demonstrated that the mental computation instructional

program had not disadvantaged the students’ mathematical learning. The Strategy

Categorisation Framework used as the basis of the program had not been trialled in a

classroom before and in this study it was shown that it provided an effective

structure for the mental computation instructional program for the class for the whole

170


school year. The students were systematically exposed to a range of computation

strategies and the related number sense understandings to enable them to show

accuracy, efficiency and flexibility in their application of strategies to complete

computations. This program could be utilised further by mathematics education

advisors as the basis of professional development for classroom teachers and whole

school groups with a focus on changing their approach to the teaching of

computation.

The successful use of the Interconnected Model of Professional Growth to

represent the classroom teacher’s professional growth was another outcome of this

study. Clarke and Hollingsworth (2002) described three different ways this model

could be used in relation to studying professional growth. The first use of the model

was as an analytical tool to categorise teacher change data. The second use was as a

predictive tool to promote particular change sequences and growth networks through

planned professional development. The final use was as an interrogatory tool to

frame theories and research questions about teacher professional development. In

this study the model was only used in the first way to summarise and represent the

classroom teacher’s professional growth as growth networks resulting in change in

her Personal Domain or her Domain of Practice.

The utilisation of the Interconnected Model of Professional Growth in the

other ways outlined by Clarke and Hollingsworth (2002) could contribute to

education advisor practice. When planning and facilitating teacher professional

development, an education advisor could use the model to predict the likely changes

in teachers’ knowledge and practices. It could also be used to frame research

questions about teacher professional development so that data could be gathered by

education advisors to judge the effectiveness of their support for teachers.

Considering this model could see education advisors specifically planning for

reflection and the enactment of changes in knowledge and practices presented in

professional development sessions. Sequences of changes could be planned to

increase the likelihood that the stimulus professional development (change in the

External Domain) could lead to long-term change and professional growth in

teachers. The model could also be used as it was in this study to summarise and

represent the professional growth of teachers who attend professional development

171


activities, becoming an evaluation tool likely to be more effective than feedback

sheets given to participants at the end of a professional development workshop.

The modified version of the Interconnected Model of Professional Growth

developed in this study to represent my professional growth provided an addition to

the existing literature on supporting the professional growth of teacher educators

(education advisors) who provide professional development for teachers. In this

study this new version of the model was used in the same way as the original to

summarise and represent the professional growth of the education advisor. A further

implication for education advisors could be to utilise this modified model in the same

three ways that Clarke and Hollingsworth (2002) intended for their original model to

predict, prepare for and research the improvement of their own knowledge, beliefs,

attitudes and practices when working to provide professional development for

teachers.

The general professional development process utilised during this study

incorporated the use of demonstration lessons in the classroom followed by

reflection and discussion of the outcomes of the lesson, practices incorporated and

concepts covered provide a professional development process worth considering as a

way of working for other education advisors. The co-teaching and co-planning were

significant aspects of the way of working which developed during this study. In my

future work as a private mathematics education consultant I intend to utilise this

structure for professional development I offer.

6.5 LIMITATIONS OF THE STUDY

This study was conducted by one education advisor who worked with one

classroom teacher in one school for one year. The classroom teacher and I co-

planned and co-taught an instructional program for the one year level, Year 3, in one

aspect of mathematics, mental computation. This narrow focus meant that the results

of study are very specific to the context in which it was conducted. The narrow

focus, however, did allow for the collection of a large quantity of data enabling both

the classroom teacher’s and my professional growth to be investigated and analysed

in great depth.

172


The context in which the study was conducted included many factors that

helped to ensure the professional development was successful. These factors

included: the support of my supervisor so I could spend an extended period of time

supporting one classroom teacher when my role involved the support of a large

number of teachers from many schools; the collegial and personal rapport built

between the classroom teacher and myself that facilitated open, informal reflections

and discussions; the classroom teacher’s familiarity with technology and her

willingness to use email for communication and reflections, enabling deep of

reflection producing rich data for identification and analysis of examples of

professional growth; and the support of the school administration and parents of the

students to allow the study to be conducted in one classroom in their school and with

their children. The results of the study confirmed the effectiveness of the method of

professional development, and showed that professional growth and the application

of learning by classroom teachers as a result of professional development is possible

when a close to ideal context is able to be used. It would be unlikely that these

factors could all be present in other contexts. Each school context is different. It is

the potential differences between contexts in which this method of professional

development could be applied that highlights a potential limitation of the application

of this study to other contexts.

In this study I was the education advisor and the researcher and as such I was

a subject of the study along with the classroom teacher and the students. While this is

not a limitation in itself, the results of this study are relative to me in the context in

which I chose to conduct this study. All of the interpretations of the evidence

gathered during the study were made by me as the researcher. The strength of this

approach is the depth of understanding that I had from my first-hand experiences of

all aspects of the study. I was able to consider change in my knowledge and beliefs

because they were mine. I was able to reflect on events and changes in my practice in

terms of aspects which may have been more difficult to elicit from another subject

other than myself. Although it is inevitable that as researcher I viewed the

experiences with my personal lens, I attempted to make sense of those experiences

by reflecting in relation to the literature, by using an analytical tool in the form of the

Interconnected Model of Professional Growth and by referring my analysis back to

173


the classroom teacher for her to read over and comment on, particularly in terms of

my interpretation of change in her knowledge.

The use of the Interconnected Model of Professional Growth to summarise

and represent the classroom teacher’s professional growth, and the modified version

of this model to summarise and represent my professional growth, provided a

consistent way to conceptualise the changes that occurred during the study. The

similarity between the models allowed for comparisons of the resulting

representations to identify patterns in the relationships between the domains. A

limitation of the use of this model was that it framed my thinking as researcher in a

particular way that could have influenced my view and interpretation of the data.

However, having the model with its domains assisted in the identification of

examples of change.

An overall limitation of this study was the method of professional

development utilised. Although successful, the professional development was time

and personnel intensive. Implementation of this method of professional development

with a larger number of teachers would require a significant commitment of

personnel and time due to the professional development involving ongoing

classroom visits and maintenance of email contact. The classroom teacher

maintained contact with me after the year in her classroom and she stated how her

practice was still using the knowledge and practices we developed together during

this study. While this method of professional development led to professional growth

and lasting change at the classroom level. for an education system to utilise this

method of professional development for all its teachers would not be a viable option.

6.6 IMPLICATIONS FOR FURTHER RESEARCH

The classroom-based nature of the professional development explored in this

study, in particular the use of demonstration lessons, co-teaching and co-planning,

reflection in person and utilising electronic means, and ongoing expert support could

be investigated in other contexts in relation to student learning outcomes, classroom

teacher professional growth or education advisor professional growth. As this study

focussed on only one teacher in one classroom, it would be interesting to see if the

successful practices identified in this study were able to be utilised in another

174


classroom context or for more than one teacher. The extended length of time taken

for this study may not have been needed. Most of the examples of the classroom

teacher and my professional growth had occurred by the middle of the year. Further

studies could also compare results gathered in a similar study with one classroom

teacher but conducted over a shorter period of time.

The mental computation instructional program was successful and the

Strategy Categorisation Framework on which it was based could be investigated

further as an organiser for classroom computation programs and whole school

computation programs. Student learning as a result of this program could be another

area for further study. The mental computation instructional program developed and

implemented in this study was only for one school year level (Year 3). The students

were introduced to strategies for addition and subtraction of two-digit and some

three-digit numbers as well as multiplication and division of basic facts (single digit

examples) and some carefully chosen two-digit numbers. The implementation of a

school program based on the Strategy Categorisation Framework that could allow for

the development of these strategies across year levels, involving larger numbers,

decimals, common fractions and other applications like percentage and proportion

problems has the potential for further research.

This study identified student learning, the professional growth of the

classroom teacher and the professional growth of the education advisor, as well as

the relationships between these elements. The analysis and representation of the

professional growth of the classroom teacher was possible using an existing model

which was modified to analyse and represent the professional growth of the

education advisor. This modified version of the existing model provided a tool that

was not evident in the existing literature. The use of both of these models in this

study was different to the way it had been used in other research. The use of these

models to represent the sequence of events and changes by numbering the arrows

connecting the domains allowed for the pattern of changes for each example of

professional growth to be represented and compared. Other studies could further

investigate patterns in the sequences of changes leading to professional growth of

classroom teachers or education advisors.

175


6.7 PROFESSIONAL GROWTH: A PERSONAL PERSPECTIVE

This study was initiated by my interest in changing my way of working as an

education advisor to facilitate greater change at the classroom level. The focus on

mathematics and in particular mental computation was also initiated by personal

interest in mental computation strategies rather than the traditional algorithms in

primary schools. While I was able to achieve change in the teaching of mental

computation strategies in the classroom in which I worked for this study, it was the

concept of professional growth and my role as an education advisor supporting the

professional growth of teachers that developed as the thesis in this study. I expected

to change my practice but I did not expect the other learnings about my knowledge,

beliefs, attitudes and practices as an education advisor. As a result of this study my

intention is that I will no longer conduct one-off professional development sessions.

My work with teachers will be characterised by co-teaching, co-planning, reflection

and ongoing support to see the changes we aim for implemented as enduring change

– professional growth. In this study it was the way the classroom teacher and I

worked together that made this study successful. The preparation of this thesis has

been a long journey. I can only hope some of the findings I have made will support

others who work in similar roles.

.

176


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Appendices

APPENDIX 1: STRATEGY CATEGORISATION FRAMEWORK

Table A1 Categorisation of Mental Computation Strategies (Hartnett 2007)

Strategy category

Strategy sub-categories Examples of efficient use of strategies

45+3: 46, 47, 48 Count on to add 45+39: 45+30=75, +5=8-, +4=84 (used with no. line) 65–4: 64, 63, 62, 61 Count back to subtract 65–49: 65–40=25, –5, –4 = 16 (used with no. line)

Count on to subtract 65–49: 49 +1, +10, + 5: +16 (used with no. line)

Count On and/ or Back

Count on to multiply 15x5: 15, 30, 45, 60, 75 45+39: 45+40=85; –1 = 84 65–49: 65–50=15, +1 = 16 24x18: 24x20,–(2x24) = 480–24= 432

Adjust one number and compensate

475÷5: 475÷10= 47.5, x2= 95 45+39: 40+40 = 80, +5 = 85, –1 = 84 Adjust two numbers and compensate 65–49: 70–50=20, –5, + 1 = 16 45+39: 44+40 = 84

Adjust and Compensate

Adjust two numbers 65–49: 26–20: 6 45+39: 40+30=70; 5+9=14; 70+14 = 84

Break up two numbers using place value 65–49: 60–40=20, –9=11, +5=16 65–49: 60–40=20; 5–9=-4; 20–4=16 45+39: 30+30=60, +15=75, +5=80, +4 =84 Break up two numbers using compatible

numbers 65–49: 50+15–30+15+4: 50–30=20; 15–15=0; –4=16 45+39: 45+30=75; +9 = 84 65–49: 65–40=25, –9: 16

Break up one number using place value

24x8: 20x8 + 4x8 = 160+32 = 192 45+39: 45+15=60, +15=75+5=80,+4=84 65–49: 65–30=, 35–10=25, –5=20,–4= 16 28x4: 25x4+3x4 = 100+12 = 112

Break Up Numbers:

Break up one number using compatible

numbers

344÷8: 320÷8 + 24÷8 = 40+3 = 43 Use a double or near double to add 20+21: 20+20=40 so +1: 41 Use a double or near double to subtract 50-25: double 25 is 50 = 25 Double to multiply by 2 4x2: double 4 : 8 Double, double to multiply by 4 12x4: 24, 48 Double, double, double to multiply by 8 15x8: 30, 60, 120 Halve to divide by 2 10÷2: half of 10: 5 Halve, halve to divide by 4 42÷4: 21, 10.5 Halve, halve, halve to divide by 8 148÷8: 124, 62, 31

Double and/ or Halve

Double and halve 14x5: 14x10, ÷2: 70 Think in multiples of ten 30+60: 3tens+6tens = 9 tens : 90 Use Place

Value Focus on relevant places 134+20: think 2 more tens: 154

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APPENDIX 2: STRATEGY CATEGORISATION FRAMEWORK STRATEGY CATEGORY ABBREVIATION CODES

Table A2 Strategy Categorisation Framework codes

Strategy Categorisation Framework Strategy

category

Strategy sub-categories Strategy code

Count on to add COA Count back to subtract CBS Count on to subtract COS

Count On and /or Back

Count on to multiply COM Adjust one number and compensate AD1C Adjust two numbers and compensate AD2C

Adjust and Compensate

Adjust two numbers AD2 Break up two numbers using place value BU2P Break up two numbers using compatible numbers BU2C Break up one number using place value BU1P

Break Up Numbers

Break up one number using compatible numbers BU1C Use a double or near double to add or subtract UD Double to multiply by 2 DM2 Double, double to multiply by 4 DM4 Double, double, double to multiply by 8 DM8 Halve to divide by 2 HD2 Halve, halve to divide by 4 HD4

Double and / or halve

Halve, halve, halve to divide by 8 HD8 Think in multiples of ten PVT

Use Place Value Focus on relevant places PVR

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APPENDIX 3: EXAMPLES OF BEISHUIZEN’S MENTAL COMPUTATION STRATEGIES LINKED TO THE SCF

Table A3 Beishuizen strategy categories with examples (in Klein, Beishuizen & Treffers, 1998) and

links to Strategy Categorisation Framework (Hartnett, 2007) for addition examples

Strategy label (Beishuizen)

45 + 39

Alignment to Strategy Categorisation Framework (Hartnett, 2007)

1010 40+30=70;5+9=14; 70+14=84 Break Up 2 Numbers using Place Value u1010 5+9=14;40+30=70; 14+70=84 Break Up 2 Numbers using Place Value N10 45+30=75; 75+9=84 Break Up 1 Number using Place Value uN10 45+9=54; 54+30=84 Break Up 1 Number using Place Value N10C 45+40=85; 85–1=84 Adjust 1 Number and Compensate A10 45+5=50; 50+34=84 Break Up 1 Number Using Compatibles 10s 40+30=70;70+5=75; 75+9=84 Adjust 2 Numbers and Compensate

Table A4 Beishuizen strategy categories with examples (in Klein, Beishuizen & Treffers, 1998) and

links to Strategy Categorisation Framework (Hartnett, 2007) for subtraction examples

Strategy label (Beishuizen)

65 – 49

Alignment to Strategy Categorisation Framework (Hartnett, 2007)

1010 60–40=20; 5–9=4 (false reversal); 20+4=24 (false answer) 50–40=10; 15–9=6; 10+6=16

Break up 2 Numbers Using Place Value Break up 2 Numbers Using Compatibles

u1010 15−9=6; 50−40=10; 10+6=16 Break up 2 Numbers Using Compatibles N10 65–40=25; 25–5=20; 20–4=16 Break up 1 Number Using Compatibles uN10 65−9=56; 56−40=16 Break up 1 Number Using Compatibles N10C 65–50=15; 15+1=16 Adjust 1 Number and Compensate

65–5=60; 60–40=20; 20–4=16 Break up 1 Number Using Compatibles A10 49+1=50; 50+10=60; 60+5=65 Answer 1+10+5=16 (adding on)

Count on to Subtract (Break Up 2 Numbers Using Compatibles)

10s 60–40=20; 20+5=25; 25–9=16 Adjust 2 Numbers and Compensate

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APPENDIX 4: END OF YEAR INTERVIEW QUESTIONS

Mental computation

• Personally, what mathematics did you learn during the year we worked together? (mental computation, maths processes, connections…)

The computation program

• Did they way the program was structured using the strategy categories each term work for you and the class?

• Did the Strategy Categorisation Framework make sense to you as a way of organising the teaching of mental computation strategies?

• Do you think the amount of learning time spent on computation that year was too much, not enough or just about right.

• How did you feel about the lack of the written algorithms in this year’s computation program?

• The program was planned as an overview at the start of the year and then we developed the detail as the lessons progressed. Were you comfortable with this way of planning or would you have preferred to have all the details planned before we started?

• Do you foresee yourself using the structure, strategies, activities etc that we used in the mental computation program in your teaching of maths next year and beyond?

Teaching computation

• How do you feel about the exclusive focus on the development of mental strategies for the computation program this year? I remember you had some hesitations on this exclusive focus at the beginning. Now that we have taught the full school year what are your thoughts about this focus?

• What have you learnt as part of us working together in relation to the connection between number sense and mental computation strategies?

• Did you learn anything about teaching maths by watching me teach lessons in your classroom to your students?

• This program focussed on getting children to discuss their strategies and share them aloud and to share their solution methods aloud with their classmates. How did you find this aspect of the program?

• Does the use of informal recording methods, which the students are being encouraged to use, give you as teacher, valid and reliable information to judge their progress? When they use an algorithm you can see what they have done. How is this approach different?

• We directly taught the strategies to the students. How do you think this worked as a teaching approach?

• Do you think the resources like the number boards and number lines added to the program’s effectiveness?

Student learning

• How does the students’ development of computation this year compare to previous years when you have taught Year 3?

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• How would you describe the student’s understanding of numbers, their number sense after the use of this learning program this year?

• Were you able to identify benefits for the students, as a group or individually, from the focus on mental computation or the way it was taught?

• I tried to focus on the development of flexibility in computation with the students, even though we taught a particular strategy each term. Do you think the students understood the importance of flexibility with mental computation?

• Do you think the students liked the teaching focus on the computation strategies? • Are there any students who you think would have preferred a traditional

algorithm focus like you have used in years past? • How has the focus on strategy based computation worked for the variety of

learners in your class?

Professional development

• As a professional development activity how do you rate this year of us working together compared to other professional development activities you have participated in? What if anything, was particularly beneficial for you and why do you think this?

• Are there any particular factors that you can identify that made this professional development activity work so well?

• How do you believe that you learn best in relation to your beliefs about teaching mental computation and maths in general and your teaching practices?

• In the end of year interview the classroom teacher commented on how the education advisor had worked with her to co-plan the details of the sequence of the program based on the Strategy Categorisation Framework categories.

• What other types of professional development have you participated in recently how do they compare to this experience?

• If you had an opportunity to participate in another one to one professional development activity in your classroom would you do it again, why or why not?

The study • How did you feel having me in your classroom one morning per week for a

whole year? • Did you see me as a visiting expert / co-teacher / outsider… other/ combination

of these? • We did a lot of reflecting and corresponding using email this year. This

correspondence was needed as data for the study. Did you find this process a burden?

• We ended up getting along very well. Do you think this positive relationship had an influence on the success of the teaching and the professional development activity?

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APPENDIX 5: TESTS TO MONITOR STUDENT LEARNING

Researcher developed tests

The items for the computation tests that I designed were chosen to provide a

range of difficulty for Year 3 students but in particular they were chosen so as to

offer opportunities for the use of a range of potentially efficient mental computation

strategies. The test items were designed with the strategy categories from the

Strategy Categorisation Framework in mind. There was not an expected ‘best

strategy’ for any question, although there were strategies that were considered

inefficient or not practical for particular questions. The strategies that were

considered potentially efficient for each item were described in terms of the Strategy

Categorisation Framework and are provided for each question in Appendix 5a and

5b.

The tests designed and administered in the study were: the Pre/Post-test of 24

items (12 addition, 12 subtraction); Mid-Year/Short-test of 12 items (8 addition, 4

subtraction). Some items were common to the Pre/Post Test and the Mid-Year/Short-

test. Each test contained only addition and subtraction items. The items are listed in

Table A5.

Table A5 Items on each of the researcher designed tests

Addition Items

Pre/Post Test

addition items

Mid-Year/Short

addition items

Subtraction items

Pre/Post Test

subtraction items

Mid-Year/Short subtraction

items 32+32 64–10 23+17 44–22 35+10 56–19 36+4 60–42

19+12 51–25 59+42 $10.00–$3.76 20+21 57–20 49+7 26–9

15+100 100–36 28+15 32–18 99+48 150–64

3+54+36 405–276 9+6 23–19

14+3 tens 45–27 37+40

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The items were presented horizontally in symbolic form without context,

similar to the study by Klein, Beishuizen and Treffers (1998) who found that

problems presented as a numerical expression only did not evoke variation in

solution procedures compared to those presented in a contextualised form. One

subtraction question involved money but there was no story to create a particular

context for that item. The questions were presented with an accompanying space for

the students to show their thinking or working out. Figure 3.1 shows the format of

the test.

32+32

20+21

23+17

49+7

Figure 3.1 Format of Pre, Post, Mid-year and Short tests

State-wide tests

During the year of the study, all students in Years 3, 5 and 7 in Queensland

sat state-wide mathematics tests. The tests had been conducted in Queensland

schools for five years prior to the study. As the students in this study were in Year 3

they had not have participated in this type of test previously. The test took place in

August. It comprised 36 questions and included items from each of the strands of the

mathematics syllabus. Reports were generated by the test administrators and were

made available to schools toward the end of the school year. Schools received a

whole of school report, a report for each class and an individual report for each

student that was sent home to parents. Data on the class report was used for this

study. This report included an analysis for each item per student showing their

responses, a scaled score for each strand and an overall scaled score. The scaled

scores enabled comparison across strands of the test as well as comparison of student

results across the years when they completed other tests (Years 3, 5 & 7). State

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average scores as percentage correct were also available per item together with a

state average scale score for each strand as well as an overall average for the test.

In the year of this study the state-wide Year 3 test had six specific mental

computation questions. These questions were presented orally and the students were

only permitted to record their answers on the student booklet. The instruction to

record only the answers was deliberate to ensure the focus was on the computations

being done mentally and to eliminate the possibility of students using written

methods, particularly traditional written methods. It was not possible to identify the

strategy chosen by the students to complete these computations as only the answers

were recorded. The test also had a variety of questions from across all the strands of

the Mathematics syllabus. The questions on the test are provided in Table A6.

Table A6 State-wide test mental computation questions.

Item No. Question 1. What is double 8? 2. Thirteen [take/subtract/minus] nine 3. Take twenty from forty-six 4. Ninety [take/subtract/minus] fifteen 5. 4 sixes 6. One hundred [take/subtract/minus] thirty-five

Data from the state-wide test reviewed for the students in the class and

contributed to the classroom teacher’s confidence that the mental computation

instructional program was not leading to any detrimental effects for the students in

the class. The class performed above the state average on all the computation

questions as well as on the test overall.

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APPENDIX 5A: LIST OF POTENTIALLY EFFICIENT STRATEGIES FOR EACH ADDITION ITEM ON THE ASSESSMENT INSTRUMENTS.

Item Instruments Potentially efficient strategies linked to Strategy Categorisation Framework

32+32 Pre/Post BU1P; BU1C; BU2P; BU2C; UD

23 + 17 Pre/Post AD2; BU1P; BU1C; BU2P; BU2C

35 +10 Pre/Post COA (in tens); PVT; PVR

36+4 Pre/Post COA; BU1P

19+12 Pre/Post AD1C; AD2C; AD2; BU1P; BU1C; BU2P; BU2C

59+42 Pre/Post AD1C; AD2C; AD2; BU1P; BU1C; BU2P; BU2C

20+21 All COA (in tens); BU1P; UD; PVR

49+7 All AD1C; AD2; BU1P; BU1C;

15+100 All PVR; BU1P

28+15 All AD1C; AD2; BU1P; BU1C; BU2P; BU2C

99+48 All AD1C; AD2C; AD2

3+54+36 Pre/Post BU1P; BU1C; BU2P; BU2C

9+6 Mid/Short AD1C; AD2C; AD2

14+3 tens Mid/Short COA (in tens); PVR

37+20 Mid/Short COA (in tens); BU1P; BU1C; BU2P; BU2C

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APPENDIX 5B: LIST OF POTENTIALLY EFFICIENT STRATEGIES FOR EACH SUBTRACTION ITEM ON THE ASSESSMENT INSTRUMENTS.

Item Instrument Potentially efficient strategies linked to Strategy Categorisation Framework

64–10 Pre/Post CBS (in tens); PVT; PVR

44–22 Pre/Post COS; UD; BU1P; BU1C; BU2P; BU2C

56–19 Pre/Post COS; AD1C; AD2; BU1P; BU1C

60–42 Pre/Post COS; AD1C; AD2; BU1P; BU1C

51–25 Pre/Post COS; AD1C; AD2; BU1P; BU1C;UD

$10.00–$3.76 Pre/Post COS; AD1C; AD2; BU1P

57–20 Pre/Post CBS (in tens); COS; PVT; PVR

26–9 Pre/Post AD1C; AD2; BU1C

100–36 Pre/Post COS; BU1P

32–18 All COS; AD1C; BU1P; BU1C

150–64 All COS; BU1P; BU1C

405–276 Pre/Post COS; BU1P; BU1C

23–19 Mid/Short COS; AD1C; AD2; BU1P; BU1C

45–27 Mid/Short COS; AD1C; AD2; BU1P; BU1C

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APPENDIX 6: OVERVIEW OF MENTAL COMPUATION INSTRUCTIONAL PROGRAM

Strategy Term / Week

Lesson focus /Content

1/1 Pre test Count on and back (number sense focus)

1/2 Count on and back orally in 1s, 2s, 10s (on the decade)

Count on and back (number sense focus)

1/3 Count on and back in 10s on and off the decade

Count on and back (number sense focus)

1/4 Addition and subtraction of multiples of ten linked to counting on in tens

Counting on to add Counting back to subtract

1/5 Count on and back in tens and ones using a number board and alternative number boards

Counting on to add 1/6 Game developed to link counting on and back in tens and ones to

Counting on to add Counting back to subtract

1/7 Addition and subtraction by counting on and back in tens and ones using a number board

Counting on to add 1/8 Addition of 9 and 11 using a number board –(9 as 10-1 and 11 as 10+1)

Counting on to add 1/9 Counting on in tens and ones to add using an empty number line to record jumps

Adjust two numbers (addition)

2/10 Adding 9 and a one-digit number using two ten frames and counters to make ten

Adjust two numbers (addition)

2/11 Adding 2 two-digit numbers by adjusting both (do opposite to each number) and why

Adjust two numbers (subtraction)

2/12 Investigate the difference in the strategy between addition and subtraction

Adjust one number and compensate (addition)

2/13 Introduce strategy for numbers with 9 in ones place focus on adding the closest multiple of ten and then adjusting

Adjust one number and compensate (addition)

2/14 Use number boards to highlight which way to adjust after a compensation

Adjust one number and compensate (subtraction)

2/15 Investigate this strategy for two-digit subtraction

Adjust two numbers and compensate

2/16 Investigate adjusting two numbers and compensating for both adjustments

2/17 Mid-year test Break Up One Number using Place Value (addition and subtraction)

3/18 Revisit number boards as for Counting On/Back to add and subtract two-digit numbers broken into tens and ones. Use variety of worksheets for different students

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Strategy Term / Week

Lesson focus /Content

Break Up Two Numbers using Place Value (addition)

3/19 Investigate breaking up two numbers into place value parts and adding the parts then adding the totals

Break Up Two Numbers using Place Value (addition)

3/20 Investigate recording breaking up two numbers using place value for addition using arrows

Break Up Two Numbers using Place Value (subtraction)

3/21 Include subtraction examples with no bridging ten for this strategy. Investigate what happens when bridging ten is needed

Break Up One Number using Place Value

3/22 Compare break up two numbers for subtraction to break up one number (subtrahend) for efficiency and discuss

Break Up One Number using compatible numbers (addition and subtraction)

3/23 Investigate breaking up numbers using compatible numbers for both addition and subtraction examples

Break Up Two Numbers using Place Value and Use doubles

3/24 Introduce double-digit addition doubles (e.g. 24+24) and strategies involving breaking up the two numbers, doubling the parts and working out the total

3/25 Year 3 test Use doubles (x2) 3/26 Link doubling (addition) to multiplication

(x2) using one-digit examples (basic facts) Use doubles (x4) – double double

3/27 Link double doubling with x4. Show with basic facts and extend to two-digit numbers

Use doubles (x8) – double, double doubling

3/28 Extend doubling to x8. Allow students to choose any number to multiply by 8 using doubling e.g. 10001x8

Multiple strategies 4/29 Provide many two-digit number examples for addition and subtraction and students choose a strategy and share their choices

4/30 Short 2 test Use doubles (÷2) – halve 4/31 Link halving to division by 2 to find half Use doubles (÷4, ÷8) – halve, halve, halving

4/32 Extend halving to divide by 4 and 8. Including some examples where result is .5

Multiple strategies / revision

4/33 Rotation of games used during the year for different strategies

Multiple strategies / revision

4/34 Play “I have… who has” whole class games based on different strategies.

4/35 Post test

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APPENDIX 7: SAMPLE LESSON AND COMMUNICATION BETWEEN THE CLASSROOM TEACHER AND EDUCATION ADVISOR

The following description is of a demonstration lesson conducted by the

education advisor. The test of the lesson outline was sent to the classroom teacher by

email prior to the demonstration lesson. Below the lesson is some text from email

communications during the week following the demonstration lesson. This sample

lesson shows the detail of lessons sent to the classroom teacher and which formed the

basis of the mental computation instructional program throughout the year.

WEEK 8: (TERM 2)

Strategy in focus: Adjusting and Compensating – Adjusting two numbers to add single digit numbers using tens frames. Conceptual focus:

• To introduce the Adjust Two Numbers strategy • To link to number sense understandings relating to single-digit numbers that total 10

to the use of this strategy Materials: per student – 1xt blank ten frame, ten + ten frame for addition (one full ten frame + empty ten frame), ten frames for addition (two empty ten frames with + between), counters of 2 different colours (10 of each colour), dice, scrap books. Activities: Part 1: • Introduce the students to tens frames (if not already familiar). Make ten in different ways

using coloured counters (quick intro) • Using counters students model simple additions where a single digit number is added to

a full tens frame for example, 10+4=14 using the full ten frame + empty ten frame activity sheet. To help them to see the connection to place value concepts as well as addition with 10. I will get them to roll a die to randomly generate a single digit to add to the full ten frame.

• Discuss relationship between adding a single digit to one ten. Focus on them seeing how easy these additions are to work out.

Part 2: • Use ten frames for addition worksheet to model single-digit additions using a number

close to ten i.e. 9. Students will show additions like 9+6 with 9 counters of one colour in one ten frame and 6 of another colour in the other.

• Discuss how to make this computation easier.. look for links to Part 1 activity. Help students to see 9+6 is the same as 10+5 by sliding one counter from the 6 to ‘make the ten’ leaving +5.

• Try some other examples (all single digits) include some using 8+ if they show good understanding

• I might get them to record some of these adjustments as equations i.e. 9+6=10+5. I’ll see how the lesson goes whether we get to this.

Possible Follow up activities: • More examples of 9+ single digit and 8+ single digit using ten frames if needed (or if

desired, the students might really like using these if they are novel)

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• Moving to double-digit examples like 19+6. I have a worksheet with a full ten frame and an empty one (for teen numbers) + an empty ten frame (see attached files) that could be used for this.

• Progress to other two-digit numbers ending in 9 or 8. Focus without number boards. Hopefully they will be confident with the strategy enough to see the generalisability from the single digit numbers. Any students who have any trouble with the larger numbers can stick to single digits or teen nos as needed.

Reflection (by education advisor via email): The first part of the lesson took very little time which would be expected for Yr 3 students with examples only using single-digits. The use of the ten frames was probably not completely necessary but it was a nice to use some hands on to start with to get the ones who like to fiddle something to manipulate. The connection to place value seemed to make sense to many of them.. those at Paul’s table and Eve’s table in particular. Some of the boys thought that using the tens frames and counters was too easy for them although I felt that using helped on some of the slower kids like Michelle, Rachel, Linda and Lincoln. I think them seeing all the class working with the materials made them more comfortable. There will be some students who will probably require the tens frames for further lessons on this strategy (these ones and probably some others) (Education advisor, Tuesday, Week 8) Reflection (by classroom teacher via email): This lesson seemed so simple. I am a little amazed that you stuck with it, especially as some of the brighter ones were saying how easy they thought it was. The ten frames were powerful and because it was treated a bit as a game they put up with it. They did seem to like manipulating the counters and rolling the dice. I actually think the ten frame did help the girls working at the table near my desk (Michelle, Rachel and Lisa) particularly well. The follow up lesson ideas sound good. I will get them to record the equations as you suggested in the lesson but didn’t get to do. I expect they will all be able to manage double-digit +9 ones by the end of the week. (Classroom teacher, Tuesday, Week 8) Reflection (by classroom teacher via email): Today [Wednesday] I gave them some examples to work on that were 18+? and 19+?. We used the ten frames again and as we thought some of them didn’t need these and were able to work out the adjusted additions easily. One group stayed with the single digit numbers. … [teacher aide’s name removed] worked with these students. I think they are seeing that the benefit is making ten then adding. The students I worked with recorded the equations. This wasn’t as easy as I thought. They were a bit amazed that you could write equations with additions on each side. (Classroom teacher Wednesday, Week 8) Communication (by education advisor via email): I have made some simple matching cards for the students who are still working with the ten frames. They show pictures of the ten frames as 9+ and then the corresponding 10+ addition. These should help those who are still using the physical ten frames to perhaps imagine the counters moving rather than actually moving them. They can match them, play snap or whatever. I suggest you copy the 9+ set onto one colour paper and the 10+ ones onto another colour so they see they are different. Then they know they have to match one of each colour. File is attached. (Education advisor, Wednesday, Week 8) Communication (by classroom teacher via email): The matching cards worked really well. I didn’t get a chance to use them until today. I think they helped. I copied enough so the whole class could use them. Even though many didn’t need this they all wanted to play the game. The coloured paper was great. I might copy some on the same colour for those who don’t need the matching clues. (Classroom teacher, Friday, Week 8)

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APPENDIX 8: STUDENT LEARNING DATA

Making thinking visible

Data about the students’ ability to record their thinking and strategy use were

gathered from each of the assessment instruments designed by the education advisor

across the year. On each instrument the students were instructed to show, in writing,

how they worked out each item. On the Pre-test, out of all the individual item

responses, only 24 were recorded in a way that enabled identification of the

strategies used. Many of the other students had attempted to show or describe their

thinking but the strategy used for the computation was not identifiable. Further

analysis of the data indicated that these 24 item responses with clearly described

strategies came from just nine students and that the greatest number of clearly

described strategies by any student was three.

The responses made by the students on the assessment items were also

analysed to identify the types of responses made. These response types were

categorised as: no response, where the question was left completely unattempted;

just answer, where the student had recorded the result of their computation but made

no other recordings; strategy identifiable, where the strategy used was clearly

described and discernable; and strategy not identifiable, where the student had

attempted to record the strategy they had used but it was not possible to elicit which

strategy had been used. Each individual response to an item was analysed and

allocated to one of these categories. In Figure A1 data is presented showing the

percentage of each of the response types for the Pre-test and the Post-test to show the

change across the year of the mental computation instructional program.

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Figure A1 Percentage of response types on the pre and post tests.

A further analysis of the range of different ways the students chose to

describe their strategies revealed three particular strategy recording approaches.

Some recordings showed use of the resources and methods demonstrated in lessons,

including number lines and number boards. Other responses showed use of the

Strategy Categorisation Framework strategy labels to describe their strategies while

others had used their own words to describe how they had solved a computation.

Figure A2 shows two examples of student responses where there was evidence of

strategies and resources which had been explicitly taught during lessons. The first

example shows use of the empty-number line and the second shows use of a number

board. Figure A3 shows two examples of students using the Strategy Categorisation

Framework category labels and Figure A4 shows student responses where the

explanation is in words or a format devised by the student.

Figure A2. Use of recording matching methods or tools demonstrated or discussed during lessons (i.e. empty number line and number boards).

Response Types

0 10 20 30 40 50 60 70

No response Just answer Strategy identifiable

Strategy notidentifiable

perc

enta

ge

Pre test Post test

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Figure A3. Student use of Strategy Categorisation Framework labels to describe strategies used

Figure A4 Students’ own description of the thinking and strategy used.

Once the students were more consistently able to record their thinking, from

the mid-year test onwards, the responses on assessment items and during lessons also

provided ways to identify students who understood the computation strategy and the

related number sense concepts as well as those who were using a strategy incorrectly

or who had other misconceptions. This provided another observable change for the

classroom teacher in relation to her knowledge of the students and their abilities.

Student accuracy

Student accuracy was measured as a count of the number of correct responses

to the items on the tests designed by the education advisor, as well as the Year 3

state-wide test. Data were collected from each of the instruments designed by the

education advisor and implemented across the school year. The accuracy data

comprised a count of the total number of correct responses for the whole class. The

count of accurate responses was converted to a class percentage out of the total

number of responses. Students who did not provide an answer were considered as

incorrect for this set of results. Table A7 shows the accuracy data as the percentage

of items answered correctly for each of the formal instruments across the year.

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Table A7 Percentage of items answered correctly on all instruments

Pre-test % Mid-year % Short 1 % Short 2 % Post-test %

Addition items 74.7 76.3 80.2 88 90.4 Subtraction items 28.2 44.6 46.6 47 55.7 All items 51.4 65.8 69 74.3 72.7

From this data it was clear that the students were progressing and their ability

to answer the range of items on the instruments continued to improve. The data

reassured the classroom teacher that the students were progressing in relation to the

instructional focus of the mental computation strategies program. The classroom

teacher considered the progress of the students based on the accuracy data and

compared them with other Year 3 classes she had taught in previous years.

As a further check on student accuracy, data from the Queensland state-wide

test completed by all Year 3 students in the state in August of the year of the study

was examined. A score for each strand was identified in the data provided to the

school, as well as an overall numeracy score (see Figure A5). The class scored

higher on average than the state average for each strand as well as overall.

Figure A5 State test data for all content strands and overall numeracy scores.

The state-wide test also contained six specific mental computation questions

completed with no recording so as to ensure the answers were found using mental

strategies. These items were provided in Table A6 in Appendix 5. The class average

accuracy, taken as the percentage correct, was higher overall than the state average

(see Figure A6) on all of the questions including this more difficult question.

Year 3 State-wide test

0 100 200 300 400 500 600 700

Number Measurement & Data

Space OverallNumeracy

Strands

Class State

Ave

rage

scal

e sc

ore

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Figure A6 Year 3 state-wide test percentage correct for each of the mental computation items.

Overall, the state-test data showed that the mental computation strategies

focussed instructional program being used in this study was not disadvantaging the

students in relation to their ability as measured by this instrument and as compared to

students across the state.

Student efficiency

Efficiency of strategy use is a difficult construct to measure without bias. To

analyse the student responses in terms of efficiency in this study, decisions needed to

be made categorising the strategy choice for each item as efficient or inefficient.

There was not considered to be a ‘best’ strategy for any item. Appendices 5a and 5b

list all items from each instrument and the strategies considered potentially efficient

for that item described as sub-categories of the Strategy Categorisation Framework.

There were particular strategies for each item which were considered inefficient.

Counting On to Add (in ones) for any addition beyond add four (e.g., 23+17) was

considered inefficient. Some other strategies were not considered appropriate for

Year 3 students. For example, while it is possible to complete 32–18 using the Break

Up Two Numbers Using Place Value strategy by considering 30–10=20 and then 2–

8= -6, the students in Year 3 involved in this study would have had no experience

with negative numbers and would not be likely to consider this strategy. Therefore

this strategy was not listed as efficient for this item.

The strategy category Use Place Value was not explicitly taught to the

students during the year of this study. However it was considered an efficient

Year 3 state-wide

0 20 40 60 80 100

1 2 3 4 5 6

Mental computation question

% c

orre

ct

Class State

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strategy for some of the items due to the use of multiples of ten which could prompt

students to either Think in Multiples of Ten or Use Relevant Places. For example, for

35+10 students could think 35+10 is 35 and 1 more ten, so they just need one more

in the tens place, 45 (Use Relevant Places) or 57–20 is 5 tens and 7 ones – 2 tens

which is 3 tens and 7 ones, 37 (Think in Multiples of Ten). Some students did choose

to use the Use Place Value strategies and presented their work in a way it was

possible to identify their use.

The number of students who chose strategies that matched the list of

potentially efficient strategies were counted and compared to the total number of

students whose strategy was identifiable from their response. Table A8 displays

these data, noting the number of efficient strategies per instrument, the total number

of clearly identifiable strategies the efficient strategies as a percentage of those

identifiable. These data show an increase in the number of efficient strategies. Data

are shown for addition items and then for the subtraction items on all instruments.

Table A8 Efficient strategies chosen of the identifiable strategies

Instrument

Number of efficient

strategies chosen

Number of clearly

identifiable strategies

Percentage

Pre-test 10 20 50 Mid-year 81 144 56.3 Short 1 79 137 57.7 Short 2 79 139 56.8 Post-test 170 223 76.2

Addition items

Pre-test 1 4 25 Mid-year 27 56 48.2 Short 1 34 72 47.2 Short 2 24 59 40.7 Post-test 97 195 49.7

Subtraction items

A difficulty experienced in identifying use of efficient strategies was that the

strategies used needed to be identifiable before they could be categorised as efficient

or not. It could not be assumed that because a student’s strategy choice was not

identifiable that they were using either an efficient or inefficient strategy. The

number of identifiable strategies was close to 70% from the Mid-year test to the

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Post-test. The percentage of efficient strategies increased on the Mid-year test and

remained similar for addition items with an increase in percentage of efficient

strategies on the Post-test. For subtraction the increase by the Mid-year was

maintained through to the Post-test. As the students’ experience with mental

computation strategies increased across the year so did their choice of efficient

strategies.

Accuracy and efficiency

A further consideration explored was whether the choice of efficient

strategies resulted in greater accuracy of the solution. In the Pre-test, of the strategies

which were identifiable, many involved the inefficient use of counting by ones. The

efficiency data was combined with the accuracy data to examine whether the use of

inefficient strategies led to correct or incorrect responses. A count was made of the

number of efficient strategies resulting in a correct or incorrect response and the

same was done for the inefficient strategies for each instrument. Table A9 shows the

accuracy of efficient and inefficient strategies for addition items and Table A10

shows the data for the subtraction items. The tables also include the number of

clearly identifiable strategies as this provides a picture of the total number of

strategies being considered. If a strategy was not identifiable these decisions were

not possible.

Table A9 Accuracy of efficient and inefficient strategies for addition items

Instrument

Number of efficient

strategies chosen

Number of inefficient strategies

chosen

Number of clearly


Pre-test 9 1 8 2 20 Mid year 80 1 58 5 144 Short 1 78 1 53 5 137 Short 2 75 4 56 4 139

Addition items

Post-test 169 1 50 4 224 correct response incorrect response

Table A10 Accuracy of efficient and inefficient strategies for subtraction items

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Instrument

Number of efficient

strategies chosen

Number of inefficient strategies

chosen

Number of clearly


Pre-test 1 0 1 2 4 Mid year 22 5 16 13 56 Short 1 20 14 20 18 72 Short 2 19 5 14 21 59

Subtraction items

Post-test 76 21 43 55 195 correct response incorrect response

Student flexibility

Flexibility was described in Section 2.5.4 as the possession of a repertoire of

strategies and knowing that there is more than one possible way to perform a

computation. The flexibility of the students was investigated by examining the

number of different strategies used by the class as a whole group and by individual

students on the instruments administered across the year.

Flexibility of the class

The flexibility of the class as a whole was measured by counting the number

of different strategies used for each item on each instrument by the class as a whole.

For this analysis, strategies were described using the sub-categories of the Strategy

Categorisation Framework. Each item appeared on at least two instruments, with

some items on every instrument. This repetition of items allowed for analysis of the

development of flexibility across the whole year by comparing strategies used on

particular items on different instruments. Data in relation to the flexibility of the

class for the addition items is presented in Table A11 and for subtraction items in

Table A12. The number of different strategies identified for each item across all

instruments was also tallied. The number of times a particular strategy was chosen by

students is not considered here, just the number of different strategies per item. It

needs to be noted again that the count of the number of strategies used on each item

was limited to the items where the strategy used was identifiable from the students’

recordings.

Table A11 Different strategies evident in addition items for each instrument and in total

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Question Pre-test Mid Year Short 1 Short 2 Post-test Across all instruments

32+32 2 – – – 5 5

23+17 3 – – – 2 3

35+10 1 – – – 6 6

36+4 1 – – – 4 4

19+12 1 – – – 5 4

59+42 0 – – – 5 5

20+21 1 4 6 5 5 6

49+7 1 5 5 6 6 7

15+100 1 4 4 6 3 6

28+15 0 3 5 6 5 7

99+48 0 4 6 5 5 6

3+54+36 0 – – – 2 2

9+6 – 6 4 6 – 6

14+3 tens – 5 5 5 – 6

37+40 – 4 6 5 – 6

The ability of the students to record their thinking and strategy use in an

identifiable manner increased notably between the Pre-test and the Mid-year test.

The data in Table A1 shows that from the Mid-year test the number of different

strategies being used by different students on all the addition items varied. This

variety clearly shows the class was flexible in their strategy use. It is also noted that

it is likely that not all the strategies used were particularly efficient.

For the subtraction items the pattern is similar. Table A10 shows the

flexibility of the whole class for the subtraction items. The range of strategies used

by the students increased after the Mid-year test when their recording of strategies

also increased. The other notable observation of the data in both Table A11 and

Table A12 was that the greatest variety of strategies evident across all items was

seen on the Post-test indicating that the goal of the program to introduce the students

to a wide range of possible computation strategies had been successfully achieved.

Table A12 Different strategies evident in subtraction items for each instrument and in total

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Question Pre-test Mid Year

Short 1 Short 2 Post-test Across all instruments

64–10 0 – – – 5 4

44–22 0 – – – 3 3

56–19 0 – – – 6 6

60–42 1 – – – 3 3

51–25 0 – – – 5 5

$10.00–$3.76 0 – – – 3 4

57–20 1 – – – 6 7

26–9 2 – – – 5 6

100–36 0 – – – 2 2

* 32–18 0 4 5 5 3 7

* 150–64 0 5 4 4 3 6

405–276 0 – – – 2 3

23–19 – 7 6 6 – 7

45–27 – 6 5 4 – 6

Flexibility of the individual students

The flexibility of each student was measured by counting the number of

different strategies used by that student on each of the instruments. The data in the

tables below also include the total number of different strategies that each student

used across all the instruments as a measure of their personal flexibility. Table A13

shows the individual flexibility data for the addition items and Table A14 shows the

data for the subtraction items. The column “ability” shows a rating of each student’s

mathematical ability provided by the classroom teacher. The class teacher was asked

to rate each student’s mathematical ability in the end of year interview. It was not

felt that a test score would provide an overall rating of their ability and the education

advisor and classroom teacher did not want to subject the students to another formal

assessment. Therefore, the classroom teacher was asked to base her rating on her

experience with the students across the year of the study as well as her many years of

experience teaching Year 3 students. The overall mathematical ability of each

student was described as low, medium or high.

A further consideration in relation to the data presented in Table A13 and

A14 was that for a strategy to be counted as different it needed to be identifiable by

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the researcher. As reported previously, the students’ ability to describe their strategy

use increased by the Mid-year test and remained quite stable for the rest of the year,

allowing identification of individual flexibility after the Pre-test.

Table A13 Individual student flexibility for addition items per instrument

Student Pre-test Mid year Short 1 Short 2 Post test Total Ability Rachel 0 3 0 1 1 3 L Michelle 1 2 2 1 2 3 L Lisa 0 1 2 3 1 5 L Linda 0 2 2 3 3 6 L Warren 0 2 0 1 3 4 L-M Melissa 1 3 2 a 3 3 L-M Lauren 0 1 1 2 1 2 L-M Lincoln 0 2 2 2 3 5 L-M Lenny 0 3 0 0 1 3 L-M Elizabeth 1 3 2 2 3 4 L-M Sarah 0 5 2 3 1 6 M Maddison 1 2 3 4 3 5 M John 0 a 2 2 2 3 M Ian 0 2 2 0 2 4 M Eleanor 0 3 3 4 1 5 M Eliza 1 0 1 2 4 5 M David 0 3 3 1 a 6 M Chris 0 4 3 3 3 6 M Amber 0 a 3 4 5 5 M Eve 0 2 3 4 3 5 M-H Bridget 1 3 3 a 3 6 M-H Wayne 2 3 4 2 4 5 H Tim 0 4 3 3 5 5 H Trevor 0 4 3 3 3 5 H Paul 1 1 1 a 0 2 H Oscar 0 2 4 3 4 5 H Owen 2 1 2 3 4 6 H Mark 0 3 4 3 3 6 H Lucas 1 3 5 3 3 6 H

a indicates student was absent

When flexibility is considered to be the possession of a variety of strategies

the addition data in Table A14 shows that the individual students in this class

generally were flexible. Of note is that the total number of different strategies used

by students across all instruments did not align with the ability level as described by

the class teacher. In this study there was no evidence that flexibility was related to

mathematical ability.

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Table A14 Individual student flexibility for subtraction items per instrument

Student Pre-test Mid year

Short 1 Short 2 Post test Total Ability

Ruby 0 1 0 0 1 1 L Michelle 0 0 1 1 0 1 L Lisa 0 0 1 1 1 3 L Linda 0 1 1 1 0 3 L Warren 0 1 0 1 0 2 L-M Melissa 0 1 0 a 1 2 L-M Lauren 0 1 0 1 1 1 L-M Lincoln 0 1 1 1 5 5 L-M Lenny 0 1 0 1 3 4 L-M Elizabeth 0 1 2 2 3 3 L-M Sarah 0 3 2 2 1 4 M Maddison 0 0 1 1 0 1 M John 0 a 2 2 2 3 M Ian 0 1 2 0 0 2 M Eleanor 0 1 2 1 3 6 M Eliza 0 1 1 1 5 5 M David 0 2 1 0 a 3 M Chris 0 2 3 1 2 6 M Amber 0 a 2 3 6 5 M Eve 0 2 2 1 3 5 M-H Bridget 1 1 2 a 3 5 M-H Wayne 0 1 2 1 3 3 H Tim 0 3 2 2 2 3 H Trevor 2 1 2 3 3 3 H Paul 0 1 1 a 0 1 H Oscar 0 0 3 3 6 7 H Owen 0 1 1 1 3 3 H Mark 0 2 2 2 4 5 H Lucas 0 3 2 2 2 3 H

a indicates student was absent

Another opportunity to measure the individual students’ flexibility as use of a

variety of strategies occurred through the administration of the Short- test twice in

the middle of Term 3, with a one week interval. This double administration of the

same instrument provided an opportunity to analyse the students’ flexibility by

assessing whether they made different strategy choices on the same items, presented

in exactly the same format with only a short time between. The assumption was that

a student who was less flexible would use the same strategy on the same items

presented in the same format. To enable a comparison the strategy used by each

student needed to be identifiable on both assessments. This was not the case with all

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responses. Examples of the use of the same strategy for the same item are shown in

Figure A7 and those who used a different strategy for the same item as presented in

Figure A8.

Figure A7 Student use of the same strategy for the same item on Short-test 1 and 2

Figure A8 Student use of the different strategy for the same item on Short-test 1 and 2

For cases when a strategy was identifiable for an item on both instruments, a

count was made whether each student used the same or a different strategy for the

same item. Table A15 shows these data. Students not accounted for in this table were

either unclear in their response on one or both of the items, just recorded an answer

on one or both items or omitted the item on one or both of the instruments.

Table A15 Strategy use on Short-test 1 and Short-test 2 instruments

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Item No. of students who used same strategy on both

instruments

No. of students who used a different strategy on

each instrument

9+6 11 5

14+3 tens 7 4

37+40 9 9

20+21 10 3

49+7 9 5

15+100 3 4

28+15 2 4

99+48 4 1

23–19 8 6

45–27 8 7

32–18 6 2

150–64 4 1

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