AN INVESTIGATION OF THE EFFECT OF THE SHELTERED INSTRUCTION OBSERVATION PROTOCOL (SIOP) MODEL IN TEACHING

MATHEMATICS: A CASE STUDY AT A TONGAN SECONDARY SCHOOL

By

Tamaline Wolfgramm Tuʻifua

A thesis submitted in fulfilment of the requirements for the degree of Master of Arts.

Copyright © 2014 by Tamaline Wolfgramm Tuʻifua

School of Education Faculty of Arts, Law and Education The University of the South Pacific

June 2014

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DECLARATION

Statement by Author I, Tamaline Wolfgramm Tuʻifua, declare that this thesis is my own work and that, to the best of my knowledge, it contains no material previously published, or substantially overlapping with material submitted for the award of any other degree at any institution, except where due acknowledgment is made in the text. Signature …………………………… Date………………………… Name ………………………………………………………………… Student ID No. ……………………………………………………… Statement by Supervisor The research in this thesis was performed under my supervision and to my knowledge is the sole work of Tamaline Wolfgramm Tuʻifua. Signature…………………………….. Date ……………………….. Name..……………………………………………………………….. Designation ………………………………………………………….

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ABSTRACT

The performance and achievement of Tongan students in secondary mathematics has been of

concern for some time. In the last five years, external examination results have been declining

both in the number and the quality of passes. This trend has been observed at the high school

where this study is based. In the search for ways to enhance students’ learning and teachers’

classroom performance, the school decided to adopt the Sheltered Instruction Observation

Protocol (SIOP) Model, a model of teaching that was initiated to enhance the linguistic needs

of second language learners in schools in the United States of America (US). The model has

been successfully used to provide high-quality instruction that enhances meaningful learning

for students. The hallmark of this model is the quality of practices and lessons that

systematically develop students’ content knowledge and academic skills while also promoting

their English language skills. This is done through integrating language objectives into the

content subjects. The SIOP model engages teachers across the curriculum to pay attention to

the language of the subject: how it is used and how it supports and sustains learning.

Developing students’ language skills is an essential element of mathematics teaching and

learning. This is particularly critical for Tongan students and teachers who are second

language users of English and have to learn in English. Both groups benefitted from the

sheltered instruction strategies which presented mathematics in English in ways that enabled

them to comprehend content and develop their English language skills simultaneously. The

model required the development of language skills to be a consistent part of daily lesson plans

and delivery. While the general belief at this Tongan secondary school was that the SIOP

Model had made significant difference in the knowledge and practice of mathematics teachers

as well as the performance of students, there is no tangible data to support this. This study

was designed to fill that gap by providing professional development on the model and then

investigating its effect through an in-depth study of both teachers’ and students’ experiences

in mathematics lessons.

The research used a qualitative design and the Case study approach to get an in-depth

understanding of this phenomenon. The participants were three mathematics teachers at the

school and 12 randomly selected students. Data was collected from video recording of three

classroom observations of each teacher for a total of nine classroom observations, individual

talanoa with the mathematics teachers, and a group talanoa with the 12 students. The eight

components of the SIOP Model - Lesson preparation, Building Background, Comprehensible

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Input, Strategies, Interaction, Practice and Application, Lesson Delivery and Review and

Assess - were tested in the nine classroom observations. .

When asked about major challenges in mathematics learning, most students pointed to (i) the

mathematics teacher and (ii) word problems. The mathematics teacher was the number one

challenge and factor. Students compared different teachers and described why they preferred

one over another. There was preference for teachers who took time to explain things clearly

and simply. Students enjoyed engagement in mathematical activities and quizzes in

comparison to boring lectures. They all enjoyed group work because it enabled them to talk

freely to each other and learn from each other. Confusion and lack of understanding of word

problems was a constant challenge and students felt that their teachers did not sufficiently

help them in this regard. Teaching mathematics in the Tongan language and code-switching

were common practices in Tongan classrooms and students agree that these should continue

and proposed as reasonable practices. The SIOP Model provided frequent opportunities for

student interaction, increased student engagement, enabled teachers to use clear and

appropriate language that enhanced students’ comprehension, increased waiting time for

students to show their work, and enhanced teachers’ classroom performances.

All teachers were observed three times and over the three sessions there was notable

improvement in their performances. Their lessons were recorded on video and played back to

them after each session. An area that teachers found challenging was questioning skills –

asking the probing why and how questions and not succumbing to the temptation of answering

their own questions. Another was to minimise on the lecture style of teaching and to use more

innovative strategies that involved students more. Relinquishing that control was not easy but

there was obvious improvement. Other areas that were attended to included writing good level

objectives, posing good thinking questions, practising ‘wait time’ to allow students to answer

(and not answering their own questions), facilitating group work, providing good feedback,

practising good motivational strategies and correct explanations. Altogether, the results were

supportive of the ability of the SIOP model to enhance teaching and learning, primarily

through enhancement of the teachers’ own pedagogical skills.

The researcher is a keen SIOP practitioner who has advocated for the SIOP model as a means

towards high-quality instruction to enhance meaningful learning not only in mathematics but

in other subjects and disciplines. The model has been successfully implemented in some

schools in the Pacific including American Samoa, the Federated States of Micronesia, Guam,

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Hawaii, the Republic of Palau, and the Republic of the Marshall Islands. In all countries,

mathematics is a status subject which, together with English, serves as a filter to higher

education and many sections of the workforce that are highly technological. The findings of

this study have important implications for improving English language skills, raising

mathematical performance and achievement in other Tongan schools and the country as a

whole, raising performance and achievement in other subjects, ensuring ongoing professional

development of teachers, curriculum development, and teacher education.

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ACKNOWLEGMENT

The completion of this thesis was made possible with the support and contribution of so many

people. I would like to give my first and foremost gratitude to my Heavenly Father for all the

divine help he has provided for this work. I would not be able to finish this project without

His divine guidance.

I was blessed to have great supervisors who spent valuable time in providing constructive

feedback and were patient in guiding me through this piece of work. To Dr. Salanieta

Bakalevu and Dr. ‘Ana Hauʻalofaʻia Koloto, “Mālō ‘aupito” for all that you have done.

Without your contribution, I would not have been able to complete this work.

To the other supervisors at USP Tonga campus: Dr. Seuʻula Johansson Fua, Dr. Moʻale

‘Otunuku and Dr. Masasso Paunga, thank you for all your support and patience with me and

our MA cohort. To Dr. Ruth Toumuʻa, thank you also for your valuable feedback, and to

‘Ana Heti Veikune thank you for proof reading the final draft of this thesis.

I am grateful to the Principal and the administrators of the study school in allowing this study

to be done at their school and for providing sole funding for this study. That financial support

was critical.

To the teachers who participated in the study as well as the 12 students, mālō ‘aupito for your

lototō, mamahiʻi meʻa, ‘ofa, tauhi vā, and support that you have showed which enabled the

completion of this research.

I would like to acknowledge special friends and colleagues who assisted me in either

transcribing data or proofreading various pieces of this work: Mele Taumoepeau, Lesieli Nai,

Mele Faʻoliu, Kahealani Nau, ‘Amelia Uata, Kalolaine Nuku, and ‘Ana Matileti Maʻu. Mālō

‘aupito e tokoni mo e ‘ofa.

I thank the staff of the Examination Unit - Kasa Kilioni and Uinimila Kakapu for your kind

assistance; TIOE staff - Liuaki Fusituʻa and Siofilisi Hingano for your great support; CEO-

Secondary schools, Manu ‘Akauʻola and Hēpeti Takeifanga for your quick response; and

‘Evaline Haʻangana for your time and support during the data collection period for this

project.

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I would like to say “Mālō ‘aupito” to my husband Lolomanaʻia Tuʻifua and our five children

Siunipa, Lolomanaʻia Jr, Penisimani, Latai, and ‘Ofa-he-lotu for all their support, prayers and

patience while I was pursuing this study. My daughter, Latai always asked me the same

question every day “Mom, when are you going to finish so you will spend time with me?”

Now that I have finished, her wish is granted. I love them with all my heart. The completion

of this project has made history for my family.

I also extend my gratitude and appreciation to my parents who had instilled in me a desire to

have faith and pursue a good education. To my brothers, sisters, and my in-laws, thank you all

for your prayers and support.

‘Ofa atu.

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Contents

Chapter 1: INTRODUCTION AND OVERVIEW .......................................................................................... 1

1.1 Introduction ................................................................................................................................... 1

1.2 Mathematics and Mathematics Teaching ..................................................................................... 2

1.3 Context of the Study ...................................................................................................................... 3

1.3.1 Mathematics Curriculum Development ................................................................................. 3

1.3.2 Mathematics Teachers ........................................................................................................... 4

1.4 Mathematics Achievement in Tonga ............................................................................................. 5

1.5 Heilala High School, Tonga ............................................................................................................ 6

1.5.1 Professional Development of Teachers .................................................................................. 7

1.5.2 Mathematics Teaching and Learning at Heilala High School ................................................. 7

1.5.3 The SIOP Model at Heilala High School .................................................................................. 9

1.6 Rationale for the Study .................................................................................................................. 9

1.6.1 Need for Quality Professional Development and Teachers’ Collaboration ......................... 10

1.6.2 Lack of research relating to mathematics education in Tonga ............................................ 11

1.7 Aims and Objectives of the study ................................................................................................ 11

1.8 Research Questions ..................................................................................................................... 12

1.9 Overview of chapters .................................................................................................................. 12

Chapter 2: LITERATURE REVIEW ........................................................................................................... 14

2.1 Introduction ................................................................................................................................. 14

2.2 Mathematics and Mathematics Teaching ................................................................................... 14

2.2.1 What is mathematics? .......................................................................................................... 15

2.2.2 Teachers’ perceptions of mathematics ................................................................................ 16

2.2.3 Relating mathematical philosophy with teaching and learning ........................................... 17

2.2.4 Teaching Mathematics Effectively ....................................................................................... 19

2.2.5 The teacher as a Facilitator .................................................................................................. 21

2.2.6 Teachers’ pedagogical and content knowledge ................................................................... 23

2.2.7 Professional Learning Development for Mathematics Teaching ......................................... 24

2.3 The Sheltered Instruction Observation Protocol (SIOP) model of teaching................................ 25

2.3.1 Introduction .......................................................................................................................... 25

2.3.2 Background of the SIOP Model ........................................................................................... 27

2.3.2 The SIOP Model .................................................................................................................... 29

2.3.3 Components of the SIOP Model ........................................................................................... 30

2.3.4 Relationship of the components .......................................................................................... 31

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2.3.5 SIOP Model Implementation ................................................................................................ 31

2.3.6 Strength of the SIOP Model .................................................................................................. 32

Chapter 3: METHODOLOGY .................................................................................................................. 34

3.1 Introduction ................................................................................................................................. 34

3.2 Research Design .......................................................................................................................... 34

3.2.1 Case Study ........................................................................................................................... 35

3.3 Theoretical Framework ............................................................................................................... 37

3.3.1 Social Constructivism ............................................................................................................ 37

3.3.2 Social Constructivist views of mathematics learning ........................................................... 37

3.3.3 Social Constructivist view of mathematics teaching ............................................................ 38

3.3.4 Implications of social constructivism for research ............................................................... 39

3.4 The Kakala Research Framework ................................................................................................ 40

3.4.1 Teu: The Planning or Preparation Phase .............................................................................. 41

3.4.2 Toli: Data Collection Methods .............................................................................................. 44

3.4.3 Tui: Data Analysis ................................................................................................................. 49

3.4.4 Luva: Dissemination of findings ........................................................................................... 50

3.4.5 Mālie ..................................................................................................................................... 50

3.4.6 Māfana ................................................................................................................................. 51

3.5 Ethical Considerations ................................................................................................................. 51

3.6 The Researcher ............................................................................................................................ 52

3.7 Limitations and challenges .......................................................................................................... 52

3.8 Summary ..................................................................................................................................... 53

Chapter 4: RESULTS AND DISCUSSION - MATHEMATICS TEACHING STRATEGIES ................................ 54

4.1 Introduction ................................................................................................................................. 54

4.2 Data Analysis ............................................................................................................................... 54

4.3 Teaching Strategies ..................................................................................................................... 55

4.3.1 Demonstrations and follow-up exercises ............................................................................. 55

4.3.2 Group work ........................................................................................................................... 56

4.3.3 Asking Good Questions ........................................................................................................ 58

4.3.4 Bell work ............................................................................................................................... 59

4.3.5 Lecture .................................................................................................................................. 60

4.3.6 Other Strategies ................................................................................................................... 61

4.4 Effective Mathematics Teaching: Teachers’ Views ..................................................................... 62

4.4.1 Facilitation of students’ learning .......................................................................................... 63

4.4.2 Motivation of students’ learning .......................................................................................... 65

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4.4.3 Preparing lessons well .......................................................................................................... 65

4.4.4 Enhancing students’ understanding ..................................................................................... 65

4.5 Effective Teaching strategies: Students’ view ............................................................................. 66

4.5.1 Group work ........................................................................................................................... 66

4.5.2 Clear definition, Explanation, and Demonstration ............................................................... 69

4.5.3 Hands-on, fun and relevant activities .................................................................................. 71

4.5.4 Provide mathematical problems .......................................................................................... 72

4.5.5 Effective classroom management ........................................................................................ 72

4.5.6 Use of the Tongan language ................................................................................................. 73

4.6 An Effective Mathematics Teacher: Students’ View ................................................................... 74

4.7 Challenges in Learning Mathematics: Students’ Views ............................................................... 75

4.7.1 Mathematics Teacher ........................................................................................................... 75

4.7.2 Word Problems ..................................................................................................................... 76

4.8 Challenges in teaching mathematics ........................................................................................... 76

4.8.1 Lack of basic mathematic skills and negative attitude ......................................................... 76

4.8.2 Lack of skills in application to real life problems .................................................................. 77

4.8.3 Language .............................................................................................................................. 77

4.9 Bringing the ideas together ......................................................................................................... 78

Chapter 5: RESULTS AND DISCUSSIONS - EFFECT OF THE SIOP MODEL................................................ 81

5.1 Introduction ................................................................................................................................. 81

5.2 Data Analysis ........................................................................................................................... 81

5.3 Case study 1: Perceived development of MT1 through SIOP ..................................................... 82

5.3.1 Quality of Lesson Planning and Preparation ........................................................................ 83

5.3.2 Advance Organizer ............................................................................................................... 85

5.3.3 Student Activities and Engagement ..................................................................................... 86

5.3.4 Language and Questioning Skills .......................................................................................... 88

5.3.5 Assessment of Students’ Understanding ............................................................................. 90

5.4 Case study 2: Perceived development of MT2 through SIOP ..................................................... 90

5.4.1 Quality of Lesson Planning and Preparation ........................................................................ 91

5.4.2 Advance Organizer ............................................................................................................... 94

5.4.3 Student Activities and Engagement ..................................................................................... 95

5.4.4 Language and Questioning Skills .......................................................................................... 96

5.4.5 Assessment of Students’ Understanding ............................................................................. 98

5.5 Case study 3: Perceived development of MT3 through SIOP ..................................................... 98

5.5.1 Quality of Lesson Planning and Preparation ........................................................................ 99

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5.5.2 Advance Organizer ............................................................................................................. 101

5.5.3 Student Activities and Engagement ................................................................................... 101

5.5.4 Language and Questioning skills ........................................................................................ 103

5.5.5 Assessment of Students’ Understanding ........................................................................... 104

5.6 Summary: Effectiveness of the SIOP Model .............................................................................. 104

5.6.2 Increase in student engagement ........................................................................................ 107

5.6.3 Use of clear and appropriate language .............................................................................. 108

5.6.4 Increased ‘Wait’ time ......................................................................................................... 109

5.6.5 Guidelines for Lesson Preparation ..................................................................................... 109

5.6.6 Enhance teacher’s performance in the classroom ............................................................. 110

Chapter 6: CONCLUSIONS AND IMPLICATIONS ................................................................................... 111

6.1 Introduction ............................................................................................................................... 111

6.2 Summary of major findings ....................................................................................................... 111

6.2.1 Typical ways of teaching mathematics ............................................................................... 112

6.2.2 Effective mathematics teaching & the Effective mathematics teacher ............................. 113

6.2.3 Challenges in learning and teaching mathematics ............................................................. 114

6.3 The SIOP Model, the successes and challenges ........................................................................ 115

6.3.1 Successes of the SIOP Model .............................................................................................. 115

6.3.2 Challenges of the SIOP Model ............................................................................................ 116

6.4 Implications of the study .......................................................................................................... 117

6.4.1 Implications for improved mathematics teaching ............................................................. 118

6.4.2 Implications for teacher development ............................................................................... 118

6.4.3 Implication for changing classroom practices .................................................................... 119

6.4.6 Implications for further research ....................................................................................... 120

REFEReNCES ........................................................................................................................................ 122

Appendix A: Classroom Observation Schedule ............................................................................... 130

Appendix B: Schedule for the talanoa sessions ............................................................................... 131

Appendix C: Schedule for the SIOP In-service Trainings ................................................................. 132

Appendix E: Questions that guide the talanoa sessions ................................................................. 134

Appendix F: Consent Letter to the Ministry of Education and Training .......................................... 135

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List of Tables

Table 1: Qualification levels of mathematics teachers in government secondary schools, 2013 ........... 4 Table 2: Qualifications of Mathematics Teachers at Heilala High School ............................................... 8 Table 3: Percentage of Student Engagement for MT1's Class Observations ........................................ 87 Table 4: Percentage of Student Engagement for MT2's Class Observations ........................................ 95 Table 5: Percentage of Student Engagement for MT3's Class Observations ...................................... 102

List of Figures

Figure 1: Tonga Mathematics External Results for the past five years ................................................... 5 Figure 2: Pass Rates for Heilala High School Mathematics External Examination 2007 - 2011 ............ 10 Figure 3: The Researcher's interpretation of the SIOP Model .............................................................. 29 Figure 4: The relationship of the SIOP components .............................................................................. 31 Figure 5: Taken from S11’s note book, cubic graph. ............................................................................. 68 Figure 6: A student work on calculating angle of an octagon through splitting of the polygon ............ 98

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CHAPTER 1: INTRODUCTION AND OVERVIEW

1.1 Introduction Mathematics knowledge is an essential element in understanding and interpreting entities

around us (Anthony & Walshaw, 2007). It is fundamental to individuals, families, societies,

governments and nations as a whole because it is through mathematics that “… technological,

industrial, military, economic and political complexes have developed” (D’Ambrosio, 2008;

p. 37). Hence, a solid background in mathematics knowledge is a vital asset in many fields of

study, including science, economics, medicine, engineering, the Arts, mechanics, and

computing.

The way we teach mathematics plays an important role in building student competency in

mathematics. Mathematics teaching and learning are inseparable (Ernest, 1991) and it is

imperative that teachers understand how students learn mathematics in order to effectively

teach it. Zevenbergen, Dole, and Wright (2004, p. 2) are conscious of the ‘new times’ when

“technololgy, globalisation, the information age and very different patterns of family, leisure

and work have brought changes to society, work, schools and life”.

The students need to develop mathematical ways of seeing and interpreting the

world, they need to develop strong problem solving skills, they need to be

numerate and, most importantly, they must have a disposition towards using

mathematics to solve the problems they confront. School mathematics needs to

adopt pedagogies that will cater for diversity within a classroom (Zevenbergen et

al., 2004; p. 3).

The students in classrooms are diverse in many ways. They vary in cognitive, physical and

social development and abilities. They come from different cultures, backgrounds and family

structures, and they speak different languages. Their interests and learning styles also vary.

The teacher’s role is to help every child to learn mathematics and develop to their maximum

potential (Reys, Lindquist, Lambdin, & Smith, 2012). Discussing the changing face and

demand of mathematics in the new times, Zevenbergen et al. (2004) identified (i) the

changing theories of how students learn, and (ii) the changing perceptions of mathematics as a

discipline, as two key movements that are having an impact upon mathematics students,

mathematics education and schools.

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Indeed, there is the urgent need to develop new pedagogies in teaching mathematics

(Zevenbergen et al., 2004; Anthony & Walshaw, 2007; Anderson, 2010) and to move away

from the traditional transmission style of lectures and guided discussion towards a more

student-centred, process-oriented approach (Cooney & Wiegel, 2003). The argument is that

the transmission model does not allow for the interaction between prior and new knowledge.

Students’ existing knowledge rarely features in traditional teaching. Ball (1993) asked critical

questions that relate to this dilemma:

How do I create experiences for my students that connect with what they now

know and care about but that also transcend the present? How do I value their

interests and also connect them to ideas and traditions growing out of centuries of

mathematical exploration and invention? (Ball, 1993; p. 375)

1.2 Mathematics and Mathematics Teaching The word mathematics has many meanings and interpretations (Cooney & Wiegel, 2003) and

all Mathematics teachers hold their own interpretations of mathematics according to what

they know and believe. In the ‘new times’, there is agreement that mathematics is more than

just numbers or the subject studied at school that consisted of fixed objects. Zevenbergen et

al. (2004, p. 8) proposed several definitions of mathematics, of which two are particularly

useful for this project: (i) mathematics is a way of thinking, seeing and organising the world,

and (ii) mathematics is a language. Mathematics as a thinking subject, ties in with the idea of

“the mathematics of the mind, both the teachers’ mind and the students’ mind” (Cooney &

Wiegel, 2003; p. 796). This is consistent with constructivist thinking that knowledge is

developed through the individual’s own experiences both inside and outside the classroom.

Mathematics as a language implies mathematical terminologies with specific mathematical

meanings and interpretations. This means that for one to fully understand mathematics, one

has to fully comprehend its terminology and its associated concepts and principles. Through

this deep understanding one will be able to identify patterns and relationships between various

variables. In particular, the language factor has immense implications for teachers and

students who are requried to teach and learn mathematics in English as a second language. All

mathematics lessons are a language lesson first (Echevarria et al., 2008; 2010); second-

language learners will first try to comprehend the English language used for instruction and

then must put in the effort to understand the mathematical language. This will be discussed at

length in the chapters that follow.

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‘New times’ call for a rethinking of the conceptions of mathematics which underpin

classroom practices. Schifter (1998) proposed a reform mode of teaching where teachers

interacted with mathematics in such a way that they reflected on their own understanding as

well as their students’ understanding. “Teachers must come not only to expect, but to seek,

situations in their own teaching in which they can view the mathematics in new ways

especially through the perspectives that their student bring to the work” (Schifter, 1998; p. 8).

The old adage that “teachers teach as they were taught, not how they were taught to teach”

has merit because most mathematics teachers in the context studied appeared to repeat the

same methods of teaching as they were taught in school.

1.3 Context of the Study This study was carried out at a secondary school in Tonga, the island kingdom located in the

South Pacific region. In this project, the pen name Heilala High School will be used to

represent the study school. Tonga has an education system that provides free compulsory

primary education for all children between the ages of six years to 14 years before students

enter secondary school for another seven years. There are 41 secondary schools in Tonga, of

which ten are owned by the government and 31 are either Church Schools or private schools.

Heilala High School is one of the church schools. In 2004, only 28 percent of students were

enrolled in government schools, while the larger proportion, 72 percent, studied in either

church schools or private schools (Tonga Ministry of Education, 2004). These numbers

indicate the critical role that the churches and church schools play in the country’s education

system.

The Ministry of Education and Training (MET) administers the education system in Tonga. It

is undergoing reform in order to “provide and sustain relevant and quality education for the

development of Tonga, and her people” (Palefau, 2007; p. 6).

1.3.1 Mathematics Curriculum Development

The mathematics curriculum for Tonga secondary schools has undergone major revision. The

mathematics syllabus for Form 1 to Form 3 was reviewed in 2012. The Form 4 syllabus was

reviewed in 2013 and is on trial in 2014. The researcher for this study was part of the

Mathematics Advisory Committee for that review and provided feedback for the Form 4

syllabus in October 2013. The syllabus for primary mathematics has been reviewed and

approved by Tonga’s Minister of Education to be launched in all primary schools.

4

Tonga’s Curriculum Development Unit has designed the mathematics syllabus to ensure the

continuity of each topic strand from Class 1 (Year 1) up to Form seven (Year 13). The

revisions had incorporated a more student-centred approach where teachers and students work

together to create meaningful learning contexts, and make mathematics learning less formal

and more fun. These have required a change of ideology and greater reflection on the part of

mathematics teachers. The Curriculum Development Unit (CDU) will be reviewing the

mathematics syllabus for Forms 5, 6, and 7 in 2015.

1.3.2 Mathematics Teachers

The quality of teachers is an important component of a successful education system. In

Tonga, the ability of the Ministry of Education to develop and retain good mathematics

teachers is an ongoing challenge. There are a good number of mathematics graduates from the

tertiary institutions but unfortunately they do not find teaching an attractive career and most

have opted for greener pastures. In Tonga, one of the features of the Education Policy

Framework is to ensure that all teachers are qualified, dedicated, knowledgeable and have the

willingness to inspire and instil in students a love for learning mathematics. In teacher

education, the Tonga Institute of Education (TIOE) has a well established teacher education

programme that provides formal teaching qualifications as well as ongoing professional

development programmes for practicing teachers. TIOE is currently reworking its

programmes and re-writing its courses. MET is continually exploring other ways of

improving and maintaining the knowledge and expertise of its teachers so that they can

remain vibrant and in tune with the developments in the system. The qualification levels of

mathematics teachers in the government secondary schools are provided in Table 1.

Table 1: Qualification levels of mathematics teachers in government secondary schools, 2013

School Maths Graduate Math Diploma Total Tonga College 2 10 12 Tonga High School 5 13 18 Vavaʻu High School 2 12 14 Haaʻpai High School 2 6 8 ‘Eua High School 4 7 11 Niuatoputapu District High School

1 4 5

Niuafoʻou District High School

1 3 4

Source: Secondary Schools Division

5

There are seven government secondary schools in Tonga. As shown in Table 1: there are 72

mathematics teachers in the government schools of which 17 had graduated with Bachelors

degrees in Mathematics and 55 graduated with teaching diplomas majoring in Mathematics.

The Secondary School Mathematics Teachers Association deserves mention in its efforts to

partner with MEWAC, the Teachers College and the schools in the continuous development

of mathematics teachers. The Association is being revived and expected to actively pursue a

nationwide membership drive soon and proceed to awareness and development activities soon

after that. The researcher is the current president of the Association.

1.4 Mathematics Achievement in Tonga Mathematics achievement in Tongan secondary schools has been an issue of concern for

some time. Recent statistics indicate a decreasing trend in the percentage of passes for the past

five years in all external mathematics examinations except for Form 7 Mathematics. The

major external examinations are the Tonga School Certificate (TSC) at Form 5, the Pacific

Senior Secondary Certificate at Form 6 (PSSC), and the South Pacific Form Seven Certificate

(SPFSC). At Form 7 level there are two mathematics strands: Form 7 Mathematics with

Calculus and Form 7 Mathematics with Statistics. Mathematics results for the past 5 years in

the external examinations are shown in fig 1 below:

Figure 1: Tonga Mathematics External Results for the past five years

Source: Tonga Examination Unit, 2013

Pass % Pass % Pass % Pass % Pass %2007 2008 2009 2010 2011

Tonga School Certificate 68 61 64 66Pacific Senior Secondary

Certificates 85 52 80 80 74

Form 7 Mathematics withCalculus 48 54 57 74 45

Form 7 Mathematics withStatistics 50 57 52 80 38

0102030405060708090

% P

ass

Rate

s

Tonga Mathematics External results for the past 5 years

6

The results for the Tonga School Certificate (TSC) examination in 2011 have not been

included. However, available data indicated that overall, TSC mathematics performance has

followed a fluctuating downward trend. In 2007, 68 percent passed, compared to 61 percent in

2008. This increased in 2009 by 3 percent but decreased again by 2 percent in 2010.

The overall results of the PSSC Mathematics examinations for the five year period 2007-11

showed a downward trend. In 2007, 85 percent passed, followed by a sharp drop to 52 percent

pass rate in 2008. The percentage of succesful students then increased in 2009 to 80 percent,

remained constant in 2010, then decreased by six percent in 2011.

The Form 7 Mathematics with Calculus examination results have shown an upward trend in

the percentage of passes from 48 percent in 2007 to 74 percent in 2010, but with a huge

decrease to 45 percent in 2011.

By comparison, the Form 7 Mathematics with Statistics results indicated that the percentage

of passes fluctuated over the five year period. There was an upward trend in the percentage of

passes from 50 percent in 2007 to 80 percent in 2010 but was followed by a sharp drop in

2011 to 38 percent.

1.5 Heilala High School, Tonga Heilala High School was established in 1947 by the Board of Education of the Church of

Jesus Christ of the Latter Day Saints (LDS) which has its headquarters in the United States of

America. The LDS schools have spread throughout the United States of America and

internationally. In the Pacific region, there are LDS headquarters and schools in New Zealand,

Australia and many islands of the Pacific. Currently there are thirteen LDS church schools in

the Pacific region, seven of which are located in the Kingdom of Tonga. Heilala High School

is the largest of the church schools in the region.

The student body at Heilala High School includes students from all over the Pacific region

and abroad. They come from Kiribati, Samoa, Fiji, Vanuatu, Papua New Guinea, Tahiti, New

Zealand, Australia and the United States. The enrolment at the high school for students in

Forms four to seven (Year 10 to Year 13) was 1127 in 2012. The lower levels (Years 7 to 9)

make up the middle school.

Heilala High School offers English, Mathematics, Accounting, Economics, and Science

(founding principles for Chemistry, Physics, and Biology) classes as core subjects. Technical

and Vocational education subjects (TVET) such as Design Technology, Drafting, Woodwork,

7

Mechanics, Welding, Electrical engineering, Arts, and Home-Economics classes have also

been included to offer alternative study pathways for the diverse student body. The school

also offers strands in Piano, Choir, Physical Education and the School Band. The wide range

of subjects provides new avenues for the development of specialist knowledge and practical

skills that can prepare students for a meaningful and sustainable future in Tonga as well as the

international community.

1.5.1 Professional Development of Teachers

Heilala High School has always considered the ongoing professional development of teachers

an important school activity because of the effect of teachers on students, the school and its

community. International research supports this (Coxon, 2000):

International research clearly shows that the essence of a good school is quality

teaching and that effective schools are only effective to the extent that they have

effective teachers. It is through the quality of teaching that schools can make a

difference to the life chances of their students. Therefore, teacher effectiveness is

the key to improved educational outcomes (Coxon, 2000; p. 389)

The mathematics teachers at the school have an organised programme of professional

development. All teachers undergo pre-service training for the first week of every school year

and in-service training once a week with one of the school administrators, or internally with

the head of the department. Important items of discussion include reflection on practice and

identification of new strategies and pedagogies. Teacher reflection is a popular activity for all

teachers and there have been positive signs and indications of changing practice and the

adoption of new ideas. The mathematics teachers also had sessions where they tried to

implement some of the ideas given at the workshop sessions. They were also encouraged to

work together to observe each other’s classes and provide useful constructive feedback at the

end. They have shown great improvement, and teaching is more student-centered as a result

of these professional developments and the peer teaching.

1.5.2 Mathematics Teaching and Learning at Heilala High School

Mathematics teachers have changed their perspectives toward mathematics teaching. They

now believe that teaching should be student oriented rather than teacher oriented, and that

they may achieve this by incorporating the SIOP Model in their classroom. Teachers saw the

8

need for implementing new teaching approaches for the ‘new times’ by planning and

delivering mathematics lessons in such a way that students will actively engage in 90 percent

to 100 percent of the class. It is a journey for Heilala High School to consistently implement

new strategies that will enable students to think and own their mathematics learning.

The main language of instruction at Heilala High School is English although the local Tongan

language is often used in a kind of bilingual exchange. All teachers are expected to teach

mathematics explicitly in English in their classrooms. Manu (2005) had written about the two

challenges encountered by local teachers and students as they have tried to go through this

process. The challenges include the difficulty of trying to understand mathematics with its

specialist English vocabulary as well as the challenge of trying to understand the language

used for instruction. These are common challenges among schools in Tonga like Heilala High

School. However, the mathematics teachers have embraced the vision to teach mathematics

explicitly in English and have made every possible effort to simplify their instruction in

English to the level of students’ understanding. Still they always see the need to restate their

points in Tongan so that students can comprehend the instructions and the mathematics

vocabulary.

There are 6 Mathematics teachers at Heilala High School and their qualifications are recorded

in Table 3:

Table 2: Qualifications of Mathematics Teachers at Heilala High School

Code Qualification Years of Experience Gender (M or F) MT1 BA in Mathematics and

History

19 M

MT2 Diploma in Accounting and Economics.

9 M

MT3 BA in Mathematics and

Chemistry.

7 F

MT4 MA in Mathematics. 10 M

MT5 BSc in Mathematics

and Science.

25 F

MT6 Diploma in Mathematics and

Accounting.

3 F

9

1.5.3 The SIOP Model at Heilala High School

The SIOP Model was first implemented at Heilala High School in 2007 and considered an

effective approach toward teaching in the ‘new times’ (Zevenbergen et al., 2004). The model

was chosen for very good reasons and on proven results. Firstly, based on recent practical

research, the model has been proven effective particularly for schools where English is the

second language. Secondly, the model has been adopted worldwide and proven successful

(Echevarria et al., 2008; Guerino, Echevarria, Short, Shick, Forbes, & Rueda, 2001).

Even though the model has been used at the school for the last eight years, most of the

teachers are still not familiar with its features due to lack of training, continuing staff turnover

and new staff recruitment. Fig 2 on page 10 shows the external Mathematics examination

result for Heilala High School for 2007 - 2011. The data for Tonga School Certificate shows a

declinging trend. There is a great need to review the model and conduct studies on the

different features and elements of the SIOP model, and to re-orient the mathematics teachers

toward its features.

The extent to which this model of teaching mathematics at Heilala High School is effective

has yet to be investigated. This research study attempts to begin this investigation of teaching

and learning mathematics at the Form 5 level.

1.6 Rationale for the Study In order to improve teaching mathematics in these “new times” there is a need to investigate

the effect of new teaching models; therefore this research study has chosen to investigate the

effect of the SIOP Model in teaching mathematics at Heilala High School. The remainder of

this section discusses the rationale for this study.

While the SIOP Model was established with the expectation that it would provide support for

teachers in their teaching and which would then improve students’ academic performance,

this has not really been proven. In fact, there has been a decreasing trend in the school’s Form

5 Mathematics external results for the past 5 years. This trend is shown in the graph on Figure

2 below:

10

The data showed that the percentage pass rates have been declining for the past five years. The

researcher was part of the teaching group and was particularly concerned about this trend for Form

5 examination results. This problem was the motivating factor behind this study - to investigate the

effectiveness of the SIOP Model in teaching mathematics at Heilala High School, particularly at the

Form five (5) levels.

1.6.1 Need for Quality Professional Development and Teachers’ Collaboration

One of the key components of collecting data for this study is the training of mathematics

teachers and re-orienting them towards the SIOP Model. There is a great need for quality

professional development in relation to the SIOP Model and for collaboration among

mathematics teachers of the school. Even though the model was officially implemented in

2007, not all teachers are expert or have fully applied the model in their classrooms.

Therefore, an important part of this study is the attempt to provide quality professional

development for the Heilala High School mathematics teachers on the SIOP Model and to

invite teachers to collaborate in sharing and taking turns in observing each others’

Figure 2: Pass Rates for Heilala High School Mathematics External Examination 2007 - 2011

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mathematics classrooms. This is designed to enable mathematics teachers at the School to be

informed about current educational issues and to learn new skills and teaching practices

which will have an impact on students’ performances (Echevarria, Short, & Vogt, 2008;

Rogers, 2007).

1.6.2 Lack of research relating to mathematics education in Tonga

There is also a great need for research in mathematics education here in Tonga, especially

regarding mathematics teaching and learning. Koloto (1995) carried out a study entitled

Estimation in Tongan Schools (1995) while Manu investigated the role of code switching in

mathematics understanding in the Tongan classroom (2005). Fasi (1999) investigated the

effect of bilingualism and learning mathematics in English as a second language in Tonga.

Together, these research studies have formed the basis of literature specifically relating to

mathematics education in Tonga.

It is hoped that the current study will generate valuable information for the school itself, the

Ministry of Education, as well mathematics teachers and educators in Tonga and the wider

Pacific. The effect and possible use of the model in the context of other Pacific island

countries is anticipated. It will also have implications for teaching Mathematics at Heilala

High School, and the results will serve to enhance future teaching practices.

1.7 Aims and Objectives of the study This project will use the SIOP Model to teach Mathematics at Form 5 at Heilala High School

over a period of three months. An assessment of the effect of the model will be made using a

variety of tools including nine classroom observations and talanoa with both student and

teachers. The effect of the model was assessed through the perceived development of the

mathematics teacher at Form 5 over the period of the observations. The main intention will be

to investigate the experiences of the mathematics teachers in using the SIOP Model.

The objectives of this study are:

1. To investigate the experiences of mathematics teachers in the use of SIOP in teaching

Form 5 mathematics at Heilala High School.

2. To highlight the major challenges in teaching and learning Mathematics at Heilala

High School.

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3. To investigate the effectiveness of the SIOP Model in enhancing the learning and

teaching of Form 5 Mathematics.

1.8 Research Questions The main research questions are:

1. What are the common ways of teaching Mathematics at Form 5 level at Heilala High

School?

2. What are the major challenges of teaching Mathematics at Form 5 level at Heilala

High School?

3. How effective is the SIOP Model in enhancing the learning and teaching Form 5

mathematics at Heilala High School?

The Assumptions

This study assumes that the development of teachers’ observed use of SIOP techniques

equates to ‘enhancement’ of teaching and learning of mathematics. It is assumed that the

SIOP techniques, which include the facilation of peer learning, deliberate teaching of key

vocabulary, use of effective teaching strategies aligned to both the content and language

objectives of the lesson and provision of regular feedback on both, will constitute good

effective teaching practice and bring about effective learning. The observation of the use of

SIOP techniques in the classroom equates to observation of ‘enhancements’ in teaching and

learning mathematics in the study context.

1.9 Overview of chapters This research study consists of six chapters.

This chapter has introduced the study, including the context and rationale of the study, the

aims and objectives of this research paper, and the research questions.

Chapter 2 will contain the literature review. The literature review will discuss literature on

mathematics teaching and learning as well as the SIOP Model and its key components, the

implementation of the model, and the strengths of the model.

13

Chapter 3 will discuss the methodology used in this study, the tools that have been used to

collect the data, how the sample and the participants were selected, ethical considerations for

this study, and how data are being treated and analysed.

Chapters 4 and 5 will consist of a presentation of the results and discussions from the study,

including emerging themes in relation to the research questions, common teaching practices,

challenges in teaching and learning Mathematics, and the effect of the SIOP Model on

teaching and enhancing learning mathematics at Heilala High School.

Chapter 6 will present the summary of this study including the implications for mathematics

teaching and learning as well as future research on this topic.

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CHAPTER 2: LITERATURE REVIEW

2.1 Introduction This project investigated the quality and effectiveness of the Sheltered Instruction

Observation Protocol (SIOP) teaching model in enhancing mathematics learning at Heilala

High School. The model is an American invention that grew out of a need to serve the

linguistic and educational requirements of immigrant and non-immigrant second language

learners in schools in the United States of America in 1998. The desire to conduct this study

stemmed from concerns raised at Heilala High School about declining achievement rates in

mathematics and students’ disenchantment with the subject. There is no doubt that

mathematics is highly valued in society by employers, politicians and parents. Unfortunately,

however, many students do not share this view because of negative learning experiences in

the mathematics classroom. They cannot see the relevance or purpose of the subject in real

life or experience it in meaningful ways (Reys et al., 2012). The researcher believes that

teachers and teaching can make a big difference because they “have powerful influence over

what and how students learn” (Zevenbergen et al., 2004, p.6).

This review chapter is divided into two main sections which are closely related. The first

section is about mathematics and mathematics teaching while the second section is devoted to

discussion of the details of the SIOP model of teaching. The first section is subdivided into

seven subsections: (1) What is mathematics? (2) Teachers’ perceptions of mathematics; (3)

Relating mathematical philosophy with teaching and learning; (4)Teaching Mathematics

Effectively; (5) The teacher as a facilitator; (6) Teachers’ pedagogical and content

knowledge; and (7) Professional Development for mathematics teachers.

2.2 Mathematics and Mathematics Teaching Mathematics is an essential tool for understanding the world (Reys et al., 2012). The authors

asked this question of mathematics teachers: What is your vision of the mathematics you will

be teaching? They explained the importance of this question as follows:

How you view mathematics will determine how you view teaching mathematics. If you

view mathematics as a collection of facts to learn and procedures to practice, then you

will teach that to your students. If you view mathematics as a logical body of

knowledge, you will adopt teaching strategies that let you focus on guiding children to

make sense of mathematics (Reys et al., 2012; p. 5)

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The perceptions of mathematics teachers about mathematics, and mathematics learning and

teaching are likely to be reflected in their teaching practices in the classroom. Teachers’

perceptions are critical factors in shaping the attitudes of students towards mathematics and

their learning behaviour in the mathematics classroom (Anderson, 2010; Rogers, 2007;

Ernest, 1995). Hence, to understand why teachers teach the way they do, it is significant to

understand their perceptions and belief systems. How they perceive mathematics has

implications for how they teach it. These discussions are pertinent to the focus of this study.

2.2.1 What is mathematics?

Many people think of mathematics as the subject they learnt at school. Others equate it to

specific areas such as arithmetic, numbers, measurement and calculations that are elements of

primary mathematics or with a collection of Algebra, Geometry, Statistics and Trigonometry

if they did secondary mathematics. These views are narrow, limited and outdated.

Mathematics has not been stagnant but is continually going through change as new ideas are

created and new uses discovered (Reys et al., 2012). Technology and the way it has

revolutionised the way we learn and teach is one such example.

Reys et al. (2012) presented five views of mathematics that have broadened the scope of how

mathematics should be perceived. They view mathematics as: (i) a study of patterns and

relationships, (ii) a way of thinking, (iii) an art, characterised by order and internal

consistency, (iv) a language that uses carefully defined terms and symbols, and (v) a tool.

Zevenbergen et al. (2004) added a sixth dimension, which was mathematics “as power” (pp.

9). This view is concerned with the political aspect of mathematics as the status subject that

opens doors to positions of power and prestige. As such, the subject is privy only to the ‘elite’

and therefore unattainable to the masses. All definitions, however, indicate an all-

encompassing view of mathematics as both content (knowledge) and process and are

therefore useful. The definition that “mathematics is a language” has important implications

for the use of the Sheltered Instruction Observation Protocol (SIOP) teaching model in

enhancing mathematics learning at Heilala High School, which is the focus of this study.

Some researchers describe mathematics as a language which has its own terminology and its

own meanings, and which is more than just numbers (Echeverria et al., 2010; Zevenbergen et

al., 2004).

In order to comprehend mathematics, one has to fully understand the meaning of

mathematical terminologies, because sometimes in mathematics, words have a different

16

meaning from the one that is used in everyday language (Fasi, 1999). It is important that

mathematics teachers are aware of the different usage and double meanings of mathematical

language. This is a challenge for both teachers and students, especially in schools where

English is not the majority of the students’ first language, but is the approved mode of

instruction.

Manu (2005) investigated this challenge in the relationship between mathematical

understanding and language switching in the context of a bilingual classroom. He concluded

that using a second language as the means of instruction for teaching mathematics initiates

two challenges for both students and teachers: the difficulty of understanding mathematics

and its terminologies, and the need to understand the language used for instruction itself.

Similarly, Neville-Barton and Barton (2005) studied the relationship between the English

language and mathematics, and investigated the tensions experienced by Chinese students

who speak Mandarin as their first language in New Zealand schools. They wanted to identify

the disadvantages that could arise from students’ lack of English proficiency. Neville-Barton

and Barton conducted two mathematics tests, one version was in English and the other in the

Mandarin language. One half of the students sat the English version of the test while the other

half sat the Mandarin version. The authors noted that, on average, students scored lower

marks by 15% on the English version as compared to the other version. They also found that

the teachers of these students were not aware of the cause of the students’ misunderstandings.

2.2.2 Teachers’ perceptions of mathematics

Thompson (1984) defined conceptions as a general mental structure, encompassing beliefs,

meanings, concepts, propositions, rules, mental images, preferences, and the like. It is,

according to Brown (2004, p. 3), an “organizing framework by which an individual

understands, responds to, and interacts with a phenomenon.” Linking up the two terms,

Brown proposed that “all pedagogical acts, including teachers’ perceptions of and evaluations

of student behaviour and performance are affected by the conceptions teachers have about the

act of teaching, the process and purpose of assessment, and the nature of learning among

educational beliefs” (ibid).

It is critical to identify what counts as mathematical knowledge in relation to the definition of

mathematics. Ernest (1995) differentiated two main views of mathematical knowledge - the

absolutist and the fallibilist views. The Absolutist view of mathematical knowledge is that it

17

is certain and absolute. It cannot be changed nor altered. Ernest (1995) argued that this view is

one of the factors that has contributed to the negative image of mathematics held by the

public, where mathematics is viewed to be “rigid, fixed, logical, absolute, inhuman, cold,

objective, pure, abstract, remote, and ultra-rational” (p. 13). He explained that giving students

irrelevant, routine mathematical tasks, where they are required to learn specific steps of fix

mathematical procedures, is an example of implementing an absolutist view in the classroom.

This example hinders mathematics learning, where the learner does not have the opportunity

to communicate, explore and have fun in the learning process, but finds maths to be a boring

subject (Anderson, 2010).

On the other hand, fallibilists view mathematics as the outcome of social processes (Ernest,

1995). Mathematical knowledge is understood to be corrigible and fallible. Kitcher and

Aspray (1998) describe it as a ‘maverick’ tradition in the philosophy of mathematics that

emphasises the practice of mathematics and its human side. This position is associated with

constructivist and post-modernist thought in education (Glasersfeld, 1995). The

constructivists believe that mathematical knowledge is personally constructed through the

individual’s experiences, before school, during school and after school (Ernest, 1991; Begg,

1995). The mathematical knowledge can be modified or changed as new ideas and

experiences emerge. So, mathematical knowledge is not absolute and certain but rather it is a

product of social processes (Ernest, 1995).

For students in this study and the Pacific countries, the social component of learning is

important. This element is better accounted for in the social constructivist viewpoint than the

individual meaning-making in constructivism (Cobb, 1994; Crawford, 1996). Social

constructivists still accept the importance of individual meaning-making but also view the

‘social’ as essential in the learning process (Cobb, 1994). The social settings of the classroom

have great impact in the process of constructing personal mathematical knowledge. This

perception drives most mathematics educators to find new ways of teaching mathematics

where the social setting is more student-centered than teacher-centered (Ecchevarria et al.,

2010; Anderson, 2010; Roger, 2007).

2.2.3 Relating mathematical philosophy with teaching and learning

It is widely accepted that “all mathematical pedagogy, even if scarcely coherent, rests on a

philosophy of mathematics” (Thom, 1973; p. 204). Research has confirmed that “teacher’s

views, beliefs, and preferences about mathematics do influence their instructional practice”

18

(Thompson, 1984; p. 125). Ernest (1995) has argued that teachers’ personal philosophies of

mathematics, which are part of their overall epistemological and ethical framework, have an

impact on their conceptions of teaching and learning mathematics. Following this logic, one

can trace a link between the absolutist views and the ‘separated’ (Gilligan, 1982) position that

champions rules, reason and analysis, abstraction and being objective. The fallibilist

conception of mathematics is linked to the ‘connected’ view (ibid) that values relationships,

connections, being holistic, and displaying human-centredness.

Notwithstanding, Ernest (1991) argued that what happens in the classroom does not strictly

follow one philosophical stance, because other factors such as time, school culture,

curriculum and assessment come into play. He proposed that it was possible to “associate a

philosophy of mathematics with almost any educational practice or approach” (Ernest, 1995,

p. 20). He identified possible ‘crossing over’ between philosophies and teaching practices.

For example, most mathematicians, as well as many mathematics teachers and students, like

the absolutist image of mathematics. They see its absolute features as powerful, giving it

status. Then there are mathematics teachers who hold an absolutist view of mathematics but

prefer to adopt a more humanistic approach to teaching.

Mathematics teaching and learning are interconnected. Good mathematics teachers are

interested in how students learn and will want to learn from it. They will either acquire new

ideas on the mathematical concept being taught or think of a better approach to teach that

particular lesson. In the same way, learning mathematics greatly depends on how the

mathematics lesson has been delivered.

Many mathematics teachers would say that they always plan their lessons. The particulars

and details of teachers’ plans, such as who they are planning for and what the details are

based on, are important considerations. Then there are teachers who do not plan at all but

simply walk into a classroom with the mathematics textbook open to certain pages with

selected exercises. Those teachers who plan may be able to produce a lesson plan with a list

of objectives to be achieved by the end of the lesson, and the objectives would begin with the

statements like: ‘At the end of the lesson students should be able to’ and would be followed

by a list of measureable and observable outcomes - what the teacher believes students should

know and be able to do.

Zevernbergen et al. (2004) have argued for more than that. They proposed that to be able to

plan how to teach mathematics effectively, “there needs to be some understanding of how

19

students learn mathematics” (p. 21). This implied that it was not the practices that teachers

use in the classroom that matter, but why they do what they do. Teachers cannot plan in a

vacuum, but they must have some notion of learning theory to guide their planning and

practice. “By having an idea of how students’ learn, teachers are better able to plan for and

anticipate in particular ways and to create learning environments to facilitate better learning”

(Zevernbergen et al., 2004; p. 22).

According to Wheatley (1991), “constructivists view learning as the adaptations children

make in their functioning schemes to neutralize perturbations that arise through interactions

with our world” (p. 13). This means that learning is attained through finding one’s way

through and making sense of one’s own experiences. These self- experiences are vital to

mathematics learning. The teacher’s role in creating such experiences in the classrooms is

significant in supporting students’ mathematics learning because without such experiences,

students will lose focus and interest in studying mathematics (Chinnappan & Cheah, 2012).

For example, Chinnappan and Cheah investigated how one mathematics teacher was able to

actively engage her students in her mathematics classes through implementing activities that

were more of a practical nature on the topic of Fractions. These experiences enabled her

students to focus and think mathematically about the activities that they were doing in their

class. Even though this was the experience of a primary teacher, the same activities could be

adapted for secondary mathematics classrooms, because the experiences of students in

mathematics classrooms are vital to mathematics learning and to becoming proficient in the

mathematical concepts taught.

In addition, students make sense of mathematics learning when classroom activities require

students to use prior knowledge and experiences from their own environment (Begg, 1995;

Roger, 2007; Echevarria et al., 2010). By allowing students to draw on prior experiences and

knowledge, they are more likely to organize their experiences in a way that makes sense.

2.2.4 Teaching Mathematics Effectively

George Polya who is credited with being the father of mathematical problem solving had this

advice for mathematics teachers:

A teacher of mathematics has a great opportunity. If he fills his allotted time with

drilling his students with routine operations he kills their interest, hampers their

intellectual development, and misuses his opportunity. But if he challenges the

20

curiosity of his students by setting them problems proportionate to their

knowledge, and helps them to solve their problems with stimulating questions, he

may give them a taste for, and some means of, independent thinking. (Polya, cited

by Boaler, 2008; pp. 26).

It is generally agreed that mathematics teaching should move from traditional methods to new

reform pedagogies which include the more collaborative teaching methods (Roger, 2007;

Zevenbergen et al., 2004; Zeverbergen, 2009; Anderson, 2010; Echevarria et al., 2008, 2010;

Hunter & Anthony, 2012; Attard, 2011). The old models of mathematics teaching that are

still very prevalent in many mathematics classes today are the teacher-centered and subject-

oriented methods that include lecture, drill and practice, and providing students largely with

individual work in the classroom. The teacher’s role in such approaches is to deliver the

lesson then quiz students for the correct answers (Hunter & Anthony, 2012). Zevenbergen et.

al. (2004) argued that these old models of teaching did not promote a positive learning

environment, but instead contributed to the mathematical incompetence of students

throughout their school years.

The new emphasis in mathematics teaching is on students’ cognitive development where

students will be able to justify, adjust and critique solutions, and be able to make connections

among mathematical concepts rather than checking for correct answers (Hunter & Anthony,

2012; Attard, 2011; Klein, 2012). The focus now is on construction of knowledge and making

sense of self experiences rather than simply absorbing information (Klein, 2012). As a result

of such teaching, mathematics proficiency will improve and sustainable deep conceptual

understanding of mathematical concepts should be achieved (Klein, 2012). This movement

reflects a social constructivist view where students learn better when they construct their own

meaning and knowledge through meaningful interaction either with their peers or with their

teacher or with the materials within the learning environment (Bennison & Goos, 2007).

When these aspects are absent in mathematics teaching then mathematics teachers are often

unsuccessful in preparing the students for the agenda of the 21st century (Klein, 2012).

Mathematics discussion is one of the core elements of effective mathematics teaching where

students will be able to interact with their peers, or with the teacher, or with the mathematics

material itself (Echevarria et al., 2008; 2010; Zevenbergen, 2009; Hunter & Anthony, 2012).

However, a productive mathematics discussion is focussed on developing the reasoning skills

and thinking of the students, rather than checking for correct answers (Hunter & Anthony,

21

2012). These mathematical discussions give opportunities for students to argue and justify

their solutions, seek for understanding from their peers, interpret mathematical reasoning, and

it allows students to construct their own mathematical knowledge. Anthony and Walshaw

(2009) used ‘mathematical argumentation’ for such classroom discourses.

Klein (2012) argues that students’ mathematics proficiency will only develop to the extent

that mathematics discourses facilitate the mathematics learning. If the classroom discussion

is only to throw in some responses for solutions, then the proficiency developed will only be

that relating to basic mathematics principles. She describes proficiency from a

poststructuralist notion, where proficiency is created in “participation, a form of agency”. It is

vital for mathematics teachers to produce a learning environment where productive

mathematics discussion emerges and gives the agency to the students to exchange meanings,

discuss mathematical procedures and processes, and provoke mathematical argumentation, in

order for them to become proficient in mathematics.

Another aspect of productive classroom discussions is the use of mathematical language. In

order for one to fully understand mathematics, s/he must be able to understand the

mathematical terms and their meanings (Zevenbergen et al., 2004; Echevarria et al., 2008;

2010; Anthony & Walshaw, 2009). Mathematics teachers must explicitly explain the meaning

of mathematical terms in other contexts as well as in mathematics (Echevarria et al., 2008;

2010) and be able to foster classroom discussions where students used these terms correctly

and appropriately.

2.2.5 The teacher as a Facilitator

The role of a mathematics teacher during the mathematics lesson is to become a facilitator and

allow students to think and engage in classroom discussions on their own. Anthony and

Walshaw (2009) elaborate on this role, where the teacher attentively listens to the students’

discussion, and discerns when to step in or out of the discussion, when to settle students’

argumentations, when to elaborate for understanding, and when to shift the discussions to

another problem. By doing so, the teacher provides the student with the agency to explore

mathematics on his/her own and have confidence in mathematics because the students are

able to contribute to the process of mathematics teaching.

Several researchers explored how some mathematics teachers gradually changed from a

traditional way of mathematics teaching to develop the habit of becoming a facilitator

22

(Anderson, 2010; Hunter & Anthony, 2012; Rogers, 2007; Chinnappan & Cheah, 2012). For

the teachers involved, it was difficult at first for them to hold back when students experienced

some difficulties, but they committed to allowing the students to explore the problem and

seek to make sense of mathematics by developing thinking that lead to the answer.

Other researchers have examined several other aspects of meaningful mathematical tasks

(Anthony & Walshaw, 2007; Grootenboer, 2009). Anthony and Walshaw (2007) explain two

aspects of mathematical tasks as elements of effective mathematics teaching: the task must be

worthwhile, and enable students to make connections among various mathematical concepts

as well as to real life experiences.

Based on the Maths in the Kimberleys (MITK) project, Grootenboer (2009) used ‘academic

and intellectual quality’ to describe worthwhile tasks where students are able to explore,

critique and justify solutions, provoke mathematical argumentation, and make conclusions.

He also agreed with Anthony and Walshaw (2007) that mathematical tasks must enable

students to make connections with various mathematical concepts and links to real practical

problems. He added four more aspects which are: group work, extended engagement, catering

for diversity and multi-representational tasks. Sullivan (2009) used ‘open-ended tasks’ as

similar to ‘multi-representational’ tasks where students have varieties of approaches toward

the solution. These tasks foster collaborative learning where students peer teach and exchange

ideas.

By providing meaningful tasks, students are enabled to establish a purpose in their

mathematics learning (Roger, 2007; Anderson, 2010) and make sense of mathematics,

because mathematics is all about making sense (Wheatley, 1991). Hence, a well-designed

task is likely to foster deep mathematics learning (Zevenbergen, 2009).

There are various definitions of ‘engagement’ but this paper pursues the definition by Attard

(2011) where “engagement occurs when students are procedurally engaged within the

classroom, participating in tasks and ‘doing’ the mathematics... ” (p. 69). Noddings (cited by

Anthony & Walshaw, 2007) posits that most effective teachers highly impact their students’

learning through expecting high student engagement in their classes. These teachers create a

learning environment free of fear, where students feel competent to participate and engage in

mathematical discussions. They also have high expectations where all students are actively

engaged in mathematics discussions and follow directions for about 90 percent to 100 percent

of the class time (Echevarria et al., 2008; 2010).

23

In such classrooms, all students contribute to the classroom discussions either by negotiating

meanings, critiquing solutions, exploring several approaches toward the answer, asking

questions for clarification or seeking explicit explanation of key words prior to solving the

problem. Students in such environment rediscover themselves as ‘insiders’ within their

classroom, and their morale is boosted because they feel important and valued as they engage

in and contribute to the mathematics classroom discussion (Attard, 2011; Anthony &

Walshaw, 2007).

2.2.6 Teachers’ pedagogical and content knowledge

“Effective teaching begins with teacher knowledge” (Anthony & Walshaw, 2007, p. 81).

Shulman (1987) proposed three categories of teacher knowledge as important for effective

teaching: subject matter knowledge, pedagogical content knowledge and curricular

knowledge. Therefore, mathematics teachers need good grounding and competence in

mathematical content knowledge, which includes the key facts, concepts, principles and

explanatory frameworks as well as the rules and proofs within the discipline. All teachers

need pedagogical knowledge, which is usually defined as the knowledge of the profession or

the art of teaching. Pedagogical content knowledge is about representing the subject in ways

that make sense to students. So the mathematics teacher has a dual responsibility in the

classroom: firstly as the primary source of student understanding of mathematics and

secondly, in communicating to students what is central about the subject and what is

peripheral (Shulman, 1987). In view of the diversity of students, the mathematics teacher

must have a flexible and multifaceted comprehension in order to be able to share alternative

explanations and viewpoints. The combined effect of the three knowledge bases influences

what a teacher does in the classroom and how effectively it is done (Anthony & Walshaw,

2007).

An empirical study by Askew, Brown, Rhodes, Johnson, and William (cited by Anthony &

Walshaw, 2007) on Effective Teachers of Numeracy in the UK have found out that high

levels of formal qualifications in mathematics teachers actually had a negative relationship

with their students’ level of performance. They found out that students of teachers who were

able to make mathematical knowledge more meaningful performed better than those in

classes of teachers who could not. Anthony and Walshaw (2007) concluded that unless

teachers are able to make mathematical concepts more meaningful, students will continue to

struggle in trying to make sense of mathematical ideas and principles taught in the

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classrooms. Moreover, Anthony and Walshaw (2007) explained that in order to teach

mathematics effectively in a diverse classroom, it demands that the teachers are confident in

their content knowledge as well as their pedagogical knowledge. This will allow teachers to

be creative and use the most appropriate techniques that will best serve the needs of each

student in the classroom.

2.2.7 Professional Learning Development for Mathematics Teaching

Teacher professional learning is defined by Roger (2007) as “those processes and activities:

formal and informal designed to enhance the knowledge, skills, and capacity of staff” (p.

631). The purpose of these activities is to generate a change in the attitude and classroom

performances of teachers which will enhance the learning of students. The two most

fundamental aspects of mathematics teaching are content and pedagogical knowledge

(Zevenbergen et al., 2004). It is through effective professional development that these two

aspects of mathematics teachers can be promoted.

Roger (2007) has explicitly explained the impact of a professional learning project on one of

the mathematics teachers’ performances where, at the end, there were changes of perspectives

toward teaching mathematics. The teacher shifted from her ‘old way’ of teaching, which was

more teacher-centered, into being a facilitator at the students’ side while giving opportunities

for students to solve mathematics problems on their own. This is similar to a study by

Anderson (2010) where she highlighted the impact of professional development on a teacher’s

perspective, and how she organized her mathematics teaching. Through collaboration with the

researcher and by using a pedagogical approach initiated by the Critical Mathematics

Education (Anderson, 2010), the teacher changed her pedagogical approach into giving the

agency to and empowering students to seek and solve their mathematical problems using their

own approaches.

In addition, both Roger (2007) and Anderson (2010) highlighted the significance of

“reflective practice” where the teacher would look back and evaluate his/her own teaching

and be able to adapt and change thier pedagogical approach. According to Roger (2007),

reflective practice is “one of the key components within effective teaching and professional

development” (p. 633).

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2.3 The Sheltered Instruction Observation Protocol (SIOP) model of teaching

2.3.1 Introduction

The Sheltered Instruction Observation Protocol (SIOP) model of teaching was initiated to

enhance the linguistic need of second language learners in schools in the United States of

America (US). The SIOP Model is a research based model. There have been studies on its

effect on teaching of various subjects including mathematics and these studies were

conducted mainly in the United States of America. The areas covered by these studies have

included:

� Effects of the SIOP model.

Short, Fidelman, and Louguit (2012) investigated the effects of the SIOP Model on

development of academic language performance of students in both middle and high school in

two district schools in New Jersey. The main research question was “will ELLs (English

language learners) in one district with teachers who received professional development in the

SIOP model show significantly higher achievement in reading, and oral proficiency in

English on a standardized measure than ELLs in a comparable district with teachers who had

no SIOP professional development?” (p. 339). The result showed, in year 1(2003-2004), the

treatment students (whose teachers were trained with the SIOP model) had scores (from New

Jersey state test on reading, writing, and oral language) below the scores received by the

comparison students (whose teachers were not trained with the SIOP model). However, in

year 2 and 3 (2004-2005 and 2005-2006), the scores received by the treatment students

exceeded those gained by the comparison students.

These researchers also proved their hypothesis to be true, namely that “both language learning

and content learning can take place in subject area classes” (p. 353). This was significant in

classrooms where the majority of the students use English as their second language. The SIOP

model integrates strategies that develop and enhance students’ academic vocabularies, which

are a crucial stepping stones to understanding of the core content areas. Rather than relying on

the English classes to teach vocabulary, the SIOP model can bridge these learning gaps for the

minorities. The SIOP model engages teachers across the curriculum to pay attention to the

language of the subject: how it is used and how it supports and sustains learning.

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Another similar study by Short, Echevarria, and Richards-Tutor (2011) explored the effects of

the SIOP model on the academic literacy development of English language learners. They

implemented three successive studies on how teacher behaviour changes with regard to the

model, affected student performance on standardized assessments and researcher-developed

measures. The results from the three studies were similar to the findings by Short, Fidelman

and Louguit (2012). The treatment teachers taught mathematics, science, history, English,

language arts, special education, and technology, whereas comparison teachers taught

mathematics, science, history and English. Treatment teachers received SIOP professional

development, while the comparison teachers taught the same curriculum without any SIOP

training. The results from New Jersey tests in reading, math, social studies, and science

showed a significance difference in the achievement of the treatment students as compared to

the comparison students, where treatment students performed better than the comparison

students. There is promising success with the model, as indicated by the results of students in

the content areas (Echevarria et al., 2008, 2010; Himmel, Short, Richards, & Echevarria,

2009; Guerino et al., 200; Echevarria, Short, & Powers, 2006).

A five year study was done by Himmel, Short, Richards, and Echevarria (2010) on the impact

of the SIOP model on middle Science. They found that students whose teachers received

SIOP training performed better in their Science studies than students with teachers who had

not received SIOP training. They also found out that the implementation of the SIOP had a

significant positive relationship with students’ success.

These studies have proven through research (Echevarria et al., 2004, 2008, 2010; Himmel,

Short, Richards, & Echevarria, 2009; Guerino et al., 2001; Echevarria, Short, & Powers,

2006) that the SIOP model contains the best teaching practices that, if implemented with

fidelity and consistence in the classroom of any content area subjects, results in students’

performance progressing and comprehension of academic vocabularies being enhanced along

with their conceptual understanding.

� How effective is the SIOP model as a tool for effective professional development?

All these studies (Echevarria et al., 2004, 2008, 2010; Himmel, Short, Richards, &

Echevarria, 2009; Guerino et al., 200; Echevarria, Short, & Powers, 2006; Short, Echevarria,

and Richards-Tutor, 2011; Short, Fidelman, & Louguit, 2012) have been designed in similar

ways, where teachers from the sample schools received professional development on the

SIOP model, coached, observed, and received feedback from SIOP mentors, while teachers of

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other sample schools did not receive such training in the model. The results from these studies

were all similar. There was a significant relationship between the teachers’ performance and

their students’ performance, where students whose teachers received training on the model

outperformed those whose teachers did not receive any training at all in the model.

In addition, students of teachers who strongly implemented the SIOP model performed better

than those students of teachers who had implemented the model at a low level. However,

Short, Fidelman, and Louguit (2012) suggested that providing professional development on

the SIOP model without any follow-up was not sufficient for progress in both teachers’ and

students’ performance. It required consistency in the professional development, time being

given to the teachers to implement and practice the model with fidelity, and opportunities for

them to be coached and receive productive feedback on their performance. Under these

conditions, the desired result will be achieved.

The next section will provide the background and history of the model, its eight key

components, describe the implementation of the model, and discuss the strengths of the

model.

2.3.2 Background of the SIOP Model

Sheltered Instruction (SI), also known as Specially Designed Academic Instruction in English

(SDAIE) can be described as strategies used for content area subjects and presented in

English in a way that enable non-English speaking students to comprehend the content and

develop their English language skills simultaneously (Hansen-Thomas, 2008; Crawford,

Schmeister, & Biggs, 2008). It was specifically designed to serve the linguistic and

educational needs of immigrant and non-immigrant second language learners in the U.S

schools (Echevarria, Vogt, & Short, 2008, 2010; Hansen-Thomas, 2008). It consists of good

teaching practices and instruction that have been found to be effective in the classroom.

Hansen-Thomas (2008) listed five features of Sheltered Instruction:

1. Use of cooperative learning activities with appropriately designed heterogeneous student

groups;

2. A focus on academic language as well as key content vocabulary;

3. Judicious use of ELLs’ first language as a tool to provide comprehensibility;

4. Use of hands-on activities with authentic materials, demonstrations, and modelling; and

5. Explicit teaching and implementation of learning strategies (p. 166).

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Sheltered Instruction is underpinned by Vygotsky’s theory of proximal development (1978)

and Gardner’s multiple intelligences theory (1989), where collaboration, peer tutoring, and

scaffolding assist the student to excel and progress from one stage to another. Due to

increasing challenges faced by non-English learners, there was the need for the professional

development of educators and teachers teaching these students to provide them with high-

quality instruction in the classroom (Echevarria et al., 2008).

Through in-service training, workshops, and classroom observation, Echevarria, Short, and

Vogt (2008) found that there were no common perspectives on what effective sheltered

instruction should look like during a lesson. This problem led to the research study of

Sheltered Instruction and Observation Protocol (SIOP) Model by Echevarria, Vogt, and Short

from 1996 to 2003(Echevarria et al., 2008, 2010). This project was implemented through

collaboration with middle schools teachers in four large urban district schools. The end

product of this project was the SIOP Model, which was originally designed as an observation

protocol for the project, in order to determine how well teachers had included the key features

of SI. However, through feedback and reports from middle school teachers, the Model was

found to be an effective approach for lesson planning and delivery (Echevarria et al., 2000,

2008). Thus, SIOP is a research-based model that has been field-tested in the classrooms and

refined to its present form.

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2.3.2 The SIOP Model

Figure 3: The Researcher's interpretation of the SIOP Model

As Figure 2 indicates, the SIOP Model is not a hierarchy model or a step by step approach but

all its components are interrelated (Short, Fidelman, & Louguit, 2012). The key aspects of the

SIOP model are:

� Based on years of nation-wide (USA) research, the model brings to light fundamental

aspects and strategies of teaching as well as providing a framework for teachers to use

lesson planning and delivery.

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� It focuses on developing students’ language skills with a particular attention to their

academic vocabularies in the content areas while at the same time boosting their

comprehension of the subject matter. Every class/subject is a “language class”

(Echevarria et al., 2004, 2008, 2010).

� This model includes best teaching practices that have been field tested and

recommended for high-quality instruction, such as cooperative learning strategies,

reading comprehension strategies (Genesse, Lindholm-Leary, Saunders, & Christian,

cited by Short, Fidelman, & Louguit, 2012), activating of prior knowledge, developing

background knowledge, increasing wait time for students’ responses, using

manipulative activities to a high degree, explicit instruction for the academic task at

hand, and differentiated classroom instruction (Echevarria et al., 2008, 2010; Hansen-

Thomas, 2008).

� It incorporates both language and content objectives in every class of any subject area.

This provides new insights into how teachers plan and deliver their lesson.

2.3.3 Components of the SIOP Model

The SIOP Model consists of eight key components which are interrelated. They are: (1)

Lesson preparation, (2) Building Background, (3) Comprehensible input, (4) Strategies, (5)

Interaction, (6) Practice and application, (7) Lesson delivery, and (8) Review and Assess.

Altogether the components have 30 features as shown in Figure 2. The first component is

mandatory while the other seven components are equally vital for effective teaching. Each of

the eight components of the model involves basic teaching practices/strategies which have

emerged from professional literature regarding best practice for delivery of sheltered

instruction as well as strategies recommended for high quality instruction (Echevarria et al.,

2008, 2010; Echevarria, Short, & Powers, 2006; Guerino, Echevarria, Short, Shick, Forbes, &

Rueda, 2001; Himmel, Short, Richards, & Echevarria, 2010).

Although the eight components of the model are not new to teaching, the evidence has shown

that there is an absence of many of these components in typical classroom instruction, even at

Heilala High School. Thus, the SIOP model provides a guideline for lesson planning and

lesson delivery so that teachers ensure that most of these components are implemented during

the lesson delivery.

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2.3.4 Relationship of the components

This diagram illustrates the relationship between the eight components.

As shown in figure 3, the implementation of the other components depends on the quality of

the lesson preparation. Lesson delivery is the execution of the lesson plan, in which the

teacher implements a range of strategies to activate students’ prior knowledge and builds on

their background experiences, enabling students to engage in classroom discourses, providing

practical problems for students to practice on and use the new knowledge learnt, as well as to

assess their performance throughout the class regarding whether the content and language

objectives were achieved or not. It is perceived by many teachers that when the other seven

components are successfully implemented, students will be able to comprehend the classroom

instructions and understand academic tasks thoroughly.

2.3.5 SIOP Model Implementation

The model has been implemented across schools and universities in the United States. Some

schools in the Pacific countries including American Samoa, the Federated States of

Micronesia, Guam, Hawaii, the Republic of Palau, and the Republic of the Marshall Islands

have also implemented it. All LDS schools worldwide including those in the Pacific region

have adopted the SIOP Model. The model has been adapted to fit the context of the various

schools and used as a new paradigm for effective teaching.

Figure 4: The relationship of the SIOP components

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2.3.6 Strength of the SIOP Model

The SIOP Model has been designed to support teachers to provide high-quality instruction in

order for learning to be more meaningful for students (Echevarria, Vogt, & Short, 2000, 2008;

Short & Echevarria, 1999; Hansen-Thomas, 2008). The hallmark of this model is the quality

of practices and lessons that systematically develop students’ content knowledge and

academic skills while also promoting their English language skills (Short, Fidelman, &

Louguit, 2012). This is done through integrating language objectives into all content subjects.

Developing students’ language skills is an essential element of teaching and learning

mathematics which needs to be a consistent part of their daily lesson plan and delivery.

It is believed that an effective SIOP lesson can construct a high level of student engagement

and interaction, either with the teacher or with other students, which then generates

meaningful discourses such as explaining processes, justifying solutions, as well as analysing

word problems, and which ultimately lead to promoting critical thinking. The model embeds

features of high-quality instruction based on current knowledge, a literature review of best

practices, and through collaboration and constructive feedback from practicing teachers

(Echevarria et al., 2008, 2010).

The SIOP Model has provided a way for teachers to reflect on their own teaching and be able

to improve their classroom instruction. Some of the schools that have implemented the model

have used peer coaches. The coach models a lesson and the others observe and provide

feedback, and then the rest will try to follow by each taking turns to model a lesson. Hence,

the model “provides a common language and conceptual framework from which to work and

develop a community of practice” (p. 204). Many administrators have reported that the

features of the model have provided a variety of ideas and techniques through professional

development for their teachers to incorporate into their teaching practices, such as operating

under the knowledge of differentiated classrooms and multiple intelligences.

In addition, research has shown that students whose teachers have been trained in the SIOP

Model demonstrate higher performance than those students whose teachers have not been

trained with the model (Guarino, Echevarria, Short, Schick, Forbes, & Rueda, 2001;

Echevarria et al., 2008, 2010). For example, Lela Alston Elementary School at Phoenix,

Arizona, was a new school in 2001. They implemented the SIOP Model at their school in

2002. The students’ performances in the state test showed a tremendous improvement in their

results for Mathematics, Reading, and Writing from 2002 to 2004 (Echevarria et al., 2008).

33

Moreover, principals have testified to the great impact of the model on the teaching practices

of teachers within their schools. There are more teachers teaching on their feet and trying to

engage students with their learning, which reflects the effort of these teachers in their lesson

preparation. Hence, students were actively engaged, and their performances were enhanced.

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CHAPTER 3: METHODOLOGY

3.1 Introduction This study investigated the effect of the Sheltered Investigation Observation Protocol (SIOP)

Model on students’ mathematics learning and understanding. This chapter discusses the data

collection methods used in the study. The overarching theme of this chapter is Thaman’s

Kakala Research Framework (2009) that is used to frame the research process. Kakala is a

holistic approach used to gain in-depth understanding of a phenomenon and it is appropriate

for this work in Tonga and the participation of Tongans in this study.

This chapter is divided into seven sections as followed: (1) Research Design; (2) Theoretical

framework; (3) Kakala Research Framework; (4) Ethical consideration; (5) The researcher;

(6) Limitation and Challenges; and (7) Summary of the chapter.

3.2 Research Design A research design is the plan and structure of an investigation used to obtain evidence to

answer questions (McMillan & Schumacher, 1997). The research design describes the

procedures for conducting the study, including the timing and collection of data, the people

involved and the constraints within which the researcher is expected to operate (Walsh, 2005).

Planning a research design, as will be discussed in this chapter, depends on the scope as the

“domain of inquiry, the coverage and reach of the project.” (Richards & Morse, 2007, p.75)

In line with the purpose of the study, the humanistic or qualitative approach (Attride-Stirling,

2001) was considered appropriate because the research is context specific. The researcher was

part of and involved with the subjects, and thus, the beliefs, knowledge and attitudes of the

researcher were impacted both by the process and the reporting of the studies. Humanistic

enquiry centred on describing in depth, the complexities of human interaction in given

settings and, as a result, the analysis of language used in learning and teaching had an

important role to play. While qualitative research encompasses several approaches that are in

some respects quite different from one another, all of them have two things in common:

i. They focus on phenomena that occur in natural or real life settings, and

ii. They involve studying those phenomena in all their complexities.

(Leedy & Ormrod, 2005, p. 135)

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According to Peshkin (1993), qualitative research studies serve one or more of the following

four purposes:

� They are descriptive and reveal the nature of certain situations, settings, processes,

relationships, systems, or people;

� They are interpretive in enabling a researcher to learn about a phenomenon, develop

theoretical perspectives about it and discover the problems that exist within;

� They verify, through allowing the researcher to test the validity of certain

assumptions, claims, theories, and generalisations within real-world contexts;

� They are evaluative in providing a means through which a researcher can judge the

effectiveness of particular policies, practices or innovations.

The current study was seen to serve the fourth purpose. It was interested in the complex cause

of student behaviour within the learning context in which it occurred. The qualitative

approach was holistic and allowed the participants to voice their perspectives freely in their

own way using both their native Tongan language and English, as it suited them. It also

allowed the researcher to form a forum for the participants through which quality and reliable

information as well as new insights emerged.

3.2.1 Case Study

The research documented in this study was generally interpretive in nature and mostly aligned

with the humanistic paradigm. It consisted of a case study of lessons taught to a class of Year

11 students at Heilala High School. The implementation of the Sheltered Instruction

Observation Protocol (SIOP) as a new teaching approach and how students and teachers

reacted to it were the main sources of information. The professional development of teachers,

the observation of student activities, the analysis and assessment of work samples and on-

going interviews provided information that was used to describe students’ understanding of

mathematical concepts during the intervention. The researcher was also the form teacher for

the period of study of the Sheltered Instruction Observation Protocol (SIOP), as defined by

the project planning.

36

The main purpose of the case study was to seek some understanding of how events occurred

and why they occurred. The approach was especially suitable because the SIOP model had

never been tested at the school; this study was able to generate information about the changes

brough about by its implementation at the school over time. Leedy & Omrod (2010) propose

that the case study is used for the purpose of “learning more about an unknown or poorly

understood situation” (pp. 108). It is also useful for investigating how a program or individual

changes over time, perhaps as a result of interventions. This was exactly what was needed for

the SIOP model; since very little was known about its effectiveness, a case study approach

was considered suitable for the investigation that would reveal the reality of the case

(mathematics teaching using the model) in the real situation at the school. The case study also

supported the effort to understand the cultural context of the system of actions being studied.

Leedy and Omrod (2010) also said that the researcher in a case study collects extensive data

on what is being studied, in this case the effectiveness of a programme. These data come

through various tools including observations, interviews and documents. In a case study, the

researcher also recorded details about the context surrounding the case (ibid, p. 137). So the

empirical enquiry looked within the real-life context and attempted to use multiple sources of

evidence. In this study, the phenomenon under investigation was the learning of students

given the sheltered instruction approach, and the complexity of the factors at work in the

classroom, considering that students’ learning is closely entwined with the context in which

they are learning. The aim of the researcher is to describe exactly the events according to the

facts from data collection, and seek to understand the phenomenon from the perspectives and

experiences of the participants (Groenewald, 2004).

The limitation of case studies concerns their reliability and external validity. The former

concerns the consistency of results and the extent to which other studies conducted in a

similar fashion would obtain similar results. External validity is particularly important for

case studies where a single case is involved, as was the situation in this study, because there is

no certainty that the findings can be generalised to other situations. A response to this would

be to stress that the case study involves multiple sources of data and does not rely on a single

form of measurement. The use of triangulation to compare multiple data sources in search of

common themes would also support the validity of the findings. Meanwhile, the potential

strength of case studies is their internal validity, which concerns the accuracy of the

information and how it matches reality. Every effort was made to maintain the normal routine

of instruction for the class.

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3.3 Theoretical Framework The theoretical framework for this study is grounded in the Social-Constructivism theory of

learning, which assumes that learning is personally constructed, and at the same time

recognises the impact of the social environment of the learner. Given that this research is

conducted in the Tongan context, the ‘Kakala research framework (Thaman, 2009) is used to

frame the methodology. This section discusses the main assumptions of the Social-

Constructivist theory and its implication for mathematics teaching and learning as well as

research.

3.3.1 Social Constructivism

Social constructivism takes into account that the construction of personal knowledge is due to

the influence of the social environment of the learner (Cobb & Yackel, 1996; Cobb, 1994).

Social interaction and negotiation of social norms are crucial elements of this theory and this

is referred as social constructivism (Koloto, 1995). The main assumption of this theory is that

“cognitive processes are subsumed by social and cultural processes” (Cobb, 1994, p. 14).

This focuses on the experiences people share with others, and the cultural context that

underpins their actions, which also influences how people construct individual knowledge

(Crawford, 1996). Thus, the educational institution is a part of the wider culture that

influences the performance and development of both their teachers and students.

3.3.2 Social Constructivist views of mathematics learning

The implications of Social Constructivism on mathematics learning are drawn from the

Constructivist theory. Constructivism is a learning theory (Begg, 1995) that emerged from the

work of Piaget (1937 cited by Cobb, 1994) and was further developed by von Glasersfeld

(1989). It is a common theory which underpins most studies in mathematics education (Cobb,

1994) and it builds on two main principles, firstly that knowledge is constructed by the

individual through personal experiences and is not passively received, and secondly, that

thought is developed and can be adapted through personal experiences and not because of

some existing truth that have not yet been discovered (Begg, 1995; Koloto, 1995).

The first principle implies that active construction of knowledge is a personal matter and it

cannot be merely transmitted from one person to another (Koloto, 1995; von Glaserfeld,

1991). The learner is solely responsible for construction of his/her mathematical skills and

ideas through his or her experiences (Koloto, 1995).

38

The second principle implies that thoughts are organised in a way that fits the thinker’s

personal experiences rather than in a way governed by an existing traditional idea of

knowledge (Begg, 1995; Koloto, 1995). Hence, people interpret meanings and what they

believe to be true through their unique personal experiences (Wheatley, 1991).

Moreover, culture is a vital phenomenon in learning mathematics, as portrayed by this theory,

because the social-cultural aspects are a feature of mathematical knowledge (Crawford, 1996).

Therefore, the learner must be able to connect their mathematical skills and knowledge to

their environment in order to construct meaningful and sustainable mathematical skills and

knowledge.

3.3.3 Social Constructivist view of mathematics teaching

Social constructivism emphasizes the social and cultural nature of mathematical activity

(Cobb, 1994). Vygotsky refers to the term ‘activity’ as the involvement of the groups or

individuals along with their commitment and intention. It is crucial as a teacher to know the

factors that underpin the cognitive structures of students, such as;

Needs and purposes of people, their actions and the meanings that they attach to an

activity, their relationships with other people in the socio-cultural arena in which

they think, feel, and act, and the presence of culturally significant artefacts.

(Crawford, 1996, p. 44).

These factors also affect how individuals construct meaning regarding the reality of their

experience and how this guides their decisions in later activities.

Since knowledge is actively constructed by the individual and is affected by the learning

environment, teachers play a significant role in setting the learning environment in a way that

invites students to explore, discover, invent, discuss and reflect on their experiences in the

mathematics classroom (Wheatly, 1991). Through these experiences, students will construct

meaningful mathematical knowledge and skills. Hence, students will be able to effectively

store mathematical knowledge in their long term memory and appropriately use it in the

future (Ormrod, 2011).

In addition, since interaction is an important element of this theory, it has essential

implications for the teaching of mathematics. As a mathematics teacher for 14 years, the

researcher believes that teachers must appropriately select their mathematical activities and

39

create such a learning environment that enables students to engage in classroom discourses

with deep sense of understanding and also to have fun in the learning process.

Language is a vital tool that assists students in constructing mathematical knowledge and

skills (Bakalevu, 2007). This implies that when activities are implemented which enable

students to utilize and develop all the four language skills in the classroom, students will tend

to negotiate and adapt meanings as they share and discuss solutions with their peers and

construct meaningful mathematical knowledge from involvement in meaningful experiences.

Moreover, an effective mathematics teacher will provide problems that relate to the students’

context and environment in order for them to make meaningful connections. However, most

times, students learn mathematics out of context, and this leads students to find mathematics a

boring subject (Anderson, 2010). The context of mathematical problems is central to students’

learning (Begg, 1995; Crawford, 1996; Anderson, 2010). This implies that mathematical

problems have to be meaningful and relate to the students’ real environments. Thus, teachers

need to constructively select problems where students can apply and explore mathematics in

their own environment. This will produce a meaningful learning environment in which

students are able to make connections to real life contexts.

3.3.4 Implications of social constructivism for research

The social constructivist view of knowledge is that it must be personally constructed, taking

into account the impact of the social norms of the environment. This understanding led this

study to utilize a social constructivism methodology.

The ultimate goal of this study is to seek understanding of the effect of the SIOP Model on the

teaching of mathematics at Heilala High School. In order to achieve this goal, there needed to

be a holistic understanding, and for the meanings of both teachers’ and students’ experiences

in the mathematics classroom to be found. Both teachers and students develop subjective

meanings of their experiences, and these meanings are negotiated and can be adapted through

social interaction with others. This implies that research should use research tools in which

both teachers and students will interact and allow them to negotiate meaning through their

interaction and experiences. This led the researcher to use group work in the in-service

training as well as to increase teacher collaboration by enabling mathematics teachers to

observe a colleague and provide constructive feedback. At the end of each in-service training

session, an exit ticket (feedback) was collected, where each teacher gave an evaluative

40

feedback of what they had learnt in the training sessions. The teachers’ feedback was drawn

from their experiences as well as their learning from discussion with the other mathematics

teachers.

Through these collaborations, teachers were able to adapt their teaching strategies via the

implementation of the SIOP Model, as well as through negotiating the meanings of the

mathematics concepts. These acts of social interaction through peer coaching have developed

both the content and pedagogical knowledge of the observer.

This also implies that research should use research tools that allow the researcher to

understand what is going on in the minds of the participants. This led the researcher to

implement classroom observations and video record them in order to fully understand the

meanings of both teachers’ and students’ experiences in the mathematics classroom.

The talanoa session was implemented as well, in order to provide the researcher with relevant

and holistic information in pursuing an understanding of the meanings of both teachers’ and

students’ experiences of the SIOP Model.

3.4 The Kakala Research Framework

A research framework identifies the key variables that guide how to conduct and go about a

study. One’s framework makes assumptions about the nature of society and of individuals,

and the relationships between them (Carrington & Macarthur, 2012). The authors add that a

framework is the basis for explaining how things work, what is defined as the problem and

the kind of ideas there are for solutions to those problems.

The Kakala research framework defined the key components that guided this study. It was

developed by renowned Pacific researcher Helu-Thaman (1997) as a fundamental instrument

for carrying out educational research in the Pacific. Kakala is a Tongan word that

encapsulates the fragrant flowers and the process of making a flower garland, an important

activity in Tonga and many Pacific countries. Thaman’s original framework developed in

1997 had three stages: Toli, Tui, and Luva. Since then, other Pacific island researchers have

utilized the Kakala research framework and testified to its usefulness in explaining the

research process clearly and successfully (Johansson-Fua, 2006, 2009; Manuʻatu, 2001;

Taufeʻulungaki, Johansson-Fua, Manu, and Tapakautolo, 2007). As the Kakala framework

continues to be used, there have been proposals for expanding it by adding another three

stages: Teu, Mālie, and Māfana (Manuʻatu, 2001; Johansson-Fua, 2009).

41

The Kakala Research Framework now comprises six phases: Teu, Toli, Tui, Luva, Mālie, and

Māfana (Manuʻatu, 2001). Teu is about preparing the necessities such as the basket for the

collection of flowers, as well as the thread and needles for the kakala. Toli is the process of

picking varieties of fragrant flowers to be used for making the kakala. Tui is the actual

process of making the kakala using only fresh, fragrant flowers specially selected to thread

the kakala. Luva is the presentation of the kakala to the person whom the kakala has been

fashioned for. Mālie and māfana are the expressions of appreciation and acceptance that the

receiver experiences after the kakala is presented to him/her. It is natural that the receiver will

have a warm feeling toward the product while at the same time admire its structure and

fragrance. The phases of the Kakala framework will be discussed next in relation to the

requirements of the current study.

3.4.1 Teu: The Planning or Preparation Phase

Teu or the planning process included thinking about and formulating the following: (i) Aims

and objectives of the study, (ii) Why the study was necessary and (iii) What the study was to

achieve. These elements dominated the initial research proposal and are included in Chapter

one. In the context of this study, teu included identifying and deciding on the objectives of the

study, formulating the key research questions, developing the data collection tools, choosing

research participants, and request participation in the study. These items are discussed in more

detail in the next section:

The objectives of this study are to:

1. Investigate the experiences of mathematics teachers in the use of SIOP in teaching

Form 5 mathematics at Heilala High School.

2. To highlight the major challenges in teaching and learning Mathematics at Heilala

High School.

3. To investigate the effectiveness of the SIOP Model in enhancing the learning and

teaching of Form 5 Mathematics.

Research Questions

The main research questions were:

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1. What are the common ways of teaching Mathematics at Form 5 level at Heilala High

School?

2. What are the major challenges of teaching Mathematics at Form 5 level at Heilala

High School?

3. How effective is the SIOP Model in enhancing the learning and teaching Form 5

mathematics at Heilala High School?

Research Participants

This study is based at Heilala High School (HHS), one of the largest mission schools in

Tonga. The school has an enrolment of about 1100 students. The researcher teaches at the

school and is therefore an insider-researcher. The insider-researcher understands the context

very well and has the conveniences of time and accessibility for data collection, which are

additional pluses for the researcher.

i. The mathematics teachers (MT1, MT2, MT3, MT4, MT5 and MT6) of HHS were

selected as participants. While all six were included in the interview, the classroom

observations were focused only on the Form 5 mathematics classes (MT1, MT2, and

MT3). Each Form 5 mathematics teacher taught six periods a day. One period was

selected for classroom observation. Table 3 (page 8) shows the basic information of

the mathematics teachers.

Code Qualification Years of Experience Gender (M or F) MT1 BA in Mathematics and

History

19 M

MT2 Diploma in Accounting and Economics.

9 M

MT3 BA in Mathematics and

Chemistry.

7 F

MT4 MA in Mathematics. 10 M

MT5 BSc in Mathematics

and Science.

25 F

MT6 Diploma in Mathematics and

Accounting.

3 F

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ii. Four students were randomly selected through the use of simple random selection

methods from the different Form 5 class groups of MT1, MT2, and MT3. Altogether

there were twelve students in the student research sample and comprised six boys and

six girls. The number of student participants was limited to 12 to enable the researcher

to monitor data more effectively.

Data collection tools and processing

This study used (i) classroom observation, (ii) group talanoa, and (iii) individual talanoa to

collect student information during learning.

Classroom observations

Three classroom observations were planned for each mathematics teacher and gave a total of

nine classroom observations altogether. Each classroom observation was recorded on video so

that maximum information was captured to enrich the data analysis and the discussions. The

use of the video camera was a powerful tool for prompting teacher self-reflection and

discussion afterwards.

The rationale of limiting the classroom observations to a total of nine is to enable the

researcher to manage the data being observed. First, there were 12 classroom observations

planned but time was too tight according to the school schedule, and therefore 12

observations were not manageable. Refer to Appendix A for the dates scheduled for the

classroom observations.

Talanoa

There are various interpretations and understandings of talanoa. Fua (2009) explained that

talanoa is a conversation based on concepts or ideas given to the participants to “muse, to

reflect upon, to talk about, to critique, to argue, to confirm and to basically conceptualise what

he/she believes the topic to be” (p. 209). Vaioleti (2006) explains talanoa as a personal

encounter where people tell stories about their experiences of reality, concerns, and ambitions.

Halapua (2003) adds that talanoa comprises “open dialogues where people can speak from

their hearts and where there are no preconceptions” (p. 18). The researcher listens carefully

while the participants’ talanoa about their perceptions and beliefs about the question being

asked or topic being discussed.

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To create a talanoa, the researcher used “an open technique...where the precise nature of

questions has not been determined in advance, but will depend on the way in which the

talanoa develops” (Vaioleti, 2006, p.26). However, the purpose of the talanoa must be clear

at the beginning in order to frame the encounter between the two parties. At times, the

researcher began with friendly chatter to establish a sense of being at ease and trust that was

essential for succesful talanoa. It was important to remove any barrier or confusion between

the researcher and the participants so that they can express their honest thoughts and opinions

on the main subject of the talanoa.

This study implemented the kind of talanoa sessions with the participants as that described by

Fua (2009), Vaioleti (2006), and Halapua (2003).

Types of talanoa sessions.

Talanoa was divided into two sessions:

1. Individual talanoa with teachers, and

2. Group talanoa with students.

Appendix B has the schedules of the talanoa sessions that were used for students and

teachers, as well as questions that guided these talanoa sessions. The talanoa sessions

addressed the first and the second research questions:

1. What are the common ways of teaching Mathematics at Form 5 level at Heilala High

School?

2. What are the major challenges of teaching Mathematics at Form 5 level at Heilala

High School?

These talanoa sessions were conducted in room C1 on the school campus because it was

easier for both the researcher and the participants to meet.

3.4.2 Toli: Data Collection Methods

Toli is the process of picking varieties of fragrant flowers for the formation of the kakala. In

the context of this study, toli is the process of collecting all necessary data, and it occurred in

five phases as follows:

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Phase 1: Professional development for mathematics teachers

Phase 2: Pre-observations interview

Phase 3: Classroom observation

Phase 4: Post-observation talanoa

Phase 5: Individual talanoa with all mathematics teachers

The five phases of Toli are discussed next.

Phase 1: Professional Development for Mathematics Teachers

Although the SIOP Model has been implemented at the school since 2007, for various reasons

many teachers at HHS were not familiar with its details and potential. The continuing high

turnover of staff at the school, the number of new teacher recruitments as well as new

administrative staff posed administrative challenges. These related factors made it necessary

to include staff training in the SIOP model as a component of this research. The support of the

Principal was very useful as it gave the researcher the time and space to collect all the relevant

information while also carrying out her normal teaching duties.

Departmental meetings were held every morning for 30 minutes from 8 o’clock. The training

on the SIOP model was conducted during these departmental meetings. Appendix C shows

the schedule of the training sessions that were conducted.

Each session ran for about 30 minutes and was held in room C6 at the mathematics wing of

the school. The sessions covered the details and implementation of the eight components of

the SIOP model. There were extra days for discussion on how to improve classroom practices

and identify best practices advocated by the model. The teachers were able to stay for the

whole session each time. The researcher conducted the training using the SIOP Model to

guide the delivery of each session and provide hand-outs during each training session. The

following section gives details of this phase:

1. A typical day’s session would be as follows: Prayer and spiritual thought, bell work in

which teachers were put on task with a question or activity that tested their prior knowledge,

oral discussion of the bell work, introduction of the content and language objectives for the

46

session, a brief lecture on the features of the component, then teachers would pair up to

discuss a specific feature of the component and report on their discussion to the whole group

of mathematics teachers.

2. During each session, teachers were encouraged to share and show what they knew of the

model and how they had implemented its features in their classes. There was a feedback

session at the end where participants gave an evaluation of their learning and development

from the session activities.

3. After the training sessions, teachers were given time to practice the model and implement

what they had learnt from the in-service training.

4. The teachers also had the opportuntity to sit in at a lesson conducted by the researcher as an

expert SIOP practitioner. At the end of the observation, they sat down with the researcher and

gave feedback and shared insights on the modelled lesson. The researcher was open to the

teachers’ comments and questions. All these were also recorded for later analysis.

Phase 2: Pre- observation interview

The researcher and each teacher met one day prior to each classroom observation. In these

pre-observation meetings, the teacher discussed and shared his/her lesson plans, which

included the components to be modelled in class.

Phase 3: Classroom observation

The researcher carried out all the class observations. They were done at the school and each

period lasted 50 minutes. Appendix D shows the Observation Schedule form that was used to

record the required details of each lesson. The criteria of the observations as shown on the

Observation Schedule were the main items of the SIOP model. How the teacher met each

criteria and utilised it to meet the lesson objectives was the focus of the observations. Since

each teacher was observed successively three times each, the results of previous observations

provided impetus for the next ones that followed. In this way, maximum improvement and

performance was expected and ensured.

Bell work is a quick task provided at the beginning of the class to get students on task and get

them to focus on the rest of the class. It can either be a form of revision, where students relate

to their prior learning, or an introductory task for a new topic. It may also be a fun activity to

47

boost the enthusiasm of the class at the beginning, or it may be any short activity designed by

the teacher to engage students immediately. This is more like an advance organizer (Ausubel,

1968) where students get to organize their thinking at the beginning of the class.

During the classroom observation, the researcher was a participant-observer, studying both

teachers and students while also helping students by clarifying tasks or questions and taking

field notes for further analysis.

The lessons were recorded on video by a helper. The recording tried as best as it could to

capture the four participant students in each class as they enganged and were involved in

class. This was important for later analysis at post-observation sessions.

This protocol addressed the third research question:

3. How effective is the SIOP Model in enhancing the learning and teaching Form 5

mathematics at Heilala High School?

Phase 4: Post – observation Talanoa

At the end of each classroom observation, the researcher and the teacher set a specific time for

a talanoa session to review the class session. Again, these talanoa sessions were flexible and

informal. They would start with a good ice-breaker to ease the tension between the researcher

and the participant (Vaioleti, 2006). The researcher made sure that these sessions were

conducted with mutual respect (fakaʻapaʻapa), humility (lototō), love (‘ofa), and

committment for a good purpose. When the participants were at ease, the talanoa was led

towards a review of the lesson. The video was replayed and the teacher was given the

opportunity to view, reflect and describe how they felt about the lesson. In between that, the

researcher asked and probed in an effort to get to the partcipants’ feeling, beliefs and

understanding. The question and answers were important in clarifying intentions and the

reasons behind them. This enabled both parties to arrive at the positive and negative elements

of each class and what needed to be improved in the next sessions. Identifying the strengths,

challenges and ways of improvement were important elements of the discussion.

At the end, the participants were thanked for their participation. While the Tongan tradition

would require gifting to show appreciation, that protocol did not take place because the

researcher was an insider and also participant in the study. So it was “Ko e ‘alu pē mei he fale

ki he fale” (Going from one house to another as one family).

48

Group Talanoa with Students

These sessions were done in three groups: MT1’s four students in one group, MT2 students

in the second group and MT3’s students in the third group. The group talanoa was done at the

end of the third lot of classroom observations.

The group talanoa sessions started with some friendly talk to allow students to settle

comfortably and establish some sense of maheni (familiarity) so that they could feel free to

share their thoughts and stories. Once the researcher felt that the environment was confortable

and free of tension and the group was māfana (experiencing warm feelings), the purpose of

the talanoa was discussed and time was given for participants to ask questions and seek

clarification on doubts and confusion. Then the participants were invited to tell their story.

Questions were more informal but the researcher occasionally probed for clarification and

examples so that the research questions were attended to.

At the post-observation talanoa sessions, the groups viewed their classroom observation

videos and were replayed back and forth to facilitate students’ thinking and reflection on what

had transpired during the class. The talanoa sessions were audio recorded. Where students did

not attend the group talanoa, an individual talanoa was organised for them. At the end of

these talanoa sessions, the researcher provided a simple lunch for each of these students as a

token of appreciation for their contribution to this project.

At all the talanoa sessions, the participants, both teachers and students were allowed to speak

in the language they were most comfortable with. It was noted that there was a lot of code

switching (Manu, 2005) between the Tongan and English languages during the talanoa.

Some responses reported in the next chapter use the vernacular language, which the

researcher had translated into English. The researcher was interested in the views of both

students and teachers regarding mathematics teaching strategies and the effect of the SIOP

Model in teaching and learning mathematics.

Phase 5: Individual talanoa with all mathematics teachers

The individual talanoa sessions with each mathematics teacher was conducted at the end of

the third lot of classroom observations.

All mathematics teachers of the school were invited to a one-to-one talanoa session with the

researcher for this part of the study. Five individual sessions were held with the five teachers.

49

All sessions were audio recorded for maximum capture of the details. One of the teachers,

MT4, was out of the country at the time and the questions were sent to him via email. His

responses were compiled along with the others recorded on site.

These talanoa were easy to fakahoko (implement) because the researcher and teachers already

had a sense of maheni (familiarity) and feangai (being used to each other) at work. However,

the researcher took time to welcome the teachers and expressed words of appreciation for

their time and willingness to part. Depending on how the talanoa developed, the researcher

allowed the participants to freely share their points of view on the SIOP model and teaching

and learning mathematics. The researcher listened intently to the discussions and probed with

further questions for more in-depth insights whenever the need arose.

The questions sheet in Appendix E was used to guide the talanoa sessions and keep the focus

on the key research questions.

3.4.3 Tui: Data Analysis

While proper data analysis will be covered fully in the next chapter, for this study it was

important to include a little of the data analysis since tui completed the kakala framework that

was used for data collection. Tui is the process of weaving or creating the kakala where only

the best and most fragrant flowers are selected for the formation of the kakala itself. In the

context of this study, this is the process where all the relevant information that had been

collected was sorted and the best pieces woven together into the perfect garland. The data

collected in this study was mainly qualitative and came from staff development activities,

classroom observations, and talanoa with both students and teachers. Some initial quantitative

data came from the external mathematics examination results.

The main activities in Tui were the (i) transcription of data, (ii) coding of the material, and

(iii) the proper analysis.

i) Transcribing Data

All data were recorded on audiotape and videotape of the classroom observations and talanoa

sessions. Three teachers at the school helped the researcher with the transcription of the data.

The researcher held an orientation for these teachers by working with them on a sample of the

transcribed materials and the guidelines of how the transcribed data will be used. This

50

transcribed data was edited and reviewed by the researcher before they were given to the

participants for confirmation.

While all the recordings were transcribed, they were not all translated into the English

language. Due to various constraints, the decision was made to translate only the parts which,

on initial listening, were regarded useful and relevant to the present study. All transcribed data

was stored for further analysis.

(ii). Coding the material.

In order to highlight emerging themes and identify patterns shown by the data, the researcher

used different coloured highlighters to code similar issues and themes that arose from the text

itself. The purpose was to dissect the textual data into meaningful and manageable chunks to

facilitate meaningful data analysis.

(iii). Analysis of the data.

After coding, the researcher went through the coding text segments and identified the

common, salient and fundamental themes that emerged from the coded segments. Analysis of

data was based on the three key research questions.

3.4.4 Luva: Dissemination of findings

Luva is the presentation of the final product (kakala) to the people for whom the kakala was

fashioned. In the context of this study, this is the presentation of the final copy of this thesis to

the examiner for the examination process. After the completion of this project the thesis

should also be luva or presented to the important stakeholders namely Heilala High School,

the Ministry of Education, and the University of the South Pacific.

3.4.5 Mālie

This study will be examined by people of status in mathematics and mathematics education as

the supervisor determines fit. If the study passes the test and and meets all requirements after

examination, a grade will be formally given in recognition of the work done. In the tradition

of the kakala, a loud applause and exclamation of ‘mālie’ will also be heard in the

community.

51

Since the findings of this study will be useful and relevant to key educational staeholders,

then cries of mālie will also be heard from those to whom the wider luva process disseminates

these findings.

3.4.6 Māfana

This stage is a response to the kakala. Once the audience groups such as the Ministry of

Education, the host school and the mathematics teachers at large perceive this study to be

mālie, the results should inspire them to change their teaching practices and implement useful

aspects of this work in their classes and schools. Best practices can be adopted while the

recommendations will ideally move them into action and change the mathematics teaching

practices in high schools in Tonga. As a result, the researcher will feel more māfana and

committed to implement these recommendations in her own mathematics classroom, and

future research in mathematics teaching and learning.

3.5 Ethical Considerations Prior to beginning the study, the researcher sought human ethics approval from the University

of the South Pacific’s Ethics Committee. That approval was granted. Then, Consent letters

were issued to the head of Heilala High School (Appendix H), the Ministry of Education

(Appendix F), participant teachers and students, and students’ parents (Appendix G). A verbal

request was made to the school and the participants to videotape the classroom observations

and this request was granted. The participants were informed of their right to withdraw

whenever they wanted. They all verbally expressed a strong desire to be part of the study.

An approval letter was circulated by a senior Ministry official to the Examinations Unit, the

Tonga Institute of Education, and the Curriculum Development Unit, requesting that the

researcher be given access to use the documents sought and assisted in data collection for this

study. This support was greatly appreciated and very useful.

Information to participating students

Prior to the classroom observation, the researcher took time to go in to the selected classes

and explain to both students and teachers the purpose and objectives of the investigation, as

well as the procedure for data collection, including the need to capture on video the classroom

observations. Students asked questions concerning the video camera and they were assured

that these videos would be kept confidential and used only for the purpose of this study. They

52

were notified that all data would be secured securely and would be destroyed six weeks after

the study had been approved. All consent forms were approved prior to the collection of data

for this study.

3.6 The Researcher The researcher is an insider-researcher who has worked as a mathematics teacher at HHS for

15 years, and held the role of Head of the Mathematics department for over 6 years. She was

also President of the Tonga Secondary Schools Mathematics Teachers Association in 2013.

The researcher has also participated in the marking of mathematics external exams both from

the South Pacific Board of Educational Assessment (SPBEA) and from the Tonga Ministry of

Education.

The researcher believes that one of the key roles of a mathematics teacher, and which is the

cornerstone of facilitating students’ learning in mathematics, is the responsibility to establish

and nurture a good relationship with students in the mathematics classroom. If the teacher

fails to establish such relationships, then effective mathematics teaching will be frustrated,

and students will have negative experiences in the class. The SIOP model has helped the

researcher to gain such perceptions and experiences in the mathematic classroom.

The role of the researcher in this study was to facilitate the mathematics teachers’ learning

and implementation of the SIOP model in their classes. The researcher was a participant

observer during the classroom observations and facilitated students with their activities and

making sense of questions.

3.7 Limitations and challenges Several difficulties were encountered during the journey of this study. Firstly, the video

recording of classroom observations revealed some recording gaps. This was a technical

challenge that was likely due to only one camera being used for all the observations and the

the poor focus of the camera. Secondly, the in-depth case study at only one school could not

allow generalisation to mathematics teaching in other schools. However, the depth of

information collected can be useful infomation for other teachers as they strive to enhance

learning and teaching mathematics and improve acheievement levels for their students and

schools. Thirdly, some of the data sought from the Ministry of Education and other high

schools was not received. However, the researcher was grateful for the limited data received

and was also aware of the sensitivity of some of the information sought. Fourthly, this study

53

demanded much from the researcher who needed to balance the requirements of a full

teaching load and the needs of a researcher at the time of the study. She is grateful to the

management of the school and colleagues at the school for all the assistance rendered to her.

As an insider, the researcher found the timeline for the many meetings in between data

collection very demanding and challenging. To manage both her teaching load and the

demand of this study, the researcher set a timetable for after working hours when work on this

study was undertaken. This gave the opportunity for the researcher to fulfil her everyday

teaching load while also meeting the timelines of completing this project. However, on my

role as a Head of Department that could affect the results of this observation.

3.8 Summary This chapter discussed the procedures and processes used for to carry out the study and

collect data. The case study approach was used because of the site and context at Heilala High

School. The participants were from the school. The main research tools used were classroom

observations, group talanoa, and individual talanoa. These tools collected much data which

would be analysed through a triangulation process. That process was expected to increase the

quality of data analysis and provide validity and reliability of the findings.

All classroom observations were video recorded and all talanoa sessions were audio recorded

in order to facilitate the participant’s reflection and the transcription of the data. There were

periods of review and discussion in-between the three observations. All tapes would be kept

safely locked in a filing cabinet and used only for the purpose of the study.

The Kakala research framework was used to guide the key components of this study. Its six

phases: Teu, Toli, Tui, Luva, Mālie, and Māfana (Manuʻatu, 2001) have been discussed in

detail to indicate how well they met the design and methodology of data collection and

analysis. The Kakala research framework was considered suitable for this study because of its

nature and context.

The analysis and discussion of the findings will be presented according to the main key

research questions in the next two chapters.

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CHAPTER 4: RESULTS AND DISCUSSION - MATHEMATICS TEACHING STRATEGIES

4.1 Introduction

This chapter presents the first set of results and discussions for this project. The second set

will be presented in the next chapter, Chapter 5. The set of results and discussions to be

analysed and discussed in this chapter respond to Questions 1 and 2 as follows:

1. How effective are the common ways of teaching mathematics at Heilala High School?

2. What are the major challenges of teaching mathematics at Form 5 level at Heilala High

School?

Data for these questions was collected from the following activities:

� talanoa sessions with each of the six mathematics teachers at Heilala High School,

� observation of a total of nine lessons or three Form five lessons of each of the three Form 5 mathematics teachers,

� talanoa with three groups of four Form 5 students who belong to the Form 5 classes that were observed

� discussion with each of the three Form 5 mathematics teachers who were observed in class.

4.2 Data Analysis Data Analysis in this chapter is done in line with the two major research questions being

addressed. Question 1 which asked about ‘the effectiveness of the common ways of teaching

matheamtics’ was subdivided into four subparts:

� Commonly used teaching strategies

� Teachers’ definitions of effective teaching

� Students’ views about effective teaching

� Students’ views of the effective teacher

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The analysis of data under these subheadings was important to compare the teachers’ views

with those of students, and also to reconcile what the teachers said with what was observed in

their classes. The next sections will highlight some interesting comparisons.

Responses to Question 2 on ‘the major challenges of teaching mathematics’ were collected

from both teachers and students. So, the data analysis was also done under the two sets of

response. The next sections record the analysis and discussions of the data collected.

4.3 Teaching Strategies

The most commonly used teaching strategies identified by the six teachers included: the

lecture, demonstration of activities and follow-up exercises, group work, and bell work and

questioning. They did say that at most times they used a combination of two or more

strategies and the most common combination was of demonstrations, group work, and follow-

up exercises. The teachers’ explanations of their preferred methods and how they used them

are recorded next.

4.3.1 Demonstrations and follow-up exercises

Five teachers said that they regularly used demonstration and follow-up exercises in their

classroom. At most times, the demonstration of an activity or a particular method of solving a

problem was alternated with student activities, which were usually undertaken in groups. The

sequence of events would normally begin with the teacher saying a few words, followed by a

demonstration of an activity. Teacher MT5 described it this way:

“I demonstrate the mathematical activity then let the students do the

exercises. When I come back to check the answers I would show how to get

the right answers on the blackboard”. [MT5]

MT6 discussed a similar process and added that if her students did not appear to understand

the first demonstration, she would provide further examples and demonstrations followed by

some follow-up exercises. Most would get it the next time around. This method is

synonymous with the “practice makes perfect” adage of the drill and practice method of

learning, which has roots in the behaviourist theory of learning (Resnick & Ford, 1981).

Two of these teachers reported this strategy during the talanoa sessions, and all three Form 5

mathematics teachers used demonstration in their lessons.

56

For example, MT1 used demonstrations twice in a particular classroom observation: (i) how

to sketch a complete probability tree, and (ii) how to calculate probability from the probability

trees. Follow-up exercises were given after the second demonstration, to assess the students’

understanding of sketching probability trees. Likewise, MT2 demonstrated how to use the

Pythagoras theorem to calculate the missing side of a right angle triangle in one of his

observed classes. MT3 demonstrated how to sketch a cubic curve using the factorizing form

in one of her classroom observations and followed this up with exercises for the students.

The five teachers talked about demonstration as a scaffolding method or a tool to enhance

students’ understanding as well as their procedural knowledge and skills. It is a tool for

mastering learning where students will master the mathematical concept through these

demonstrations and repeated practice.

In the traditional Tongan context, “fakatātā” (demonstration) is one of the main teaching

strategies in informal learning practices. Literally, “fakatātā” means that the teacher will

perform the learning in the first place while the learner observes. Then the learner tries to

follow exactly the performance or practices that were performed by the teacher. The process

is repeated until the learner has mastered the desired learning outcome.

This activity is commonly used in dance practices and the learning of new skills such as

weaving, cooking, the making of tapa cloth, and any new skills that need to be transmitted

from the elders to the young people or from mother to daughter or father to son. The practice

of learning from and imitating one’s elders is still used in Tongan society today as both a

teaching and learning strategy.

4.3.2 Group work

Five out of the six mathematics teachers shared how they used group work in their

mathematics classes and its benefits. The advantages of group work which were highlighted

included that it allowed students the opportunity to engage freely with each other and propose

possible solutions. This allowed then to think for themselves and debate, and also enhance

their confidence in their ability. Three teachers (MT6, MT4, and MT2) specifically discussed

how the group work went in their classes and how it benefited students. MT6 explained that

he organised activities for student pairs or groups and “students enjoyed it a lot”.

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MT4 discussed that through group work “students learned to interact among themselves” and

they always look forward to “sharing what they have got from their small group to me and the

whole class”.

MT2 also had a strong preference for group work. In all three observed classes, he used group

work to get students to answer questions, complete an activity or prepare a presentation. In

one of his classes, students were divided into groups of three called “home groups”. The class

was divided into five home groups and the student members were assigned numbers; Number

1, Number 2 and Number 3. All students who were assigned as Number 1 came together to

form another group called the “Expert group number 1; all Number 2 students formed

“Expert group number 2” and likewise, the last group of students formed “Expert group

number 3”. Each expert group focused on a specific activity to solve the problem.

In this particular exercise, the class was required to find the sum of the interior angles of

different polygons. Expert group number 1 used the protractor to find the answer; expert

group number 2 had to divide the polygon into triangles and used that to find the answer;

expert group number 3 used the formula (n – 2) * 180 to calculate the answer the total interior

angles. Each expert group selected a group leader to lead the discussions and plan towards the

solution. Once that was done, each member would go back to their home group and take turns

to teach the particular method learnt from their expert group. Thus MT2 used a very

sophisticated group work strategy usually referred to as the ‘jig saw’ technique as a tool to

enhance students’ understanding of the mathematical concepts and activities.

Both MT1 and MT3 also used group work in their classes. For example, MT1 used group

work throughout all his observed lessons. He grouped students into groups of fours and

students were asked to work on a question then prepare to report back to the whole class.

MT4 and MT3 also followed this type of reporting back to the class as a form of assessment

of students’ learning and understanding.

In the traditional Tongan context, “ngāue fakakulupu” is the equivalent of group work and it

is one of the main tools of learning used in society and the communities. For example,

“kulupu toungāue” would involve a group of men with common goals who then gather

together, create some rules and guidelines to frame their association, then put their combined

effort into working at each other’s plantations or whatever tasks need to be done. Performing

in groups makes work easier, lightens the load, and is more enjoyable for participants. The

members motivate each other and strengthen the relationships and solidarity in the

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community. The goals are those of the community and the common good. Women in Tonga

also form similar work groups for attending to important activities such as weaving or

“kulupu toulālānga” and making tapa cloth.

The idea of “ngāue fakakulupu” is still strong in the villages where women and men gather

together to work for a common goal for the benefit of families and the community. The same

underlying principles for group work in the communities is applicable to group work in the

mathematics classroom today, where sharing ideas and supporting each other has the potential

to promote a more positive image of the subject.

4.3.3 Asking Good Questions

Three teachers talked about asking good questions and using good questioning skills as

important components of effective teaching. Two of them (MT3 and MT2) cited this skill

during the talanoa sessions, while MT1 was the only one who used questioning in his classes.

MT3 pointed to the lack of student response as a challenge: “I gave them the question...after

two minutes, I said, okay, this is how it is done”. Many teachers face this dilemma and solve it

by answering their own questions. They realise that it is not helping students but do not see

any other way out. MT3 continued “It’s more like I spoon feed them...they just rely on the

teacher, ‘ikai ha taimi ia (there was no time) to let them think”. She acknowledged that, “I

am still using a method I used at the middle school to teach the new entrees from the primary

school – we showed them what to do”.

MT1 used a variety of questions in his classes. This was clearly seen in the video clips of his

three classroom observations. In one of them the question came in the bell work or advance

organiser activity:

Bell work question: Calculate the size of the unknown angles. Give a reason for your answer.

After he gave some time to the students to work on this, he discussed the answers with them:

MT1: What is the size of the angle?

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Ss: Seventy six degrees.

MT1: What is your reason for giving seventy six degrees to angle A?

S1 Add all the angles inside, it will be equal to one hundred and eighty.

MT1: S1 said to add all the angles inside it will be equal to one hundred and eighty. But

the question is why angle A is seventy six degrees?

S2: The base angle must be the same.

MT1: Very good. He said the two base angles must be the same.

He asked a lot of “why” questions that required students to verify and justify their answers –

this would confirm that they understood and were not merely guessing. This form of

questioning was evident in all three classes.

4.3.4 Bell work

Three teachers were observed using bell work as one of their typical ways of teaching

mathematics. Bell work is a quick task provided at the beginning of the class that get students

on task immediately and therefore set the tone for the rest of the class. This work can either

be a form of revision where students relate to their prior learning, or an introductory task for a

new topic, or it can be a fun activity to boost the enthusiasm of the class right at the

beginning, or it can be any short activity designed by the teacher to engage students

immediately. This is more like an advance organizer (Ausubel, 1968) where students get to

organize their thinking at the beginning of the class. MT1 said “my usual method is

providing a bell work” for the students. In all of his three classroom observations, he has used

a leading question either to activate students’ prior knowledge or guide students’ thinking to a

new topic. For example, one of his bell work activities involved a revision question

concerning their previous topic, preparing the students to be able to calculate angles on

parallel lines which was the main objectives for that lesson.

Bell work was also a common strategy for MT2 and MT3’s classroom practices and was

observed during each of their three classroom observations.

MT2’s classroom observation revealed bell work which involved the students calculating the

sum of the interior angles of a polygon. The following observation excerpt illustrates this.

Bell work: On the bell work section on the board, he provided the definition of a polygon as: “closed figures made up of straight lines”. In pairs, discuss and

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explain why these two figures are not polygons (One shape was a cone, and the other was an open figure).

The students were seated in pairs and discussed why the two figures given were not

polygons. This was an introduction to their topic for the day, which was based on the

calculation of the total angles of polygon.

Similarly, MT3’s bell work was also written on the board as a revision question, where

students were able to link their thinking to their previous lesson:

Bell work: Draw the parabola y = (x + 3)² + 4. Students were seated in groups of 4 or 5, and discussed and reviewed how to draw this

parabola by working out the main points necessary to sketch the shape of the graph, such as

the x and y - intercepts and the vertex of the graph. This bell work helped activate students’

prior learning regarding sketching parabola, preparatory to sketching the cubic curves which

was the main objective for the day.

Bell work is similar to “ngāue he tā ‘a e fafangu” where work will start exactly on time or

when the bell rings. The context that this applied to in Tongan society is, when churches

begin on Sunday. When the bell rings, the church will start no matter how many people are

attending at that time. If half of the congregation are late, it does not matter. In the context of

the classroom, bell work at Heilala High School is expected to set students on task on time in

order to maximize students’ learning time.

All students who were interviewed strongly supported the idea of bell work. They always

looked forward to the activities.

4.3.5 Lecture

In the talanoa sessions, three teachers identified lectures as a typical method of teaching

mathematics. MT4 acknowledged that the lecture is “the traditional way of teaching” and he

used it “as a format for teaching”.

MT5 and MT1used lectures frequently in their classes to teach mathematics. In the

observation of his classes, MT1 did not use the lecture method that much. His explanation

was that

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“I used lecture a lot in the old days but now I have changed to new methods

of teaching mathematics”. [MT1]

The lecture is the most common method of teaching where the lecturer is ‘the sage on the

stage’ talking down to the students and giving them instructions. The assumptions are that the

teacher knows everything and is there to fill up students’ minds which are like empty vessels.

Many mathematics teachers perceived that while the lecture is not entirely unproductive, it is

best used with other strategies including student activities and questioning.

The lecture is similar to “fakahinohino” in the Tongan context where the teacher gives

instructions, procedures, and the expectation so that the learner will absorb all the

information. The teacher in the traditional context is the parent or a knowledgeable person in

the village or community. The teacher has the knowledge, experience and skills that the

young student can learn from. In time, the learners/listeners will have learned enough to give

him/her the right to give the “fakahinohino”. During the process of “fakahinohino”, the

learner listens, imitates and grows in knowledge. The learner’s behaviour is an act of

fakaʻapaaʻpa (respect).

This learning context exists in Tongan society today. However, as part of the international

community, we are seeing many changes both in society and in the formal classroom where

students are encouraged to ask questions and give their views freely. In the modern

classroom, students are the centre of learning and the tasks and activities are presented to

encourage them to think and express an opinion. Effective teaching encourages student

autonomy and creativity. The teacher guides at the side while students dominate the

classroom activity. The focus has shifted to the students who are encouraged and supported to

plan, think for themselves and arrive at solutions.

4.3.6 Other Strategies

Storytelling and gallery walks were also mentioned by one teacher.

MT4 talked about telling stories as a tool to boost students’ interest and enthusiasm for

mathematics

“Sometimes, I would tell a story that is related to the mathematical topic or

problem being discussed. I like to relate the history of a mathematical

concept and the mathematician or an important event that is related to

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specific mathematics issues. I would do this when I see students struggling

and facing a lot of pressure in investigating or solving mathematical

problems. The stories relieve some of the pressure and make mathematics

real and fun. When they laugh even for a short time, it can be an incentive

to push on.” [MT4]

Telling stories is equivalent to “talanoa” in the context of the Tongan culture. It is a

common strategy that elders use to share their knowledge to the younger generation in a way

where they bring reality into the learning environment. For example, when the elders taught

the concept of fakaʻapaʻapa, they would tell a story in which fakaʻapaʻapa was demonstrated

and how the moral of the story enhance the concept of fakaʻapaʻapa. Tongans love telling

stories and they make connections through the concept of talanoa.

In today’s mathematics classes, “talanoa” is a useful strategy to help place mathematical

ideas in practical, local contexts that are familiar to students. Sharing stories of mathematical

exploits and discoveries bring life and reality to the otherwise abstract lessons and enables

students to make connections and apply the ideas in their daily life experiences.

MT6 was the only teacher who talked about gallery walks:

“I think that is a nice one, because students get to walk around in each

gallery and pick a problem to solve”. [MT6]

She explained how she divided the classroom walls into different stations and then posted

various questions on the station walls. Students would walk around each station, pick a

problem from each one and settle down to work at solutions. The questions in each station

varied from easy questions to difficult ones. This activity gave students the opportunity to

choose the questions they were prepared to solve, whether it was the easy ones or those that

offered more challenge. With time, students could use gallery walks as a context for

developing problem solving skills, confidence and maturity.

4.4 Effective Mathematics Teaching: Teachers’ Views The six teachers in the sample were asked for their definition of effective mathematics

teaching. The definitions are grouped into four main categories:

� facilitation of students’ learning

� preparing lessons well

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� motivation of students’ learning

� Enhancing students’ understanding

The definitions overlap and are interrelated.

In describing effective mathematics teaching, some teachers also illustrated the characteristics

of an effective mathematics classroom. MT1 described the nature of an effective classroom as

one where “students are actively working and co-operating with each other...and they know

(ilo’i) what the teacher is explaining as well as the activities for the day”.

MT6 agreed and said,

“When you go to his/her classroom, all students are enlightened and

involved in discussions and in doing their works. They look like they

understand which shows through what they write and their responses to the

teacher’s questions...and even when they ask a lot of questions. This shows

that they are learning something and want to know more” [MT6]

Both teachers focused on all students engaging in the class rather than only some of the

students. MT1 used the words active and co-operate and MT6 elaborated by saying that in

such classrooms, students were involved in the class discussions, doing their work,

responding to teacher’s questions and asking questions. She continued on to say that such

classrooms produced an environment where the students “want to know more”, which will

lead them to explore, experiment, and dig for deeper understandings which result in effective

mathematics teaching.

The teachers’ views of what constitutes effective teaching are discussed next.

4.4.1 Facilitation of students’ learning

Two teachers believed that ‘effective mathematics teaching’ has to do with the ability of the

teacher to facilitate students’ learning.

MT2 explained facilitation as “when the teacher allows students to spend time on the

activities and discuss amongst themselves rather than the teacher dominating the class”.

He saw facilitation as the teacher allowing and enabling students to do the work and to

discuss freely amongst themselves and be actively engaged with the mathematical tasks. The

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teacher as facilitator places himself/herself on the side and gives centre stage to students who

become the focus of classroom activities.

MT2 demonstrated this belief in his classes that were observed. In one of his classes, most of

the activities were done by students, and he was keenly observing them. An important point

for this teacher was his lesson plan and learning objectives. He had differentiated the

following content learning objectives (CLO) and language learning objectives (LLO) and

written them on the board at the beginning of the class:

CLO: Students will be able to: LLO: Students will be able to:

� Identify polygons and names of polygons

� Calculate sum of interior angles of a polygon using the formula (n – 2) *180

� Calculate the sum of interior angles of a polygon by measuring it with the protractor

� Calculate the sum of interior angles of a polygon by dividing into triangles

� Share and discuss

condition of a polygon,

and the different methods

of how to calculate the

sum of interior angles of

a polygon.

All objectives were achieved through group work which was discussed earlier.

MT2 moved around, listened, observed, and assisted students when needed. The class was a

hive of activity and students were talking and discussing freely amongst themselves. They

discussed the strategy while MT2 provided support to enhance students’ understanding.

MT6 described such a facilitator as “one who is prepared and he teaches the students not the

lesson”. She believed that a successful teacher “knows the need of each students, their

weaknesses and confusions, and is able to discern when students are confused and don’t

understand”. Through personal experience, this is significant because most of the time,

mathematics teachers focus more on teaching the lesson and completing the syllabus rather

than focusing on the students’ understanding of mathematics and meeting their learning

needs.

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4.4.2 Motivation of students’ learning

Two teachers talked about the teacher’s role in the motivation of students’ learning. MT1 saw

effective mathematics teaching as “my ability as a teacher to motivate the students to work on

their own”. According to him, this began from “the ability of the teacher to clearly explain

the lesson’s objectives at the beginning” which will set the tone for the rest of the class.

As a motivation of students’ learning, MT4 believes that the mathematics teacher will “also

change the lives of the students by his or her examples and behaviour in and outside the

classroom”. MT4 believed that such a teacher will influence students to commit themselves

to do better both in school and in the community.

4.4.3 Preparing lessons well

MT3 perceived that effective mathematics teaching “begin from the lesson plan and continues

to how the lesson is delivered to meet the students’ learning needs”. She added that delivery

of the lesson must include a “variety of activities” to cater for the needs of the students.

The researcher noted that MT3 was better prepared in her last two observed lessons than in

her first one. Her students (S31, S32, S33, and S34) mentioned in the group talanoa that they

enjoyed the activities in the last two observations better than the first one. S34 said

“I enjoyed the second observation than the first observation because the

activities were fun and it helped me to understand the exercise faster”.

[S34]

The result of this talanoa session is discussed from the students’ view of effective teaching

strategies on page 69. This is a significant view raised by the students because well organized

activities are the result of a well-planned lesson. The researcher emphasized here that the

essence of an effective lesson started from effective lesson preparation. A quality time spent

on organized and well planned activities resulted in students’ actively involved in the

mathematical tasks as well as a positive attitude toward the subject.

4.4.4 Enhancing students’ understanding

Two teachers believed that ‘effective mathematics teaching’ must enhance students’

understanding.

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MT5 thought that “after teaching the mathematics exercise, the student is able to

understand”. This perception focused on whether the student has understood what was taught

or not. If students’ understanding was not enhanced, then the teaching has not been effective.

MT4 explained that the teacher “makes the hard concepts easy” and the teaching “is based

on how to find the answer instead of what is the answer”. Students’ understanding will be

enhanced when teaching is based on the process of finding the answer rather than the answer

itself.

4.5 Effective Teaching strategies: Students’ view The data presented in this section was collected from the 12 students who were asked for their

views on what constitutes (i) effective teaching and (ii) an effective teacher. The students

were part of the classes that were observed - four from MT1’s class, four students from

MT2’s class, and the third group of four students from MT3’s classes. The students were

shown video clips of their classes and were then asked for their views of (i) effective

teaching, and (ii) the effective teacher.

The students’ views were interesting when compared to what their teachers said. The students

listed (i) group work, (ii) demonstration, clear definitions and explanation, (iii) hands-on

activities and mathematical problems, and (iv) effective classroom management as important

components of effective teaching. For the students, group work was perceived the most

effective strategy and all students said that they learn best from group work.

4.5.1 Group work

All 12 students identified group work as an effective teaching strategy.

S13: ‘Oku tokoni lahi ‘aupito ‘a e ngāue fakakulupu keu toe mahinoʻi ange ai ‘a e fika.

S13: Working in groups has helped me to understand maths better.

S21: Ko e taimi koia naʻa ne vahe mai ki he ngāue fakakulupu pea mo e tautau tokoua, naʻe toe lava ke u mahinoʻi ange ai hono faikoia ‘a e foʻisiakale.

S21: When he divided us into groups and in pairs, I tended to understand better how to do the circle.

S22: Sai ‘aupito ‘a e taimi ‘oku mau ngāue fakakulupu ai, ‘oku mau fevahevaheʻaki ai mo e tamaiki ‘a e ngaahi meʻa koē‘oku ‘ikai mahino. ‘Oku lava ke hanga ‘e he tamaiki ‘o tokoniʻi au ‘i he meʻa ‘oku

S22: It is better when we worked in groups because we shared the things that we don’t understand. The students helped me with the things that I

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‘ikai ke u ‘ilo. did not know.

S31: Kia au, mahalo ‘oku sai ange ngāue fakakulupu...kemau lava ai ke fevahevaheʻaki ‘ete mahino mo e mahino koē ‘a e niʻihi kehe pea te lava ai ‘o tokoni kiate kinautolu he ngaahi meʻa ‘oku nau faigataʻaʻia ai. Pea nau lava ‘o tokoni kiate kita he meʻa koē ‘oku te faingataʻaʻia ai pea te fengāueʻaki mo kinautolu hono kumi e moʻoni.

S31: To me, working in group is better...we are able to share our understanding and I get to help those who struggle. They also helped me in the things that I struggled with, and we worked together to find the answer.

The students’ responses used terms that describe the nature of group work and how it

enhanced learning and understanding:

mahinoʻiange: to better understand

tokoni: help

tokoniʻi:

fetokoniʻaki:

act of helping

helping one another

fevahevaheʻaki: act of sharing

fengāueʻaki: working together

Fevahevaheʻaki (sharing) of various ideas, fetokoniʻaki (helping one another), fengāueʻaki

(working together) within the group underpinned students’ perception of why this strategy

enhanced their understanding and improved their social skills.

S14 said that the members of the group influenced him to work,“They make me want to

work”.

For S22, working in groups is a win-win situation. “When I helped others, I was able to gain

more knowledge myself”.

Through talking and discussing with others, the student learned a lot, something he would

have missed had he stayed on his own. The values of group work are best represented in the

following conversation between S11, S1, S13 and S2in one of MT3’s classes. The teacher had

divided the class into groups of 4 and they were to discuss how to sketch non-linear graphs.

Each member in the group had a particular graph to work on.

S11 was teaching how to sketch a parabola:

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S11: This (pointing to the vertex) is the half of these two (pointing to the two x-

intercepts). Do you understand?

The rest: Yes

S11: Then we continue

Now it is S13’s turn to teach. She is teaching how to sketch y= (x + 2) (x + 1) (2 – x)

S11: What is that number? (asking for the x-intercepts

S13: It’s negative two and negative one

S1: So this is positive two?

S13: Yes

S1: Re-do this again...(pointing to the y-intercepts)

S13: Do another example

She wrote down Y = (x + 3) (x + 2) (1 – x) and they all worked together in finding the x-

intercepts.

S11: It’s negative three, negative two and positive one

S13: Then replace X with zero finding the y-intercepts). Zero plus three, zero plus two

and one minus zero ((0 + 3) (0 + 2) (1 – 0).

S1: Get six

This is a copy of their final graph, taken from S11’s note book:

Group #3 has been actively engaged in discussing, explaining, and demonstrating

understanding in the task of sketching cubic curves. S2 was attentive during the group

discussion but the remaining three were more vocal. S2 would only talk when he had a

question. Everyone focused, asking questions for another demonstration, and S13 was able to

provide another equation for another example in order to enhance the group’s understanding.

Figure 5: Taken from S11’s note book, cubic graph.

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4.5.2 Clear definition, Explanation, and Demonstration

Most students agreed with this strategy but 5 of them were more vocal and their responses are

discussed below. S11, S12, and S13 were all in one group, S22 was in the second group, and

S34 was in the third participant group.

As group one watched their classroom observations, S11 and S13 emphasized the significance

of explaining the mathematical terms at the beginning of the class. He made reference to one

of their classes that was about “Angles on Parallel lines”.

S11 said that he was able to understand angles on parallel lines, “because he (MT1) first

explained the meaning of the key words”.

S13 agreed and said, “I understand after the class how to do the parallel line (angles) and I

liked how he defined the key words right at the beginning”.

In the video, MT1 explained the following key words: transversal line, parallel lines, and

supplementary angles, straight after the bell work:

Today, you should be able to learn three relationships caused by a line

crossing two other lines. We call this line a transversal line...but the two

lines we are focusing today are called parallel lines. So when a transversal

line crosses two parallel lines it forms three angles relationships. The first is

corresponding angle... [MT1]

The conversation continued but MT1 have repeatedly explained the meaning of a transversal

line and demonstrate these angle relationships on the board. He repeated:

There are three important words that you need to be aware of...transversal

line, parallel line, and supplementary. Supplementary angles occur when

two angles are adding up to one hundred and eighty [MT1]

Explicit explanation of the notes and key mathematical words enhance mathematical

understanding (Echeverria et al., 2008).

Moreover, S34 emphasized the effectiveness of this strategy while watching the video of their

classroom observations. S34 referred here to MT3 and stated:

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“In observation one, she didn’t really explain it but she gave us some

exercise, but in observation two, she explained it to us in English and in

Tongan, and she gave us some key word so we can understand it better.”

[S34]

In the video, the main focus of the lesson was on probability. Along with her objectives on the

board, MT3 wrote down her key vocabulary for the lesson, which included: ‘probability’,

‘event’, ‘outcomes’, ‘impossible’, and ‘certain’. After discussing her bell work she introduced

her objectives for the day by describing:

The teacher counted that 12 out of 17 hands were raised. She wrote this as a fraction on the

board. The discussions continued by using the students’ background to clarify the key words

for the lesson and she used code switching (Manu, 2005) to clarify the meaning of the key

vocabularies.

The pace of the teacher talk is very important for students and was picked up by S22. As the

group was watching the video, S22 said this of his teacher MT2: “I really liked it because he

was speaking slowly, not too fast, when he was explaining”.

In the video, MT2’s pace of talking suited the learning needs of S22, particularly his English

proficiency level.

In addition, S11, S12, and S13 perceived that providing a lot of demonstrations enhanced

their mathematical understanding. S12 pointed out “MT1 provides a lot of examples and I

understand math better compared to last year”.

MT3: What are words you can find from the word probability?

Some: Probable

Some: Ability

MT3: Look outside. Do you think it will rain today?

Ss: Yes

MT3: Why?

Ss: There are dark clouds outside...No sunshine...cloudy

MT3: Raise your hand if you think it will rain today

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In the video, it was noted that MT1 provided three demonstrations on how to sketch a

probability tree and use it to calculate the probabilities.

The researcher asked if they understood the objective after the first example.

S11 responded “I didn’t understand it” while S12 “I understand some and some I didn’t

understand”. However they all said that things became clearer after the second example.

S13 said, “I started to understand how to do probability tree”. It became obvious that the

third example was for the benefit of students like S13.

4.5.3 Hands-on, fun and relevant activities

Five students talked about hands-on activities and ‘fun’ in mathematics.

As group one watched the video of their third classroom observation, S11, S13, and S14

reported that they best appreciated methods of learning mathematics involving use of their

hands. In reference to that video, S11 said,

“I really like this strategy which is working out math using my hands, for

example, using the protractor to measure the angle in the parallel line”.

[S11]

At the same time, S14 added:

S14: Na’a ku saiʻia mo au he konga ko

ia naʻe ngāueʻaki ai ‘a e

protractor. We all participated.

I also liked that part when we used the

protractor. We all participated.

The same student stressed during this talanoa session, the need for the activity to be relevant

and connected to the students’ background. He referred to MT1 and said, “MT1 used a lot of

activities that relates with us and really helped us to understand”.

MT1 has responded to a significant point; the need for mathematical activities to be relevant

and meaningful (Echevarria, 2008, 2010; Anthony & Walshaw, 2007; Brootenboer, 2009) in

order for students to make connections and be actively engaged, which results in deeper

learning.

However, S31, preferred mathematical games. She said,

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“I like it better if there will be games, games about math problems, may be

student will understand them better, we all participated”. [S31]

As the third group watched the video of their second classroom observations, S34 said,

“I enjoyed the second class more than the first because the activities were

fun and it helped me to understand the exercise better”. [S34]

According to S34, making the activities enjoyable, and at the same time meaningful, is

significant to mathematics learning because she is able to understand the exercise faster when

this occurs.

4.5.4 Provide mathematical problems

Two students saw mathematical problems as an effective strategy to enhance their critical

thinking skills.

S31 reported:

“I love to go there (maths class) in order to figure out math problems, I

always love it as well as having lot of exercises and working time, as well as

doing works”. [S31]

She elaborated that her teacher the previous year often gave them an activity where she:

“listed numbers on the board. The students picked any number and each

number corresponds to a mathematical problem. Once we picked a number,

the teacher gave us the corresponded mathematical problem then we figure

out the problem” [S31]

S31 obviously enjoys solving problems, but these mathematical problems have to be

meaningful and relevant to students’ learning. S11 expressed similar sentiments earlier when

she said that she preferred that the teacher uses games with the mathematical problems.

4.5.5 Effective classroom management

Two students thought that effective classroom management was important for effective

mathematics teaching and learning.

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For S34 this element is demonstrated when the teacher

“can manage the class well in order for them to be quiet and pay attention

solely to what the teacher is explaining so the class can understand it”.

[S34]

S31 expressed a similar view, that “the math teacher to work on time and not to easily give in

to students when they complaint about having lots of work to do”. She gave the example of a

mathematics teacher from a previous year:

“When we get to her class, bell work was ready, we would do the bell work

on time, there would be activities such as copied notes then she would

explained them. There were times provided for us to work, time to mark

them, and then we were given home work. If we didn’t do our homework the

next day, we would be in trouble. Not only did we enjoy it but students

would focus in math” [S31]

According to the students, a well organised teacher will be able to organize the transition of

activities so students will participate. They wil also provide ample time for students to work,

and follow up with exercises or homework rather than leaving it to be marked another day.

4.5.6 Use of the Tongan language

One student said that he strongly preferred that their mathematics teachers teach using the

Tongan language, because things are clearer then. Student S22 reported:

“I prefer a teacher that teaches us in Tongan because he makes the work

clearer and we can understand. When we ask questions, he will explain

easily also”. [S22]

S22 said that his English proficiency is not high and when her teacher teaches in English, she

hardly understands it. This is the reason why she prefers mathematics teachers who teach in

Tongan. For second-language speakers of English, this student’s request makes sense. It was

important to mention this item because the English language is a big challenge for students

and many struggles with it in all their classes. As most mathematics teachers believed, a

mathematics problem is first and foremost a language problem.

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4.6 An Effective Mathematics Teacher: Students’ View The researcher was aware of the sensitive nature of this section and took care that the

discussions were general and not about particular teachers. Students identified certain

characteristics they thought useful to have in a mathematics teacher who would best facilitate

learning; their thoughts have been grouped into three categories as follows:

� Is active, helpful, motivating, encouraging, and fair;

� Provides many activities, uses Tongan language, provides mathematical problems for

practice, and clear explanations; and

� Practices effective classroom management.

Active and helpful nature

Five students preferred having a mathematics teacher who was active and at the same time

helpful:

S11: Kou fiemaʻu ha faiako koē ‘oku longomoʻui ko e ‘uhi ke longomoʻui ai pē mo e kalasi pea mo ha faiako ‘oku fie tokoni.

S11: I want a teacher who is active so the class will be active as well and a teacher who is helpful.

S13: Fiemaʻu ‘e au ha taha ‘oku ngaungaue ka e ‘oua ‘e ha’u pe ki he kalasi ‘oku ‘ikai ke fuʻu ngaungaue ia hangē ai pē ‘oku fakatupu fiemohea pea te fie hola ai pe kita he kalasi, pehē mo ha taha ‘oku fietokoni.

S13: I want someone who is active because if he is not active, the class will be sleepy too and it makes me want to run away from class. Also someone who is helpful.

S31: Kou saiʻia he ‘alu koē ki ha faiako ‘oku ngaungaue ma’u pē, ‘ikai ke mai pe ‘ekisesaisi pea tuku ai pē ia ‘o toki fakatonutonu pē ha ‘aho.

S31: I love to go to a teacher that is active always, but not to give us exercises and leave it to mark on some other days.

S11 and S31 preferred a teacher who was fietokoni (helpful).

S12 and S14 describe this teacher as one who“comes and asks if I understand, and if I say no,

he will explain it until I understand, then he can move on to the next topic”.

An effective mathematics teacher is expected to ngaungaue (move around) and longomoʻui

(be active), helping students with their work and being accessible to facilitate students’

learning. If the teacher is not active, unintended consequences occur. For example, students

get sleepy and want to run away or become a nuisance.

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Motivating, encouraging, and fair

One student wanted a mathematics teacher who treated everyone equally, and had the ability

to motivate and encourage the students to do well in mathematics.

S32 said,

“I love the math teachers to ‘Be themselves’. I also prefer them to be fair,

and he/she treat everyone equally. I also prefer teachers that encourages

and motivates the students to do their work, do the bell work”. [S32]

This is similar to MT1’s discussion on page 64 when he described how effective mathematics

teaching should include the ability of the teacher to motivate his/her students to be

independent learners.

4.7 Challenges in Learning Mathematics: Students’ Views The students identified two key challenges in mathematics learning:

(i) The mathematics teacher, and

(ii) Word problems.

4.7.1 Mathematics Teacher

Six students acknowledged that the mathematics teacher themself was their number one

challenge.

S11 talked about the language and quality of the discussions: “explanations...particularly in

using of vocabularies” that did not help his understanding. S12 shared the same view: “even

though my teacher last year clearly explained the notes, I find that this year’s teacher

explained better and I understand better now”.

The teacher has a significant impact on the students’ desire to love mathematics and the

teacher’s use of difficult or incorrect vocabulary could affect understanding. The researcher is

aware that this is a common challenge for all second-language speakers and learners.

Unfortunately, as the students shared, the frustration could lead to students’ strongly disliking

mathematics and/or the mathematics teacher. (Attride-Stirling, 2001)

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4.7.2 Word Problems

The second challenge relates to word problems. S11 repeatedly expressed this challenge, and

traced it simply to a lack of understanding of mathematical terms. This is similar to the reason

given by S21 and S24 as to why they found word problems in mathematics hard to solve. S21

elaborated, “My weakness in math is simplifying the problem”.

S21 and S24 were doing some word problems using Pythagoras theorem and they tried but

could not solve the first problem. When probed for the reason, S24 expressed,

“MT2 gave us the exercise in a different way/method... given in words, not

using numbers. It was hard to convert from words to numbers” [S24]

While S21 said, “There were some words that I didn’t understand”.

S24 added, “If my English is good...calculation of math stuff will be much easier”.

S24 believed that if a student’s English proficiency was good, s/he would experience the

solving of word problems as a much easier task. Thus, for students, language is one of the

main barriers to unpacking the meaning of word problems in order to solve them.

4.8 Challenges in teaching mathematics The challenges faced by mathematics teachers were: (i) students’ lack of basic mathematics

skills and a negative attitude, (ii) lack of skills in application to real life problems, and (iii)

language. These challenges are discussed next.

4.8.1 Lack of basic mathematic skills and negative attitude

Teachers learned from their experiences, and 3 teachers reported experiencing the lack of

basic mathematical skills and attitudinal issues as challenges in their classroom. Both MT1

and MT2 saw the lack of basic mathematical skills as the number one challenge in teaching

mathematics. MT1 gave an example:

“I can say for example multiplication, children’s knowledge for multiplication is always

below average even though they are expected to memorize it” [MT1]

while MT2 reported; “if we asked them something in form 4 or form 5, basic terms, they don’t

know it”.

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As a result of lack of basic skills, MT1 believed “his interest in math decline” and “if he had

more knowledge of basic math skills, it will help him to love math”. MT3 added:

“In teaching mathematics, there are some students that come to class who

already have negative attitudes towards maths”. [MT3]

She identified these students and said: “I go back and give them some work that they should

have done from the middle school, some basics”. As a result, “there are some improvements

in them when I give them simple works like that”.

4.8.2 Lack of skills in application to real life problems

MT4 noticed the problem of lack of skills application in his classroom, and reported:

“The major challenge that I faced with when I taught Calculus, Maths, and Physics is the mismatch between curriculum objectives demand and the cognitive ability of the students. Students can calculate the point of intersection, they can do the hard calculus of integration but when they were asked to apply to their real world they couldn’t do that right. These two needs to be match in order to avert the future occurrence of dismal performance of students in classroom and National exams in relate to the students’ experience world” [MT4]

Drawing from his experience, he believed that this is a major challenge for the teaching of

mathematics in Tonga. Students can calculate straightforward exercises but when it comes to

application, “they couldn’t do that right”. This result is consistent with students’ perceptions

and experiences with word problems, which was identified as a challenge to their learning of

mathematics.

4.8.3 Language

The teachers, like their students, realise that teaching in the English language presents serious

challenges for all parties in the classroom. According to MT5,

“I believe the major challenge is language. There are times that you explain

to the students and you think they understand but they don’t. What you do is

try to simplify the explanation or you asked them questions so they can

understand”. [MT5]

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She believed that language is one of the major challenges, with students having a hard time

understanding explanations in English. As a result, she has to go back and simplify her

explanation to the language level of the students.

This is similar to MT2’s perspective. MT2 identified students’ lack of knowledge of basic

mathematical terms as a source of their lack of understanding. MT6 gave an example which

supports this notion:

“The first time I taught here, a lot of time like the mathematics language, a

lot of students didn’t really get it. When you say expand, a lot of them

factorize and when you say factorize, they expand. They didn’t really get

those vocabularies”. [MT6]

4.9 Bringing the ideas together The main findings in this section relate to the definition of effective mathematics (EMT)

teaching, the notion of group work, and the significance of language and the mathematics

teacher to students’ mathematics learning.

Effective mathematics teaching involves the ability of the teacher to motivate, facilitate, and

enhance students’ mathematical learning. This type of teaching is student-centred rather than

teacher centred. From students’ perspectives, they preferred mathematics teachers who

motivated and encouraged them through their mathematical learning.

Through observations, the researcher believes that teaching which focuses on students’ needs

rather than the teacher’s plans is critical to effective mathematics teaching. The researcher

believes that in order for mathematics teachers to have the motivation to incorporate best

practice into their everyday teaching practices, s/he must at least teach with ‘ofa (love), tauhi

hono vā mo ‘ene fānau ako (nurturing of his/her relationships with his/her students) through

fakaʻapaʻapa (respect), mamahiʻi meʻa (committed), faitotonu (integrity), lototō (humility),

and tui (faith). Once these attributes are woven into the character of a mathematics teacher,

s/he will be able to believe in the ability of all of his/her students to do mathematics, and

motivate and encourage them to do better. As a result, the teacher creates hope and dreams for

his/her students, which raises their expectations and causes them to persevere in their

performance to achieve those dreams.

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Group work emerged from students’ perspectives as their most preferred teaching strategy.

All of the 12 students preferred group work because they were able to fengāueʻaki (work

together), fevahevaheʻaki (share), and fetokoniʻaki (help each other). The second most

preferred teaching strategy was clear definition, explanation, and demonstration. This is

critical to the learning of Tongan students because this nature of working in groups is closely

related to the attributes of nurturing relationships - one of the core cultural values of Tongan

culture which underpins the indigenous Tongan education system (Thaman, 2009). This is

related to the finding by Johannson-Fua (2007, cited by Thaman, 2007) in which she

implicated that demonstrate (fakatātā) and working together with the student (kaungā ala) are

found to be vital teaching strategies to Tongan students’ learning style.

This is parallel to the notion of the zone of proximal development by Vygotsky (1978) where

students are able to reach their actual potential through some sort of scaffolding. Group work,

in this context, is one form of scaffolding used by teachers, where students explore other’s

ideas, ask questions, justify and critique solutions, and simplify ideas. Klein (2012) perceived

that such experiences helped students to make sense of their learning and create in-depth

conceptual understandings of mathematical concepts, and by doing so, this enhanced

students’ cognitive development (Hunter & Anthony, 2012; Attard, 2011). This reflects a

social constructivist’s view, as discussed in the literature review, where students learn better

when they construct their own personal meaning through productive and meaningful

interaction with others (Bennison & Goos, 2007).

Another issue arising from this data was the impact of the mathematics teacher in facilitating

mathematics learning. Most of the students identified that the teacher was their number one

challenge in learning mathematics. From their classroom experiences, most students reported

that the classroom strategies used were boring or the explanations were not clear or

understandable. Sometimes it was simply the attitude of the teacher in the classroom that was

not favourable to students.

Teachers should be aware of their significant impact on students’ learning, which in turn

impacts upon their perspectives and attitudes towards mathematics learning and mathematics

as a subject. The researcher strongly agreed that teachers’ beliefs, values, and attitudes toward

mathematics teaching impacts their performance in the classroom (Brown, 2003; Ernest,

1995). If the teacher perceives that all students can learn mathematics then this should be

reflected in his/her classroom. The researcher believes that if the teacher cannot establish a

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connection with his/her students and the students with the content, then students will perceive

them to be their challenge for learning mathematics. In such connections, the teacher

encourages students to have vision for themselves and he/she teaches the student rather than

the lesson, as stressed by MT6 earlier. As a result, students favoured the class which initiated

a desire within them to come to class and “want to learn more” (MT6), even when they

perceived mathematics to be hard.

In addition, the researcher particularly emphasizes the students’ reports that they prefer a

mathematics teacher who gives him/her immediate feedback rather than collecting homework

or tests and marking it at some later date. Marking of the test paper immediately and returning

it to the students is critical to mathematical learning, so that students can evaluate their own

performance and make improvements. This issue needs to be addressed by mathematics

teachers at Heilala High School.

Moreover, language is a key challenge for both the teaching and learning of mathematics.

MT5 elaborated that she had to simplify her English explanations in order for students to

understand. Some problems emerged such as students’ lack of application skills for applying

mathematics concepts in real life problems, as perceived by MT4. This is parallel with the

students’ perspectives, where most expressed that they found word problems to be hard

because of a lack of knowledge of mathematical terms. This is similar to the finding by Manu

(2005) in which he emphasized that using their second language presented a double challenge

for both students and teachers; first the student must decode the meaning of the classroom

instructions, and then try to comprehend the meaning of the task itself.

This finding emphasized the significance of explicit explanation of mathematical terms in

order to understand the mathematical content (Echevarria et al., 2008, 2010). As illustrated by

S11 and S14, the students understood the concepts better when the teacher explicitly

explained the key mathematical terms prior to the main content of the lesson. The researcher

emphasizes this point because most mathematics teachers are not aware that language is a

barrier to the understanding of mathematical concepts (Neville-Barton & Barton, 2005) and

they mostly focus on the content of the subject rather than language first, which seems to be

the foundation of mathematical understanding.

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CHAPTER 5: RESULTS AND DISCUSSIONS - EFFECT OF THE SIOP MODEL

5.1 Introduction This project has two data chapters. The previous chapter, chapter four, analysed the data

covering the common ways of teaching mathematics at Heilala High School and their

perceived effectiveness, as well as the challenges that both teachers and students face in the

teaching and learning of mathematics. This chapter follows on from the previous one and

answers the final research question:

Q3: How effective is the SIOP Model in enhancing the learning and teaching Form 5

mathematics at Heilala High School?

The answer to this question is based on the assumption that the development of teachers’

observed use of SIOP techniques equates to ‘enhancement’ of teaching and learning of 5th

Form mathematics.

Data for this section came from the classroom observations that the researcher carried out in

the classrooms of the three Form five mathematics teachers at Heilala High School (MT1,

MT2, and MT3). Each teacher was observed three times over a period of six weeks, giving a

total of nine observations. The researcher used an Observation Schedule to guide the

observation process. A copy of this can be found in Appendix D. The school timetable

allocated five mathematics periods per week for Form five. Prior to each classroom

observation, the researcher met with each teacher to agree on the observation process and also

look at the lesson plans. After each classroom observation, the researcher met with the

teachers again to discuss the observed lessons and highlight any issues for consideration. All

nine classroom observations were recorded on video to capture critical components of the

lesson that would enrich the analysis as well as the presentation of the data.

5.2 Data Analysis

The data is analysed and presented differently in this chapter. In order to capture the

effectiveness of the SIOP model in enhancing learning and teaching, it was considered best to

do this using the Case Study format. For this, the development of each teacher over the three

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observations is presented in the format of three case studies. The discussions in each Case

study are based on the following key elements that make up the Observation Schedule:

� The Quality of Lesson Planning and Preparation, Use of the Advance Organizer,

� Students’ activities and engagement,

� Language and Questioning skills, and

� Assessment of students’ understanding.

The last section of the chapter will bring the three case studies together and present what has

emerged as the effectiveness of the SIOP Model.

5.3 Case study 1: Perceived development of MT1 through SIOP MT1 has been teaching mathematics at Heilala High School for the last 19 years. In this time,

he spent 12 years teaching mathematics in the middle school which included Year seven,

Year eight, and Year nine, and the last seven years teaching Form five at the high school. He

holds a Bachelor of Arts degree in Mathematics and History, is highly regarded by colleagues

and is one of the senior teachers at the school. MT1 was first introduced to the SIOP model in

2007, but confesses that he still has not mastered all the components of the SIOP model.

This case study will record the perceived development of this teacher over the period of the

observations. Each classroom observation demonstrated one of the components of the SIOP

Model. The first classroom observation was focused on the sixth component of the model -

Practice and Application. The second classroom observation was based on the third

component - Comprehensible input, and the third classroom observation demonstrated the

fourth component - Strategies. As addressed in the literature section, the SIOP Model is not a

hierarchy model or a step by step process as shown in the classroom observations.

In the three lessons that were observed, the lesson topics were:

Lesson 1: Sketching non-linear graphs

Lesson 2: Probability tree

Lesson 3: Angles on parallel lines

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5.3.1 Quality of Lesson Planning and Preparation

Lesson planning and preparation is an important part of classroom practice. In addition to

preparing a Lesson Plan, teachers also prepare lesson notes, exercise sheets for student

activities, teaching aids such as charts, cards or visual aids, and generally prepare the

classroom well for the lesson. For this element of the SIOP model, the researcher was

interested in the following items:

� Level of objectives

� Quality of planned activities

� Planned assessment activities

� Time allocation

Level of Objectives

In all three classroom observations, MT1 posted both the Content and Language Learning

Objectives on the board and discussed them well so that his students were aware of what to

expect from the lesson. There was a better mix of objectives and a general improvement in

the level of objectives as the observed lessons progressed. As an example, the objectives of

the second lesson were a marked improvement from the objectives of the first lesson, as

shown here:

Lesson 1: Students will be able to:

� Review in practice how to draw each of the five non-linear graphs

� Discuss how to draw each of the non-linear graphs, i.e. parabola, cubic graph,

hyperbola, exponential graphs, circles.

Lesson 2 : Students will be able to:

� Draw a tree diagram to show the outcomes of an event from several attempts

� Calculate the probability of an event from the outcomes on a tree diagram

� Describe how to draw a probability tree diagram and use it to calculate the

probabilities.

Quality of Activities

The Bell work activities were attempted using a mixture of whole class discussion, group

work, and individual tasks. The activities were in line with the content and language learning

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objectives. For example, in the first lesson MT1 used “home groups” and “expert groups” to

complete the activities. The class was divided into several “home groups” comprising three

students each. Then each home group member went to join an “expert group”; there were

five expert groups altogether, each one focused on graphing a particular non-linear graph. The

members of each expert group worked together to learn everything about graphing their type

of equation. At the end of the allocated time, the students returned to their “home groups”

where they shared with their home group members what they had learned from their

respective expert groups. This activity was as different as it was lively and rewarding for

everyone. This is the part that answers the question: How effective is the SIOP Model in

enhancing the learning and teaching Form 5 mathematics at Heilala High School? MT1 has

implemented several features of the SIOP Model such as: frequent opportunities given to

students to interact among themselves, students being engaged for about 90 to 100 percent of

the time, and various strategies were implemented to make the mathematical concepts more

understandable. Thus, students were able to learn from their peers, took ownership of their

learning, asked questions, justified solutions, and clarified explanations. Hence, teaching and

learning mathematics were enhanced.

Planned Assessment

The planned assessments were mostly formative and in the form of probing questions,

observation of individual work, group discussions, and one-on-one discussions. These forms

of assessment were common throughout the three observed lessons. MT1 frequently asked

questions to assess students’ understanding and the achievement of the learning outcomes.

Whenever the students were involved in group work and discussions, the teacher moved

around and observed what was happening. He facilitated effectively – he listened in on the

groups’ discussions, paused and checked their work, and frequently smiled and nodded to

indicate approval and satisfaction regarding students’ progress. Frequent feedback and asking

probing questions about students’ work helped students to identify and solve confusion and

create critical mathematical thinking, hence enhancing both teaching and learning of

mathematics.

Time Allocation

Time allocation improved greatly as the observations progressed. In the first lesson, the

allocated time for the first group work activity was only five minutes and he was advised to

increase that to allow for students to process their thinking, discussions and to be able to

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complete the set work well. Time allocation for the different parts of the lesson improved in

the second and third lessons.

The researcher’s major discussion points with MT1 about Lesson Planning concerned the

quality of the activities planned, how effective they were in enhancing learning and their

alignment with the Objectives. The researchers noted that the only teaching resource used was

the protractor in the third lesson. The use of teaching resources including charts and artefacts

was an item to be emphasised. Overall, MT1 did well in planning and preparing for the class,

and there were no major problems with these aspects. The third lesson was very good and

showed much improvement. The third lesson also flowed better and there was more

discussion and engagement in class. Through his own experiences, MT1 developed the pacing

of his lesson more appropriately to match his students’ ability levels, which allowed his

students’ time to process their thinking and show their working. Thus, appropriate pacing

enhances students’ thinking process and therefore the learning of the mathematics content.

5.3.2 Advance Organizer

MT1 used Bell work in all three lessons to catch students’ attention and get them on task from

the beginning of the class. Two of these were revision questions where students were required

to connect their thinking to previous lessons. The first Bell work question was at the level of

Recall and tested their memory of previous work.

Bell work question Lesson 1:

Name the type of graphs for each function below:

1. y = x²

2. y = x³

3. xy = k

4. y =

5. x² + y² = r²

The bell question in the third classroom observation was much more successful in requiring

students to think and formulate a reason for their answer. It set the tone well for the lesson

that followed.

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Bell work question Lesson 3: Calculate the size of the unknown angles. Give a reason for

your answer.

These types of questions promote higher-order thinking skills, which is a feature of the SIOP

Model. Hence, teaching and learning mathematics at MT1 class were both enhanced through

enhancing the depth of thinking.

In the discussion after the second observation, the researcher suggested the need to ask

application questions that linked the mathematical concept to the students’ everyday

experiences. The application of probability and the probability tree to everyday occurrences

was a classic example. MT1 later shared how he changed the bell work activity - he invited

two boys and three girls up to the front and used them to demonstrate the possibilities of

selecting two students out of five students. He found that students’ performance and response

improved greatly in the final classes compared to the earlier ones. MT1 implemented features

of Building Background as one of the components of the SIOP Model. Thus, students were

able to make sense of the probability tree through building meaningful background

experiences which enhanced both the teaching and learning of this mathematical concept.

5.3.3 Student Activities and Engagement

There were high levels of student engagement in every lesson. Students were engaged as they

listened to the teacher. This was evident in the expression on their faces and their note taking.

When there were group work activities, there was moving and talking and students were

involved with each other and the activities. One of the features of the SIOP Model is that

students are engaged for about 90 percent to 100 percent of the class time (Echevarria et al.,

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2008, 2010). In an attempt to measure this feature, the researcher used the following table to

calculate the percentage of students’ engagement. The following table shows the level of

student engagement for the first two observations.

Table 3: Percentage of Student Engagement for MT1's Class Observations

Observation 1

Task Duration (minutes) Total number of

students (15).

Percentage of engagement

Bell work 5 mins 13 87

Oral Discussions 5 mins 15 100

Group work 5 mins 13 87

Oral Discussion 8 mins 15 100

Expert Group 10 mins 15 100

Home Group 17 mins 15 100

Observation 2.

Task Duration (minutes) Total number of

students (15).

Percentage of engagement

Bell work 5 mins 12 80

Oral Discussions 5 mins 15 100

Demonstration 5 mins 14 93

Group work 5 mins 15 100

Oral Discussions 3 mins 25 100

Group work 8 mins 15 100

Oral Discussions 5 mins 15 100

Group Competition 4 mins 15 100

Group work 10 mins 15 100

The first column shows what transpired during the lesson, the second column records the

allocated time for each activity, the third column records the number of students participating

in each activity out of the total students in the class, and the last column shows the percentage

of student engagement. In order to calculate the percentage of engagement for bell work in

observation one, the calculation was = 86.67. However, all calculations

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were rounded to the nearest whole number. All percentages of student engagement were

calculated in the same way.

During the first classroom observation, two out of the six activities involved only 13 students.

The other two students were not on task initially and had to be directed by the teacher to move

to the task. The other four activities recorded 100 percent student engagement.

MT1’s ability to engage the students improved in the second observation, where eight out of

the nine activities recorded above 90 percent student engagement except for the Bell work

activity with 80 percent (12 out of 15 students). This result was similar to the level of student

engagement in the third classroom observation.

MT1 used mostly group work tasks. The students listened intently, asked a few questions,

justified solutions and clarified ideas during the group tasks. There was ample opportunity to

practice all four language skills: reading, speaking, writing, and listening. These features

emphasized the role of the SIOP Model in enabling students to enhance their mathematics

understanding while at the same time improving their language skills. As a result of these dual

goals, mathematics learning and teaching was enhanced. At the end of the observed lessons,

the researcher stressed the importance of finding good attention-grabbing questions for the

Bell work.

5.3.4 Language and Questioning Skills

MT1 used language that was appropriate to the proficiency level of his students in all of his

classes. He taught mathematics using English but occasionally used Tongan for certain

mathematical terms to drive home important descriptions. The use of proper and relevant

language to clarify concepts and enhance understanding is an important item in the SIOP

model.

MT1’s explanations were clearer and easier to understand when he used Tongan to clarify

meanings. This practice was also welcomed by students. Three students (S11, S12, and S13)

commented on that particular strategy.

S13 reported

“After that, I understand how to find the angles when he explained well. I

liked how he defined the key words right at the beginning”. [S13]

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MT1’s questioning skills were much better in the last two lessons. In the first classroom

observation, he divided the class to work in groups while he facilitated their learning. In the

process, he assisted some of the group with their work and asked questions, but ended up

answering his own questions.

For example, one of the groups discussed how to sketch the hyperbola that has the equation

represented by:

MT1 saw that they needed support so he stepped in and guided the discussion that attempted

to work out the coordinates of points on this graph. Certain x-values were proposed and the

group then worked on finding the corresponding y-values, then plotting those on the x-y axis:

It was not easy to follow this discussion and MT1’s input did not help greatly. His questions

were not real questions but a mixture of prompting and answering. The teacher needed to ask

questions that helped students to think and organise their thinking well enough to give a

response. This skill improved in the last two classes, where the teacher asked questions, gave

students time to think, and modified the question to a simpler one to guide students’ thinking

toward the answer. The following discussion between MT1 and his students took place during

the third classroom observation. The discussions were based on solutions for bell work

activity 3 that was discussed earlier in 5.2.2 and based on question (b):

MT1: What is the size of angle A?

Ss: Seventy six degrees

MT1: What is your reason for giving seventy six degrees?

Amy: You add all the angles inside and they must equal one hundred and eighty degrees

MT1: Good … but why is angle A seventy six?

Ss: Because the base angles must be the same.

MT1: Like one and what?...One and six, one of them must be negative.

S1: Negative two.

MT1: Negative two and...

S1: Positive three...

MT1: Okay, how about if we put negative six and positive one? After that, then we plot the

graph.

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MT1’s questioning skills had improved. He asked both thinking (WHY?) and higher order

questions. As students worked, he asked short probing WHY and HOW questions that caused

students to stop, look at each other for support, talk some more and come up with answers.

The exercise in thinking as individuals and also together was rewarding. This is a prominent

feature of the SIOP Model which greatly enhances students’ conceptual thinking processes;

hence, both teaching and learning were enhanced. The researcher emphasised this point again

at the end of the lesson, the need to support students to think and solve things on their own.

5.3.5 Assessment of Students’ Understanding

MT1 used similar assessment tasks in all three classes. As discussed earlier, he asked

questions at the beginning and also during group work when students were carrying out

activities. As the students worked on their activities, MT1 walked around the class, observed

and listened to the group discussions, checked the group reports and provided some feedback.

On a few occasions he shared some of his observations with the whole class. The group

reports of their answers, which they gave at the end, formed an important part of their

understanding and the achievement of the learning objective, and this was evidence used to

answer the third reseach question.

The researcher suggested to MT1 after the second observation that peer assessment was also

important, that students or the groups be made to check each other’s work as another form of

providing feedback. This gave students greater responsibility and control of their learning. A

group approach to this activity is an effective learning experience for students, where the team

decides together what to assess and how to assess it. As they do this, they themselves become

better learners. MT1 liked the suggestion but felt he needed more time to prepare for such an

activity.

5.4 Case study 2: Perceived development of MT2 through SIOP MT2 has been a teacher at Liahona High School for nine years and has been a mathematics

teacher for the past six years. He graduated with a Diploma in Accounting and Economics

from the Tonga Institute of Education (TIOE) and was introduced to the SIOP Model in 2007.

However he shared during the SIOP Model training that,

“I have heard of SIOP but have not really given it much thought. The

training that we are getting now has helped me to know more about it, its

features and what it can do for me”. [MT2]

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This case study will record the perceived development of MT2 over the period of the

observations which assumed to provide evidence to answer the third research question: How

effective is the SIOP Model in enhancing the leanring and teaching Form 5 matheamtics at

Heilala High School? The first classroom observation demonstrated the sixth component of

the SIOP Model – Practice and Application, the seventh component of the model was

demonstrated during the second classroom observation, and the first component was

demonstrated during the third classroom observation. The topics for MT2’s three lessons

were:

Lesson 1: Graphs of circles in the form of and

Lesson 2: Angles of a polygon

Lesson 3: Using Pythagoras to calculate missing sides of a right-angle triangle.

5.4.1 Quality of Lesson Planning and Preparation

MT2 showed improvement in the quality of his lesson planning and preparations through the

three consecutive classroom observations which is evidence that the SIOP model enhances

teaching and learning mathematics at Form 5 level. As with the other teachers, lesson

planning and preparation was assessed under the following elements:

� Level of Objectives

� Quality of planned activities

� Planned assessment activities

� Time allocation

Level of Objectives

In all three classes, all objectives were posted on the board and explained clearly to students

so that they were clear about what was expected from the lessons and what they would learn.

While the objectives were clear and simply worded, they were mostly at the recall and

comprehension levels, which are considered too low for this level of study (Form 5). The

following showed both the content and language learning objectives for lesson two and lesson

three.

First lesson Objectives: Students will be able to:

� Identify names of polygons

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� Calculate the sum of interior angles of a polygon by using the formula (n – 2)

by measuring with a protractor, by dividing the polygon into triangles from a fixed

point.

� Share and discuss the different methods of how to calculate the sum of interior angles

of a polygon.

Second lesson Objectives: Students will be able to:

� Define Pythagoras theorem and hypotenuse

� Calculate the missing sides of a right-angle triangle using the Pythagoras theorem

� Discuss and share the names of the three sides of a right-angled triangle

� Discuss and share steps to follow when using the Pythagoras theorem to calculate the

unknown side of a right-angled triangle.

The researcher and MT2 agreed that more thought should be put into developing objectives at

higher levels especially at application level and above. The first objectives in both lessons

could be removed as they would be more relevant at Form three. Application objectives that

linked the theory to real life contexts for polygons and right-angled triangles would be very

useful. Similarly, analyses objectives could be of the component angles of complex or 3D

shapes. In lesson three the teacher gave a word problem and asked students “read and apply

the Pythagoras theorem” to solve it. The researcher suggested at the end of the classroom

observation that MT2 could have expanded that objective to ‘read, analyse, and apply the

Pythagoras theorem to calculate the unknown side of a right-angled triangle’.

In addition, MT2 used a variety of supplementary materials, which is very important in

helping students to make meaning. In Lesson One, he used five mini white-boards and white-

board markers for the group activities; in Lesson Two, he gave a hand-out with a picture of

three tiles forming a right-angled triangle to be used in developing understanding of the three

sides of a right-angled triangle and the formation of Pythagorus theorem. Use of these

supplementary materials helped students to visualize the relationships among the three sides

of a right-angled triangle and understand how the Pythagorus theorem came about. Thus, the

Pythagorus Theorem was more meaningful to students. This is evidence that helps to answer

the third research question because the use of supplementary materials to a high degree is a

feature of the SIOP Model. Not only that, but MT2 had prepared hand-outs for all lessons and

they had short notes, exercises and questions to facilitate the discussions.

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Quality of Activities

MT2 planned to use Bell work, whole class discussion (oral discussion), group work, group

competition, try-outs and individual tasks. He obviously believed in students’ learning and

planned for students to be fully involved and that they learn from the activities. All of the

group activities engaged students in discussions, asking questions, and teaching one another,

which enabled them to express their thoughts, share their ideas, and respect their peers. This

was evidence of the Interaction component of the SIOP Model, and how it enabled students to

enhance their understanding of mathematical concepts as well as develop teaching.

A new and interesting activity used was group competitions. The plan was to engage

individuals to represent their groups in solving problems. The idea of individual tasks and

group tasks was good in the sense that it catered for the different needs of students. This

aligned with classroom differentiation as an aspect of the SIOP Model. Thus, using various

strategies to cater for the diversity of the learners’ needs in the classroom is evidence of the

effect of the SIOP Model in learning and teaching mathematics.

Planned Assessment

The planned assessments used for all three observations were all formative assessments,

including: observation of students, quick check of understanding by a show of hands,

providing questions, and listening to group discussions. These forms of assessments are

important elements of the eighth component of the SIOP Model which is “Review and

Assess”. Through consistent feedback on students’ work, teachers can evaluate the students’

performance, and decide whether to provide another demonstration or move on with the class

discussion. It also allowed the students’ to be on the right track in solving problems. Hence,

both learning and teaching mathematics are enhanced.

Time Allocation

Time allocation for the different parts of the lesson was important for MT2’s classes because

he wanted to do different things in each class. The advice was to consider the best activities

for each lesson and maybe not to crowd any lesson with too many activities. In the first

lesson, the group competition was planned for 10 minutes but the activities and questions did

not appear to require that much time. The advice to MT2 was to take fewer activities, develop

good exercises and questions that require thinking and discussion and allocate time not only

for students’ activities but also for the teacher to provide occasional feedback and

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confirmation of students’ responses. His planning and timing skills improved greatly in

Lessons 2 and 3, which enhanced his teaching as well as students’ learning of mathematics.

5.4.2 Advance Organizer

MT2 used Bell work questions as his lesson starter in all three classes. For example, the

second bell work question provided here enhanced students’ understanding of the word

polygon, before exposing them to calculating its total interior angles. This was evident in the

clarification identified by the students in the following bell work:

Question for Bell work 2:

Definitions of polygon: “Polygon is a close figure made up of straight lines” In pairs, discuss

and explain why these two figures are not polygons.

This bell work was shared by pairs. S24’s partner shared:

Okay, my pair and I discussed the reason why number one is not a polygon

is because the shape is not a close figure and number two, it is because of

the curve part, it is not straight [S24’s partner]

The rest of the pairs reported likewise. This was similar to the bell work in the third

observation, where students were asked to define the meaning of ‘hypotenuse’ and discussed

the relationships between the three sides of a right-angle triangle. This activity enhanced

students’ ability to identify the hypotenuse in application of the Pythagoras theorem. Hence,

the defining of key mathematics vocabulary is a further piece of evidence of the efficacy of

the feature of the SIOP Model which clarifies meaning and enhances both teaching and

learning.

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5.4.3 Student Activities and Engagement

MT2 used Bell work, whole class discussion (oral discussion), group work, group

competitions, try-outs, and individual tasks. While some of the activities required students to

participate, not everyone was engaged. In Lesson One, one of the activities was group

competitions where each member of the group was given an assigned number. When a

number was called out, the group member with that number ran to the front to represent their

group in a competition. The teacher wrote an equation on the board, each competitor worked

their solution on the mini board and showed their solution to the class all at the same time.

During this activity, the competitors as well as the other members of the groups should have

been working to get the required solution. However, it was found that some students were not

engaged at all. The following table showed the level of students’ engagement for the first

observation and the progress made in the second observation:

Table 4: Percentage of Student Engagement for MT2's Class Observations

Observation One:

Task Duration (minutes) Total number of

students (20)

Percentage of engagement

Bell work 5 mins 16 80

Oral Discussion 5 mins 16 80

Demonstration 5 mins 20 100

Group work 8 mins 16 80

Group competition 10 mins 15 75

Try outs 17 mins 18 90

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Observation Two:

Task Duration (minutes) Total number of

students (16)

Percentage of engagement

Bell work- work in

pairs

3 mins 16 100

Oral Discussion 2 mins 16 100

Expert group 10 mins 16 100

Home group 10 mins 16 100

Try out 4 mins 16 100

Oral Discussion 3 mins 16 100

Home group 8 mins 14 88

Sum-up 4 mins 16 100

Individual Task 6 mins 16 100

The percentage of student engagement increased in lesson two as compared to lesson one

which is an evidence of high student engagement, a feature of the SIOP Model. This is

equated in this study to enhancement of teaching and learning mathematics. After lesson one,

the researcher suggested reducing the number of students per group and increasing the

number of groups. This would ensure that every student got a chance to participate and MT2

tried these suggestions in lesson two and the percentage of engagement increased as shown

above. In Lesson three, one of the activities was a group competition where the level of

student engagement was very low, only 56 percent (10 students out of 18 students). MT2 was

concerned when he saw the video of this lesson and promised to allow time for group

discussions after every attempt. That way the activity was not only about winning but also

about learning the mathematical concept and working as a group.

5.4.4 Language and Questioning Skills

MT2 knows the students and used simple language suitable for the students’ level of English

proficiency and understanding. He simplified concepts and made sure to explain them in the

context of activities or scenarios that relate to students’ prior knowledge and their everyday

life. His questioning and discussion skills improved from lesson to lesson. The following are

excerpts from Lesson 1 and Lesson 2:

Students were asked to sketch the graph of .

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MT2: If we are given relation in this form, then our centre of origin will be...

Ss: Zero, zero (0, 0)

MT2: Therefore, this is your X and Y axis (sketching them on the board), this is (0, 0) then

this number (pointing to 16), you are going to...

Ss: Square root

MT2: You square root this number, you get a...

Ss: Four

The kinds of questions asked were greatly improved in the second lesson as followed.

Lesson Two

Here the students were asked to calculate the total interior angles of an octagon. Two students

were asked to show their working on the board; one used the formula (n – 2) ; the other

used splitting the octagon into triangles.

MT2: Are the two workings the same?

Ss: Yes

MT2: Okay, how many triangles in here?

Ss: Eight

MT2: And we multiply it with one hundred and eighty. Why do we multiply it with one

hundred and eighty?

Ss: Because it is the sum of the interior angle of a triangle.

Here the questions were slightly better. However, MT2 is still providing part of the answer in

the last part when he should have planned more questions and left the answering to students.

MT2’s dilemma is very typical of many mathematics teachers who cannot resist providing

answers and leaving little for students to discover. However, MT2’s development of asking

higher-order thinking questions enhanced the students’ critical thinking. This is one of the

features of the SIOP Model, and its presence is evidence of enhancement of mathematics

teaching and learning.

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5.4.5 Assessment of Students’ Understanding

MT2’s assessment of students’ understanding improved during the last two classroom

observations. He used oral questioning and presentation to assess his students’ understanding

in the last two classroom observations.

The best example of this improvement is a snapshot of his third classroom observation. The

activity demonstrated here was discussed in 5.2.4. However, after the discussion, most of the

students understood only how to use the formula but not the splitting of the polygon into

triangles as shown by one of the student’s work below:

After MT2 discussed the solution on the board, he asked the students “Hands up if you have

got it right?” Some students raised their hands indicating that others got it wrong. In

response, the teacher said, “It seems that some of you still don’t get it. Now, I want you to go

back to your home group and discuss again how to do the splitting of the polygon into

triangles and use it to find the total interior angle”. MT2 explained that students learnt best

from each other and that he was encouraging that. This is in line with the SIOP Model’s

requirement for teachers to provided regular feedback to students. In this case, the students

were able to re-discuss the solutions on how to split the polygons into triangles from a fixed

point. So both the teaching and learning mathematics were enhanced.

5.5 Case study 3: Perceived development of MT3 through SIOP MT3 taught mathematics for five years at the middle school (Years Seven, Eight, and Nine)

and has been teaching at Heilala High School for the past two years. She was introduced to

the SIOP Model in 2007 when she was at the middle school. She graduated with a Bachelor of

Figure 6: A student work on calculating angle of an octagon through splitting of the polygon

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Science in Mathematics and Chemistry and was also trained at the Tonga Institute of

Education.

This case study recorded the perceived development of MT3 over the period of her three

classroom observations. The first classroom observation focused on Building Background,

which is the second component of the SIOP model, the second observation demonstrated the

fifth component - Interaction, and the third observation focused on the eighth component -

Review and Assess. The lesson topics for the three observed lessons were:

Lesson 1: Sketching of cubic curve (factorizing form)

Lesson 2: Calculating Probability

Lesson 3: Solving word problem using Pythagoras Theorem.

5.5.1 Quality of Lesson Planning and Preparation

MT3 has shown tremendous improvement in the quality of her lesson plans and preparation

throughout her classroom observations based on the following elements:

� Level of objectives

� Quality of planned activities

� Planned assessment activities

� Time allocation

Level of objectives

During the first observation, MT3 did not place any objectives on the board, but discussed the

purpose of the lesson with the students. At the end of the observation, the researcher

suggested that she share the learning objectives with students in writing so that they know

what is coming and can be part of the learning process. MT3 improved on this in the second

and third observations. The objectives for lesson two and lesson three are shown here:

Lesson 2 Objectives: Students will be able to:

� Define probability

� Calculate probability

� Provide examples of probability from everyday lives

� Apply the rules taught for calculating probability

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Lesson 3 Objectives: Students will be able to:

� Solve word problems using Pythagoras theorem

� Write word problems in mathematical symbol

� Use Pythagoras rules to solve word problems

The Language Objectives appeared to be repeating the content objectives. Language

objectives are literacy tasks which are observable and have to do with reading, speaking,

listening, and writing (Echevarria et al., 2008). In lesson two, the students were able to create

and write their own examples of a certain event and an impossible event then share it with the

whole class. The language tasks were present in the class but MT3 needed more scaffolding

regarding creating the language objectives. At the end of the second observation, the

researcher facilitated MT3 in differentiating the content and language objectives. She made an

improvement on the third lesson, in which one of her language objectives was to “write word

problems in mathematical symbols”. This development in MT3’s skills enabled her to plan

strategies aligned with this objective. During the class observation, she asked the class to read

the word problem together. They then identified the key words and translated them into

mathematical symbols. This exercise helped students to comprehend the word problem,

consider its context, and then solve it using the appropriate mathematics rule. This is evidence

of ‘enhancement’ of mathematics teaching and learning.

Quality of planned activities

The most common forms of planned activities used by MT3 were Bell work, whole class

discussion, students presenting solutions on the board, and individual tasks. MT3’s planned

activities improved in the last two observations. In Lesson 1, there was little distinction

between the activities intended for the groups and individual tasks causing confusion. The

activities and related questions greatly improved in the second and third lessons.

Planned Assessment

MT3 had questions and observations as forms of assessment. She targeted students’

discussion, feedback from one-on-one teaching and the students’ work on the board, as

indications of their thinking and learning. These forms of assessment enhanced both teaching

and learning mathematics because MT3 was able to evaluate her students’ work and make key

decisions about whether they needed help or how she could better facilitate their learning. At

the same time, the students were able to frequently get feedback on their output which greatly

enhanced their understanding.

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5.5.2 Advance Organizer

MT3 used Bell work in all lessons. These were mainly in the form of revision questions that

aimed to link students’ understanding of the previous lessons and what they already knew

with the lesson that was coming up. The first Bell work question for Lessons 1 and 3 are

given here. The question in Bell work 1 asked for details of the parabola, in anticipation of the

new lesson on sketching cubic graphs. The question for Bell work 3 was a real life application

question on the Pythagoras theorem.

Question Bell work 1;

Draw the parabola y = (x + 3)² + 4. The third bell work provided here has prepared the

student to work on application of the Pythagoras theorem.

Question Bell work 3:

Sione uses a ladder that is four meters long to get to the roof that he wants to clean. The

ladder leans on to a wall. The foot of the ladder is two meters away from the wall. What is the

length of the wall from the ground to the top?

For Bell work 3, the teacher guided students well:

“...read the question carefully, think about it, then draw and label the

picture...and last you substitute into the Pythagoras theorem”. [MT3]

She repeated and modelled this procedure throughout the class in order for students to easily

solve the word problem. This task instruction was clear, simple, and appropriate to the level

of her students, which is a feature of the SIOP Model. Clear instructions helped students to

understand mathematics better, and ultimately enhanced both mathematics learning and

teaching.

5.5.3 Student Activities and Engagement

The following table shows the levels of student engagement in the second and third

observations of MT3’s classroom.

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Table 5: Percentage of Student Engagement for MT3's Class Observations

Observation two

Task Duration (minutes) Total number of

students (17).

Percentage of engagement

Bell work 5 mins 16 94

Oral Discussion 5 mins 16 94

Work in pairs 5 mins 17 100

Oral Discussion 6 mins 15 88

Group work 10 mins 17 100

Demonstration 5 mins 16 94

Group work 14 mins 16 94

Observation three

Task Duration (minutes) Total number of

students (16).

Percentage of engagement

Bell work 5 mins 15 94

Oral Discussion 7 mins 15 94

Give me five 4 mins 16 100

Group work 7mins 16 100

student

presentation

7 mins 16 100

Group work 5 mins 16 100

student

presentation

5 15 94

Group work 10 16 100

The percentages of students’ engagement improved during the third observation due to

feedback provided in the first two observations. The feedback on the first observation was to

reduce the number of students per group and increase the number of groups. This was done in

the second class observation and the percentages of student engagement were greatly

improved. Then the feedback provided after the second observation was to improve the pace

of the lesson because the duration of some activities was longer than it should be. MT3

improved the pace of her lessons and engaged most of her students in different ways during

the third observation. The researcher was interested in the ability of the teacher to engage the

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students with the word problem because that was a challenge in their learning of mathematics

identified in the previous chapter. Group configuration and appropriate pace of the lesson are

crucial features of the SIOP Model and these features were enacted by MT3 in the third

lesson, resulting in the enhancement of mathematics learning and teaching.

The comment by Student S34 about this class is important for consideration:

I enjoyed the second observation better than the first observation because

the activities were fun and it helped me to understand the exercise faster

[MT1]

Students’ evaluation comments such as this are important for teachers, because if students are

happy and having fun in mathematics, it is an excellent start to meaningful learning which

enhance both mathematics learning and teaching.

5.5.4 Language and Questioning skills

MT3 spoke clearly and simply. She asked good questions that students were able to follow. A

group’s discussion of the Bell work question three was discussed earlier in section 5.4.2 (page

101). It is recorded again here to support MT3’s improvement:

Group three discussion of the bell work:

S1: Ko e tuʻunga S1: The ladder

S2: ...’Oku ‘alu ki ‘olunga mita ‘e fā S2: ...It goes up four meters

S3: ‘Oku mita ‘e fā ‘a e loloa ‘o e tuʻunga pea mita ‘e ua ‘a e wall

S3: The length is four meters and the wall’s length is two meters

S3 misinterpreted the statement “the foot of the ladder is 2 meters away from the wall” to be

the height of the wall. She was confused and moved across to ask Group two for their

solution. The language of this application question was difficult for students to understand

and they had difficulty labelling the parts of their diagrams. MT3 challenged them to think of

the real situation of a ladder leaning against a wall, and then linked the parts of the question to

their picture of the activity. The question was good but the students’ language problems

created confusion in making meaning. Once MT3 helped with that, the solution process was

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easier to get through. As discussed earlier in 5.4.2, MT3 repeated a simple procedure and

consistently modelled the procedure for solving word problems through application of the

Pythagorus theorem. The modelling of simple procedures is a learning strategy described in

the SIOP model which helps students to organize their thinking process and enhances their

understanding. Thus, use of this SIOP Model enhanced mathematics learning and teaching.

5.5.5 Assessment of Students’ Understanding

The highlight of MT3’s performance was in the third class where she used a quick “thumbs-

up” to assess students’ understanding and selected students to present their solutions on the

board. As students’ worked on the activity, she urged them on and said: “When you

understand the solution, just give me the thumbs-up”. She even demonstrated how to do that.

Several students picked up on this easily and gave her the required signal. Another assessment

that she used was to get students to present their solution on the board. As this happened, she

and the class provided feedback to the students at the board and also helped the rest of the

class. Thus, constant feedback on the students’ work (a feature of SIOP) helped students to

evaluate their work, re-direct their thinking and discussions, gain confidence in their

performance, and enabled them to enhance their understanding.

5.6 Summary: Effectiveness of the SIOP Model This section is a summary of the learnings which emerged from the three case studies of how

they helped to answer the third research question:

Q3: How effective is the SIOP Model in enhancing learning and teaching in Form 5

mathematics?

All three teachers had been introduced to the SIOP model at an earlier date but the training

they received in this project was obviously more meaningful. These quotes were identified

through the one-on-one talanoa session:

I have heard of SIOP Model but I didn’t take it seriously but these in-

services has helped me to gain more in-depth understanding of its

components and each features [MT2]

These in-services reminded me to prepare my lessons more effectively. I

have also learnt from other teachers when they shared experiences in their

classroom [MT1]

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The process of prompting self awareness and feedback through watching the video recordings

of their teaching had the desired effect – they saw themselves and their performance through

the eyes of their students and were determined to do better as a result. The feedback from the

researcher also helped them to observe the components of the SIOP Model more closely and

somehow enhanced their classroom practices. MT3 shared during the talanoa session:

“I may say that what I used before...was more like spoon feeding

them...once I give them the problem, may be after two minutes, then I said to

them, okay this is how you do it...ko u toki hanga ‘e au ia ‘o liliu mai ki mui

lolotonga ‘eta (I have changed it during our) classroom observations...I

give them the problem, allowed time for them to solve it on their own and

they do it” [MT3]

As discussed in each case study, each classroom observation demonstrated a component of

the SIOP model. In MT1’s case, the first lesson was focused on the sixth component of the

model (Practice and Application), the second class was based on the third component

(Comprehensible input), and the third class demonstrated the fourth component (Strategies).

The second teacher, MT2 demonstrated the sixth component of the SIOP Model (Practice and

Application) in the first lesson, the seventh component of the model was demonstrated in the

second class and the first component during the third classroom observation. MT3 focused on

the second component (Building Background) in her first lesson, the fifth component

(Interaction) in the second lesson and eighth component, Review and Assess, in the final

lesson. As indicated in the class observations, the SIOP Model is neither hierarchical nor a

step by step process.

There is no doubt that the project process and the model has helped all three teachers in many

aspects of their teaching. They learned to prepare better, to design effective lessons and

activities, to help students make meaning, to observe them during learning and be able to

follow their thinking and progress, to interact and engage with students and help them to work

effectively amongst themselves and to be a good facilitator. A good facilitator creates

opportunities for students to discover things for themselves and be able to take charge of their

own learning. The following key elements emerged from the experiences of these teachers

through the use of the SIOP Model:

� Increased opportunities for student interaction

� More student engagement

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� Use of clear and appropriate language

� Good questions and increased wait time

� Guideline for lesson preparation

� Enhance teachers’ performance in the classroom

These elements of the SIOP Model have helped students gained better mathematics

understanding, hence mathematics teaching was improved. These were evidences of the effect

of the SIOP Model in enhancing mathematics learning and teaching which directly addressed

the third research question.

5.6.1 Increased student interaction and engagement

This feature was common amongst the three case studies. As the teacher developed and

progressed from one observation to the other, s/he planned and implemented strategies that

allowed students to engage more amongst themselves and with the teacher.This is directly

related to the fifth component of the SIOP Model which is Interaction. This was possible

through the use of group work, peer discussion and whole class discussion. Throughout the

nine classroom observations, the teachers provided frequent opportunities for students to

engage in meaningful discourse where students taught each other, observed each other’s

work, listened, asked question for clarification, justified solutions and even challenged ideas

and thoughts. On occasion, some students were more vocal than others but the others listened

and gained from the proposed ideas and answers. Through these interactions, students learned

to nurture their relationships among themselves by sharing their ideas, helping each other, and

even motivating each other to be part of the group work.

The three teachers shared the following views about “effective mathematics teaching”:

“The teacher allows students to spend time on the activities and discuss

amongst themselves rather than the teacher dominating the class” [MT2]

MT3 agreed, stating:

“the classroom should be student centred and the students should dominate

the discussion and class activities” [MT3]

Similarly MT1 reported, “I like to use group works a lot because it is where students help and

learn from each other”.

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These views were evident in the three case studies where the teacher provided frequent

opportunities for students to discuss ideas amongst themselves. These teachers have

implemented several features of the SIOP Model, including: using of variety techniques to

make the concepts clearer (directly related to the third component “Comprehensible Input”),

given frequent opportunities for students to interact amongst themselves (directly related to

the fifth component of the model “Interaction”), and these activities have integrated all four

language skills (reading, listening, speaking, and writing) directly related to the sixth

component “Practice and Application”. There was clear indication of elements of the SIOP

Model being incorporated and featuring in the classes observed – these were planned prior to

the class and also practised.

5.6.2 Increase in student engagement

One of the key elements that emerged from lesson observations was the increase in student

engagement during the class. All three teachers had planned activities for individual work,

group work and whole class, and also allocated time for these activities. This is directly

related to the seventh component of the SIOP Model - “Lesson Delivery”, where students

were actively engaged with the mathematical tasks for about 90 to 100 percent of the time.

This aspect improved as the lessons progressed. When the class began, the Bell work activity

(question) was already posted and students went straight to these at the beginning of lessons.

The sequence of the activities was well planned and students were actively engaged with the

mathematical tasks for about 90 percent to 100 percent of the time which is shown by tables

4, 5, and 6, as discussed earlier. MT6 had this to say about an “effective mathematics

classroom”:

“When you go to the classroom, all students are engaged and involved in

discussions and in doing their work. They appear to understand and are

following the lesson. They are writing and responding to the teacher’s

questions...and they also ask a lot of questions. It shows that they are

learning something and want to know more”. [MT6]

MT6’s view fitted the description of the classes that were observed, and how MT1, MT2, and

MT3 organised their lessons and helped their students to engage actively and learn from each

other and the class discussions. MT1 shared how this aspect directly related to SIOP:

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“I had used grouping before but not very often...ka ‘i he haʻu ko ‘eni ki he

(but coming now to) SIOP, I preferred group works more because it allowed

the students to engage and learn from each other”. [MT1]

Thus, targeting to engage all students with the mathematical activities either through group

work or individual tasks enable students to study and learn from each other which enhanced

both learning and mathematics teaching.

5.6.3 Use of clear and appropriate language

The three teachers had taken into consideration the proficiency level of the students and made

sure that the way they presented the mathematical tasks and activities was in language that

was clear, simple and easy to comprehend. Where necessary, they used the Tongan language

to clarify concepts, definitions and instructions that were vague and confusing.

According to MT1,

“One thing I got from SIOP was how I deliver my lesson...usually I speak

fast and am not concerned about students’ language ability. After learning

and using SIOP ideas, I have learned to speak slower and think of my

students”. [MT1]

This change was noticed in the second and third lessons.

MT5 supported MT1 and said

“SIOP has reminded me to try to make my explanation clearer. I copy the

key vocabularies on the board and take time to discuss them so students can

understand better”. [MT5]

Taking time to discuss concepts and terms was evident in the three case studies and teachers

displayed more of this activity in the final three classes. It became clearer to the teachers that

students could only understand the mathematical concepts if they understood the language

clearly. All three teachers took time to develop Language Learning Objectives along with the

Content Learning Objectives, posted them on the board and discussed them well. They then

planned activities that matched both sets of objectives for the lesson. The Language Learning

Objectives helped the students to achieve the Content Learning Objectives.

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For example, MT1’s Content and Language Learning Objectives for Lesson 1 were:

Content Learning Objectives: Students will be able to:

� Review and practice how to draw each of the five non-linear graphs.

Language Learning Objectives: Students will be able to:

� Discuss and share how to draw each of the non-linear graphs.

The Language Learning Objectives framed the activities and enabled the students to discuss

and share. They enabled the students to master the content and but at the same time develop

their English proficiency through writing, speaking, listening, and reading which is the

hallmark of the SIOP Model (Echevarria et al., 2010).

5.6.4 Increased ‘Wait’ time

It was evident in all cases that all three teachers had learned to hold back the urge to answer

their own questions. They learned to exercise ‘wait’ time to allow students to process their

thinking then respond to the question. They also learned to help students to think by

rephrasing the questions and giving smart hints. MT1 developed this ability in his final

lessons, as discussed on page 89, and reported:

“SIOP taught me to provide some opportunities for children to discover

things for themselves. Before, I would ask a question and if the students are

slow to respond, I ended up answering the question myself”. [MT1]

Allowing ample time for students to complete activities was another achievement. Since all of

the teachers prepared for the activities in all lessons, they found out that time allocation for

these and for reporting and feedback at the end were all important and needed to be catered

for in time allocation during lessons. In this context, aspects of the SIOP model applied in

lesson planning and during the lesson delivery allowed students time to comprehend the

mathematical concepts and gain in-depth knowledge and skills either from their peers or from

the teacher, which ultimately enhanced both learning and teaching mathematics. This is an

aspect of components five and seven of the SIOP Model: Interaction and Lesson Delivery.

5.6.5 Guidelines for Lesson Preparation

MT1 saw SIOP as a “teacher preparation method which enables the teacher to effectively

teach the lesson”. The components of the SIOP Model helped the teachers to take into

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consideration each of the components of the SIOP model as essential elements for effective

teaching. MT3 agreed that the SIOP Model is a “guideline for teaching” and MT2 added that

the use of the SIOP Model assisted him to “know exactly the sequence of my lesson in order

to deliver it well”. These perceptions were demonstrated in the observed classes.

5.6.6 Enhance teacher’s performance in the classroom

All three teachers expressed strong support for the SIOP Model because it helped them to

change their teaching practices to be more student-centred through using of more

collaborative strategies. MT4 explained

“This model helped me to mould my teaching strategies. I used to mostly

teach in the traditional way of teaching using the lecture. Now I have turned

to get students involved by doing group work, discussing and sharing. This

method also helps students to build their confidence”. [MT4]

MT4’s perception was similar to the performance of MT1, MT2, and MT3 over the period of

the classroom observations. As the three teachers took into consideration the key components

of the SIOP Model, they were able to simplify their language to the level of students’

proficiency, provide frequent opportunities for student interaction, be aware of giving enough

wait time for the students to process their thinking, enable students to engage for maximum

time, and train themselves as effective facilitators moving around the class.

In summary, when teachers planned and implemented opportunities for student to interact

amongst themselves, had high expectations of student engagement during the lesson, used

clear and appropriate language in demonstration and instructions, provided ample time for

students to process their work, and used the SIOP Model as guideline for their lesson

preparations, both learning and teaching were enhanced.

Altogether this study showed clear evidence of improvements that teachers could put into

their teaching using the lessons learned from the SIOP model. The students’ input is just as

important. They were able to clearly say what worked and what did not. The last chapter

discusses the various implications of this study.

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CHAPTER 6: CONCLUSIONS AND IMPLICATIONS

6.1 Introduction The Sheltered Instruction Observation Protocol (SIOP) is an American model of teaching that

was introduced at Heilala High School, Tonga in 2007 to improve the quality of instruction

and enhance meaningful learning of students. Over the years, the school has continued to re-

emphasise the SIOP components through ongoing professional development and it was

believed to be making a positive contribution to mathematics education. This project was an

opportunity to formalise another set of SIOP professional development sessions with a select

group of mathematics teachers and to follow up with formal investigation of the effect of the

model in improving the knowledge and practice of the teachers as well as the preformance of

students. The researcher is an ardent SIOP practitioner and was both researcher and

participant in the study. She provided five weeks of professional development for the teachers

on the eight components of the model: (1) Lesson preparation, (2) Building background, (3)

Comprehensible input, (4) Strategies, (5) Interaction, (6) Practice and application, (7) Lesson

delivery, (8) Review and assess. The effect of the SIOP Model was then investigated through

talanoa with both students and teachers, video recording of nine classroom observations and

after-class discussions with the researcher. The following key research questions guided the

study:

1. How effective are the common ways of teaching mathematics at Heilala High School?

2. What are the major challenges of teaching mathematics at Heilala High School?

3. How effective is the SIOP Model in enhancing mathematics learning and teaching at

Heilala High School?

This final chapter reviews the research findings of the study and examines some broad

implications of the study for mathematics teaching and learning.

6.2 Summary of major findings Traditionally, mathematics teaching has followed a very rigid approach that has been resistant

to change. The history of the discipline, the nature of mathematics and the lack of confidence

in many mathematics teachers have caused mathematics to be taught in typical ways, research

indicating that there are much better ways to teach (Grootenboer, 2009).

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6.2.1 Typical ways of teaching mathematics

The common ways of mathematics teaching at Heilala High School were probably no

different from most other places. An important aspect of this study was to measure how the

teachers’ beliefs, knowledge and practices changed through the period of professional

development and testing. As articulated in the interviews and demonstrated in the observed

lessons, different combinations of Bell work, lecture and discussion, demonstration, follow-up

exercises and group work were identified as particular ways of teaching mathematics.

(i) Bell work or advance organizer (Ausubel, 1968) to introduce the lesson and get students

on task was popular with both students and teachers and was seen to improve in all cases over

the three observations. For example, MT1’s first Bell work question was at the level of recall

of the previous lesson and merely tested memory of the previous lesson. However, the Bell

work question in the third lesson required more thinking and asked for application of learning

to everyday occurrences. In addition, MT1 did not simply ask a question to be answered but

used students to demonstrate the activity. This was in line with Building Background, the

second component of the SIOP model.

(ii) All teachers identified the lecture as a typical method of teaching mathematics. However,

they also agreed that it was not entirely productive and needed to be mixed with question and

answer sessions, discussions and activities. MT4 acknowledged lecture as the traditional way

of teaching that he used only to introduce lessons. MT1 said that he no longer used this

method much as he had adopted new methods. The role of the teacher as the facilitator of

learning rather than the ‘sage on the stage’ was emphasised. All mathematics teachers viewed

effective mathematics teaching to be about the “act of facilitation and motivation of students’

learning, and the ability to enhance students’ understanding”. Anthony & Walshaw (2009)

discussed the role of the teacher as facilitator who listened attentively to students’ discussions,

discerned their learning needs and knew exactly when to step in or out of the discussions,

when to provide more scaffolding to enhance understanding, and when to shift the discussion

to the next topic.

(iii) Group work was used extensively in all observed lessons and both students and their

teachers showed support for this strategy. All teachers noted how group work helped students

to open up, have a say and interact with others, which were uncommon in ‘normal’

mathematics lessons. In group work, students felt safe amongst their peers and gained

confidence to share their ideas. More importantly, the sharing with peers created a fun

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environment where students were seen to enjoy mathematics. MT2 used a sophisticated group

work strategy that had ‘home groups’ and sub-groups called ‘expert groups’ which were

responsible for different aspects of a particular problem. The two levels of participation not

only ensured that more members had tasks to perform but that good leadership and

coordination were needed to arrive at finding solutions.

Students identified group work as the best and most effective strategy. S14 said the group

influenced him to work while S22 said that the strategy not only helped his own learning but

also gave him the opportunity to assist others’ learning. A few examples of the mathematical

discourse in group work were captured in chapter 4. The discussions were done in a bilingual

manner using both the Tongan and English languages as students asked questions and

negotiated meaning. Not only did they learn in the group activities but they also enjoyed the

engagement and discussions amongst themselves.

As Grootenboer (2009) proposed, group work is about students bringing strength and

knowledges to the table that will be different from their peers but collectively, these

differences enable them to grow. The value of helping one another, sharing and working

together stems from the social component in the learning process (Cobb, 1994). This is in line

with the Tongan communal way of learning and living together. Through the SIOP Model, the

social learning environment is emphasized when teachers plan and focus on achieving the

element of Interaction (the fifth component of the model). In doing so, they must at least use

a variety of teaching strategies (the fourth component), clear explanations of mathematic tasks

(Comprehensible Input- the third component), provide opportunities for students to practice

and apply the new concepts taught (Practice and Application - the sixth component), provide

feedback on students’ work and discussions (Review and Assess – the eighth component), and

these concepts must relate to the students’ background experiences (Building Background-

the second component). The effectiveness of lesson delivery is based on the effectiveness of

lesson preparation and the consideration of all the SIOP elements in harmony with each other.

6.2.2 Effective mathematics teaching & the Effective mathematics teacher

The teachers’ and students’ views of what constituted effective mathematics teaching and a

definition of the effective mathematics teacher provided some interesting comparisons. While

the teachers’ views were broader and more encompassing, the views of students were local

and specific to what facilitated learning in the actual lessons. Two related items were

described by most teachers to describe effective teaching: (i) effective facilitation of students’

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learning and (ii) effective support and motivation of students. As MT4 explained, the

effective teacher “makes the hard concepts easy” and effective teaching “is based on how to

find the answer instead of what is the answer.

From the students’ perspective, group work is the most effective teaching strategy that helped

them to understand. This has been discussed fully in section 6.2.1. The students’ comments

that group work enhanced their learning through “helping each other”, “sharing”, “working

together” indicated that they took responsibility for their learning. This meant that group work

shifted the responsibility from the teacher to the students themselves, which was what

teachers had hoped for in effective facilitation of learning. Linked also to group work is the

students’ ability to communicate mathematically. This group of students obviously preferred

to use the Tongan language to negotiate meaning. For them, clear definitions, explanations

and demonstration were important for making meaning. Yet again, the components of the

SIOP model were being highlighted in different ways.

6.2.3 Challenges in learning and teaching mathematics

In the discussion of challenges in learning mathematics, the students pointed to the

mathematics teacher as the main challenge and word problems as the second. The teachers on

the other hand identified a lack of basic mathematic skills and negative attitudes, a lack of

skills in application to real life problems, and language as the key challenges to teaching

mathematics. In other words, the teachers were pointing to a lack in the students while the

students laid the blame for their lack of interest and engagement on their teachers. The

researcher is of the view that teachers needed to shift their approach from focusing on what

students do not know to what they know and to build on that. This will be discussed fully

under section 6.4.

If there is one aspect where both the teachers and the students agreed it was about the

challenge of language in mathematics learning. Students faced difficulties with making sense

of word problems and which could be the cause of their lack of application skills to real life

problems. This is similar to the findings by Manu (2005) that teaching in a second language

initiated two challenges for both teachers and students: first the challenge of understanding

the language itself, and the challenge of understanding the mathematical task. This means that

for non-English speakers the problem of understanding is firstly a language problem and then

a mathematical one. This challenge also emerged from the classroom observation where the

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students could not get to solve the mathematical problem took time because they did not

understand the meaning of the problem itself.

6.3 The SIOP Model, the successes and challenges The design of this study in providing support for the mathematics teachers at Heilala High

School through (i) the initial 5 weeks of awareness and professional development in the SIOP

model, (ii) capturing their observed lessons on video, and (iii) regular feedback on the

observed lessons were effective in enabling teachers to improve on their practices and

highlight the important role that the SIOP Model has in positively influencing the teaching

and learning of mathematics. All eight components of the SIOP Model were tested in the nine

classroom observations and the results were overwhelmingly supportive of the ability of the

SIOP model to enhance teaching and learning, primarily through enhancement of the

teachers’ own pedagogical skills.

6.3.1 Successes of the SIOP Model

The eight key components of the SIOP Model: Lesson preparation, Building Background,

Comprehensible Input, Strategies, Interaction, Practice and Application, Lesson Delivery, and

Review and Assess were reviewed in lessons that were observed in this study. The details of

the results were discussed fully in chapters 4 and 5. They are summarised below.

In lesson preparation, there was marked improvement in the way teachers incorporated both

the content and language objectives into their lessons and discussed them well with students.

This was the essence of the SIOP model that teachers presented mathematics in English in

ways that enabled students to undertstand the mathematics and develop their English language

skills simultaneously. The model required the development of language skills to be a

consistent part of daily lesson plans and delivery. The other important development was the

alignment of the objectives with the lesson activities and the assessment items such as the

questions asked. The use of Bell work as an advance organizer to get students’s attention was

important in mathematics learning where the usual practice was to begin with a serious lecture

and/or exercises. As the lessons progressed, the teachers increasingly got better at explaining

and clarifying matters using simple words and phrases. There was a notable intention to

highlight key vocabulary items and discuss them at the beginning of the class. They were also

careful with their discourse and used clear and appropriate language that fitted students’

English proficiency levels.

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The observed lessons showed the teachers’ attempt at using a variety of scaffolding

techniques such as demonstrations, explanations and asking questions to promote thinking.

This component catered for the diversity of students in a class. The teachers were mindful of

the need to engage their students for about 90 percent to 100 percent of the time. This

component assisted teachers in planning good activities and processes that would engage all

students fully. Throughout the activities, teachers were teaching on their feet; they practised

questioning skills that included good wait time for students to process their responses.

Creating an environment with a high level of student engagement is crucial to effective

mathematics teaching (Attard, 2011; Anthony & Walshaw, 2007).

6.3.2 Challenges of the SIOP Model

While the study highlighted many positives of the SIOP Model in enhancing students’

mathematics learning, there is no denying that ongoing staff development and strengthening

of the SIOP components will be critical. Three areas that would need continuous attention are:

� effective planning strategies

� development of thinking skills through high-order questions and activities, and

� providing students with useful feedback

Ongoing work will be required to get teachers to develop good thinking and probing questions

that would provide clearer undertstanding of students’ thinking including their confusions and

misconceptions. Continual assessment and review of students’ learning to ascertain whether

to re-teach a topic or move on to the next stage was another challenge. Throughout this

project, teachers mostly used formative assessment either by questioning, observing and

listening to students’ discussion, one-on-one checking of exercises, or a quick thumbs-up to

assess if there was a need to re-teach the content or move on to the next topic. These

experiences were similar to the experience of teachers who have implemented the SIOP

Model elesewhere (Echevarria et al., 2008; 2010; Hansen-Thomas, 2008; Guerino et al.,

2001). In those cases, their performance gradually changed from traditional ways of

mathematics teaching towards becoming a facilitator of learning (Hunter & Anthony, 2012;

Rogers, 2007; Chinnappan & Cheah, 2012).

The need for teachers to keep the Objectives in mind and review them in the middle and end

of the lesson was still a challenge. It was noted that teachers usually discussed both the

content and learning objectives at the beginning of the class and then forgot about them. The

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use of supplementary materials and teaching resources needed to be given greater attention.

The researcher believed that this was a consequence of poor planning and preparation.

Bearing in mind that the SIOP model is a foreign development in response to a particular

need, it will be important to contextualize the processes and activities as well as the language

and vocabulary requirements so that the local students can make the necessary connections

and apply their meaning to the mathematical problems at hand. This could mean that the

Language Learning Objectives would need to be inculcated with the students’ background

experiences rather than the textbook examples of the foreign contexts. Teachers will need to

contextualize the vocabulary so students will be able to connect and apply their meaning to

the mathematical problems at hand.

The use of the SIOP model in the mathematics classroom will enhance the nurturing

relationships (tauhi vā) both between the teacher and the students and among the students

themselves. Tauhi vā is one of the core traditional Tongan values that are reflected reflect in

the SIOP mathematics classroom where students are engaging in the mathematical activity

through fetokoniʻaki (helping each other), fevahevaheʻaki (sharing), and fengāueʻaki (working

among them). These aspects of Tongan traditional values exist and are practised in the way of

life of Tongan families and communities. The research has shown that students could transfer

and use these aspects of Tongan cultural contexts in the mathematics classroom through

group working. The use of the SIOP model can also mould the teacher into longomo’ui

(active) during his/her lesson delivery, which portrays her mamahi’i me’a (commitment to

her/his job), caring, love for the students, and knowing for sure (‘iloʻilo pau) his/her immense

impact on the lives of students. S/he is on his/her two feet demonstrating, explaining, helping,

encouring, and facilitating the students into responsible learners. It is through these practices

that effective mathematics teaching in a Tongan mathematics classroom can occur. This is

primarily due to the ability of the teacher to establish and nurture healthy relationships with

the students and establish connections between the student and the mathematics content.

6.4 Implications of the study The underlying intention of the Sheltered Instruction Observation Protocol (SIOP) model of

teaching to enhance the linguistic need of second language learners in schools made it

relevant for learning in all classrooms in Tonga as well as other Pacific Island countries. With

the ongoing, immense challenges that our students face when studying in English as a second,

third and even fourth language, the promise of the SIOP model is worth considering. In

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addition to language support, the SIOP model has been found to positively impact on the

teaching and learning in various subjects including science and mathematics. The results of

studies discussed in chapter 3, indicated enhancement in students’ academic vocabularies

along with their conceptual understanding. The current study had demonstrated this very well

for mathematics teaching at Heilala High School. The lessons learned by the teachers and

echoed by their students, proposed plausible implications for language learning, mathematics

learning and teaching, ongoing teacher development and future research.

6.4.1 Implications for improved mathematics teaching

This study had tested best practices that mathematics teachers of non-English speakers could

use to enhance their students’ mathematics learning and achievement. The three case studies

in chapter 5 detailed the development of three teachers in specific areas of the model. All

teachers showed strong support for the model. Their teaching became more student-centred as

demonstrated in the way they communicated, interacted with and allowed students to

participate freely and take charge of their learning. In turn, the teachers trained themselves to

be effective facilitators of learning. The students noticed the change in their teachers and they

talked about mathematics classes as engaging, meaningful and fun. These were profound

observations not usually recorded in mathematics classes. There was no fear of mathematics!

Another important item of consideration from this study was the teachers’ identification of the

perceived lack or deficiency in students as the major challenge in mathematics teaching. Their

complaint was that students lacked the prerequisite knowledge on which to build learning.

Teachers needed to shift their interest from what students did not know to what they know as

well as their strengths and then to build from that. From that starting point, they should be

able to provide appropriate strategies that will, in turn, enable students to build new

understandings.

6.4.2 Implications for teacher development

The professional development of teachers must be ongoing and consistent to bring about

growth and change in teachers’ values and vision as well as their knowledge, skills and

practice (Roger, 2007). A follow-up component that was often lacking was the assessment of

this growth and change. This study demonstrated a sequence of steps and phases that

contributed to its success and are worth noting. Firstly, there was an intensive period of staff

development where the teachers were carefully taken through the elements of the SIOP model

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by the researcher who was the expert. This was followed by the observation of the teachers in

class. This assessment benefitted from three sources of assessment information from the

observer and the two groups observed. The critical assessment of the researcher as expert

practitioner was balanced by the honest reflections of the teachers and the very pointed

comments of the students. This study had a determination to see improvement and was able

to achieve that through intensive staff development and follow-up.

Teacher education programmes prepared trainee teachers for practice. Many of the elements

of the SIOP model of teaching are similar to those that teacher education programmes

emphasise and instil in teachers during training. However, when teachers enter the world of

work they are often confronted with school environments and people who have not changed

much because of various factors. The new graduates are not always able to bring about the

change they envisioned and unfortunately, they could be forced to become like everyone else.

Heilala High School had embarked on a change process in 2007. At the time of this study,

there was a need to revisit that process and put in measures to effect change. The study did

this very well and showed that ongoing staff development, including the measurement of

change processes in teaching and classroom practice will be critical to introduce new ideas,

challenge current thinking and engage teachers in action research for change. Listening to

students as partners in learning and teaching will be important.

6.4.3 Implication for changing classroom practices

Mathematics teaching and learning are inseparable (Ernest, 1991) because the teaching cannot

be effective unless the teacher knows how the students learn best (Zevenbergen et al., 2004).

The results of this study highlighted effective teaching strategies such as group work where

students were able to fengāueʻaki (engage in the act of helping each other) and fevahevaheʻaki

(engage in the act of sharing). This implied that mathematics teachers need to integrate more

group work strategies where students feel comfortable discussing mathematical activities and

learning from their peers (Grootenboer, 2009). It also implied that the classroom setting must

support a social learning environment. A very important outcome of the study was the

effective use of hands-on activities that helped students to make sense of their learning and

enjoy mathematics rather than experience it as a boring subject (Wheatley, 1991). That

students can be seen to be very involved in the class activity, be talking excitedly and having

fun in the mathematics class was a great achievement. The ‘fun’ element was unusual in

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mathematics classes and obviously the teachers will want to continuously see that in many

more lesssons.

Quality engagement with students was seen to be an important component of learning. Setting

clear language objectives that flowed well into the content objectives was an important part of

the planning process. When the teachers kept track of this dual focus, they were able to

support students to talk and use mathematical discourse. Together, they negotiated meaning in

a bilingual manner where a mix of both the Tongan and the English language were used. The

free discussions in the groups allowed students to arrive at possible solutions. Another

important implication that was often neglected but which was demonstrated in this study was

the need to provide immediate and clear feedback on the students’ work. It was demonstrated

clearly that the quality planning was critical to the effective delivery of a lesson (Echevarria et

al., 2008, 2010).

Mathematics teachers needed to have high expectations for students’ engagement in their

classes (Anthony & Walshaw, 2007). If the teacher believes that everyone can learn

mathematics and they can all engage in the mathematical activities, then such a belief will be

reflected in classroom practice.

6.4.6 Implications for further research

An investigation of a teaching model such as that done in this project served to provide

relevant information and further directions for research so that a sound conceptual basis could

be established. The analysis in the study should be the beginning for more investigations.

Further in-depth study using a larger sample would allow for greater representativeness of the

findings. The findings of the current study suggest that the SIOP model of teaching is a viable

option for use to enhance learning in mathematics classess. There is scope to replicate the

elements of this study to extend it to other subject areas. Otherwise, there is scope for

conducting small action research studies that could investigate the effectiveness of each of the

components of the SIOP model of teaching. The findings of those studies could add weight

and support to the current study.

The enhancement of learning and teaching must be an ongoing aim of schools and teachers.

Schools must be supported to develop and maintain a research culture that will support the

assessment and evaluation of current practices as well as the introduction and testing of new

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ideas. This study indicated that teachers at Heilala High School and elsewhere are well placed

to do that successfully.

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Appendix A: Classroom Observation Schedule

Code Observation #1 Observation #2 Observation #3

MT1 8th of August, 2012 14th of August, 2012 29th of August, 2012

MT2 2nd of August, 2012 14th of August, 2012 31st of August, 2012

MT3 9th of August, 2012 16th of August, 2012 11th September, 2012

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Appendix B: Schedule for the talanoa sessions

Teachers’ talanoa sessions

Code Time Date Place MT1 9:00a.m 13/11/2012 Room C1 MT2 11:00a.m 10/10/2012 Room C1 MT3 11:00a.m 13/11/2012 Room C1 MT4 left Email received

15/1/2013

MT5 10:00a.m 5/4/2013 Room C1 MT6 7:00p.m 28/5/2013 Room C1

Group Talanoa Session with the Students.

Focus Group Number

Code # Date Time Teacher Code

1 S11 20th of Sept, 2012

9:00 a.m MT1 S12 S13 S14

2 S21 19th of Sept, 2012

9:00 a.m MT2 S22 S23 S24

3 S31 19th of Sept, 2012

2:00 p.m MT3 S32 S33 S34

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Appendix C: Schedule for the SIOP In-service Trainings

Date

SIOP Model Component

22rd of May, 2012 Lesson Preparation

24th of May, 2012 Building Background

29th of May, 2012 Comprehensible Input

31st of May, 2012 Strategies

12th of June, 2012 Interaction

18th of June, 2012 Practice and Application

20th of June, 2012 Lesson Delivery

25th of June, 2012 Review and Assess

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Apppendix D: Observation Schedule Form

The purpose of this observation schedule was to assist the researcher in examining the context of the mathematics classroom with the SIOP Model. The focus is both on students and

teachers.

Teacher’s Name:..................... Topic:...........Date:..............Period #:..........Duration:............

Item observed What actually happens Duration Feedback Quality of Lesson Planning and Preparation.

1. Level of Objectives 2. Quality of Planned Activities 3. Planned Assessment Activities 4. Time Allocation

Advance Organizer - Level of questions asked

Student Activities and Engagement - quality of the activity - opportunities for interaction

Types of Activities used and number of students participated

Language and Questioning Skills - use of clear and appropriate language - defining of key vocabularies

Assessment of Students’ Understanding - frequent feedback on students’ work

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Appendix E: Questions that guide the talanoa sessions

Talanoa with Teachers

Questions that guide the talanoa with math teachers were based on the main key research questions.

a. What are teaching strategies that you have used in your classroom in teaching mathematics?

b. What are the challenges you have face in teaching mathematics?

c. What is effective mathematics teaching to you?

d. What is SIOP Model to you?

e. How effective is the SIOP Model in teaching and learning mathematics?- You can specify a particular component that has helped you.

Talanoa with Students

Questions that guide the talanoa with the student samples.

a. What are strategies that the teacher have used in this period that have helped you to understand the class today?

b. Can you describe the nature of an effective mathematics teacher that helped you learn mathematics better?

c. What are challenges you have face in learning mathematics at form 5 level?

d. What are some strategies that you preffered matheamatics teachers to use in the classroom and why?

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Appendix F: Consent Letter to the Ministry of Education and Training Matangiake, Tongatapu, 29th/ 5/ 2013. Mrs ‘Emeli Moala Pouvalu, CEO for Education, Ministry of Education & Training, Nuku’alofa, Tongatapu.

Dear Madam,

I am currently a student at the University of the South Pacific (Tonga campus) and doing my thesis for my Master of Arts in Education. My thesis is on: “Investigating the effect of the

SIOP Model in teaching mathematics: A case study at a Tongan secondary school”. Dr

Salanieta Bakalevu and Dr. ‘Ana. H. Koloto are my supervisors for this study.

I am asking that your Ministry will kindly assists me in access to the following data which has been recommended by my supervisors as they enrich the literature review of this study, particularly:

� Mathematics Curriculum use by Tonga’s Secondary Schools in the past 10 years, and

any major changes in the past decade. Chief Examiners’ reports on students’

performance in mathematics. � Key Trends in Students’ mathematics achievement in form5, 6, and 7 exams � Secondary schools’ Mathematics teachers- gender, age, qualifications, number of

years teaching mathematics. � Professional Development programmes for Mathematics teachers in the secondary

schools. � Pre-service Teacher Education programmes at TIOE, mathematics and mathematics

education courses, and the types of approaches used for preparing mathematics teachers.

The Ministry’s effort in providing me with the above data would be greatly appreciated. Dr Koloto is kindly assisting me in trying to get the above data and we are happy to meet with relevant staff in your Ministry to clarify our information if required.

Yours sincerely,

___________________

(Tamaline. Tu’ifua).

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Appendix G: Participant Consent Form

The University of the South Pacific

Institute of Education

PARTICIPANT CONSENT FORM

Name of the Project: Investigating the effect of the SIOP Model in teaching Mathematics: A

case study at a Tongan Secondary School.

I have read and understood the Information Sheet describing the above-named project. I agree to participate as a subject in the project. I consent to publication of the results of the project/the information given to me on the understanding that my anonymity is preserved. I understand that at any time I may withdraw from the project, as well as withdraw any information that I have provided. I note that this project has been reviewed and approved by the University Research Ethics Committee at the University of the South Pacific. Name (please print) Signature Date

(where appropriate) I am signing this Consent Form on behalf of

to allow her/him to participate in this project.

Age ( years)

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Appendix H: Consent Letter to the Head of School

Institute of Education

University of the South Pacific

Suva, Fiji.

Heilala High School Administrators,

P. O. Box 60.

Tongatapu, Tonga.

6/8/2012.

Dear Sir,

I am hereby requesting a permission to carry out an educational research at Heilala

High School during the academic year 2012 on the topic “Investigating the effect of the SIOP

Model in teaching Mathematics: A case study at a Tongan Secondary School” particularly at

the form 5 level.

The vision of the school for this year is to “Save the one”. I believe as a mathematics

teacher, the finding of this study will be able to save some students in the field of mathematics

in the coming future. The researcher is currently teaching at the school and my vision is to

upgrade the standard of mathematics at Heilala High School through an in-depth investigation

of what is actually happens in a mathematics classroom through the implementation of the

SIOP Model. This is an American model and it was from a different cultural context.

However, the finding of this study will inform the school, and other mathematics teacher on

the effect of this model in the context of a Tongan mathematics classroom.

I have selected to do the investigation in form 5 mathematics and there are 3 teachers

that share the load of form 5 mathematics. The tools that I will use to gather my data are

document analysis, classroom observation and interviews. I am also asking for permission to

video record the classroom observations. This video tape will be secured and kept

confidentially in order to protect the anonymity of the participants. This will enable the

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researcher to replay the scene in the classroom in order to get an in-depth understanding of the

topic.

This study has two phases which are training of the mathematics teachers by the

researcher on the features of the SIOP Model, and then teachers are continually monitored,

coached and observed on the implementation of the model. There will be a copy of the final

thesis once it approved send to the administrators as a token of appreciation for granting the

opportunity to do this research study at Heilala High School.

Sincerely yours.

Tamaline Wolfgramm Tu’ifua.