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Exploring the Challenges of Teaching Similarity of Triangles, the Case of Areka Town Primary Schools, Ethiopia

by

BEREKET TELEMOS DORRA

submitted in accordance with the requirements for the degree of

DOCTOR OF PHILOSOPHY IN EDUCATION

in the subject

CURRICULUM STUDIES

at the

UNIVERSITY OF SOUTH AFRICA

SUPERVISOR: Prof. ZMM Jojo

October 2021

ii

DECLARATION

Name: Bereket Telemos Dorra

Student number: 57654433

Degree: Doctor of Philosophy in Education (Curriculum Studies)

I declare that the above thesis is my own work and that all the sources that I have used

or quoted have been indicated and acknowledged by means of complete references.

I further declare that I submitted the thesis to originality checking software and that it falls

within the accepted requirements for originality.

I further declare that I have not previously submitted this work, or part of it, for examination

at Unisa for another qualification or at any other higher education institution.

13th October 2021

SIGNATURE DATE

iii

DEDICATION

This research is dedicated to my father, Telemos Dorra. Even though he is not alive, his

paternal advice during my early school days had huge impact on my current academic

carrier.

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ACKNOWLEDGEMENT

Many individuals contributed to my PhD journey. Despite their immense contribution

things would not have been so had my Lord Jesus not protected me and channelled my

feet along safe ways. Therefore, first I would like to extend my heart-felt thanks to God

for his gracious support in every aspect of my life.

I would like to express my sincere thanks to my advisor Prof. ZMM Jojo for her valuable

advice and guidance throughout the course of my research. I am also grateful to her

precious and constructive comments and suggestions during the compilation of this

thesis.

I would like to thank all the teachers who participated in this study. I also thank Ashebir

Sidelil (PhD) and Mesfin Mekuria for their close assistance and advice which was

available whenever I needed it.

I thank UNISA and Wolaita Sodo University for financial and material support.

Furthermore, I extend my thanks and appreciation to friends in South Africa for their

constant support during my stay in that country.

Still, this would not have been achievable without the affection and support of my family.

Thank you, my wife, Mekdes Minjar for the love, support, and willingness to fill the entire

gap for our sons’ and accompany them in my absence. Also, thank you all my brothers,

sisters, and colleagues; as usual, you were with me.

Lastly, thanks to my sons Yaya and Malalia for being my inspiration to do well, to be

diligent, and to work ahead of expectation.

v

ABSTRACT

Similarity and its related concepts are central components of geometry. It is an important

spatial-sense and geometrical concept that can facilitate students’ understanding of

indirect measurement and proportional reasoning. Many see geometry as a significant

subject in mathematics and the similarity of a triangle is found to be a key concept within

geometry, but there is very little research done on teachers’ challenges on teaching

similarity of triangles. Thus, this study focused on exploring the challenges of teaching

similarity of triangles in Grade 8 classes and how those challenges could be minimised.

A qualitative and exploratory case study design was used. In addition, purposive sampling

was used to select 5 mathematics teachers from Areka Town primary schools, in Ethiopia.

In this regard, the data of this study were collected using classroom observation, semi-

structured interview, and teachers’ questionnaire. The data were coded manually and

categorised into four themes.

Based on the data, the findings in this study indicated that teachers faced mathematical

knowledge and pedagogical knowledge challenges. In relation to this fact, students’ poor

background knowledge, resources, and the mathematics curriculum were also among the

challenges teachers faced in teaching the similarity of triangles. The teacher-student

interaction was minimal, and the teaching approach was dominated by teacher talk and

chalk.

Based on the literature reviewed, a theoretical framework that underpinned this study,

and empirical data obtained, the researcher proposed a model for meaningful teaching

on the similarity of triangles and used it to minimise the challenges of teaching the

similarity of triangles. A meaningful teaching similarity of triangles refers to providing an

activity that offers an opportunity for students to connect the similarity of triangles to their

real-life experiences and has a goal to connect the similarity of triangles to real-life

situations. Through the interventions using the prosed model, the participants were able

to explain models of similar figures, polygons, and the application of the similarity of

triangles in the real life. Furthermore, the teaching approaches used by the participants

have come to show van Hieles’ five phases of instruction for teaching of similarity and in

this regard, participants had known the van Hieles’ theories and its importance for

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meaningful teaching similarity of triangles.

This study further recommends the reform of pre-service teachers’ education,

incorporating continuous professional development, revising the geometric contents in

the existing syllabus, and including the different theories related to teaching and learning

geometry in the mathematics syllabus.

KEY TERMS: Challenges; Meaningful teaching; Mathematical knowledge challenge; Pedagogical

approaches; Pedagogical knowledge challenges; Phases of instruction: Phases of

instruction for teaching similarity of triangles; Similarity; Similarity of triangles; Student

background; Teacher-Students Interaction; van Hieles’ theory.

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Lists of tables Table 1.1 The nature of mathematics content in Grade 8 mathematics syllabi in Ethiopia .......... 2

Table 1.2 Grade 8 students’ average correct geometry domain items in TIMSS ........................ 3

Table 1.3 PECRE students’ Mathematics result ......................................................................... 5

Table 2.1 The nature of geometric contents and expected outcomes in Grade 8 mathematics

curriculum of Ethiopia (MoE, 2009) ...........................................................................................32

Table 4.1 Characteristics of interpretivism .................................................................................88

Table 5.1 The initial and final observation guide after a pilot study .......................................... 105

Table 5.2 The Initial and final Interview questions after a pilot study ....................................... 106

Table 5.3The Initial and final questionnaires items after a pilot study ...................................... 107

Table 5.4The teachers’ demographic information .................................................................... 110

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Lists of Figures Figure 2.1 Similar Triangles..................................................................................................................... 27

Figure 2.2 �ℎ� ��ℎ�� ℎ�� 2 + �2 = �2 ......................................................................... 28

Figure 2.3 �� .......................................................................... 28

Figure 2.4 Application of similar triangles ............................................................................................. 30

Figure 2.5 Static approach to similarity adopted from impacting teachers’ understanding of

geometric similarity .................................................................................................................................. 36

Figure 2.6Translation of triangle ABC to A'B'C' ................................................................................... 37

Figure 2.7 Reflection ................................................................................................................................ 38

Figure 2.8 Rotation................................................................................................................................... 38

Figure 2.9 Enlargement ........................................................................................................................... 39

Figure 2.10 Reduction ............................................................................................................................. 39

Figure 2.11 Transformations-based approach to similarity adopted from impacting teachers’

understanding of geometric similarity.................................................................................................... 40

Figure 2.12 Similar triangles .................................................................................................................. 41

Figure 2.13 Example of similar triangles .............................................................................................. 42

Figure 3.1 Two sets of similar triangles ................................................................................................ 65

Figure 3.2 Examples of similar plane figures ....................................................................................... 73

Figure 3.3 Similar triangles .................................................................................................................... 74

Figure 3.4 Network relations of similar triangles ................................................................................. 76

Figure 3.5 Proposed model for teaching similarity of triangles, implication of van Hieles’ phases

of learning .................................................................................................................................................. 79

Figure 5.1 The data coding process .................................................................................................... 109

Figure 5.2 Theme-1: Importance of learning geometry .................................................................... 111

Figure 5.3 Theme-2: Phase of the instruction in teaching similarity of triangles .......................... 112

Figure 5.4 Theme-3: Challenges teacher faced in teaching similarity of triangles ...................... 113

Figure 5.5 Theme-4: Minimising the challenges of teaching similarity of triangles ...................... 114

Figure 5.6 TE’s example on enlargement of triangles ...................................................................... 117

Figure 5.7 TE’s classroom work on enlargement of triangles ......................................................... 118

Figure 5.8 TD’s Examples of two similar rectangles ......................................................................... 119

Figure 5.9TB’s connection of similarity with congruence ................................................................. 121

Figure 5.10 TC’s explanation of similar triangles .............................................................................. 124

Figure 5.11 TC’s example on similarity of triangles .......................................................................... 124

Figure 5.12 TB’s work on similar triangles ......................................................................................... 127

Figure 5.13 TD’s examples on theorem of similarity of triangles .................................................... 128

Figure 5.14 TA’s explanation of similar triangles .............................................................................. 130

Figure 5.15 TA’s examples of similar triangles .................................................................................. 133

Figure 5.16 TB’s example of similar triangles .................................................................................... 134

Figure 5.17 TB’s examples of similar triangles .................................................................................. 135

Figure 5.18 TC’s Initial phase of the lesson ....................................................................................... 138

Figure 5.19 TD’s initial phase of the lesson ....................................................................................... 140

Figure 5.20 TE’s initial phases of the lesson ..................................................................................... 141

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Figure 5.21 non-similar figures ............................................................................................................. 142

Figure 5.22 TE’s examples of similar triangles .................................................................................. 143

Figure 5.23TA’s demonstration of similar figures .............................................................................. 145

Figure 5.24 Student participation in TD classroom ........................................................................... 146

Figure 5.25 Student participation in TB classroom ........................................................................... 146

Figure5.26 TC’s work on tests of similarity of triangles .................................................................... 148

Figure 5.27 TD’s work on similarity theorem ...................................................................................... 149

Figure 5.28 TE’s work on enlargement by using coordinate ........................................................... 150

Figure 5.29 TB’s demonstration of triangles ...................................................................................... 155

Figure 5:30 Geometric figures on the walls of school A ................................................................... 158

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ACRONYMS

ATA: Areka Town Administration

CCSSM: Common Core Standards for Mathematics

MoE: Ministry of Education

NCTM: National Council of Teachers of Mathematics

PECRE: Primary Education Completion Regional Examination

RLA: Regional Learning Assessment

SMASEE: Strengthening Mathematics and Science Education in Ethiopia

SNNPRS: Southern Nations, Nationalities and Peoples Regional State

TIMSS: Trends in International Mathematics and Science Study

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Table of Contents

DECLARATION ...................................................................................................................................... ii

DEDICATION ......................................................................................................................................... iii

ACKNOWLEDGEMENT ....................................................................................................................... iv

ABSTRACT ............................................................................................................................................. v

Lists of tables ........................................................................................................................................ vii

Lists of Figures .................................................................................................................................... viii

ACRONYMS ........................................................................................................................................... x

Table of Contents .................................................................................................................................. xi

CHAPTER ONE .......................................................................................................................................... 1

OVERVIEW OF THE STUDY ............................................................................................................... 1

1.2. BACKGROUND TO THE STUDY ................................................................................................ 4

1.2.1 Geometry ................................................................................................................................... 6

1.2.2 Studies on Similarity ................................................................................................................ 7

1.2.3 Purpose of the study ................................................................................................................ 9

1.3 THEORETICAL FRAMEWORK .................................................................................................. 10

1.4 LITERATURE REVIEW .............................................................................................................. 11

1.5 STATEMENT OF THE RESEARCH PROBLEM ...................................................................... 11

1.6 THE AIM OF THE STUDY ........................................................................................................... 12

1.7 THE OBJECTIVES OF THE STUDY .......................................................................................... 12

1.8 THE RESEARCH QUESTIONS ................................................................................................ 12

1.9 SIGNIFICANCE OF THE STUDY ............................................................................................... 13

1.10 DELIMITATIONS OF THE STUDY ........................................................................................... 13

1.11 RESEARCH METHODOLOGY AND DESIGN ....................................................................... 14

1.11.1 Research paradigm.............................................................................................................. 14

1.11.2 Research methodology ....................................................................................................... 14

1.11.3 Research Design .................................................................................................................. 14

1.12 POPULATION AND SAMPLING TECHNIQUES ................................................................... 15

1.12.1 Participants ........................................................................................................................... 15

1.13 DATA SOURCES AND COLLECTION TECHNIQUES ......................................................... 15

1.13.1 Observation ........................................................................................................................... 16

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1.13.2 Semi-structured interview ................................................................................................... 16

1.13.3 Questionnaire ....................................................................................................................... 16

1.14 DATA ANALYSIS AND INTERPRETATION ........................................................................... 17

1.15 VALIDITY AND RELIABILITY.................................................................................................... 17

1.16 PILOT STUDY ............................................................................................................................. 18

1.17 ETHICAL PROCEDURES ......................................................................................................... 19

1.18 CHAPTER OUTLINE .................................................................................................................. 19

1.19 DEFINITIONS OF CONCEPTS ................................................................................................ 20

2.20 Summary....................................................................................................................................... 21

CHAPTER TWO ....................................................................................................................................... 22

LITERATURE REVIEW ..................................................................................................................... 22

2.1 THE HISTORY OF GEOMETRY ................................................................................................ 22

2.2 THE IMPORTANCE OF LEARNING GEOMETRY .................................................................. 26

2.3 THE CONCEPT OF GEOMETRIC SIMILARITY ...................................................................... 34

2.3.1 Traditional approach of similarity ......................................................................................... 34

2.3.2 Transformation approach of similarity ................................................................................. 36

2.3.2.1Translation ............................................................................................................................. 36

2.3.2.2 Reflection ............................................................................................................................. 37

2.3.2.3 Rotation ................................................................................................................................ 38

2.3.2.4 Dilation .................................................................................................................................. 38

2.3.3 Similar triangles ...................................................................................................................... 41

2.4 TEACHING OF GEOMETRY ....................................................................................................... 42

2.5 THE ROLE OF TEACHERS IN TEACHING GEOMETRY ...................................................... 47

2.6 GEOMETRY CLASSROOM ........................................................................................................ 48

2.7 CLASSROOM INTERACTIONS.................................................................................................. 49

2.7.1 Communication in teaching geometry ................................................................................. 49

2.8 TEACHING GEOMETRY THROUGH TECHNOLOGY ........................................................... 51

2.9 CHALLENGES OF GEOMETRY TEACHING ........................................................................... 53

2.10 STRATEGIES THAT CAN MINIMISE THE CHALLENGES OF THE TEACHING OF

GEOMETRIC SIMILARITY ................................................................................................................. 55

2.10.1 Promoting teachers’ mathematical knowledge for teaching similarity ......................... 57

2.10.2 Effective instructional practices in geometric similarity teaching .................................. 58

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2.11 Conclusion .................................................................................................................................... 59

CHAPTER THREE ................................................................................................................................... 60

THEORETICAL FRAMEWORK ......................................................................................................... 60

3.1 INTRODUCTION ........................................................................................................................... 60

3.2 THE VAN HIELES’ MODEL ......................................................................................................... 62

3.2.1 The van Hieles’ levels of geometric thinking ...................................................................... 63

Level 1: Visualisation ....................................................................................................................... 64

Level 2: Analysis ............................................................................................................................... 65

Level 3: Abstraction ......................................................................................................................... 66

Level 4: Formal deduction ............................................................................................................... 66

Level 5: Rigour .................................................................................................................................. 67

3.2.2 Properties of the van Hieles’ models ................................................................................... 67

Property 1: Intrinsic versus Extrinsic ............................................................................................. 68

Property 2: Sequential ..................................................................................................................... 68

Property 3: Distinction or Linguistic ............................................................................................... 69

Property 4: Separation or Mismatch .............................................................................................. 70

Property 5: Attainment ..................................................................................................................... 70

3.2.3 Criticism of van Hieles’ theory .............................................................................................. 71

3.2.4 Van Hieles’ phases of instruction ......................................................................................... 72

Phase 1: Information/Inquiry ........................................................................................................... 72

Phase 2: Directed Orientation ........................................................................................................ 73

Phase 3: Explication ........................................................................................................................ 74

Phase 4: Free Orientation ............................................................................................................... 75

Phase 5: Integration ......................................................................................................................... 76

3.2.5 VAN HIELES’ LEARNING MODEL FOR THE TEACHING OF SIMILARITY ............... 76

3.3 FISCHBEIN’S THEORY OF FIGURAL CONCEPTS ............................................................... 80

3.4 DUVAL’S THEORY OF FIGURAL APPREHENSION ............................................................. 81

3.5 Conclusions .................................................................................................................................... 82

CHAPTER FOUR ..................................................................................................................................... 83

RESEARCH DESIGN AND METHODOLOGY ................................................................................ 83

4.1 INTRODUCTION ........................................................................................................................... 83

4.2 RESEARCH AIMS AND QUESTIONS .................................................................................... 83

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4.3 RESEARCH PARADIGMS ........................................................................................................... 84

4.3.1 Positivist paradigm ................................................................................................................. 85

4.3.2 Critical paradigm ..................................................................................................................... 86

4.3.3 Pragmatic paradigm ............................................................................................................... 86

4.3.4 Interpretivist paradigm ........................................................................................................... 86

4.4 RESEARCH DESIGN ................................................................................................................... 88

4.5 RESEARCH METHODOLOGY ................................................................................................... 89

4.5.1 The qualitative method .......................................................................................................... 89

4.6 POPULATION AND SAMPLES ................................................................................................... 90

4.7 DATA SOURCES AND COLLECTION TECHNIQUES ........................................................... 92

4.7.1 Observations ........................................................................................................................... 92

4.7.1.1 Participant observation ....................................................................................................... 93

4.7.1.2 Non-participant observation .............................................................................................. 93

4.7.1.3 Classroom organisation and resources ........................................................................... 94

4.7.1.4 Teacher activity ................................................................................................................... 94

4.7.1.5 Teacher-Student interaction .............................................................................................. 95

4.7.1.6 Teacher-language ............................................................................................................... 95

4.7.2 Interviews ................................................................................................................................. 96

4.7.2.1 Open-ended interview ........................................................................................................ 96

4.7.4.3 Semi-structured interview .................................................................................................. 96

4.7.3 Questionnaires ........................................................................................................................ 98

4.8 PHASES OF DATA COLLECTION ............................................................................................. 98

Phase: 1 ............................................................................................................................................. 98

Phase: 2 ............................................................................................................................................. 98

Phase: 3 ............................................................................................................................................. 99

4.9 TRUSTWORTHINESS ................................................................................................................. 99

4.9.1 Member checking ................................................................................................................... 99

4.9.2 Triangulation ........................................................................................................................... 99

4.9.3 Peer-debriefing ..................................................................................................................... 100

4.9.4 Prolonged stay in the field ................................................................................................... 100

4.10 VALIDITY OF THE DATA ........................................................................................................ 100

4.11 RELIABILITY OF THE DATA .................................................................................................. 101

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4.12 ETHICAL CONSIDERATIONS ................................................................................................ 102

4.12.1 Informed consent ............................................................................................................... 103

4.12.2 Confidentiality ..................................................................................................................... 103

4.12.3 Data anonymity ................................................................................................................... 103

4.13 CONCLUSION ........................................................................................................................... 103

CHAPTER FIVE ..................................................................................................................................... 104

DATA PRESENTATION AND ANALYSIS ...................................................................................... 104

5.1 INTRODUCTION ......................................................................................................................... 104

5.2 PILOT STUDY .............................................................................................................................. 104

5.3 DATA ANALYSIS PROCESS .................................................................................................... 108

5.3.1 Background Characteristics of Participants ..................................................................... 110

5.4 RESULTS OF THE THEME ANALYSIS .................................................................................. 110

5.4.1 Theme 1: Importance of learning geometry .................................................................... 115

5.4.1.1 Reasons for studying geometry ...................................................................................... 115

5.4.1.2 Geometry in relation to daily lives .................................................................................. 116

5.4.2 Theme 2: Phases of the instruction in teaching similarity of triangles ........................ 119

5.4.2.1 Concepts related to similarity .......................................................................................... 119

5.4.2.2 Similarity of triangles ........................................................................................................ 122

5.4.2.3 Teaching approaches ....................................................................................................... 129

5.4.3 Theme 3: Challenges teachers faced in teaching of similar triangles .......................... 147

5.4.3.1 Mathematical knowledge challenges ............................................................................. 147

5.4.3.2 Pedagogical knowledge challenges ............................................................................... 153

5.4.3.3 Students poor background knowledge .......................................................................... 157

5.4.3.4 Resources .......................................................................................................................... 158

5.4.3.5 The Mathematics syllabus and other challenges ......................................................... 159

5.4.4 Theme 4: Suggested strategies to minimise the challenges of teaching similarity of

triangles ........................................................................................................................................... 160

5.4.4.1 Strategies to minimise the challenges ........................................................................... 161

5.5 Conclusion .................................................................................................................................... 163

CHAPTER SIX ........................................................................................................................................ 164

DISCUSSION OF FINDINGS ........................................................................................................... 164

6.1. INTRODUCTION ........................................................................................................................ 164

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6.2. DISCUSSION .............................................................................................................................. 164

6.2.1 The importance of learning geometry ............................................................................... 164

6.2.1.1 Reasons for studying geometry ...................................................................................... 164

6.2.1.2 Geometry in relation to daily lives .................................................................................. 165

6.2.2. Phases of the instruction in teaching similarity of triangles .......................................... 166


6.2.2.2. Similarity of triangles ....................................................................................................... 167


6.2.3 Challenges teachers faced in the teaching of similar triangles ..................................... 176


6.2.3.2 Pedagogical Knowledge Challenges ............................................................................. 178

6.2.3.3 Students’ poor background knowledge ......................................................................... 179

6.2.3.4 Resources .......................................................................................................................... 180

6.2.3.5 The Mathematics syllabus and other challenges ......................................................... 181

6.2.4 Suggested strategies to minimise the challenges of teaching the similarity of triangles

.......................................................................................................................................................... 182

CHAPTER SEVEN ................................................................................................................................. 184

SUMMARY, RECOMMENDATIONS AND CONCLUSION ......................................................... 184

7.1. INTRODUCTION ........................................................................................................................ 184

7.2. SUMMARY OF THE STUDY .................................................................................................... 184

7.3 SUMMARY OF THE RESEARCH METHODOLOGY ............................................................ 185

7.4 SUMMARY OF FINDINGS FROM THE STUDY .................................................................... 185

7.4.1: Challenges teachers faced in the teaching of similar triangles .................................... 185


7.4.1.2 Pedagogical knowledge challenges ............................................................................... 186

7.4.1.3 Learners’ poor background knowledge ......................................................................... 187

7.4.1.4 Resources .......................................................................................................................... 187

7.4.1.5 The mathematics syllabus and other challenges ......................................................... 188

7.4.2 The importance of learning geometry ............................................................................... 188

7.4.3 Phases of instruction in teaching similarity of triangles .................................................. 189


7.4.3.2 Similarity of triangles ........................................................................................................ 189

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7.4.3.3.1 Teacher-learner interaction in teaching similarity of triangles ................................ 191

7.5 PEDAGOGICAL APPROACHES WHICH PROMOTE MEANINGFUL TEACHING OF

SIMILARITY ........................................................................................................................................ 192

7.6 RECOMMENDATIONS .............................................................................................................. 194

7.6.1 Recommendation to the Education department and College or University ................ 194

7.6.2 Recommendation for further research .............................................................................. 194

7.7 CONCLUSION ............................................................................................................................. 195

REFERENCES ....................................................................................................................................... 196

APPENDICES ..................................................................................................................................... 220

1

CHAPTER ONE

OVERVIEW OF THE STUDY

1.1. INTRODUCTION

This chapter presents the overview of the study as it unfolded. First, an introduction to the

study is outlined, this is followed by a background to the study, and a summary of the

purpose of this study is given, as well as an account of the context in which it took place

and of its significance for the reader. An outline of the summary of relevant literature

together with the theoretical framework underpinning this study is presented. The problem

is stated together with the aims and objectives of the study. There is also a summary of

research methodology, research design together with data analysis and interpretation

used in this study. Lastly, this chapter presents the chapter outlines of the whole study

and definition of concepts.

Geometry is the most intuitive, concrete, and reality-based area in mathematics education

and according to researchers (French, 2004; Mammana and Villani, 1998), it has

developed over two thousand years. According to the National Council of Teachers of

Mathematics (NCTM), (2000), there are at least three reasons for teaching geometry.

These include the fact that geometry uniquely (i) connects mathematics with the real

physical world (ii) enables the visualization of ideas from different fields of mathematics,

while it non-uniquely (iii)provides an example of a mathematical system. Learning

geometry continues to be significant in the 21st century. This is because the advancement

of technology such as computer graphics and multimedia has greatly expanded the scope

and power of visualisation in every field to benefit from the learning of geometry (Jones,

2002). Teaching geometry well can therefore mean, enabling more students to be

successful in their entire mathematical understanding and competencies. However, the

performance of students in mathematics and geometry at all levels is weak.

In Ethiopia, geometry constitutes about 45% of the national mathematics curriculum for

primary education. Accordingly, students are supposed to do geometric problems in equal

comparisons to other components of mathematics like, algebra, arithmetic, statistics and

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trigonometry (Ministry of Education (MoE), 2009). At present, the Grade 8 mathematics

syllabus is composed of several aspects of geometric content (see Table 1.1). This study

is not concerned with the teaching of the whole geometry, but it focused on the teaching

of “Similar figures” and its related concepts in the Grade 8.

Table 1.1: The nature of mathematics content in Grade 8 mathematics syllabi in Ethiopia

Similarity and its related concepts are central components of the middle school geometry

curriculum throughout the world (Common Core State Standards for Mathematics

(CCSSM), 2010; Cox & Lo, 2012; MoE, 2009; NCTM, 2006; Seago, Jacobs, Driscoll &

Nikula, 2013).

Similarity is a visual representation of many topics throughout mathematics involving

proportional reasoning and it serves as a building block for more advanced studies in

trigonometry and calculus (Chazan, 1988; Lappan & Even, 1988). Moreover, many events

in daily life provide us experience with similar figures, for example, sun shadows, mirrors,

photos, and copying machines and while other examples can be identified throughout

physics and other sciences. This makes the learning and teaching of similarity likely to

have profound effects on students’ ability to learn a broad variety of mathematical

concepts. For example, ratio, proportion, trigonometry, projective geometry and calculus.

Thus, it is an important area to research.

The Trends in International Mathematics and Science Study (TIMSS) is an assessment

of 4th and 8thgrade students’ achievement in mathematics and science across the world.

Mathematics content domains

Total annual

periods

Percentage (%)

Unit 1 Squares, square roots, cubes and cube roots 20 12

Unit 2 Further on working with variables 25 15

Unit 3 Linear equation and inequalities 30 18

Unit 4 Similar figures 25 15

Unit 5 Circles 20 12

Unit 6 Introduction to probability 15 9

Unit 7 Geometry and measurement 30 18

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The mathematics assessment items for each grade are organised around two

dimensions: content dimension and cognitive dimension. The content domains and topic

areas within them are different for 4th and 8th grades but the cognitive domains are the

same (Chrostowski, Gonzalez, Martin & Mullis, 2003). Geometric similarity is one of the

topics amongst the mathematics assessment items for Grade 8 in the past decades in

TIMSS assessment. Ethiopia has not been amongst TIMSS participating countries in past

decades. In Table 1.2 below, the average correct percentage that students achieved in

geometry content domain items is indicated:

Table 1.2: Grade 8 students’ average correct geometry domain items in TIMSS

From the perspective table and figures, it can be observed that the highest correct

response in the geometry domain was 42% in 2003 and the lowest was 37% in 1999 and

2015. Overall students’ performance in TIMSS from 1999 to 2015 in the geometry domain

in Grade 8 appears to be below the average 50% across the world. It seems that the

geometric skills of students in Grade 8 have not been adequately developed. This may

indicate that there is a challenge in the learning and teaching of geometry.

Portnoy, Grundmeier, and Graham (2006) argue that students’ geometric misconceptions

and low academic achievement in geometry are due to teachers limited geometrical

knowledge. In addition, studies (Hill, Rowan& Ball, 2005) indicate that teachers have a

key influence on students’ learning and achievement through aspects such as teachers’

mathematical knowledge, pedagogical content knowledge (PCK), teaching beliefs about

mathematics and instructional practices. Teachers’ insufficient mathematical knowledge

and pedagogical knowledge, impede their abilities to use curriculum materials effectively,

interpret and respond to students’ work, choose correct representations, tools, and

Year Content Domain International average corrects

2015 Geometric shapes 37



2003 Identify similar triangles 42

1999 Similar triangles 37

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reinforcement activities within a lesson. On the other hand, students need to understand

middle school concepts in geometry to have success in high mathematics. For students

to get this understanding, their teachers need to have strong mathematical knowledge

and PCK in geometry.

Lobato and Ellis (2010) indicate that the learning and teaching of similarity is a problem

for both learners and teachers. For example, similarity items of dimensional growth or

reduction (from a smaller shape to a larger shape or vice versa) appear to be the most

difficult for students. In addition, Seago, Jacobs, Heck, Nelson and Malzahn, (2014)

indicate that USA middle school teachers performed poorly on similarity items on

geometry and faced challenges on teaching similarity in the classroom. Some of the

challenges teachers faced include lack of experience, mathematical knowledge,

pedagogical content knowledge and professional development necessary to improve

students’ learning. In Ethiopia, currently, there are very few studies addressing teachers’

challenges in the teaching of geometry and particularly geometric similarity and ideas on

solutions for challenges in the teaching of similarity at all levels. Thus, this study will

explore the challenges teachers face in the teaching similarity of triangles and its

approach in 8th-grade primary schools.

1.2. BACKGROUND TO THE STUDY

This section describes the context in which this study has been conducted. Southern

Nations, Nationalities and Peoples Regional State (SNNPRS) is one of the regional states

in the Federal Democratic Republic of Ethiopia. It shares borders with Kenya in the south;

Oromia Regional State in the north, southeast and northwest; South Sudan in the

southwest; the Sidama Regional State to the east and Gambella Regional State in the

northwest. Administratively, the region is divided into 16 zones, 7 special districts, 152

districts and 74 town administrations. There are 4,370 Kebele Administration councils

(3853 rural localities and 517 urban localities) that represent the lowest administrative

entities (SNNPR Education Bureau, 2020). Areka Town is one of the town administrations

in the SNNPR State. The capital city of the region is Hawassa, and it is a fast-growing

city both in terms of size and economic activity. Among the regions comprising the

Ethiopian Federation, SNNPR holds several nations nationalities and peoples who speak

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their own language, their own culture, values, and beliefs. There are more than 55 ethnic

groups comprising about 60 indigenous languages. According to the regional report in

2020, SNNPR had a total of 5917 primary schools (Grades 1-8). Currently, in Ethiopia,

primary education is done through an eight-year course leading to the award of Primary

Education Completion Regional Examination (PECRE) and is split into primary first cycle

(Grades 1-6), and primary second cycle (Grades 7-8). The secondary education is done

a four-year course (Grades 9-12). Students are streamed either into university education

or into Technical and Vocational Education and Training (TVET), based on the

performance of secondary education completion certificate examination.

Mathematics is one of the core subjects studied by all students till the tertiary levels of

education in Ethiopian. In primary school mathematics, students learn to observe, depict,

and investigate patterns and correlations in social and physical occurrences, as well as

between mathematical objects. This knowledge, combined with a comprehension of the

subject, influences decision-making in all aspects of our life (Eshetu, Dilamo, Tsefaye &

Zinabu, 2009; MoE, 2009).

In the SNNPRS there is a noticeable continuous poor achievement of students in

mathematics. Mathematics has remained the only subject in the PECRE where more than

half of the students are unable to achieve 50%, which is the minimum score to move from

one level to the other. In Table 1.3 below, the average regional correct percentages that

students achieved in mathematics are indicated:

Table 1.3: PECRE students’ Mathematics result

Year % of students that scored 50% and above

2020

2019

2018

2017

2016

34

30

32.27

7.3

7.25

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Overall, the mathematics result in PECRE from 2016 to 2020 shows an upward trend in

students’ achievement. From the perspective table, there is an upward trend from 7.25%

in 2016 to 34% in 2020. Overall, students’ mathematics achievement is below 50%. In

addition to this, findings from of Regional Learning Assessment (RLA) carried out in 2014

at Grade 8 in SNNRP show that only 6.42% of students scored 50% and above in

mathematics. Studies (RLA, 2014; Strengthening Mathematics and Science Education in

Ethiopia (SMASEE), 2014) indicate that some variables responsible for the inadequate

performance of students in mathematics are (i) most of the teachers appear to lack

competence to teach, (ii) teachers’ insufficient level of teaching skills, (iii) absence of

active learning, (iv) unqualified teachers in the system, and (v) traditional mode of

teaching in Ethiopian schools. Accordingly, teachers faced a challenge to teach

mathematics.

1.2.1 Geometry

According to NCTM (2000), students can identify, describe, compare, and classify

geometric shapes. According to Toptas (2007), by building, drawing, measuring,

visualizing, comparing, and classifying, learners develop spatial intuition and uncover

correlations between geometric shapes. Similarly, geometry is a natural area in which

students' reasoning, judgment skills and proving geometric theorems develop (Ersoy,

2003). For this reason, geometry is one of the most important subjects in the mathematics

curriculum.

In Ethiopia, geometry is included in early grades as one of the five stands of mathematics.

The geometric content constitutes about 40%of the national mathematics curriculum for

primary education and categorised in two sections according to students' psychological

and cognitive development (see Table 2.1). The geometric curriculum in the first cycle of

primary education is based on the realistic or practical approach and includes the study

of properties and representation of 2-dimensional shapes and 3-dimensional objects and

the part they play in their everyday life. Students at primary second cycle can reason with

and are familiar with properties of geometry through division, connection deformation of

shapes, and describe the forming and unfolding of 3-dimensional figures. Second cycle

primary school provides a vital step in the learning process of geometry when students

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begin to develop more sophisticated ideas of shape and learn to reason about

geometrical concepts in terms of short-chain of deduction (MoE, 2009). It is significantly

important to develop geometric reasoning at the middle school level to bring the students

into their comfort sector for secondary level.

Similarity is among the content which has been studied since the beginning of geometry

and it is as old as geometry itself. Similarity provides a way for students to connect spatial

and numeric reasoning and provides the basis for advanced mathematical topics such as

projective geometry, calculus, slope, and trigonometric ratio (Chazan, 1988; Lappan &

Even, 1988). The application of similarity includes surveying, as well as map and model

making.

The similarity between two shapes in the curriculum is introduced for the first time to the

8th grade primary school in Ethiopia; the teaching at 8th grade mainly emphasises the

comprehension of the concept of similar plane figures, similar triangles, learning the

properties of similar shapes such that students can use these properties to solve real-life

problems (MoE, 2009).

1.2.2 Studies on Similarity

Similarity and its related concepts are central components of geometry. It is an important

spatial-sense, geometrical concept that can facilitate students’ understanding of indirect

measurement and proportional reasoning. While many see geometry as a significant

subject in mathematics and similarity is a key concept within geometry, there is very little

research done on teachers’ challenges of teaching similarity. Many of the research

studies that investigated learning similarity focused on school-age children.

Studies (Chazen, 1987; Lamon, 1993; Hart, 1998; Lobato & Ellis, 2010; Swoboda & Tocki,

2002; Martin et al. 2003) indicate that students have difficulty in understanding similarity.

Chazen (1987) identified three difficulties for students in learning similarity. These include,

(i) notations of similarity, (ii) proportional reasoning, and (iii) dimensional growth

relationships. He also indicated that the use of the term ‘similar’ might deceive students

who have strong allied images with the term usual in a non-mathematical context. For

example, some students might think that all triangles are similar because they are

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generally alike, and all have three angles and three sides. In geometry, all triangles are

not similar. Studies (Lehrer, Strom & Confrey; 2002; Swoboda & Tocki, 2002) indicate

that one of the methods to elucidate these circumstances to learners, treat “similarity” as

a unique technique of classifying shapes. For example, if a dilated or scaled image of one

figure can be rotated, and/or translated to exactly match another figure, the two figures

are similar. Similar figures are distinguished from each other by a change of scale.

Davis (2003) asserts that the nature of pre-service teachers’ ideas of “what and how to

teach” similarity (p. 34) (i) focus on procedural generalisations conveying meanings (ii)

should contend with prospective teachers’ backgrounds and belief structures (iii)

substantive and syntactic knowledge of prospective teachers are limited and fragmented.

He argues that there are areas of promise in prospective teachers’ subject matter

knowledge of similarity and areas that need attention. This study focuses on in-service

teachers' challenges of teaching similarity and its related concept at 8 grade level in

Ethiopian primary schools.

Seago, Jacobs, Heck, Nelson and Malzahn (2014) assert that the Learning and Teaching

Geometry (LTG) professional development designed to impact middle school teachers’

mathematical knowledge in the domain of similarity and geometric transformation, made

the intended teacher knowledge outcomes, including the gain in geometry content

knowledge along with the knowledge to apply the understanding about content in

mathematics instruction. Accordingly, another take-away message is that there is still

important work that remains to be carried out in order to support a transformation-based

understanding of similarity in line with the CCSS. Baseline data from our pre-assessment

of both teachers and students showed that they did not have a strong grasp of this content

area. The teachers, on average, solved about 65% of the items on the pre-assessment

correctly. Those who took part in the PD saw, on average, a gain of approximately 10

percentage points.

Studies directed towards teaching similarity stress several important reasons for the study

of this topic. Clements and Battista (1992, p. 157) suggest that to improve the instructional

practices of geometry mathematics educators need teaching-learning research that leads

students to the construction of robust concepts through a meaningful synthesis of

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diagrams and visual images on the one hand, and through verbal definitions and analyses

on the other. Within this context, the current national interest of the Ethiopian government

is improving the instruction of mathematics in schools by developing a new pedagogical

approach and hence, exploring teachers’ challenges in teaching geometry, for which

similarity is a part that deserves more attention (MoE, 2009; SNNPR Education Burea,

2014).

The researcher found very few studies addressing the challenges experienced whilst

teaching similarity at all educational levels in Ethiopia. Much the same thing can be said

for studies with solutions for such challenges. Thus, this study focuses on teachers'

challenges in teaching the similarity of triangles closely related to teachers’ interaction

ability with learners; it identifies pedagogical approaches that can promote meaningful

teaching of the similarity of triangles and designs a strategy to minimise the challenges

of teaching the similarity of triangles.

1.2.3 Purpose of the study

The purpose of this study was to explore teachers’ challenges in the teaching of similar

triangles and their approaches in the 8th-grade primary school. The learning-teaching of

similarity are a problem for students and teachers. The focus on similarity is motivated by

that (i) similarity connects a wide variety of critical mathematical topics, such as

proportional reasoning, scale factor, linear functions, modelling and transformations, and

(ii) the poor achievement of students in mathematics is a challenge that exists and

improving the instruction of mathematics to increase students’ achievement in

mathematics is necessary.

Teachers often lack pedagogical content knowledge, mathematical fluency to make

instructional decisions and professional development necessary to improve students’

learning of similarity (Seago et al., 2014). Helping learners to understand similarity means

teachers must be able to engage them in geometric thinking and encourage them to apply

geometric similarity to solve problems, as opposed to relying on memorisation of the

concepts. Thus, this study is important since it tried to investigate the challenges teachers

faced in the teaching of geometry in particular similarity and identified pedagogical

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approaches that could promote meaningful teaching of similarity in primary schools in

Ethiopia.

1.3 THEORETICAL FRAMEWORK

The study was based on the fundamental theories in the teaching of geometry. The

literature in mathematics education reveals that studies in the school geometry education

context mainly refer to the three theories in the teaching and learning of geometry. Those

theories and models include van Hieles’ (1985) theory, the theory of figural concepts by

Fischbein (1993), and Duval’s (1995) theory of figural apprehension. There is a brief

discussion in the next section that covers the theoretical framework.

According to van Hieles’ theory, teachers should systematise their teaching in five

different phases when they teach geometry, following the phases:

1. Information: The teacher and learners engage in conversations and geometric

activities that determine what previous knowledge their learners have on geometric

concepts.

2. Directed Orientation: Teachers should purposely organise sequential activities for

geometric problems and direct learners to explore the uniqueness of each geometry

topic through hands-on manipulations.

3. Explication: Learners should be aroused to communicate and partake their

thoughts with the teacher and classmates by using appropriate mathematical

language.

4. Free Orientation: Teachers provide geometric problems for learners that can be

solved in numerous ways and encourage them to think and solve the problems.

5. Integration: Teachers should help learners to discuss, what they have observed,

manipulated, and solved, and connect the crucial geometric concepts.

This is a meaningful approach and teachers should be aware of the important

pedagogical areas of concern such as the ways of teaching, organisation of instruction,

content and material used in the teaching of geometry, for which similarity is a part. The

development of students’ geometric thinking levels is discussed in Chapter 3 that covers

the theoretical framework.

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1.4 LITERATURE REVIEW

The next chapter reviews the relevant literature on the history of geometry, the importance

of learning geometry, the concept of geometric similarity, teaching geometry, the role of

teachers in teaching geometry, geometry classroom, classroom interactions, teaching

geometry through technology, challenges of teaching geometry, and strategies to

minimise those challenges. In this chapter, only a summary of the literature reviewed is

provided.

According to Jones (2002), as one of the oldest disciplines, the learning of geometry is

an important aspect of developing intuition in mathematics. In this regard, it helps to

develop students spatial reasoning, visualising skills, and to derive conclusion by

deduction (Jones, 2002). Similarity and its related concepts are central components of

geometry. It is an important spatial-sense, a geometrical concept that can facilitate

students’ understanding of indirect measurement and proportional reasoning.

According to Fujita and Jones (2002), the pedagogical approaches used in the teaching

geometry continues as a basic challenge in mathematics education. For example,

according to the Royal Society's study on geometry teaching (2001), “the most significant

contribution to improvements in geometry teaching will be made by the development of

effective pedagogy models, which will be supported by well-designed activities and

materials” (p.30). This means that the current pedagogies emphasise on memorisation of

geometrical concepts and learners lack experience to develop a conceptual

understanding of geometry.

1.5 STATEMENT OF THE RESEARCH PROBLEM

The poor achievement of learners in geometry is a challenge that exists and it is

necessary to improve the instruction of geometry in schools by developing a new

pedagogical approach. Currently, in Ethiopia learners are not learning similarity as they

need or are expected to. There is little evidence documented that teacher apply active

teaching methods as suggested in the mathematics syllabus in Ethiopia. This could be

an existing scenario since most teachers lack experience, pedagogical content

knowledge, and mathematical fluency to teach similarity. They are also not successful in

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engaging their learners or using appropriate geometry instruction. Rather the emphasis

is on how much the learners can remember and less on how well they can apply their

thinking.

Euclidean geometry constitutes 45% of the Grade 8 curriculum and the bulk of it is

similarity. Currently, there are very few studies addressing teachers’ challenges in the

teaching of similarity and solutions for the challenges in the teaching of similarity in

Ethiopia. Due to this, the researcher was motivated to explore the challenges of teaching

similarity. Thus, this study explores the challenges teachers faced when teaching

similarity of triangles and suggests a strategy to minimise the challenges faced by

mathematics teachers to promote meaningful teaching of similarity of triangles.

1.6 THE AIM OF THE STUDY

The main aim of this study was to explore how the challenges of teaching similarity of

triangles in Grade 8 class can be minimised.

1.7 THE OBJECTIVES OF THE STUDY

To unpack the challenges that teachers encounter in teaching similarity of triangles in

Grade 8, the following objectives were suggested. The study seeks to:

1. Identify teachers’ challenges in the teaching of similar triangles.

2. Determine the teacher’s interaction ability to promote meaningful teaching of the

similarity of triangles.

3. Identify pedagogical approaches that can promote meaningful teaching of the


4. Suggest strategies that can be applied to minimise the challenges of teaching the


1.8 THE RESEARCH QUESTIONS

To address the objectives stated above, the following main research question and sub-

questions were set.

Main research question: How can the challenges of teaching similarity of triangles to

Grade 8 students be minimised?

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Sub-questions:

What are the challenges faced by mathematics teachers in teaching similarity of

triangles?

How do teachers interact with learners in the teaching of similarity of triangles?

Which pedagogical approaches can promote meaningful teaching of similarity of

triangles?

How can the strategies be applied such that the challenges in teaching similarity

of triangles are minimised?

1.9 SIGNIFICANCE OF THE STUDY

A need for different pedagogical approaches in geometry instruction to promote

meaningful teaching of geometry and increasing geometry achievement of students has

been realised by mathematics researchers (Choo, Rafi, Mohamed, Hoon & Anuar 2009;

Besana et al., 2002; Eshetu, Dilamo, Tsfaye & Zinabu, 2009; Fujita & Jones, 2001; Jones,

Fujita & Ding, 2006; MoE, 2009; NCTM, 2000; SNNPR Education Burea, 2014; Royal

Society, 2001). Therefore, this study is hoped to enhance the instruction in the teaching

of geometry particularly in the similarity of triangles which in turn may increase successful

progress of the teaching-learning of geometry, and consequently improve the

achievement of mathematics. Furthermore, this study adds value to mathematics

education community since it provides them with compiled data about teachers’

challenges of teaching similarity of triangles. In addition, the study provides an alternative

pedagogical approach for the teaching of similar triangles and indicates areas for further

study.

1.10 DELIMITATIONS OF THE STUDY

The study was confined to the government primary schools in Areka Town SNNPR state

of Ethiopia. This study aimed to focus on Grade 8 mathematics teachers challenges of

teaching similarity of triangles. Only mathematics teachers in Grades 8 were studied. This

study was limited to the challenges teachers faced in teaching similarity of triangles.

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1.11 RESEARCH METHODOLOGY AND DESIGN

A summary of the research paradigm, methodology and design is provided in this section.

In Chapter 4, they will be presented and discussed in detail.

1.11.1 Research paradigm

Creswell (2014) considers “the philosophical assumptions (ontology, epistemology,

axiology and methodology) as key premises that are folded into interpretative frameworks

used in qualitative research” (p.22-23). Thus, an interpretive social constructivist

paradigm is used as the philosophical framework for this study.

1.11.2 Research methodology

A qualitative approach was used to investigate the challenges of teaching similarity of

triangles. According to Harwell (2011), a qualitative approach focuses “on discovering

and understanding the experiences, perspectives, and thoughts of participants through

various strategies of inquiry”(p. 56). In addition, Creswell (2010) asserts that “a qualitative

method involves detailed exploration with a few cases or individuals rather than to search

for causal relationships” (p. 156). In this study, a qualitative approach assisted the

researcher to explore the challenges teachers faced in the teaching of Grade 8 similarity

of triangles.

1.11.3 Research Design

According to Mouton (2006), design refers to a blueprint for how researchers will perform

their research. According to the author, a research design “provides the structure for data

collection and analysis, as well as the procedures to be followed” (p.55).

According to Yin (2003), an exploratory case study is a research design used to explore

a contemporary phenomenon that is inseparable from the context in which it exists. The

researcher aimed to explore the challenges of teaching similarity of triangles in Grade 8

and how they can be minimised. Thus, an exploratory case study design was chosen as

the research design. In Chapter 4, the research design and methodology for this study

was discussed in depth.

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1.12 POPULATION AND SAMPLING TECHNIQUES

Areka is in the Northern part of Wolaita Zone in SNNPRS of Ethiopia. According to the

statistical information obtained from Areka Town Administration Education Office, there

are 9 primary schools (5 government and 4 non-government schools). Each school had

one Grade 8 mathematics teacher. Thus, the population of this study constitutes 9 Grade

8 mathematics teachers.

To obtain accurate and valid data, purposive sampling technique was used because it

provided accurate representation of the study population. According to Patton (1990),

“the logic and power of purposeful sampling lies in selecting in formation-rich cases for

study in depth. Information-rich cases are those from which one can learn a great deal

about issues of central importance to the purpose of the research, thus the term

purposeful sampling can be used for obtaining accurate representation of the target

population” (p. 169). The sample in this study composed of 5 Grade 8 mathematics

teachers from 5 government primary schools purposely selected.

1.12.1 Participants

Since each school only had one Grade 8 mathematics teacher, one teacher was chosen

per school. The focus was on the challenges teachers faced on the teaching of similarity

of triangles, teacher-student interaction, and pedagogical approaches they used in

teaching the similarity of triangles.

1.13 DATA SOURCES AND COLLECTION TECHNIQUES

Data sources such as observations, semi-interviews and questionnaires were utilised in

the collection of data used in answering the research questions. In qualitative research,

the investigator is the primary data collection instrument (Creswell, 2010). The

investigator identified and collected data from sources in order to carry out this research

inquiry.

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1.13.1 Observation

Observations involve collecting qualitative information about human actions and

behaviours in social activities and events in a real social environment, such as classroom

teaching and learning (Cohen, Manion & Morrison, 2011). Participant observation and

non-participant observation are the two basic observation techniques (Cohen, Manion &

Morrison, 2011; Creswell, 2010). Participant observation is when the researcher becomes

part of the group under study and participates in everyday social activities of that social

system to obtain the actual feelings and experiences of the phenomena, while at the same

time taking notes of the actions and behaviours of the participants (Cohen, Manion &

Morrison, 2011). In contrast, a non-participant observation technique involves the

researcher sitting or standing on the side while social activities like teaching and learning

are taking place, in and out of the classrooms (Cohen et al., 2011).

During data collection, the researcher functioned as a non-participant observer. The

classroom observation was conducted to get a clear picture of how teachers interact with

learners in the teaching of similarity and what pedagogical approaches teachers were

used in the teaching of similarity. The classroom events were video-recorded and

subsequently transcribed. The use of a video camera provides detailed and accurate

information about the instructional sessions.

1.13.2 Semi-structured interview

Data on the challenges faced by mathematics in teaching similarity of triangle was

gathered through semi-structured interviews. The inquest was conducted in the form of

an interactive and in-depth study that used one-on-one procedures to collect data from

participants in each school classroom (McMillan & Schumacher, 2010, p.360). The

interview held with grade mathematics teachers were audio recorded.

1.13.3 Questionnaire

In this qualitative exploratory study, open-ended questionnaires were employed to

triangulate data obtained from classroom observations and interviews. The

questionnaires in this study comprised 14 open and closed-ended items. It consisted of

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two parts; the first part about demographic information of the respondent, had three

questions. The second part addressed the teachers' challenges, reviews, and reflections

of teaching similarity of triangles; it consisted of 11 questions.

1.14 DATA ANALYSIS AND INTERPRETATION

According to Hatch (2002), analysis means “organizing and interrogating data in ways

that allow researchers to see patterns, identify themes, discover relationships, develop

explanations, make interpretations, mount critiques, or generate theories” (p. 148).

Furthermore, Miles, Huberman, and Saldana (2014) describe qualitative data analysis as

“three concurrent flows of activity: (i) data condensation, (ii) data display, and (iii)

conclusion drawing/verification” (p. 12). In this research, prior to analysis, preparation,

and transcription of data from the interview audiotapes, lesson observation frameworks

and the episodes in the videos were done.

The researcher used coding to condense a large volume of data into manageable units

during data analysis. According to Elliot (2018), researchers use a coding process to

develop new categories and themes from the data collected. It is here that irrelevant

information is discarded and set aside for future use if the researcher has to re-examine

data previously deemed useless. Researchers use coding to categorise data relevant to

a theme instead of following the sequences in which the participants responded to

research instruments. Elliot (2018) also emphasised the importance of coding as it assists

the researchers to source meaning that speaks to the category of research questions.

Figure 5.1 summarizes the coding procedure used in this research.

In this study, the coded data were categorised into 4 themes. These are:(i) importance

of learning geometry, (ii) phases of instruction in teaching similarity of triangles, (iii)

challenges teachers faced in teaching similarity of triangles, and (iv) suggested strategies

to minimise the challenges of teaching similarity of triangles.

1.15 VALIDITY AND RELIABILITY

Validity and reliability are key aspects of all research. According to Creswell (2010), if an

instrument measures what it claims to measure, it is considered as valid. In qualitative

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research, they are seen in terms of increasing the re-applicability of research design and

the verifiability of research outcomes (Scott & Morrison, 2006). However, in qualitative

research, where the focus for investigation is one or a small number of cases, the

application of such measures is difficult (Scott & Morrison, 2006). This implies that the

researcher should be aware that, the strategies used to address validity and reliability in

qualitative research are not the same as in quantitative research. Moreover, there are

different techniques for validation in qualitative research.

Triangulation is one of the techniques to validate data in qualitative research (Creswell,

2010). As stated by Scott and Morrison (2006), triangulation means collecting study data

from multiple sources or the uses of more than one data collection instruments. In line to

this, observation, interviews, and questionnaires were utilized to triangulate the data

collected.

According to Creswell (2010), reliability is defined as the capacity to use the same

instrument at different periods and produce consistent findings. To ensure the reliability

of the observation protocol, semi-structured interviews, and questionnaires were piloted

with Grade 8 mathematics teachers in a government primary school not participating in

this research.

1.16 PILOT STUDY

A pilot study was conducted at a selected government primary school before the actual

data collection. The school look like the schools employed in the research in terms of

location and the challenges teachers faced in the teaching of similarity in geometry. The

objective of the pilot study was to see whether the instruments were reliable and get more

understanding on the challenges teachers faced in the teaching similarity of triangles.

The purpose of the pilot study was to see if participants could correctly address the given

interview questions and for enabling an investigator to rephrase them as needed, as well

as to identify if the participants required additional clarification. The pilot study's findings

revealed which questions were unclear, and these were paraphrased so that participants

could respond appropriately.

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1.17 ETHICAL PROCEDURES

In compliance with the Unisa research ethics policy, the researcher sought and obtained

informed permission from the Areka Town education office, the school principals, and the

participant teachers to conduct the research. The researcher also obtained consent from

Grade 8 mathematics teachers who were taking part in the study. The participants were

informed of the confidential nature of the research, that participation was voluntary, and

the participants had an option of not participating in the research and could leave the

study at any time and not be penalized. After this, an ethical clearance certificate was

granted to the investigator by Unisa’s Ethics Committee to conduct the research.

1.18 CHAPTER OUTLINE

There are seven chapters in the research report:

Chapter 1: Overview of the Study

In this chapter, an introduction to the study is outlined, this is followed by a background

to the study, and a summary of the purpose of this study is given, as well as an account

of the context in which it took place and of its significance for the reader. An outline of the

summary of relevant literature together with the theoretical framework underpinning this

study is presented. The problem is stated together with the aims and objectives of the

study. There is also a summary of research methodology, research design together with

data analysis and interpretation used in this study.

Chapter 2: Literature Review

Chapter 2 presents a detailed review of the history of geometry, the importance of learning

geometry, the concept of geometric similarity, teaching geometry, the role of teachers in

teaching geometry, geometry classroom, classroom interactions, teaching geometry

through technology, challenges of teaching geometry, and strategies to minimise those

challenges.

Chapter 3: Theoretical Framework

Chapter 3 was dedicated to the theoretical framework, van Hieles’ (1985) theory, the

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theory of figural concepts by Fischbein (1993), and Duval’s (1995) theory of figural

apprehension.

Chapter 4: Research Design and Methodology

This chapter outlines how the qualitative research was designed and conducted. It also

presents the research paradigm, design, methodology, population and sample, data

sources and collection techniques, and phases of data collection. Further, Chapter 4

provides how trustworthiness, validity, reliability, and ethical considerations of results

were ensured in this study.

Chapter 5: Data Presentation and Analysis

This chapter aimed to present, and interpretation of the data collected to answer the

research questions. The first part of this chapter presents how the pilot research was

carried out and the second section presents data analysis and interpretation.

Chapter 6: Discussion of Findings

This chapter provide a discussion of the main findings concerning literature reviewed

together with the theoretical framework lens and a phase of instruction or model

suggested for teaching the similarity of triangles.

Chapter 7: Summary, Recommendations and Conclusions

This chapter summarises the findings, conclusions and recommendations of the study

based on the data collected and analysed. Further, Chapter 7 presents the outlines on

how the proposed model assisted the research participants and recommendations for

future studies. It also includes conclusions which are important to further researchers in

the teaching and learning of geometry as well.

1.19 DEFINITIONS OF CONCEPTS

The operational definitions of concepts used in this research are provided in this

section.

Challenge: Any difficulty teachers faced in the teaching of geometry.

21

Classroom interaction: Is the interaction between the participants of a classroom,

among a teacher and the learners or amid the learners themselves

Geometry: The branch of mathematics that deals with the position, size, and shape of

figures.

Pedagogical approach: This refers to certain strategies of instruction, or the strategies

used in the process of teaching geometry.

A meaningful geometry teaching-learning refers to providing an activity that offers an

opportunity for learners to connect geometry to their experiences and has a goal to

connect geometry to further study.

Similar figures: Figures that have the same shape but not necessarily the same size.

Teaching geometry: Knowing how to recognize intriguing geometrical problems,

respecting the historical evolution and cultural context of geometry, and comprehending

the various and different uses to which geometry is applied are all essential skills for

effective geometry instruction.

2.20 Summary

This chapter presented a summary of the study by presenting and discussing aspects

such as the study's background and overview, purpose, statement of the problem, aims

and objectives of the study. The next chapter presents a review of related literature.

22

CHAPTER TWO

LITERATURE REVIEW

In the previous chapter, the overview of the study presented. This chapter presents the

literature reviewed regarding the teaching of similarity of triangles in primary schools. The

chapter discusses the history of geometry, the importance of learning geometry, the

concept of geometric similarity, teaching geometry, the role of teachers in teaching

geometry, geometry classroom, classroom interactions, teaching geometry through

technology, challenges of teaching geometry, and strategies to minimise those

challenges. The chapter concludes by outlining the summary of the chapter.

2.1 THE HISTORY OF GEOMETRY

According to Greenberg (1973, p.6), the word “Geometry” comes from the Greek

geometrein. It was originally the science of measuring land. The Greek historian

Herodotus (5th century B.C.) credits Egyptian surveyors with having originated the

subject of geometry, but other ancient Mesopotamian, Babylonian, Hindu, and Chinese

civilisations also contributed to the origins of geometric information. Greenberg (1973)

further asserts that the motivation for the development of Egyptian geometry was the

desire for quick and accurate methods for surveying the farmers’ fields. In response to

those simple demands, the Egyptians then developed a simple geometry of mensuration,

the part of the geometry that consists of the techniques and concepts involved in

measurement. For example, the Egyptians had the approximation �~ ��

��

�

~3.1604.

They found the correct formula for the volume of a frustum of a square pyramid, a

remarkable accomplishment. A frustum is a portion of solid (a cone or pyramid) that lies

between one or two parallel planes cutting it. It is formed by a clipped pyramid; frustum

culling is a method of hidden surface determination. Egyptian geometry was not a science

in the Greek sense. Egyptian mathematics had no structure to their geometry, just a

collection of rules and solutions aimed at specific circumstances. They did not use

deductive reasoning to uncover geometric techniques from the first principles. Instead,

they used trial and error and, if a solution was not readily available, used trial and error to

arrive at an approximation. As the similarity of triangles is part of geometric content,

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teachers must be aware of the historical development of the content for meaningful

teaching of the concept.

The Mesopotamians had a much deeper understanding of numbers and of the techniques

of computation than the Egyptians. Therefore, they developed approximations of

solutions that were far more accurate than those of their Egyptian counterparts. This is

especially true in algebra, but it is also true that some of the geometric problems that they

solved were more advanced than those studied in Egypt. For example, they calculated

the sides of the right-angled triangle by using the Pythagorean theorem. The Pythagorean

theorem states that, “in a right triangle the square of the length of the hypotenuse equals

the sum of the squares of the two remaining sides” (MoE, 2009, p.251). The

Mesopotamians understood the Pythagorean theorem at a much deeper level and could

solve a variety of problems associated with it. Ancient geometry was an empirical subject

in which approximate answers were usually enough for practical purposes. For example,

the Mesopotamian and Egyptian computed the volume of an object that had the shape of

a city wall a three-dimensional (3D) form with straight sides that is thicker at the bottom

than at the top, but their emphasis was on the mud-brick wall, not the abstract form. Thus,

ancient geometry was a collection of rules of thumb procedures arrived at through

experimentation, observation of analogies, guessing, and occasional flashes of intuition.

However, now in the 21st century, experimental geometric concepts are taught at

foundation levels school mathematics curriculum and geometric problems solved by

deductive reasoning using axioms, postulates and theorems done at later higher levels.

Geometry continues to change and evolve.

The Greek approach to mathematics was different. It was more abstract and less

computational. Greek mathematicians investigated the properties of classes of geometric

objects. Geometric shapes were sorted into classes according to defined geometrical

properties, such as number and relationship of sides, shapes of faces, and surfaces,

including, equal and parallel sides; nature of angles, four right angles of a rectangle.

Nowhere is this emphasis more easily seen than in the work of the Greek philosopher

and mathematician Thales of Miletus (650- 546 B.C). For example, Thales suggested a

geometric shape of a triangle, a plane figure with three straight sides and three angles.

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Consequently, based on the angle measurement, triangles are classified into three types;

acute, a triangle with all three angles less than 90o; right, a triangle that has one angle

that measures exactly 90o; and obtuse, a triangle that has one angle measure more than

90o. Moreover, based on the measure of the length of their sides, triangles are classified

into three types; scalene a triangle that has all three sides of different length; isosceles,

a triangle that has two sides of the same length and the third side of a different length and

equilateral, a triangle which has all the three sides of the same length.

Thales of Miletus insisted that geometric statements be established by deductive

reasoning rather than by trial and error. Deductive reasoning is the process of reasoning

from general principles to specific instances. Because the world of mathematics is all

about facts, deductive reasoning is relied on instead of inductive reasoning to produce a

correct conclusion about mathematical concepts. This is significant for this study

because, on the teaching of similar triangles, it is necessary to standardise the

understanding within the class by stating a definition of the similarity of two triangles. All

mathematicians today work by beginning with known principles and then deriving new

facts as logical consequences of those principles, but Thales was the first to apply this

method rigorously. When proving the similarity of triangles students used the deductive

approach after they had learned the definition of similar triangles; the definition of

similarity of triangles will be discussed in Section 2.3. For example, to test the similarity

of two triangles, students should use the Angle-Angle (AA) similarity theorem and prove

their similarity by deductive reasoning. The AA-similarity theorem states that, “if two

angles of one triangle are congruent to the corresponding two angles of another triangle,

then the two triangles are similar” (MoE, 234). Here it is sufficient to show the two angles

of a triangle are congruent with corresponding angles. The other definition of similarity of

triangles will be discussed in Section 2.3. Although the Greek approach to mathematics

was deductive, logical, and, in many ways, very modern, the way that the Greeks

expressed their results was different from what most of us are accustomed to today.

The systematic foundation of plane geometry by the Pythagorean school was brought to

a conclusion around 400 B.C. Euclid is one of the best-known mathematicians in history,

or to be more precise, Euclid has one of the best-known names in the history of

25

mathematics. Euclid was a disciple of the Pythagorean school. Around 300 B.C. he

produced the definitive treatment of Greek geometry and number theory in his 13-volume

Elements. The first book is an introduction to the fundamentals of geometry and the

remaining 12 volumes survey many of the ideas that were most important to the

mathematicians of the time. Element is a remarkable textbook that is still worth reading.

For example, Ethiopian schools still use Euclid’s work as a textbook, and even today most

plane geometry textbooks are modelled on parts of the Elements. Euclid begins the very

first section of the first book of the Elements with a long list of definitions, a sort of

mathematical glossary, and then follows this list with a shortlist of axioms and postulates.

Euclid places the axioms and postulates at the beginning of his work because they are

so important. The axioms and postulates are the basic building blocks of his geometry.

An axiom is a “self-evident proposition, requiring no formal demonstration to prove its

truth, but received and assented to as soon as mentioned” (Hutton, 2012, p. 3). A

postulate is a claim to take for granted the possibility of simple operation. For example, a

straight line can be drawn between any two points. A postulate is a simple problem of

self-evident nature, distinguished from the axiom. Euclid made a distinction between the

axioms, which he believed were obvious and universally applicable, and the postulates,

which were narrower in scope. Both the axioms and the postulates served the same

function (Martin & Stutchens, 2000).

Euclid’s approach to geometry has dominated the teaching of the subject for over two

thousand years. Moreover, the axiomatic method used by Euclid is the prototype for all

of what we now call "pure mathematics." His method consists of assuming a small set of

intuitively appealing axioms and deducing many other propositions and theorems from

these. Today, mathematicians tend to use the word geometry to describe any system of

deductive knowledge that is concerned with relationships between points, lines, planes,

and other geometric objects.

The similarity of triangles is one of the concepts analysed under Euclidean geometry. The

similarity in triangles involves a comparison of sides and angles. All the corresponding

sides are proportional, and the corresponding angles are congruent in two similar

triangles. Similar triangles are identical in shape but not necessarily in size. When

26

teachers deal with the similarity of triangles in Grade 8, they should start the lesson by

revising the congruency of figures. Two figures are congruent if they have the same size

and shape. The teachers should provide the experiences for the learners in the process

of developing an understanding of similarity that could lead to other definitions of

similarity.

2.2 THE IMPORTANCE OF LEARNING GEOMETRY

The importance of learning and teaching geometry is extensively documented in the

literature, and it is emphasized in modern mathematics curriculum not just as a

standalone mathematical topic but also as a way of developing other mathematical

concepts (Gagatsis, Sriraman, Elia & Modestou, 2006; Kurina, 2003; Clements, Sarama

& Wilson, 2004). Learners should learn more about geometric shapes and structures, as

well as how to analyze their properties and relationships, through the study of geometry

(NCTM, 2000). They should also progress from recognising distinct geometric shapes to

geometry reasoning and problem solving (Daher & Jaber, 2010). In particular, the

Ethiopian education curriculum prescribes that the students should learn the geometric

concepts of similar figures, circles, and measurements. The similar figures content

occupies 15% of the Grade 8 curriculum (MoE, 2009).

Geometry is the most intuitive and reality-based part of mathematics education (Franch,

2004). It is intuitive in the sense that it is characterised by the high connection with reality,

the frequent use of manipulatives of different kinds such as folding cards, strips, and cords

with the aim of leading students to their own discoveries of the geometrical shape

properties. Geometry is concerned with surveys, measurements, areas, and volumes. In

essence, the similarity of plane figures can be connected with reality by a visual

representation of many topics throughout mathematics involving proportional reasoning.

For example, when students learn the similarity of triangles (see Figure 2.1 below), they

are supposed to understand the proportionality between the reduced/enlarged

corresponding sides of the two triangles. The teaching of similar plane figures can also

facilitate the students' understanding of proportional reasoning between the sides of the

plane figures. Proportional reasoning refers to the ability to make comparisons between

objects using multiplicative thinking instead of additive thinking. This means that instead

27

of describing something as “smaller than” or “bigger than”, students learn to think about

relationships in terms like double, half, three-times. In Figure 2.1 below, the sides of the

two triangles are proportional and their corresponding angles are congruent. Thus,

∆ABC is similar to ∆DEF.

Figure 2.1: ��

According to NCTM (2000:72), geometry connects mathematics with the physical world.

For example, an enlarged photograph is a similar figure to the original one. The new

geometric object is the “same shape” as the old one but has all of its parts reduced or

enlarged in size or “scaled” by the same ratio. In geometry, two figures that have the

same shape but not necessarily the same size are said to be similar to each other, a more

precise geometrical definition of similarity is discussed in Section 2.3. The use of

geometry to maintain daily life chores can be regarded within the scope of practical

activities. However, the attempt to learn and teach the similarity of triangles when not

associated with the daily life experiences of students would create problems in

understanding the geometric meaning of similarity. Students would be interested in

learning and the effectiveness of learning would be enhanced when similarity is taught in

the context of daily life activities.

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Figure 2.2: �ℎ� ��ℎ�� ℎ�� + �� = ��

Geometry is a language that discusses shapes and angles blended in algebraic terms.

For example, in Figure 2.2, a right-angled triangle uses the Pythagorean theorem which

states that “the area of the square whose side is the hypotenuse, the side opposite to the

right angle, is equal to the sum of the areas of the square on the other two sides” (MoE,

2009, p.251). The algebraic explanation of a right-angled triangle is �� + �� = ��. In

addition, geometric regions and shapes are useful for developmental work with the

meaning of fractional numbers, equivalent fractions, ordering of fractions, and computing

with fractions. For example, a circle is a geometric shape. The circle can be divided into

2 equal parts, 3 equal parts and 4 equal parts as shown in Figure 2.3 below. The shaded

regions are the visual representation of fractional numbers �

�,

�

� ��

�

�.

Figure 2.3: ��

Geometry is an orderly way to describe and represent our inherently geometric world.

Basic to the understanding of geometry is the development of relating to or involved in

the perception of relationship in space, objects, and an intuitive feeling for our real

environments. Intuition refers to the acquisition of knowledge without inference. This

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implies that it is a principle of an analysis of simple facts of perception, which might in

some cases be supported by the knowledge of the construction of our sense organs.

Geometric intuition is the skill of being able to identify geometric figures and solids, create

and manipulate (Fujita, Jones & Yamamoto, 2004). For example, if two points are on

opposite sides of a line, the segment joining them crosses the line. Furthermore, spatial

capabilities appear in everyday life and are important for success in mathematics

(Southwest Educational Development Laboratory (SEDL), 2002). Spatial ability refers to

the capacity to mentally generate, transform, and rotate a visual image and thus

understand and recall a spatial relationship between real and imagined objects. For

example, it is the capacity to understand and remember the spatial relations among

geometric shapes, equilateral triangles, right-angled triangles, rectangles, and squares.

Students who develop a strong sense of spatial relationships and master the concepts of

similarity are better prepared to learn geometric concepts as well as other related

advanced mathematical topics. For students to understand the similarity between two

triangles they must explore the relationship of different attributes of the triangles or

change one characteristic of shape-preserving others. The shape refers to the length of

the sides of the triangles. Learners should examine the direction, orientation, and

perspective of objects in space; and the relative shapes and sizes of figures and objects.

This skill is relevant to their understanding of the similarity of shapes. For example, in

Grade 8, learners are needed to learn the properties of similar triangles such that they

can use those properties to mitigate the current practical problems.

Geometry is a field that is done with describing and objectivising the concepts abstracted

from real cases, and most of the descriptions, which are regarded as meaningful

according to experiences, emerge visually. For example, Eiffel (1832-1923) was a French

engineer who specialised in revolutionary steel construction. He used thousands of

triangles, some the same shape but different in size to build the Eiffel Tower because

triangular shapes result in a rigid construction. According to Jones (2002), geometry is

not only considered as the most important component of the school mathematics

curriculum but also, as one of the most important elements of mathematics itself. The

reasons for teaching-learning geometry are countless and include providing opportunities

for students on consolidating knowledge and comprehension for the capability to make

30

use of geometrical properties to solve problems in real life. For example, in Figure 2.4,

Jose is wondering how far apart two docks are on the other side of a river. He knows that

the river is 300 yards across at all points in this section of the river. He has a measuring

tape to measure distances on his side of the river. How can he use similar triangles to

find out the distance between the two docks? The activity could be implemented in 8th-

grade and 9th-grade classrooms when students are learning geometry concepts such as

the similarity of triangles.

Figure 2.4: ��

Jose could stand back several yards from the shore in a place where dock # 1 is directly

across the river from him. He can look at dock # 2 from this location and see what point

on the shore lies directly between him and the second dock. Two similar triangles have

now been formed, one is the triangle with Jose as one vertex and the two docks as the

other two vertices, and the other is the triangle with Jose as one vertex, and the points on

the shore that are directly between Jose and the two docks. The two triangles are similar

because the riverbanks are parallel. Jose can measure all side lengths of the smaller

triangle, and he knows the side length of one side of the larger triangle because it is 300

yards plus the distance from where he is standing to the river’s edge. The corresponding

side lengths of Jose to the river’s edge and Jose to the dock that is directly across the

river can be compared with the scale factor for the two similar triangles, and then that

scale factor can be used to find the missing side length between the two docks. Therefore,

geometry skills are beneficial not only in the classroom but also outside of it.

According to Jones (2002), geometry will continue to be significant in the 21st century.

This is because the advancement of technology such as computer graphics and

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multimedia has greatly expanded the scope and power of visualisation in every field to

benefit from the learning of geometry. Ben-Chaim, Lapan, and Houang (2004) drew

attention to the role of visualisation in the development of inductive, deductive, and

proportional reasoning. Geometry allows students to understand the world by comparing

shapes, objects and their connections. Goos, Stillman and Vale (2012) argue that the

development of visualisation and reasoning is part of mathematical thinking. Moreover,

Duval (1998) states that geometric thinking involves the cognitive processes of

visualisation and reasoning. Visualisation and reasoning are those essential mental skills

required for mathematics (Battista, Wheatley & Talsma, 1989), and these cognitive

processes are interconnected and promote students’ success in geometry (Duval, 1998).

Furthermore, visualisation is a skill that helps students to recognise shapes, create new

shapes or objects, and reveal relationships between them (Arcavi, 2003). Battista

(2007:843) asserts that geometric reasoning refers to the act of “inventing and using

formal conceptual systems to investigate shape and space”. The conceptual system

refers to a system that is composed of non-physical objects such as ideas or concepts.

Visualisation and reasoning skills can be improved through the instruction methods

(Arýcý, 2012; Goos et al., 2012; Jones, 2002). Furthermore, the NCTM (2000) also

recommends the use of Dynamic Geometry Software (DGS) to promote reasoning skills

and geometric understanding. Moreover, according to van Hiele’s (1986) theory,

visualisation is the first level and a necessary one in the hierarchy of geometric thinking.

It is a necessary means of geometrical concept formation. In developing the concept of

similarity of triangles, students need to visualise the same shapes with a particular kind

of transformation, enlargement, and reduction. This section will be discussed under

Section 2.3 on geometric similarity. Thus, besides including geometry topics in the school

curriculum, to get the most benefit out of it, effective methods of teaching and learning

geometry are desirable for mathematics education.

In Ethiopia, the curriculum for both primary and secondary schools was revised in 2009.

Within the mathematics curriculum, geometry is included in early grades as one of the

five stands of mathematics, with geometric similarity beginning in Grade 8 (MoE, 2009).

The mathematics curriculum includes algebra, arithmetic, statistic, geometry and

trigonometry at primary school levels (MoE, 2009). In the discipline of mathematics, there

32

are many categories of geometric concepts. These include Hyperbolic geometry,

Projective geometry, Euclidean geometry, non-Euclidean geometry, Analytic geometry,

Plane geometry and Vector geometry. This study is confined to Euclidean geometry.

Euclidean plane geometry constitutes about 45% of the National mathematics curriculum

for primary second cycle education in Ethiopia. Accordingly, students are supposed to do

geometric problems in equal comparisons to other components of mathematics like

algebra, arithmetic, statistic and trigonometry (MoE, 2009). Table 2.1 presents geometric

contents and expected outcomes of learners in the Grade 8 mathematics curriculum of

the country.

Table 2.1: The nature of geometric contents and expected outcomes in Grade 8 mathematics curriculum of Ethiopia (MoE, 2009)

Grade The nature of geometry contents The outcomes expected of learners at each geometric

concept

8 Similar figures (25 periods)

Similar plane figures Illustration and definition of similar figures Scale factors and proportionality Similar triangles Introduction to similar triangles Tests for similarity of triangles (SSS, SAS and AA) Perimeter and area of similar triangles

Circles (20 periods)

Further on circles, central angle and inscribed angle Angles formed by two intersecting chords Cyclic quadrilaterals

Geometry and measurement (30 periods)

Theorems: Euclid's Theorem, the Pythagoras' Theorem

Introduction to trigonometry The trigonometric ratios The values of sine, cosine and tangent Solid figures Pyramid

Know the concept of similar figures and related

terminologies

Understand the condition for triangles being

similar

Apply tests to check whether two given triangles

are similar or not

Have a good know-how on circles.

Realise a connection among lines and circles

Apply basic facts about central and inscribed

angles and angles formed by intersecting chords

to compute their measures

In the current school mathematics curriculum of Ethiopia, most geometry topics covered

in elementary and secondary schools fall under Euclidean geometry, which stems from

Euclid’s classical book of Elements, which was written around 300BC. Even when middle

school curricula incorporate foundational geometric concepts such as similarity of

triangles, teachers frequently lack the experience and professional development to use

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these materials with the mathematical fluency necessary to improve student learning

(Clements, 2003). Moreover, researchers Dündar and Gündüz (2017) reveal that

prospective teachers had difficulty in justifying challenges associated with the daily life

examples of congruence and similarity in triangles. The situation is similar because

Ethiopia is a part of the world. Literature reveals that learners’ performance in geometry

is below the expected level. Researchers (Cox & Lo, 2012; French, 2004; Lo, Cox &

Mingus, 2006; Fujita & Jones, 2002; Seago, Jacobs, Driscoll & Nikula, 2013) also

emphasise that teachers face challenges in teaching geometry. Some of the challenges

include lack of experience, mathematical knowledge, pedagogical content knowledge,

and professional development necessary to improve students’ learning. Moreover, it is a

common activity that geometric topics are usually included in the last part of the textbooks

which may cause a problem in content coverage. This could result in a failure to grasp

the basic concepts of geometry by the learners. For example, the concept ‘similarity of

triangles’ is in the 5th chapter of the Grade 8 mathematics curriculum, and learners are

supposed to learn the same topic in Grade 9 in the 6th chapter. In most rural area schools

in Ethiopia, schools suffer due to a shortage of teachers and late commencement of the

academic calendars. Usually, the content in the last parts of the textbook is not covered.

In Ethiopia, the concept of similarity between two shapes is introduced in the 8th-grade

primary school and extended to in the 9th-grade of secondary school, with special

emphasis on the similarity of triangles. Students are expected to develop a strong

background in this concept at the elementary level. Similar geometric figures are a central

component amongst the geometric contents of the 8th-grade mathematics syllabi (see

Table 2.1). The teaching of similarity in 8th-grade primarily concerns on the

comprehension of the concept of plane figures, shapes, similar triangles, learning the

properties of similar shapes, and application to real-life problems (MoE, 2009).

Despite the importance of geometry in today’s world with similarity being the central part

of geometry, the performance of learners in mathematics, at all levels is weak. Thus, this

study explored the challenges of teaching geometry, particularly the similarity of triangles

to identify pedagogical approaches that can promote meaningful teaching of similarity.

The next section reviews literature on similar plane figures and the similarity of triangles.

34

2.3 THE CONCEPT OF GEOMETRIC SIMILARITY

Researchers, (CCSSM, 2010; Cox & Lo, 2012; Lo, Cox & Mingus, 2006; MoE, 2009;

NCTM, 2000; Seago, Jacobs, Driscoll & Nikula, 2013) assert that similarity is an important

concept taught in middle school geometry curriculum throughout the world. Similarity is a

visual representation of contents throughout mathematics involving proportional

reasoning and it serves as a building block for more advanced study in trigonometry and

calculus (Chazan, 1988; Lappan & Even, 1988). According to (Chazan, 1988; Lappan &

Even, 1988) similarity provides a way for learners to connect spatial and numeric

reasoning and provides the basis for advanced mathematical topics such as projective

geometry, calculus, slope, and trigonometric ratio. For example, measurement of a similar

figure including length, perimeter, and area requires the integration of numerical and

spatial thinking. Investigative tasks in geometry and measurement provide opportunities

for students to analyse mathematically their spatial environment, to describe

characteristics and relationships of geometric objects, and to use number concepts in a

geometric context. Moreover, many events in daily life provide us experience with similar

figures, for example, sun shadows, mirrors, photos, and copying machines while other

examples can be identified throughout physics and other sciences. This makes the

learning and teaching of similarity likely to have profound effects on learners’ ability to

learn a wide range of mathematical concepts and an important area of research. In the

next section, the concept ‘similar figure’ is discussed.

2.3.1 Traditional approach of similarity

Baykul (2009) notes that the concept ‘similarity’ in a school geometry context is more than

one hundred years old, and it has gone through a considerable change. The definition of

similarity has also changed to:

Two similar figures are distinguished from each other by a change of scale

(Evans,1922, p.147)

Two figures are similar when any three points in one form a triangle similar to the

triangle formed by the three corresponding points in the other (Evans,1922, p.147).

35

Similarity is defined as a relationship between two shapes, where the two shapes

have the "same shape," yet are not the same size (Chazen, 1988, p.12).

Two polygons made of line segments are similar if their corresponding angles are

congruent and corresponding sides are proportionate (Chazen, 1988, p.13).

Seago, Jacobs, Heck, Nelson and Malzahn (2013) note that traditionally similarity has

been defined as the same shape, not necessarily the same size. However, this definition

is not precise, and it likely appears to produce defective conceptions. For example, see

Figure 2.11, pairs of shapes that are similar or even congruent but oriented differently

may confuse learners. Orientation refers to the relative arrangement of the points after a

transformation. Reverse orientation means that the points are opposite to the original

shape. The same orientation means that the points are a reflection and in perfectly the

same order as the original figures. Learners may think the figures do not look like the

‘same shape’ because they do not recognise that rotated images are congruent to the

original figure. The second conceptualisation of similarity is a numeric relationship

between two figures. For example, one might say that the corresponding side lengths for

similar triangles are proportional. The authors label this definition as a “static” approach

to similarity because it focuses on setting up and solving proportions that are not

connected with geometric meaning. For example, see Figure 2.5, the two triangles have

side length ratios of �

� , and the side length of the triangle on the right is 2 times the side

length of the triangle on the left. Thus, the two triangles are similar. However, students

are unable to set up the correct proportions when applying this definition to a problem-

solving context.

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Figure 2.5: Static approach to similarity adopted from impacting teachers’ understanding of geometric similarity

2.3.2 Transformation approach of similarity

A geometric transformation involves the movement of an object from one position to

another on a plane. The movement is accompanied by a change in position, orientation,

shape, or even size. Some examples of transformations are translation, reflection,

rotation, and dilation. Before looking at similarity in the form of transformation let us look

at each of the types of geometric transformations.

2.3.2.1Translation

In Euclidean geometry, a translation is a geometric transformation that moves every point

of a figure or space by the same distance in a given direction. It is a transformation that

involves one-to-one correspondence between two sets of points or mapping from one

plane to another. For example, in Figure 2.6 below, the translation of triangle ABC to its

new position A′B′C′ is defined by describing the movement from A to A′ or from B to B′ or

from � �� ′. These three displacements are parallel, and they are called translation

vectors.

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Sticky Note

i'm concerned about the definition of similarity for this section. It is interesting to get similarity and how the properties of similarity are explained here.

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Figure 2.6: Translation of triangle ABC to A'B'C'

From Figure 2.6 above, given that the initial coordinates points of the triangle �� are

�(−3,0), �(−3, −2) and �(−1, −2) the shifting occurred by 4 points to the right and 3

points up such that the new coordinates of triangle �′�′�′ are ��(1,3), �′(1,1) and �′(3,1).

2.3.2.2 Reflection

A reflection is a transformation in which the object turns about a line, called the mirror line

(Umbel, 2012). In so doing, the object flips, leaving the plane and turning over so that it

lands on the opposite side. In the reflection illustrated in Figure 2.7, the triangle on the

left is the object and the triangle on the right is the image. The mirror line is the vertical

line. The image has a different orientation to the object and is said to be flipped or laterally

inverted. If we try to slide the object across the mirror line to fit on its image, it will not

match, we must turn it over to fit exactly over its image. In a reflection, the perpendicular

distance between an object point and image point from the mirror line is the same. This

property enables us to locate the image as a reflection.

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Sticky Note

This part is also interesting on how this study explains similarity.

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Figure 2.7: Reflection

2.3.2.3 Rotation

A rotation is defined as a geometric transformation in which an object is turned or rotated

about a fixed point, called the centre of rotation (Umbel, 2012) as illustrated in Figure 2.8.

The size of the turn is specified by the angle of rotation. The direction of the turn can be

anticlockwise, or clockwise.

Figure 2.8: Rotation

2.3.2.4 Dilation

Dilations can be of two major types of enlargement and reduction (Umbel, 2012). When

the scale factor (�) is greater than one, the image is larger than the object (enlargement)

and when the scale factor is less than one the image is smaller than the object (reduction).

The scale factor (�) is the ratio of the corresponding sides; it usually expresses as,

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Sticky Note

The similarity stems heavily on the scale factor, hence, proportional reasoning is essential.

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� =��ℎ �� ℎ� �� ℎ� ��

��ℎ �� ℎ� �� ℎ� ��

An enlargement (� > �) of the image, triangle A′B′C′ is larger than the object, triangle

ABC. The shape is preserved but size changes, image, and object are similar see Figure

2.9.

Figure 2.9: Enlargement

When the reduction factor lies between 0 and 1, (0 < � < �) the image triangle A′B′C′ is

smaller than the triangle ABC see Figure 2.10. Again, the shape is preserved but the

size changes.

Figure 2.10: Reduction

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A geometric transformation focuses on enlarging or reducing figures proportionally to

create a class of similar figures (Seago, Jacobs, Driscoll, Nikula, Matassa & Callahan,

2013). Transformation is an operation that maps, or moves, a figure onto an image.

Although both static and geometric transformation approaches to similarity are

mathematically correct, a transformations-based approach may be more strongly

constructed and clarifies the corresponding parts of rotated or dilated figures (Seago,

Jacobs, Heck, Nelson & Malzahn, 2013). However, teachers and learners are unable to

apply transformations to similarity tasks and their poor performances on similarity are well

documented.

Figure 2.11: Transformations-based approach to similarity adopted from impacting teachers’ understanding of geometric similarity

As shown in Figure 2.11, the transformation-based approach focuses on rotating,

reflecting, and translating to determine congruence and enlarging or reducing figures

proportionally dilating to create a class of similar figures. Thus, the geometric

transformations-based definition of similarity is as follows: a figure is similar to another if

the second can be obtained from the first by a sequence of rotations, reflections,

translations and dilations (Seago, Jacobs, Driscoll, Nikula, Matassa & Callahan, 2013, p.

76).

Son (2013) asserts that solving similarity items problems requires: (1) understanding the

concept of similarity, (2) recognising the proportionality embedded in similar figures by

comparing length and width between figures or by comparing the length to width within a

Highlight

Sticky Note

This section explains similar figures and align them to enlargement and and reducing. i agree with the authors. Hence, i raised some concerns on the above sections of transformation.

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figure or determining a scale factor, (3) representing the relationship between two similar

figures using a ratio, a proportion and (4) carrying out related procedures. To improve

knowledge of the proportional relationship between the figures, teachers should use a

variety of simple and sophisticated related figures.

2.3.3 Similar triangles

On the previous definition of similar polygons, we learnt that any two polygons that have

the same shape but not necessarily the same size are similar. Triangles are a special

type of polygons and therefore the conditions of similarity of polygons also hold for

triangles. In Ethiopia, the primary and secondary school mathematics textbook's definition

of similar triangles includes these properties of similar polygons. Thus, two triangles are

similar if, (1) corresponding sides are proportional, and (2) corresponding angles are

congruent (MoE, Grade 8 mathematics 2009, p.112)

Figure 2.12: Similar triangles

We say that ∆�� is similar to ∆�� and denote it by writing ∆��~∆�� (Fig. 2.12)

The symbol ‘~’ stands for the phrase “is similar to”.

If ∆��~∆��, then by definition

∠� ≡ ∠�, ∠� ≡ ∠�, ∠� ≡ ∠� ��

��=

��

��=

��

��

The teacher should provide an activity before defining the similarity of triangles, to have

the information about the corresponding sides of triangles and show learners how they

calculate scale factors. For example, as shown in Figure 2.13, the teacher will ask

students the similarity of the two triangles. The teacher should let the classroom for

42

discussion about the similarity of triangles then correct the activities. The teacher should

ask a question if ∆�� ∆�� then find the values of � �� . Following their

reply the teacher can start the discussion by defining similar triangles as similar triangles

are identical in shape but not necessarily in size.

Figure 2.13: Example of similar triangles

2.4 TEACHING OF GEOMETRY

As a teacher of mathematics at a school, the researcher noted that some teachers still

believe in the traditional way of teaching mathematics, especially geometry. The reason

they give for teaching in the way they do is that it saves them time and that they are able

to cover many works in a short period, thus are left with more time for revision. The

traditional approach of geometry instruction is based on the transmission of axioms and

theorems formulated by other mathematicians. These are recorded in texts for students

to study. Students are not given the opportunity to question and understand them. This

creates the impression that geometry comprises the sequence of facts and formal proofs

that should be followed as they are. Gourgey (2001) argues that the use of this method

encourages students to expect to be told what to do and believe that they cannot discover

on their own. Gourgey (2001) further states that the explain-memorise teaching method,

which is prominent in traditional mathematics classrooms, promotes memorisation and

not understanding. Understanding is essential and crucial for success in mathematics,

especially geometry.

The teaching of geometry continued as a basic challenge in mathematics education

(Jones, 2002; Jones & Fujita, 2002). For example, according to the Royal Society's study

43

on geometry teaching (2001), “the most significant contribution to improvements in

geometry teaching will be made by the development of effective pedagogy models, which

will be supported by well-designed activities and materials” (p.30). This means that some

of the current pedagogies emphasise on memorisation of geometrical concepts because

mathematics teachers do not have the appropriate skills, mathematical knowledge, and

in addition the pedagogical content knowledge necessary to be effective in a mathematics

classroom. Thus, teachers require ongoing professional development in order to improve

and update subject knowledge and methodology in the teaching of geometry, hence

facilitating the teaching and learning of geometry using ICT tools. In mathematics

classrooms, the use of technology helps learners and teachers to perform better

calculations, analyse data and enhances the exploration of mathematics concepts, thus

leading in long-term and efficient mathematics learning (Akgul, 2014). Effective teachers

optimise the potential of technology to develop learners’ understanding, stimulate their

interest, and increase their proficiency in mathematics. Technology is the tool to facilitate

the process of bringing the real-life application of geometry to abstract geometry thinking

and to challenge the cognitive process in problem-solving.

Among the challenges identified in teaching geometry, the major problem is the dual

nature of geometry, of which one is geometric figures and the other is verbal

communication (Fujita and Jones, 2002; Laborde, Kynigos, Hollebrands & Strasser,

2006). Geometric figures mean the perception of the physical world, an image, a picture,

or a model. Verbal communication refers to axiomatic geometry, reasoning, and proving.

Fujita & Jones (2002) additionally showed that the dual nature of geometry is helping

teachers in connecting its concepts to students’ real-life environments though, in practice,

for numerous students’ this dual nature is witnessed as a challenge to overpass.

Successful reasoning in geometry may be related to the synchronisation between figural

and conceptual constraints. This implies that all mathematics classes should provide

ongoing opportunities for students’ capabilities with reasoning and sense-making.

Ding and Jones (2006) investigate the geometry instruction at the lower secondary school

level in Shanghai, with particular attention to the relationship of the teaching phases

organised by teachers with learners’ thinking levels demonstrated in classrooms and

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examination papers at Grade 8 (learners aged 14). Analysis of data from the pilot study

suggested that an essential teaching strategy used by the Chinese teachers was mutually

reinforcing visual and deductive approaches in order to develop students’ geometric

intuition in the learning of deductive geometry.

In addition, Gunhan (2014) investigates a case study on 8th-grade learners’ reasoning

skills on geometry. The finding revealed that students have insufficient geometrical

knowledge, visual perception and do not know the requirement for the formation of a

triangle. The triangle inequality theorem states that the “sum of any 2 sides of a triangle

must be greater than the measure of the third side” (MoE, 2009, p. 34). Moreover, studies

(Türnüklü, 2009; Alatorre, Flores & Mendiola, 2012) revealed that students and teachers

experienced difficulties in solving the triangle inequality theorem. Learners should be

presented with problems that allow them to use different reasoning skills and exploratory

activities on the triangle inequalities can be conducted. The similarity of the triangle deals

with either enlarging or reducing the sides of the triangle. The triangle inequality theorem

verifies the possible measure for the sides of a triangle. Learners should use the triangle

inequality theorem on solving problems of similar triangles of unknown side value.

Yilmazer and Keklikci (2015) compared the effect of using the puppets method and

traditional approaches on learners’ success in geometry. The finding revealed that

geometry instruction through traditional methods does not have a positive influence on

learners’ success in learning 8th-grade geometric shapes. On the contrary, geometry

instruction via the use of a puppet built by the researcher has been determined to

positively affect learners’ success in learning geometric shapes and led towards a

statistically significant difference in terms of learners’ success. Yilmazer and Keklikci

(2015) argue that the puppets method instructional approach should improve the

achievement of learners in mathematics. Learners learn geometry meaningful through

exploration.

Erson and Guner (2014) investigate the teaching of congruence and similarity through

creative drama. The research was conducted with 42 learners studying at 7th-grade level.

Within the scope of implementation, 21 learners were taught the subject through creative

drama practices, while the other 21 learners completed the process via the traditional

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method. The research revealed that the learners in the creative drama group could learn

the concepts of congruency and similarity between the triangles, create congruent and

similar polygons, and derive polygons and the stages of forming polygon similar to a

polygon better than the traditional group learners. As a result of the creative drama

method, it became apparent that the concepts of congruency and similarity of triangles

were better understood, and the stages of forming and deriving congruent and similar

polygons actualised. Creative drama refers to animation and representation of any

subject with a group utilising the improvisation and role-play techniques and using mainly

the experiences of the group members (Adlguzel, 2013). In addition, Debreli (2011)

asserts that the teaching based on creative drama is meaningful as learners actively

participated in the lessons and it allowed for working in cooperation and for self-

awareness and consequently led to better performance.

Koo, Ahmad, Teoh and Khairul (2012) propose a pedagogical guide for geometry

education based on literature. A pedagogical guide refers to “the teaching and learning

of geometry through the relationship of the real world surrounding and the real application

of geometry in real life” (Koo, Ahmad, Teoh & Khairul, 2012, p. 35). In Ethiopia, the

educational policy document recommends a problem-based approach to teaching

mathematics and science (MoE, 2009). Problem-based learning is an instructional

method that challenges learners to “learn to learn” working cooperatively in groups to

seek solutions to real-world problems. Learning to learn implies organising learners’ own

learning including the effective management of time and information, both individually and

in the group. Problem-based learning prepares students to think critically and analytically

and to find and use appropriate learning resources. According to van Hieles’ (1986), a

significant element in many teachers' failure to create meaningful understandings in

geometry is their inability to match instruction to their learners' levels of geometric

thinking. Teachers need to organise a problem-based approach in teaching geometry to

promote a meaningful learning environment and to attain the desired instructional

objectives of geometric contents.

Islksal, Koç and Osmanoglu (2010) assert that 8th-grade students have difficulty in solving

problems, in demanding a conceptual understanding of reasoning, and in measuring the

46

surface area and volume of cylinders. This implies that classroom instruction is mainly

focused on memorising the formulas to solve problems requiring a low level of cognitive

demand rather than fostering a conceptual understanding of the surface area and volume

measurement. Moreover, Battista (2007) emphasised that teachers need to understand

learners’ thought processes in order to provide them with meaningful teaching. Learners

experience issues in problem-solving due to poor reasoning. Poor reasoning involves

unfounded and hasty reasoning processes resulting from an insufficient understanding of

the subject in question. Mukucha (2010) asserts that most learners lacked a conceptual

comprehension of mathematical concepts and reasoning skills in problem-solving.

Similarly, Arslan (2007) noted that learners in 6th, 7th and 8th grades exhibited low-level

reasoning skills. Researchers (Pilten, 2016; NCTM, 2000; Aineamani, 2011; Briscoe &

Stout, 2001; Lithner, 2000) suggested that different methods and techniques are

necessary for students to develop reasoning skills. Some of these are metacognition-

based education, cooperative learning and communication skills. Teachers should be

more aware of students’ possible misconceptions in any new piece of knowledge and be

prepared to incorporate that in their instructional considerations.

According to Hartshorn and Boren (2005), one way to strengthen learners’ understanding

of mathematics is using manipulatives. Moreover, findings by (NCTM ,2010; Suydam &

Higgins, 2003; Sowell, 2000; and Thomson, 2003) indicate that the use of concrete

models consolidates teaching and learning mathematics at all levels. Furthermore, the

NCTM’s Curriculum and Evaluation Standards (2010) for Grades 5 through 8 focuse

these models in representing mathematical concepts and processes. NCTM further notes

that learning should be grounded in the use of concrete materials designed to reflect

underlying mathematical ideas (p.87). Scholars in mathematics have emphasized the

importance of involving students in analyzing, measuring, comparing, and contrasting a

wide range of shapes to build crucial learning abilities (NCTM, 2010). Therefore, it is

important to use the concrete model of similar triangles when teaching at Grade 8 level

for student meaningful learning of similarity of triangles.

Researchers (Fennema, 2004; Szendrie, 2011) argue that learners can learn better if

their learning environment incorporates geometric encounters with models that are

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appropriate for their cognitive development. As learners move through elementary school,

real representations may be substituted by symbolic models to aid understanding of

abstract mathematical concepts (NCTM, 2010). Learners can only learn with symbols at

the concrete operational stage of cognitive development if the symbols represent

behaviors that they have already experienced (Fennema, 2004). Thus, the teacher needs

to introspect about the concrete teaching aids and fashion them according to the

development stage of the students (NCTM, 2010)

Fielder (2013) outlined some selection criteria for concrete models such that: the concrete

models should (1) serve the purpose, for which they were intended, (2) be multipurpose

if possible, (3) allow for proper storage and easy access by teachers and learners,(4)

prompt the proper mental image of the mathematical concept, (5) be attractive and

motivating, (6) be safe to use, (7) offer a variety of embodiments for a concept, (8) be

durable, and (9) be age-appropriate in size, and model of real problem-solving situations.

The teacher must be aware of the criteria to select the models of similar geometric figures

and use them properly for meaningful teaching of similar triangles. Students should be

able to select from a variety of models to discover one that is appropriate for their

developmental stage (Fennema, 2004). Furthermore, Elswick (2005) argues that

manipulatives can assist pupils acquire confidence in their abilities to think mathematically

in the long run. The use of manipulatives in the classroom can assist teachers teach

similar figures in a more relevant way, as well as boost teachers' confidence and

competence in the classroom. Teachers need to be more creative and innovative in

carrying out teaching and learning approaches or strategies so that learners could acquire

knowledge effectively. In the next sections, the teachers' role in teaching geometry is

discussed.

2.5 THE ROLE OF TEACHERS IN TEACHING GEOMETRY

The teacher's responsibility is to develop activities that encourages students actively

participate in their learning (Frobisher, 2010). Further, the New Educational Policy

National Curriculum Statement (MoE,2019) of Ethiopia envisages a teacher who acts as

councillors, analyser, designer of learning programs and resources, as well as a leader.

The present education policy in Ethiopia has undergone a full paradigm shift from earlier

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traditional techniques, which were 'teacher-centred' to a teacher who acts as a learning

facilitator.

According to Faulkner, Littleton, and Woodhead (1998), a classroom situation in which

an emphasis is based on neatness, order, and exact replication of shown techniques is

considered be traditional. A new policy requires learners to demonstrate activities to think

rationally and analytically. Students are also needed to get skill in transferring talents

from known to unknown conditions (MoE, 2009, 2019).

2.6 GEOMETRY CLASSROOM

The learning environment has a variety of effects on students. According to Chaplain

(2003), the quality of teacher-student interaction largely depends on classroom layout,

sitting arrangement, classroom atmosphere and fragrance. Mathematics students

creative thinking is a crucial point to develop their cognitive ability and this further brings

an environment which is full of ideas, experiences, motivation, and teaching resources

that can stimulate students’ creativity (Craft, Jeffrey & Leibling, 2001). In a geometry

class, pictures, and three-dimensional items are useful to relate the existence of

geometric concepts, like referring to students’ homes and environments. Students

experience geometry through drawings of the actual objects that they see in their

neighbouring environments.

Clements (2003) argue that the geometry classroom is expected to be characterised by

the following criteria: (1) appropriate activities to support the connection between prior

understanding to new learning and developing logical thinking abilities, (2) investigative

tasks/real-world problems to support developing logical thinking abilities and spatial

intuition, (3) use technology, visual representations, and interpretation of mathematical

arguments, and (4) employ collaborative learning.

In conclusion, the classroom organisation has effects on both the nature of teacher-

student interaction and their relation in a given classroom and may consequently affects

teachers’ lesson objectives.

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2.7 CLASSROOM INTERACTIONS

Instructional support refers to how teachers effectively support students’ cognitive

development and language growth. A meaningful geometry teaching-learning refers, to

providing an activity that offers an opportunity for students to connect geometry to their

experiences and has a goal to connect geometry to further study.

Englehart (2009) argues that teacher- learner interaction does not take place in a vacuum.

It occurs within a very complicated meticulous socio-cultural environment. Similarly,

Bruce (2007) asserts that mathematics teachers face challenges in facilitating high-quality

teacher- learner interaction. Some of those challenges are: (1) the way of teaching

mathematics, (2) lack of mathematics content knowledge, (3) prerequisite for facilitation

skill and concentration to classroom dynamics, and (4) lack of time. Researchers and

teachers should provide strategies to minimise these challenges to improve teacher-

learner interaction create opportunities like professional development and teacher

education.

Good interaction between teachers and learners will create positive relationships in the

classroom and contribute to meaningful teaching and learning of geometry. Studies (Way,

Reece, Bobis, Anderson & Martin, 2015; Ayuwanti, Marsigit, Siswoyo, 2021) indicate that

teacher interactions with learners vary in quality and have appreciable effects on

mathematics achievement outcomes. Moreover, (Pianta & Hamre, 2009; Pianta, 2016)

argued that teacher-learner interactions are malleable features of classroom

environments and have been the focus of international efforts to raise mathematics

achievement. There are few studies specific to the teacher- learner interaction in

geometry. However, as geometry is one of the sections in mathematics, effective teacher-

learner interaction will improve learners’ academic achievements.

2.7.1 Communication in teaching geometry

Communication refers to the transmission of ideas, both in discourse and in writing, of

socially constructed knowledge (Cobb, Boufi, McLain & Whitenack, 2010; Lampert, 1990).

It has been a central theme in the reform of mathematics classrooms due to its role in

facilitating learning through discourse (Cazden, 2010; Knuth & Peressini, 2001). Thus,

50

mathematics can be viewed through its structure, syntax, and cultural meaning (Pimm,

2007). The act of interacting with other learners while communicating in the mathematics

classroom has been described as “organising and consolidating ideas, thinking

coherently and clearly, analysing and evaluating strategies, and expressing ideas

precisely” (NCTM, 2000, p. 60). Such interactions in the classroom where learners are

communicating and defending their proofs are essential for the development of a more

rigorous understanding of the basic constitution of proof. Geometrical proof is defined as

a formal way of expressing particular kinds of reasoning and justification. Communication

should be viewed as both an instructional idea and a geometrical idea. Instructional ideas

refer to small, routine segments of instruction that specify how the teacher and learners

will participate, interact with materials and content. The importance of communication in

the mathematics classroom makes it imperative to focus on how varied communication

strategies can be utilised in teaching the similarity of triangles.

Researchers (Lithner, 2000; Briscoe & Stout, 2001; Aineamani, 2011) argue that

communication skills are important for the development of learners’ reasoning skills. For

example, both teachers and learners should be in the habit of asking ‘why?’, as this

question is essential for learners to develop their mathematical reasoning skills (Mansi,

2003). Information about a learner’s reasoning skills helps the teacher to develop an

opinion regarding the learner’s thoughts, based on which he or she can review the

procedures and techniques used in learning processes, where necessary.

Van Hieles (1986) emphasises the use of proper language by the mathematics teacher

when teaching geometry. The language of the teacher should be simple and accessible

to the learners. Precise and unambiguous use of language and rigour in the formulation

are important characteristics of mathematical treatment. Quite often, people are unable

to grasp or follow each other's cognitive processes. This condition is adequate to explain

why teachers sometimes fail to assist learners in understanding geometry. Learners and

teachers speak their own languages, and teachers frequently utilize language to

communicate with them which learners do not understand. For example, the teachers

should properly differentiate the concept, similarity in geometry different from the similarity

of colloquial speech. “Similar” means looking or being almost, but not exactly the same.

51

For example, John is very similar in appearance to his brother. Whereas similarity in

geometry refers to have the same shape but not necessarily the same size.

2.8 TEACHING GEOMETRY THROUGH TECHNOLOGY

Technology is the tool to facilitate the process of bringing the real-life application of

geometry to abstract geometry thinking and to challenge the cognitive process in

problem-solving (Kesan & Caliskan, (2013). According to Laborde et al. (2006), geometry

teaching generally is based on two fields, namely, diagrams and language. Traditional

geometry teaching usually puts theoretical properties or principles into diagrams and

learners solve geometrical problems and are shown theoretical concepts from diagrams.

Thus, how to successfully present diagrams to help learners understand and construct

geometry theories or principles becomes a key issue for geometry teaching.

Technology materials, such as computer software with electronic whiteboards, provide

the best solution for visual representations because computers can show dynamically the

manipulations and interactions between the geometric figures and learners (Laborde et

al., 2006). Throughout the world, there are several popular technology resources for

exploring geometry such as Logo driven Turtle Geometry, Geogebra, and Dynamic

Geometry Environments (DGE), including the programs of Cabri-3D and the Geometer’s

Sketchpad (GSP). Studies (Hwang, Chen, Dung & Yang, 2007; De Lisi & Wolford, 2002;

Laborde, et al., 2006; Vincent, 2003; Wu, 2013) indicate that dynamic geometry software

can have positive effects on geometry learning for students. However, it needs to be used

properly. For example, Vincent (2003), notes that special attention should be given to

teaching proof, the use of dynamic geometry is not to take away the motivation for proof.

The researcher argues that the use of computer software helps in teaching the similarity

of triangles.

In addition, the NCTM has developed a position statement, which provides a framework

to utilize technology in mathematics education. The NCTM statement endorses

technology as an essential tool for effective mathematics learning. Using technology

appropriately can extend both the scope of content and range of problem situations

available to learners. NCTM recommends that learners and teachers have access to a

52

variety of instructional technology tools, teachers are provided with appropriate

professional development, the use of instructional technology be integrated across all

curricula and courses, and that teachers make informed decisions about the use of

technology in Mathematics instruction (Johnson, 2002). Acknowledging and responding

to the varied learning styles of learners is a critical component of effective inquiry-oriented

standards-based geometric instruction. According to Johnson (2002), effective strategies

for differentiating geometry instructions include rotating strategies to appeal to learners’

dominant learning styles, flexible grouping, individualising instruction for struggling

learners, compacting giving credit for prior knowledge, tiered assignments, independent

projects, and adjusting question level.

Although computer software has been designed for some topics, teachers still need

content knowledge and pedagogical knowledge to choose the best style when they use

these technology materials in the classroom. Studies (Hwang, Chen, Dung & Yang, 2007;

Jonassen, 2000; Laborde et al., (2006) suggest teachers should play a mediating role to

develop correspondence between geometric knowledge and knowledge developed by

interactions between computers and students. For example, teachers should know how

to connect the geometric content and the geometric knowledge and concepts while

students are interacting with the computer tasks, they should make decisions about the

time of accessing each task, and they should identify students’ misconceptions about

solving geometric problems with computer software. Although using computer software

and IT materials should help students’ learning outcomes, it is still not common in

Ethiopian schools. Thus, it is important and attention-demanding to use technology in

mathematics classrooms to improve mathematics achievements in Ethiopia.

Research (Chazan, 1988; Denton, 2017; Edwards & Cox, 2011) has consistently

highlighted that geometric similarity is a mathematical topic with which both learners and

teachers encounter difficulties. Simultaneously, some studies do suggest that carefully

designed dynamic mathematical technology (DMT), might help learners and teachers to

overcome difficulties and misconceptions about geometric similarity as the dynamic and

visual nature of digital technology offers such skills. For example, dragging, visualisation,

measurement to explore the underlying concepts and discover the embedded variant and

53

invariant relationships. Invariant refers to a property that does not change after a certain

transformation. For example, the side length of a triangle does not change when the

triangle is rotated. Such opportunities might enable teachers and learners to experience

and examine the dynamic nature of geometric similarity in more tangible ways. For

example, teachers can exploit the affordances of digital technology to help learners build

connections between geometric transformations and geometric similarity so that learners

understand how to use translations, reflections, rotations, and dilations to determine if two

figures are similar. Additionally, making use of technology in a dynamic environment

where learners can formulate, test, and verify mathematical conjectures, teachers can

support learners to surmount their misconceptions about the ideas of geometric similarity,

particularly those who make the incorrect use of an additive strategy as the learner in

Son’s (2013) study. In the following subsection, we will look at the challenges teachers

faced when teaching geometry.

2.9 CHALLENGES OF GEOMETRY TEACHING

Studies (Adolphus, 2011; Choo, Eshaq, Hoon & Samsudin, 2009; Aydogdu & Kesan,

2014; Das, 2015; French, 2004; Kambilombilo & Sakala, 2015; Jones, Mooney & Harries,

2002; Jones, 2002; Sitrava & Bostan, 2016) are some of those conducted to explore the

challenges of teaching geometry. Accordingly, the main challenges teachers faced are:

(1) lack of pedagogical knowledge, (2) teachers may not have adequate content

knowledge, (3) poor foundation of mathematics teachers, (4) teaching and learning

environments are not conducive, and (5) lack of commitment to geometry. In this next

section, some of the studies are discussed. There are few studies on the concept of

similar figures and the similarity of triangles, discussed in Chapter 1.

Das (2015) studied the challenges faced by mathematics teachers when teaching

geometry. The result of the study showed that mathematics teachers' experiences lacked

pedagogical knowledge and may not have adequate content knowledge. Moreover, Choo

et al. (2009) studied teachers’ perception of geometry and geometry teaching

approaches. The result of the study indicates that teaching geometry is not easy; the

challenges are making it easier, more interesting, more practical based on real-life

examples, and more accessible to all students. Teachers need professional development

54

to minimise the challenges faced on teaching geometry and experience sharing cultures

across schools on the regular ground to minimise the challenges, which will help learners’

achievements.

Kambilombilo and Sakala (2015) explored the challenges in-service mathematics

student-teachers face in understanding transformation geometry. The findings revealed

that in-service mathematics student teachers encounter challenges in transformation

geometry; use of instruments such as protractor and compass; dealing with reflection in

slant lines; writing the equation of lines reflection; inadequacies in rotation geometry and

limitation on van Hieles’ levels III and IV. Consistent with the previous study, Gomes

(2011) conducted an exploratory study to evaluate pre-service elementary teachers’

content knowledge on geometric transformations. The findings revealed that pre-service

teachers were found to have a lack of understanding of geometric transformations. They

lacked the necessary understanding to teach this subject, and they struggled with three

geometric translations, translation, reflection, and quarter-turn rotation. The teacher

education colleges are supposed to evaluate the courses and enrich the required

contents during teacher training to minimise the difficulties.

Adolphus (2011) investigated the problems of teaching geometry. The findings revealed

that the problems of teaching and learning geometry are that the foundation of most

mathematics teachers in geometry is poor; the teaching and learning environment is not

conducive and teachers lack the commitment to teach geometry. Minimising the problems

of teaching and learning geometry for the teachers at the foundation needs special

consideration for students and further success in mathematics.

Fujita and Jones (2006) investigated primary trainee teachers’ geometry content

knowledge related to defining and classifying quadrilaterals. The results indicated that

although trainee teachers could draw the figure of quadrilaterals, they could not provide

their definitions. Besides, they did not have enough knowledge about the hierarchical

relationship between quadrilaterals. Consistent with the previous studies, Jones, Mooney

and Harries (2002) reported that trainee primary teachers’ confidence in geometry and

their geometric vocabulary knowledge was poor. Particularly, they had difficulties in

calculating the area and the volume of geometric figures. For learners' success in

55

mathematics education to be achieved, the teaching and learning of geometry need

special attention.

According to French (2004), teaching and learning geometry requires a lot of planning

than teaching algebra because geometry is less procedural for problem-solving and more

dependent on intuition. For example, learners may confuse lines with line segments

because the teachers cannot draw an infinitely long line on the blackboard for introducing

the concept of a line. A line is an endless straight, continuous path made up of a

continuous collection of points whereas the line segment is a part of a line. It has a

beginning point and an ending point. In addition, studies (Cox & Lo, 2012; Kao, Roll &

Koedinger, 2007) emphasise that sources of difficulty on geometric learning that teachers

should be aware of are that learners usually have difficulty solving multistep geometry

area problems. For example, given a complex diagram that consists of a large

parallelogram and a small interior rectangle with the area between the large shape and

the rectangle shaded, students are instructed to determine the shaded area. It is

important to explore the challenges that the teacher faced in the teaching of the similarity

of triangles. The next section discusses strategies that can improve the teaching of

geometry.

2.10 STRATEGIES THAT CAN MINIMISE THE CHALLENGES OF THE

TEACHING OF GEOMETRIC SIMILARITY

Geometry, an important branch of mathematics, has a place in education for the

development of critical thinking and problem-solving. Furthermore, geometrical shapes

are parts of our lives as they appear almost everywhere; geometry is utilised in science

and art as well. Being a specific content, geometric similarity should be treated as a

concomitant to any subject involving analysis and reasoning. Concomitant means it

requires visual analysis of the between shapes like enlarged/reduced, rotated, and

translated. It is often observed that some students are unable to visualise geometric

figures possibly because of the lack of logical reasoning. For example, students do not

properly visualise the image of an isosceles triangle if they lack the properties of an

isosceles triangle. Therefore, understanding the properties of an isosceles triangle can

help students to visualise the shape of an isosceles triangle. Carrol (1998) and Fuys,

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Geddes, and Tischler (1988) observed that in many middle and high schools, learners

lack experience in reasoning about geometric content. Lack of reasoning among learners

may be because of the sense of failure, and mathematics anxiety. However, Mistretta’s

(2000) survey found out that most learners changed their attitudes towards geometry

through the proper use of van Hiele’s (1958) theory. The students said that the hands-on

exercises proposed by Van Hiele made geometry more entertaining, fascinating, and

easier to learn. Teachers should be familiar with van Hiele's phases of instruction and

how to use them when teaching geometry. Van Hiele’s theory will be discussed in the

next chapter. Thus, teaching geometry according to van Hiele’s phase-based instruction

is considered a meaningful approach to improve learners’ geometrical thinking.

Seago, Jacobs, Heck, Nelson and Malzahn (2014) developed a professional development

(PD) material for impacting teachers’ understanding of geometric similarity in the US,

based on Common Core State Standards (CCSS) for the teaching of geometric similarity

of plane figures. Furthermore, Cohen and Hill (2000), and Smith (2001) argue that a

practice-based approach may help teachers to examine the mathematical skills and

explore instructional practices that support student learning. Thus, teachers need a

planned PD material to promote meaningful teaching of similar triangles.

Researchers (Herbst, 2006; Jones, Fujita & Ding, 2004; Jones & Herbst, 2012) suggest

that to promote geometrical reasoning teachers are supposed to use various instructional

techniques and strategies. Teachers should be aware of the different instructional

methods and how to apply them in their mathematics classrooms. According to Biggs

(2011), teacher-centered approach focusses on the activities that mathematics teachers

do to bring concepts to the learners while learner-centered approaches emphasises on

the tasks that students do to understand the concept. Mathematics teachers are

continuously faced with the task of identifying and implementing the most effective

teaching strategies that will improve academic accomplishment while also catering to the

diverse needs of their students (Jayapraba, 2013; Visser, McChlery & Vreken, 2006). The

authors further argue that teachers must become more aware of their own teaching styles,

as well as their students' learning styles, to provide effective instruction. Teaching style

means a teaching method that comprises the principles and methods used by the teacher

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to enable student learning. Learning style refers to the preferential way in which the

learner absorbs, processes, comprehends and retains information. Researchers argue

that improving teacher’s mathematical knowledge and sharing the experience on effective

teaching strategies of geometry can improve the challenges that teachers face in teaching

geometry.

2.10.1 Promoting teachers’ mathematical knowledge for teaching similarity

Teachers need opportunities to gain a specialised type of content in geometry:

“mathematical knowledge for teaching” (MKT) (Ball, Thames & Phelps, 2008, p. 34),

which includes not only a deep understanding of geometric transformations and similarity

but also the knowledge and fluency to make instructional decisions that support students’

learning of this content. The mathematical knowledge necessary to teach effectively is

recognised as being a complicated issue than simply needing an understanding of subject

knowledge (Franke & Fennema, 1992). To make similarity meaningful for the learners,

teachers must be provided with the opportunity to utilise geometrical concepts and

language to make connections between representations and applications, algorithms,

and procedures (Sowder, 2007). Training programs that provide geometrical experiences

and allow teachers to work together to explore mathematics can help them gain

confidence in their abilities to develop understanding.

Clark-Wilson and Hoyles (2017) explored the impact of 40 secondary mathematics

teachers’ engagement with professional development (PD) and classroom teaching on

their mathematical knowledge for teaching geometric similarity. Their study explored the

teachers’ starting points teaching the definition of similar figures using data collected

through survey-items, PD tasks, and lesson plans. Key to the design of the PD were

several tasks for teachers that required them to closely analyse hypothesised student

responses whilst engaging with a particular dynamic mathematical technology (DMT),

and ‘Cornerstone Maths’ (CM). Clark-Wilson and Hoyles found that the combination of

PD activity focused on geometric similarity and classroom teaching involving DMT led to

notable improvements in teachers' MKT in relation to the geometric similarity. According

to the researchers, the use of learners’ work created in the DMT environment encouraged

the teachers to think deeply about the “within ratio” invariant property, properties that do

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not change during transformation. Having engaged with the task in the DMT environment,

they were able to successfully articulate the underlying mathematical ideas related to the

property that, for similar shapes, the ratios of the side lengths for any pair of

corresponding sides within the shape is invariant.

Son (2013) and Seago et al. (2014), Cunningham and Rappa (2016) also investigated

mathematics teachers’ ability to solve geometric similarity problems. The researchers

surmise that, like Seago et al., when teachers introduce a transformations-based

approach together with a static-based approach when teaching geometric similarity,

students are likely to understand the underlying ideas of geometric similarity more deeply.

Therefore, they assert that it is important to investigate teachers’ mathematical knowledge

of geometric similarity from both perspectives because the teachers’ specific geometric

subject matter understanding could play a crucial role in the process of learners’ learning.

2.10.2 Effective instructional practices in geometric similarity teaching

Sabean and Bavaria (2005) have synthesised a list of the most significant principles

related to geometry teaching and learning. Some of these principles are teachers’

expectations, teachers’ questioning, learners’ prior knowledge and experiences, learners’

problem-solving strategies and problem-based activities and the geometric curriculum.

Furthermore, Sabean and Bavaria (2005) assert that the effective instructional approach

in geometry classroom involves: (1) learners’ engagement is at a high level, (2) tasks are

built on learners’ prior knowledge, (3) scaffolding takes place, making connections to

concepts, procedures, and understanding, (4) high-level performance is modelled, (5)

students are expected to explain thinking and meaning, and (6) students self-monitor their

progress. Teachers should apply these principles and suggestions to improve learners’

understanding of similar triangles.

The researcher attempted to review literature on the strategies to minimise the challenges

teachers faced on the teaching of similarity. However, there are few studies on teachers'

challenges of teaching and learning of similar triangles in the literature.

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2.11 Conclusion

The first section of this chapter reviewed literature on the historical foundation of

geometry. The second section reviewed literature on the importance of learning of

geometry. This was followed by the review of the concept of geometric similarity and its

importance on the middle school geometry curriculum. The fourth section reviewed the

teaching of geometry. This was followed by Sections 2.5, 2.6, and 2.7 which reviewed

literature on the role of teachers in teaching geometry, the geometry classroom and

classroom interaction, respectively. Sections 2.8 and 2.9 reviewed literature on teaching

geometry through technology, challenges of geometry teaching, respectively. Finally,

strategies that can minimise the challenges of the teaching of geometric similarity were

reviewed and summed up in two themes. In the following chapter the theoretical

framework that underpinned this will be discussed.

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CHAPTER THREE

THEORETICAL FRAMEWORK

3.1 INTRODUCTION

The theoretical framework that underpins this research is presented in this chapter.

According to Grant and Osanloo (2014), a theoretical framework refers to the ‘blueprint’

or guide for research. It is based on an existing theory in a field of inquiry that is related

and/or reflects the hypothesis of a study. It is a blueprint that is often ‘borrowed’ by the

researcher to build his/her own house or research inquiry (Grant & Osanloo, 2014, p.13).

The two scholars further state that a theoretical framework consists of concepts together

with their definitions and reference to relevant scholarly literature. Moreover, Ravitch and

Carl (2016) argue that the theoretical framework assists researchers in situating and

contextualising formal theories into their studies as a guide. Therefore, a theoretical

framework serves as the focus for the research, and it is linked to the research problem

under study. The structure and vision for a study are thus unclear without a theoretical

framework.

The literature in mathematics education reveals that, studies situated in the school

geometry education context mainly refer to the three theories in teaching and learning of

geometry. Those theories and models include van Hieles’ (1985) theory, the theory of

figural concepts by Fischbein (1993), and Duval’s (1995) theory of figural apprehension.

Each of these frameworks contains theoretical resources to aid study into the

development of geometrical reasoning in learners, as well as associated aspects of

visualisation and construction in teaching-learning geometry. The three theories are

discussed in the next sections.

Concerning the teaching and learning of geometry, the van Hieles’ developed an

influential theory on levels of geometric thinking. In discussing the profound impact of

Pierre and Dianne van Hieles’ theory in mathematics education, Clements (2003, p. 151)

concludes, van Hieles’ theory gave educators and researchers a model that promoted the

understanding of the important conceptual based level of thinking. It is also a model of

synergistic connections among theory, research, the practice of teaching, and students’

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thinking and learning. Conceptual geometric thinking refers to the ability to critically

examine factual information, relate to prior knowledge to the existing ones, to see

patterns, and connect the geometric concepts (NCTM, 2000). Moreover, Arbaugh,

Mcgraw, and Patterson (2019, p.160) define synergy as “the interaction of elements that

when combined produce a total effect that is greater than the sum of the individual

elements”. Synergetic pedagogy creates a scientifically valid approach for converting

theoretical knowledge into research competences. For example, synergistic effects-

based theory and practice relationships among quadrilaterals in the teaching-learning of

geometry used to improve students’ learning of parallelogram. Thus, the van Hieles’

theory is a model of synergistic effects based on the practice of teaching-learning

geometry and provides a framework to conduct research for which instruction can be

planned and evaluated by mathematics educators, researchers and teachers.

This study draws on van Hieles’ theory of phases of instruction for teaching geometry to

explore the challenges of teaching geometric similarity at Grade 8 primary schools in

Areka Town. According to van Hieles’ (1986), an effective way of learning geometry does

not go in line with teaching and learning other mathematics topics, such as statistics, and

arithmetic. The teaching-learning of geometry remains as a challenge in mathematics

education. This is because geometry is dual, it has geometric figures and verbal

communication (Fujita & Jones, 2002; Laborde et al., 2006). Geometric figures mean the

perception of the physical world, an image, a picture, or a model. Verbal communication

refers to an axiomatic geometry, reasoning and proving. Geometry is extremely rich and

requires more time to understand and prepare to teach effectively by teachers. To teach

geometry effectively, mathematics scholars need to design lessons in the way that

students can learn from the classroom environment and utilise the resources of geometry

through manipulation. Successful reasoning in geometry may be related to the

synchronisation between figural and conceptual constraints. This implies that all

mathematics classes should provide ongoing opportunities for learners’ capabilities with

reasoning and sense-making.

According to NCTM (2000), geometry enables the interplay and interconnection between

mathematical language and the language of pictures. Mathematical language is the

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system used by mathematicians to communicate mathematical ideas among themselves,

while the language of pictures includes visual language. It uses simple pictures to

represents both concrete and abstract ideas. Geometry is a unifying theme to the entire

mathematics curriculum and as such is a rich source of visualisation for an arithmetical,

algebraic, and statistical concept (Jones, 2002). For example, geometric regions and

shapes are useful for developmental work with the meaning of fractional numbers,

equivalent fractions, ordering of fractions, and computing with fractions. Van Hieles’

(1986) argue that geometry is a conceptual system where geometric concepts are entirely

connected and sequential from each other. They used it to demonstrate the pedagogical

application of the theory. To better describe the van Hieles’ (1986) theory and how it has

been applied used in the mathematics education, the following section provides the

historical background and a general description of the theory.

3.2 THE VAN HIELES’ MODEL

What has become known as the van Hiele level theory was developed by Pierre Marie

van Hiele and his wife Dina van Hiele-Geldof in separate doctoral dissertations at the

University of Utrecht, Netherland in 1957. The couple were greatly concerned about

difficulties their students encountered with studies of geometry and this investigation led

to the creation of the van Hiele theory of geometric thought. The theory has three “aspects

namely, (1) levels of geometric thinking, (2) properties of the levels and (3) phases of

learning which offer a model of teaching that teachers could apply to promote their

learners’ levels of understanding in geometry” (van Hieles’, 1986, p. 165). These three

aspects of the van Hieles’ model discussed in the next section.

According to Clements (2003), the van Hieles’ theory is not only used to find out the

learners’ geometric thinking levels, but the theory may also be taken for designing

meaningful geometric education and dealing with the challenges of teaching-learning

geometry. Meaningful geometric instruction should focus on mathematical reasoning,

communication of ideas, and connections between geometry and related disciplines; use

everyday life experiences and use technology. Its planning involves considering the

learning activities that take into account students’ interests and abilities, then the learning

goals and objectives of the lesson. Some of the problems encountered in teaching

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geometry are: (1) teachers may not have adequate content knowledge, (2) teachers lack

pedagogical knowledge, (3) poor foundation of mathematics teachers, (4) teaching and

learning environments are not conducive, and (5) lack of commitment to geometry. Also,

the learning of geometry becomes challenging when learners do not properly attain the

geometric thinking levels required, and teachers do not support their teaching by using

manipulative and information communication technologies. Van Hiele (1986) assert that,

“if meaningful learning geometry means learning to think and being able to attain the

highest possible level of conceptualization thus, all mathematics teachers should be well

versed in the nature of good geometry teaching” (p.151). Good geometry teaching refers

to promote learners’ geometric thinking level from one level to the next level required

through teachers' assistance in the processes of exploration and reflection.

3.2.1 The van Hieles’ levels of geometric thinking

According to the van Hieles’ (1986) theory, learners’ progress through five sequential and

hierarchical levels of thinking. The levels are said to be sequential because understanding

geometric concepts requires a learner to pass through each level in order. Also, the levels

are hierarchical since “to function successfully at a particular level, a learner must have

acquired the strategies of the preceding levels” (Crowely, 1987, p. 7). These “levels are,

Visualisation, Analysis, Informal Deduction/ Order, Deduction, and Rigor”. Van Hieles’

(1986) assert that these levels of thinking are linked with the types of geometrical activities

the learners experienced rather than to the learners’ age. This implies that teachers are

supposed to implement the instructional strategies which help learners to attain the van

Hieles’ levels and to pass through each level. Researchers (Gutierrez, Jaime, & Fortuny,

1991; Usiskin & Senk, 1990; Wilson, 1990) argue that the nature of those levels, together

with assigning of learners to a specific level and the application of the model has been

the most troublesome characteristics of the van Hieles’ theory. It has been difficult since

students acquire multiple van Hieles levels at the same time and mastering a single level

might take months or even years.

The van Hieles considered the levels to be discrete but other researchers (Battista, 2007;

Burger & Shaughness, 1986; Crowely, 1987; Usiskin, 1982) reveal that the levels are

dynamic and continuous. According to those authors, the levels are dynamic because

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learners may move back and forth between levels. They are also continuous since

concept formation in geometry may well occur over long periods and require specific

instructions. As a result, assigning a learner to a certain level, especially those in transition

from one level to the next, is particularly difficult. For example, a learner may exhibit

different chosen van Hieles’ levels of reasoning on different tasks, particularly phenomena

observed between Level 2 and Level 3. Some learners may switch between levels on the

same task. Considering this argument, mathematics educators would need to be familiar

with van Hieles' theory and how learners construct their grasp of geometric concepts to

meet their pupils at their current level of comprehension or inside their conceptual

schema. However, from the researcher’s experience at one of the Ethiopian teacher

education colleges for primary school mathematics teachers, teaching the courses

“methods of teaching mathematics” there is no content about the van Hieles’ theory.

Teachers in Ethiopia do not know or have experience of the van Hieles’ theory and its

application in geometry instruction. Consequently, the in-service teachers lack knowledge

about identifying a learner’s level of geometric thinking and van Hieles’ phases of

instruction.

Originally the van Hieles’ numbered the levels from 0 to 4 and the names used for the

levels were first used by Hoffer (1979) as the van Hieles did not name the levels. In 1986,

the van Hieles started to use the 1 to 5 scales and consequently most researchers today

use the same scale.

The model described by Usiskin (1982) with Levels 1-5 is adopted in this study. These

levels are further discussed below.

Level 1: Visualisation

At this level, the learner reasons about geometric shapes utilising visual considerations

of concrete examples (Usiskin, 1982). A learner who is reasoning at Level 1 recognises

certain shapes holistically without paying attention to their component parts (Crowley,

1987). Based on their visual characteristics, students can identify triangles. For example,

given two similar triangles, students can identify them as two triangles because they “look-

alike”. Those students can tell that one triangle is bigger or that one is smaller, but they

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will not conceptualise the properties of similarity. They can identify parts of the triangles

and even line up the two triangles to see that the angles are the same size, but they do

not analyse a figure according to its components.

Level 2: Analysis

At this level, learners reason about geometric shapes according to their properties

(Crowley, 1987). They can separate right, obtuse, and acute triangles amongst different

classes of triangles. They can identify triangles that are equilateral as having three

congruent sides and equal angles and isosceles triangles as having two congruent sides

together with two base angles equal. Given two similar triangles, those learners can

identify them as having the same shape. Going further, they can compare the size of the

angles and match the corresponding angles in the two triangles. They can also measure

the sides to see that the corresponding sides of similar triangles are proportional. For

example, as illustrated in Figure 3.1, Level 2 students can separate sets of similar

triangles into classes, understanding that all triangles are not similar and that there are

different sets of similar triangles. Those learners can compare triangles according to their

relationship, but they cannot generalise how properties are interrelated. They cannot, for

instance, reason that if corresponding angles are congruent then corresponding sides are

proportional.

Figure 3.1: Two sets of similar triangles

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Level 3: Abstraction

At this level, learners can identify the relationship between classes of geometric shapes

(Usiskin, 1982). For example, given any pair of similar triangles, learners can understand

that similarity results in corresponding angles being congruent and corresponding sides

being proportional. Moreover, at this level, those learners are expected to be able to

develop an understanding of relationships between and among properties and

demonstrate a greater ability to apply the “if-then” reasoning (Crowley, 1987). They can

conclude that “if two triangles are similar then the corresponding angles are congruent”.

Furthermore, they also may conclude that if corresponding angles in two triangles are

congruent, then corresponding sides are proportional. Those learners can construct

informal arguments to show that equilateral triangles are always similar and that isosceles

triangles are sometimes similar. However, those learners are not ready for axiomatic

structures of deductive reasoning.

Level 4: Formal deduction

Learners who operate at this level are expected to be able to extend their examination of

the properties of shapes. At this level, the significance of deduction as a way of

establishing geometric theory within an axiomatic system is understood (Crowley, 1987).

The interrelationship and role of undefined terms, axioms, postulates, definitions,

theorems, and the proof is seen. In geometry, formal definitions are formed using other

defined words or terms. There are words in geometry that are not formally defined. These

words are point, line, and plane. They are building blocks, used to define other geometric

concepts. At this level students should be able to reason about the similarity of triangles

abstractly through formalising deductive arguments that reach a logical conclusion. They

can, for example, develop proofs from axioms and theorems. For example, if two pairs of

corresponding sides are proportional, and the included angles are congruent, then the

correspondence is a similarity. Learners can also understand and demonstrate the

necessary and sufficient conditions for similarity.

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Level 5: Rigour

At this stage, learners should be able to evaluate systems of axioms and investigate non-

Euclidean geometries such as projective geometry (Crowley, 1987). Projection is a

transformation of points and lines in one plane onto another plane by connecting

corresponding points on the two planes with parallel lines. Projective geometry refers to

the study of geometric properties that are invariant with respect to projective

transformations. Invariant refers to the property of remaining unchanged regardless of the

change in conditions of measurement. However, this study is confined to Euclidean

geometry. At this level, learners should be able to learn, establish, and build a deductive

system. Smart (2008) argues that most of the students who have fitted in this level

become professionals in geometry, so they can carefully develop the theorems in different

axiomatic geometric systems. Therefore, learners at high school do not attain the 5th level

and it is usually assigned to college or university students in higher education. There is a

limited number of studies on learners’ van Hieles’ geometric thinking levels in Ethiopia at

all school levels. The researcher recommends, mathematics educators and researchers

investigate learners’ van Hieles’ geometric thinking levels in Ethiopia.

It is not the intent of this study to examine learners’ van Hieles’ levels of geometric

thinking. However, this study recognises that, in implementing instruction based on the

van Hieles’ phases of learning of similar triangles, teachers should recognise and

understand the van Hieles’ levels of their learners. Teachers need to help their learners’

progress through these levels in preparation for the axiomatic deductive reasoning that is

required in high school geometry.

3.2.2 Properties of the van Hieles’ models

The van Hieles identified five properties that characterise the model. Crowley (1987)

argues that these features are important because they offer educators with useful

assistance when making instructional. Teachers are supposed to decide the instructional

activities such as the participation of learners, content and how they interact with content

in the instructional process. The following section describes the properties of van Hieles’

model.

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Property 1: Intrinsic versus Extrinsic

According to (Crowley,1987), when a geometric concept is essential to learning it is said

to be intrinsic to that learning once the learner has understood the geometric concept at

that level. It becomes external (extrinsic) for the new learning of concepts. For example,

at Level 1 only the form of a triangle is perceived by learners while at Level 2, the triangle

is defined according to its properties and components (Crowley,1987). The properties are

there at visualisation level, but the learners are not yet consciously aware of them until

the analysis level. Thus, a geometric curriculum to promote progression through the levels

needs to exhibit a logical development in its content and process. Logical development

refers to planning curriculum across the grade levels from kindergarten through high

schools, building upon instructional based upon standards. However, from the

researcher's experiences in Ethiopia, the geometric curriculum lacks logical development.

This is because at all levels geometric contents are placed in the last chapters of the

mathematics syllabus. Mostly, those contents in the last chapters in the mathematics

syllabus are usually not taught. They are generally neglected by teachers. For example,

if we look at the concept ‘similarity’ it is in the 4th chapter of Grade 8 out of 7 chapters and

5th chapter at Grade 9 out of 7 chapters in a textbook. The important concepts, ‘congruent

figures’ for understanding geometric similarity are found in the 5th chapter of Grade 7 out

of the 5 chapters. Therefore, the researcher recommends the placing of geometric

contents vertical aligned and linked with the knowledge contained within the mathematics

curriculum across the grade levels. For example, locating geometric similarity at the

beginning of Grade 8 mathematics syllabus may result in: (1) extension of revision time

for the topics which are not covered in previous grades, (2) content learnt in previous

grades, and (3) vertical sequence alignment of the geometric curriculum.

Property 2: Sequential

Researchers (Gutierrez, Jaime & Fortuny 1991; Mayberry, 1983; Usiskin, 1982) argue

that the sequential character of the van Hieles' levels is the most important of the

attributes. All the other properties evolved from the sequential nature of the levels. For

example, some of the features are the intrinsic/ extrinsic nature, the significance of

language at each level, and the difference in perception for different levels. Usiskin (1982)

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argues that a learner cannot be at van Hieles’ level � without having gone through level

� − 1. Furthermore, Hoffer (1981) argues that learners cannot achieve one level without

passing through the previous level and will have mastered large chunks of previous levels

to perform effectively at one of the higher levels. Moreover, the sequential nature of levels

is supported by the investigation into the learning and understanding of congruences

carried out by Nasser (1990). She reveals that in general learners performing at a certain

level were successful in tasks demanding lower-level performances. Congruence is a

special kind of similarity of figures. Understanding congruency may help students to

recognise similar figures. In Ethiopia, learners are supposed to learn congruency at Grade

7 in the 5th chapter out of 5 chapters and similarity of figures in the 4th chapter of the

Grade 8 mathematics curriculum. The geometric curriculum in the Ethiopian primary

schools should be revised to consider the sequential nature of the van Hieles’ levels to

minimise the challenges of teaching similarity of triangles to Grade 8 learners.

Property 3: Distinction or Linguistic

According to Crowely (1987), the distinction or linguistic property refers that the van

Hieles’ geometric thinking “has its own linguistic symbol and system of connecting

relations” (p.5). Distinction refers to the ability to use and understand the vocabulary

associated with the level (Crowely,1987). For example, a learner at Level 1 will recognise

“a square as a square and a rectangle as a rectangle but not a square as a rectangle”

because they have yet to start analysing the properties of each figure. Thus, a 'correct'

relationship at one level can be changed at a higher level. Moreover, for example, a

geometric figure can have multiple names. “A square is also a rectangle and a kind of a

parallelogram” too. Once a learner has progressed to Level 2, they can begin to realise

so as “a square is a rectangle” because a square also has all the properties that make a

rectangle.

The van Hieles (1986) argue that as students move through the geometric thinking levels,

it is important that they conceptualise their newly acquired knowledge in their own

language. They further explain the importance of language as, (1) learners will orally

express and communicate to others, (2) learners will discuss and listen to others, and (3)

learners language development is specific to the geometric thinking levels and essential

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for the development within the level. Without the availability and use of appropriate

language learners cannot use verbal expressions and teachers cannot communicate to

learners.

Property 4: Separation or Mismatch

According to Crowely (1987), separation or mismatch is defined as the inability of two

people who are at different levels of geometric thinking to understand each other about

geometric concepts. Researchers (Usiskin, 1982; Mayberry, 1983; Senk, 1989) argue

that it is this property that explains why most secondary school learners fail to succeed in

learning geometry. Because the material for secondary school geometry is at Level 3, if

a learner has not attained that level of understanding geometry, then they will not advance

to the next level. For example, Usiskin (1982) quoted a student who explained to his

instructor, “I can follow a proof when you do it in class, but I can’t do it at home.” (p.5).

This learner is probably at Level 2 and the teacher at Level 4. The learner may fail to

understand what is taught by the teacher. Teachers are supposed to use language

appropriate at learners’ levels when verbally communicating. For example, the definition

of similar triangles is misunderstood by learners at different levels. A learner’s (Level 2)

definition may be vastly different from that of the teacher (Level 4) while the teacher

presents a drawing of two similar triangles and defines similarity on the chalkboard.

Therefore, the teacher needs to develop instructional activities based on rotating,

translating, or dilating the triangles to define the similarity of triangles. Learners would

then explore the definition from the instructional activities. The geometric thinking levels

of their learners are intended to be known by their teachers.

Property 5: Attainment

According to Crowley (1987), attainment can be explained as progress through the van

Hieles’ geometric levels of learners that are more dependent on the instruction they get

than on their age. Furthermore, the author argues that the development of learners’

geometric thinking levels needs an exploratory activity and should be placed by

considering the learning phases of van Hieles’ theory. Learners cannot do well at one

level unless they have mastered the preceding levels. Thus, the pedagogy and the

organisation of instruction, as well as the geometric content and materials applied during

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the teaching and learning process are important areas of pedagogical concern for

learners’ geometric thinking levels. The van Hieles’ addressed the attainment of the

geometric thinking levels by proposing phases of learning and hypothesised about the

way geometric concepts may be acquired. Teachers hold the key to this transition from

one level to the next and need to recognise and understand the van Hieles’ levels at which

their learners operate.

3.2.3 Criticism of van Hieles’ theory

The van Hieles’ theory has been, firstly, criticised for emphasising that the development

of geometric thinking should sequentially take place. However, researchers (Battista,

2009; Bleeker, Stols & van Putten, 2013; Bruce & Hawes, 2015; Gagnier, Atit, Ormand &

Shipley, 2017; Sinclair & Bruce, 2014) argue that the same learner may possess different

van Hieles’ levels for different geometry concepts simultaneously. For example, based on

the learning activities experienced on solid geometric shapes a Grade 2 learner may

reach at geometric thinking Level 2, this is possible for Grade 7 learners only. Solid

geometry is concerned with three-dimensional shapes. Some of the example three-

dimensional shapes are cubes, rectangular solids, prisms, cylinders, spheres, and

pyramids. Moreover, Ness and Farenga (2007) argue that it is difficult to identify the van

Hieles’ level for learners as they attain different levels for different geometry concepts.

Secondly, van Hieles’ theory does not take into account how the geometric concepts are

developed, rather it is an attempt to locate the misconceptions about geometry concepts

at different stages in a diagnostic manner (Gunčaga, Tkacik & Žilková, 2017). This implies

that van Hieles’ theory focuses on remedial materials. Thirdly, Guven and Baki (2010)

argue that van Hieles’ theory does not account for any developmental trajectory for non-

Euclidean geometries; it focuses on the development of concepts of the Euclidean

geometry.

Finally, researchers (Clements, Swaminathan & Sarama, 1999) argue that many learners

at visualisation level do not reason in a completely holistic fashion, but may focus on a

single attribute, such as the “equal sides of a square or the roundness of a circle”. They

have proposed renaming this level the syncretic level. Syncretic level refers to a level in

which learners classify the shapes of geometric figures both by comparing them to visual

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prototypes and by paying attention to the property attributes. For example, learners may

do such activities “this geometric figure has three sides, but it does not look like triangles,

thus it is not a triangle”. Other modifications have also been suggested (Battista, 2009)

such as defining sub-levels between the main levels. However, none of these

modifications have yet gained popularity. In the next section, the five phases of instruction

are discussed.

3.2.4 Van Hieles’ phases of instruction

As indicated above, van Hieles (1986) emphasized that the development of geometric

thinking is more dependent on the instruction they get than on the psychological

developments of the learners. Psychological developments refer to the development of

learners' cognitive, emotional, intellectual, and social capabilities. The van Hieles’ theory

proposed that teachers should arrange their teaching in five different phases when they

teach geometry to direct learners from one level to the next (van Hiele, 1986). The phases

are information or inquiry, guided or directed orientation, explication, free orientation, and

integration. For this study, the phases of instruction are summarised as follows:

Phase 1: Information/Inquiry

At this initial stage, the teacher and learners should engage in conversation and activity

about similar geometric figures. Furthermore, observations of similar figures are made,

questions about similar figures are raised, and level-specific vocabulary is introduced by

teachers to learners (Hoffer,1983, p.208). For example, as shown in Figure 3.2, the

teacher may start the lesson by discussing the concept of similar figures using models of

figures, object-like, photographs, polygons having the same shape but different in size.

The teacher could also provide activities that require the learners to choose pairs of

similar figures from given models or pictures. Crowley (1987) asserts that “teachers

engage with activities at this initial phase so that students learn what prior knowledge they

had about the topic while they learn what direction further study will take” (p.5). This

implies that teachers may have information about learners’ prior knowledge of similar

figures. The teacher should conclude this stage in the discussion by using examples of

similar and non-similar figures. Similar figures like those illustrated in Figure 3.2 can be

used.

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�ℎ��ℎ �� ℎ� �� ?

�ℎ��ℎ �� ℎ� �� ?

Figure 3.2: Examples of similar plane figures

Phase 2: Directed Orientation

In the direct orientation phase, the mathematics teacher should purposely organise

sequential activities for geometric problems and direct learners to explore the uniqueness

of each geometric topic through hands-on manipulation (Crowley,1987).

In the directed orientation phase, the mathematics teacher is required to get deeply

involved in the learning process so that students can be directed where and how they

should approach selected problems. For example, as illustrated in Figure 3.3, the teacher

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may provide some activities that require the learners to identify similar triangles for a

cluster of different triangles given in a figure. He/she can also make shadows of a triangle

that were purposely prepared to compare the relationship between the image and pre-

image under enlargement. Learners may also use measuring tools to compare the length

of the corresponding sides and measures of the corresponding angles and an

enlargement associated with constant of proportionality or scale factors.

Figure 3.3: ��

Phase 3: Explication

Building on their previous experiences, learners may acquire knowledge to verbalise their

understanding of the geometric concept and its connections (Crowley,1987). Learners

become more aware of the new geometric concept and communicate these in appropriate

geometrical language. In this regard, the teacher is needed to make sure that the learners

can master appropriate geometrical terms namely, similar triangles, corresponding sides,

and corresponding angles, dilation/enlargement when learning the similarity concept. The

interaction between the teacher and learners is important in supplying learners with a

necessary and sufficient amount of help so that they can achieve the maturation essential

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for the growth to the next level. Without the use of appropriate language, learners cannot

verbally express and exchange ideas they have been exploring in learning similarity

concepts. Good interaction between the teacher-learners must be established. The

teacher should often after class ask him/herself about the responses of the learners and

attempt to understand their meanings.

Phase 4: Free Orientation

In the free orientation phase, the teacher may provide geometric problems for learners

that can be solved in numerous ways and encourage learners to master the network of

the relationships (Crowley,1987). Learners may gain experience in finding their own ways

of resolving the learning tasks. Clements and Battista (2004, p. 431), argues that one of

the important roles of teachers at this phase is selecting appropriate similarity activities

and problems that need specific levels of thinking to solve geometric problems.

Appropriate activities and geometric problems in similarity include models of similar

figures, examples, and non-examples of similar triangles, paper folding activities, and use

of technology. In this phase, the teacher’s role is minimal and provides the geometric

activities appropriate for the level and students shall find their own method of integrating

themselves into the network of relationships to complete the assigned tasks. The network

of relations in completing a problem like, in Figure 3.4 illustrates an exploration of a pair

of similar triangles. In this activity, learners should first find the names of the two triangles.

Next, they must state all the corresponding angles and corresponding sides of the two

triangles. In the problem, learners may develop the network relationship from the two

parallel lines �� ∥ ��, the congruent angles ∠� ≡ ∠�, ∠� ≡ ∠�, and ∠� ≡ ∠�. Learners

could find the ratio of the corresponding sides Then based on the proportionality constant

ratio and congruent angles learners could then prove that the two triangles ∆��~∆��

are similar.. Learners can mention the three congruent corresponding angles and the

proportional corresponding sides in their proof. They would now be familiar with the

learning of similarity of the triangle, corresponding side ratio, and corresponding angles.

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Figure 3.4: Network relations of similar triangles

Phase 5: Integration

According to Crowley (1987), in the phase of integration, students can construct an

overview of the similarity of triangles learned and the teacher should help the learners

gain an overview of the similarity concepts. Similarly, students’ summaries their

comprehension about of similarity of triangles and integrate the appropriate language for

the new higher geometric thinking level. Learners should be made to understand the

smaller and larger right-angled triangles, their corresponding sides, corresponding

angles, constant ratio, and congruent angles. At this phase, it should be remembered that

those summaries only include what the learners already knew. At this time, no new

material should be introduced. The aim of the activity in this phase should be evident to

the students, so the teacher should provide less and less support. The learners could

collectively do a review; the work is done, and the observations made in the first four

phases, and they create a summary that provides an overview of the new concepts. Using

the example given in Phase 4, the learners may summarise the similarity of triangles.

After this phase, learners would have attained this level of understanding.

3.2.5 VAN HIELES’ LEARNING MODEL FOR THE TEACHING OF SIMILARITY

Geometry is still a difficult subject to teach and learn in mathematics education (Jones &

Fujita, 2002). For example, according to the Royal Society's study on geometry teaching

(2001), “the most significant contribution to improvements in geometry teaching will be

made by the development of effective pedagogy models, which will be supported by well-

designed activities and materials” (p.30). This means, the current pedagogies emphasise

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on memorisation of geometrical concepts and teachers have little experience to develop

a conceptual understanding of geometry. Teachers lack professional development

programs to minimise the challenges faced in teaching geometry.

Researchers (Al-ebous, 2016; Atebe & Schafer, 2008; Cofie & Okpoti, 2018; Crowley,

1987; Ding & Jones 2006; Muyeghu, 2008) argue that van Hieles’ theory is used as, (i)

one of the ways of dealing with the problems of teaching geometry, (ii) the most significant

theoretical framework to understand learners’ learning processes, and (iii) to improve

learners’ geometric thinking levels. Other scholars have undertaken similar studies,

particularly using van Hieles' approach to teach geometry, technique to teaching

geometry. However, there are some distinctions in this study's problem perspective,

research approach, topic chosen, study environment (curriculum, school context,

resource availability), sample population, and instruments utilized. In the first place, there

is no research found which integrated exploring teachers' challenges and identified a

pedagogical approach that can promote meaningful teaching of geometry. Moreover, in

Ethiopia, there are no studies found that are underpinned by van Hieles’ theory and

applied van Hieles’ phase-based instruction to enhance the teaching-learning of

geometry. Educational technology and accompanying infrastructure, on the other hand,

are the most recent developing initiatives in mathematics education. Researchers

(Clements, Battista & Sarama, 2001; Clements, 2003; Ding & Jones 2006; Korenova,

2017; Venturini & Sinclair, 2017) argue that the van Hieles’ instructional approach has

been integrated with technologies within the dynamic geometry environments such as

Logo and GeoGebra to promote the development of geometry concepts. But contrary to

this demand and experience still, a large part of the world population, including this study

area lacks the use of educational technology in learning geometry at the primary and

secondary school levels.

The researcher used van Hieles’ theory to explore how the challenges in the teaching of

similarity of triangles in Grade 8 class can be minimised in Areka Town primary schools.

This study contributed to mathematics education body of knowledge by designing

instructional activities based on van Hieles’ five phases of learning and explored the van

Hieles’ phases of learning as a pedagogical approach to minimise the challenges of

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teaching similarity of triangles in Areka Town primary schools. As shown in Figure 3.5,

this study designed instructional activities on the five phases of instruction for teaching of

similarity of triangles; learners should be presented with a variety of geometrical

experiences. To develop a learner’s geometric thinking levels, teachers are supposed to

use the activities and examples in the phases of instruction. Based on van Hieles’ theory,

the researcher concludes that the five phases of instructions are a more meaningful

approach for the teaching of similarity of triangles. As discussed in the instructional phase

section, teachers should be aware of each of the five instructional phases and their

instructional activities, and examples for learners' geometric thinking. Moreover, teachers

should be aware of the important pedagogical areas of concern such as, the ways of

teaching, organisation of instruction, content, and material used in teaching-learning of

geometry, for which similarity of triangles is a part. Teachers must recognise and

comprehend their students' van Hieles levels, and they must assist them in progressing

through them in preparation for the axiomatic, deductive reasoning that is required in high

school geometry. In the next sections, the theory of figural concepts by Fischbein (1993),

and Duval’s (1995) theory of figural apprehension are discussed.

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Figure 3.5: Proposed model for teaching similarity of triangles, the implication of van Hieles’ phases of learning

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3.3 FISCHBEIN’S THEORY OF FIGURAL CONCEPTS

Fischbein (1993) proposed that geometric figures are not solely concepts, but they have

an intrinsic figural nature. That is, a geometric figure is a figure and a concept

simultaneously. For example, a geometric shape triangle can be described as having

intrinsically conceptual properties and it is not solely a concept, it is an image too. An

intrinsic property is a property that an object has of itself, independent of other things,

including its content. Fischbein (1993) argue that all geometrical figures represent mental

constructs which possess, simultaneously, conceptual, and figural properties. According

to this notion of figural concepts, geometrical reasoning is characterised by the interaction

between these two aspects, the figural and the conceptual. A difficulty in conceptualising

a geometric figure may arise if the figural properties are not in accordance with the

conceptual properties of the figure. And this tension may give rise to prototypical figural

concepts (Fujita, 2012). That is, learners may not recognise a rectangular quadrilateral

as a parallelogram even though they have knowledge of conceptual properties of a

parallelogram (Fujita & Jones, 2007; Walcott, Mohr & Kastberg, 2009).

According to Fischbeins (1993), learners should learn similar figures through mentally

manipulate geometric objects and at the same time to apply operation with similar figures,

logical correlation, and operation. These mental activities may involve the following tasks,

(1) drawing an image of similar triangles by unfolding geometrical objects, (2) identifying

similar geometric figures that can be created by enlargement, and (3) asking learners

which sides of the similar triangles is enlarged or reduced. The theory of figural concept

considers the conceptual development of geometry concepts as merely cognitive

concepts with no mention of the role of language and teachers’ instructional methods in

learners’ geometric conceptual developments. However, according to van Hieles’ theory,

learners’ conceptual development in geometry is sequential and depends on the

instructional process. The theory of figural concept does not mention the instructional

procedures in the teaching of geometry; teachers are supposed to know the figural theory

when they teach geometry for learners’ conceptual development. Geometric figure is a

figure and a concept.

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3.4 DUVAL’S THEORY OF FIGURAL APPREHENSION

According to Duval (2017), a given geometrical figure can be recognised in several

distinctive ways depending on the set of rules applied for visual representations. Duval

(1995, p.145) provides a model for analysing the semiotics of geometric drawing. In his

model, he identified four types of “cognitive apprehension”. These are: (1) perceptual

apprehension: this is what is a geometric figure recognised at first glance, (2) sequential

apprehension: this is employed when constructing or describing a figure, (3) discursive

apprehension: perceptual recognition depends on discursive statements because

mathematical properties represented in a drawing cannot be determined solely through

perceptual apprehension, some must first be given through speech, and (4) operative

apprehension: this involves operating on the figure, either mentally or physically, which

can give insight into the solution of a problem(p.145).

Duval (1995) reveals that, students may face a “conflict between perceptual apprehension

of a figure and mathematical perception: difficulties in moving from perceived features of

a figure can mislead students as to the mathematical properties and objects represented

by a drawing and can obstruct appreciation of the need for the discovery of proofs” (p.

155). For example, the definition of similar figure in geometry differs from the colloquial

meaning of “similarity” likeness.

Duval (1995) argues that “operative apprehension does not work independently of the

others; indeed, discursive and perceptual apprehension can vary often and obscure

operative apprehension (p. 155). In teaching geometry, Duval (1995, p.155) argues that

special and separate learning of operative as well as of discussive and sequential

apprehension are required. He suggests that geometric software may support the

development of sequential apprehension and operative apprehension. He concludes that

a mathematical way of looking at figures only results from co-ordination between separate

processes of apprehension over a long time (Duval,1995, p.155).

Duval (1995) suggests that geometry involves three kinds of cognitive processes working

together, namely: (a) visualisation processes regarding space representations, (b)

construction processes by tools, and (c) the reasoning processes. According to him, any

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activity in geometry should involve communication between these three processes, even

though the different processes can be performed separately. He argues that the cognitive

processes are intertwined, and their cooperation is required for geometric reasoning

proficiency.

3.5 Conclusions

Each of the three theoretical frameworks supports research in geometry education. These

are not the only theoretical frameworks, but they are among the most significant in

geometric education research. Therefore, the characteristics of the process by van Hieles’

levels can be used for teachers’ instruction, or textbook structuring, or for design tests for

checking learners’ knowledge of geometry in different grades. Fishbein points out that

one of the main tasks of mathematical education, in particular, of geometry is to create

different types of didactic-methodology situations that would systematically seek strict

cooperation between the two, images and concept aspects. Duval points out that, any

geometric activities should incorporate and communicate the three cognitive processes.

He argues that the cognitive processes are intertwined, and their cooperation is required

for geometric reasoning proficiency. The three theoretical frameworks indicate that

learners do not see and distinguish geometric figures. Learners see geometric activities

as blended and structured in a series of procedures, as a result, students' geometric

thinking levels are poorly linked. Geometry teaching and learning require more research

in the field of mathematics education.

In this chapter, the three well-developed frameworks for clarifying the development of

geometrical reasoning are meant to give a quick overview of the theoretical framework

that can be used in teaching geometry research. The chapter discussed the implications

of van Hieles’ levels of geometric thinking; these were explored in the context of the

challenges teachers faced in the teaching of geometry. In chapter 4, how the qualitative

research was designed and conducted that underpinned this study will be discussed.

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CHAPTER FOUR

RESEARCH DESIGN AND METHODOLOGY

4.1 INTRODUCTION

The previous chapter focused on theories of the teaching and learning of geometry. The

literature in mathematics education reveals that studies in the school geometry education

context mainly rely on three theories. Each of those theories was reviewed and provided

to assist the researcher in addressing the research question.

This chapter outlines how the qualitative research was designed and conducted to explore

the challenges of teaching similarity of triangles in the Grade 8 primary schools of Areka

Town. It also presents the research paradigm, design, methodology, population and

sample, data sources and collection techniques, and phases of data collection. The

chapter also presents the research aims, research questions and explains why the

qualitative exploratory case study approach was used. Finally, trustworthiness, validity,

reliability, and ethical considerations of the data are discussed.

4.2 RESEARCH AIMS AND QUESTIONS

This qualitative exploratory case study was intended to explore the challenges that

teachers encounter in teaching similarity of triangles in Grade 8 using the following

general and specific research questions:

How can the challenges of teaching similarity of triangles to Grade 8 learners be

minimised?

Sub-questions:

What are the challenges faced by mathematics teachers in teaching similarity of

triangles?

How do teachers interact with learners in the teaching of similarity of triangles?

Which pedagogical approaches can promote meaningful teaching of similarity

triangles?

How can the strategies be applied such that the challenges in the teaching of

similarity of triangles are minimised?

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4.3 RESEARCH PARADIGMS

A detailed know-how on the concept of the research paradigm is gained by an

examination of literature in the field. The word paradigm has become commonplace in

educational research and social theory since its use by Thomas Kuhn in his seminal: The

Structure of Scientific Revolutions (1971). Kuhn (1970, p.75) asserts that “the concept

paradigm is defined as the entire constellation of beliefs, values, techniques, shared by

members of a given scientific community”. According to Punch (1998), a paradigm is a

complex term. The author asserts that a paradigm is a set of assumptions about the social

world, and about what constitutes proper techniques and topic of inquiry. It is a means of

a view of how science should be done, and it encompasses elements of epistemology,

theory, and philosophy, along with methods (p. 28). Thus, a research paradigm describes

an investigator’s point of view, which is led by the paradigm's assumptions, beliefs, norms,

and values.

In educational research, the term paradigm refers to a researcher's 'worldview'

(Mackenzie & Knipe, 2006). According to Mackenzie and Knipe (2006), a paradigm is a

set of abstract beliefs and concepts that determine how a researcher perceives the world,

as well as how he or she interprets and behaves in it (p. 234). It is the frame of reference

through which a researcher examines the world. It is the conceptual prism through which

the researcher evaluates the methodological aspects of their research topic in order to

select the research methods to be employed and the data to be analysed. Furthermore,

Denzin and Lincoln (2005) argued that paradigms are human constructs that deal with

basic principles or ultimates that indicate where the researcher is coming from when

constructing meaning from evidence. Thus, paradigms are significant because they

provide beliefs and mandates that determine what should be examined, how it should be

studied, and how the study's results should be understood by researchers in a discipline.

According to Neuman (2011), a paradigm is best described as a whole system of thinking.

In this sense, a paradigm refers to the established research traditions in a discipline, or a

philosophical framework, as Collis and Hussey (2009, p.55) opine. In particular, a

research paradigm would include the accepted theories, traditions, approaches, models,

frame of reference, body of research and methodologies; and it could be seen as a model

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or framework for observation and understanding (Creswell, 2014, p. 56; Babbie, 2010, p.

123; Rubin & Babbie, 2010, p. 32). As a result, a paradigm is a fundamental set of beliefs

that govern action. Creswell and Piano Clark (2007) underline that all educational

research should be based on a framework of theoretical assumptions. The paradigm

determines the researcher's philosophical orientation, which has ramifications for every

decision made during the research process, including technique and methods selection,

as well as how meaning will be derived from the data collected.

In educational research, different paradigms have been developed which are, because of

their constituent components, incommensurable. One such grouping is as follows:

positivist, interpretive, critical and postmodernist. Each paradigm is based upon a sharply

different assumption about epistemology. It is based on how knowledge is generated and

accepted as valid; it is about the purpose of the research. This study lays its foundation

on the second of the major paradigms identified, specifically, the interpretive paradigm.

The next sections discuss these paradigms.

4.3.1 Positivist paradigm

Auguste Comte (1798 – 1857), a French philosopher, was the first to propose the

positivist paradigm. According to Comte (1856), testing, observation, and reasoning

based on experience should be the basis for understanding human behaviour. It is the

preferred perspective for research that attempts to explain observations in terms of

realities or quantifiable entities (Fadhel, 2002). Furthermore, positivism is a social theory,

which views the natural sciences as the paradigm for social inquiry, a major belief in

naturalism. It is applied to quest for causal connection in natural surroundings. Thus,

positivism may be an approach to social research that seeks to apply the natural science

model of research as the point of departure for investigations of social phenomena and

explanations of the social world (Denscombe, 2008, p.14). To arrive at results,

researchers in this paradigm use deductive logic, hypotheses development, hypothesis

testing, operational definitions and mathematical equations, computations,

extrapolations, and expressions. However, this paradigm is not followed in this study

because the aim was to explore the challenges that teachers encounter in teaching


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4.3.2 Critical paradigm

The Critical paradigm positions its research in social justice issues and seeks to address

the political, social and economic issues, which lead to social oppression, conflict,

struggle, and power structures at whatever levels these might occur (Kivunja & Kuyin,

2017). In this regard, Myers (2009) assumed that a transactional epistemology which

supports the researcher to interact with the study participants, an ontology of historical

realism, especially as it is related to operation, a methodology that is dialogic and axiology

which gives position to cultural norms. Critical researchers also assume that social reality

is historically constituted and that it is produced and reproduced by people (Myers, 2009).

4.3.3 Pragmatic paradigm

The pragmatic paradigm was developed by philosophers who discovered that “it was not

feasible to obtain the truth of the real world solely by one scientific approach, as the

positivist paradigm claimed, nor was it possible to discern social reality”, as the

interpretive paradigm claimed (Alise & Teddlie, 2010, p.234). A mono-paradigmatic

orientation research was not enough for this scholar, but this philosopher (Alise & Teddlie,

2010; Biesta, 2010; Teddlie & Tashakkori, 2003; Patton, 1990) claimed that what was

required was a global view that could provide research methods that were deemed most

appropriate for investigating the phenomenon at hand. As a result, these theorists sought

for more realistic and pluralistic ways to research that would allow for the use of a variety

of methods.

4.3.4 Interpretivist paradigm

According to Guba and Lincolin (2005), the main endeavor of the interpretive paradigm

is to understand the subjective world of human experience (p.78). This approach tries “to

get into the head of the subjects being studied’ to speak and understand and interpret

what the subject is thinking or the meaning s/he is making of the context (Guba & Lincolin,

2005, p.78). Every effort is made to try to understand the viewpoint of the subject being

studied, rather than the researcher. Emphasis is placed on understanding the individual

and their interpretation of the world around them. Hence, the basic principle of the

interpretive paradigm is that the truth built from the society (Creswell, 2014). Due to this

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fact, an interpretive paradigm has been termed as the constructivist paradigm. In this

paradigm, the theory does not come before the research, but it follows the ground of the

data generated by the research act. This research paradigm considers a subjectivist

epistemology, relativist ontology, a naturalist methodology, and balanced axiology and

these elements are discussed below.

According to Punch (2009), subjectivist epistemology is defined when the researchers

create meaning from the give data through their own thinking and cognitive processing of

data informed by their interactions with the research participants. Punch (2009) further

argue that an understanding of the researcher will consolidate knowledge social due to

the fact that h/she had personal experience on real-life with in the natural environment

researched. There is also a belief that an investigator and his/her respondents can be

engaged in interactive processes in which they synthesise, argue, question, listen, read,

write, and record the study data. When there is an assumption in which multiple realties

can be explored and reconstructed through human interaction between the researcher

and the respondents, we call it relativist ontology (Chalmers, Manley & Wasserman,

2009). According to a naturalist methodology, an investigator uses the data, which is

gathered through interviews, discourses, text messages, reflective sessions, and works

with details before generalisation (Creswell, 2014). Balanced axiology on the other hand

assumes that the result of a study will show the values of an investigator trying to present

a balanced report of the findings.

Constructivist researchers often address the process of interaction among individuals.

They also focus on the specific context in which people live and work in order to

understand the historical and cultural settings of the participants (Creswell, 2014). In this

study’s, context, for example, the teachers' training programme for elementary school,

the teaching-learning environment, background knowledge, professional development,

the mathematics curriculum and availability of the resources are explored. The researcher

intended to make sense or interpret the meanings teachers have about the challenges

they faced in the teaching of similarity of triangles to Grade 8 learners and how such

challenges could be minimised. Thus, this study is situated in the interpretive paradigm.

In this study, as shown in Table 4.1 below, the characteristics of interpretivism used in

this study are categorised into the purpose of the research, the nature of reality (ontology),

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nature of knowledge and the relationship between the inquirer and the inquired-into

(epistemology) and the methodology used (Creswell, 2010).

Table 4.1: Characteristics of interpretivism

Characteristics Explanation

Objective of research Explore teachers’ challenges when teaching of similarity and its

approaches in the 8th-grade primary school.

Ontology What are the challenges teachers faced in the teaching of similarity of

triangles?

The challenges teachers faced can be explored, and minimised through

the investigation of teachers' knowledge, views, and experiences.

Discover how teachers make sense of their own teaching-learning of

similar triangles in primary schools by means of classroom observation,

and their interaction with learners in the classroom.

Epistemology An investigator is not expelled out but get parts in the process of research

and set out the meaning that is explored.

Methodology Processes of data collection by observations, semi-structured interviews,

and questionnaire.

The researcher describes in detail the context of the study.

A research paradigm is a set of common assumptions, attitudes, values, and practices

that a community of researchers hold regarding research. Since this research was about

exploring how students learn similarity of triangles it was vital to think of a research design

that would produce the greatest results. (Creswell, 2010). The next section introduces the

research design.

4.4 RESEARCH DESIGN

Mouton (2006) defines research design as a plan for conducting a study. In this regard,

he further explained the concept of research design as “it focuses on end product,

formulates research problems as a point of departure and emphasises on the logic of the

research” (p.55). Mouton (2006) further claimed that a research design shows a

framework for data collection and analysis, as well as the technique to be followed.

Based on the argument of David and Sutton (2004, p.133), the aim of a design is to give

a structure for collecting and analysing the data in the way that improves the validity of

the research investigation. In this study, an exploratory case study design was used to

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explore the challenges teachers faced in teaching similarity of triangles. The rationale that

triggered an investigator to use the research design is that it lets the investigator to

concentrate on the challenges teachers faced when teaching similarity of triangles.

According to Yin (2009), a case study is utilized to obtain a deeper understanding of a

real-life event; however, this understanding must include key contextual factors that are

extremely relevant to the study's phenomenon, and they are believed to be classroom

context, which are associated to teaching similarity of triangles. A classroom context

refers to appropriate resources such as models of similar figures, images of triangles, and

technology. Thus, in this study, the case study was chosen as Creswell (2010) state that,

it may allow the researcher to have a better idea of the situation.

A further essential point to note is that a case study focuses on few cases of analysis,

usually just a person, a team, or an organization, that are closely investigated (Welman

& Kruger, 2001, p. 105; Creswell, 2010, p.125). The Grade 8 primary school mathematics

teachers in ATA are the cases of the unit in this study. The five primary schools and one

teacher from each school were selected through the purposeful sampling technique

discussed in the sampling section. The following section introduces the research

methodology.

4.5 RESEARCH METHODOLOGY

The research method is a strategy of enquiry, which moves from the underlying

assumptions to research design, and data collection (Creswell, 2014). Although there are

other distinctions in the research modes, the most common research methods are

qualitative and quantitative (Creswell, 2010). At one level, qualitative, and quantitative

method refer to distinctions about the nature of knowledge, how one understands the

world, and the ultimate purpose of the research (Creswell, 2014). On another level of

discourse, the terms refer to research methods, that is, how data are collected and

analysed, and the type of generalisations and representations derived from the data

(Creswell, 2014). Thus, this study used the qualitative method.

4.5.1 The qualitative method

According to McMillan and Schumacher (2010) qualitative research as an analysis of

people’s individual and collective social actions, beliefs, thoughts and perceptions and

Highlight

Highlight

Sticky Note

The purposiveness of selecting the participants needs to be explained. In the results section, the teachers cannot demonstrate similarity. The selection of the participants in this study is essential concerning the characteristics of the participants and why they were deemed fit to be part of this study.

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are primarily concerned with understanding the social phenomena from the participant’s

perspective in addition to this definition (p. 320). Furthermore, according to Creswell

(2010), the aim of a qualitative research study is to engage in research that probes for a

deeper understanding of a phenomenon rather than searching for causal relationships (p.

56).

The qualitative method was used in this study to explore the challenges teachers faced

when teaching similarity of triangles. The approach was followed because it enables

researchers to get understanding into individuals' inner experiences, to establish how

cultural meanings are produced, and to uncover instead of test characteristics (White,

2005 p.81; Corbin & Strauss, 2008 p.12).

As a qualitative researcher, it is essential to explain the setting where this research was

carried out. According to Terre-Blanche, Kelly, and Durrheim (2006), in every qualitative

study where a researcher is physically present, the context is accepted in a realistic

manner. In the context of this study, the argument is that it will be the most useful for

recognizing the challenges teachers faced when teaching of similarity of triangles. Thus,

it is better to go to the schools and examine teacher-learner interactions and the

pedagogical approach teachers used in the teaching of similarity of triangles. As a result,

the participants and the data obtained are thought to be influenced by the context.

Moreover, Merriam (2009, p.13) agrees herewith situation and include that affair can be

realised if they are viewed in the natural settings. Therefore, in this study, the researcher

observed the actual teaching and learning process of similarity of triangles in all the

selected government primary schools under Areka Town Administration (ATA). This

exploratory case study was unique in the sense that, no such research has been

conducted in the Areka Town Administration (ATA) at Grade 8 level, in particular those

that explored the challenges faced by teachers in the teaching of similarity of triangles.

4.6 POPULATION AND SAMPLES

The first task in selecting a sample size is defining the population of the study. According

to Fraenkel, Wallen and Hyun (2012) a population can be any size, and that it will have

at least one and/or sometimes several characteristics that set it off from any other

population. In educational research, the population of interest is usually a group of

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teachers, students, or other individuals who possess certain characteristics (Creswell,

2014). Thus, in this study, the population constitutes 9 Grade 8 primary school

mathematics teachers in Areka Town Administration. According to the education office of

ATA, there are 9 primary schools (5 government and 4 non-government). There was one

Grade 8 mathematics teacher in each school.

According to Creswell (2010), the process that helps the researcher to take a set of

participants from the target population is said to be sampling and the researcher uses it

for selecting a subset of a population for inquiry. A sample is examined to learn more

about the population from which it was selected. Bryman (2012, p.416) argues that

describing the sample is not the prime purpose, however, it is a technique to assist the

researcher in describing aspects of the population. Moreover, McMillan and Schumacher

(2010, p.129), further described sampling that it is a way of determining individuals from

whom the data is collected.

Methods of selecting samples are typically divided between probability sampling and non-

probability sampling, where the former uses a group’s size in the population as the sole

influence on how many of its members will be included in the sample, while the latter

concentrates on selecting sample members according to their ability to meet specific

criteria. Marshall and Rossman (2006) provide an example of sampling four aspects:

events, setting, actors and artifacts. Researchers may sample at the site level, at event

or process level, and the participant level. This study sampled all the government primary

schools of ATA. To collect the most comprehensive source of data possible, sampling

decisions are made. Smaller sample sizes are common in qualitative research than in

quantitative research. In qualitative research, sampling is dynamic, and it frequently

continues until no new themes emerge from the data collection process, a process known

as data saturation (Creswell, 2010).

According to Patton (1990), the logic and power of purposeful sampling lies in selecting

in formation-rich cases for study in depth. Information-rich cases are those from which

one can learn a great deal about issues of central importance to the purpose of the

research, thus the term purposeful sampling can be used for obtaining accurate

representation of the target population (p. 169). In this regard, the total population the

study was 9 Grade 8 mathematics teachers in ATA Primary schools. The sample in this

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study then comprised 5 Grade 8 mathematics teachers from 5 government primary

schools (4 male and 1 female Grade 8 mathematics teachers). All the sample teachers

had a qualification of BSc degree in mathematics, qualified to teach the grade level, but

the schools were known to practice challenges and were underperforming in teaching of

mathematics. Therefore, this study laid its focus on the challenges teachers faced in

teaching of similarity of triangles, teacher-student interaction, and the pedagogical

approach they used in teaching similarity of triangles.

4.7 DATA SOURCES AND COLLECTION TECHNIQUES

To conduct this study, the research found and selected data sources (White, 2005,

p.186). Data sources such as classroom observations, semi-structured interviews and

questionnaires were used to collect data in this study. According to Creswell (2010, p.78),

in qualitative research the major data collection device is the researcher. Since the

researcher is a tool, no research can be done without him or her. Over a period, the

researcher gathered data at the school's local setting. The data was collected in three

stages.

4.7.1 Observations

Creswell (2010) explains observation as a systematic process of recording the

behavioural patterns of participants, objects, and occurrences without necessarily

communicating with them (p. 83). Observation is an everyday activity whereby our senses

(seeing, hearing, touching, smelling, tasting) are used but also our intuition to gather bits

of data. McMillan and Schumacher (2010, p.208) claim that observation is used to

describe the data that are collected, regardless of the technique employed in the study.

Observational research methods also refer, however, to a more specific method of

collecting information that is very different from interviews or questionnaires. The

observational method relies on a researcher’s seeing and hearing things and recording

those observations, rather than relying on subjects’ self-report responses to questions or

statements. De Vos (2001, p.278) describes observation as a typical approach to data,

which implies that data cannot be reduced to figures. In the observation of participation,

the emphasis is thus both on one’s own and on the participation of others.

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Denzin and Lincoln (2000, p.673) assert that researchers observe both human activities

and the physical setting in which such activities take place. Observation is used to

describe the data that are collected, regardless of the technique used in the study.

Classroom observations were the most important tools of this qualitative research as the

researcher witnessed all the processes of teaching in a natural setting (McMillan &

Schumacher, 2010, p.208). The degree of observer participation can vary considerably.

According to (Frankel, Wallen & Hyun, 2014), there are four different roles that a

researcher can take, ranging on a continuum from complete participant to complete

observer. Those roles are: (i) complete participant (ii) participant-as-observer (iii)

observer-as-participant and (iv) complete observer (p.231).

4.7.1.1 Participant observation

In participant observation studies, researchers participate in the situation or setting they

are observing. When a researcher takes on the role of a complete participant in a group,

his identity is not known to any of the individuals being observed (Frankel, Wallen & Hyun,

2014). The researcher interacts with members of the group as naturally as possible.

When a researcher chooses the role of participant-as-observer, he participates fully in the

activities of the group being studied, but also makes it clear that he is doing research

(Frankel, Wallen & Hyun, 2014). Participant observation can be overt, in that the

researcher is easily identified and the subjects know that they are being observed; or it

can be covert, in which case the researcher disguises his or her identity and acts just like

any of the other participants.

4.7.1.2 Non-participant observation

In a non-participant observation study, researchers do not participate in the activity being

observed but rather “sit on the side-lines” and watch; they are not directly involved in the

situation they are observing (Frankel, Wallen & Hyun, 2014, p. 446). On the contrary,

when a researcher chooses the role of observer-as-participant, s/he identifies

herself/himself as a researcher but cannot be considered as a member of the group s/he

is observing (Frankel, Wallen & Hyun, 2014). Finally, the role of a complete observer is

at the extreme opposite from the role of complete participant (Frankel, Wallen & Hyun,

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2014). The researcher observes the activities of a group without in any way participating

in those activities.

During data collection, the researcher functioned as a non-participant observer. As he

visited the primary schools and undertook informal one-on-one observations with Grade

8 mathematics teachers, the researcher pretended to be a non-participant observe. In

each school, the researcher sat in the back of the classroom, taking field notes and

utilizing an observation guide to describe the classroom activities. The classroom events

were video-recorded and after that, the transcription of each episode was done. The use

of a video camera provided detailed and accurate information about the instructional

sessions. The classroom observation was conducted to get a clear picture of how

teachers interacted with learners in the teaching of similarity of triangles and what

pedagogical approaches were used. The researcher observed the following: classroom

organisation and resources used, teacher activity, teacher-learner interaction, and

teacher-language.

4.7.1.3 Classroom organisation and resources

Many teachers and educators appreciate the value of concrete materials in teaching and

learning Geometry. Van Heiles’ (1958) theory reveals the fact that the use of concrete

experiences in the geometry classroom may improve the conceptualisation of abstract

ideas about geometry. Teaching techniques presented by the van Hieles allow learners

to learn geometry through hands-on activities. In so doing, learners can combine their

concrete experiences with problem-solving strategies and reach higher order thinking

skills at an abstract level (Fuys, et al., 1988). The classroom that lacks resources will be

a barrier for the teaching of similar triangles.

4.7.1.4 Teacher activity

According to van Hieles’ (1986) theory, the development of geometric thinking is more

dependent on the instruction received. The phases of instruction presented by the van

Hieles allow learners to learn geometry through hands-on activities. In so doing, learners

can combine their concrete experiences with problem-solving strategies and reach the

higher order thinking skills at an abstract level. In this aspect, the responsibility of the

teacher is crucial. Therefore, students would be hampered in their learning if their teacher

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lacks the necessary language or skills to communicate specially if the classroom

instruction is in the foreign language. This was one of the major impediments for Ethiopian

students’ and as result pedagogical approaches to improve teaching the of similarity of

triangles should be established.

4.7.1.5 Teacher-Student interaction

According to Van de Walle (2007, p.30), in the learning environment, teacher-student

interaction is successful if students are involved on developing mathematical knowledge

and comprehension. The students learn through hand-on manipulation and thus assist in

the discovery of solutions to problems. Schunk (2004, p.412) in this regard verified that

learners built geometrical knowledge as it is suggested by teachers. As a result, teachers

are not seen providing students with remedies to activities; rather, they assist students in

finding these solutions. Success acquired in this manner would pique students' interest

in learning.

4.7.1.6 Teacher-language

Various research findings also emphasise on the use of proper language by the

mathematics teacher, which should not be too pedantic. The language of the teacher

should be very simple and understood by the learners. Precise and unambiguous use of

language and rigour in the formulation are important characteristics of mathematical

treatment. Quite often, people cannot understand each other or follow the thought

process of each other. This situation is sufficient to explain why at times teachers fail to

help learners in geometry learning. The learners and teachers have their own languages,

and often teachers use language which learners do not understand. This reason is well

noticed in the studies made by Van Hieles (1958).

Observations provide a holistic view of the research problem. The researcher also gains

knowledge from his own observations and reflections, that are integrated into the final

analysis. Unofficial observation data was supplemented with semi-structured interviews

conducted throughout his time in the context, as well as factual data verification through

document interrogation as appropriate. In this study, the researcher observed the lesson

plan, the teaching environment, and the pedagogical centres in the schools sampled.

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4.7.2 Interviews

According to Creswell (2010, p. 181), it is claimed that an interview takes place if the

researchers ask one or more participants general, open-ended questions and record their

answers (p. 217). In qualitative research, there are three types of interviews: open-ended

interviews, semi-structured and structured interviews (Creswell, 2010, p.181).

4.7.2.1 Open-ended interview

The researcher's aim in an open-ended interview is to learn about the participants'

thoughts, feelings, beliefs, and attitudes toward issues. Open-ended questioned

interviews usually last a long time and are made up of several interviews.

4.7.2.2 Structured interview

In the structured interviews, as in survey research, questions are detailed and gathered.

It is usually used in larger sample groups or many case studies. This is done to maintain

uniformity.

4.7.2.3 Semi-structured interview

During the interview, an interview protocol was used by employing English as a channel

of discussion in an interview. To this effect, the researcher summarized data gathered

through individual interviews. Every question was answered by all the participants.

Personal interviews provide a lot of flexibility and adaptability, but they are also expensive

and time taking (Welman & Kruger 2001). Furthermore, the authors argued that semi-

structured interviews cannot be done in an anonymous manner; thus, researchers must

avoid saying anything that could be regarded as the desired response and instead utilize

open-ended questions. It is further noted here that there are two main advantages of

personal interviews in that it gives the researcher control over responses. The

respondents focus on the researcher's control on the interview scenario through human

interaction, which allows both an investigator and respondents to build confidence in

interviewing evasive responses in the way that shows incomplete or imprecise responses,

resulting in rich data. The study subjects can be more interested in sharing what they

have known before in the interview if the researcher is physically present.

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The semi-structured kind of interview is commonly used in research projects to

corroborate data found in other resources. It often takes long and requires the participant

to answer a series of prearranged questions. A semi-structured interview guide is drawn

to define the line of questioning. As suggested by Lincon, and Guba (2006), semi-

structured interviews clarify concepts and problems and allow the formulation of possible

answers and solutions. Furthermore, by probing further into the participant's perspective,

semi-structured interviews allow different aspects of the problem to emerge. Semi-

structured interviews were chosen by the researcher because they would allow him to

describe the pedagogical approach teachers practise and challenges teachers faced in

the teaching of similarity. After conducting classroom observations with five Grade 8

mathematics teachers, interviews were conducted in Grade 8 classes at the school site.

The interviews lasted around thirty to forty minutes and took place during school hours.

After receiving permission to conduct study from the Areka Town Administration

Education office, the interviews were done school hours.

The researcher requested the participants for permission to audiotape the interviews after

observing the lessons. Individual interviews with the teachers were done by the

researcher. The interviews were conducted in a semi-structured manner with Grade 8

mathematics teachers. In order to find the meaning and context of the interview sessions,

the interviewer listened several times for the respondents; consequently, all the interview

data were tape-recorded and changed in to written form by the researcher.

The researcher used probes for participants to provide further information as advised by

Leedy, and Ormond, (2005) and Creswell (2010, p.81). The researcher kept a diary where

he recorded his reflections during the interviews. He stored the transcribed data in a safe

place. The principles given by Creswell (2010, p.87) were used to create an interview

guide.

Throughout the interview, the whole data were noted during the interview sessions

considering both oral and facial expressions. After the interviews were completed, the

researcher asked all participants if they had any questions in case the researcher had

missed something. The researcher thanked all the participants for their time, their

contribution, and for agreeing to take part in this research.

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4.7.3 Questionnaires

According to Cohen et al., (2011), questionnaire is used to gather primary data from the

respondents. The data from observations and interviews were triangulated using open-

ended questionnaires in this qualitative exploratory study. The questionnaire in this study

was composed of 14 open and closed-ended items. They have two sections, the first

section is about the demographic information of the respondent, which consists of three

questions. The second addressed the teachers' challenges, reviews, and reflections of

teaching similarity of triangles which consists of 11 questions.

4.8 PHASES OF DATA COLLECTION

This research was carried out in three phases. These were observations of the lessons,

semi-structured interviews, and questionnaire administrations. Data was collected for five

weeks, one week at each of the sampled schools. A pilot study was conducted by the

researcher prior to the actual data collection. The pilot study is discussed in the next

chapter.

Phase: 1

During the first phase, the researcher used observation protocols to in order to acquire

which pedagogical approaches teachers used when teaching similarity of triangles, the

how the teachers interacted with the students in the classroom, and also what challenges

teachers faced in teaching similarity of triangles. Sitting in the back of the class, the

research takes notes and videotaped the actual instruction. At each school, the

researcher observed three lessons of similarity of triangles. Fifteen mathematics lessons

were observed.

Phase: 2

The researchers employed semi-structured interviews in the second phase (see Appendix

D) after the classroom observations in schools. The interviews lasted approximately

twenty to thirty minutes. The researcher interviewed all the teachers to confirm what he

had seen during the observation.

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Phase: 3

During the third phase, the researcher triangulated the data obtained by observation and

interview through a questionnaire. The questionnaire was administrated after observation

and interview. The participants were requested to fill in the questionnaire alone. All the

teachers filled the questionnaires collected on the following days. Moreover, during the

data collection phase, instructional materials such as student books, teacher lesson

plans, and students exercise books were collected by the researcher.

4.9 TRUSTWORTHINESS

Creswell (2014) argue that in qualitative research, trustworthiness has become an

important concept because it allows researchers to describe the virtues of qualitative

terms outside of the parameters that are typically applied in qualitative research. The

concepts of generalisability, internal validity, reliability and objectivity are reconsidered in

qualitative terms. To assure the reliability of the data utilized in this study, the researcher

used the following strategies: “member checking, triangulation, peer debriefing, and a

long stay in the field” (Creswell, 2014, p. 236).

4.9.1 Member checking

The data was transcribed, organized into cases, and analyzed after it was collected. The

researcher returned to the participants with the cases to ensure that they had been

properly captured. In cases where inaccuracies or misconceptions were discovered, they

were corrected.

4.9.2 Triangulation

Triangulation in qualitative research has come to mean a multimethod approach to data

collection and data analysis (White, 2005, p.89). In a qualitative investigation, researchers

tend to use triangulation as a strategy that allows them to identify, explore, and

understand different dimensions of the units of the study, thereby strengthening their

findings and enriching their interpretations (Creswell, 2014). The data collected through

observations, semi-structured interviews and questionnaires were triangulated. Before

the data collecting processes began, the data collection instruments were pilot tested. A

Highlight

Sticky Note

Rephrase the sentence. it is misleading.

Sticky Note

This part can be taken to the next chapter to see how the triangulation was applied. The application needs to show how the three instruments data were merged and triangulated to get the findings.

Sticky Note

The trustworthiness of data is best understood by when synthesizing data.

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pilot study guides the development of the study plan, as the smaller study informs and

gives feedback to the larger (final) study. Based on this feedback, the researcher can

adjust and refine the research instruments before attempting the final study. According to

Frankel, Wallen and Hyun (2014), a pilot study is a small-scale trial of the proposed

procedures. In addition, researchers (Sampson, 2004; van Teijlingen & Hundley, 2001)

define it as a specific tool for pre-testing of research instruments, including

questionnaires, observation guide and interview schedules associated with a quantitative

approach. In fact, the importance of pilot study has been expanded to the qualitative

inquiry where it is carried out as preparation for the major study. Kim (2010) and Padgett

(2008) argue that there is a measurable lack of research on pilot studies in general and

on pilot studies in qualitative research. Crossman (2007) argues that pilot studies are

much more common for quantitative studies than for qualitative ones. Regardless of the

paradigm researchers (Teddlie & Tashakkori, 2003; van Teijlingen & Hundley, 2002)

provide the reasons for performing a pilot study in quantitative and qualitative research.

Those reasons are the data collection process, resources management and the scientific

process. The pilot test will be discussed in the next chapter.

4.9.3 Peer-debriefing

The investigator discussed his findings with a friend who is professional in mathematics

education and took criticism that allowed him to add further details.

4.9.4 Prolonged stay in the field

To collect reliable data, a researcher must spend enough time in the field. Data was

collected in three phases over the course of five weeks. Following that, the researcher

spent a week performing member checks. According to Guba and Lincoln (2006),

member checking is a technique that is used in qualitative research to establish

trustworthiness.

4.10 VALIDITY OF THE DATA

Validity refers to the appropriateness, meaningfulness, and usefulness of the inferences

researchers make based specifically on the data they collect, while reliability refers to the

Highlight

Sticky Note

this section is about triangulation not validity. The highlighted explanation is about validity, please take note and adjust.

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consistency of these inferences over time, location, and circumstances (Frankel, Wallen

& Hyun, 2014). Quantitative researchers’ constructs of reliability and validity are

problematic for qualitative researchers in part because they represent rules of a research

game that qualitative researchers cannot possibly play. Qualitative research is based on

subjective, interpretive, and contextual data, whereas quantitative research attempts to

control those data (Auerbach & Silverstrein, 2003). Thus, the positivist viewpoints of

validity and rigour that are applied to quantitative research are not entirely applicable to

qualitative research (Maxwell, 1996).

Maxwell (1996) reveals five categories to judge the validity of qualitative research. These

are descriptive validity, interpretive validity, theoretical validity, generalisability validity and

evaluative validity. Descriptive validity means the accuracy of recorded information. The

data must accurately reflect what the participant has said or done. Moreover, the reporting

of the data must also reflect the same accuracy, which means that the transcription is an

accurate account of what was said, or the transcription of the video records portrays the

unfolding of the events in an accurate manner (Maxwell, 1996). Descriptive validity forms

the base on which all the other forms of validity are built upon. In this study, the researcher

returned to the participant with the cases to ensure that they had been accurately

collected. Interpretive validity refers to how well the researcher reports the participants’

meaning of events, objects and or behaviours (Maxwell, 1996). Theoretical validity seeks

to evaluate the validity of the researcher’s concepts and the theorised relationship among

the concepts in the context of the phenomena. Generalisability means the ability to apply

the study result universally (Auerbach & Silverstrein, 2003). For qualitative research,

generalisability is problematic. This study was delimited to Areka Town primary schools’

mathematics teachers. Finally, evaluative validity refers to the validity that moves away

from the data itself and tries to assess the evaluations drawn by the researcher (Maxwell,

1996). Validity in qualitative study is mainly concerned with description and explanation,

particularly whether a given explanation matches to a given description.

4.11 RELIABILITY OF THE DATA

In the field of research, reliability is broadly described as the dependability, consistency,

and or repeatability of a research’s data collection, interpretation, and analysis. Reliability

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is viewed very differently in qualitative research from how it is viewed in quantitative (Guba

& Lincoln, 2006). In the quantitative domain, reliability is specifically characterised as the

extent to which multiple researchers arrive at similar results when they engage in the

same study using identical procedures (Frankel, Wallen & Hyun, 2014). In these

conditions, differences in results are described as measurement errors. Therefore, from

a quantitative perspective, reliability is specifically defined, sought, and measured, and it

is accepted as an essential indicator of a study’s quality.

In contrast, because of the paradigmatic and methodological diversity of approaches that

comprise the field, reliability has not been described with such uniformity in qualitative

research (Creswell, 2014). Whereas many qualitative researchers describe parallel

concepts such as credibility, dependability, confirmability, and consistency as appropriate

qualitative correlates to reliability, others avoid the purposeful quest for reliability

altogether (Creswell, 2014). Three of the commonly cited indicators of credibility and

dependability are, (1) methodological coherence, which refers to the appropriate and

thorough collection, analysis, and interpretation of data, (2) researcher responsiveness,

which refers to the early and ongoing verification of findings and analyses with study

participants, and (3) audit trails, which refer to transparent descriptions of all procedures

and issues relative to the research project. The researcher employed the three strategies

to ensure the reliability-related issues.

4.12 ETHICAL CONSIDERATIONS

Before addressing other procedures, the researcher requested and took ethical clearance

of approval from UNISA (see Appendix 1). Primary schools in the study area are

administered by the town education offices and in this regard the researcher quested the

town education office (Mayor office) and received consent letter to carry out the study

(see Appendix B). After receiving the letter of consent, an investigator went to the selected

primary schools at ATA and obtained each school principal and informed about the study.

Following this procedure, he contacted each of Grade 8 mathematics teachers who were

chosen for the research and let them to sign the consent form. To secure the participants'

identities, codes (TA, TB, TC, TD and TE) were used instead of their real names and

addresses.

Sticky Note

it would help to explain how the reliability of data was achieved in this study of the limitations to achieve the reliability of data for this study.

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4.12.1 Informed consent

Overall, the researcher assured that written consent was obtained (Appendix C). The

purpose of the study was explained to participants, and they were ensured of their

anonymity. They can leave the study at any moment. According to Bless, Higson-Smith,

and Kagee (2006), the researcher should inform participants about the study and what

they must do to participate. An informed consent form was requested of each participant,

indicating that they had fully comprehended the information presented to them. The

importance of voluntary engagement cannot be overstated. The researcher described the

study's aim and why they were included in the sample.

4.12.2 Confidentiality

To maintain participant confidentiality, the information they gave, particularly personal

information, was kept private and was promised not to expose to someone else. Through

a written notice, all participants were ensured of their anonymity. To protect their identities

and ensure confidentiality, they were given aliases. In this study, to ensure confidentiality,

pseudonyms were used for the five mathematics teachers using the letters of the alphabet

TA, TB, TC, TD and TE. To ensure confidentiality, the participating schools were coded

using the letters of the alphabet, from A to E.

4.12.3 Data anonymity

The research data collected from the respondents should not be disclosed to any external

body and due to this fact, it should be confidential. The researcher promised for all

respondent, the raw data would be open only to the researcher and the researcher's

supervisor. The names of the participants and the names of the schools were not included

in the transcribed raw data.

4.13 CONCLUSION

In this chapter, the research design and methodology which underpinned this study was

presented. It also included the research paradigm, design, qualitative approaches,

population and sample, data collection technique and phases of data collection. Further,

it examined how trustworthiness, validity, reliability, and ethical issues were considered.

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CHAPTER FIVE

DATA PRESENTATION AND ANALYSIS

5.1 INTRODUCTION

In Chapter 4, the focus was research paradigm, design, qualitative approaches,

population and sample, data collection technique and phases of data collection. This

chapter aimed to present, and interpretation of the data collected to answer the research

questions. The data were collected from five Grade 8 mathematics teachers using non-

participant observations, semi-structured interviews, and questionnaires. The instruments

were piloted before performing the final study. The first section of this chapter explains

how the pilot study was conducted, followed by data analysis and interpretation.

5.2 PILOT STUDY

Pilot research was done at one school prior to the main fieldwork. In the pilot study, an

observation guide, semi-structured interviews, and questionnaires were used as data

collection tools. Two Grade 8 math teachers from a school with similar resources to those

in the main study took part in the pilot study.

My first visit to the school was a non-official meeting with the two mathematics teachers.

The purpose of this non-official meeting was to introduce myself to them and explain the

purpose of my research and get the timetable for classroom observation. In the meeting,

I emphasised the consent to their participation in the pilot study and shared experiences

on the teaching-learning of geometry. The pilot study was conducted after informed

consent from the participants had been obtained, and it addressed issues such as

confidentiality and the opportunity to resign from the study at any time. The pilot study

was conducted for one week. The pilot study was conducted in three phases. The first

phase of the pilot study was classroom observation, two lessons on the similarity of

triangles were observed. The researcher functioned as non-participant observer. The

lessons were video-recorded and after that, the transcription of each episode was done.

The use of a video camera provided detailed and accurate information about the

instructional session. The observation guide was used qualitatively to observe the

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classroom teaching and learning on the similarity of triangles. After the pilot study, some

changes were made to the observation guide. Table 5.1 below lists the initially planned

observation guide as well as the changes made to them after the pilot study.

Table 5.1: The initial and final observation guide after a pilot study

Before the pilot study After the pilot study

1. Which pedagogical approach teachers’ uses in teaching of similarity? Changes: adding, of triangles 1.1 How do teachers explain the concept similarity? Changes: adding, of triangles

Multiple perspective for the concepts Essential feature of similarity: static

nature /transformation nature Connection of the concepts

Changes: adding, in other geometry or mathematics, topics/ratio/proportion/slope/ graph of linear function

Relate with real life/environment of learners

Choose definition and common examples 1.2 Teachers teaching strategies for similarity Changes: adding, of triangles

Teacher use different diagrams picture Does the teacher give home and class work

give feedback? Does teacher encourage learners to use

hands-on manipulative activities? Inductive approach Deductive approach

Changes: removed, inductive and deductive approach Whole class approach, small group, as pair

and individual 2. How teachers interact with learners in the teaching-learning process? Changes: adding, of triangles

Interaction in the classroom Teacher-learner interaction Student-learner interaction Response to students’ questions Language

3. How the lesson plan prepaid? What are the methodologies suggested? Assessment techniques

4. How the teachers identify learners’ learning difficulty? Changes: adding, in similarity of triangles 5. What are the challenge teachers’ faces in teaching similarity? Changes: adding, of triangles 6. What strategies they adopt to solve these challenges?

1. Which pedagogical approach do teachers use in the teaching of similarity of triangles? 1.1 How do teachers explain the concept similarity/ similarity of triangles? a. Multiple-perspective for the concepts b. The essential features of similarity: static nature /transformation nature c. Connection of the concepts in other geometry or mathematics

topics/ratio/proportion/slope/graph of linear function

d. Relate with real-life/environment of learners e. Choose definition and common examples 1.2 Teachers’ teaching strategies for similarity of triangles a. Teacher uses different diagrams, pictures. b. Does the teacher give home and classwork and give feedback? c. Does the teacher encourage learners to use hands-on manipulative activities? d. Whole class approach, small group, as pair and individual 2. How do teachers interact with learners in the teaching-learning of similarity of triangles process? a. Interaction in the classroom b. Teacher- learners interaction c. Learners - learners interaction d. Response to learners’ questions e. Geometric Language 3. How was the lesson plan prepared? a. What are the methodologies suggested? b. Assessment techniques 4. How do the teachers identify students’ learning difficulty in the similarity of triangles? 5. What are the challenges teachers face in the teaching of similarity of triangles? 6. What strategies do they adopt to solve these challenges?

From the Table 5.1 above, some of the observation guide questions were modified for the

final study. After classroom observation, the researcher obtained consent from the

participants to audiotape the semi-structured interviews. In this regard, about twenty-five

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minutes was spent during the interview. Some of the interview questions were refined

and reworded. Table 5.2 below lists the initially planned interview questions as well as

the changes made to them after the pilot study.

Table 5.2: The Initial and final Interview questions after a pilot study


1. Do you explain the importance of learning geometry in general and the concept similarity in particular at Grade 8 mathematics syllabus?

Changes: Q1 refined and separated 2. Can you briefly tell me about the concept of geometry your learners learned before similarity? 2.1. Do you know what learners learn after similarity? Changes: reworded 3.Why (not) use instructional materials in teaching similarity? Changes: refined 4. Is there any difference between your teaching method/activities/ for similarity and other geometry topic? Changes: refined 5. Can you explain the common definition and or essential features of similarity? Give an example Changes: reworded 5.1 Do you explain the static and transformational approach of similarity? Changes: reworded 6. When you teach similarity, do you arrange any particular teaching environment? Changes: reworded 6.1Why did you conduct certain activities/examples/ in a class during the teaching of similarity? Changes: merged with Q6 7. What educational theories related to geometry do you know? Changes: adding another question 8. Do you have any factors that affect your interaction with learners in teaching similarity? How? Changes: reworded, adding mention and explain them 9. For you, what kind of teaching methods (pedagogical approach) are best for teaching similarity? Why? 10. What are the challenges you faced in the teaching of similarity? Why? Changes: adding, explain

1. Explain the importance of learning geometry in general.

2. What is the importance of learning the concept similarity of triangles in particular at Grade 8 mathematics syllabus?

3. Can you briefly tell me about the concept of geometry your learners learned before similarity of triangles? 3.1 What other topics are informed by the

knowledge of similarity of triangles? 4. Which instructional materials do you use when

teaching similarity of triangles? 5. Do you use a different teaching method/activity

when teaching similarity? Explain. 6. What is your understanding of the common

definition and or essential features related to similarity? Give examples. 6.1 Explain static and transformational approach

of teaching similarity? 7. What educational theories related to geometry do

you know? 8. How do these theories inform your teaching of

similarity of triangles to Grade 8 learners? 9. Do you have any factors that would affect/ impact

your interaction with learners while teaching similarity of triangles? Mention them and explain.

10. For you, what kind of teaching methods (pedagogical approaches) are best for teaching similarity of triangles? Why?

11. What are the challenges you faced when teaching the similarity of triangles? Explain.

As it is illustrated in Table 5.2 above, the researcher modified and restate some of the

interview questions before the main study. In this regard, some drawbacks of interview

items were omitted. Finally, the questionnaires were administrated for the 2 teachers, and

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some questions were modified. Table 5.3 below lists the initially planned questionnaire

items as well as the changes made to them after the pilot study.

Table 5.3: The Initial and final questionnaires items after a pilot study


A. Personal background (please circle the answer) 1. How old are you? i. Less than 30 ii. 30-35 iii. 36-40 iv. 41-50 v. 51-60 2. Are you female or male a. Female b. Male 3. What is your qualification? i. Certificate ii. college diploma iii. BSc iv. BEd 4. How many years will you have been teaching Grade 8 mathematics? Changes: refined and tabulated B. Teachers challenges review and reflection 1. Have you attended a professional development related to teaching similarity in the past 3 years? Changes: refined, labeled Q3 and added Q4 a. Never b. once or twice c. 3-5 times d. more than 5 e. other specify____________ 2. What is the importance of learning similarity at Grade 8? Changes: labeled Q1 3. What educational theories related to geometry do you know? For examples, the development of Piaget’s geometric concepts, and the development of van Hiele couple’s geometric thinking levels, please briefly describe some theory you had known below its importance for teaching similarity? Changes: reworded and adding, Fischbein’s theory, and Duval’s theory. Explain their importance for teaching similarity of triangles? labeled Q2 4. What mathematical/geometrical concepts must students have experience before they can truly understand similarity? Changes: refined labeled Q5 5. Can you tell me the concept of geometry your students had learnt before similarity? After similarity? Changes: labeled Q6 and added, of triangles 6. What factors affects your interaction with your learners? Why? Changes: labeled Q7 7. For you, what kind of teaching method (activity) is best in teaching similarity? Why? Changes: labeled Q8 8. What analogies, illustration, example or explanation do you think are most helpful for teaching similarity? How? Changes: labeled Q9 9. What are the challenges you faced in teaching similarity? Why? Changes: refined and separated as Q10& Q11

A. Demographic information

Please indicate/ fill below as appropriate: GENDER:

Male Female

Number of years teaching mathematics (in years)

< 5 6 – 10 11 – 15 16 – 20 21– 25

>25

Highest level of academic qualification

Diploma B Ed/ B Sc.

MEd/M Sc Other (specify)

B. Teachers Challenges Review and Reflection 1. What is the importance of learning similarity at Grade 8? 2. Which educational theories related to teaching and learning geometry do you know? For example, van Hieles’ geometric thinking levels theories, Fischbein’s theory, and Duval’s theory. Explain their importance for teaching the similarity of triangles? 3. Have you attended a professional development related to teaching similarity of triangles in the past 3 years? 4. If yes (in 3), what topics were covered? 5. What mathematical/geometrical concepts must learners understand before they can truly understand the similarity of triangles? 6. Can you tell me the concept of geometry your students had learnt before the similarity of triangles? After the similarity of triangles? 7. What factors affect your interaction with your learners? Why? 8. For you, what kind of teaching method (activity) is best in teaching similarity of triangles? Why? 9. What analogies, illustrations, examples or explanations do you think are most helpful for teaching similarity? How? 10. Name the challenges you faced in teaching the similarity of triangles? 11. How did you overcome each of them?

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From Table 5.3, some of the questionnaire’s items were modified and restated for the

final study.

5.3 DATA ANALYSIS PROCESS

According to Hatch (2002), analysis refers to “organizing and interrogating data in ways

that allow researchers to see patterns, identify themes, discover relationships, develop

explanations, make interpretations, mount critiques, or generate theories” (p. 148). A

qualitative data analysis is a continuous process which involves synthesis, evaluation,

interpretation, categorization, hypothesizing, comparison, and pattern finding (Creswell,

2014, p. 156).

According to Creswell (2010), the data that emerge from a qualitative study are

descriptive. That is, data are reported in words (primarily the participant's words) or

pictures, rather than in numbers (p. 195). The aim of data analysis is to get synergetic

effects of collected data in the way that gives sound conclusions and findings based on

set criteria and standards. Moreover, Miles, Huberman, and Saldana (2014, p.12) further

describe qualitative data analysis as three concurrent flows of activity: (i) data

condensation, (ii) data display, and (iii) conclusion drawing/verification. In this research,

the data collected from five mathematics teachers were subjected to qualitative methods

through the process of data analysis, a process of ordering, structuring, and generating

themes to a set of data collected (Merriam, 2009).

The researcher used coding to condense a large volume of data into manageable units

during data analysis. According to Elliot (2018), researchers use a coding process to

develop new categories and themes from the data collected. It is here that irrelevant

information is discarded and set aside for future use if the researcher has to re-examine

data previously deemed useless. Researchers use coding to categorise data relevant to

a theme instead of following the sequences in which the participants responded to

research instruments. Elliot (2018) also emphasised the importance of coding as it assists

the researchers to source meaning that speaks to the category of the research problem.

In this study, preceding to analysis preparation, the transcription of data from the lesson

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observation of the episode in the video records and the interview audiotapes was done.

The coding processes followed in this study are summarised in Figure 5.1.

Figure 5.1: The data coding process

During data collection, field notes were taken, and transcripts of classroom observations,

semi-structured interviews, and questionnaires were examined, synthesised, and

critically analysed in order to recognise trends and their corresponding categories (see

the Figure 5.1). The data collected from the classroom observations, semi-structured

interviews and questionnaires were coded manually then, coded data were categorised

into themes, which were used to draw conclusions and develop assertions on exploring

the challenges of teaching similarity of triangles to Grade 8 students and the pedagogical

approaches to promote meaningful teaching of similarity of triangles. The emerging

categories and themes should address a phenomenon related to a specific set of

research questions discussed in the following sections. The data theme analysis was

done in the sections that follow starting with the demographic characteristics of the

participants.

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5.3.1 Background Characteristics of Participants

Table 5.4: The teachers’ demographic information

GENDER

Male

4

Female

1

Number of years teaching

mathematics (in years)

< 5 6 – 10 11 – 15 16 – 20 21– 25 >25

- 1 2 1 1

Highest level of academic

qualification

Diploma B Ed/ BSc. MEd/M Sc Other (specify)

- 5 - -

From Table 5.4, the result of part A of the questionnaire shows that there were 4 male

and 1 female Grade 8 mathematics teachers in the sampled school. In addition, Table 5.4

above indicates that teachers’ teaching year range from 6 to 22. All the teachers had good

experiences in mathematics teaching. From the Table 5.4, all the teachers were qualified,

which was an excellent reflection of what a qualified teacher cohort should look like. In

general, there was no reason to doubt the qualifications of teachers as a potential cause

of challenge in teaching similarity of triangles in primary schools. The results of theme

analysis are presented in the section below. In this study, pseudo-names were used for

the five mathematics teachers for ethical reasons. The following pseudo-names were

used to identify the teachers from the five schools who participated in this study: TA; TB;

TC; TD and TE represented Areka Mulu 1 Dereja; Del-Behiret; Addis Fana; Dubo Mulu;

and Wormuma Primary Schools, respectively.

5.4 RESULTS OF THE THEME ANALYSIS

Themes and the related categories which arose from the codes are presented in the

following sections. In this study, the coded data were categorised into 4 themes. These

are:(i) importance of learning geometry, (ii) phases of instruction in teaching similarity of

triangles, (iii) challenges teachers faced in teaching similarity of triangles, and (iv)

suggested strategies to minimise the challenges of teaching similarity of triangles. The

following figures show how the coding was done, the subcategories, categories and

Highlight

Sticky Note

This part needs to be explained to show the meaning of qualified in this study. example, the category of highest qualification. The result is B. Ed./BSc and the explanation is that they were qualified. In which sense? This affects what the teachers can do when they get to classrooms.

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themes that emerged. Each of the themes and categories is discussed in the next

sections.

Figure 5.2: Theme-1: Importance of learning geometry

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Figure 5.3: Theme-2: Phase of the instruction in teaching similarity of triangles

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Figure 5.4: Theme-3: Challenges teacher faced in teaching similarity of triangles

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Figure 5.5: Theme-4: Minimising the challenges of teaching similarity of triangles

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5.4.1 Theme 1: Importance of learning geometry

As illustrated in the Figure 5.2, Theme 1 emerged from the data analysis carried out from

non-participant observation, semi-structured interviews, and questionnaires. The data

revealed that the teachers in the sampled schools were cognizant of the importance of

teaching geometry in general and the teaching of similarity of triangles. In the next section,

I expanded further on Theme 1 and reported the following categories that emerged from

the codes that support teacher’s awareness of the importance of learning geometry: (i)

reasons for studying geometry and (ii) geometry in relation to daily lives.

5.4.1.1 Reasons for studying geometry

As it has been argued on the teacher’s awareness of the importance of geometry, the

data from observations, semi-structured interviews and questionnaires revealed that

teachers responded that learning geometry improves students’ geometric and cognitive

skills. For example, during the interview session, TB was requested to explain the

importance of teaching and learning geometry, in this regard TB reported that:

“The importance of learning geometry is ehh...it helps us to decide on what materials showed we use and what designs showed we make; it also playsa vital role in a construction and it also helps to understand shapes, the measurements of three-dimensional objects, such as cubes, cylinders, pyramids and spheres can be computed using geometry, so this is the importance of learning geometry. Geometry is the study of measurement.”

Moreover, TE explained the importance of geometry as it is part of mathematics. During

the interview, she was asked to explain the importance of teaching and learning

geometry, and she said that:

“Geometry is a part of mathematics and when we teach geometry, its importance is, … the concept geometry is derived from ‘geo’-, which means ‘earth’ and -‘metry’ means ‘measurement’, especially our subject teaches about measurement, and it is also found in engineering, and the students get this basic knowledge in their future building professions, and others too. It has this much importance.”

Some of the teachers further explained the importance of learning similarity. During the

interview, TA was asked to explain the importance of learning similarity, and he claimed

that:

Highlight

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“Similarity is an important topic to be taught in Grade 8 mathematics because learning similarity helps to understand the geometric definition of similarity of figures, such as triangles and polygons.”

Teachers TC, TD and TE also responded in the same way on the importance of similarity

of triangles in questionnaires. However, TB further explained the importance of similarity

as a visual representation of mathematical concepts such as ratio and slopes. He said

that: “similarity is the visual representation of concepts such as ratio, proportion and slope

of triangles”. Participants responded to the question on the importance of geometry by

relating geometry to the real world.

5.4.1.2 Geometry in relation to daily lives

During the interview session held with TA, he argued on relating the importance of

geometry to this real-world as follows:

“The importance of geometry can be seen in various ways. In a real-world, there are many things related to the geometry that we can see in our naked eyes; there are many tactile issues related to it…when we simply see our school compound’s fence, it has a rectangular shape. uhh…geometry in other ways, injera that we bake to eat has a circular shape. if we raise the concept “how to bake injera”, we can attach our students to the geometry concept…so we can talk geometry in our daily life activities…there are many things that can be connected to geometry. So, we describe geometry in various ways, and it is essential.”

TA explained that many things found in our surrounding environment are related to

geometry. He related the importance of learning geometry to our daily life experiences,

while some of the teachers related the importance of learning similarity to the real world.

They further explained that learning similarity may help for daily life activities such as sun

shadow, copying machines, mirrors and photos. Some of their arguments are presented

below.

TC said that:

“Learning similarity is important. Similarity helps to know the concept of enlarged

photographs, maps, cars, persons and copying machine, etc.”

TB said that:

“Learning similarity for the students improves their geometrical knowledge; it helps to measure the area of the land; it helps to understand the relationship between

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points, angles, surfaces, and solids. Similarity helps to find similar plane figures by using criteria. To have knowledge of similar triangles and their properties or theorems which made the two triangles similar.”

TD said that:

“Similarity is connected to many events in our daily life. For example, sun shadow of objects, we observed sometimes enlarged and reduced in size. Similarity may help to understand such phenomena easily.”

During my observation, I also observed that some teachers used examples on the

enlargement of triangles; however, they seemed to lack to mention the application of

similarity related to real life. For example, TE used the example on the enlargement of

triangles. She drew triangle �� on the chalkboard and tried to read out what is displayed

in Figure 5.6.

Figure 5.6: TE’s example on the enlargement of triangles

After she had written an example on the chalkboard, TE argued that:

“ይሄ ምንን ይገልጻል ይች እንዚህ ትራያንግሎች enlarged triangle �� with scale factor 2 and central

enlargement, 2 ከየት ይወታል? እንዴት ይመጣል? በፎሩሙላ ውስጥ ሆኖ እዛ ቅድም የስኬል ፋክተሪ ፎሩሙላ

እንዳየነዉ k=2”

(What does this describe? This is about enlarged triangle �� with scale factor 2

and central enlargement. From where does 2 emerge? How is it found? As we

have already seen in the formula of scale factor, � = 2).

It was again observed that TE language is not understandable. She further explained that:

“We have 2 from the previous calculation of the proportionality constant”. I did not observe

TE finding the value 2. However, as illustrated in the Figure 5.6 above, 2 is the given

scale factor value to enlarge triangle ��. TE lacked the understanding of example.

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She continued in Amharic by saying:

“ለምሳሌ ይህ ትራይንግል equilateral ነው”

(For example, the given triangle is an equilateral.)

Here, it was also observed as an example that TE argued that the given triangle is an

equilateral. However, from Figure 5.6, there is no evidence indicating that the triangle

�� is an equilateral triangle.

She started to copy the solution from her notebook and wrote on the chalkboard while

speaking (see the Figure 5.7.).

Figure 5.7: TE’s classroom work on enlargement of triangles

As illustrated in the Figure 5.7, TE wrote without explanation of how each of the sides of

triangle �� was enlarged. At no stage did she mention that the enlarged triangle �′�′�′

and the ratio of the corresponding sides of triangle �′�′�′ is 2. It is my opinion that TE

may lack the knowledge of how to enlarge and reduce triangles by relating to objects or

figures in our surroundings. Moreover, the explanation lacks proper use of geometric

language.

It was also observed that TD used the following example (see the Figure 5.8) during his

explanation of similarity. The same figure was also observed in TB and TC lessons.

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Figure 5.8: TD’s Examples of two similar rectangles

As illustrated in Figure 5.8, TD used smaller and larger rectangles to explain the similarity

of plane figures. However, he could not mention similar plane figures can be obtained by

enlargement/reduction of the same plane figure. It was also observed that TA, TC, and

TD did not use examples that related similarity to the real environment.

As the questionnaire results revealed, teachers’ arguments favoured the importance of

learning similarity of triangles. This is what they said:

TB: “Similarity is a fundamental property and great importance in retrieval and

categorisation tasks alike. Similarity is important to enlarge and reduce figures by

a scale factor k.”

TD: “Similarity helps to solve some real-life problems and to prove certain

geometrical properties. It also helps to construct a similar figure”.

TC: “Similarity helps to develop students’ sketching ability of figures.”

5.4.2 Theme 2: Phases of the instruction in teaching similarity of triangles

As illustrated in the Figure 5.3, Theme 2 emerged from the codes that yielded the

following categories: (i) concepts related to similarity, (ii) similarity of triangles and (iii)

teaching approaches. In the next section, I expand further on Theme 2 and report on the

following related categories to elaborate the actual instructional process on the sampled

schools.

5.4.2.1 Concepts related to similarity

As illustrated in Figure 5.3 above, the concepts related to similarity emerged from the

observations, semi-structured interviews and questionnaires. The codes in the first

section were categorised under concepts related to similarity. They were further

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subcategorised concepts learned before similarity and concepts learnt after similarity of

triangles. The analysis revealed that some of the teachers were aware of the connection

of similarity with other mathematical and geometric concepts. To understand teachers’

awareness of the concepts related to similarity, the data from the semi-structured

interviews and questionnaires were used. Some of the teachers had an awareness of the

concepts to be learned before similarity. During the interviews, when TE was asked to

explain the concepts of geometry that her learners had learnt before and after similarity.

She said that:

“First, we should teach the students about the proportionality, or enlargement before going to teach them about similarity because the students get full understanding when these concepts are first introduced. Sorry, I don’t know what they will learn after similarity.”

TE did not mention the concepts learnt after similarity during the interview. However, she

responded on the questionnaire concepts like “theorems of similarity, SSS, SAS and AA”

students’ to be learnt after similarity.

TA also explained the connection of similarity with other concepts during the interview.

He said that:

“Similarity is related to concepts of shapes and size of geometric figures. I hope if students understand about shapes and size they will easily understand about the similarity of triangles.”

He further responded on the questionnaire that the concepts learners learned before and

after similarity. This is what he said:

“The students learnt about naming of 3-sides polygons as triangles, 4-sided polygon as square or rectangle or parallelogram or rhombus, or kinds of lines by saying as line segment, line or ray, a point or denoted as (.). But after similarity, the students learn Tests i.e. (SSS, SAS, AA) similarity theorems which are used in order to decide whether the given triangles are similar or not and the relationship between the perimeter, side and area of the given similar polygon.”

TB explained the connection of similarity with the measurement of plane figures. He said

that:

“Before similarity, they understand the concept geometry as the measurement of

figures and lines, angles and solids figures.”

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He further responded to the questionnaire about the connection of concepts with

similarity. This is what he said:

“Students must learn concepts of geometry before similarity such as plane figures like triangles, rectangles (area and perimeter) of rectangles, and quadrilateral. Also, they learnt about measurements like angles, types of angles, etc. After similarity they learn about similar triangles, rectangles and other polygons, and also the area and perimeter of similar polygon.”

During observation, it was noted that TB connected similarity with congruence. He wrote

on the chalkboard, “congruence is a similarity where the constant of proportionality is 1.

If two triangles are congruent, then they are similar.” He further said that: “not all similar

figures are congruent”. He wrote an example on the chalkboard at the same time

speaking (see the Figure 5.9).

Figure 5.9: TB’s connection of similarity with congruence

TB started the explanation in Amharic, and he said that:

“እነዚህ ትራያንግሎች ኮንግራንት መሆን የሚችሉት መች ነው? ፕሮፕሪሽናል ኮንስታንታቸዉ 1 ስሆን ነዉ፣ ስለዚህ ይህን ማሳየት ይጠበቅብናል,”

(We say the triangles are congruent if their proportionality constant is 1. So, we need to show it).

Then he started the explanation, he said that:

“When we say proportionality constant is 1, we are saying simply side �� ≡��, �� ≡ ��, �� ≡ ��.”

TB: For example, if the value of the length �� 2, then

Students: ��, �� 2 (responded as a whole class)

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TB: therefore ��

��=

��

��=

��

��= 1.

TB: the angles are also congruent, ∠� ≡ ∠�, ∠� ≡ ∠� �� ∠� ≡ ∠�

It was observed that TB tried to connect similarity with the congruence of triangles.

TC explained the connection of similarity with proportionality. During the interviews, when

TC was asked to explain the concepts of mathematics or geometry that his learners learnt

before and after similarity. He said:

“Before teaching the students about similarity, the students should be given proportionality concepts i.e enlargement. Sorry, I do not know what they will learnt after.”

It was observed that some of the teachers’ TA, TC, TD lessons lacked connection of

similarity with other concepts. The data showed that most of the participants may lack the

concepts to be learnt after teaching the similarity of triangles.

5.4.2.2 Similarity of triangles

As illustrated in the Figure 5.3, the similarity of triangles category emerged during the

actual teaching of the concept, similarity of triangles. During the instructional process,

teachers try to explain the concepts by drawing and giving examples from the students’

textbook. The following section illustrates some of the lessons on the similarity of

triangles, and examples teachers used to explain similar triangles.

TC started the lesson on the similarity of triangles by writing on the chalkboard “similar

Triangles” and questioned the learners. TC wrote notes on the chalkboard while speaking,

“If ∆��~∆��, what does it mean? He also asked the same question in Amharic,

“ትራይንግል ��ከትራይንግል ��ጋር ስሚላሪ ነዉ ማለት ምን ማለት ነዉ? (What do we mean if we said

triangle ABC similar to triangle DEF)” then,

TC: ∆��~∆�� ehh….

Learners: their corresponding sides are proportional (one learner replied).

TC: wrote on the chalkboard, corresponding sides are proportional

TC: the next is what?

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Learners: their corresponding angles are congruent (the same student replied)

TC: wrote on the chalkboard, corresponding angles are congruent

TC: so, we say that the two triangles are congruent their corresponding sides are proportional and corresponding angles are congruent.

He shaded on the angles which are congruent on the chalkboard (see the Figure 5.10

below). TC continued the explanation on the chalkboard.

TC: first, corresponding sides are proportional what do we mean? ehh…

Learners: silent

TC: 1. ��

��=

��

�� then, the next.

It was observed that TC drew the triangles after he wrote the ratio of the corresponding

sides. He continued the explanation:

TC: 2. the other sides ��

��=

��

��

TC: This is what we mean sides are proportional, pointing at the ratio of the sides

TC: The next is angles are congruent, and that means which angle is congruent to

which one?

TC: Angle A is congruent to what?

Learners: Angle D (as a whole class)

Then, TC wrote on chalkboard while speaking.

TC:1. ∠� ≅ ∠� �� (∠�) = �(∠�)

TC: 2. ∠� ≅ ∠� �� (∠�) = �(∠�),

TC: 3. ∠� ≅ ∠� �� (∠�) = �(∠�).

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Figure 5.10: TC’s explanation of similar triangles

It was observed that TC labelled the angles on each triangle after the explanation. TC

continued the lesson by saying, in Amahric, “አንድ ምሳሌ መስራት ይቻላል, (‘We can do one

example)”, Then he wrote Example 1 from student textbook on the chalkboard at the

same time speaking (see the Figure 5.11).

Figure 5.11: TC’s example on the similarity of triangles

After writing the above example illustrated in the Figure 5.11, TC argued about the

example in Amharic. He argued that:

“አሁን እዚጋ የተጠየቅነዉ ምንድነዉ? (Here what were we asked?); using the given condition and definition we can find the side length of �� and ��.”

Then, he pointed on Figure 5.11 illustrated above, then he wrote on the chalkboard:

TC: Solution

TC: From this, using the definition, angles congruence and proportionality we know that triangle �� is similar to what?

Learners: triangle �� (one learner)

TC: No! ∆�� (He wrote ∆�� on the chalkboard then removed it)

TC: Triangle ��

TC: From side proportionality ⇒��

��=

��

��=

��

��

Learner: Which is equal to � (one learner)

TC: Wrote � on the ratio equation

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TC: From this, we can find �� and �� (pointing on the chalkboard and shaded the points)

TC also said in Amharic, “ስለዚህ �� እና �� ነዉ የምንፈልገው(so, we need to find BC and XZ) ”. Then

continued his explanation.

TC: We can substitute the value of �� (he checked learner textbook; then, corrected the ratio equation).

TC was unassured of what he was explaining in the classroom. Then,

He corrected the ratio equation as:

TC: ��

��=

��

��

Learners: They were seen murmuring with each other (on what the teacher wrote on the chalkboard)

TC: �� is given 6cm, over �� or �� is also 5cm, ��

��=

��

��=

��

��

Students: teacher…. teacher (loudly) �� is not given, the given is ehh….�� (2 students were found participant in the class)

TC: removed 6cm from the ratio

Learners: �� is the given which is 14cm (one learner)

TC: a) �� = �� + ��

TC: ��=?

Learner: 8�� + 6�� (as a whole class)

TC: �� = 14��, let find the unknown value of by using the ratio equation

TC: ⇒��

�=

��

�=

�

��= �

TC: Now we can find the value of ��, the equation becomes what?

TC: ��

�=

�

��

TC: By cross multiplication, the value of �� is equal too……

Learner: 42 = 14 ∗ �� (one earner)

TC: 14 �� = 42

Learner: We divide 14 on both sides (as a whole class)

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TC: �� =��

��

Learner: We can cancel by 2 the equation

TC: Ok the value of �� =3cm

TC: We can check the value by finding the value of �, now we do not have time to check.

TC: � =��

��

It was observed that two learners participated during the lesson while they were correcting

the teacher’s work. It was also observed that the rest of the learners did not participate in

the lessons. They were talking to each other. The teacher might have missed the

important knowledge on what was given on the example, and what was required. He

missed writing the corresponding sides properly of the two similar triangles. Then, to find

their proportionality constant, he was found poorly prepared for the lesson. The following

section is TB lesson observation.

It was observed that TB began lesson by writing on the chalkboard “similar triangles” at

the same time speaking. Then he said that:

TB: Triangles are three-sided polygon yes! then in Amharic (ትሪያንግሎች ሶስት ሳይድ

ያላቸው ፖልጎን ናቸዉ)

TB: Triangles are three sides and three angles, and their measure of interior angles

is what?

Learners: 180 degrees (the whole class)

TB: የትሪያንግል ስሚላሪትን በሁለት እና ያለን በአንግልና በሳይዳቸው:: ትሪያንግሎች በስንት ይከፈላሉ? (We

can see the similarity of triangles in two ways, -in angle and sides. Triangles can

be classified into?)

TB: Based on angles, triangles can be divided into what?

Learners: Two (the whole class)

However, the teacher did not correct the students’ response which was wrong.

TB: Right angle, obtuse and acute (he wrote on the chalkboard at the same time

speaking)

127

TB: Depending on their sides, a triangle can be classified into……

Learners: Silent

TB: Equilateral, scaler and isosceles triangles. Now, let us see their similarity.

TB drew two triangles ∆�� and ∆�� at the same time speaking (see the Figure 5.12.)

He labelled the angles of each triangle with their corresponding angles.

Figure 5.12: TB’s work on similar triangles

TB: �� ∆��~∆��, what are the criteria?

Learners: Silent

TB: i) Corresponding angles are congruent, ∠� ≡ ∠�, ∠� ≡ ∠�, ∠� ≡ ∠�

Learners: Talking after the teacher wrote on the chalkboard

TB: The next criteria is what?

Learners: Silent

TB: ii) Corresponding sides are proportional, ��

��=

��

��=

��

��= �

TB: We will see that there are theorems for similarity of triangles and, then he wrote

AA similarity, SAS similarity, SSS similarity.

It was observed that TB started the lesson by reminding learners about triangles. As

illustrated in Figure 5.12, TB did not label the corresponding sides of the two triangles.

He labelled the corresponding angles. He put the ratios; however, it could not be seen by

the learners at the back of the classroom. It was observed that TB did not remind his

128

students about the criteria of similarity of plane figures before explaining the similarity of

triangles.

On the contrary, TD began the lesson by writing on the chalkboard at the same time

speaking “Tests for similarity of triangles SSS, SAS, AA”. Then, he wrote “Theorem: AA

similarity theorem”. He shaded the theorem then; he wrote the definition on the

chalkboard at the same time speaking.

TD: If two angles of one triangle are congruent to the corresponding two angles of

another triangle, then the two triangles are similar.

After he wrote the theorem, he said, “let us see an example” and he drew the Figure 5:13

below.

Figure 5.13: TD’s examples on the theorem of similarity of triangles

He started the explanation by saying:

TD: Angle � is similar to or congruent to angle � (he looked at his notes then wrote)

∠� ≅ ∠�

TD: Another one is, ∠�� is congruent to or similar to angle… ∠�� ≈ ∠��(see

the Figure 5.13)

TD: Pointing on the chalkboard on the figure angle � and ∠� are corresponding

angles

TD: This angle is congruent to this angle (pointing on the figure ∠�� ∠��)

because they are vertically opposite angles.

129

It was observed that TD could not explain the two similar triangles. The language for

“similar to or congruent” used by TD the symbol “, ≈” was not correct. TD’s explanation

lacks the condition for similarity of triangles; that is, the proportionality of corresponding

sides of the two triangles and the congruence of their corresponding angles. He did not

properly locate or show the angles ∠��, ∠�� on the chalkboard from the figure. He

just talked. TD could not mention why the two triangles were similar if the two angles were

congruent. TD may lack the knowledge of the importance of the theorem. He could not

mention the two similar triangles. It was observed that the theorem is not explained for

learners properly. TD may face geometric language and mathematical knowledge

challenges.

TA’s explanation and examples on the similarity of triangles are presented in the next

section, the teaching approaches.

5.4.2.3 Teaching approaches

As illustrated in the Figure 5.3, the teaching approaches category emerged from the

coded data. Under the teaching approach category, subcategories such as teacher-

student interaction and hands-on manipulative activities emerged from the coded data. In

support of predominated teacher-talk pedagogical approaches, the results from the

lesson observation data analysis revealed that teachers used chalkboard as the directed

classroom activities through writing the definition of similarity of triangles then followed by

a demonstration of examples using the chalk and talk approach.

For example, during the observation, one of the teachers, TA as illustrated in the figure

below wrote the definition of similarity of triangles on the chalkboard then directed

learners to copy the note from the chalkboard. The following section presents the actual

classroom observation.

TA wrote on the chalkboard “similar triangles (p.112)” then he said that “I am going to

write the note or the definition for you, you should have your exercise book, pen and

textbook, be ready to write the definition after I wrote on the chalkboard.” then he started

to write the definition on the chalkboard (see the Figure 5.14).

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Figure 5.14: TA’s explanation of similar triangles

According to the notes depicted in the above Figure 5.14, it took 9 minutes to write the

definition of similarity of two triangles on the chalkboard. TA then waited for the learners

to copy the note from the chalkboard. The classroom was made noisy. Some of the

learners were busy talking with each other. Some learners copied the definition from the

chalkboard. After 2 minutes, TA started the discussion on the definition of similarity of

triangles and it had been presented as follows:

TA: To find out whether the given two triangles are similar, he pointed on the

chalkboard on the figures of the given two triangles ∆�� ∆��.

TA pointed on the chalkboard, saying:

TA: Two basic conditions for the similarity of triangles. The first condition shows

that their corresponding sides are proportional. He reminded the discussion on

the previous lesson about the proportionality constant �; then, started to find for

the value of �.

Learners: Listening.

TA started writing on the chalkboard at the same time saying:

TA: The value of � can be determined from the relationship written, ��

��=

��

�� ,

��

��=

��

�� ,

��

��=

��

�� pointing on the chalkboard.

TA wrote at the same time saying:

TA: � =��

�� =

��

�� where � ≠ 0, � ≠ 0.

131

Then, he started simplifying on the portion of the chalkboard for the value of � = 32� . TA

concluded by saying:

TA: Since we have obtained the proportionality constant �, which is constant for

the two triangles then the two triangles are similar.

In all of TA’s explanations, there was no mention of the checking of triangles'

corresponding angles.

However, later TA continued on the second condition by pointing to the chalkboard,

“their corresponding angles are congruent”. He continued the explanation:

TA: If we measure the degree measure of the angle < � and the degree

measure of angle < � by using protractor they are equal. Since we do not have a

protractor, we have to assume the two angles are congruent. Angle � �� < �

are congruent, �� < � �� < � are congruent and angle < � �� < � are

congruent.

TA continued his explanation by giving an example from his mind, writing on the

chalkboard, and talking at the same time.

TA: If the degree measure of angle �(< �) = 30,

Learners: The degree measure of angle �(< �) = 30, (Responding as a whole

class)

Then he said, “the same is true for the others”.

TA said that:

If the degree measure of �(< �) = 60, what is the degree measure of

angle(< �) = 60?”.

Learners: The degree measure �(< �) = 60. (Answering as a whole class).

TA: You can determine the remaining angle from the properties, the sum of the

degree measure of the triangle is 180.

132

TA did not specifically conclude here that the two given triangles are similar from the

definition of similarity. That is, the sides of the two triangles are proportional, and their

corresponding angles are congruent. It was observed that the TA classroom is traditional

teacher-centred, learners were found busy copying notes from the chalkboard. They were

passive listeners during the discussion. TA continued to write notes on the chalkboard.

It was also observed from TA that the note continued, “We have an example, I will write

an example for you then, you have to copy it from the chalkboard after I write it”. TA wrote

down the example with the answer from his notebook see Figure 5.15. TA also took 9

minutes to write the example with its answer on the chalkboard. Learners were busy

copying the example from the chalkboard. TA moved around the classroom as the

learning were writing the notes.

TA started to read the example from the chalkboard by pointing on the chalkboard and

then gave explanations. He had also shaded on the points on the chalkboard. He

continued the discussion in the classroom by saying:

TA: “Let us discuss; this example is useful, because it gives as clues for the

exercises which are written in your textbook.” He was reading the example from

the chalkboard and pointing on the chalkboard, “let ∆�� be similar to ∆��; then,

find angle �(< �) , angle �(< �) and the length of ��”. He continued the

discussion by pointing on and reading the solution from the chalkboard. He made

shadow on the points. TA said, “from the figure �(< �) = �(< �), again

�(< �) = �(< �) and �(< �) = �(< �). " Some of the students participated as

the whole class.

133

Figure 5.15: TA’s examples of similar triangles

It was observed that the lessons lacked the connection of similarity with real life

environments, the connections of concept with geometry and other mathematical

concepts. TA’s instructional approaches were observed while he was directing learners

to copy from the chalkboard followed by talk.

During the interviews when TA was asked to explain the kind of pedagogical approach

best for teaching similarity of triangles, he said that:

TA: “I use demonstration to show them and to make them use the methods I

demonstrated; showing the figures, similar figures; the next is discussion,

discussion now is used to develop ideas run by demonstrations. Open discussion

is good to develop the students’ next academic status.”

It was also observed that TA lessons lacked demonstration approaches of teaching

similar figures. The lessons were dominated by copying notes from the chalkboard. TA

lacked pedagogical knowledge on how to apply demonstration approach for teaching

method. I could not observe discussion on the similarity of triangles in the classroom

during the initial phases of the lesson. However, I observed that TA showed the maps of

Ethiopia from the student textbook. The uses of different models were discussed in the

next section. During the initial phases of the lesson, learners were subjected to copying

notes from the chalkboard; they were passive listeners.

The practices of writing notes on the chalkboard followed by explanation was also

observed in TB, TC, TD and TE. However, how TB and TC worked the examples on the

chalkboard differed although the lessons were teacher directed. The following

presentation shows TB’s instructional approach.

TB began the lesson by writing on the chalkboard at the same time speaking “Similar

figures”. Then, he continued the lesson by writing on the chalkboard at the same time

speaking “Any plane figures are similar if and only if what?”

TB: What are the criteria?

TB: The first is corresponding sides are proportional and the second one is what?

Highlight

Sticky Note

there is not relevance of this sentence to your argument, and it is left hanging.

134

Learners: The corresponding angles are congruent (whole class responded)

TB continued the discussion by drawing two triangles from his mind at the same time

speaking. He labelled each side and angles of the two triangles (see the Figure 5.16).

Figure 5.16: TB’s example of similar triangles

After drawing the above similar triangles on the chalkboard, TB argued that:

TB: Now we want to show the two triangles are similar, what are the criteria?

He started to state the criteria,

TB: The first criteria is corresponding sides are proportional

Learners: Corresponding sides are…proportional (speaking out, reading from

chalkboard)

TB: From given ∆�� ∆��, ��

��=

��

��=

��

��= �, this � said to be

proportionality constant.

TB continued the discussion by asking “What is the second criteria?”. He is pointing on

the angles on chalkboard, then he wrote at the same time saying:

TB: angle � ≡ ��, � ≡ ��, � ≡ ��. Thus, ∆��~∆��. We say the triangles are

similar.

TB continued the discussion by writing an example from student textbook (see the

Figure 5.17). He gave about 5 minutes for learners to copy from the chalkboard.

135

Figure 5.17: TB’s examples of similar triangles

As illustrated in Figure 5.17 above, TB started the discussion by pointing the figures on

the chalkboard. Then, he read the examples after he wrote the example.

TB: “We want to show the two triangles are similar, so for triangle ∆�� the

given condition are ∠� = 55, ∠� = 75, there is unknown angle, pointing on

chalkboard on angle ∠�, he put question mark on angle �.”

TB also put a question mark on the unknown angle of the triangle ∆��, on the

angle ∠�. Then he said that “First we have to find the unknown values of the angles”.

TB wrote on the chalkboard at the same time speaking:

TB: Note that, the sum of interior angles of any triangle is what?

Learners: 180 degrees, (as a whole class)

TB: By using the rule, 75� + 55� + �(∠�) = 180�

TB: What is the sum of 75 �� 55?

Learners: 130

TB: So, 130 + �(∠�) = 180�

TB: Therefore, the measure of angle �(∠�) = 180� − 130�

Learners: Speaking at the same time what the teacher writing on the chalkboard

TB: The measure of angle �(∠�) is equal to?

Learners: 50 degrees as whole class

136

TB: Yes, 50 degrees. After we determined the degree measure of the angle, we

will go to similarity.

TB: In order to show the similarity, the first criteria of the corresponding sides are

what?

TB: Proportional,

Learners: Proportional, after the teacher spoke it.

TB: Therefore, triangle ∆��~∆�� if and only if, the first their sides are

proportional

It was observed that TB wrote the corresponding sides ratios as follows:

TB: ��

��=

��

��=

��

��

TB: What is the length of ��?

Learners: 5 centimetres, as a whole class,

TB: What is the length of ��?

Learners: 2 centimetres (responding as a whole class)

TB wrote the ratio on the chalkboard at the same time speaking, “ ��

��=

�.��

��=

��

��=

�”. Learners are answering as a whole class, by saying “5 cm, 2cm…” for each of the

sides when the teacher was asking them. Then TB said, “We cancel centimetre by

centimetre from each ratio and the values of � = 2.5”.

TB: The value of � is unitless because it is a constant number. What are the next

criteria?

Learners: Sides are congruent (as a whole class)

TB: What about angles?

Learners: The angles are congruent, (as a whole class)

TB: The corresponding angles must be congruent.

TB continued the explanation by writing and speaking at the same time,

TB: The measure of angle �(∠�) = �(∠�) this implies that if the degree

measure of angle ∠� is equal to 55, then what is the value of angle ∠�?

137

Learners: 55 degree

TB: Again, what is the next angle?

Learners: Angle �, as a whole class

TB: �(∠�) = �(∠�) that is equal to?

Learners: 75 degrees, (as a whole class)

TB: �(∠�) = �(∠�) = 75�, what is next?

Learners: Angle �, (as a whole class)

TB: �(∠�) = �(∠�) this angle what we find it

Learners: It is 50 degrees

TB: �(∠�) = �(∠�) = 50�,

It was observed that TB said that “after this, the two triangles are similar”. TB instructional

practices were talking then followed by doing examples. Learners participated as a whole

class during the lesson’s discourses.

During the interview, I asked TB to explain the factor affecting or impacting his interaction

with learners while teaching the similarity of triangles. He said that:

“Some of the factors includes the students’ prior knowledge of similarity; material

problems, such as stationeries, teaching areas, the students’ less motivated to

learn geometry.”

I observed that TC's initial phases of the lessons were different from TA and TB. It was

also observed that TC started the lesson by writing “what does similar figures mean”, then

he said, “who can define?”. The learners did not respond to the question. TC continued

the lesson, by writing on the chalkboard (see the Figure 5.18 below).

138

Figure 5.18: TC’s Initial phase of the lesson

After he wrote on the chalkboard, TC continued the explanation “let us see the two

triangles”. He drew two triangles ∆�� ∆�� on the chalkboard at the same time

speaking.

TC: Triangle ∆�� is similar to….

Learners: ∆�� (whole class)

TC: ∆��~∆�� since these two triangles are similar, we say that the angles

are congruent.

TC: What do we mean the angles are congruent?

TC: Which angles are congruent?

Learners: Angle � is congruent, (as a whole class)

TC: Angle � is congruent to which angle?

Learners: Silent

TC: ∠� ≅ ∠�, since these two angles are the first angle. What next?

He labeled on the angles the figures.

TC: ∠� ≅ ∠�. The third angle is……

Learners: Angle �

TC: Angle � is congruent to….

139

He Labeled angle �

Learners: Angle � (as a whole class)

TC: ∠� ≅ ∠�

TC: we have said corresponding sides are proportional. What do we mean?

TC: side �� proportional to ��, we mean that their corresponding side ratio is

constant, ��

��=

��

��=

��

��= �

Learners: Talking as a whole class

TC said that: “we use this relation to determine when two similar triangles given with

unknown variables”, TC continued the discussion by writing “what are figures that are

always similar” on the chalkboard. Students were silent. Then, TC wrote at the same time

speaking, “any two circles, the degree measure of any circles is 180 degree. Any two-line

segments, the degree measure of any straight line is 180 degree. Any two equilateral

triangles and any two circles, the degree measure of any circles is 180 degree”.

In coarse of teachers interview, I interviewed TC to brief factors affecting his interaction

with learners. He said that:

“Some factors that spoil student-teacher interaction are students’ prior knowledge,

language and students’ eagerness to know the concepts these are some of the

factors that affect my interaction with students.”

TD’s initial phases of instruction were different from TC’s. TD started the lesson by using

examples of similar and non-similar figures (see the Figure 5.19 below). As illustrated in

the figure below, TD started the lesson by pointing on the chalkboard saying, “the two

hexagons are similar”. Then he asked the question on the similarity of the two figures. He

asked, “Are the two figures similar?”. Learners responded that “yes similar figures”. TD

did not mention why the figures are similar or not. He never mentions the criteria for the

similarity of the two polygons.

140

Figure 5.19: TD’s initial phase of the lesson

TD, continued the discussion by drawing three figures, square, rectangle, and rhombus

on the chalkboard (see the Figure 5.19 above), then asked learners:

TD: Are the three figures similar?

Learners: Not similar. (Whole class)

TD said, “Ok, the figures are not similar because the angles are not congruent, and the

sides are not congruent.” He spoke and pointed to the figures on chalkboard. He said

that:

“Squares are not similar to rectangles because square and rectangle have the

same congruent angle and different sides. A rectangle is similar to another

rectangle.”

It was observed that TD presented the initial phases of the lesson by discussing examples

of similar figures. However, TD lacked mathematical knowledge to mention the reasons

for the similarity of two hexagons and the other figure. He also did not mention the

proportionality ratios on the sides of the rectangles, square and rhombus. As illustrated

in the Figure 5.19 above, the figures are not properly drawn and labeled on the

chalkboard.

When the researcher asked TD to explain about factors affecting his interaction with

learners, he said that:

141

“Lack of similar figures, and students do not understand the concepts easily. In

addition to this, lack of time to give additional tutorial classes.”

The following section presents TE, lesson observation transcription. It was observed that

TE, wrote “similar figures” on the chalkboard, and then she started to explain similar

figures in Amharic. She said that:

“ስሚላሪ ፕሌን ፊገር የሚንላቸው ተመሳሳይ ቅርጽ ያላቸው ነገሮች ናቸዉ ተመሳሳይ ቅርጽ ያላቸዉ ነገሮች አንዴ ትንሽ ሆነው ሊታዩ ይችላሉ ሌላ ጊዜ ትልቅ ሆነው ሊታዩ ይችላሉ ለምሳሌ: በመጻፋችሁ የኢትዮጵያ ካርታ ትልቅ

እና ትንሽ ሆኖ ይታያል (Similar figures are figures having similar shapes, similar figures

can be viewed as small and larger in sizes. For example, in your textbook there

are two maps of Ethiopia, one is small in size and the other is large.)”

She spoke in Amharic for 5 minutes. TE did not show the maps from the learner textbooks.

She defines similar figure as follows:

“Similar plane figure means geometrical the same shape, equal corresponding

angle and proportional the same.”

TE did not mention the corresponding sides of the plane figures. She continued to explain

similarities by drawing figures. She drew two figures on the chalkboard (see the Figure

5.20).

Figure 5.20: TE’s initial phases of the lesson

TE continued the explanation by pointing on the chalkboard at the two figures, she said

that:

“The two figures are similar, because their shapes are similar, and their sizes are

proportional.”

142

TE did not mention which sides of the two figures were proportional. It was observed

that TE, continued the discussion by drawing two figures (see the Figure 5.21).

Figure 5.21: Non-similar figures

As illustrated in the Figure 5.21 above, TE used the figures to explain non-similar figures.

She said, “the two figures (rectangle and square) are not similar”. It was observed that

TE did not mention the proportionality of the sides of the rectangle and square.

Furthermore, it was observed that TE continued the lesson by defining the similarity of

triangles and she wrote on the chalkboard the definition at the same time speaking on

the given topic.

After she wrote the two triangles on the chalkboard (see the Figure 5.22 below), she

said that:

“Let us find the unknown values for the sides of the two triangles

∆�� ∆��.”

She spoke in “Wolayttatto” when labeling the sides of the two triangles. She said that,

“Ha na’’u bootan immettibeenaageeta demmanaayyo bessees (We need to find

the unknown sides of the two triangles.)”

Wolayttatto is the mother tongue for the learners and the teacher. It is not the instructional

language at Grade 8 level. English is the instructional language. However, teachers

sometimes code-switch to make the given lesson clearer to the learners.

143

Figure 5.22: TE’s examples of similar triangles

As illustrated in Figure 5.22 above, the side length of ��, �� are � �� ,

respectively. TE did not label the corresponding sides of each triangle. Moreover, she did

not label the corresponding angles and the names of the similar triangles as the given

condition to find the unknown values of � �� .

It was observed that TE started the answer by copying from her notebook. She wrote on

the chalkboard at the same time speaking.

TE: Solution, the corresponding sides are what? Proportional.

TE: We can find the length of ehhh…..., ��

TE: We can find the length of �� (by pointing on the figure and

underlining �� )

TE: Triangle ∆�� ≡ ∆�� (the triangles are similar), after this

TE: �� over what?

Learners: Silent

TE: ��

��=

��

��=

��

�� from this, you can find,

She looked at her notes at the same time write on the chalkboard.

TE: �� = 7��, �� = 21��, �� = 6��, �� = � from this…...

TE: ��

��=

��

�=

��

� let find �, to find we criss-cross the equation

TE wrote the equation:

144

TE: 7�� × � = 21�� × 6�� to find � we divide the equation by 7. Therefore,

the value of � = 18��.

Learners copied from the chalkboard.

TE: Let us find the value of �,

TE spoke by pointing the equation ��

��=

��

�=

��

� on the chalkboard, then she said

that: “We can use the value of � to find �. Therefore, the equation above reduced to, ��

��=

��

�”. She wrote on chalkboard at the same time speaking.

TE: Therefore, we crisscross to find �, � × 6�� = 18�� × 9��, the value of � is

Learners: � =��×��

�� (responding as a whole class)

TE: Ok, by cancelling 6 cm, we can get the value of � = 27��.

It was observed that TE’s instructional approach was teacher talk and note copying. As

illustrated above, TE lacks the pedagogical approach to teach the similarity of triangles.

She could not use the proper symbol for the similarity of triangles. She used the symbol

“≡” for the similarity of the triangles.

To further unpack how teachers’ pedagogical approaches manifested during lesson

observations, teachers’ use of different diagrams and learners’ learning on hands-on

manipulation observation were interrogated. The result of analysis established that one

of the teachers supported his teaching of similarity of triangles by showing similar figures

from learners’ textbooks. The data also showed that all the lessons observed lacked

hands-on manipulative activities. The next section presents the lesson observation

transcriptions on the use of different diagrams and hands-on manipulative activities.

It was observed that only TA discussed classroom Activity 4.1 from students Grade 8

mathematics textbooks to illustrate the similarity of plane figures before defining similarity

of triangles. He started the activity after he wrote it on the chalkboard.

TA started the lesson by writing Activity 4.1 “Which of the following maps are similar?”

from Grade 8 mathematics student textbook. Before discussing the activity, he revised

the criteria for the similarity of triangles. He said that:

145

“Two triangles are said to be similar if: (i) they have the same shapes, (ii) their

corresponding angles are congruent and(iii) their corresponding sides should are

proportional.”

He wrote on the portion of the chalkboard at the same time speaking. It was observed

that, TA continued the demonstration by using the learners’ textbook.

TA: “As you see the maps, Map ‘a’ belongs to Ethiopia and Map ‘b’. Therefore,

Map ‘a’ and Map ‘b’ are not similar. Whereas Map ‘a’ and Map ‘c’ are similar.”

Figure 5.23: TA’s demonstration of similar figures

As illustrated in Figure 5.23 above, I observed that, TA tried to show the maps from the

textbook. He is pointing to one of the figures in the textbook. However, the maps could

not be seen by the learners who could not differentiate the two maps of Ethiopia because

of distance. The demonstration of maps required closer attention.

TD’s teaching approach was dominated by teacher talk and chalk, and the chalkboard

used as instruction materials. Learners were passive listeners in most of the lessons

observed. In one of the lessons, teacher TD wrote an activity (see Figure 5.24 below) on

the chalkboard. He invited individual learners to come up to the chalkboard and

demonstrate this. Once the learners had grasped the concept, then TD demonstrated it

to them.

146

Figure 5.24: Student participation in TD classroom

In one of TB’s lessons, he wrote classwork activity (see Figure 5.25 below) on the

chalkboard. He invited individual learners to come up to the chalkboard and demonstrate

this. Once the learners had grasped this, then TB explained the reason why the two

triangles were similar.

Figure 5.25: Learner participation in TB’s classroom

I observed that during the lesson, TB was correcting his learner demonstration. The

analysis revealed that the lessons observed lacks the use of different diagrams and

hands-on manipulative activities.

Teachers responded in the questionnaire about the factors affecting their interaction with

learners while teaching the similarity of triangles. This is what they said:

TA: “My interaction with my students is affected by misbehaving, mis concepts

towards the lesson or education because currently the students do not have good

vision towards their education.”

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TB: “Students’ misbehavior, teaching learning materials, students’ individual

problems etc affects the interaction with students and also economic problems and

students’ interest to learn geometry.”

TC: “Lack of teaching aids because in our school there is no much enough teaching

aids to show similarity of triangles in pedagogical centre.”

TD: “Lack of examples, teaching aids in the school pedagogical centre.”

TE: “Students lack knowledge about plane figures and lack of plane figures in our

school pedagogical centre.”

5.4.3 Theme 3: Challenges teachers faced in teaching of similar triangles

As illustrated in Figure 5.4, Theme 3 emerged from the codes that were categorised as

(i) mathematical knowledge challenges, (ii) pedagogical knowledge challenges (iii)

students background knowledge (iv) resources (v) the mathematics syllabus, and (vi)

other. In the next section, I expand further on Theme 3 and report on the following related

categories to elaborate the challenges teachers faced in the teaching of similar triangles.

5.4.3.1 Mathematical knowledge challenges

As illustrated in Figure 5.4, the coded data from the transcription of observation, semi-

structured interviews and questionnaires revealed that teachers may lack mathematical

knowledge in the teaching of similarity of triangles. Participants’ mathematical knowledge

challenge manifested during the observation of the lessons, when teachers were working

on the examples. The lesson observation of the participants is presented below.

TC started a discussion after writing the theorem AA Similarity theorem as shown in

Figure 5.26 below.

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Figure5.26: TC’s work on tests of similarity of triangles

TC then started the discussion after he read the theorem from the chalkboard. He wrote

the example at the same time speaking then started the discussion, by saying, “Let us do

this example” pointing to the figure on the chalkboard.

TC: solution, he wrote since ∆��~∆�� (see the Figure 5.26 above).

As illustrated in the Figure 5.26 above, it was observed that, the similarity of the two

triangles ∆��~∆�� is not the given condition on the example. However, TC used as

given condition. TC continued his discussion by saying that:

TC: ∠� ≅ ∠� ⇒ ��

TC: ∠�� ≅ ∠�� ⇒ ��

TC: ∆��~∆�� ⇒ �� ℎ��

Here also it was observed that, the congruence of the two angles ∠� ≅ ∠� is the given

condition in order to show the similarity of triangles ∆��~∆��. Further from the

expression, the corresponding angles ∠�� ≅ ∠��, did not correctly name the vertical

opposite angles. He must name the angles as ∠�� ≅ ∠�� because angles and

corresponding sides correctly corresponded.

It was observed that, TC could not mention from the definition of the sum of the degree

measure of the interior angles of triangle the remaining angles of the two triangles are

congruent ∠� ≅ ∠�. Moreover, TC could not explain about the sides of the triangles’

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proportionality since from the definition of similarity of triangles’ proportionality of the

corresponding sides of the triangles is one of the criteria to determine similarity. TC may

lack the understanding to connect the theorem with the definition of similarity of triangles.

He never mentioned the importance of the theorem then after AA similarity theorem would

be used to show for the similarity of triangles. The same challenge was observed in TD’s

lesson on the same example during observation and it was presented under Section

5.4.2.2. Moreover, I also observed that mathematical knowledge challenge in TD’s

lessons during explanation of SAS similarity theorem.

TD wrote the theorem 4.2 SAS Similarity Theorem: “if two sides of one triangle are

proportional to the corresponding two sides of another triangle and their included angles

are also congruent, then the two triangles are congruent” on the chalkboard. TD then

drew Figure 5.27 as an example to elaborate the theorem. He labelled the figure.

Figure 5.27: TD’s work on the similarity theorem

As illustrated in Figure 5.27 above, TD started the explanation by reading the theorem

from the chalkboard and pointing on the figure on his right hand.

He said that:

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TD: So, angle �, this angle is included (pointing on the figure).

TD: Angle ∠�� ≅ ∠�� (without pointing on the figure, he read from the note).

TD: The two corresponding sides are proportional. Then this theorem is side angle

side theorem.

It was observed that, TD could not mention the sides that included angle �. It was also

observed that TD did not show the two triangles from the given figure. He also did not

locate the two proportional sides of the triangles and that the included angles are

congruent. It was observed that the lesson lacked proper explanation of the theorem and

its importance to use for checking similarity of triangles. Learners could not understand

the lesson.

The mathematical knowledge challenge was also observed in TE’s lesson presentation.

TE wrote on the chalkboard “enlargement by using coordinate” as shown on Figure 5.28.

The next section presents her discussion.

Figure 5.28: TE’s work on enlargement by using coordinate

It was observed that she started her explanation by reading “enlargement by using

coordinate” from chalkboard. Then, she said that in Wolayttatto:

“Koordieetiya giyoogee naa”u cachchati woykko naa’’u naxibeti gaytiyoosaa giyoogaa.

Hegaa gishawu, hagaa wode quxxuree immetin gaytiyoosaa oychchanawu danddayettes

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woykko gaytyoosay immettin quxxuriyaa koyaawu daddayettees. Hegaa gishawu diccidi

biyaageeti awan gaytiyaakko koyanawu dadayettees”

(Coordinate means the place where two points meet. If we have given points located in

coordinate plane, we can find their corresponding values, and then by joining the points

we can find the vertex of the given figure).

She started to plot the points without explanation and connected to form of the larger

triangle as shown on Figure 5.28. In the same way, she plots the smaller triangle. Then

she said that:

TE: You can find the points �, � �� .

Learners: Talking to each other

TE: This is the original point and the bigger one is the enlarged.

TE continued her explanation by saying that:

TE: What we said �, � is the enlargement of the triangle ��′ with coordinate of

its vertex �� (she copied from her notebook then wrote on chalkboard)

She could not say anything about the �, the enlargement before. Then she wrote the

points “�(3,3) �(7,3) �(5,1)" as shown in Figure 5.28 on chalkboard from her note. Then

she said that, “what is the co-ordinate of �′?” She pointed to the figure, then wrote (6,0).

In the similar way ��(7,8)�� (4, 4). She could not explain how she got the coordinate

values of A�, B�and C�. As illustrated in the Figure 5.28 above, the points were not located

appropriately on the coordinate plane. TE lacked the skill of drawing the points on the

chalkboard.

It was observed that the presentation lacked proper explanation of the examples, what

was the given condition and what was the required. It was observed that TE was reading

from her notebook during talking and the lesson lacked proper presentation of how to

enlarge the smaller triangle, by scale factor. It was also observed that, TE wrote the

coordinate of the vertex of the smaller triangle after she plot the triangle. Moreover, the

triangles could not be located on the chalkboard properly. In my opinion, TE could lack

the understanding of what enlargement means on the coordinate plane. Furthermore, the

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language used by TE is not appropriate. She could not explain the lessons well and lacked

geometric language.

TA and TB’s mathematical challenges were also observed on that poor explanation on

the definition of similar figures, lessons and working on the theorems (see Section 5.4.2.3

on teaching approaches).

Teacher’s mathematical knowledge challenges manifested in their interviews. During the

interviews, when TA asked to explain his understanding on the static and transformational

approach of the definition of similarity of triangles, he said that:

“I don’t know the static and transformational approach. For me, we say two triangles are similar if their corresponding sides are proportional and if their corresponding angles are congruent. I never heard this approach before.”

However, TC tried to answer and said:

“Let me try it. In similarity, there is an arithmetic definition and an algebraic definition. In arithmetic definition, we see its proportionality, for example, the similarity of sides, when we describe them in number and ratio; when we come to the geometric concept, we draw pictures, we use protractors, rules, and we do teach in these ways. I hope I tried to answer.”

During the interview, I asked TB to explain the static and transformational or geometric

approaches of the definition of similar triangles. He said that:

“Static approach is explained by which sides are corresponding and

transformational approach is based on the definition of similarity; for example, by

helping students to visualise corresponding sides of transformational figures.”

Further, TD and TE may lack the understanding of static and transformational approaches

or geometric approaches of the definition of similar triangles. This is what they said:

TD: I don’t know sorry.

TE: Uhh…Ok. I cannot get it.

Most of the respondents did not understand the static and transformational approaches

definition of similarity of triangles. The teachers were not supposed to differentiate the

concept, similarity in geometry is different from similarity in colloquial speech. Moreover,

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teachers responded on the questionnaires that, geometry is not appropriately included in

the teacher education program for primary school teachers. This is what they said:

TA: I am not interested to teach geometry; this is because students have fewer

attitudes to geometry; unavailability of instructional materials in our schools, and it

needs more preparation of teaching aids compared to algebra.

TB: I have taken only two geometry courses and the way I learned is not

interesting. Due to this, I am not interested to teach the geometry part of Grade 8

mathematics.

TC and TD responded the same way as the teachers on the geometric contents they

learned. The teachers in the sampled schools could be found lacking in mathematical

knowledge to teach the similarity of triangles.

5.4.3.2 Pedagogical knowledge challenges

As illustrated in Figure 5.4, the coded data from the transcription of observations, semi-

structured interviews and questionnaires revealed that teachers may lack pedagogical

knowledge to teach the similarity of triangles. The pedagogical knowledge challenges

manifested in the teachers’ presentation of the lesson. It was observed that the lessons

lack the connection similarity with learners’ real life. The following section presents the

lesson observation.

For example, TB started the lesson on similar figures by saying:

TB: When we say ehh.. two similar ehh…figures are similar?

Learners: Corresponding sides are proportional and ehh……. (Loudly as a whole

class)

TB: Please, one of you raise your hands, then yes learner (pointing to the student)

Learner: If corresponding sides are proportional and corresponding angles are

congruent then the triangles are similar.

TB: (Pointing to another student) You can add more, if two triangles are similar,

yes!

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Learner: The corresponding angles are congruent, and sides are proportional (the

student pointed by teacher)

TB: Good!!

TB continued the discussion:

TB: Similar figures are similar, their shape is similar their only difference is their

size

TB: Let’s take the Ethiopian map from your textbook yes, one is larger and the

other is smaller. (He could not show the maps from the student textbook or

command the students to look from the textbook).

I observed after the lesson that some learners had the textbook i their hands.

TB: They have what? By what there are similar?

Learners: Silent

TB: By shapes they are similar, by what their difference?

Learners: Silent

TB: By size

Learner: By sizeeeee… (Murmuring)

TB: Ok, one is larger and the other is smaller.

TB continued his explanation on the similar figure.

TB: We can see figures like the map of Africa, cars, etc.

As the teacher said “car”, one of the learners asked: “Teacher, is car a polygon?”, then

TB said, “Yes! it is polygon”. I observed that there was a picture of two similar cars in the

learners’ mathematics test book p.102.

TB continued his explanation by saying, “Let us see similar triangles”.

TB: What do I mean by triangle?

Then he took, a model of a triangle as shown in Figure 5.29.

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Figure 5.29: TB’s demonstration of triangles

TB: what is the name of this triangle?

Learner: right angle triangle (one student), then right-angle triangle (loudly as a

whole class)

TB: Yes

Then TB wrote on the chalkboard at the same time speaking, “∆��~∆��" then said

that:

TB: Let as take this, then he wrote on the chalkboard at the same time speaking

TB: ∠� ≅ ∠� � �� (∠�) = �(∠�), ∠� ≅ ∠� � �� (∠�) = �(∠�), ∠� ≅

∠� � �� (∠�) = �(∠�),

Then he pointed to the points he had written on the chalkboard and said, “this shows that

their corresponding angles are congruent”.

TB continued his explanation said that “Their corresponding sides are proportional.” He

wrote the ratio corresponding sides, “ ��

��=

��

��=

��

��.”

It was observed that the TB had two triangles as illustrated in the Figure 5.29 above, the

smaller and larger triangles, which he never used to show the corresponding sides and

angle of similarity of triangles. Moreover, he could not draw two similar triangles on that

chalkboard during his explanation of angles’ congruence. TB did not appropriately

connect similarities with real figures; he had difficulty explaining the similarity of models

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of cars and the application of similarity in the real life of students. TB faced pedagogical

knowledge challenges of teaching the similarity of triangles. It was observed that the

lessons in TC, TD, and TE did not show the maps of Ethiopia, the model of similar figures

as illustrated in learner textbook. In line with this, the pedagogical challenges of TA, TC,

TD and TE have been presented in the section above under Theme 2. For example, TA’s

pedagogical approach was writing notes on the chalkboard and then explanation.

Learners copied notes from the chalkboard. It was observed that the lesson presented

lacked the explanation of the similarity of triangles by using the transformation approach,

rotation, reduction, and enlargement. Moreover, the lesson observation revealed that in

all lessons, teachers did not use problem-based approaches to teaching geometry. All

the lessons were teacher-centred, and learners were passive listeners. The teachers did

not look at what the learners were doing in the classroom. I observed that in TE’s class

learners were sitting at the back of the class doing the homework given by the English

teacher.

During the interview with TB, I asked TB to explain the educational theories related to the

teaching-learning of geometry, and how those theories informed the teaching of similarity

of triangles. He said that:

TB: As to me, this is the first time when I listen from you; I don’t know these

theories.

All the teachers responded in the same way; they do not know the theories related to

teaching geometry.

Further, teachers responded to the questionnaire challenges they faced in the teaching

of similarity of triangles. This is what they said:

TA: Currently across the world there are different software that help the teaching

and learning of geometry, here in Ethiopia there is no such type of technology that

helps the teaching of similarity of triangles. Most mathematics teachers could not

be involved in professional development programs. I could not get any training on

the teaching of geometry, in particular, the similarity of triangles.

TB: I am sorry to say, I do not like to teach geometry. Geometry teaching is a

challenge. We need training on the various methods of teaching like active learning

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method, problem-solving method as well as production of appropriate teaching

aids by collaborating with nearby colleges and universities.

TC: Teaching geometry is a challenge. It takes more time during preparation as

one needs to search for different models and diagrams. However, there are no

such materials near our schools.

TD: In our school, there is a school pedagogical centre but there are no relevant

geometrical models, figures, mathematical instruments. We face problems in

showing geometrical figures and models to our students

TE: I am not interested in teaching geometry because students have attitudes to

geometry; there is also the unavailability of instructional materials in our schools,

and it needs more preparation of teaching aids compared to algebra.”

All the teachers responded on the questionnaire that, they could not be involved in a

professional development training in the last there years.

5.4.3.3 Students poor background knowledge


structured interviews, and questionnaire revealed that students' geometric background

knowledge is one of the challenges teachers faced in the teaching of similarity of triangles.

During the interviews with TB, said that:

“Students’ prior knowledge is one among the challenges I faced in teaching

similarity of triangles. Students’ language and eagerness to know the concepts are

also some of the challenges I faced.”

In addition to this TA said that:

Students lack the motivation to learn the similarity of triangles.

I further asked TD about the challenges he faced in teaching similarity of triangles. He

said that:

“There are many challenges I faced as a teacher. Some of these are students do not easily understand the similarity of triangles because there are no real/ models to show. In addition to this, students do not give attention to learn geometry and they think or consider the topic is difficult to understand.”

Moreover, TC said that:

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In my class students do not participate during geometry lessons, they lack prior

knowledge. In addition to this as you observed the classroom lack geometrical

pictures, and the chalkboard is not appropriate to use.

TE also said that “Students lack knowledge about plane figures. They are poor to visualise

the enlarged/reduced triangle.” During my observation of the lessons, students do not

participate in the class, only two or three students were participating in the lesson

observed. The analysis revealed that learners’ background knowledge is one of the

challenges teachers faced when teaching similarity of triangles.

5.4.3.4 Resources


structured interviews, and questionnaire disclosed the lack of resources among the

challenges teachers faced in the teaching of similarity of triangles. During my observation,

I found that all the schools lack pedagogical centres. Most of the classrooms were lacking

in geometric figures and models on the walls. I observed in one of the schools the

following Figure 5.30 on the outside of the classroom.

Figure 5:30: Geometric figures on the walls of School A

As illustrated in the Figure 5.30, I observed the geometric figures in one of the schools.

In the coarse interview with participants, I interviewed TA to explain an importance of

using geometric resources in the teaching of similar triangles and their availability in the

schools. TA said that:

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“I use the figures from the student textbooks, the small figure and the enlarged figure. The photo of small and the enlarged percentage of that photo. In our school, there is no pedagogical centre to produce instructional materials. The only instructional material is the student textbook.”

Further, TB said that:

“Yeah, there are several materials-instructional materials in teaching similarity. For example, we use pencil, pen, graphs, diagrams, textbooks, ruler, rubbers, and extra materials. In my school, there is no environment suitable for teaching similarity since there are different problems in teaching aids-stationeries and an economic problem and learning environments. There is a lack of instructional materials like models, similar figures in our school. I am using a student textbook.”

The teachers, TC, TD and TE also responded on the questionnaire that lack of

instructional materials is one of the challenges in the teaching of the similarity of triangles.

This is what they said:

TC: The challenge I faced in the teaching of similarity is the lack of textbooks,

teachers' guides and syllabus materials.

TD: There are not enough teaching aids in our school. This is because of

inadequate budgets.

TE: There are no real objects or models of similar figures to show students from

the laboratory.

It was observed that most of the schools lack similar geometric figures inside their

pedagogical centers.

5.4.3.5 The Mathematics syllabus and other challenges


structured interviews and questionnaire revealed that the place of geometry in

mathematics syllabi was among the challenges teachers faced in the teaching of similarity

of triangles.

During the interview with teachers, some of the teachers explained the nature of

mathematics syllabi when I asked them challenges related to the mathematics syllabus.

This is what they said:

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TA: “As I mentioned earlier, students lack geometric knowledge to me related to

the place geometry in the mathematics curriculum. Students do not get enough

time to cover for geometric contents in the last chapters, I think they are not

covered. This makes a challenge for students on the next class level.”

TD: “Learning geometry is essential in the Ethiopian context, but books prepared

starting from Grade 5 hold geometry concept in the last part of the textbooks and

teachers couldn’t do enough with geometry.”

TC: “Most of the contents in elementary school particularly in Grades 5, 6 and 7

are not covered. As for me, this is one of the challenges of teaching geometry in

Ethiopia. Most teachers do not teach the content in the last chapters.”

The data from the questionnaire revealed that other factors such as lack of supervision

from the school principals and teachers’ economic problems were the challenges

teachers faced in the teaching of similarity of triangles. This is what TA said:

“Lack of feedback from vice directors of a school and walking more than 2 hours

on foot until to this school are the challenges I faced when teaching similarity of

triangles”.

Teachers in the study area proposed strategies to minimise the challenges they faced in

the teaching of similarity of triangles. The following section presents the proposed

strategies.

5.4.4 Theme 4: Suggested strategies to minimise the challenges of teaching

similarity of triangles

As illustrated in Figure 5.5, Theme 4 emerged from the codes that categorised strategies

to minimise the challenges teachers faced in the teaching of similarity of triangles. The

coded data revealed some of the so subcategories such as (i) pedagogical approaches,

(ii) reform on pre-service teachers’ education, and (iii) continuous professional

development. In the next section, I expand further on Theme 4 and report on the following

related category to elaborate the strategies to overcome the challenges in the teaching

of similarity of triangles.

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5.4.4.1 Strategies to minimise the challenges

As illustrated in Figure 5.5, the coded data from semi-structured interviews and

questionnaires revealed that teachers proposed strategies to minimise the challenges in

the teaching of similarity of triangles. The data analysis from the open questionnaire item

revealed that pedagogical approaches such as active learning methods, reform on pre-

service teachers training and continuous professional development related to geometry

education, in particular similarity of triangles will minimise the challenges. During the

interviews, I asked teachers to explain what kind of methods or pedagogical approaches

are best for teaching the similarity of triangles and why? This is what they said:

TA: I use demonstration to show them and to make them use the methods I

demonstrated; showing the figures, similar figures; the next is discussion,

discussion now is used to develop ideas run by demonstrations. Open discussion

is good to develop the students’ next academic statuses.

TB: To me the pedagogical approaches best are the student-centred method or

active teaching methods are very important to teach similarity because it

participates all students during the teaching-learning process, and it motivates

students actively for participation in the subject matter.

TC: I believe it is better to utilize instructional materials for example, photos, and

maps during teaching similarity of triangles.

TD: First, before teaching similarity, I should give an awareness of the concept

similarity. At least, I should bring pictures, and I should show this is that, or that is

these concepts. Mere talk is not essential, so I should bring two similar objects and

I should show them, and the students can get more knowledge about the similarity

in these ways.

TE: “The best teaching method is the one my students participated actively. As a

teacher, I adjust my methods and strategies in the response to my students’ ability

to learn the materials being presented. Student-centred is important because in

these approaches both teacher and learner equally participate in the teaching-

learning process”.

Furthermore, TA and TB responded on the questionnaires that analogies, illustrations, or

examples are most helpful for the teaching of similarity. This is what they said:

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TA: As I think the analogy most helpful for teaching similarity of triangles is showing

the original figure or maps or car etc and the enlarged figure or maps or car, etc.

As a result, the students may get the base concept of similarity. Examples: - the

students must have a chance of looking at the original map and the enlarged map

and consider even though the size of the map differs the shape of the maps is the

same. The illustrations are helpful in teaching similarity as there are small photos

and the enlarged photos of the given persons, animals, birds, objects, polygons

figure, etc.

TB: Using analogy in the classroom is an effective strategy as students tend to find

it easier to understand a lesson when teachers form connections between new

topics and what has already been thought. Such analogies are any two equilateral

triangles, squares and circles that are similar. For example, any congruent figures

are similar but not any similar figures are congruent.

As illustrated in Figure 5.5 above, some of the teachers proposed strategies to overcome

the challenges they faced in the teaching of similarity of triangles. This is what TA said:

TA: a. By informing the concerned body (example, supervisors, Woreda Education

Office).

b. Giving counseling service according to their misbehaving to improve their

behaviour.

c. Motivating the students by considering their feelings.

d. Attending a professional development program if I get a chance to participate.

e. Informing the school directors, supervisors, education office and other

stakeholders.

f. Struggling to solve economic problems by upgrading my education.

TB also suggested strategies to overcome the challenges he faced. This is how he

responded to the questionnaire.

TB: The problems/challenges are two types. The first one is the problem that I can

overcome and the second on the challenges which I can’t overcome at my level. I

can overcome the material problems by preparing the teaching aid/instruction

materials from locally available materials. Secondly, the other challenges which

are solved or overcome financially, I inform the school directors and management

committee in the school.

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TC, TD and TE did not respond to the questionnaire on the strategies to overcome the

challenges of teaching the similarity of triangles.

5.5 Conclusion

This chapter aimed to present data and analysis of the data collected to answer the

research questions. Furthermore, chapter 5 presented how the pilot research was carried

out. In Chapter 6, the main findings of this study will be discussed.

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CHAPTER SIX

DISCUSSION OF FINDINGS

6.1. INTRODUCTION

The previous chapter focused on the data presentation and analysis. The report also

showed that similarity and its related concepts are central components of geometry. It is

an important spatial-sense, a geometrical concept that can facilitate students’

understanding of indirect measurement and proportional reasoning. Many see geometry

as a significant subject in mathematics and the similarity of a triangle is found to be a key

concept within geometry, but there is very little research done on teachers’ challenges on

teaching similarity of triangles. As it has been depicted from various sources, many of the

research studies that were carried out on learning similarity issues focused on school-

age children. The purpose of this chapter, however, is to provide a discussion of the main

findings concerning literature reviewed together with the theoretical framework lens and

a phase of instruction or model suggested for teaching the similarity of triangles.

6.2. DISCUSSION

The study findings are discussed under each of the four themes forwarded to put

synthesised ideas together with interpretation and analysis. The following section

presents the discussion on each of those themes.

6.2.1 The importance of learning geometry

Much has been argued on presentations of the previous chapter sections, and in this

regard, several conclusions can be drawn from the results presented in Chapter 5 as

connected to the first theme. The observation, semi-structured interviews and

questionnaires revealed that teachers at the sampled schools were cognizant of the

importance of teaching geometry in general and the similarity of triangles.

6.2.1.1 Reasons for studying geometry

Results reveal that participants note that learning geometry improves learners’ geometric

and cognitive skills. In connection to this datum, NCTM (2000) also acknowledged that

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through the study of geometry, learners should learn more about geometric shapes and

structures, as well as how to analyze their properties and relationships, through the study

of geometry. They should also progress from recognising distinct geometric shapes to

geometry reasoning and problem solving (Daher & Jaber, 2010). Moreover, participants’

data further revealed that geometry, as a part of mathematics helps in studying

measurements.

Respondents during the interviews revealed that similarity is an important topic to be

taught in Grade 8 and it is a visual representation of concepts such as ratio, proportion,

and slope. This concurs with the views by researchers (CCSSM, 2010; Cox & Lo, 2012;

Lo, Cox & Mingus, 2006; MoE, 2009; NCTM, 2000; Seago, Jacobs, Driscoll & Nikula,

2013) who assert that similarity is an important concept taught in middle school geometry

curriculum throughout the world. In this regard, participants related the importance of

geometry to the real world. The model for teaching similarity of triangles also suggests

teachers discuss the reason for learning similarity of plane figures.

6.2.1.2 Geometry in relation to daily lives

Respondents during the interviews revealed that many objects found in our environment

are related to geometry. This concurs with NCTM (2000), who contends that geometry

connects mathematics with the physical world. The participants further argue that the

learning of similarity could help in learners’ daily life phenomena, such as sun shadow

and copying. However, the participants seem to lack experience in mentioning the

connection of similarity to real-life during the actual teaching of the concepts. For

example, it was found that in Figure 5.8, TD’s example of the smaller and larger rectangle

was used to explain similar plane figures. He did not mention that similar figures can be

obtained by enlargement/reduction of the same figure. This study again revealed that the

participants lacked to relate similarity by using examples such as an enlarged photograph

is a similar figure to the original one. The new geometric object is the “same shape” as

the old one but has all of its parts reduced or enlarged in size or “scaled” by the same

ratio. This finding concurs with Dündar and Gündüz (2017) who argue that prospective

teachers had difficulty in justifying challenges associated with the daily life examples of

congruence and similarity in triangles. The use of geometry to maintain daily life chores

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can be regarded within the scope of practical activities. However, the attempt to learn and

teach the similarity of triangles when not associated with the daily life experiences of

students would create problems in understanding the geometric meaning of similarity.

Those authors further assert that learners would be interested in learning geometry and

the effectiveness of learning would be enhanced to establish similarity between the

subjects and daily life.

The model of teaching similarity of triangles proposed in this study suggests that teachers

and learners should engage in classroom conversations and activities about the

importance of learning similarity (see Figure 3.5). This would help teachers to easily relate

the similarity with students’ daily lives. At the information phase of instruction, the teacher

is supposed to show smaller and larger maps of Ethiopia and Africa. After the observation

of similar figures has been made, the teacher is supposed to define the geometric

meaning of similarity of two plane figures. This engagement provides opportunities for the

teachers to explore learners’ prior knowledge. Then, he/she will get information about

learners who understood similar and non-similar figures. As suggested in the model, the

teacher classroom discussion with the learners creates more awareness of the

importance of learning of similarity in relation to daily lives. It provides, the connection of

similarity with geometric and mathematical concepts such as proportion, enlargement,

and slope.

6.2.2. Phases of the instruction in teaching similarity of triangles

As presented in Chapter 5, the phases of instruction in teaching similarity of triangles that

come out from the coded data in relation to similarity of triangles and teaching approaches

are discussed under the sub-themes: concepts related to similarity and similarity of

triangles.


The data in Chapter 5 revealed that some of the participants were aware of the

connections of similarity with other mathematical and geometric concepts.

According to the reports, some of the participants were aware of the concepts to be

learned before similarity. In this regard, they mentioned during the interviews that

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concepts such as proportion, enlargement, shapes, sizes of geometric figures, plane

figures, like triangles, rectangles, and congruence are the pre-knowledge necessary for

students to learn before they are introduced to similarity. It was also observed that some

of the teachers’ lessons; for example, TB’s lesson connected similarity of triangles with

congruence. However, it was found that most of the observed lessons lacked a

connection of similarity with other concepts. In line with this, the data further revealed that

most of the participants were not aware of the concepts to be learnt after the similarity of

triangles. This refutes Chazan (1988) and Lappan and Even (1988) who argue that

similarity provides a way for learners to connect spatial and numeric reasoning and

provides the basis for advanced mathematical topics, such as projective geometry,

calculus, slope, and trigonometric ratio. For example, measurement of a similar figure

including length, perimeter, and area requires the integration of numerical and spatial

thinking. Moreover, according to these authors, investigative tasks in geometry and

measurement provide opportunities for learners to analyse mathematically their spatial

environment to describe characteristics and relationships of geometric objects, and to use

number concepts in a geometric context. The proposed model for teaching similarity

suggests teachers to provide conversation activity about concepts to be learnt before and

after similarity. Moreover, the connection of similarity with other geometric and

mathematical concepts is essential. This provides ways of connecting spatial and numeric

reasoning for learners. The model provided opportunities for teachers to minimise the

challenge related to connecting similarity with other mathematical concepts.


The results of classroom observation presented in Chapter 5 revealed that none of the

participants reminded their students about the similarity of any other two polygons before

defining the similarity of triangles. However, triangles were explained as a special type of

polygon and therefore the conditions of similarity of polygons also stands true for

triangles. The model of teaching similarity of triangles suggests that teachers should

remind the definition of similar figures before defining the similarity of triangles.

From the classroom observation of participants’ lessons presented in Chapter 5, TC

started an explanation of similar triangles by writing “If ∆��~∆��, what does it mean?”.

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Observations also revealed the lack of activities before the definition of the similarity of

triangles, information about the corresponding sides of triangles, and how learners could

calculate the scale factors were lacking. For example, as shown in Figure 5.11, TC asked

learners the similarity of the two triangles. Then, he allowed for the discussion in the

classroom about the similarity of triangles and then corrected the activities. It was

expected that he asks a question with regard to ∆�� being similar to ∆�� then, such

that the TC could find the values of �� . Following the learners’ classroom

response, TC was seen starting the discussion by defining similar triangles as similar

triangles that are identical in shape but not necessarily in size.

It was also found that the participants’ chalkboard drawing as shown in Figure 5.10 of the

two similar triangles could create a misconception for students. This concurs with Chazen

(1987) who identified three difficulties for students in learning similarity. These include:

“notations of similarity, proportional reasoning, and dimensional growth relationships” (p.

134). As shown in Figure 5.10, the two triangles are congruent in size. However, similar

figures do not mean figures are always congruent. The similar triangles should be

properly represented on the chalkboard either enlarged/reduced in size or rotated see

Figures 2.1 and 2.12. Learner were found to easily understanding the enlarged triangles

when visual representations were used. All their corresponding sides and angles should

have been labelled.

The data in Chapter 5 revealed that participants did not properly use the symbol for

similarity. This was because teachers did not define the similarity of triangles as

represented in Section 2.3.3 and represent the symbol of similarity of triangles.

It was also revealed that participants lacked an awareness of properly using the symbol

‘~’ which stands for the language “similar to” during explaining the similarity of two

triangles. Moreover, participants lacked proper representation of the corresponding sides

and angles of similar triangles from the given examples. In this regard, Son (2013)

acknowledged that solving similarity of items requires: (1) understanding the concept of

similarity, (2) recognising the proportionality embedded in similar figures by comparing

length and width between figures or by comparing the length to width within a figure or

determining a scale factor, (3) representing the relationship between two similar figures

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using a ratio, a proportion and (4) carrying out related procedures. However, in this study,

the data showed that participants lacked the knowledge to define similar triangles and

draw appropriately two similar triangles.


In support of predominated teacher-centred pedagogical approaches, the results of data

analysis presented revealed that participants used chalkboard as the directed classroom

activities through writing the definition of similarity of triangles, and then followed by a

demonstration of examples using the chalk and talk approach. The data further revealed

that the teaching approaches used by participants missed the van Hieles’ phase of

instruction for the teaching of geometry in general and the phases of instruction for the

teaching the similarity of triangles.

As presented in Chapter 3, in the model for teaching the similarity of triangles, it was

suggested that participants should arrange their teaching in five different phases during

teaching similarity of triangles and to guide learners from one level to the next (van

Hieles’, 1986). The phases of learning are information, guided or directed orientation,

explication, free orientation, and integration.

According to the model for teaching similarity of triangles, the first phase was information.

In this first phase, the teacher, and learners should be engaged in conversation and

activity about similar geometric figures. Furthermore, observations of similar figures were

made, questions about similar figures were raised, and level-specific vocabulary was

introduced by teachers to learners. The teacher is supposed to start the lesson by

conversation and engagement about similar figures using the model of figures, objects-

like, photographs, polygons having the same shape but different in size. In doing those

activities and conversations learners’ prior knowledge was explored. Then, learners

learned similar and non-similar figures.

Contrary to the information phase of the model, data revealed that the participants began

the lesson in writing about the definition of similar triangles, which correspond to the initial

phases of the lesson. Learners were subjected to copy notes from the chalkboard, and

they were passive listeners. The data revealed that the lessons lacked the engagements

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and conversations about similar figures phases. For example, TC’s initial phases of the

lessons would start by asking, “What does a similar figure mean?”. The observed lessons

revealed that the instructional practices included talking then followed by doing examples.

The lessons did not use activities and models of similar figures during questioning of

similar triangles. This concurs with the studies by Fennema (2004) and Szendrie (2011)

who argue that learners may be able to learn more effectively if their learning environment

offers opportunities to interact with models that are appropriate for their cognitive level.

Moreover, the TA, TB, TD and TE lessons observed were dominated by teacher talk and

lacked engagement. The participants did not get information about learners’ prior

knowledge of similar figures. This refutes Crowley (1987, p.5) who asserts that teachers’

basic attention lies on engaging with activities in the way that discovers the learners’

earlier knowledge about similar triangles and where the direction of the lesson further go.

The observed lessons lacked information on the initial phases of instruction for teaching

similarity of triangles. The challenges observed were due to the participants' lacked

knowledge of the van Hieles’ phase of learning geometry. However, since the model for

teaching the similarity of triangles includes the activities, the researcher argues that the

participants' challenges will be minimised by using the model as a meaningful tool for the

teaching of similarity of triangles. This is in line with researchers (Howse & Howse, 2015;

Mostafa, Javad & Reza, 2017; Muyeghu, 2008; Ramlan, 2016) who argue that van Hieles’

theory is used as one of the methods for coping with the challenges of teaching geometry

and improve learners’ geometric thinking levels.

As illustrated in Chapter 5, after some of the participants wrote notes of similar triangles

on the chalkboard then they started explaining by using examples. For example, TA, TB,

TC, TD and TE lessons revealed that the classroom teaching lacked some activities that

required the learners to identify similar triangles from a cluster of different triangles.

However, according to the model for the teaching of similarity of triangles, the second

phase was directed orientation and teacher should be actively engaged in the teaching-

learning process and direct the learners where and how they should approach the

problems. The teacher is supposed to define the similarity of triangles on the arithmetic

and transformational approach and provide sequential activities which encourage

learners’ hands-on manipulation (see Figure 3.5). However, the data presented in

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Chapter 5 revealed that most of the lessons lack directed orientation phases of learning.

For example, TA, TB, TC, TD and TE lessons observed revealed that the participants

used chalkboard as direct classroom activities through writing the definition of similar

triangles then followed by a demonstration of examples using the chalk and talk approach.

Learners were subjected to copy notes from the chalkboard. They were passive listeners.

The researcher argues that lessons lacked the direct orientation phase, due to the

reasons that the participants do not know the van Hieles’ phase of learning, or their

practice lacked components of the mode. Therefore, the model will minimise the observed

challenges related to the phase when an intervention is based on the model provided for

the participants.

On the other hand, the classroom observation data also revealed that some of the lessons

were poorly presented, and explication was found in the third phase of instruction for

teaching similarity of triangles. According to the model for teaching the similarity of

triangles, during this phase, students learn and verbalise their understanding of the

similarity of triangles and its connection. Learners become more conscious of the similar

figures and the similarity of triangles expresses these in accepted geometrical language.

On the contrary, the data collected revealed that most of the lessons lacked the

connection of similarity with other geometrical and mathematical concepts. Moreover, the

mastering of the correct geometrical language such as similar triangles, and

corresponding sides and angles, dilation/enlargement of similar triangles was found

lacking in the observations made. In line with this, some of the participants, for example,

TC, TD and TE lacked the use of proper geometric language. Without the use of

appropriate language, learners could not verbally express and exchange ideas they had

been exploring in the learning of similarity of triangles.

Furthermore, according to the model for teaching similarity of triangles at this phase

teacher supposed to establish good interaction with students. The interaction among the

teachers and the learners is important in supplying them with necessary and enough

support so that the students can achieve the maturation essential for the growth to the

next level. However, the finding in this study revealed that the teacher and student

interaction was minimal. Only two students were found participating in TA, TB, and TC

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classrooms. In most lessons it was observed that the participants were talking, and

students were taking notes. Thus, good interaction between the teacher-students must

be established.

In the free orientation phase, according to the model for teaching similarity of triangles

teachers supposed to provide geometric problems for students that can be solved in

numerous ways and encourage students to master the network of the relationships. In

contrary, the finding in this study indicated that only in two lessons, TB and TD provided

class work for learners. The data revealed that TA, TC, TD and TE would solve examples

by copying from their notebooks. According to the model for teaching similarity of

triangles, in this phase, the teachers' role was observed to be minimal and provided the

geometric activities appropriate for the level (see activities in the Figure 3.4). Students

were also recommended to get independent and put themselves in the network of

relations to fulfil activities of similarity of triangles. However, the data revealed that the

activities were deficient of networks. They were not appropriate for the free orientation

phase of the instruction for teaching similarity of triangles. Moreover, the data revealed

that the lessons lack the free orientation phase.

According to the model for teaching the similarity of triangles, students ought to construct

an overview of the similarity of triangles learned and the teacher should help the learners

to gain an overview of the similarity concepts. Similarly, students’ summaries their

comprehension about of similarity of triangles and integrate the appropriate language

(terminology) for the new higher geometric thinking level. However, the data revealed that

the lessons lack the integration phase of teaching similarity of triangles. The participants

did not summarise their explanation by using appropriate examples like those presented

in the Figure 3.5. In addition, the lessons lacked the phases of teaching the similarity of

triangles. Participants’ responses in the interviews and questionnaires revealed that they

did not know the van Hieles’ theory and its importance for teaching and learning geometry,

in particular the similarity of triangles. The researcher argues that the model of teaching

the similarity of triangles minimises the challenges teachers faced in the teaching of the

topic. To develop learners’ geometric thinking levels, the participants were supposed to

use the activities and examples in the model. Students should be presented with a variety

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of geometric experiences. Moreover, the participants should be aware of each of the five

instructional phases, their instructional activities and examples for learners’ geometric

thinking. Teachers should be aware of the important pedagogical area of concern such

as the ways of teaching, organisation of instruction, content and materials used for

teaching similarity of triangles.

Furthermore, the classroom lessons observations revealed that some of the participants,

for example, TC, TD and TE’s initial phases of the lessons discussed the similar figure

after the teacher drew similar polygons on the chalkboard. However, the participants

displayed poor mathematical knowledge in explaining the reasons for the similarity of

polygons. This concurs with several research studies (Adolphus, 2011; Choo, Eshaq,

Hoon & Samsudin 2009; Aydogdu & Kesan, 2014; Das, 2015; French, 2004;

Kambilombilo & Sakala, 2015; Jones, Mooney & Harries, 2002; Jones, 2000; Sitrava &

Bostan, 2016), which identified teachers’ inadequate content knowledge and poor

foundation of mathematics. Furthermore, research by (Chazan, 1988; Denton, 2017;

Edwards & Cox, 2011) has consistently highlighted that geometric similarity is a

mathematical topic with which both learners and teachers encounter difficulties. The

lesson observation revealed that participants did not label the corresponding sides and

angles when working on the examples of similar triangles. The literature review provided

that for learners to understand the similarity between two triangles, they must explore the

relationship of different attributes of the triangles or change one characteristic of shape -

preserving others.

To further unpack how the participants’ pedagogical approaches were manifested during

lesson observation it was found that teachers’ use of different diagrams and learners’

learning on hands on manipulation observation were interrogated. The result of analysis

established that TA supported his teaching of similarity of triangle by showing similar

figures from learners’ textbooks (see Figure 5.23). However, the maps used could not be

seen by the learners. Learners could not differentiate between the two maps of Ethiopia

because they could not see the figures. TA’s demonstration of maps required much

attention. Furthermore, TB also supported the explanation of triangles by showing a

model of triangle (see Figure 5.29). However, he could not show the corresponding sides

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and angles of the triangles. The data also revealed that all the lessons observed lacked

the use of manipulative activities. This refutes Hartshorn and Boren (2005) who argue

that manipulatives are one approach to help students improve their mathematical

comprehension. Moreover, studies (Suydam & Higins, 2003; Sowell, 2000; Thomson,

2003) indicate the importance of the use of concrete models in teaching and learning at

all grade levels. NCTM (2010) has also encouraged the use of concrete models for

teaching mathematics at all levels. The NCTM’s Curriculum and Evaluation Standards

(2010) for Grades 5 through 8 emphasise the use of concrete models in representing

mathematical concepts and processes. NCTM further notes that “learning should be

grounded in the use of concrete materials designed to reflect underlying mathematical

ideas” (p.87). Mathematics educators underlined that engaging learner in examining,

measuring, comparing, and contrasting a wide variety of shapes to develop essential

learning skills (NCTM, 2010). Therefore, it is important to use the concrete model of

similar triangles when teaching at Grade 8 level for meaningful learning of similarity of

triangles.

The classroom observations revealed that in TB and TD’s lessons learners were

observed working on the classwork activities. This implies that the teaching approach was

dominated by teacher talk and chalk, and the chalkboard was used as instructional

materials. In the same token, learners were passive listeners in most of the lessons

observed. However, in Ethiopia, the educational policy document recommends a

problem-based approach to teaching mathematics and science (MoE, 2019). Problem-

based learning prepares students to think critically and analytically and to find and use

appropriate learning resources. According to van Hieles (1986), a significant reason in

many teachers' failure to create meaningful comprehension in geometry is their inability

to match instruction to their learners' geometric thinking levels. Teachers need to organise

a problem-based approach in teaching geometry to promote a meaningful learning

environment and to attain the desired instructional objectives of geometric contents.

Data revealed that teachers were found unable to use proper geometric language. This

is in line with the findings of (Jones, Mooney & Harries, 2002) who noted that teachers’

geometric vocabulary knowledge was poor. However, van Hieles’ (1986) theory

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emphasises the use of proper language by the mathematics teacher when teaching

geometry. The language of the teacher should be very simple and understood by the

learners. Precise and unambiguous use of language and rigour in the formulation are

important characteristics of mathematical treatment. Quite often, people cannot

understand each other or follow the thought process of each other. This situation is

enough to explain why at times teachers fail to help learners in geometry learning. The

learners and teachers have their own languages, and often teachers use a language

which learners do not understand. For example, the teachers should properly differentiate

the concept, similarity in geometry from similarity of colloquial speech. “Similar” means

looking or being almost the same, but not exactly the same. For example, John is very

similar in appearance to his brother. Whereas similarity in geometry refers to have the

same shape but not necessarily the same size.

Lessons observed further revealed a lack of interaction between teachers and learners.

The act of interacting with other learners while communicating in the mathematics

classroom has been described as “organising and consolidating ideas, thinking

coherently and clearly, analysing and evaluating strategies, and expressing ideas

precisely” (NCTM, 2000, p. 60). Such interactions in the classroom where learners are

communicating and defending their proofs are essential for the development of a more

rigorous understanding of the nature of proofs.

Results revealed that there were factors that affect teacher-learner interaction during

teaching-learning about similarity of triangles. These included: (1) learners’ misbehaviour,

(2) learners lack knowledge on plane geometry, (3) learners lack interest in learning

geometry, and (4) lack of resources. Concerning this, Englehart (2009) asserts that

teacher- learner interaction does not take place in a vacuum. It occurs within a very

complicated meticulous socio-cultural environment. This is in line with Bruce (2007) who

asserts that mathematics teachers face challenges in facilitating high-quality teacher-

learner interaction. Some of those challenges are: (1) the way of teaching mathematics,

(2) lack of mathematics content knowledge, (3) prerequisite for facilitation skills and

concentration on classroom dynamics, and (4) lack of time. Moreover, researchers (Way,

Reece, Bobis, Anderson & Martin, 2015; Ayuwanti, Marsigit & Siswoyo, 2021) indicate

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that teacher interactions with learners vary in quality and have appreciable effects on

mathematics achievement. Furthermore, (Pianta & Hamre, 2009; Pianta, 2016)

acknowledge that teacher- learner interactions are malleable features of classroom

environments and have been the focus of international efforts to raise mathematics

achievement. Good interaction between the teachers and learners usually creates a

positive relationship in the classroom and contributes to meaningful teaching and learning

of the similarity of triangles.

The data also revealed that only a few learners were participating in TB, TC, TD and TE’s

classrooms during the lessons observed. Frobisher (2010) advises that it is the teacher’s

concern to make activities that consolidate students to get involved in their learning.

Furthermore, the Educational Policy National Curriculum Statement (MoE, 2019) of

Ethiopia envisages a teacher who acts as a councilor, analyser, designer of learning

programs and resources, as well as a leader. The present education policy has seen a

complete paradigm change from Ethiopia's previous traditional approaches, which were

'teacher-centred' to a teacher who acts as a learning facilitator. However, the teaching

and learning of the similarity of triangles observed were still teacher-centred. The findings

concur with how Faulkner, Littleton, and Woodhead (1998) verified a traditional class as

teacher-centred where the emphasis is on neatness, order, and exact replication of

shown techniques.

6.2.3 Challenges teachers faced in the teaching of similar triangles

The data presented in Chapter 5 revealed that participants faced challenges in the

teaching of the similarity of triangles. Those challenges include: (1) mathematical

knowledge, (2) pedagogical knowledge challenges, (3) learners’ poor background

knowledge, (4) resources, and (5) mathematics syllabus and other challenges. The

following section presents the challenges the participants faced.


The data from the classroom observation revealed that participants faced mathematical

knowledge challenges in the teaching of the similarity of triangles. In particular, it was

observed that participants lacked (1) understanding and apply the test of similarity of

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triangles theorems, (2) showing the similarity of triangles by using the given condition that

is “two angles of one triangle are congruent to the corresponding two angles of another

triangle then the two triangles are similar”(MoE, 2009,p. 230); (3) locating the angle, and

corresponding sides of the similar triangles, (4) mentioning the sides that include the

given congruent angle of the two similar triangles, (5) explaining the theorems and their

importance for checking similarity of triangles, (6) locating the vertex of triangles on the

coordinate plane, (7) the knowledge of how to get the enlarged triangle coordinate value,

and (8) using geometric language during the explanation of similar triangles.

Furthermore, the data on classroom observation illustrate that some of the participants

might have missed the important knowledge on what was given on the examples and

what was required during explaining examples of similar triangles. The finding related to

(Türnüklü, 2009; Alatorre, Flores & Mendiola, 2012) acknowledgement in that learners

and teachers experienced difficulties in solving the triangle inequality theorem. In addition,

Kambilombilo and Sakala (2015) acknowledge that in-service mathematics teachers

encounter challenges in transformation geometry; use of instruments such as protractor

and compass; dealing with reflection in slant lines; writing the equation of lines reflection;

inadequacies in rotation geometry and limitation on van Hieles’ levels III and IV. The

finding in the current study illustrated that the participants faced mathematical knowledge

challenges observed by (Adolphus, 2011; Das, 2015; Gomes, 2011) who assert that

teachers’ geometric knowledge was not adequate to teach geometric transformation. This

was also consistent with the previous studies, Jones, Mooney, and Harries (2002) who

reported that primary teachers had difficulties in calculating the area and the volume of

geometric figures. Moreover, Adolphus (2011) acknowledged that the foundation of most

mathematics teachers in geometry is poor. The teaching and learning of the similarity of

triangles need to give special attention to learners’ success in mathematics education.

The data from interviews and questionnaires revealed that participants did not understand

the static and transformational approaches definition of the similarity of triangles.

Moreover, the participants responded that geometry is not appropriately included in

teacher education programs for primary school teachers in Ethiopia. The finding of the

current study concurs with (Seago, Jacobs, Heck, Nelson & Malzahn, 2014) whose study

indicated that USA middle school teachers performed poorly on similarity items on

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geometry and faced challenges on teaching similarity in the classroom. Moreover, Fujita

and Jones (2006) also acknowledged that trainee teachers could not define geometric

content knowledge related to classifying quadrilaterals.

6.2.3.2 Pedagogical Knowledge Challenges

The data also revealed that participants may face challenges with their pedagogical

knowledge. The data obtained through classroom observations revealed that: (1) lessons

presented lacked the connection of similarity with real-life; (2) participants did not show

map and models of similar triangles as illustrated in the learners’ textbooks; (3)

participants had difficulty to explain about models of similar figures and polygons; (4)

participants had difficulty to explain the application of similarity of triangles in real-life; (5)

the pedagogical approaches were centred on writing notes on the chalkboard and then

explanations; (6) the lesson presented lack the explanation of the similarity of triangles

by using transformation approach, rotation, reduction and the enlargement; (7)

participants did not use problem-based approaches of teaching similarity of triangles and

(8) participants lacked classroom management skills. Due to these facts, it seems

possible to reflect that the participants had faced challenges of pedagogical knowledge.

In connection with scenarios, the Royal Society's study on geometry teaching (2001)

argued that “the most significant contribution to improvements in geometry teaching will

be made by the development of effective pedagogy models, which will be supported by

well-designed activities and materials” (p.30). This means that some of the current

pedagogies emphasise memorisation of geometric concepts because mathematics

teachers did not have appropriate skills, content knowledge, as well as the pedagogical

knowledge necessary to be effective in a mathematics classroom. In this regard, studies

carried out by scholars (Adolphus, 2011; Choo, Eshaq, Hoon & Samsudin, 2009;

Aydogdu & Kesan, 2014; Das, 2015; French, 2004; Kambilombilo & Sakala, 2015; Jones,

Mooney & Harries, 2002; Jones, 2000; Sitrava & Bostan, 2016) which were conducted to

explore the challenges of teaching geometry. Accordingly, the main challenges teachers

faced are: (1) lack of pedagogical knowledge, (2) teachers may not have adequate

content knowledge, (3) poor foundation of mathematics teachers (4) teaching and

learning environments are not conducive, and (5) lack of commitment to geometry.

Furthermore, the finding of this study is in line with Fujita and Jones (2002) who

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acknowledged that the pedagogical approach used for meaningful teaching geometry

continues as the main challenges in mathematics instruction.

Participants’ responses to semi-structured interviews and the data obtained through

questionnaires revealed that participants did not know theories related to teaching and

learning geometry. They could not get professional development programs related to the

teaching of the similarity of triangles. The result of the current study reflected the

arguments put forward by Das (2015) and Seago et al. (2014) since teachers often lack

good pedagogical content knowledge, mathematical fluency to make instructional

decisions and professional development necessary to improve learners’ learning on

similarity. Moreover, participants responded that they did not like to teach geometry and

teaching geometry is a challenge. The finding concurs with the ideas of Choo et al. (2009)

who state that teaching geometry is not an easy task and consequently making its

pedagogy easier, more interesting, more practical, based on real-life examples, and more

accessible to all learners is not easy. Moreover, Adolphus (2011) acknowledges that

teachers lack the commitment to teach geometry.

6.2.3.3 Students’ poor background knowledge

Participants’ data obtained during interviews and questionnaires revealed that students’

prior knowledge and language are challenges faced in the teaching of the similarity of

triangles. Participants also indicated in the semi-structured interviews and questionnaires

that learners lack knowledge about plane figures, and they are poor to visualise the

enlarged/reduced triangles. Gunhan (2014), in this regard, acknowledged that learners

have insufficient geometrical knowledge and visual perception, and do not know the

requirement for the formation of a triangle. In addition, Mukucha (2010) also

acknowledged that most learners lacked a conceptual understanding of geometrical

concepts and reasoning skills in problem-solving. Similarly, Arslan (2007) noted that

students in 6th, 7th and 8th grades exhibited low-level reasoning skills. Furthermore,

Türnüklü, (2009) and Alatorre, Flores and Mendiola (2012) revealed that students and

participants experienced difficulties in solving the triangle inequality theorem. The

analysis revealed that learners’ background knowledge is one of the challenges teachers

faced when teaching similarity of triangles.

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6.2.3.4 Resources

The data analysis revealed that lack of resources was among the challenges teachers

faced in the teaching of the similarity of triangles. The data from the observation, semi-

structured interviews, and questionnaires further cleared that all schools lack pedagogical

centres, and most of the classrooms lacked geometric figures and models. Clements

(2003) also argued that geometry classroom is expected to be characterised by the

following criteria: (1) appropriate activities to support the connection between prior

understanding to new learning and developing logical thinking abilities, (2) investigative

tasks/real-world problems to support developing logical thinking abilities and spatial

intuition, (3) the use of technology, visual representations, and interpretation of

mathematical arguments, and (4) employing collaborative learning.

The data revealed that all the schools lack pedagogical centres. Further, most of the

classrooms lacked geometric figures and models on the walls. Participants also

mentioned that lack of resources was among the challenges they faced in the teaching of

the similarity of triangles. Studies (Suydam & Higins, 2003; Sowell 2000; Thomson 2003)

indicated that concrete models are important in teaching and learning mathematics at all

grade levels. In a geometry class, pictures, and three-dimensional objects on display are

useful to relate the existence of geometric concepts, like referring to learners’ homes and

environments. Learners experience geometry through drawings of the actual objects that

they see in their neighbouring environments. The use of manipulatives, according to

Hartshorn and Boren (2005), is one technique to improve learners’ mathematical

knowledge.

Researchers (Fennema, 2004; Szendrie 2011) argue that learners could learn better if

their learning environment contains encounters with models that are appropriate for their

cognitive development. The teaching and learning environment full of concepts, learning

experiences, fascinating materials, and geometric resources can stimulate creativity

(Craft, Jeffrey & Leibling, 2001). NCTM recommends that learners and teachers have

access to a variety of instructional technology tools, teachers are provided with

appropriate professional development, the use of instructional technology be integrated

across all curricula and courses, and that teachers make informed decisions about the

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use of technology in mathematics instruction (Johnson, 2002). Although using computer

software and IT materials should help learners’ learning outcomes, it is still not common

in Ethiopian schools. Thus, it is important and attention-demanding to use technology in

mathematics classrooms to improve mathematics achievements in Ethiopia.

6.2.3.5 The Mathematics syllabus and other challenges

Participants responded in the interviews that the place of geometric content in

mathematics syllabi was among the challenges they faced in the teaching of the similarity

of triangles in Ethiopia. Some of the participants mentioned that learners did not get

enough time to cover geometric contents in the last chapter, and mostly they are not

covered. It is a common activity that geometric topics are usually included in the last part

of the textbooks which may cause a problem in content coverage in Ethiopia. This result

is associated with the failure to grasp the basic concepts of geometry by the learners. For

example, TA mentioned that the concept of the similarity of triangles is in the 5thchapter

of Grade 8 mathematics curriculum, and learners were supposed to learn the same topic

in Grade 9 in the 6th chapter. In most rural situated schools in Ethiopia, schools suffer due

to a shortage of teachers, being late to start the academic calendars. Usually, the content

in the last parts of the textbook is not covered.

The data collected from the questionnaires also revealed that other factors such as lack

of supervision from the school principals and teachers’ economic problems were the

challenges they faced in the teaching of similarity. Research by Dalawi, Zakso, and

Radiana (2019), recommends the need for academic supervision by school supervisors

to improve teacher professionalism. However, in this research, some of the participants

mentioned that there is lack of supervision from the school principals and supervisors.

Ramadhan (2017) here also acknowledges that the implementation of academic

supervision by school supervisors and school principals has a significant influence on

teacher performance. Thus, in the study area, teachers should be supervised in teaching

mathematics for students’ meaningful learning on similarity of triangles.

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Participants in the study area proposed strategies to minimise the challenges they faced

in the teaching of the similarity of triangles. The following section presents the discussion

on the proposed strategies.

6.2.4 Suggested strategies to minimise the challenges of teaching the similarity of

triangles

The data revealed that the participants proposed strategies to minimise the challenges

they faced in the teaching of the similarity of triangles. Those strategies were, pedagogical

approaches such as active learning methods, reform on pre-service teacher training and

continuous professional development related to the teaching-learning similarity of

triangles.

Participants who responded to the interviews and questionnaires reflected those

pedagogical approaches such as student-centred or active learning approaches are

believed to be the best methods for teaching the similarity of triangles. Researchers

(Herbst, 2006; Jones, Fujita, & Ding, 2004; Jones & Herbst, 2012) also suggested that to

promote geometrical reasoning, teachers are supposed to use various instructional

techniques and strategies. Teachers should be aware of the different instructional

methods and how to apply them in their mathematics classroom. According to Biggs

(2011), teacher-centered teaching methods focus on the activities that mathematic

teachers do to bring the concepts to the students, whereas student-centered pedagogies

focus on the activities that the students do to understand the concept. Mathematics

teachers are supposed to use the instructional strategies that favour learners’

understanding of the contents rather than merely finishing the lesson time.

The participants also reported that the training in teacher’s education institutions or the

university at large lacks geometric contents and different theories related to teaching and

learning geometry. Another challenge put forward by the participants is that they are not

involved in professional development programs related to the teaching of the similarity of

triangles which is likely to be recommendable as a need to minimise the challenges they

faced in the teaching of the similarity of triangles. This concurs with (Cohen & Hill, 2000;

Smith 2001) who argue that a practice-based approach may help teachers to examine

the mathematical skills and explore instructional practices that support student learning.

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Thus, teachers in the study area need a planned professional development material to

promote meaningful teaching of the similarity of triangles.

Ball, et al. (2008) asserts that mathematic teachers need opportunities to gain a

specialised type of content in geometry: “mathematical knowledge for teaching” (MKT)

(p.34) which includes not only a deep understanding of geometric transformations and

similarity but also the knowledge and fluency to make instructional decisions that support

students’ learning of this content. The mathematical knowledge necessary to teach

effectively is recognised as being a complicated issue than simply needing an

understanding of subject knowledge (Franke & Fennema,1992). In order to make the

similarity of triangles meaningful for the learners, teachers must be provided with the

opportunity to utilise geometrical concepts and language to make connections between

representations and applications, algorithms and procedures (Sowder, 2007). Training

programs that provide geometrical experiences and allow teachers to work together to

explore mathematics can help them gain confidence in their abilities to develop

understanding.

The researcher argues that the proposed model is one of the ways to deal with the

challenges and a meaningful approach for teaching the similarity of triangles. To be more

certain, the model needs further investigation. That is preparing an intercession and test

the feasiblity. Chapter 7 comprises the conclusions and recommendations of this study.

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CHAPTER SEVEN

SUMMARY, RECOMMENDATIONS AND CONCLUSION

7.1. INTRODUCTION

This chapter summarises the findings of the study based on the data collected and

analysed. It also gives an overview of the study aims and objectives in addition research

questions. The chapter further presents the outlines on how the proposed model assisted

the research participants and recommendations for future studies. It also includes

conclusions which are important to further researchers in the teaching and learning of

geometry as well.

7.2. SUMMARY OF THE STUDY

The study’s summary is presented in this section. Chapter 1 presented the overview of

the study, background to the study, purpose of the study, statement of the research

problem, and significance as well as delimitations of the study. Then, the study also

attempted to answer the following research question.

How can the challenges of teaching similarity of triangles to Grade 8 learners be

minimised? Based on this general research question, the following sub-questions were

answered:

I. What are the challenges faced by mathematics teachers in teaching similarity of

triangles?

II. How do teachers interact with learners in the teaching of similarity of triangles?

III. Which pedagogical approaches can promote meaningful teaching of similarity of

triangles?

IV. How can the strategies be applied such that the challenges in teaching similarity

of triangles are minimised?

To answer the above research questions, aims and objectives were set. The literature

reviewed addressed the research questions presented.

Chapter 2 focused on the literature reviewed regarding the teaching of similarity of

triangles in primary schools. This chapter covered a presentation on the history of

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geometry, the importance of learning geometry, the concept of geometric similarity,

teaching geometry, the role of teachers in teaching geometry, geometry classroom,

classroom interactions, teaching geometry through technology, challenges of teaching

geometry, and strategies to minimise the challenges of teaching similarity of triangles. In

addition, Chapter 3 was dedicated to the theoretical framework in which the van Hieles’

(1985) theory, the theory of figural concepts by Fischbein (1993), and the Duval’s (1995)

theory of figural apprehension were discussed. In connection to this, Chapter 4 also

emphasis on research methodology employed to collect data, data analysis and all

modalities of the study. Then Chapter 5 focused on data presentation, and analysis while

Chapter 6 focused on the discussion of the findings.

7.3 SUMMARY OF THE RESEARCH METHODOLOGY

This study laid its foundation on the interpretive paradigm. The study used an exploratory

case study design, and qualitative methods. Classroom observations, semi-structured

interviews, and questionnaires were used to collect data. During the data collection, field

notes were taken, and transcripts of classroom observation, semi-structured interviews,

and questionnaires were examined, synthesised, and critically analysed to recognise

trends and their corresponding categories. The data were categorised into four themes.

7.4 SUMMARY OF FINDINGS FROM THE STUDY In this section, the summary of the findings based on the main themes identified before

the participants applied the model proposed in this study are presented under the

following headings: (i) challenges teachers faced in the teaching of the similarity of

triangles (ii) importance of learning geometry, and (iii) phases of instruction in the teaching

of the similarity of triangles.

7.4.1: Challenges teachers faced in the teaching of similar triangles

Results showed that participants faced challenges in teaching the similarity of triangles.

These challenges have been classified as: (i) mathematical knowledge, (ii) pedagogical

knowledge challenges, (iii) learners’’ poor background knowledge, (iv) resources, and (v)

mathematics syllabus and other challenges.

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Data revealed that the participants faced mathematical knowledge challenges in teaching

the similarity of triangles. In particular, it was observed that the participants lacked (1)

understanding and applying the test of similarity of triangles theorems, (2) showing the

similarity of triangles by using the given condition for similarity of two triangles, that is two

angles of one triangle are congruent to the corresponding two angles of another triangle

then the two triangles are similar; (3) locating the angles, and corresponding sides of the

similar triangles, (4) mentioning the sides that include the given congruent angles of the

two similar triangles, (5) explaining the theorems and its importance for checking similarity

of triangles, (6) locating the vertex of triangles on the coordinate plane, (7) the knowledge

of how to get the enlarged triangle coordinate value, and (8) using geometric language

during the explanation of similar triangles. Furthermore, the participants:

Displayed poor mathematical knowledge in explaining the rationales for the


Did not label the corresponding sides and angles when working on the examples

of similar triangles.

The lessons observed lacked the use of manipulatives activities. The teacher is

supposed to define the similarity of triangles on the arithmetic and

transformational approach and provide sequential activities which encourage

learners’ hands-on manipulation.

Missed the important knowledge on what was given on the examples and what

was required during explaining examples of similar triangles.

Static and transformational approaches were found very important in teaching

similarity, but the data showed that participants did not understand the static and

transformational approaches on the definition of the similarity of triangles.

7.4.1.2 Pedagogical knowledge challenges

The data also pointed out that the participants had faced challenges in their pedagogical

knowledge. In particular, it was also observed that: (1) lessons presented lacked the

connection of similarity with real-life and were poorly presented, (2) participants did not

show maps and models of similar triangles as illustrated in the learners’ textbooks; (3)

187

they had difficulty to explain about models of similar figures, polygons, and the application

of similarity of triangles in real-life; (5) the pedagogical approaches were centred on

writing notes on the chalkboard and then explanations; (6) the lesson presented lack the

explanation of similarity of triangles by using transformation approach, rotation, reduction

and the enlargement; (7) they did not use problem-based approaches of teaching

similarity of triangles, and (8) participants lacked classroom management skills.

Furthermore, the model of teaching similarity of triangles explored the following findings

on the pedagogical challenges participants faced in the teaching similarity of triangles.

Those challenges include: (1) the lessons missed conversation and models of similar

figures; (2) the lessons were dominated by the teacher talking and lacked hands-on

engagement, (3) learners’ prior knowledge of the similarity of triangles was not explored;

(4) the lessons lacked the direct orientation phase; (5) the lessons poorly presented and

lacked connection of similarity of triangles with other geometric and mathematical

concepts, (6) participants lacked to use appropriate geometric language; 7) the lessons

lacked geometric problems that can be solved in numerous ways and network

relationship; and (8) the lessons lacked the integration phase. In general, the teaching

approaches used by the participants missed the van Hieles’ phase of instruction for the

teaching of geometry. In this regard, participants did not know van Hieles’ theory and its

importance for teaching and learning geometry and the similarity of triangles.

In addition, the participants could not get professional development related to teaching

and learning geometry. Thus, the participants did not like to teach geometry because

teaching geometry is a challenge to them.

7.4.1.3 Learners’ poor background knowledge

According to the data obtained from the participants, learners lack knowledge about plane

figures, and they were poor to visualise the enlarged/reduced triangles, and their

background knowledge is one of the challenges participant teachers faced in the teaching

and learning of the similarity of triangles.

7.4.1.4 Resources

Participants also mentioned that lack of resources was among one of the challenges they

faced in the teaching of the similarity of triangles. All the sampled schools lack

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pedagogical centres, and most of the classrooms did not have geometric figures and

models. Lack of supervision from the school principals and economic problems were the

challenges participants faced in the teaching of similarity.

7.4.1.5 The mathematics syllabus and other challenges

Results revealed that geometry was not appropriately included in the teacher education

program for primary school teachers in Ethiopia. The place of geometric content in

mathematics syllabi was among the challenges participants faced in the teaching of the

similarity of triangles in Ethiopia.

Some of the participants mentioned that students did not get enough time to cover

geometric contents in the last chapter, and mostly they are not covered. It is a common

activity that geometric topics are usually included in the last part of the textbooks which

causes a problem in content coverage in Ethiopia. This result is associated with the failure

to grasp the basic concepts of geometry by the learners.

7.4.2 The importance of learning geometry

Based on the empirical data obtained through classroom observation, semi-structured

interviews, and questionnaires, the data revealed that teachers at the sampled schools

were found cognizant of the importance of teaching geometry in general and the similarity

of triangles. This was after the intervention and the use of the proposed model in this

study.

Results further reveal that participants note that learning geometry as part of mathematics

improves students’ geometric and cognitive skills and helps them in studying

measurements. Moreover, during the interviews, respondents revealed that similarity is

an important topic to be taught in Grade 8 and it is a visual representation of concepts

such as ratio, proportion, and slope. Furthermore, interviewees revealed that many

objects found in their environment are related to geometry. In this regard, participants

related the importance of geometry to the real world. The model for teaching the similarity

of triangles also required teachers to discuss the reason for learning the similarity of plane

figures. The teachers’ classroom discussion with learners should relate more awareness

of the importance of learning similarity in relation to daily lives. This would help teachers

to easily relate similarity to students’ real world. The participants also mentioned the sun,

189

shadow and copying as some of the examples in learners’ daily lives that could be used

in the meaningful teaching of similarity.

7.4.3 Phases of instruction in teaching similarity of triangles

From the phases of instruction in teaching similarity of triangles that come out from the

coded data related to similarity and teaching approaches, the researcher drew the

following summary of findings.


According to the reports, some of the participants were aware of the concepts to be

learned before similarity. In this regard, they mentioned during the interviews that

concepts such as proportion, enlargement, shapes, sizes of geometric figures, plane

figures, like triangles, rectangles, and congruence are the pre-knowledge necessary for

learners to learn before they are introduced to similarity. It was also observed that some

of the teachers’ lessons; for example, TB’s lesson connected similarity of triangles with

congruence.

The data further revealed that most of the participants were not aware of the concepts to

be learnt after the similarity of triangles. This refutes Chazan (1988), Lappan and Even

(1988) who argue that similarity provides a way for learners to connect spatial and

numeric reasoning and provides the basis for advanced mathematical topics, such as

projective geometry, calculus, slope, and trigonometric ratio.


The results of classroom observation presented in Chapter 5 showed that none of the

participants reminded their learners about the similarity of any other two polygons before

defining the similarity of triangles. However, triangles were explained as a special type of

polygon and therefore the conditions of similarity of polygons also stand true for triangles.

Thus, the similarity concept was only taught and confined to triangular figures only.

It was also found that figures that were drawn by TC, TD and TE on the chalkboard for

the two similar triangles could create misconceptions for learners. For example, as shown

in Figures 5.10, 5.17 and 5.22, the two triangles are congruent in size. However, similar

figures do not mean figures are always congruent. This concurs with Chazen (1987) who

identified three difficulties for learners in learning similarity. These include: notations of

190

similarity, proportional reasoning, and dimensional growth relationships. Furthermore, it

was also found that participants lacked an awareness of properly using the symbol ‘~’

which stands for the language “similar to” during the explanation of the similarity of two

triangles. Moreover, participants lacked proper demonstration of the corresponding sides

and angles of similar triangles from the given examples.


In the next section, the researcher drew the summary of the findings from the data

obtained through classroom observations, semi-structured interviews, and questionnaires

on participants’ teaching approaches.

The results revealed that participants used chalkboard as the directed classroom

activities through writing the definition of similarity of triangles, and then followed by a

demonstration of examples using chalk and talk approach. The data further revealed that

the teaching approaches used by participants missed the van Hieles’ phase of instruction

for the teaching of geometry in general and the phases of instruction for the teaching of


Data indicated that the participants began the lesson by writing on the chalkboard about

the definition of similar triangles, which correspond to the initial phases of the lesson.

Learners were subjected to copy notes from the chalkboard, and they were passive

listeners. The data again reflected that the lessons lacked the students’ engagements

and conversations about similar figures. For example, the TC’s initial phases of the

lessons were questioning by saying “What does a similar figure mean?”. The observed

lessons revealed that the instructional practices included talking then followed by doing

examples. The participants did not use activities and models of similar figures during

questioning of similar triangles.

The finding in this study, therefore, indicated that only in two lessons, TB and TD provided

classwork for learners, and the data also reported that some of the participants would

solve examples by copying from their notebooks.

In addition, the classroom lessons observation also revealed that some of the

participants; for example, TC, TD and TE’s initial phases of the lessons were discussed

on similar figures after the teachers drew similar polygons on the chalkboard. However,

191

the participants displayed poor mathematical knowledge in explaining the reasons for the

similarity of polygons.

To further unpack how the participants’ pedagogical approaches were manifested during

lesson observation, it was found that teachers’ use of different diagrams and students’

learning on hands-on manipulation observation were summarised. The result of the

analysis established that TA supported his teaching of similarity of triangles by showing

similar figures from learners’ textbooks. However, the maps used could not be seen by

the learners. Hence, learners could not differentiate between the two maps of Ethiopia

because they could not see the figures. Thus, TA’s demonstration of maps required much

attention. Furthermore, TB also supported the explanation of triangles by showing a

model of the triangle. However, he could not show the corresponding sides and angles of

the triangles. Ironically, all the lessons observed lacked the use of manipulative activities.

7.4.3.3.1 Teacher-learner interaction in teaching similarity of triangles

In TB and TD’s lessons, learners were observed working on the classwork activities. This

implies that the teaching approach was dominated by teacher talk and chalk, and the

chalkboard was used as instructional material. In the same token, learners were passive

listeners in most of the lessons observed.

The data also revealed that teachers were found unable to use the geometric language

since the participants lacked awareness of properly using the symbol of similarity. This is

in line with the finding of Jones, Mooney and Harries (2002) who noted that teachers’

geometric vocabulary knowledge was poor. However, van Hieles’ (1986) theory

emphasises the use of proper language by the mathematics teacher when teaching

geometry.

Lessons observed further portrayed a lack of interaction between teachers and learners.

Only a few learners were found participating in TB, TC, TD and TE’s classroom during

the lessons observed. TA’s lessons observed were dominated by copying notes from the

chalkboard. Learners were subjected to copying notes from the chalkboard. Furthermore,

they were passive listeners. The teaching and learning of similarity in the lessons

observed were still teacher-centred. The findings concur with how Faulkner, Littleton, and

Woodhead (1998) describe the traditional class as teacher-centred where the emphasis

is on neatness, order, and exact replication of shown techniques. Results thus revealed

192

that there were factors that affect teacher- learner interaction in teaching and learning of

the similarity of triangles. These were: (1) learners’ poor knowledge of plane geometry,

(2) learners’ lack of interest in learning geometry, and (3) lack of resources.

7.5 PEDAGOGICAL APPROACHES WHICH PROMOTE MEANINGFUL

TEACHING OF SIMILARITY

Based on the literature reviewed, a theoretical framework that underpinned this study,

and empirical data obtained, the researcher argues that the proposed model for teaching

the similarity of triangles was used as a pedagogical approach that promotes meaningful

teaching of similarity of triangles.

Outlines on how the proposed model assisted the research participants

According to the researcher’s current proposed model for teaching similarity of triangles,

the participants showed the following improvements:

1. The lessons presented had a connection similarity with real life.

2. The proposed model showed participants how to use maps and models of similar

triangles from the learners’ textbooks.

3. Participants were found confident enough to explain models of similar figures,

polygons, and the application of similarity of triangles in real life.

4. The pedagogical approaches used were centred on engaging students

accompanied by explanations.

5. The lessons presented contained explanations of the similarity of triangles by using

transformation approaches, rotation, reduction, and enlargement.

6. Participants defined the similarity of plane figures, polygons, and triangles using

static and transformational approaches.

7. Participants used problem-based approaches of teaching the similarity of triangles

and participants showed strong classroom management skills.

8. The lessons included conversation and models of similar figures.

9. The new proposed model helped participants to organise and direct sequential

activities about the similarity of triangles.

10. The lessons were also dominated by learner talking and depicted hand-on

engagements.

11. The learner’s prior knowledge of the similarity of triangles was also explored.

193

12. The lessons used the direct orientation phase and connected to other geometric

and mathematical concepts.

13. Teachers were seen using appropriate geometric language and they made

learners verbalise their understanding of similarity and its connection.

14. The lessons also included geometric problems that can be solved in numerous

ways and network relationships.

15. The lessons incorporated the integration phase.

16. The teaching approaches used by the participants have come to show the van

Hieles’ five phases of instruction for teaching of similarity and in this regard,

participants had known the van Hieles’ theory and its importance for the teaching

and learning geometry and similarity of triangles.

17. Moreover, the participants were given professional development related to

teaching and learning geometry. Thus, the participants showed strong interest to

teach geometry because teaching geometry was found useful to them.

18. This proposed new model laid a strong baseline for good teacher-student

interaction.

19. Participants had an awareness for using appropriate materials such as models of

similar figures, examples, and non-examples of similar triangles, paper folding

activities and technology.

20. Learners mastered the network of the relationship and gained experiences in

finding their own ways of resolving the learning tasks. They reviewed the works

done and created a summary that provided an overview of the new concepts on


To sum up, meaningful teaching and learning of similarity of triangles refers to providing

an activity that offers an opportunity for learners to connect the similarity of triangles to

their real-life experiences and has a goal to connect the similarity of triangles to further

study. Moreover, the proposed model for teaching similarity of triangles is one of the ways

to deal with the challenges and a meaningful approach for teaching the similarity of

triangles.

194

7.6 RECOMMENDATIONS

Based on the findings, summary, and conclusion, the researcher presented the following

recommendations:

7.6.1 Recommendation to the Education department and College or University

i. Reforms on pre-service teachers’ education, to incorporate continuous

professional development, to revise the geometric contents in the existing syllabus,

include the different theories related to teaching and learning geometry should be

made.

ii. Teachers should be provided with strategies to minimise the challenges to improve

teacher- learner interaction, create opportunities like professional development

and teacher education.

iii. Frequent interventions should be prepared for teachers based on the model of

teaching similarity of triangles and to test its possible effects.

iv. Teachers need to be given continuous professional development support to

improve and update their subject knowledge and pedagogy in teaching the

similarity of triangles and support the teaching and learning of geometry with

Information Communication Technology (ICT) materials.

v. All primary schools’ classrooms, pedagogical centre and learners should be

equipped and given adequate materials for learning the similarity of triangles.

7.6.2 Recommendation for further research

i. There is a limited number of studies on learners’ van Hieles’ geometric

thinking levels in Ethiopia at all school levels. The researcher recommends

that mathematics educators and researchers need to investigate learners’

van Hieles’ geometric thinking levels in Ethiopia,

ii. Studies on exploring teachers challenges of teaching geometry should be

conducted in other regions of Ethiopia,

iii. The curriculum developers, and policymakers should revise the geometric

contents places in the mathematics textbooks and the curriculum needs

modification for the place of geometry in mathematics education. For

195

example, if Grade 8 mathematics syllabi contain geometry in the last

chapter, then Grade 9 should include it in the first chapter.

7.7 CONCLUSION

The summary of the study was presented in this chapter. This was followed by the

challenges experienced by the participants before they applied the suggested model plan.

The chapter further presented the observed pedagogical approaches which promote

meaningful teaching of similarity. The chapter then concluded by suggesting

recommendations for further studies.

196

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APPENDICES

Appendix A: Research Ethics Clearance Certificates

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222

Appendix B: Letter of permission from Areka town mayor office

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Appendix C: Informed Consent Letter form for Teachers

Dear Sir/madam

Beloved Participant

My name is Bereket Telemos Dorra (BT Dorra) and I am doing research with ZMM Jojo,

a Professor in the Department of Mathematics Education towards a Doctor of Philosophy

in Education at the University of South Africa. We are inviting you to participate in a study

entitled “EXPLORING THE CHALLENGES OF TEACHING SIMILARITY OF TRIANGLES, THE CASE

OF AREKA TOWN THE PRIMARY SCHOOLS, ETHIOPIA”. The researcher conducting this study

to explore the challenges of teaching similarity of triangles and its approaches.

This study will involve the observation of the instructional process of Grade 8 similarity

lessons. I will be a passive participant who will do video recording and take field notes

while teacher and learners are busy. I would like to observe three Grade 8 similarity

lessons.

This research project will also involve semi-structured interviews with the Grade 8

mathematics teachers and it last 30-40 minutes. Moreover, the teachers will answer both

open and closed questionnaire items.

You were not obligated to participate in this study, and you were not promised any form

of incentive for doing so. If you do decide to take part, you will be given this information

sheet to keep and be asked to sign a written consent form. You are free to withdraw at

any time and without giving a reason.

Thank you.

Bereket Telemos Dorra

+251911607042

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CONSENT TO PARTICIPATE IN THIS STUDY (Return slip)

I ___________ confirm that the individual who requested my permission to participate in

this study informed me about the study's nature, process, potential benefits, and expected

inconvenience.

I consent to the observation, interview, and questionnaire being recorded in the

classroom. The informed consent agreement has been signed and returned to me.

Participant Name & Surname__________________________________

Participant Signature________________________ Date_________________________

Researcher’s Name & Surname____________________________

Researcher’s signature ______________________ Date________________________

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Appendix D: Questionnaire for teachers

Beloved respondent

This questionnaire is a part of my PhD thesis at the University of South Africa. My

research entitled “EXPLORING THE CHALLENGES OF TEACHING SIMILARITY OF TRIANGLES,

THE CASE OF AREKA TOWN THE PRIMARY SCHOOLS, ETHIOPIA”. The researcher

conducting this study to explore the challenges of teaching similarity of triangles and

its approaches. The study will propose a strategy to minimise the challenges faced by

mathematics teachers in teaching similarity of triangles to promote meaningful

teaching of similarity.

All information collected through this questionnaire will be used solely for research

purposes and will be kept private. Your participation is entirely voluntary, and you have

the option to skip any question or withdraw from the questionnaire at any time without

penalty

You confirm that you accept to participate in this study by filling out the questionnaire.

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Demographic information

Please indicate/ fill below as appropriate:

GENDER: Male Female

Number of years teaching

mathematics (in years)

< 5 6– 10 11– 15 16 – 20 21– 25 >25

Highest level of academic

qualification

Diploma B Ed/ B Sc. M Ed/M Sc Other (specify)

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Teachers Challenges Review and Reflection

1. What is the importance of learning similarity at Grade 8?

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

2. Which educational theories related to teaching and learning geometry do you know? For examples, van Hieles’ geometric thinking levels theories, Fischbein’s theory, and Duval’s theory. Explain their importance for teaching similarity of triangles? ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

3. Have you attended a professional development related to teaching similarity of triangles in the past 3 years?

------------------------------------------------------------------------------------------------------------

4. If yes (in 3), what topics were covered?------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

229

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

5. What mathematical/geometrical concepts must students understand before they can truly understand similarity of triangles?---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

6. Can you tell me the concept of geometry your students had learnt before similarity of triangles? After similarity of triangles?-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

7. What factors affects your interaction with your students? Why?------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

230

8. For you, what kind of teaching method (activity) is best in teaching similarity of triangles? Why?--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

9. What analogies, illustration, example or explanation do you think are most helpful for teaching similarity? How?--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

10. Name the challenges you faced in teaching similarity of triangles--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

231

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

11. How did you overcome each of them?---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

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Appendix E: Observation Protocols

1. Which pedagogical approach teachers’ uses in teaching of similarity of triangles? ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1.1 How do teachers’ explain the concept similarity/ similarity of triangles?----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

a. Multiple-perspective for the concepts-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

b. Essential feature of similarity: static nature /transformation nature--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

c. Connection of the concepts in other geometry or mathematics -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------topics/ratio/proportion/slope/graph of linear function---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

d. Relate with real life/environment of students---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

233

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

e. Choose definition and common examples-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1.2 Teachers teaching strategies for similarity of triangles-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

a. Teacher use different diagrams, picture-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

b. Does the teacher give home and class work give feed-back-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

c. Does teacher encourage students to use hands-on manipulative activities---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

234

d. Whole class approach, small group, as pair and individual-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

2. How teachers interact with students in the teaching-learning of similarity of triangles process? -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

a. Interaction in the classroom------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

b. Teacher-student interaction--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

c. Student-student interaction-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

d. Response to students’ questions------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

235

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

e. Geometric Language --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

3. How the lesson plan prepared?-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

a. What are the methodologies suggested -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

b. Assessment techniques ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

4. How the teachers identify students learning difficulty?-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

236

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

5. What are the challenge teachers’ faces in teaching of similarity of triangles? ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

6. What strategies they adopt to solve these challenges?------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

237

Appendix F: Semi-structured interviews guide

1. Explain the importance of learning geometry in general.

2. What is the importance of learning the concept similarity of triangles in particular at Grade 8 mathematics syllabus?

3. What is your understanding on static and transformational approach of the definition of similarity of triangles? Explain

4. Can you briefly tell me about the concept of geometry your students learnt before similarity of triangles? After similarity

5. Which instructional materials do you use when teaching similarity of triangles?

6. What educational theories related to geometry do you know?

7. How do those theories inform your teaching of similarity of triangles to Grade 8 students?

8. Do you use a different teaching method/activity when teaching similarity of triangles? Explain

9. Do you have any factors that would affect/ impact on your interaction with students while teaching similarity of triangles? Mention them and explain

10. For you, what kind of teaching methods (pedagogical approach) are best for teaching similarity of triangles? Why?

11. What are the challenges you faced when teaching of similarity of triangles? Explain

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Appendix G: Editorial Certificate