The teaching of Euclidean geometry: A Universal Design for Learning Approach by Matheko Thabo Thamae B.Sc DEGREE (NUL); PG Dip (UFS) Dissertation in fulfilment of the requirements for the degree MAGISTER EDUCATIONIS (Mathematics Education) Faculty of Education University of the Free State Supervisor: Dr M.M. Moleko December 2022 ii DECLARATION I declare that the dissertation, THE TEACHING OF EUCLIDEAN GEOMETRY: A UNIVERSAL DESIGN FOR LEARNING APPROACH, hereby handed in for the qualification of Magister Artium at the University of the Free State is my sovereign work, and I have not previously submitted the same work for a qualification at/in another university/faculty. I hereby cede copyright to the University of the Free State. ------------------------------------------------ iii ACKNOWLEDGEMENTS My sincere gratitude to: • The Lord Almighty, for this project, would not have been possible without His protection and guidance. Glory be to Him. • My supervisor, Dr. Moleko, for her selfless and immense support in guiding me throughout this project. This endeavour would not have been possible without her support and motivation. Your soft words of encouragement did wonders. • My lovely children (Theko, Mojela, and Qethekile). I want to extend my deepest gratitude to them for their prayers and unconditional support during this challenging time in my life. Their belief in me kept my spirit and motivation high. • My mother (Mathabo Jemina Lebitsa) who supported me and encouraged me not to give up on this project. I am grateful and fortunate to have you as my mother and prayer warrior. The sacrifices you made for me to complete this journey will forever be cherished. • My brother (Taki Lebitsa) for always backing me up. I know I could always bank on your support! • My friend (Masheshe Sheane) for her undying support and unwavering belief that I can achieve so much. Had it not been for your motivation and prayers, I would never have completed this project. • The team I worked with in making this project a success (teachers, students, DOE officials, and the UDL coach). Words cannot express my gratitude to them. • My colleagues (SRP Unit UFS South Campus) for their unwavering support. Their patience and emotional support cannot be underestimated. iv DEDICATION This dissertation is dedicated to my beloved people, Theko, Mojela, Qethekile, and my nieces and nephews for their prayers, unconditional love, encouragement, and support. My mother, Mathabo Jemina Lebitsa. You are my pillar of strength. Many thanks to you! v ABSTRACT Numerous studies have shown the effectiveness of Universal Design for Learning (UDL) in supporting learners with extensive needs, disabilities and planning inclusive mathematics curricula. However, little has been documented about the effects of implementing UDL in the teaching Euclidean geometry. Hence the focus of this study was to explore the implementation of UDL to enhance the teaching of Euclidean geometry. Learners’ performance in Euclidean geometry had been a concern to all stakeholders in the education system locally and internationally. Several studies affirmed that the root cause of the poor performance in Euclidean geometry were the ineffective teaching strategies which resulted in teachers not meeting the needs of the learners in their classrooms. UDL is a curriculum framework designed to address diverse learners’ needs and create a conducive and enabling learning environment for all learners. The study was qualitative in nature, underpinned by social constructivism and adopted a participatory action research (PAR) as a research design. Data were collected from a team of five mathematics teachers with experience of more than ten years of teaching from Grades 8-10, the Mathematics HOD, the subject advisor, Mathematics coordinator for the senior phase, the Curriculum Education Specialist, and the UDL coach. The coach raised awareness about the diversity in classrooms and how the principles of UDL could be implemented to teach Euclidean geometry effectively. The lesson observations and focus group discussions were used as the primary data collection instruments. Data were generated through active engagement and discussion among the co-researchers using free attitude interview technique (FAI). The findings revealed that implementing UDL principles in teaching Euclidean geometry could make geometry content accessible and perceptible to all learners. This study could add to the body of knowledge as there was little documented about how UDL could be implemented to create inclusive and flexible teaching strategies for teaching Euclidean geometry and making the content accessible to the society of diverse learners. Keywords: Approach, Euclidean geometry, diverse learners, Universal Design, Universal Design for Learning, and universal teaching. vi LIST OF ABBREVIATIONS AMESA Association for Mathematics Education of South Africa CAPS Curriculum and Assessment Policy Statement CES Chief Education Specialist CPD Continuing Professional Development DBE Department of Basic Education DoE Department of Education ET English Translations FAI Free Attitude Interview FET Further Education and Training HOD Head of Department ICT Information and Communications Technology INSET Education and Training In-Service MMAE Multiple Means of Action and Expressions MME Multiple Means of Engagement MMR Multiple Means of Representations NSC National Senior Certificate PAR Participatory Action Research PCK Pedagogical Content Knowledge PDW Professional Development Workshops PLCs Professional Learning Communities SA South Africa SC Social Constructivism SGBs School Governing Bodies vii SMTs School management Teams SPTD Senior Primary Teacher Diploma SWOT Strengths, Weaknesses, Opportunities and Threats UD Universal Design UDL Universal Design for Learning USA United States of America ZPD Zone of Proximal Development viii TABLE OF CONTENTS DECLARATION .....................................................................................................................ii ACKNOWLEDGEMENTS ..................................................................................................... iii DEDICATION ........................................................................................................................ iv ABSTRACT ........................................................................................................................... v LIST OF ABBREVIATIONS ................................................................................................... vi TABLE OF CONTENTS ...................................................................................................... viii LIST OF FIGURES ............................................................................................................. xiii LIST OF TABLES ................................................................................................................ xiv CHAPTER 1 INTRODUCTION .............................................................................................. 1 1.1 INTRODUCTION AND BACKGROUND ..................................................................... 1 1.2 PROBLEM STATEMENT ........................................................................................... 3 1.3 PURPOSE AND OBJECTIVES OF THE STUDY ....................................................... 4 1.4 RESEARCH QUESTIONS.......................................................................................... 4 1.5 THEORETICAL FRAMEWORK OF THE STUDY ....................................................... 4 1.6 RESEARCH METHODOLOGY .................................................................................. 5 1.7 RESEARCH INSTRUMENTS ..................................................................................... 6 1.8 DATA COLLECTION PROCEDURES ........................................................................ 7 1.9 DATA ANALYSIS ....................................................................................................... 7 1.10 TRUSTWORTHINESS ............................................................................................... 8 1.11 SELECTION OF THE CO-RESEARCHERS ............................................................... 8 1.12 VALUE OF THE STUDY ............................................................................................ 9 1.13 ETHICAL CONSIDERATIONS ................................................................................... 9 1.14 OUTLINE OF THE DISSERTATION ......................................................................... 10 CHAPTER 2 LITERATURE REVIEW .................................................................................. 11 2.1 INTRODUCTION ...................................................................................................... 11 2.2 THEORETICAL FRAMEWORK ................................................................................ 11 2.2.1 Historical Background of Social Constructivist Theory ....................................... 12 2.2.2 Objectives of Social Constructivism ................................................................... 13 2.2.3 Nature of Reality ................................................................................................ 14 2.2.4 Role of the Researcher as informed by Social Constructivist Theory ................. 14 2.3 DEFINITION OF OPERATIONAL TERMS ................................................................ 15 2.3.1 Euclidean Geometry .......................................................................................... 15 2.3.1.1 Van Hiele's theory of geometric thinking ........................................................ 17 ix 2.3.2 Approach ........................................................................................................... 17 2.3.3 Diverse Learners ............................................................................................... 17 2.3.4 Universal Teaching ............................................................................................ 18 2.3.5 Universal design (UD) ....................................................................................... 18 2.3.6 Universal Design for Learning (UDL) ................................................................. 18 2.3.6.1 Multiple Means of Representation (MMR) ...................................................... 19 2.3.6.2 Multiple Means of Action and Expressions (MMAE) ....................................... 19 2.3.6.3 Multiple Means of Engagement (MME) .......................................................... 20 2.3.7 Relationship between UDL Principles and Brain Networks ................................ 20 2.3.8 Definition of UDL in the Context of the Study ..................................................... 22 2.4 REVIEW OF THE RELATED LITERATURE ............................................................. 22 2.4.1 Challenges ........................................................................................................ 22 2.4.1.1 Lack of knowledge of inclusive teaching strategies for teaching Euclidean geometry ...................................................................................................................... 22 2.4.1.2 Teachers' inability to create an engaging environment for meaningful learning to take place ................................................................................................................ 24 2.4.1.3 Lack of visualisation skills .............................................................................. 25 2.4.1.4 Lack of knowledge of geometry terminology and symbolic representation ..... 27 2.4.2 Solutions to the Identified Challenges ................................................................ 28 2.4.2.1 Inclusive teaching strategy for teaching Euclidean Geometry ........................ 28 2.4.2.2 Creating an engaging environment for meaningful learning ........................... 29 2.4.2.3 Strategies to enhance Learner Visualisation Skills ......................................... 30 2.4.2.4 Enhancing knowledge of Euclidean Geometry vocabulary ............................. 31 2.5 ANTICIPATED THREATS TO THE TEACHING AND LEARNING EUCLIDEAN GEOMETRY ....................................................................................................................... 32 2.5.1 Teacher Training on how to Teach Euclidean Geometry ................................... 32 2.5.2 Lack of Knowledge of Learners and Geometric Thinking ................................... 33 2.5.3 Time Factor and Shortage of Funds .................................................................. 34 2.6 CONCLUSION ......................................................................................................... 35 CHAPTER 3 RESEARCH DESIGN AND METHODOLOGY ............................................... 37 3.1 INTRODUCTION ...................................................................................................... 37 3.2 PARTICIPATORY ACTION RESEARCH .................................................................. 37 3.2.1 Origins and Historical Background of PAR ........................................................ 38 3.2.2 Objectives of PAR ............................................................................................. 39 3.2.3 Suitability of PAR for this Study ......................................................................... 40 3.2.4 Participatory Action Research Phases ............................................................... 41 3.3 SELECTION OF CO-RESEARCHERS ..................................................................... 45 x 3.3.1 Co-researchers and their Roles ......................................................................... 45 3.3.1.1 Universal Design for Learning coach ............................................................. 46 3.3.1.2 Curriculum Education Specialist..................................................................... 46 3.3.1.3 Mathematics coordinator in the senior phase ................................................. 46 3.3.1.4 Subject Advisor .............................................................................................. 46 3.3.1.5 HOD of Mathematics ...................................................................................... 47 3.3.1.6 Mathematics teachers .................................................................................... 47 3.3.2 SWOT analysis.................................................................................................. 47 3.3.3 Mitigation of the threats and the weaknesses .................................................... 48 3.4 DATA GENERATION ............................................................................................... 49 3.4.1 Procedure .......................................................................................................... 50 3.4.2 Quality Assurance ............................................................................................. 50 3.4.3 Instrumentation.................................................................................................. 51 3.5 DATA ANALYSIS ..................................................................................................... 51 3.6 VALUE OF THE STUDY .......................................................................................... 54 3.7 ETHICAL CONSIDERATIONS ................................................................................. 54 3.8 CONCLUSION ......................................................................................................... 55 CHAPTER 4 PRESENTATION, ANALYSIS, AND INTERPRETATION OF DATA ............... 56 4.1 INTRODUCTION ...................................................................................................... 56 4.2 CHALLENGES IN THE TEACHING OF AND LEARNING OF GEOMETRY ............. 56 4.2.1 Knowledge of Flexible and Inclusive Teaching Strategies ................................. 56 4.2.2 Knowledge of Euclidean Geometry Vocabulary and Expressions ...................... 58 4.2.3 Resources and Competency for Teaching Euclidean Geometry ........................ 61 4.2.4 Factors that affect Teaching and Learning of Euclidean Geometry .................... 64 4.2.5 Visualisation Skills ............................................................................................. 66 4.3 SUMMARY ............................................................................................................... 68 4.4 COMPONENTS OF THE SOLUTIONS .................................................................... 68 4.4.1 Varied Strategies to teach Euclidean Geometry ................................................ 68 4.4.2 The Explicit Teaching of Mathematics Vocabulary and Expressions .................. 73 4.4.3 Appropriate Resources for Teaching and Learning Euclidean Geometry ........... 79 4.4.4 Promoting Learner Engagement in Euclidean Geometry ................................... 81 4.4.5 Enhancing Visualisation Skills for Euclidean Geometry ..................................... 84 4.5 FACTORS THAT IMPEDE THE EFFECTIVE IMPLEMENTATION OF UDL STRATEGIES IN TEACHING EUCLIDEAN GEOMETRY ................................................... 87 4.5.1 Workshops ........................................................................................................ 87 4.5.2 Lack of Expertise in Teaching Euclidean Geometry ........................................... 90 4.5.3 Resources ......................................................................................................... 93 xi 4.6 CONCLUSION ......................................................................................................... 95 CHAPTER 5 CONCLUSIONS ............................................................................................. 96 5.1 INTRODUCTION ...................................................................................................... 96 5.2 THE OBJECTIVES OF THE STUDY ........................................................................ 96 5.3 SUMMARY OF THE STUDY .................................................................................... 97 5.4 FINDINGS ON THE CHALLENGES PERTAINING TO TEACHING OF EUCLIDEAN GEOMETRY ....................................................................................................................... 98 5.4.1 Teachers’ Lack of Knowledge of Inclusive Teaching Strategies ......................... 99 5.4.2 Shortage of Resources .................................................................................... 100 5.4.3 Teachers’ Inability to develop Engaging Learning Environments ..................... 100 5.4.4 Teachers’ Inability to develop Learners’ Visualisation Skills............................. 101 5.4.5 Workshops do not address Teachers’ Lack of Expertise ................................. 101 5.4.6 Lack of Knowledge of Euclidean Geometry Vocabulary and Expressions........ 102 5.5 STRATEGY FOR EFFECTIVE IMPLEMENTATION OF UDL PRINCIPLES IN TEACHING EUCLIDEAN GEOMETRY ............................................................................. 102 5.5.1 Knowledge of Inclusive Teaching Strategies ................................................... 103 5.5.2 Availability of Teaching and Learning Resources ............................................ 103 5.5.3 Sustainability of the Outcomes of INSET Workshops ...................................... 104 5.5.4 Creating an Engaging Learning Environment .................................................. 105 5.5.5 Commitment .................................................................................................... 106 5.6 IMPLEMENTATION OF UDL IN TEACHING EUCLIDEAN GEOMETRY ................ 106 5.6.1 Threats that may hamper the Successful implementation of UDL .................... 106 5.6.2 Strategies that can Address the Threats .......................................................... 107 5.7 LIMITATIONS OF THE STUDY .............................................................................. 107 5.8 IMPLICATIONS FOR PRACTICE ........................................................................... 108 5.9 RECOMMENDATIONS .......................................................................................... 112 5.9.1 Recommendations for Practice ........................................................................ 112 5.9.2 Recommendations for Future Research .......................................................... 113 5.10 A FINAL WORD ..................................................................................................... 113 REFERENCES ................................................................................................................. 114 APPENDICES ................................................................................................................... 131 Appendix A1: Ethical clearance ..................................................................................... 131 Appendix A2: Approval to conduct research .................................................................. 132 Appendix A3: Request to conduct research in school .................................................... 133 Appendix A4: Information sheet for assent form (parent/child) ....................................... 134 Appendix A5: Information sheet and consent form (teacher) ......................................... 135 Appendix A6: Information sheet and consent form (HoD) .............................................. 136 xii Appendix A7: Information sheet and consent form (subject advisor) .............................. 137 Appendix A8: Information sheet and consent for (Mathematics coordinator) ................. 138 Appendix A9: Information sheet and consent form (Curriculum Education Specialist) ... 139 Appendix A10: Information sheet and consent form (UDL coach) .................................. 140 Appendix B: The training programme ............................................................................ 141 Appendix C: Ground rules ............................................................................................. 142 Appendix D1: Observation tool ...................................................................................... 143 Appendix D2: Transcripts .............................................................................................. 145 Appendix E: Proof of editing .......................................................................................... 152 Appendix F: Similarity Index .......................................................................................... 153 xiii LIST OF FIGURES Figure 2.1: Classification of UDL principles ......................................................................... 21 Figure 3.1: Stages of PAR .................................................................................................. 42 Figure 4.1: Example of two angles on a straight line ........................................................... 59 Figure 4.2: Geo-board ......................................................................................................... 69 Figure 4.3: Shapes at different orientations ......................................................................... 72 Figure 4.4: A condition for angles on a straight line to add up to 180◦ (sharing of a common vertex .................................................................................................................................. 74 Figure 4.5: Interior and exterior angles of triangles ............................................................. 74 Figure 4.6: Interior and exterior angles of a cyclic quadrilateral ........................................... 75 Figure 4.7: The sum of angles on a straight line .................................................................. 76 Figure 4.8: Radii and the chord of a circle ........................................................................... 86 Figure 5.1: Diagrammatical representation of parallel lines ............................................... 109 Figure 5.2: Parallel lines using a geo-board ...................................................................... 109 xiv LIST OF TABLES Table 1.1: Stages of PAR ...................................................................................................... 6 Table 2.1: The weighting of content areas ........................................................................... 16 Table 3.1: The phase of PAR and how they were followed in this study .............................. 42 Table 3.2: SWOT analysis .................................................................................................. 48 Table 3.3: Six stages of the thematic analysis followed in this study ................................... 53 1 CHAPTER 1 INTRODUCTION 1.1 INTRODUCTION AND BACKGROUND Euclidean geometry, the study of planes and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c.300 BCE), is one of the critical topics in the mathematics curriculum, with a certain history attached to it in the South African education system. It was included Paper 3 as an optional mathematics assessment standard for examination for Grades 10, 11, and 12 from 2006 to 2011 (Assessment Instruction 26 of 2009). It was re-introduced in Grade 10 as Paper 2 in 2012 (Dlamini, 2012). Since the inception of the Curriculum Assessment Policy Statement (CAPS) in 2012 (Maharajh et al., 2016), this topic has been made a compulsory examinable section of the Mathematics second paper in Grades 10, 11, and 12. Prior to 2012, most schools opted not to do Mathematics Paper 3 to prevent learners' results in first and second papers from being negatively affected by Paper 3 results, especially Euclidean geometry, since it was regarded as challenging with the likelihood that learners would fail. However, such a decision to opt out of Paper 3 (Euclidean geometry) negatively impacted the economic development of countries that made this decision, including South Africa, since Euclidean geometry is the bedrock of engineering, architecture and technological development (Makhubele, 2014). Euclidean geometry is one component of geometry, dealing with axioms and proofs of theorems and their converses through deductive thinking (Mamali, 2015). It is of great practical value as geometrical skills are critical in construction work, architectural design, engineering, mechanical drawing and even in deciding the shapes of houses (Alex & Mammen, 2014). Euclidean geometry is, however, challenging to teach and learn because teachers often lack content knowledge and the knowledge of inclusive strategies to teach it (Ubah & Bansilal, 2019). According to Mthembu (2007), the leading cause of poor performance in Euclidean geometry is how teachers present it, especially in Grades 11 and 12. Brodie and Borch (2004) point out that the ‘chalk and talk’ approach, which promotes rote learning, is predominantly used to teach Euclidean geometry in South African classrooms. 2 Consistent with Brodie's findings, Boggan et al. (2010) also confirm that many teachers still prefer the traditional teaching approach for geometry and other mathematics topics and are reluctant to use manipulatives to enhance metacognitive skills and strategies to enable high thinking skills. However, the traditional teaching methods are not productive since they do not encourage deep learning (Moleko & Mosimege, 2020). They also cause mathematics anxiety, stir resentment towards mathematical concepts and promote rote learning (Howie, 2003). These forms of teaching (traditional teaching methods) further lead to the poor development of learners' reasoning strategies rather than problem-solving and critical thinking skills (Snyder et al. (2008). According to Guo et al. (2019), learners generally become bored and disinterested in teaching and learning because teachers use pedagogies that promote rote memorisation. Teachers are ignorant about learners' need for equal recognition and attention when addressing their preferences and interests (Ashraf et al., 2021). Research shows that most teachers did not study Euclidean geometry at any level of their training (Machisi, 2021). They lack inclusive instructional strategies, which would promote learners’ equal access to instruction that would address their differing learning style needs (Nyahunda et al., 2020). The complexity of teaching Euclidean geometry is exacerbated by its textual nature. Moleko (2018) avows that the problems presented in the form of text are often difficult for most teachers to teach and for learners to learn. This is supported by Kutama (2002), who explains that Euclidean geometry comprises the types of problems mainly presented in the form of text, requiring teachers to develop learners' spatial knowledge and reasoning skills. However, these problems (geometry problems) are complex to teach and engage learners since teachers lack both the content and instructional knowledge. I, the researcher, am a Mathematics teacher with the experience of more than ten years. I conducted training workshops for in-service teachers. Through such interactions, I realised that the contextual nature of Euclidean geometry was not the only factor contributing to learners’ poor performance. Teachers’ lack of knowledge of flexible and inclusive teaching strategies to cater for diverse learners is another contributing factor. When I consulted the literature, Possi & Milinga (2018), Buli- Holmberg & Jeyaprathaban (2016), to mention a few, they concur with my findings. Chidziva (2021) states that unless teachers create an engaging environment wherein learners can talk, write, draw, and become practically involved, learners will continue 3 to experience challenges in learning Euclidean geometry. Hence, the study's interest was on implementing UDL as a fresh approach that caters to all learners regardless of the characteristics they bring to the classroom. Previous research has shown that poor performance in Euclidean geometry is a problem not only in South Africa, but in developing and advanced countries such as the United Kingdom (UK) (Mamali, 2015). Research indicate that mathematics teachers leave the institutions of higher education not fully equipped to deal with diversity in their classrooms (Panthi & Belbase, 2017). The study conducted on Euclidean geometry in secondary schools in Rivers State, Nigeria, revealed that the foundation of most mathematics teachers in geometry was poor (Adolphus, 2011) with teachers, lacking conceptual understanding of the components of Euclidean geometry. A study in South Africa (SA) indicates that the mathematics teachers use strategies that facilitate procedural understanding of Euclidean geometry and do not use instructional materials to enhance the learning of the concepts thereof (Howie, 2003). Although there have been attempts to address the challenges of teaching geometry, teachers struggle to teach this topic due to poor pedagogical choices influenced by economic status such as lack of teaching aids and technological tools (Panthi & Belbase, 2017). According to Shulman (1986), Pedagogical Content Knowledge (PCK) and understanding of teachers is defined as involving the relationship between knowledge of teaching materials, how to transfer the subject matter, and the knowledge of students in mathematics on algebraic functions that the subject matter may be understood by students. The fact that teachers lack knowledge of the inclusive and flexible teaching strategies to make geometry content accessible to all learners calls for exploring other teaching alternatives. Against this backdrop, this study seeks to explore the implementation of the universal design for learning (UDL) (a framework that guides inclusive and flexible teaching) to enhance the teaching of Euclidean geometry. 1.2 PROBLEM STATEMENT The application of Universal Design for Learning (UDL) has generally found immense recognition in addressing the issues of learners with different learning styles and preferences, particularly applied in mathematics instruction to assist learners with extensive support needs (Yavuzarslan & Arslan, 2020), used to plan inclusive 4 mathematics curricula (Lambert et al., 2021) and to support learners with disabilities, to mention a few (Ross, 2019). Despite its successes, little has been reported on how UDL can be implemented to enhance the teaching of Euclidean geometry, which is the focus of the study. 1.3 PURPOSE AND OBJECTIVES OF THE STUDY This study aims to explore the implementation of the Universal Design for Learning (UDL) to enhance the teaching of Euclidean geometry. In line with the purpose of the study, the objectives formulated are as follows: 1. to identify the challenges pertaining to the teaching and learning of Euclidean geometry; 2. to highlight ways in which UDL can be used in the teaching of Euclidean geometry; 3. To determine mitigating factors/circumstances against threats that may hinder the optimal benefit of using UDL in the teaching of Euclidean geometry. 1.4 RESEARCH QUESTIONS The main research question of this study is: How can the Universal design for learning be implemented to enhance the teaching of Euclidean geometry? The main research question necessitated the formulation of secondary research questions: 1. What are the challenges pertaining to the teaching and learning of Euclidean geometry? 2. How can the Universal Design for Learning (UDL) be applied in the teaching of Euclidean geometry? 3. What are the mitigating factors that can be used to circumvent the threats that may hinder the optimal benefit of the use of UDL in the teaching of Euclidean geometry? 1.5 THEORETICAL FRAMEWORK OF THE STUDY Social constructivism was adopted as the theoretical framework this study. According to Geels (2020), social constructivism is considered an eminent approach in social sciences. It promotes the notion of knowledge construction through collaborative 5 working and sharing of ideas (Rannikmae et al., 2020). For instance, Vygotskian ideas suggest that learners can create new knowledge through social interaction (Adam, 2017). Verwey (2010) states that social constructivists view learning as a social process in which new knowledge is built based on the current knowledge and experiences. Social constructivist theory is considered a suitable theoretical framework for this study because it provides the teachers with the platform to share their experiences and best practices in teaching Euclidean geometry. Through shared debates, teachers would be empowered with knowledge that will assist them in productively teaching Euclidean geometry. The shared debates, conducted through social interactions, would enable teachers to construct new knowledge (Tlali, 2013) of alternate teaching approaches and strategies and realise that the unproductive practices they were using could be replaced (Moleko, 2014). Social constructivism was thus deemed pertinent for this study because it allowed the teachers to be exposed to diverse perspectives and teamwork (Shangase, 2013) to enhance the teaching of Euclidean geometry, thus deepening their knowledge. 1.6 RESEARCH METHODOLOGY Participatory action research (PAR) was followed as an approach to generate data in this study. PAR is a qualitative research method that emphasises the collaboration of the researcher with the co-researchers (MacDonald, 2012). In the context of PAR, the participants are viewed as co-researchers because they operate at the same power level as the researcher and are also involved in all the stages of the research project (Motsoeneng & Mahlomaholo, 2015). Therefore, participants in this study are referred to as co-researchers. For a profound understanding of how to address the challenge at hand, the researcher and co-researchers worked together and operated on the same power basis. Contrary to the last few decades, where the research process assumed participants to be the 'research subject'; where their ideas and feelings were not of significance, PAR advocates the consideration of participants as influential and valuable individuals for the research process in educational settings (Sokhanvar & Salehi, 2018). PAR is deemed apposite for this study because of its potential to create a conducive environment for both the researcher and co-researchers to collectively share ideas pertinent to addressing the identified problem. It resonates with the paradigm of this study since they both advocate knowledge construction through collaborative working. 6 According to Tetui et al. (2017), the PAR approach is a cyclical process that consists of the following stages/phases (see Table 1.1): Table 1.1: Stages of PAR Stage Number Stage Name Description 1. Problem identification - this is the stage in the study wherein the problem will be collectively identified. 2. Planning - this is the step where the best strategies to address the problem were identified and prioritised to promote a positive change. The planning would include a UDL training workshop by the UDL coach with more focus on planning UDL customised lessons. 3. Implementation - this is the phase where strategies and plans are executed. This includes the implementation of UDL principles. 4. Action observation - the researcher will sit in class to observe and audio-video tape the lessons. An observation tool will be used to collect data from this stage. 5. Reflection - After the lesson's presentation, the researcher and the co-researchers will meet to reflect upon the representations and shared experiences, identifying areas of weakness and strengths. The team will also re-plan to address the identified areas of weakness - [session will be audio- recorded]. (Source Tetui et al., 2017) The above stages were used to guide data generation in this study. 1.7 RESEARCH INSTRUMENTS Data were generated through class observation, focus group discussions and reflection sessions. According to Johnson and Christensen (2012), observation is one 7 of the most effective tools for collecting data. A researcher can see and hear what is happening at the site without interacting with the co-researchers. On the other hand, focus group discussion is a technique that offers the researcher and co-researchers an opportunity to deeply explore the issue under discussion. The researcher acts as the facilitator to guide the group's meeting (Nyumba et al., 2018). The focus group discussion was appropriate for the study because it allowed for sharing and comparing of understandings and ideas and yielded more insights about implementing UDL in the teaching of Euclidean geometry. Therefore, the focus groups were essential in assisting the researcher is focusing on the issues during the discussions. The sessions also allowed the researcher to conduct group follow-up discussions and provide more clarity to the co-researchers. In addition, the reflection sessions allowed the co- researchers to reflect on their teaching practices and experiences. 1.8 DATA COLLECTION PROCEDURES A series of class observations and focus group discussions were conducted. During these meetings, the discussions were centred on the challenges of teaching Euclidean geometry, identifying suitable solutions to the challenges, and mitigating threats that may hamper the implementation of the identified solutions. The conversations/discussions were audio and video recorded. The unproductive practices observed during the lesson presentations were also discussed during the reflection sessions. A standardised lesson observation tool was used to assess geometry according to UDL principles (Moleko, 2018). The Free Attitude Interview (FAI) technique allowed for the initiation of conversations/discussions. According to Meulenberg-Buskens (2011), when FAI is used, the co-researchers get to communicate as in normal day-to-day conversations. This technique elicited as much information as possible from the co-researchers. The technique encourages open- ended questions, which enabled the co-researchers to say more than they would in surveys with close-ended questions. 1.9 DATA ANALYSIS Thematic analysis was used to analyse the collected data. Nowell, Norris, White and Moules (2017) note that thematic analysis, which consists of six stages, is fundamental for examining different co-researchers’ points of view. The thematic analysis allows flexibility in interpreting collected data and highlights similarities and differences in 8 generating unanticipated insights in response to the research question (Nowell et al., 2017). This technique, described in Chapter 3, assisted in terms of identifying and organising the emerging themes. 1.10 TRUSTWORTHINESS The trustworthiness of a study concerns the level of trust and confidence in the data, interpretation and methods used to ensure the quality of the study (Polit & Beck, 2012). In contrast to the positivistic approach where the method is structured and detailed, the PAR approach allows the flexibility of the researcher and co-researchers because it accommodates a wide range of individual contributions throughout all phases of the research process. Triangulation, which refers to multiple data generation, was used as a strategy to test the 'validity of data' through the convergence of information from various sources (MacDonald, 2012). Moreover, it was used to clarify meaning, verify the repeatability of observations, and interpret the generated data. Audio and videotape recordings, transcriptions, and the documentation of minutes by the researcher and co-researchers influenced the credibility, transferability, dependability, and confirmability of the findings and their interpretations. Member checking was done with the co-researchers. According to Birt et al. (2016), member checking is a tool or a technique that enhances the credibility and trustworthiness of results. This technique, which makes it possible for researcher and the co-researchers to agree on what exactly was discussed or pointed out without data being misinterpreted. It enabled the researcher to capture and interpret the data correctly and in context. Member checking further assisted in addressing the researcher's biases towards the data. Before analysing data, the researcher familiarised herself with the depth and breadth of data to be reported (Braun & Clarke, 2006), which enabled the researcher to write credible data and package it systematically. 1.11 SELECTION OF THE CO-RESEARCHERS In participatory action research (PAR), similar to the other modes of research, selecting co-researchers is seen as a vital step that requires a thoughtful approach as it allows the co-researchers to deliberate on their lived experiences (Alase, 2017). The study followed purposive sampling, which involved selecting co-researchers knowledgeable about the concept under study (Creswell et al., 2011). The study included ten co-researchers. Some were from one senior secondary school in the 9 Motheo district in the Free State, while others were departmental officials. The co- researchers had teaching experience ranging between ten and fifteen years. There were five mathematics teachers, one subject advisor for Mathematics, one head of the department of mathematics (HOD), 1 Chief Education Specialist (CES), one Free State mathematics coordinator, and 1 UDL coach. The purposive sampling technique was deemed pertinent for the study. All participants provided crucial information that any person not in mathematics education could not give. 1.12 VALUE OF THE STUDY The results of this study hopefully provide teachers of Mathematics, subject advisors and curriculum developers with a deeper understanding of how to teach Euclidean geometry through UDL. Teachers would consider diversity in mathematics classrooms and take learners through every level of Van Hiele's theory of geometry thinking (Bonyah & Larbi, 2021). The results could inspire teachers and learners because teaching Euclidean geometry through UDL creates a conducive learning environment for learners with diverse learning styles. This study could possibly add to the body of knowledge since there is limited literature on how UDL could be used in teaching Euclidean geometry. The study thus contributes to expanding the existing knowledge of teaching and learning theories about geometry. 1.13 ETHICAL CONSIDERATIONS The researcher applied for ethical clearance from the University of the Free State to ensure that the study adheres to the ethical principles of research. The study was ethically cleared and assigned the following reference: UFS-HSD2020/1868 (see Appendix A1). Permission to conduct the study was also sought from the Department of Education in the Free State province and the school principal, where data was collected and permission was granted (see Appendices A2 and A3). Parents were requested to sign the assent forms allowing learners below 18 years of age to participate in the research (see Appendix A4). The co-researchers (teachers, HOD, subject advisor, mathematics district coordinator, CES and UDL coach) were requested to sign the consent forms (see Appendices A5; A6; A7; A8; A9, and A10). Plowright (2013) states that confidentiality is essential to consider once the data have been collected, stored, and analysed. In line with the above principle, the researcher 10 used pseudonyms to protect the confidentiality and dignity of the co-researchers. Data will be stored in a safe filing cabinet for three years after the completion of the study, after which it will be destroyed. 1.14 OUTLINE OF THE DISSERTATION The study will consist of five (5) chapters. Chapter 1 introduced the research and outlined the background of the study: the problem statement, the purpose of the study; research questions; research design and methodology; the significance of the study; ethical considerations, and the layout of the chapters. Chapter 2 covers a review of the literature on the challenges encountered in teaching Euclidean geometry and how UDL can address these challenges. Chapter 3 outlines the research design and methodology used. Chapter 4 presents the analysis of the data and the presentation and discussion of findings on the implementation of UDL. Chapter 5 summarises the findings, draws conclusions, presents the limitations of the study, opportunities for further research, implications for practice, and offers recommendations. 11 CHAPTER 2 LITERATURE REVIEW 2.1 INTRODUCTION The study aimed to explore implementing the universal design for learning (UDL) teaching approach to enhance the teaching and learning of Euclidean geometry. In line with this purpose, this chapter first discusses social constructivist theory as a paradigm that guides the study. This is followed by the literature review section, which reviews the literature on challenges pertaining the teaching and learning of Euclidean geometry, the solutions implemented and suggested to address the identified challenges, threats that may impede the successful implementation of the strategies and ways to mitigate against risks that may hinder optimal benefits of the use of the identified strategies for teaching and learning Euclidean geometry. The chapter also provides the definitions of the operational concepts, namely, approach, diverse learners, Euclidean geometry, Universal Design, universal teaching and Universal Design for Learning, since they serve as pillars on which the study is anchored. 2.2 THEORETICAL FRAMEWORK According to Swanson and Chermack (2013), a theoretical framework is a structure that supports the theory of a research study. Alaidaros et al. (2020) state that a theoretical framework gives researchers a direction to interpret, develop tools for analysing, designing and monitoring the progress, and explain, evaluate and generalise from their findings. The theoretical framework helps a reader understand why the researcher conducts the study on a particular topic. It also gives the researcher a different perspective about the study and relevant ways to explain how and why things happen the way they do (Lynam et al., 2007). According to Kivunja (2018), the theoretical framework serves as a lens that magnifies the contents of the data. It divulges interconnections that make sense of the research questions and address the problem the researcher wants to investigate. Moleko (2018) indicates that the role of the theoretical framework is to demonstrate the relationship between new ideas and existing knowledge. Adam (2017) concurs with Moleko that creating knowledge is based on understanding the existing 12 knowledge. Moleko (2018) further states that the study's objectives should influence the choice of the theoretical framework. For the current study, the purpose is to explore the implementation of UDL to enhance the teaching of Euclidean geometry. With this in mind, social constructivism was adopted as a suitable theoretical framework that underpins the study. According to Bozkurt (2017), Social Constructivism (SC) promotes active interaction in the teaching and learning process. Rankhumise and Imenda (2014) noted that social constructivists believe that knowledge is socially constructed and that knowledge is often gained as people interact with one another. Social constructivists advocate social interaction for learning to be socially constructed. The application of social constructivism in this study would enable the researcher and the co-researchers to interact and engage in exploring the implementation of UDL to enhance the teaching and learning of Euclidean geometry. This would result in the construction of new knowledge on how UDL could be implemented to cater to diverse learners and make content accessible. According to Kivunja (2018), in a constructivist classroom, a facilitator must create a highly dynamic teaching and learning environment to allow learners to participate as partners in knowledge construction. In this study, the researcher would play a similar role to ensure that the co-researchers participate fully in the research project in a safe environment wherein their voices are heard and respected (Dold & Chapman, 2012). 2.2.1 Historical Background of Social Constructivist Theory The social constructivism theory of learning was coined by post-revolutionary Soviet Psychologist Lev Vygotsky, who believed that knowledge is mutually constructed through social interaction (Nassaji & Tian 2018). Vygotsky was a cognitivist; however, he had a different viewpoint from other cognitivists, such as Piaget and Perry, who believed that learning could be separated from social context (Bozkurt, 2017). According to Adam (2017), Piaget and Perry considered knowledge construction as a distinctive process. In contrast, Vygotsky believed that people create meaning through interacting with each other and the objects in the environment under the guidance of a more skilled peer or an adult. Bozkurt (2017) points out that Vygotsky developed the Zone of Proximal Development (ZPD) concept to explain this social and participatory learning with the more informed 13 peer. ZPD refers to the range of abilities one can carry out under the guidance of an expert but cannot execute on their own (Chang, 2021). By the same token, Knestrick (2012) considers ZPD as the difference between what a child can do independently and what they can do through scaffolding. This confirms that social interaction facilitates meaningful learning to a greater extent than what one can learn individually without interacting with others (Kim, 2001). According to Bozkurt (2017), social constructivism theory indicates that knowledge is built and constructed actively; therefore, social interaction plays a vital role in the learning of Mathematics. Social constructivist theory is adopted in the teaching and learning of Mathematics as implied as follows in the Curriculum and Assessment Policy Statement (CAPS): This can be done through observations, discussions, practical demonstrations, learner-teacher conferences, informal classroom interactions … (DBE, 2012: p. 51). Additionally, Department of Basic Education (DBE) emphasises the importance of applying active and critical approach to learning, rather than memorisation and uncritical learning of given truths (Booysen, 2018). Social constructivism theory fits precisely into the context of this study because it advocates for collaboration, knowledge sharing amongst the co-researchers, and mutual interaction to construct new knowledge. Holmes (2020) states that it is crucial for the researcher to declare his/her position in research so that they may integrate a reflexive perspective into their research. Therefore, my stance as a researcher in this study is that of a social constructivist. Similar to other social constructivists such as Akpan et al. (2020), I believe that when teachers collaborate and share knowledge with each other, they can construct new knowledge. I also believe that knowledge sharing and shared debates would provide teachers with opportunities to learn and be empowered. I think that teachers are life-long learners and that platforms such as focus group discussions and reflection sessions wherein the teaching experiences and best practices are shared, provide them with opportunities to learn and socially construct new knowledge. 2.2.2 Objectives of Social Constructivism Thomas et al. (2014) state that social constructivism focuses on revealing ways in which individuals and groups participate in knowledge construction. Thomas et al., (2014) further affirm that social constructivist theory informs the researcher on how to 14 construct a conducive environment for building new knowledge. According to Adam (2017), the theory establishes the influence of socio-cultural background on cognitive development. It highlights the crucial role played by semiotic mediation in knowledge construction. As stated by Bozkurt (2017), Vygotsky defines semiotic mediation as an investigation of how knowledge is constructed using language, various systems of counting, mnemonic techniques, algebraic symbol systems, diagrams, and mechanical drawings, to mention a few. In this study, social constructivism enables the researcher to understand how knowledge is constructed using language, various systems of counting, mnemonic techniques, algebraic symbol systems, diagrams, etc. 2.2.3 Nature of Reality Epistemologically, social constructivists maintain that knowledge is constructed through social interaction (Bozkurt, 2017). Social constructivists also believe in multiple truths since people differ according to their experiences and backgrounds (Moleko, 2014). This, therefore, means that there can never be one truth. The truth is, consequently, multi-layered. In this study, the co-researchers engaged in social interactions and discussions to explore the implementation of UDL to enhance the teaching of Euclidean geometry. Therefore, the co-researchers (teachers) shared different perspectives and experiences. Thus, there can never be only one perspective considered in the process but multiple perspectives. Tsimane and Downing (2020) point out that one way to improve teaching and learning is to allow researchers and co-researchers to work as a team to share diverse ideas and experiences in education. For diverse perspectives to be shared, researchers need to create a conducive environment where everyone is free to participate without being judged (Hlomuka, 2014). 2.2.4 Role of the Researcher as informed by Social Constructivist Theory The role of the researcher is to gather the co-researchers together and create a platform for them to engage in discussions (Tsotetsi, 2013). The discussions empower the co-researchers with the knowledge to employ the UDL principles in teaching Euclidean geometry. Additionally, the researcher should explicitly explain the aim and purpose of the study and assist in clarifying the roles of the co-researchers (Hlomuka, 15 2014). Furthermore, the researcher's role is to interpret the co-researchers’ ideas and opinions to make sense of them (Moleko, 2014). According to Moleko (2014), engaging the co-researchers enables them to manage the present situation and develop ownership of the research project's outcomes. Within the context of social constructivist theory, the researcher acts as a facilitator of the discussion. The researcher works collaboratively with co-researchers because the social constructivist approach is participatory and advocates for teamwork (Shangase, 2013). For research to succeed, Tsimane (2019) affirms that a researcher should be trustworthy, honest, friendly, patient, transparent and a team player. Shangase (2013) further points out that researchers must be compassionate, patient, and transparent as they interact with the co-researchers to allow reflexivity and humbleness among themselves. 2.3 DEFINITION OF OPERATIONAL TERMS The following sections provide the comprehensive definitions of the operational concepts that underpin this study. It is important to define these concepts according to the context of this study for the readers to be enlightened and have a deeper understanding of their meaning. They are Euclidean geometry, approach, diverse learners, universal teaching, Universal Design (UD), and Universal Design for Learning (UDL). 2.3.1 Euclidean Geometry According to Merriam Webster dictionary, Euclidean geometry is a Mathematical system contributed by Alexandrian Greek Mathematician by the name of Euclid. In his textbook Euclid's Elements, he wrote about a small set of instinctively interesting axioms and deduced many other theorems. Euclid was the first to show the applicability of theorems in deductive and logical systems. He wrote about the geometric properties of two and three-dimensional figures. Güven and Kosa (2008) define Euclidean geometry as a study of shapes and space, which can be conceptually understood if one has well-developed spatial skills. On the other hand, Mamali (2015) states that Euclidean geometry is a field of mathematics dealing with axioms and proofs of theorems through deductive thinking. Similarly, according to Artmann (2012), Euclidean geometry studies planes and solid figures based on the axioms and theorems, as engaged by the Greek Mathematician Euclid. Euclidean geometry is 16 commonly taught in secondary schools and in the South African curriculum, it has a weighting of about 33,3% in Mathematics Paper 2 (DBE, 2012, p.12). Table 2.1: The weighting of content areas Paper 2: Grades 11 and 12: theorems and or trigonometric proofs: maximum 12 marks Description Grade 10 Grade 11 Grade 12 Statistics 15 ± 3 20 ± 3 20 ± 3 Analytical Geometry 15 ± 3 30 ± 3 40 ± 3 Trigonometry 40 ± 3 50 ± 3 40 ± 3 Euclidean Geometry and Measurement 30 ± 3 50 ± 3 50 ± 3 Total 100 150 150 (Source: DBE, 2012, p.12) Brannan et al (2011) state that Euclidean geometry includes the theory of points, lines, angles and circles on a flat plane. According to Jones (2000), the aims of teaching Euclidean geometry are to: ▪ develop learners' spatial awareness and ability to visualise. ▪ encourage learners to use conjecture, deductive thinking and proofs. ▪ enable the development of learners' conceptual understanding of geometrical properties and theorems to solve problems in a real-world context. Even though various scholars define Euclidean geometry diversely, in the context of my study, I adopt definitions provided by Brannan et al. (2012) and Mamali (2015), which state that Euclidean geometry is a field of mathematics that includes the theory of points, lines, angles and circles on a flat plane as well as dealing with axioms and proofs of theorems through deductive thinking. Learners can provide solid support to a conclusion and establish geometric truth after reasoning from one or more statements to reach a logical conclusion. In the teaching of Euclidean geometry, teachers must ensure that learners are engaged in using descriptions, demonstrations and rational justifications for strategic learning and the construction of proofs of theorems. According to Machisi (2021), Van Hiele's theory-based instruction has a positive impact on students' attitudes and confidence towards Euclidean geometry. 17 2.3.1.1 Van Hiele's theory of geometric thinking Bishop (2020) declares that two mathematics teachers, Dina, and Pierre Van Hiele from the Netherlands, observed that their students had difficulties learning geometry. In the 1950s, they developed Van Hiele's theory of geometric thinking to prove that learners' structure for reasoning is crucial in teaching and learning geometry (Prayito et al., 2019). Van Hiele’s theory of geometric thinking describes how learners learn geometry, comprising five levels, namely; (1) visualisation/recognition, (2) analysis, (3) abstraction, (4) formal deduction and (5) rigor (Vojkuvkova, 2012). These are the levels that a learner must hierarchically pass through to progress from recognising figures to writing formal geometric proofs. Vojkuvkova (2012) further mentions that Van Hiele believed that instruction informs cognitive progress in geometry learning. 2.3.2 Approach Merriam Webster’s dictionary defines approach as taking preliminary steps toward a particular purpose. Therefore, the teaching approach is a method of teaching ‘something’ using learning techniques. In line with this, the current study proposed using UDL as an approach to enhance the teaching of Euclidean geometry. This teaching (the UDL approach) entailed multiple and flexible ways in which: ▪ Content is presented using various formats. ▪ Teaching is designed in a manner that seeks to remove barriers to learning. ▪ Learners are afforded opportunities to demonstrate their knowledge (or what they know or what they have learned) in diverse ways. ▪ Different (varied) strategies are used to keep learners motivated, engaged with learning and focused on the mathematical tasks. An approach in the context of this study refers to teaching that is tailor-made to cater to diverse learner populations (that is, inclusive, equitable, and accessible teaching. 2.3.3 Diverse Learners Diver learners refer to learners that are racially, ethnically, critically and linguistically different. In the context of this study, they are the learners who have different learning styles and preferences. This calls for teachers to create a classroom environment cognisant to learners’ cultural background so every learner feels safe and welcome. 18 2.3.4 Universal Teaching The term ‘universal’, as defined by Merriam-Webster dictionary, refers to ‘something that applies throughout the universe to many things and is accessible to all people.’ It is adjustable to many sizes, uses, and devices. In line with this definition, therefore, the term ‘universal teaching’ refers to the teaching suitable to meet all learners' needs in a classroom. It is a form of teaching that encompasses varied strategies to cater to all learners' needs. It is the type of teaching which provides flexibility in the ways learners access learning material, engage with it and demonstrate what they know. In the context of this study, the term universal teaching means teaching that is planned in a manner that caters to a diverse learner population or teaching that considers learner variations. 2.3.5 Universal design (UD) Jones (2014) describes UD as an environment or situation design that can be accessible to everyone regardless of age, size, ability or disability. According to Burgstahler (2009), the architect Ronald Mace invented this concept known as UD when describing barrier-free products and physical environments that integrate people with disability into the mainstream. UD in the education system is a supporting structure in designing various educational products, including a curriculum that is accessible, perceptible, simple, and intuitive to all learners (Burgstahler, 2009). In this study, UD is about designing a learning environment that caters to all learners regardless of the characteristics (visual, tactile and auditory) they bring into the mathematics classrooms. 2.3.6 Universal Design for Learning (UDL) Dalton (2017) refers to UDL as a framework for curriculum design that is informed by the values of UD. Barteaux (2014) explains that UDL is intended to create a conducive and enabling learning environment for all learners. Therefore, if learners' variability is recognised, curriculum design and instruction should thus address learners' diverse needs. Dalton (2017) states that UD aims to create a barrier-free physical environment, whereas UDL is designed to eliminate barriers from the learning environment. In addition to access to the content and information, learners need to have engagement and connection to what they are learning (Chan et al., 2014) 19 LaRocco and Willken (2013) report that UDL is a scientifically sound teaching framework that provides flexible guidance on how information is presented to learners with diverse learning styles and preferences, ways to demonstrate what they know and various ways in which they can be engaged in the process of learning. UDL reduces barriers to teaching and learning and provides apposite support to learners with different learning styles (Moleko, 2018) and assists in ensuring the accessibility of learning to all learners (Boothe & Lohmann, 2020). In addition, Stolz (2020) defines UDL as a framework that congregates flexible curriculum and pedagogy that respond to the diversity of learners. UDL has been adopted as a framework for designing and delivering barrier-free strategies for teaching and learning (Capp, 2020). Rose and Strangman (2007) indicate that research on cognitive science gives a hint of three brain networks in cognition and learning. The first brain network is for the recognition of patterns, while the second network is for planning and generating patterns. The third network is for determining which patterns are essential to learning. According to Dalton et al. (2012), these brain networks are referred to as recognition (recognition of information to be learned), strategic (application of strategies to process the information) and affective network (engagement in the learning task). Additionally, Rose and Strangman (2020) point out that the UDL framework is developed on these three brain networks, as a guide to creating a flexible curriculum that embraces learner variability. UDL comprises three core principles: namely, Multiple Means of Representation (MMR), Multiple Means of Action and Expression (MMAE) and Multiple Means of Engagement (MME) 2.3.6.1 Multiple Means of Representation (MMR) MMR provides multiple and flexible ways of presenting content and information to create a barrier-free learning environment for all learners. The presentation may be given using visual, auditory or tactile material (Dalton et al., 2012). MMR provides learners with options for perception, comprehension, language and mathematical expressions and symbols (Meyer et al., 2014). 2.3.6.2 Multiple Means of Action and Expressions (MMAE) MMAE is characterised by learners displaying more than one way to interact with different materials as they demonstrate what they have learned (Dalton 2017). There is no strategy that all learners most favour; hence, it is important to afford opportunities 20 for action and expressions to learn in a class of diverse learners. MMAE provides learners with options to use multiple formats of planning, organising, and initiating purposeful activities and how they demonstrate the mastery of the acquired knowledge (CAST, 2018). 2.3.6.3 Multiple Means of Engagement (MME) This principle enhances learning by providing multiple flexible ways for engagement in learning (Dalton, 2017). It is about different ways in which learners can be motivated. It advocates for teachers to tap into learners' meta-cognition development, which could reveal their differences in neurology, culture, personal relevance and background. There is no one means of engagement that all learners favour in all contexts. MME provides learners with options for promoting expectations and beliefs that optimise motivation, facilitating personal skills and strategies, as well as developing self- assessment and reflection (Novak & Rodriguez, 2018). 2.3.7 Relationship between UDL Principles and Brain Networks According to Balt et al. (2021), UDL is a framework based on cognitive neuroscience that concentrates on engaging multiple brain networks. UDL provides a comprehensive guide in terms of how learning takes place. Its three principles are classified in line with the three brain networks, thus providing a clear exposition of how learning occurs. Studies of the brain have long established that the three main networks are active during learning (Connell et al., 2012). The networks include affective, recognition, and strategic. Affective networks are responsible for the ‘why’ of learning. They regulate the emotional involvement with learning, such as our motivation and our ability to focus and remain engaged with tasks. Recognition networks regulate the ‘what’ of learning and are responsible for receiving information and concept formation. Strategic networks govern the ‘how’ of learning. They are responsible for planning, executing, and monitoring our actions. In line with this, Neuroscience has shown and confirmed that learner variability is the rule rather than the exception. Figure 2.1 illustrates classification of UDL principles in line with the brain networks, thus explaining how learning takes place. 21 The "WHAT" of learning The "HOW" of learning The "WHY" of learning Figure 2.1: Classification of UDL principles (Source: adapted from Balta, Supple & O'Keeffe, 2020) MMR supports recognition of brain networks. These brain networks make it possible for learners to receive, analyse information and recognise the object of learning (mathematical concept); hence it addresses the ‘what of learning’ (Ross, 2019). In a classroom setting, teachers apply the MMR principle by customising information display (that is, representing content in multiple ways and thus meeting the needs of the various learners (Parrish, 2019). Strategic brain networks. MMAE supports them. They facilitate the demonstration of the acquired knowledge using multiple and flexible physical actions and expression methods. They address the ‘how of learning’. In a classroom situation, teachers may Recognition network: Brain network 1 Make it possible to receive and analyse information. Strategic network : Brain network 2 Make it possible to generate patterns and develop strategies for action and problem solving, Affective network: Brain network 3 Fuel motivation and guide ability to establish priorities, focus attention and choose action. MMR “WHAT” MMAE “HOW” MME “WHY” Provides options for: ❖ Perception ❖ Language, mathematical expressions and symbols, ❖ Comprehension Provide options for: ❖ Physical action, expressions and communication ❖ Executive functions. Provide options for: ❖ Recruiting interest, ❖ Sustaining effort and persistence ❖ Self-regulation 22 allow learners to interact with one another and the material to demonstrate what they learned in diverse ways (Meyer et al., 2014) MME supports affective brain networks. According to García-Campos et al. (2020), affective brain networks trigger enthusiasm and interest in learning. They harness the power of emotions and motivation in learning. Affective brain networks are responsible for the ‘why of learning’. They control learners' emotional involvement with learning and motivate them to focus and remain engaged with tasks. 2.3.8 Definition of UDL in the Context of the Study Various scholars have provided definitions of the UDL concept (Barteaux, 2014; Dalton, 2017; Stolz, 2020). They all consider UDL to consist of proactive strategies to create a barrier-free learning environment. Barteaux’s (2014) definition, which states that UDL makes information and learning activities accessible to a diverse classroom, is the most appropriate one for the study's purpose. 2.4 REVIEW OF THE RELATED LITERATURE This section discusses literature based on the teaching and learning of Euclidean geometry. The subsequent sections are organised in line with the objectives and research questions of the study as highlighted in (1.3). The section, therefore, covers the following: ▪ challenges on the teaching and learning of Euclidean geometry, ▪ the solutions implemented to address the challenges and ▪ ways to mitigate against risks that may hinder optimal benefits of the use of the identified strategies for teaching and learning 2.4.1 Challenges The subsequent sections highlight some of the challenges pertaining to the teaching and learning of Euclidean geometry. 2.4.1.1 Lack of knowledge of inclusive teaching strategies for teaching Euclidean geometry According to Ubah and Bansilal (2019), teaching Euclidean geometry is difficult and complex because teachers lack the knowledge of content and inclusive teaching 23 strategies. Some Mathematics teachers lack knowledge of Euclidean geometry and how to teach proof and reasoning because they did not study the topic at any level of their studies (Machisi, 2020), especially those who did matric prior to the introduction of the Curriculum Assessment Policy Statement (CAPS) in 2012 (DBE, 2012). Machisi further articulates that teachers often use teacher-centred approaches where they copy theorems and proofs onto the chalkboard and then teach these theorems using a traditional lecture method. Learners thus copy and memorise the notes written on the board so they can reproduce them in-class tests and examinations. Learners are therefore not given the opportunity to discover the concepts independently, which impedes their ability to develop conceptual understanding. Mthembu (2007) refers to this traditional method as an ‘explain-memorise’ teaching approach. This method promotes memorisation and procedural understanding. According to Boggan, Harper and Whitmire (2010), some teachers seem reluctant to use resources such as manipulatives, making it difficult for learners' metacognitive skills to be enhanced and to develop high thinking skills. Ubah and Bansilal (2019) confirm that poor performance in Euclidean geometry is due to non-inclusive teaching strategies. It should be noted that there are learners with different learning styles and preferences in every classroom. Some are visual learners, while some are tactile (touching and doing) or auditory (through listening). This means that learners’ learning styles should be taken into consideration by the teacher in the choice of approaches and strategies. Thus, the teachers' failure to vary and use inclusive teaching strategies makes it difficult for meaningful learning to take place. Euclidean geometry is a practical topic that requires teachers to convey principles and explain concepts verbally and make demonstrations and drawings, which are embedded in levels 1 and 2 (recognition and analysis) of Van Hiele's theory of geometrical thinking. A study conducted by Ngirishi and Bansilal (2019) revealed that most learners are operating at visual (level 1) and analysis (level 2) of Van Hiele's theory of geometric thinking. This is problematic because grade 11 learners must think critically and use deductive reasoning to prove theorems as suggested by the Department of Basic Education (Machisi, 2021). This means that teachers must plan their forms of instruction aligned to learners' levels of geometric thinking and be mindful of the different ways learners assimilate information in the process. Mthembu (2007) observed that teachers consider teaching Mathematics as manipulation of 24 numbers. They overlook the importance of spatial representation and language, which are crucial in developing and communicating mathematical ideas, particularly in the teaching of Euclidean geometry. Mthembu (2007) further testifies that teachers feel that the investigative approach of teaching, which is in level 3(abstraction) of Van Hiele's theory of geometric thinking, is time-consuming and delays the completion of the syllabus. This explains why the teachers practice the drilling method in teaching Mathematics, especially Euclidean geometry. Studies conducted by Abdullah and Zakaria (2013) and Siyepu and Mtonjeni (2014) show that Euclidean geometry is taught through a traditional teacher-centred approach. According to Govender and Govender (2019), some teachers are expected to teach Euclidean geometry, yet they never had any contact with the topic. Machisi (2021) points out that in many geometry classrooms, teachers write theorems and proofs on the chalkboard and give a lecture, and then learners are asked to copy them onto their books. The learners are bound to memorise and reproduce the theorems and proofs during the class tests and examinations. Teachers' shortcomings in using inclusive teaching strategies to meet the needs of diverse learners in the mathematics classroom emanate from their Lack of knowledge of the geometric concepts. 2.4.1.2 Teachers' inability to create an engaging environment for meaningful learning to take place The engaging learning environment is the one that encourages learners to embrace their uniqueness and motivates them to strive for excellence (Boligger & Martin, 2018). The learning environment is engaging if it allows multiple flexible ways of instruction to ensure that learners can work cooperatively. Burgstahler (2009) ascertains that in cooperative learning, learners can be divided into small groups to discover and help each other understand new concepts. However, in my observation, many classrooms' seating arrangements do not encourage collaborative work since teachers are often at the front. At the same time, learners are seated facing the same direction, implying that the teacher is the only source of knowledge. This sitting arrangement does not promote an engaging environment. Burgstahler (2009) further indicates that to create an engaging learning environment, physical space in the classroom should be such that it allows interaction of learners and maximises meaningful learning for all. 25 According to Makina (2010), meaningful learning, as opposed to rote learning, is characterised by the learner's ability to gather information efficiently, sort it creatively, and come to a reliable and trustworthy conclusion. It is the ability to relate new events to already existing concepts. Bolliger and Martin (2018) point out that meaningful learning is influenced by three forms of interaction in the learning environment: learner- content, learner-instructor, and learner-learner interaction. Kutama (2002) notes that unless teachers create an engaging learning environment wherein learners can read and write with understanding, talk, draw, and show with their hands, learners will continue to experience challenges in the learning of Euclidean geometry; thus, they will fail to develop connections and conceptual understanding. Conceptual understanding is characterised by learners' ability to transfer acquired knowledge into a new situation and use it for different purposes (Alex & Mammen 2018). Teachers overlook the importance of creating a positive, engaging classroom environment for all learners. According to Sepeng and Webb (2012), this leads to productive discussions that deepen the learners' understanding of the content. Teachers who do not create an engaging environment deny learners an opportunity to interact and engage with each other and the teaching material to explore and practice high-level critical thinking skills. This makes it difficult for meaningful learning to take place and for learners to develop conceptual understanding. 2.4.1.3 Lack of visualisation skills Visualisation is a cognitive process of forming images or constructing mental representations (Klerkx et al., 2014). It enhances critical thinking, which is the most crucial skill in learning mathematics (Makina 2010). Although visualisation plays a vital role in teaching and learning, Makina (2010) indicates that teachers often present geometry lessons without understanding how learners think during problem-solving. They do not allow learners to imagine and think about the concepts that they are learning but instead, they focus on teaching them rules without engaging them in reflective and independent thinking. Strakova & Cimermanova (2018) affirm that learners' ability to think critically does not develop naturally; teachers must facilitate it. According to Mudaly and Dowlath (2016), learners who cannot visualise information have difficulty sketching their diagrams from given information. This becomes a barrier to learning Euclidean geometry because this concept includes types of problems mainly in the form of text (Kutama, 2002). These problems require learners to read 26 and create mental pictures concurrently of what they are reading, which becomes a challenge since many teachers do not incorporate strategies that help develop the learners' visualisation skills in their teaching (Moleko, 2021). Moreover, according to Djumanova (2021), teachers seem to be ignorant about the impact of learners' prior knowledge; consequently, they tend to build new knowledge on an unstable shaky foundation. Moleko (2014) affirms that insufficient prior knowledge negatively impacts learning. For example, learners cannot visualize and prove theorems if they neither know the properties of angles, lines, and shapes nor visualise these properties. Thornton (2001) states that in any classroom, there are learners who learn efficiently from either concrete, pictorial imagery (pictures in mind), pattern imagery, memory images of formulae, or kinaesthetic imagery (involving muscular activity) or dynamic (moving) imagery. Mathematics teachers are not mindful of this range of visualisation skills generated by individuals in their classrooms. Studies have been conducted on the role played by visualisation in many disciplines, including health and psychology (Makina, 2010). These studies affirm that visualisation plays an important role in making large-scale decisions in life. Making decisions based on visualisation involves cognitive processes, including thinking, knowing, remembering, judging, and problem- solving. Bradford (2004) witnesses that learners are variant in learning and have diverse preferences. Bradford further declares that about 65% of the population are visual learners. They learn best with pictures, images, and mind maps, to mention a few. About 30% of the population consists of auditory learners who learn best through hearing. Lastly, kinaesthetic learners form 5% of the population. Their best way of learning is to involve them in physical activity (Bradford 2016). Based on the above, teachers must not ignore the impact of visual representations in teaching and learning mathematics. Buentello-Montoya, Lomelí-Plascencia & Medina-Herrera (2021) ascertain that visualisation is an essential cornerstone of teaching for understanding, and critical thinking is the determining factor in understanding quality. As a matter of fact, poor visualisation skills negatively impact teaching and learning of Euclidean geometry, which predominantly affects. For instance, traditional tools such as textbooks and chalkboard methods do not support spatial visualisation skills (Alqahtani et al., 2017). According to Algahtani et al. (2017) instructors indicate that students who come to tertiary institutions are challenged to process visual objects and mental 27 images of the mathematical models. The reason could be that learners’ visualisation skill was never promoted and stimulated at the high school level. 2.4.1.4 Lack of knowledge of geometry terminology and symbolic representation Burgstahler (2009) states that learners are distinct in character because they come from various ethnic and racial backgrounds; hence they have diverse learning styles and ways in which they assimilate information. Unfortunately, for many of the learners in South Africa, English is not their first language (Mahlambi & Mawela, 2021). Kodisang (2015) describes language as a vehicle for communication and in the mathematics classroom, using appropriate terminology and symbols is crucial to enhancing meaningful understanding of concepts in the discussion. Being contextual, as stated by Makhubele at el. (2015), Euclidean geometry challenges learners because teachers fail to give proper attention to the vocabulary used and ways to unpack statements and diagrams in the question. They provide learners with information and steps on how to answer questions without analysing and understanding the given information and symbols in the question. According to Sinclair, Bartolini Bussi, de Villiers, Jones, Kortenkamp, Leung & Owens (2016), the threats of learning geometry, even from the primary level, consist of understanding of visuospatial reasoning, the use of diagrams, understanding the use of technologies, teaching and learning of geometric terminology and moving beyond traditional teaching strategies. Problems written in the form of text are complex for most teachers to teach Moleko (2018), and as Euclidean geometry includes problems mainly in the form of text and symbols, teachers need to develop learners' spatial knowledge and reasoning skills (Kutama, 2002). Learners' challenge lies in the analysis and correct interpretation of geometric statements; for example, if a learner does not know what a bisector is and what it does, it becomes challenging to understand theorems such as "a line drawn from the centre of a circle, perpendicular to the chord, bisects the chord." Terms including bisector, perpendicular, parallel, segment, cyclic quadrilateral, chord, and others make learning theorems understandable and perceivable. Appropriate terminology in any field is critical for acquiring a sound understanding of the content, hence it is imperative that learners develop this mathematics vocabulary even if it is in English. 28 Makhubele (2014) avows that the performance in Mathematics depends on language for meaningful teaching and learning and this is true with the teaching and learning of Euclidean geometry as teachers in South Africa are challenged to teach Euclidean geometry because of language-related technicalities (Mamali, 2015). 2.4.2 Solutions to the Identified Challenges The following sections highlight the solutions to the identified challenges. 2.4.2.1 Inclusive teaching strategy for teaching Euclidean Geometry Zambo and Zambo (2008) consider Professional Development Workshops (PDW) as a tool to influence teachers' content knowledge and skills required to teach Mathematics. PDW creates a platform for teachers to learn new ideas and ways to help learners (Zambo & Zambo, 2008). This is in line with the UDL framework, which advocates fostering collaboration and community amongst the teachers (CAST, 2018) to build their capacity to teach mathematical concepts. CAPS (DBE, 2012) 1.3c points out that one of the general aims of the South African curriculum is to encourage an active and critical approach to learning rather than rote learning of given truths. Furthermore, CAPS (DBE, 2012), 1.3e states that inclusivity should be central in planning and teaching; therefore, teachers should have a sound understanding of embracing diversity in classrooms. An inclusive teaching strategy requires teachers to use multiple flexible forms of instruction to cater to learners' uniqueness in learning styles and preferences. Teachers must create a learning environment that, according to the UDL framework, should afford learners options to demonstrate what they have learned and promote their expectations and beliefs that optimise motivation (CAST, 2018). This kind of learning environment is essential to promote active learning, foster critical thinking and motivate learners to own the process of learning solving activities (Chan et al., 2014). According to Mthembu (2007) learners who work in pairs or groups perform better than when they work individually. For this reason, teachers should structure their instruction on geometry so that learners could interact with one another. It means that learner involvement is critical in the teaching and learning of Euclidean geometry. In summary, teachers must know their learners and the way they learn. Burgstahler (2009) established that teachers should use equitable and flexible teaching strategies that promote accessibility and 29 maximise learning. When teachers know that learners present with different learning styles, they can design inclusive lessons to make content accessible to all learners and effectively communicate necessary information. For example, suppose a teacher wants the learners to show the validity of the theorem that says “ tangents drawn from the same point outside of the circle are equal in length.” The teacher must allow flexibility in approaching this question. Some learners may decide to accurately draw a diagram of a circle and two tangents from the same point and then measure the lengths of the tangents to check if indeed they are equal. Alternatively, others may prefer to calculate the lengths of the tangents if given the radius or diameter of the circle and the distance from the centre of the circle to the point of intersection of the two tangents. 2.4.2.2 Creating an engaging environment for meaningful learning An engaging learning environment is an interactive environment that can be developed by integrating technology into the teaching and learning mathematical concepts. Knowledge of learners and their learning styles is indispensable when creating an engaging environment because one must appreciate learners' diversity in the way they absorb and assimilate information. Lebid and Shevchenko (2020) affirm that technology improves learners' memory skills, visual-spatial skills, and multitasking abilities. Bolliger and Martin (2018) point out that in a classroom situation, collaboration involves a working relationship between a teacher and the learners and amongst learners themselves. Bolliger and Martin further state that engagement strategies allow learners to participate in collaborative group work where they can facilitate presentations and discussions and be involved in hands-on activities. As identified by Einfeld (2014), three forms of interaction in promoting engagement in teaching and learning are fundamental. They are Learner-content: Learners must engage with content, such as being involved in hands-on activities. However, not all learners would learn from hands-on activities. Bradford (2016) confirms that learners assimilate information in diverse ways. Bradford further indicates that visual learners rely on pictorial representations. They need to see pictures, diagrams, and graphs to visualise given information. A saying that a picture is worth a thousand words is valid for visual learners. For auditory learners, the best way to stimulate communication and learning is through discussions, 30 oral presentations, repetitions, and group chats, to mention a few. They prefer to discuss what they hear. Lastly, Bradford (2016) further affirms that kinaesthetic learners need to get up and be involved in the action for the information to sink into their memory. For instance, learners may create geometry vocabulary cards for conceptual understanding of geometry terminology, or they may prefer to sing and rap properties of shapes. They may use geoboards to explore the properties of shapes, angles, lines and their relationships. According to Mukunthan (2013), Geogebra was invented in the 1950s by an Egyptian Mathematician called Caled Gattegno. It is an excellent tool for exploring basic geometric concepts such as properties of 2- dimensional shapes, area, perimeter and geometry of straight lines. Learner-instructor means learners must work in collaboration with the teacher. For instance, Machisi (2020) points out that in the teaching of Euclidean geometry, a teacher may start by giving learners a brief history of its origin, reasons for offering it at the secondary school level, the role it plays in a real-life context for careers that are key in the economic development of any country. They will be motivated to study the concept and take ownership of their learning. Learner-learner means learners are encouraged to work as a team. According to Moleko (2014), learners' engagement creates a platform for them to work together in actively constructing knowledge that enhances meaningful learning. They do most of the discussion among themselves while the teacher contributes minimally. For example, some learners understand the theorem about the sum of opposite angles of a cyclic quadrilateral equal 180°, if they could draw the cyclic quadrilateral and measure the opposite angles. Khan et al. (2017) encourage teachers to target active learner involvement and user- friendly material. Moreover, teachers may engage learners in a small group discussion or assign different roles to learners in their discussion. For instance, a learner may be assigned to demonstrate that angles in the same segment are equal. This affords them avenues for content mastery, delivery and knowledge of communication. 2.4.2.3 Strategies to enhance Learner Visualisation Skills Bradford (2016) affirms that generally, a population consists of groups of people with invariant learning styles. This is also true in a classroom situation. About 65% are visual learners, about 30% are auditory and 5% are kinaesthetic learners. The skill of 31 visualisation is pivotal in teaching and learning mathematics and other areas such as maps and drawings (Klerkx et al., 2014). It is dependent on the connection between prior knowledge and new knowledge and if prior knowledge is not sufficient, the whole learning process would be negatively affected (Moleko, 2014). Similarly, Masilo (2018) confirms that considering learners' prior knowledge enables them to connect concrete and abstract levels of Van Hiele's theory of geometric thinking. For instance, Masilo is very vocal about probing to stimulate critical thinking. Using manipulatives promotes perception, encourages recognition, and ensures conceptualisation and visualisation of analysis and conjecturing (Masilo, 2018). This supports learners' ability to create pictures mentally based on the words they hear and the texts they read to critically analyse them (Palavan, 2020). According to Shatri and Buza (2017), visualisation positively influences development and increases learners' critical thinking, including analysing, interpreting, presenting and evaluating. A teacher may give learners a geometrical puzzle which can be solved simply by looking, thinking, and imagining; for example, learners may be provided with a set of 2- dimensional polygons and asked to identify, without calculating, the ones that are congruent or similar. 2.4.2.4 Enhancing knowledge of Euclidean Geometry vocabulary According to Alex and Mammen, (2018) knowledge and correct terminology are central in any field of study. The use of geometric terms such as point, line angle, parallel, perpendicular, plane, square, triangle and rectangle, to mention a few, enable one to communicate their ideas either in writing or speaking in precise forms in a diverse society. Teachers must use the correct terminology when teaching Euclidean geometry to avoid misconceptions and confusion. Peng and Lin (2019) state that the study's findings conducted in the United States of America (USA) at the University of Texas in Austin, revealed that Mathematics vocabulary made a noticeable contribution to word problems, particularly in measurements and geometry. According to Cain, A. (2014), knowledge of the terminology of a subject creates a positive classroom environment, improves attention, supports emotional regulation and reduces the anxiety of learners. The correct use of Mathematics vocabulary is critical for improved mathematics performance, and for this reason, Moleko and Mosimege (2020) encourage the explicit teaching of mathematics vocabulary to maximise learning. In 32 my experience as a Mathematics teacher, I realised that for a meaningful understanding of any topic in Mathematics, knowledge of terminology and jargon is very important. The mathematics curriculum is spiral. This means that the curriculum allows a logical progression from simple to complex ideas. Most topics build on each other with increasing comp