UNIVERSITY OF EDUCATION WINNEBA TOPIC EXPLORING THE EFFECT OF INTERACTIVE GEOMETRY SOFTWARE ON SENIOR HIGH SCHOOL STUDENTS’ UNDERSTANDING OF, AND MOTIVATION TO LEARN GEOMETRY JUATIE DOURI BENNIN JUNE, 2012 i EXPLORING THE EFFECT OF INTERACTIVE GEOMETRY SOFTWARE ON SENIOR HIGH SCHOOL STUDENTS’ UNDERSTANDING OF, AND MOTIVATION TO LEARN GEOMETRY JUATIE DOURI BENNIN (8109110027) A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS EDUCATION, FACULTY OF SCIENCE EDUCATION AND SUBMITTED TO THE SCHOOL OF GRADUATE STUDIES, UNIVERSITY OF EDUCATION, WINNEBA, IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF MASTER OF PHILOSOPHY IN MATHEMATICS EDUCATION. JUNE, 2012 ii DECLARATION STUDENT’S DECLARATION I hereby declare that this thesis, with the exception of quotations and references contained in published works which have all been identified and duly acknowledged, is entirely my own original work, and it has not been submitted, either in part or whole, for another degree elsewhere. NAME OF STUDENT: BENNIN JUATIE DOURI SIGNATURE……………………… DATE…………................................ SUPERVISORS’ DECLARATION We hereby declare that the preparation and presentation of the thesis was supervised in accordance with the guidelines on the supervision of thesis as laid down by the University of Education, Winneba. NAME OF PRINCIPAL SUPERVISOR: DR. ISSIFU YIDANA SIGNATURE………………………… DATE…………………………...... NAME OF SECOND SUPERVISOR: DR. JOHNSON NABIE DATE…………………………............. SIGNATURE………………………… ii DEDICATION To my dear wife, Amy Bening, for her love support and encouragement. iii ACKNOWLEDGEMENT I am very grateful to my supervisor, Dr. Issifu Yidana under whose direction and guidance this work has been a reality. I would sincerely like to express my heartfelt gratitude to him, for his patience, many in-depth constructive criticisms and valuable suggestions, which have immensely contributed to the success of this work. God richly bless you, sir. I am also grateful to all lecturers at the Department of Mathematics UEW, especially: Prof. E. Awanta, Prof. D. Mereku, Dr. C. Okpoti, Dr. Aseidu-Addo, Dr. P. O. Cofie, Dr. Nabie, Dr. Assuah, Mr. P. Akuyaare, and Mr. J. Apawu, whose tuition and great thoughts have contributed in no small measure in conducting this research. I cannot forget Madam Katumi Comfort Madah of the Home Economics Department of the UEW, for the pieces of advice she gave me during my work. Special thanks go to my family upon whose advice I came to pursue this course. My sincere thanks also go to the mathematics tutors and students of Kanton High School and the staff of YARO for their tremendous support in offering the needed information. To my colleague MPhil mathematics 2012 students: Nana Akosua, Rufai, Kotei, George, Moses, Stella, Baba, Daniel, John and Nii, I appreciate your company and support. Finally, I am grateful to Rebecca Bennin and my wife for their support during the period I pursued this programme. iv ABSTRACT The study sought to find out the effect of the use of interactive geometry software (IGS) on Senior High School (SHS) students’ conceptual understanding of, and their motivation to learn, plane geometry. It investigated ways in which IGS provides support for student-centred learning in a geometry class. The study was carried out in Kanton Senior High School involving 75 students, 43 in the Experimental group and 32 in the Control group. Purposive sampling was used to sample the school, while simple random sampling was used to select students to respond to the interview guide. Sequential explorative mixed methods, which was quasi-experimental and involved pretest-posttest non equivalent groups was used. The participants wrote pre-test after which IGS was used to teach the Experimental group for two weeks to improve students’ conceptual understanding. The participants then wrote a post-test as well as answered questionnaires and interview items to ascertain their experiences about the effect of IGS on their motivation and understanding. The findings indicated no significant difference in the conceptual understanding between the Control and Experimental groups in the pre-test. However, in the post-test, the findings indicated that the Experimental group had a mean score of 76.61, while the Control got 58.06. The t-test results revealed that there was significant difference in the conceptual understanding of geometry in favour of the Experimental group at P=0.001. The findings also showed that students were highly motivated to learn geometry, because they enjoyed the IGS lessons. It further revealed that the use of IGS supported student-centred learning in a number of ways; the lessons were activity based, very interactive in nature, students worked in groups and learn collaboratively through discussions. v TABLE OF CONTENTS DECLARATION .................................................................................................................ii DEDICATION ....................................................................................................................iii ACKNOWLEDGEMENT ..................................................................................................iv LIST OF TABLES .............................................................................................................. ix LIST OF FIGURES .............................................................................................................x CHAPTER ONE ................................................................................................................. 1 INTRODUCTION .............................................................................................................. 1 1.0 Overview .............................................................................................................. 1 1.1 Background to the Study ...................................................................................... 1 1.2 Statement of the Problem ..................................................................................... 7 1.3 Purpose of the Study ............................................................................................ 9 1.4 Research Questions .............................................................................................. 9 1.5 Significance of the Study ................................................................................... 10 1.6 Delimitation of the Study ................................................................................... 10 1.7 Organization of the Study ................................................................................. 11 CHAPTER TWO .............................................................................................................. 12 LITERATURE REVIEW ................................................................................................. 12 2.0 Overview ............................................................................................................ 12 2.1 Theoretical Framework ...................................................................................... 12 More Knowledgeable Other (MKO):. ....................................................................... 14 2.1.1 Main Differences between Vygotsky’s and Piaget’s Theories of Cognitive Development ............................................................................................... 17 2.1.2 Application of Vygotsky’s Theory in an IGE to Learn Geometry. ............ 19 2.3 The Traditional Method of Teaching and Learning Geometry in Ghana ......... 20 2.4 The Use of Technology in the Teaching of Geometry ...................................... 23 2.4.1 2. 5 Effects of ICT on the Teaching and learning of Plane Geometry.............. 28 The Conceptual Understanding of Mathematics Concepts ................................ 29 2.5.1 Effects of IGS on students Conceptual Understanding of Mathematics..... 31 vi 2.6 Motivation and Students Learning of Geometry ............................................... 32 2. 7 Student-centred Learning ................................................................................... 34 The Effects of IGS on Motivation and Student-centred Learning. ............. 35 2.7.1 CHAPTER THREE .......................................................................................................... 37 METHODOLOGY ........................................................................................................... 37 3.0 Overview ............................................................................................................ 37 3.1 Research Design ................................................................................................. 38 3.2 Population, Sample and Sampling Procedure .................................................... 39 3.3 Research Instruments ......................................................................................... 40 3.3.1 Pre-test and Post tests. .................................................................................... 41 3.3.2 Interview Guide for Teachers ......................................................................... 41 3.3.3 Interview Guide for Students .......................................................................... 42 3.3.4 Questionnaires ................................................................................................ 42 3.4 Intervention Tools and Procedures ..................................................................... 42 3.4.1 GeoGebra ........................................................................................................ 43 3.4.2 Design of Instructional Materials ................................................................... 44 3.4.3 Design of the Intervention .............................................................................. 45 3.5 Piloting the Instruments .................................................................................... 46 3.7 Intervention and Data Collection Procedures..................................................... 48 3.7.1 Lesson: Introduction to GeoGebra Software Lessons ................................ 49 3.7.2 Lesson on Angles and their Properties........................................................ 51 3.7. 3 Lesson on Properties of Parallel Lines........................................................ 53 3.7.4 Lesson on Triangles .................................................................................... 54 3.7.5 Summary of the Lessons Taught................................................................. 55 3.7 Data Analysis ..................................................................................................... 56 CHAPTER FOUR ............................................................................................................. 59 RESULTS AND DISCUSSION ....................................................................................... 59 4.0 Overview ............................................................................................................ 59 vii 4.1 The Effect of IGS on SHS students’ Conceptual Understanding of Plane geometry ........................................................................................................................ 59 4.2 The Effect of IGS on SHS Students’ Motivation to Learn Plane Geometry...... 72 4.2.1 The Effect of IGS on Motivating SHS Students’ to Learn Plane Geometry the Student Perspective. ............................................................................................ 72 4.2.2 The Effect of IGS on Motivating SHS Students’ to Learn Plane Geometry the Perspective of the Teacher. ........................................................................................... 75 4.3. Ways in which IGS Provided Support for Student-centred Learning in a Geometry Class ............................................................................................................. 76 4.4 Discussion of Major Findings. ........................................................................... 82 CHAPTER FIVE .............................................................................................................. 90 SUMMARY, CONCLUSION AND RECOMMENDATIONS ....................................... 90 5.0 Overview ............................................................................................................ 90 5.1 Summary of Study.............................................................................................. 90 5.2 Summary of Findings ......................................................................................... 91 5.3 Conclusion and Implications for Practice .......................................................... 95 5.4 Limitations of the Study ..................................................................................... 96 5.5 Recommendations .............................................................................................. 97 REFERENCES ................................................................................................................. 99 APPENDIX I: PRE-TEST AND POST TEST QUESTIONS ........................................ 112 APPENDIX II: RESPONSES FROM IN-CLASS ACTIVITIES .................................. 114 APPENDIX III: QUESTIONNAIRE ............................................................................. 118 APPENDIX IV: INTERVIEW TEACHERS AND STUDENTS .................................. 120 APPENDIX V: LESSON PLANS .................................................................................. 124 APPENDIX VI: APPLETS............................................................................................. 132 APPENDIX VII: WORKSHEETS ................................................................................. 136 viii Table 3. 1: Table 4. 1: Table 4. 2: Table 4. 3: Table 4. 4: group Table 4. 5 Table 4. 6: Table 4. 7: Table 4. 8: Table 4. 9: Table 4. 10: Table 4. 11: Table 4. 12: Table 4. 13: Table 4. 14: LIST OF TABLES Sequence of Data Collection ....................................................................... 49 Effect of IGS on students’ Conceptual Understanding in both the Control and Experimental Groups. .......................................................................... 60 The Development of Students in Conceptual Understanding of Geometry in the Experimental and Control groups ..................................................... 67 Means, Standard deviation and Maximum scores for Experimental group 69 Means, Standard deviation, Maximum and Minimum Scores for Control 70 Independent Samples t-test for Experimental and Control groups ............ 70 Students’ and Teachers’ views on the Effect of IGS and Conceptual Understanding of Geometry. ....................................................................... 71 The Proportion of Students’ views on the Effect of IGS and their Motivation. .................................................................................................. 72 Students’ Mean Rating on the Effect of IGS on Motivation to Learn ....... 73 Students’ Mean rating on their level of Motivation.................................... 75 Proportion of Teachers’ view on the Effect of IGS on Students’ Motivation to Learn Plane Geometry. ........................................................................... 76 Percentage Distribution of Students’ views on Student-centred Learning ................................................................................................................. 77 Mean Rating of Students views on Student-centred learning. .................... 78 Proportion of Students’ views on Student-centred learning. ..................... 79 Teachers’ view on Ways in which IGS Supported Student-centred Learning. .................................................................................................... 81 ix Figure 2. 1: Figure 2. 2: Figure 3. 1: Figure 3. 2: Figure 4. 1: Figure 4. 2: Figure 4. 3: Figure 4. 4: LIST OF FIGURES Illustration of the Zone of Proximal Development .................................... 14 The Interactions between Externalization and Internalization as a cyclic process......................................................................................................... 27 GeoGebra Interface and Construction Tools…………………………….. 44 A cross section of Students Exploring the Geogebra in the Laboratory .... 51 The Working Process of Student A’s question 3 in the Pre-test. ............... 61 The Working Process of Student B’s question 2 in the Pre-test ................. 62 The Working Process of Student A’s question 3 in the Post-test ............... 64 The Working Process of Student B’s question 2 in the Post-test. ............. 65 x CHAPTER ONE INTRODUCTION 1.0 Overview This chapter provides an introduction to the research study. The introduction includes the background of the study, statement of the problem, the purpose of the study and research questions which guided the study. It further highlights the significance, the delimitations and organization of the study. 1.1 Background to the Study Development in almost all areas of life is based on effective knowledge of Science and Mathematics. There cannot be any meaningful development in any area of life without knowledge of Science and Mathematics. It is for this reason that the education systems of countries that are concerned about their development put great deal of emphasis on the study of mathematics. It is therefore not surprising that the government of Ghana made mathematics a core subject at both the Basic and Secondary levels of Education in Ghana. The Senior High School (SHS) syllabus in Ghana is based on the notion that an appropriate mathematics curriculum results from a series of critical decisions about three inseparably linked components: Content, Instruction and Assessment (MOE, 2010). Plane geometry is one of the major content domains the mathematics curriculum covered to promote the acquisition of mathematical knowledge and skills for life. Plane geometry in the Senior Secondary School mathematics curriculum covered angles of a polygon, Pythagoras’ and circle theorems including tangents (MOE, 2010). 1 Geometry is the study of shapes and space. Drickey (2001) defined geometry as a branch of mathematics that provides a rich source of visualization for understanding, algebraic, arithmetic and statistical concepts. Geometry appears naturally in the structure of the solar system, in geological formation of some rocks and crystals, in plants and flowers, and even in animals. It is also a major part of our synthetic world such as art, architecture, cars, machines, and virtually everything humans create. The knowledge of geometry is so important that its utility is needed by everyone. Fortunately geometry is well represented in the Ghanaian mathematics curriculum at all levels of education. The applications of geometry are diverse and universal in all aspects of life. In the school, studying geometry provides many foundational skills and helps to build logical thinking skills, analytical reasoning and problem solving among others. Geometry has an applicable link to many other topics in mathematics, specifically Measurement. Consequently, a very good grip of the knowledge of geometry prepares students to adequately respond to the challenges of further mathematics in life. In the place of work, geometry is used by architects, engineers, physicists, pilots, captains of ships and land surveyors. More importantly, a teacher without a good knowledge of geometry cannot adequately convey the concepts and the beauty that comes along side its teaching to students. In order to draw the full benefits of geometry in the mathematics curriculum, classroom instructions should aim at enhancing students’ geometric thinking. Improving students’ geometric thinking levels is one of the major aims of mathematics education. This is because geometrical thinking is an important tool in many scientific, technical and occupational areas. One of the best descriptions of students’ geometric thinking level on 2 two-dimensional shapes is the Van Hiele theory of Geometric Thinking (Batista, 2007). Teaching geometry at the Senior High level should be done in ways that promotes geometric thinking. However, Mehdiyev (2009) stated that in Azerbaijan the teaching and learning of geometry tend to focus on having students learn a list of definitions and the properties of shapes. According to Mehdiyev, textbooks for use at the Senior High Schools provided only pencil and paper illustrations that are not comprehensive, because, they lack the visual description of a complete interactive process needed for the construction of geometrical concepts. These illustrations, often lead to memorization and does not target development of conceptual understanding. This situation is not different from what is happening in Ghanaian schools. Fredua-Kwarteng and Ahia (2005) observed that the teaching and learning culture of mathematics in Ghanaian schools have the following characteristics: Students learn mathematics by listening to their teachers and copying from the chalkboard rather than asking questions for clarifications and justification. Furthermore, students learn mathematics by regurgitating facts, theorems or formulas instead of probing for meaning and understanding of mathematical concepts. Students in the learning process hardly ask the logic or philosophy underlying those mathematical principles, facts, or formulas. Consequently, students learn mathematics as a body of objective facts rather than a product of human invention. Instead of memorising properties and definitions, Battista (2007) suggested that students should be made to personally develop meaningful geometric concepts and ways of reasoning that enable them to carefully analyze spatial problems and situations. This 3 calls for an alternative teaching approach where Information and Communication Technology (ICT) can be used to enhance students’ thinking and problem-solving skills. One change that has irresistibly affected education institutional delivery globally has been the introduction of technology into society which results in the explosion of computers into schools. Technology in schools, particularly the computer with its communicative abilities, has become the focus and substance of strategic planning in shaping national economies. Governments of both developed and developing nations have recognized, as a matter of urgency, the role of computer technology in redefining their economic activities (World Bank, 1998). This then calls for the integration of ICT into the teaching and learning situation in the world over. The integration of ICT into education is recognized as: providing opportunities for developing skills that has the potential to transform pedagogical practices, and for reforming curricula (Roschelle, Pea, Hoadley, Gordin, & Means, 2000). Recent advancement in communication technology has contributed immensely to minimize the effect of distance in education. Roblyers (2006) aptly describes the situation as the “death of distance”. She writes that the death of distance has given new life to education. Technology is essential in the effective teaching and learning of Mathematics. It influences the Mathematics taught in schools and enhances students learning (National Council for Teachers of Mathematics (NCTM), 2000). The integration of ICT into teaching enables students to fully grasp Mathematical concepts and link them up to other topics for further Mathematics education. Thus ICT is considered as an essential tool for developing understanding about mathematical concepts (Fitzallen, 2005). These concepts then builds strong mathematics background in students, hence they have better choices 4 about higher education, which is a gateway to more careers. According to Chief Executive Officers’ (CEO) forum, ICT has the ability to motivate both teachers and students as well as enhance the learning of current curriculum content collaboratively (CEO Forum, 2001). From the discussions above, the researcher observed that ICT is a powerful tool that can accelerate the attainment of educational goals. According to Butzin (2001), the focus of ICT in education should be on integrating technology into teaching and learning and must not be predicated on learning. This entails the application of ICT tools to facilitate the teaching and learning situation in the school. The use of ICT as suggested by Reynolds, should supplement classroom activity by accessing existing information and knowledge, rather than as an integral part of pedagogical practice (Reynolds, 2001). In Ghana, the goal of ICT is to enable every Ghanaian to be able to use ICT tools and resources confidently and creatively to develop the skills and knowledge needed to achieve personal goals and be full participants in the global economy by 2015 (MOE, 2006). The ICT for accelerated development policy document outlined some guiding principles towards the integration of ICT into classroom practice. The policy stipulated that a curriculum reform is necessary for effective integration and utilization of ICT in the classroom. It encouraged teachers to explore and use ICT tools in teaching to improve students’ learning in order to develop skills necessary for the competition in the knowledge `economy and information society. Exploration of ICT is crucial to provide best experiences for educators to incorporate this new technology into teaching (McGehee & Griffith, 2004), as the positive impact of ICT depends on how teachers use ICT in their teaching and learning activities (Galbraith, 2006). Consequently, efforts are 5 being made to integrate ICT into the Ghanaian mathematics curriculum and researchers are calling for the adoption of strategies that will make ICT integral to teaching and learning processes (Assuah, 2010; Yidana & Amppiah, 2003; Dontwi, 2001). Achieving this requires extensive research that will identify strategies applicable in the Ghanaian classroom. Mehdiyev (2009) stated in his study on students’ learning experiences using dynamic geometry software that, the teaching and learning of geometry in Dynamic Geometry Environment (DGE) established positive effects on students’ conceptualisation of Mathematics concepts. The DGE encouraged students to discuss, interact with each other and explore the content collaboratively. The students are not coerced to accept the geometrical content with absolute certainty. Rather, students are motivated to learn in a student-centred dynamic environment. Mehdiyev later in his research described the traditional teaching and learning of geometry at secondary schools as “teacher-centred”, which is at variance with learning geometry in a DGS environment. Personal experience and empirical research (Mereku, 2010) indicated that teaching in the Ghanaian classroom is teacher-centred. Thus the student is made a passive listener in the learning process, which makes the student deficient in mathematical analysis and logical reasoning. Therefore, Mathematics achievement in Kanton Senior High School is poor. It appears that something is wrong with the way mathematics is learnt and assessed in Ghana. The Trends in International Mathematics and Science Study (Asabre-Ameyaw & Mereku, 2009; Anamuah-Mensah, Mereku & Ghartey-Ampiah, 2008; Anamuah-Mensah, & Mereku, 2005), report that Ghana remained second from the bottom in 2003, 2007, out of the number of countries that participated in the examination. The reports stated that 6 students’ performance in geometry was the lowest in the five domains the test covered. Also National Educational Assessment (NEA), which is an indicator of the overall national status of Mathematical achievement in the primary school system in Ghana. The NEA reported that mean scores percent of primary 3 and primary 6 pupils in mathematics respectively of 41.8% and 39.6% was far below the average of 50% (CRDD, 2009). In Addition, the West African Examination Council Chief Examiner’s annual reports for the SSSCE & WASSCE from 2003 to 2006 observed that candidates were weak in Geometry of circles and 3-dimensional problems. The reports repeatedly indicated that most candidates avoided questions on 3-dimensional problems. Where they attempted geometry questions, only few of the candidates showed a clear geometrical understanding of the problem in their working process. Even though there are some research works on the integration of ICT into the teaching and learning of Mathematics in Ghana (Assuah, 2010; Yidana & Amppiah, 2003; Dontwi, 2001), little is known about the use of interactive geometry software in the teaching and learning process in the Ghanaian classroom. This study is therefore developed to explore the teaching and learning of geometry in an Interactive Geometry Environment (IGE) in Ghanaian classroom using Geogebra. 1.2 Statement of the Problem Students are unable to construct, visualize and justify geometrical concepts in the teaching and learning process in Ghanaian classrooms. The traditional method of teaching alone makes the student a passive listener, which makes students deficient in geometrical analysis and reasoning. This approach to teaching and learning geometry 7 emphasized how much a student can remember and less on how well the student can think and reason. The teacher dominates the classroom and turns students to mere listeners. Consequently, the students are not encouraged to discuss, interact with each other and to explore the content collaboratively (Mereku, 2010; Mehdiyev, 2009; Affum, 2001). Therefore students see learning as unnatural and seldom bring satisfaction to them. Several reports (TIMSS, NEA, WAEC) indicate that there is a persistent and consistent poor performance of Ghanaian SHS, JHS and Primary School students in the field of mathematics in general and geometry in particular. National reports show that Ghanaian students are performing poorly in Geometry (Asabre-Ameyaw, Mereku, 2009; Anamuah-Mensah, Mereku & Ghartey-Ampiah, 2008; Anamuah-Mensah, & Mereku, 2005). In addition, the West African Examination Council Chief Examiner’s annual reports for the SSSCE & WASSCE from 2003 to 2006 observed that candidates were weak in Geometry of circles and 3-dimensional problems. This poor performance may be due to the traditional method employed in the teaching and learning discourse in the Ghanaian classroom (Mereku, 2010). In the domain of ICT integration into teaching and learning, little is known about the use of interactive geometry software in the Ghanaian classroom. This study is therefore developed to explore the teaching and learning of geometry in an Interactive Geometry Environment (IGE) in a Ghanaian classroom using an Interactive Geometry Software (IGS). Such exploratory studies can enable curriculum developers in Ghana to determine what must be done to the proposed ICT for accelerated development policy as well as making curriculum implementation more pragmatic. One in which the teacher can assume the role of a guide or researcher in the lesson delivery through student-centred 8 activities to enable students in Ghana improve their conceptual understanding in the Geometry content domain. 1.3 Purpose of the Study Good knowledge of geometry is essential, because it facilitates the acquisition of other mathematical knowledge for further studies. This study therefore explored the effect of IGS on students’ conceptual understanding of plane geometry and how it motivates and supports student-centred learning. The study sought to find out: The effect of IGS on the development of SHS students’ conceptual understanding 1. of geometry 2. The effect of the use of IGS on SHS students motivation to learn geometry 3. Ways in which IGS provides support for student-centred learning in a geometry class 1.4 Research Questions In pursuance of the purposes stated above, the following research questions were formulated to guide the study: 1. To what extent does the use of IGS affect SHS students’ conceptual understanding of geometry? 2. How does the use of IGS motivate SHS students to learn geometry? 3. In what ways do IGS support student-centred learning in a geometry class? In order to answer the research question 1, the following null and alternative hypotheses were formulated. 9 HO: There is no difference in the understanding of geometry between the Control and Experimental groups. H1: There is significant difference in the understanding of geometry between the Control and Experimental groups. 1.5 Significance of the Study The study explores the effect of IGS (GeoGebra) on SHS students learning experiences in an interactive geometry environment. The findings of this study will be a resource for policy makers, teachers and other stakeholders to help improve students’ geometric reasoning in Ghana, through the use GeoGebra. It will generate information that could inform policy makers on ways of implementing the national policy on the integration of ICT into the teaching and learning of mathematics. Again, the findings of the study will serve as a resource for curriculum developers and teachers to improve students’ learning outcomes in schools especially in the Sissala East District and the nation at large. The study will also serve as a baseline document for other researchers investigating into the effects of IGS on motivation and student-centred learning. It will further make a significant contribution to existing literature. 1.6 Delimitation of the Study The study covered only students of Kanton Senior High School in the Sissala East District in the Upper West Region. Tumu is the District capital of the Sissala East District Assembly, located in the North Eastern part of Ghana. It shares boundaries with the Sissala West District to the West, in the East with Wa East and Burkina Faso to the south. The area was chosen because of its familiarity to the researcher. The choice was made 10 with the belief that population for the study would be easily accessible to the researcher. The study explored students learning experiences when using interactive geometry software. 1.7 Organization of the Study The study is organized into five chapters. Chapter one covers the introduction, background to the study, statement of the problem, research questions, purpose of the study, significance of the study, organization of the study and delimitation. The second chapter reviews related literature and discusses the theoretical framework. Chapter three deals with the research methodology; this includes the research design, population and sampling, instrumentation, procedures for gathering data and how the data were analyzed. The presentation of the results and the discussion of the findings are described in chapter four. The final chapter which is chapter five looks at the summary, conclusions, recommendations and areas for further research. 11 CHAPTER TWO LITERATURE REVIEW 2.0 Overview The study investigated the effect of IGS on SHS students’ understanding of and motivation to learn geometry. This chapter contains specific overview about the literature related to this study. The following are the headings under which both the theoretical and empirical reviews were made: 2.1 Theoretical Framework The Traditional Method of Teaching and Learning Geometry The Use of Technology in Teaching Geometry Students’ Conceptual Understanding of Geometry Students’ Motivation and Learning Geometry Student-centred Learning in a Geometry Lessons Theoretical Framework High school students usually have difficulties in understanding geometry concepts when they are taught through the traditional method alone (Mehdiyev, 2009). Several research studies have been carried out on enhancing students’ understanding of geometry (Ruthven, Hennessy, & Deaney, 2008; Hollebrands, 2003; Gawlick, 2002; Laborde, 2001). They suggested that the use of technology promotes students’ understanding of geometry and therefore, recommended an Interactive Geometry Environment (IGE) for teaching and learning of geometry. 12 However, the study by Vygotsky made a significant impact on the improvement of teaching and learning of geometry (Falcade, Laborde & Mariotti 2007; Laborde 2003). Vygotsky (1978) proposed a learning model known as Social Cognitive Development Theory for teaching and learning which aims at a more effective education. A method of teaching that has the potential of developing students’ conceptual understanding and makes them productive in the knowledge economy. The major theme of Vygotsky's theoretical framework is that social interaction plays a fundamental role in the development of cognition. The theory stipulates that the potential for cognitive development depends upon the “Zone of Proximal Development” (ZPD): a level of development attained when children are engage in social behaviour. The full development of the ZPD depends upon full social interaction. This implies that students learn better when they are guided to interact socially in an explorative environment, where they discuss their findings. This learning environment provides the opportunity for students to learn collaboratively in guided group activities. The range of skills that can be developed with an adult guidance or peer collaboration exceeds what can be attained by an individual alone. According to Vygotsky (1978), every function in the child's cultural development appears twice: first, on the social level between people (inter-psychological) and second, on the individual level inside the child (intra-psychological). This applies equally to voluntary attention, to logical memory, and to the formation of concepts. This means that in learning mathematical concepts students have the opportunity to learn better through discussions and social interactions. 13 Vygotsky's social cognitive development theory has two main tenets. These are, More Knowledgeable Other (MKO) and Zone of Proximal Development (ZPD). More Knowledgeable Other (MKO): The More Knowledgeable Other (MKO) refers to someone who has a better understanding or a higher ability level than the learner, with respect to a particular task, process or concept. Although the implication is that the MKO is a teacher or an adult, this is not necessarily the case. A child's peers or grown up child may be the individuals with more knowledge or experience. For example, one who is more likely to know more about the newest teen-age music groups, how to win at the most recent PlayStation game. In recent times, electronic tutors have been used in educational settings to facilitate and guide students through the learning process. The key to MKOs is that they must have or be programmed with more knowledge about the topic being learned than the learner does. In this study the researcher and group leaders who have good knowledge on how to explore the concepts using the software play the role of MKO. Zone of proximal development: Zone of proximal development is the difference between the child's capacity to solve problems on his own and his capacity to solve them with assistance. Figure 2. 1: Illustration of the Zone of Proximal Development 14 Figure 2.1 Illustrates the Zone of Proximal Development. This refers to all the functions and activities that a child or a learner can perform only with the assistance of someone else (MKO). The figure shows that every student comes to the learning environment with a previous knowledge. The student is then coached through guided practice to learn new ideas. This is what Vygotsky termed as the ZPD, which prepares them to solve difficult challenges later in life. The person in this scaffolding process, providing non-intrusive intervention, could be an adult (parent, teacher, caretaker, language instructor) or another peer who has already mastered that particular function. This provides an enabling platform for the learner to learn through coaching until they can do things independently. This description often leads to the actual developmental level of the child. The actual developmental level refers to all the functions and activities that a child can perform independently, without the help of anyone else. In this study the researcher created an interactive geometry environment, where students were guided first by the researcher to explore geometry concepts. This process afforded students opportunity to learn by observation, imitation and independently exploring the concepts provided in the Applets. Then later the researcher guided the students into a social behaviour where they were engaged in uninterrupted discussions and interactions thereby learning collaboratively. Vygotsky's zone of proximal development has many implications for those in the educational milieu. One is that, human learning presupposes a specific social nature and is part of a process by which children grow into the intellectual life of those around them (Vygotsky, 1978). According to Vygotsky (1978), an essential feature of learning awakens a variety of internal developmental processes that are able to operate only when the child is in the action of interacting with people in his environment and in cooperation 15 with his peers. Therefore, in learning mathematics, the authenticity of the environment and the affinity between its participants are essential elements to make the learner feel part of this environment. These elements are rarely predominant in conventional classrooms. This call for a strategy of teaching and learning that is activity based, studentcentred and one that aims at developing students’ conceptual understanding in a social environment. In social activism, learning takes place in social environment where there are collaborative activities. Through these activities, learners communicate, interact and learn from one another. These activities therefore promote meaningful learning at the school environment. Both Bruner and Vygotsky in their theory of social interaction stated that learning is shaped and affected by the learner’s background and cultural experiences (Dricoll, 2000; Roblyers, 2006). Social interaction is critical as knowledge is created in the process (Schunk, 2004). Vygotsky (1978) affirmed that, this type of social interaction involving co-operative or collaborative dialogue promotes cognitive development. According to Vygotsky (1978), a more effective education may be induced by the use of technology, as studied by a number of researchers (Falcade, Laborde & Mariotti 2007; Laborde 2003). Effective educational activities and cognitive tools improve students’ active involvements in the teaching and learning process and encourage their reflections on the concepts and relations investigated. It is claimed that usage of interactive software not only increase students’ conceptual understanding and problem solving skills but also promotes their positive attitudes towards mathematics since they provide “concrete experiences” that focus attention and increase motivation. Falcade, 16 Laborde & Mariotti (2007), are of the view that cognitive and psychological effect arises from the use of technology in education. Vygotsky was not a lone ranger in the school of cognitive development theories. Jean Piaget and Lev Vygotsky developed similar theories concurrently; however that of Vygotsky better explains the effect of IGS in student-centred social learning environment. Section 2.1.1 compares the two main theories and that explains the reasons why the researcher aligned this study with Vygotsky’s theory. 2.1.1 Main Differences between Vygotsky’s and Piaget’s Theories of Cognitive Development Source of cognitive development Piaget believed that the most important source of cognition is the children themselves. Piaget emphasised the role of an inbuilt (biological) tendency to adapt to the environment, by a process of self-discovery and play. Vygotsky emphasised the role of culture and experience. Vygotsky believed that what drives cognitive development is social interaction a child’s experience with other people. Culture shapes cognition. Stage Theory Piaget emphasised universal cognitive change. Vygotsky’s theory can be applied to all ages (not a stage theory) and emphasised individual development. Discovery Learning Piaget advocated for discovery learning with little teacher intervention. 17 Vygotsky promoted guided discovery in the classroom with the help of a MKO. Language and Thought For Piaget, language is a product of cognitive development. In other words, cognitive development determines language use. Vygotsky believed that language develops from social interactions, for communication purposes. Later language ability becomes internalised as thought and “inner speech”. Thought is the result of language. In other words, social interactions determine language use. Both Piaget and Vygotsky believed that young children are curious and are usually actively involved in their own learning, discovery and development of new understandings/schema. Piaget’s stage theory prescribed cognitive development according to the child’s age and his emphasis on discovery learning with little teacher intervention do not fully support this study which is explorative in nature and needs much teacher intervention at the early stages. Piaget emphasized self-initiated discovery whereas, Vygotsky placed more emphasis on social contributions to the process of cognitive development. According to Vygotsky (1978), much important learning by the child occurs through social interaction with a skillful tutor. The tutor may model behaviours and provide verbal instructions for the child. Vygotsky refers to this as cooperative or collaborative dialogue. The child seeks to understand the actions or instructions provided by the tutor (MKO) then internalize the information, using it to guide or regulate their own performance. In line with Vygotsky’s theoretical framework, 18 this study incorporated the use of IGS (GeoGebra) within a geometry class and explored its effect on student-centred learning, students’ motivation to learn and understand geometrical concepts. For this, the researcher developed some Applets with the IGS (GeoGebra), which represented the construction of the intended geometrical concepts to be explored by students. 2.1.2 Application of Vygotsky’s Theory in an IGE to Learn Geometry. Most of the original work of Vygotsky was done in the context of language learning in children (Vygotsky, 1962) although later applications of the framework have been broadened to include the teaching and learning of mathematical concepts (Wertsch, 1985). Several research studies on the effect of the use IGS on enhancing students’ understanding of geometry in a social learning environment have been done (Laborde, 2001; Gawlick, 2002; Hollebrands, 2003; Ruthven, Hennessy, & Deaney, 2008). These researchers concluded that the use of technology promotes students understanding of geometry and therefore, recommended an Interactive Geometry Environment (IGE) for teaching and learning of geometry. This study therefore incorporated the use GeoGebra within a geometry class to explore its effect on student-centred learning, students’ motivation to learn and understand geometrical concepts. The researcher developed some Applets with the GeoGebra which represented the construction of the intended geometrical concepts. In line with the theoretical framework, the researcher became the “More knowledgeable Other”, who created and facilitated students’ exploration of geometric concepts in a student-centred environment. The researcher coached students on how to construct geometrical objects using the GeoGebra. The researcher then put 19 students into smaller groups and created the platform for them to interact, discuss and learn cooperatively. 2.3 The Traditional Method of Teaching and Learning Geometry in Ghana The traditional approach to teaching and learning of mathematics in general involves the directed flow of information from the teacher as sage to student as receptacle. The effectiveness of this transmission is then tested by posing various exercises to the student. With regard to pedagogy, what one observes in most mathematics classrooms today is not significantly different from the traditional approach, which emphasised mainly on basic skills which are predominantly computational. Mehdiyev (2009) described the traditional method of the teaching and learning of geometry at secondary schools in Azerbaijan as teacher-centered in one word. According to Mehdiyev, the teacher dominates the classroom and turns students to mere listeners. The teaching of geometrical objects is pedagogically authoritative in nature and therefore, students are not encouraged to question the validation or construction of geometrical entities. The students are not encouraged to discuss and interact with each other the content presented to them collaboratively. The traditional approach presents mathematics as a cut-and-dry proposition, that there is only one way to do everything; the way the teacher says (Fredua-Kwarteng & Ahia, 2005). Fredua-Kwarteng and Ahia (2005) pointed out that, fifty years after independence, mathematics teaching in Ghanaian schools is still characterized by the “transmission” and “command” models. According to them the learning culture of mathematics in Ghanaian schools are such that: 20 1. Students learn mathematics by listening to their teacher and copying from the chalkboard rather than asking questions for clarifications. Consequently, mathematics is learnt by regurgitating facts, theorems or formulas instead of probing for meaning and understanding of mathematical concepts. Students hardly ask the logic or philosophy underlying those mathematical principles, facts, or formulas. 2. Students go to mathematics classes with the object to calculate something. Therefore, if the classes do not involve calculations they do not think that they are learning mathematics. So students learn mathematics with the goal to attain computational fluency, not conceptual understanding or meaning. Conceptual understanding requires students to think critically and act flexibly with what they know. Students are fond of asking, “How do you calculate that?” instead of asking, why do you calculate it, and in that way? (FreduaKwarteng & Ahia, 2005). According to Anamuah-Mensah & Mereku (2005), in the last four decades, several Ghanaian authors have been involved in curriculum development for schools. Teachers continue to teach by merely transmitting mathematical facts, principles and algorithms, and students are commanded to learn them in a passive and fearful manner. One major factor influencing the transmission model relates to the inefficiencies within the curriculum materials used in Ghanaian schools. Textbooks in schools are still based on philosophies of teaching which are no more valued globally in school mathematics. (Mereku, 2010). 21 Furthermore, even though textbooks and other teaching aids are very important tools in today’s classrooms, the supply is inadequate in Ghanaian schools (Mereku et, al. 2007). You hardly find a school where students have one to one access to mathematics textbooks. Seating and writing places are poor and inadequate, while textbooks are in short supply. The result is that students have little opportunity to practice mathematics learned. According to Mereku et, al. (2007), students do little reading on their own in the subject and are not encouraged to create their own mathematics. In addition, students do not usually pose questions or engage in problem-solving activities in order to attain both conceptual and procedural understanding of concepts taught them. Even though this is what is valued globally today in learning mathematics, in Ghana students are made to copy the algorithms teachers demonstrate and write on chalkboards and simply memorize and regurgitate them during tests or examination. The strategy employed in this method is limited to performing routine exercises. Consequently, teaching and learning mathematics in the traditional methods do not motivate students; neither does it target the development of conceptual understanding or support student-centred learning. Students are not involved in tackling problems with a number of possible alternative solutions. Hence, Mehdiyev (2009) concluded that, the school curriculum in Azerbaijan on the whole lacks collaborative learning, discussions, the use of technology and the development of conceptual understanding of non-routine problem solving strategies. This conclusion is not different from what is happening in Ghana. This therefore calls for alternative teaching approaches that will enhance students’ conceptual understanding of geometry in particular and mathematics in general. This study therefore explored the effect of IGS (GeoGebra) tool in a geometry lessons. 22 2.4 The Use of Technology in the Teaching of Geometry The integration of Information and Communication Technology (ICT) into teaching and learning entails combination of all technology parts, such as hardware and software, together with each subject-related area of the curriculum to enhance learning. It establishes the connection between subject matter and the real world. This provides opportunities for developing skills for the 21st century, with the potential to transform pedagogical practices, and playing a role in reforming curricula. Additionally, it is considered an essential tool for developing understanding about mathematical concepts (Fitzallen, 2005). Fortunately, the ICT policy for accelerated development for Ghana Education Service and the mathematics curriculum are documents that provide the platform for ICT integration in Ghanaian schools. Increasingly, efforts are being made to integrate ICT into the curriculum and researchers are calling for the adoption of strategies that will make ICT integral part of the teaching and learning processes (Assuah, 2010; Roschelle, Hoadley, Gordin, & Means, 2000). Ward (2003) goes further to claim that there is limited use of ICT in classroom practices. This indicates a need for teachers to gain an understanding of how ICT can be used to enhance students’ thinking and problem-solving skills, rather than just as a publication and research tool. Therefore, there is imperative need to employ interactive ICT tools into the classroom pedagogy. Interactive (dynamic) geometry is an active, explorative geometry carried out with interactive computer software. It enables one to visualize abstract geometrical concepts. Hershkowitz, Dreyfus, Ben-Zvi, Friedlander, Hadas & Resnick (2002) stressed 23 that interactive geometry tools like the Geometer Sketchpad, the Geometric Inventor, GeoGebra and Cabri offer more opportunities to construct and justify geometrical concepts than the pencil and paper settings. According to Hershkowitz, et al (2002), the pencil and paper environment has a limited capacity in introducing a geometrical concept with emphasis on its intrinsic properties. According to Mehdiyev (2009), this insufficient feature of the pencil and paper medium causes the tendency in students to construct a limited concept image. Interactive geometry patches up this insufficiency by providing students with the option of generating practical evidence to progress from particular cases to the general case. In addition, interactive geometry medium plays an essential role in developing the proofs of geometrical conjectures (Hershkowitz, et al. 2002). In an interactive geometry learning environment, proving the validity of geometrical concepts by means of interactive geometry software (IGS) is done, through dragging the relevant points of the constructed objects towards a situation in which they satisfy predefined conditions. IGS enables the design of such activities in which students explore the relevant properties of the geometrical objects in order to construct a more appropriate concept image (Hershkowitz, et al. 2002). Hence, learning geometry in an interactive geometry environment (IGE) can offer students opportunities to construct and manipulate geometrical figures and carry out empirical investigations. These activities are almost impossible in a static geometry environment (Laborde, 2003). According to Laborde, drawing refers to the material entity, while figuring refers to a theoretical object (Hershkowitz, et al. 2002). She made a clear distinction between drawing and figuring for the following reasons: 24 1. Some properties of a drawing can be irrelevant. For example, if a rhombus has been drawn as an instance of a parallelogram, then the equality of the sides is irrelevant. 2. The elements of the figure have a variability that is absent in the drawing. For example, a parallelogram has many drawings; some of them are squares, some of them are rhombuses, and some of them are rectangles. 3. A single drawing may represent different figures. For example, a drawing of a square might represent a square, a rectangle, a rhombus, a parallelogram, or a quadrilateral. Therefore, it is not possible to provide an adequate representation for all properties simultaneously in a pencil and paper environment. However, this is an easier task in an IGE. An IGE has a variety of tools that enable students to construct geometrical objects and visualized geometrical conjectures at a perceptive level. Also, the tools offer flexibility of the objects being constructed. The flexibility of the geometrical construction grants students the opportunity to justify, validate or refute conjectures as well as build conjectures based on empirical evidence. Thus, an IGS is a learning medium which ensures a new learning setting and new interactions, because it includes unique features that support the learning of geometry. An IGS offers tools to manipulate objects in a physical sense, and subsequently, these tools turn into psychological artefacts. A number of researchers (Arzarello, Paola, & Robutti, 2002; Gawlick, 2002 & 2005) focused on the dragging modalities (different ways of dragging) provided by the tool. Clearly, this new learning medium provides tangible experiences to learners through 25 physical interactions. This physical interaction supports the development of cognition Vygotsky (1978). Also Falcade, Laborde, and Mariotti, 2007 stated that the use of technology in education has the potential of helping students’ internalize geometric concepts. Based on the views discussed, some activities were designed for a geometry class in this study through the use of IGS (GeoGebra) in order to explore and construct the geometrical concepts. These activities are based on playing with the appropriate Applets designed by the researcher using GeoGebra. The Applet guided students to investigate the relevant properties of the geometrical objects in order to construct appropriate concept image and procedures. According to Mehdiyev (2009) the underlying point teaching the intended geometrical concepts in an IGE is based on facilitating externalization of the representations of the concepts. Usually, such representations are implicitly described in the geometry textbook call for students to use mental performances. However, the dragging on the computer screen can facilitate the externalization of implicit ideas which become visible phenomena that can be shared and discussed (Zbiek, Heid, Blume & Dick, 2007). Figure 2.2 describes the interactions between externalization and internalization as a cyclic process. This interaction only becomes possible within a social interaction model. From Figure 2.2, the externalization of the representations of geometrical concepts provides a medium for socialization which in turn ensures the internalization of the geometrical concepts. An IGS tool is assumed to support this cyclic process (Mehdiyev, 2009). 26 Externalization Socialization Internalization Figure 2. 2: The Interactions between Externalization and Internalization as a cyclic process According to Hershkowitz et, al. (2002) an interactive geometry as a medium enable students alleviate hard psychological experiences that are required for the geometrical constructions and manipulations with pencil and paper. This process is facilitated by the interactive features of the geometry software. Furthermore, the external experience supports the required internal processes needed for theoretical knowledge construction. Therefore, it is assumed that the successful integration of technology into mathematics education has the potential to bring about positive changes in the teaching and learning processes, in particular if combined with student-centred learning activities. In addition students have the opportunity to engage in mutual communications and interactions (Gilmore & Halcomb, 2004). Nevertheless, the successful incorporation of IGS in the teaching and learning of geometry may differ, depending on the social and cultural domain. Therefore, in this study, the researcher explored the effect of IGS in a geometry class under a Social and Cultural background differed from earlier research. The general objective of this study is 27 to investigate the practical changes that an IGS brings to students’ learning experiences in geometry lessons. The focus of this research is therefore on the effect of IGS on: (1) conceptual understanding, (2) motivation and (3) student-centred learning. The researcher is convinced that these aspects do not stand alone independent. Rather, they are believed to be interrelated and the consequences of change made in one aspect may affect the other. Hence, it is assumed that the IGS-based learning medium should provide support to each of these aspects. 2.4.1 Effects of ICT on the Teaching and learning of Plane Geometry The use of IGS in teaching and learning of geometry in particular and mathematics in general is believed to have a positive effect on students learning outcomes Mehdiyev, (2009). Swan, Kratcoski, Schenker, & Van (2010), investigated whether the use of interactive whiteboards affects students’ achievement. The findings indicated a slightly higher performance among students in the interactive Whiteboard group as compared to those not using the Whiteboard. The results revealed that interactive whiteboards in English language arts and Mathematics lessons improved student learning in those areas as measured by student scores on state achievement tests. Also, Hershkowitz et al. 2002 in study on the effect of IGS on students’ conception of mathematical concepts concluded that students had opportunity to develop a deeper understanding on geometrical concepts and problem solving strategies informally. Furthermore, Choi-Koh (2003) investigated the effect of using technology (GSP, Excel, Graphing calculators) to support trigonometry instruction at Middle and High School. These studies showed that the graphing calculator, Excel spreadsheets, and the Geometer 28 Sketchpad all have a very positive impact on exploring trigonometric functions. Traditional pre-calculus textbooks (Cohen, 2005) offer a sequence of instructions which delay engaging students in discovering for themselves the numerous applications of trigonometry to real world examples. However, Mehdiyev (2009) conducted a research to establish the effect of IGS on students’ conceptual understanding of geometry in Azerbaijan. She concluded that the use of the software did have any positive effect on students learning outcomes. 2. 5 The Conceptual Understanding of Mathematics Concepts Conceptual understanding of mathematics concepts refers to an integrated and functional grasp of mathematical ideas. Conceptual learning on the other hand involves students to be creative thinkers, apply content knowledge in multiple situations and develop effective learning methods. Students are expected to be meta-cognitively aware of their own learning habits and deepen their knowledge by utilizing their curiosity to create conjectures and seek out justifications for mathematical thought. The effect of IGS on Conceptual understanding is a key component of this study. According to Kilpatrick, Swafford and Findell (2001), conceptual understanding is regarded as key to grasping mathematical concepts and ideas. Conceptual understanding is an important strand in mathematical proficiency development. Students who have good conceptual understanding are able to make the interconnection between mathematical concepts and their representations. Conceptual understandings thereby provide support for students to develop insights into mathematical procedures, ideas and to competently apply them in solving non-routine mathematical problems. Conceptual understandings also assist 29 students to acquire better competencies in formulating alternative solution methods and connecting these methods with each other. Students with good conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kind of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. In addition, Conceptual understanding supports retention. Based on the elaborated definition of conceptual understanding as a mathematical proficiency strand, the researcher assumed that an IGS learning medium provides insightful experience for students in learning geometry concepts. Because students develop conceptual understanding of geometric ideas and procedures, they are expected to know ways these geometric procedures are deduced and how to apply them in solving geometry problems. According to Kilpatrick et, al. (2001), conceptual understanding provides support to develop procedural fluency. This refers to knowledge of procedures, knowledge of when and how to use them appropriately and competence in performing them accurately and flexibly. Procedural fluency alone, in the researchers’ thinking, is not desirable, nor does it precede the conceptual understanding. Students without good conceptual understanding may get better at performing procedures based on rote memorization in solving routine problems. However, when such a student comes across none-routine problems involving strategic skill she/he might be found wanting. Hence, conceptual understanding is supposed to pave a way for developing problem solving strategies or strategic competencies as mentioned in the literature. For the purpose of this research it was assumed that a student with good conceptual understanding would also 30 achieve high scores when assessed on those concepts taught. Consequently, conceptual understanding and achievement are used interchangeably in this study to mean high mathematical achievement due to sound conceptual understanding. 2.5.1 Effects of IGS on students Conceptual Understanding of Mathematics. In this age of modern technology, many learning aid tools are used to facilitate the learning process in the classroom. A study conducted by Abdul-Halim and Effandi (2011) on students’ perceptions toward the Van Hiele’s phases of learning geometry using geometer sketchpad software (GSP) indicated positive effects on students conceptual understanding. The results showed that students had better understanding of the lessons taught and their confidence was boosted to learn geometry. Hence, the use of GSP software is encouraged as an alternative strategy in learning geometry to increase the students’ understanding and geometric thinking. Also, Zittle (2004) explored the effects of whiteboard lessons on Native American elementary students learning of geometry. Zittle compared students whose teachers used interactive whiteboards with students whose teachers did not. He found statistically significant differences between the groups with the interactive white board group obtaining an average gain score of 20.76, while the Control group got 11.48. In addition, Noraini (2007) conducted a research on the effect of geometer sketchpad on Malaysian students’ mathematical achievement and the Van Hiele’s Geometric thinking level. The results showed that the pre-achievement test did not indicate a significant difference between the Control and Experimental groups at P< 0.05. However in the Post-test the Control group exhibited a mean performance of 13.08, while 31 the Experimental group was 19.65. The computed test statistic between the Control and Experimental groups was 2.78 with p=0.02. Consequently, the result showed a significant difference in mathematical achievement between the Control and Experimental groups. The significant difference in geometry achievement of Experimental groups as compared to the Control groups indicated that the geometer’s sketchpad has promising implications and the potential of using geometer sketchpad in teaching geometry at the secondary school level is imperative. The results of the study is consistent with Huitt (2001), who reported that the addition of dynamic geometry software in geometric constructions has increased her students’ interest in geometry as well as enhanced their understanding. 2.6 Motivation and Students Learning of Geometry Motivation is an internal state or condition (sometimes described as a need, desire, or want) that serves to activate or energize behaviour and give it direction (Kleinginna & Kleinginna, 1981). Motivation is one aspects of mathematics which is important to students and teachers because of its ability to affect learning outcomes. Motivation is linked with the emotion which is manifested either in positive (interest, joy) or negative (frustration, anger) emotions depending whether the situation is in line with motivation or not (Hannula, 2006). It is assumed that, in this research, students will express positive emotions when working with computers in the classes. The value of these positive emotions is also added by employing student-centred group workings. Based on the notion that the computer support student-centred instructions, the researcher assumed that students should be stimulated to interact with each other for discussions and sharing of ideas. 32 The motivation to learn is defined as a student’s tendency to find academic activities significant and worthwhile and to try to get the intended learning benefits from them (Huitt, 2001). Heafner (2004) uses the three factors of the expectancy-value model to show how the implementation of technology positively impacted students’ motivation to learn. The study engaged twenty-five high school government students, who worked in a computer laboratory to create PowerPoint slides as a political campaign advertisement for their state’s senatorial race. Students were able to search the Internet and be as creative as possible with their slides, incorporating sound bytes, video clips, pictures, text and animation. Students were excited about learning and displayed pride in the PowerPoint slides they created. Due to their familiarity with technology, students suddenly felt confident in their ability to accomplish the project and enjoyed working on a task that they viewed as challenging and engaging. All students reported enjoyment in the task because technology made their work easier and more fun to do. Based on interviews, observations, field notes and work samples, the study revealed high levels of motivation surrounding this technology project. This was in contrast to a Control group in a traditional classroom environment, where students avoided the task because they viewed the task as boring. This study illustrates the way in which technology can change the value of a task, increase student self-efficacy, and improve student worth. In line with this study, the researcher deployed GeoGebra into geometry lesson to explore its effects on students’ motivation to learn. 33 2. 7 Student-centred Learning The term student-centred learning (SCL) is widely used in the teaching and learning literature. Many terms have been linked with student-centred learning, such as flexible learning (Taylor, 2000), experiential learning (Burnard, 1999), self-directed learning and therefore the slightly over used term ‘student-centred learning’ can mean different things to different people. According to Kember (1997), for student-centred learning, knowledge is constructed by students and that the teacher is a facilitator of the learning rather than a presenter of information. Student-centred learning describes ways of thinking about learning and teaching that emphasize student responsibility for such activities as planning learning, interacting with teachers and other students, researching, and assessing learning. Rogers (1983) identified an important pre-condition for studentcentred learning as the need for a leader or person. One, who is perceived as an authority in the situation, is sufficiently secured within himself and in his relationship to others that experience an essential trust in the capacity of others to think for themselves and learn for themselves. Student-centred learning is manifest very strongly in the computer-based cooperative learning activities, when teacher’ interventions are reduced to a minimum level. In an IGE, the belief is that the use of computer technology provided the basis for the accomplishment of student-centred learning. According to Laborde (2001), the incorporation of technology into mathematics education changes the teaching system. All aspects in the classroom, such as the structure of activities and the content to be taught received new shapes. This applies also to the IGS, acting as a mediator between students and content. This mediation affected students’ learning experience, in particular the interactions and the communication. Furthermore, students interact with the tools of the 34 IGS and their activities result in representations to which they have to react. That is when interacting with IGS; students receive feedback on the basis of which they made new interactions. Hence, this interaction-feedback cycle of working is assumed to provide support for student-centred learning activities. Also, according to Gilmore and Halcomb (2004) it is unlikely to think that the use of technology based student-centred activities alone will enhance performance and collaboration among students. Rather, in order for technological integration in the classroom to be effective, the emphasis on instructional design must be increased. For this, the design of Worksheets and Applets as instructional materials were used. 2.7.1 The Effects of IGS on Motivation and Student-centred Learning. Current theories of learning emphasized the importance of actively engaging children in the learning process (Bransford, & Cocking, 1999) and recently there have been a variety of technologies designed to support active engagement in learning. One such technology is the interactive white board. Interactive white boards allow teachers and students to interact with content projected from a computer screen onto a white board surface. The combined effects of the process enhance students’ understanding and foster’s student-centred learning. Studies have shown that both teachers and students like the technology (Beeland, 2002; Kennewell & Morgan, 2003; Hall & Higgins, 2005; Smith, Higgins, Wall & Miller, 2005) and that students are more engaged and motivated to learn when IGS (whiteboards, GeoGebra etc) are employed (Smith, Hardman & Higgins, 2006; Painter, Whiting & Wolters, 2005; Miller, Glover & Averis, 2004, 2005; LeDuff, 2004; Beeland, 2002). In addition, research studies have shown that the use of 35 IGS shifted instruction from presentation to interaction and students’ focus away from teachers and onto content, making interactive Software’s lessons more student-centered than traditional ones (Miller, Glover & Averis, 2003, 2004; Painter, Whiting & Wolters, 2005; Cuthell, 2005). Finally, a research that was conducted by Rosnaini, Mohd, and Ismail (2009) on development and evaluation of a computer aided instructions (CAI) GReflect, on Students’ achievement and motivation in learning mathematics in Malaysia. The results from t-test showed that there was a significant difference in the mean scores obtained (t (67) = 10.162, p≤0.05). The treatment group was found to perform better in the test compared to the Control group. In terms of motivation, results from the questionnaire showed that the students from the treatment group were highly motivated in learning mathematics. The purpose of this study was to use GeoGebra to explore its efficacy on conceptual understanding and student-centred learning. It was used to design Applets and Worksheets as a guide which enabled students explored geometric concepts in series of group activities. The activities were meant to create a social environment to enable students engage in an intensive discussions and interaction as a means of building and enhancing cognitive development. 36 CHAPTER THREE METHODOLOGY 3.0 Overview The study sought to find out the effect of the use of IGS on SHS students’ conceptual understanding of, and their motivation to learn, plane geometry. It also investigated ways in which IGS provides support for student-centred learning in a geometry class. In pursuance of the purposes stated above, the following research questions were formulated to guide the study: 1. To what extent does the use of IGS affect SHS students’ conceptual understanding of geometry? 2. How does the use of IGS motivate SHS students to learn geometry? 3. In what ways do IGS support student-centred learning in a geometry class? In order to answer the research question 1, the following null and alternative hypotheses were formulated. HO: There is no difference in the understanding of geometry between the Control and Experimental groups. H1: There is significant difference in the understanding of geometry between the Control and Experimental groups. This chapter describes the research process under the following headings: the research design, population, sample and sampling procedures, research instruments, data collection procedure, in-class activities and data analysis. 37 3.1 Research Design The general approach chosen for this study was a sequential explorative mixed method, which employed quasi-experimental design as a strategy of enquiry. A quasiexperimental study takes place in a real life setting as opposed to only a laboratory setting. According to Vanderstoep and Johnson (2009), quasi-experiment is an empirical study used to estimate the causal impact of an intervention on its target population. The quasi-experimental design chosen for this study is the Pretest-Posttest non-equivalent group strategy. The purpose of this strategy was to use qualitative data and results to assist in explaining and assigning reasons for quantitative findings. Morgan, (1998) suggested that the mixed method design is appropriate to use when testing elements of an emergent theory resulting from the qualitative phase and that it could also be used to generalize qualitative findings to different samples. Golafshani (2003) described that, qualitative research uses a naturalistic approach that seeks to understand phenomena in context-specific settings such as real world setting in which the researcher does not attempt to manipulate the phenomenon of interest but only try to unveil the ultimate truth. Qualitative methods were used in the study in order to provide a more profound understanding of the effect of IGS on the variables (conceptual understanding of geometry, motivation and student-centred learning) been investigated. Quantitative research on the other hand, utilises experimental methods and quantitative measures to test hypotheses and generalize the outcomes. It also emphasised the measurement and analysis of causal relationships between variables (McMillan & Schaumacher, 2006; Creswell, 2003). In this study, the quantitative methods (data collection and analysis) were used to establish the relationship between the performance 38 of the students in the Pre-test and Post-test results. The researcher tested the efficacy of a supplementary teaching approach that integrated ICT into the pedagogical discourse in the classroom. The study is aligned with the curriculum topic plane geometry 1 (Angles, Triangles and parallel Lines) and carried out in the same weeks in which the regular lessons were taught. 3.2 Population, Sample and Sampling Procedure The population consisted of Senior High School Students in the Sissala East District Assembly in the Upper West Region of Ghana. Kanton Senior High School was sampled for the study because students in this school are posted from Basic Schools all over the ten regions in Ghana. Kanton Senior School is a government school which operates under the local authority. It is a grade B school rated by Ghana Education Service. Currently, the school has about 1,300 students offering various programmes. The students were purposively sampled. Creswell (2009) remarked that purposive sampling is employed because of the special characteristics of the school in facilitating the purpose of the research. In purposive sampling the unit of the sample are selected not by a random procedure, but they are intentionally picked for the study because of their unique characteristics or because they satisfy certain qualities which are not randomly distributed in the universe, but they are typical or they exhibit most of the characteristics of interest to the study. Kanton Senior High School had the basic facilities that facilitated implementation of the designed activities. It had a new computer laboratory stocked with 50 new computers, three projectors and projector screens. The school also has an ICT centre sponsored by the Action Aid Ghana, with 10 computers connected to the internet 39 at the science resource centre. Additionally, the home economics department at the school had five new computers in their workshop which were all functioning. The key to purposive sampling is that selection is intentional and consistent with the goal of the research. Simple random sampling was the second sampling technique that the researcher employed in this study. This ensured that bias was eliminated while giving equal opportunities to each sample point selected. The sample units in the population were selected by a random process, using a random number generator so that each person in the population had the same probability of been selected for the study. Thus the sample was 75 students of Kanton Senior High School, 43 in the Experimental class and 32 in the Control. This was randomly sampled using the random number generator. Seven students were also randomly sampled to respond to the questionnaires and interview items. 3.3 Research Instruments The instruments used for the data collection were: test, interview guide and questionnaires. The test were made up of 5 items, two questions were on Parallel Lines while three on Triangles and Angles at a point. Ten questions were designed and split into two groups and administered one after the other as Pre-test and Post-test. The test was written before and after the students was taken through some lessons using the GeoGebra software. The interview questions were eight (8) with open ended responses. It covered the three main variables investigated: conceptual understanding, student-centred learning and motivation to learn geometry. The interview was administered to both 40 teachers and students after they were taken through some lesson using the GeoGebra software. The questionnaire on the other hand was used to elicit information on how IGS motivated students to learn and how it supported student-centred learning. This consisted of five scale attitude tests of 15 items, designed to measure student-centred learning and students’ motivation to learn geometry. The questionnaire was divided into two parts; the first part contained 10 questions on student motivation, while the other 5 were on studentcentred learning. The aspect on motivation was a Likert attitude test scored from 5 =Strongly Agree, 4 =Agree, 3 =Neutral, 2 =Disagree and 1 =Strongly Disagree. Where the statement was negative, the interpretation was done by reversing the response to positive thus making the Comparism uniform. Five questions solicited students’ views on student-centred learning activities. On a scale of 1-4, the students rated the IGS lessons on some indicators of student-centred-learning lessons. This was administered after the students were taken through some lesson using the GeoGebra. 3.3.1 Pre-test and Post tests. A Pre-test and Post-test were conducted before and after implementing the designed activities (Appendix I). The differences between the results of the Pre-test and Post-test were analyzed to measure the changes made in students’ conceptual understanding of the geometry concepts. The analysis and comparison were then used in answering research question 1. 3.3.2 Interview Guide for Teachers After implementing the design, an interview was arranged with the teachers to obtain their reflections on the effect the IGS lessons (Appendix III). This data was used to 41 evaluate from teachers’ point of view, the effect of IGS on students’ motivation and student-centred learning activities. Therefore, this data was used to answer research questions 2 and 3. 3.3.3 Interview Guide for Students With support from the mathematics teachers, some students were interviewed after the implementation of the design (Appendix III). This source of data was used to analyze students’ reflections and views on the role of IGS in supporting their motivation, student-centred learning activities. Their answers were then compared with the other data sources and this enabled the researcher made valid conclusions. Therefore, these data were helpful in answering the research questions 2 and 3. 3.3.4 Questionnaires After implementing the design, students responded to a questionnaire (Appendix II). In order to get extensive and reliable answers, some of the questions were asked in traverse ways. The data from the questionnaire was then used to evaluate students’ views on the role of IGS in supporting their motivation and student-centred activities. Thus, this data was helpful in answering the research questions 2 and 3. 3.4 Intervention Tools and Procedures Teaching with the use of IGS was the main part of the intervention; it covered four lessons each of which covered a 90 minute period. There are numerous interactive software (IGS) products available which serves the purpose of increasing students’ interactions in geometry lessons. Interactive geometry softwares that are available in the open market includes: Cabri 2D & 3D, WinGeom, Sketchpad, Euclide, Cinderella, and 42 GeoGebra. However, softwares like WinGeom and GeoGebra are free and can be downloaded from their official websites. Notwithstanding some common commands like drawing and dragging, the tools have differences. For instance, unlike other tools, GeoGebra provides the option of making interactive Applets. Furthermore, it has controllable dynamic features and free and technical accessibility, GeoGebra 3.2.46.0 was used for the study. 3.4.1 GeoGebra The software is technically available through the internet and can be installed independent of user platform. GeoGebra requires Java 6.250.6 which was also downloaded from their official website (www.GeoGebra.com). GeoGebra has a good option of dynamic manipulations with availability of a slider motion tool. This tool enabled users to manually manipulate the drawn geometric objects and to monitor the changes interactively. The software unites algebra, calculus and geometry concepts. The possibility of making algebraic and geometric representations in the same medium is the advantage of this tool. In addition, this software package has an option of Applet construction which allows the author to determine the deepness of the interactivity of users in design time. An Applet is a Java application that uses the client's web browser to provide a user interface. The researcher constructed Applets with dynamic and visual representations these became explorative sources to students. The interface of GeoGebra and its construction tools are shown in Figure 3.1. 43 GeoGebra Interface Construction tools: Menu: Figure 3. 1: 3.4.2 GeoGebra Interface and Construction Tools. Design of Instructional Materials The researcher designed Applets with GeoGebra, which guided students to explore the intended concepts. Worked sheets were also prepared to guide and facilitate students’ activities at the group work. GeoGebra was used to develop Applets representing the geometrical concepts taught in this study. Four Applets were designed to represent the intended geometrical concepts. The first and Second Applets were on properties of Angle, the third was on Parallel Lines, while the fourth one was on Triangles. The design of all Applets provided the students with an option to transform the geometrical constructions. In the Applets, dragging a given point or variable on a slide bar had certain consequences for the shape of the geometrical constructions. By dragging the points the students generated their own data as they observed and recorded the consequences of the different values of the variable for the geometrical constructions. 44 In order to support students’ work with the Applets, four Worksheets were developed (Appendix VI). Each of the Worksheets contained Tables, and questions given based on the Applets. The tables were then filled with the appropriate values describing the different states of the geometrical constructions in the Applets. The students generated these values by measuring angles through the slide bars. The tasks and questions guided the students to develop a line of reasoning based on the recorded values in the tables. 3.4.3 Design of the Intervention The intervention took place between the months of September and October, 2011. The targeted group for this research intervention was Kanton Senior High School form three students at Tumu in the Upper West Region. The students were between the ages of 17 and 24. The researcher taught sequentially the regular geometry lessons both in the classroom and at the computer laboratory. The intervention was preceded by a Pre-test, as a baseline (Appendix I: Pre- and Post-tests). After the Pre-test, the intended classroom and laboratory activities were implemented. One week was used for the traditional classroom activities and two weeks for the laboratory lessons. The students were introduced to the Interactive Geometry Software to obtain knowledge and skills in construction and manipulation of Angles, Triangles and Parallel Lines. Thereafter, in four sessions students were introduced to the intended geometry concepts (Appendix IV: Lesson plans). The selection of the geometry concepts was consistent with the schedule and content of the school curriculum. The concepts taught were developed into a series of activities based on the Worksheets and Applets that the researcher designed as instructional materials for a classroom and laboratory intervention. The Worksheets were used to help the students in investigating 45 the Applets developed through the use of the GeoGebra. During each session the students were grouped and guided to work with the electronic Applets. At the end of the intervention, a Post-test, Questionnaire and Interviews were administered to both students and teachers to evaluate the effect of the intervention on students’ motivation and student-centred learning. 3.5 Piloting the Instruments Validity and Reliability are appropriate concepts for attaining rigor in qualitative and quantitative research works. Validity refers to the extent to which the research conclusions are authentic (Creswell, 2009). It is a demonstration that a particular research instrument measures what it purports to measure (Creswell, 2007). Validity is a measure of the extent to which research conclusions effectively represent empirical reality or whether constructs devised by researchers accurately represent or measure categories of human experience (LeCompte & Preissle, 1993). The validity measures taken in this study were based on these conceptions of the notion of validity. To validate the research instruments (test items) the researcher consulted the mathematics curriculum as well as the Textbooks. This was to help the researcher develop the instruments in line with the curriculum requirement. Thus, Textbook questions on plane geometry were used for this study. Zeller (1988) stated that establishing content validity involves specifying the domain of content. After constructing the test items, the researcher consulted teachers and lecturers to crosscheck and approve them. Durrheim (1999) suggests that the researcher approach others in the academic community to check the appropriateness of his or her measurement tools. After 46 administering the tests and the marks scored, the researcher returned to the school and discussed with students their scores. This made students to be convinced that the scores accurately represented their abilities. The process of validity just described is what Lincoln and Guba (1985) referred to as member checking, a process that has the advantage of putting the respondent on record as having said or done certain things and having agreed to the correctness of the investigator’s records of them. Joppe, (2006) defined reliability as: the extent to which results are consistent over time and an accurate representation of the total population under study is reliability, if the results of a study can be reproduced under a similar methodology, then the research instrument is considered to be reliable. Reliability refers to the extent to which a measurement instrument, a questionnaire, a test yields the same results on repeated applications (Durrheim, 1999). It means the degree of dependability of a measurement instrument. In this study, the split-half method was used to check the reliability of the instruments, because it was a “more efficient way of testing reliability” and it was less time consuming (Durrheim, & Wassenaar, 1999 p90). The spilt-half method requires the construction of a single test consisting of a number of 10 items. These items were then divided or split into two parallel halves (usually, making use of the even-odd item criterion). These test items were then piloted at Swedru Senior High School using the form 3 students. According to Lancaster, Dodd and Williamson (2004) Piloting the research instruments helps to identify inconsistencies in the research instruments (test items, interview guide and questionnaire) and gives the opportunity to the researcher to redesign and predict possible problems one may encounter in using the instruments. Students’ scores from the split half were then correlated using the Cronbach alpha 47 formula. The coefficient of reliability calculated was α = 0.798 which signified that the degree of reliability of the test items was strong. The test items used for piloting assisted the researcher to modify them for better understanding. The researcher realized that questions students could not answer were due to their low conceptual knowledge but not due to ambiguity. Also the researcher was able to ascertain the appropriate time period for both the IGS lessons and the administration of test items. 3.7 Intervention and Data Collection Procedures For the research intervention at Kanton Senior High school, the District Directorate of Education at Tumu was officially informed and the school administration duly requested to release the school laboratory for teaching the intended geometry lessons. The laboratory had 50 new computers, three projectors and a screen for the projector, which were all functioning. All the computers were brand new but not connected to the internet. The GeoGebra software was installed in all the 50 computers and the Applets were projected over the screen for students to see and draw similar ones. In addition, the Applets and the Worksheets were printed out which facilitated the group activity. The computers were placed on five long benches along the walls with 10 computers on each bench connected to one system unit. The space and the number of computers were enough for the Experimental group to work in as individuals and in groups. The students were put into groups of four, and each group was assigned to a computer. This setup was conducive for group discussions and interactions in the laboratory. The participants in the study were the SHS form three students of Kanton Senior School, numbered 43, most of them were boys. From the day of the Pre-test until the day 48 of the interview all the students participated in the sessions. The researcher took advantage of the early days of the term where most teachers were still revising test papers of the previous terms’ work. Table 3. 1: Sequence of Data Collection Date 23/09/2011 Type of Activity Pre-test Subject Matter Angles triangles and Parallel lines Periods 1:00 Hour 27/09/2011 Lesson Introduction to GeoGebra 90 Minutes 29/09/2011 Lesson 90 Minutes 5/10/2011 Lesson Properties of Angles Properties Parallel lines Triangles and their properties 7/10/2011 Post-test 14/10/2011 Administration of the interview and questionnaire 3.7.1 Angles, Triangles and Parallel lines Interview and questionnaire 90 Minutes 1:00 Hour Lesson: Introduction to GeoGebra Software Lessons The researcher introduced both students and teachers to the software in the laboratory. This was preceded by a practical demonstration on how the software works over a projector. The students followed the process step by step at their desktop computers. The researcher explained practically how the software is use to learn geometry, especially on Angles, Polygons and Parallel Lines. From the beginning of the lesson, the researcher observed that some of the students were sufficiently able to work with the software, because of their familiarity with the computer. The researcher coached students how to use the software to draw Triangles, Rectangles, Circles, Regular polygons as well as to select and manipulate plane figures. In addition, they were taught to use the 49 slider motion for the manipulation of coherent geometric parts and objects. Some of the students were quick in performing the steps and in those cases, the researcher asked those students to help others. During the session, the researcher walked along the groups facilitating the exploration done by the students and the teachers. The researcher observed that the students were working enthusiastically with the software. At this point the students were asked to draw various Polygons, Angles and Parallel Lines with given specifications. This activity seemed difficult; however, the researcher gave them time to work collaboratively. After a while most of the students managed to construct the plane figures with the motion slider. The researcher then gave the students and teachers opportunity to explore the software and draw any Polygon, Angles and parallel lines of their choice. Most students took this opportunity and drew very wonderful Plane figures with the software. In conclusion, the researcher asked the students to do a similar construction with regular polygons as homework. The researcher was impressed by the positive change in students’ attitude towards the software. They were vividly expressing that, learning geometry through the GeoGebra software started to make sense and very interesting to them. Figure 3.2 shows students on one-on-one basis exploring the GeoGebra in the computer laboratory during the IGS lessons. 50 Figure 3. 2: A cross section of Students Exploring the Geogebra in the Laboratory 3.7.2 Lesson on Angles and their Properties Students were put into groups of four and each group nominated a leader who collated their responses and presented it to the researcher for marking. The import of the group work was to create a platform for the students to discuss and interact among themselves freely before presenting their ideas for marking. All the groups took up their places at their computer desks, numbering A to J. Even though there were more than four computers for a group, one computer was booted to enable the group members concentrate on the group work. The researcher reviewed students’ previous knowledge on the introduction to the GeoGebra software. Some students demonstrated on the computer connected to the projector on how to construct Triangles and Parallel lines to the rest of the class members. The researcher then introduced the new lesson on Angles and their properties. Students were asked questions on the definition of an Angle and the properties. A few students could remember them from their previous lessons, though not in detail. For example, the type of angles was more difficult for them to remember than 51 any of the other mentioned concepts. The researcher reviewed students’ previous knowledge and found that the students did not have a good conceptual understanding of Angles at a point, although they recognized degree as the unit for measuring angles. For example many of the students could not describe two or more angles at a point. The researcher then made each group to launch the GeoGebra programme installed in their computers for the lesson to begin. Each group was given Worksheets and Applets as a guide to work with. Consequently the researcher projected the first Applet (Appendix V) on the screen for them to discuss among themselves and to construct similar ones. A brief explanation of what to do with the first work sheet and Applet A was done by the researcher. Students were to explore Applet 1 and find answers needed to fill-in the Table given in Worksheet 1 (Appendix VI). The first question on Worksheet 1, demanded students to explore and draw four different types of angles as in Applet A and measure the angles enclosed. In addition the students were asked to find the technical names of those angles they had drawn using the software. During the lesson the students were encouraged to interact with each other and with the teacher when needed. The responses developed by the groups were organised and described in Table 1 in Appendix II From Table 1 in Appendix II, it was obvious that most of the groups had similar experiences as demonstrated from the work students presented. All groups recorded the angles and names of the figures they had drawn with aid of the Geogebra. The responses showed that interacting with one another has enhanced their conceptual understanding in the concepts explored. Before the lesson, the students could not articulate their views properly on the concepts explored. The interaction gave the students opportunity to cross 52 fertilize ideas at the group level before presenting them for marking. The students were also given a second task on Worksheet 2 (Appendix VI). In this activity students were asked to explore Applet B (Appendix V), measure the angles and consequently identify the technical names of those angles enclosed. The responses from Worksheet 2 and Applet B are then presented as Table 2 in Appendix II. Students investigated the relationship between the angles enclosed in the diagrams and the technical names derived. All the students gave appropriate answers except that none of the groups came out with the expected name for Figure 4 (Angles at a Point). Even though the answer students gave was not wrong, the researcher expected another name which was synonymous to what they all mentioned. The responses showed that students had similar understanding of the concepts explored. The whole class then discussed common challenges students faced during the lesson delivery. This last activity brought the lesson to an end. The researcher then encouraged students and teachers alike to visit the computer laboratory to explore the software more during their free periods. 3.7. 3 Lesson on Properties of Parallel Lines. The researcher reviewed students’ previous knowledge on Angles and their properties. Students were asked questions on the definition of Parallel Lines. A few students could remember them from their previous lessons, though not in details. For example, the transversal was more difficult for them to remember. The review of students’ previous knowledge showed that students did not have a good conceptual understanding of the properties of parallel lines. The researcher then made each group to launch the GeoGebra programme to begin the lesson. Each group was given Worksheets and Applets as a guide. 53 Applet C (Appendix V) was projected on the screen for students to discuss and draw similar ones. The groups were asked to explore Applet C and find answers needed to complete Worksheet 3 (Appendix VI). During the lesson students were allowed to interact and discuss with each other and with the teacher where necessary. The students were asked to find the relationship that existed between the angles made by the transversal. The responses developed by the groups were presented as Table 3 in Appendix II Students investigated the relationship between the angles enclosed by the transversal and the Parallel Lines. This investigations created a platform that enabled them understand the concepts been discovered. The students grouped the Angles that fell in line with concepts such as: Corresponding angles, Supplementary angles, Alternate angles, Interior angles etc. All the groups gave appropriate answers except, group 3H B and 3H G. Their angles were not Corresponding and Alternating and did not sum up to 3600 at each of the points. The students in these groups could not conclusively establish some of the relationship that existed amongst the angles. However majority of the groups did not have that problem, but were able to link up relationship correctly. The responses of this activity were gathered and presented as Table 4 on the relationship between the angles measured in Appendix II. The researcher then led the whole to discuss issues that were challenging during the exploration of the software. This last activity brought the lesson to an end. 3.7.4 Lesson on Triangles The researcher reviewed students’ previous knowledge and discovered that the students did not have the conceptual understanding of Pythagorean triples. Students were 54 asked questions on the properties of Triangles. The researcher introduced the new lesson on Triangles and their properties. The researcher then made each group to launch the GeoGebra programme installed in their computers for the lesson to begin. Each group was given Worksheets and Applets as a guide. Applet D (Appendix V) was projected on the screen for students to explore and discuss among themselves before drawing similar ones. The researcher then gave a brief explanation of what to do with the Applet D. Students were to explore Applet D and find answers that completed the Table given in the worksheet 5 (Appendix VI). During the lesson the students were allowed to communicate with each other and with the teacher when needed. The task in Worksheet 5 demanded that students explore Applet D and then draw four different types of Triangles as in applet D and measure the angles enclosed in the Triangles. In addition the students also found the technical names of those Triangles drawn using the software. The responses developed by the groups were presented in Table 5 in Appendix II. 3.7.5 Summary of the Lessons Taught In all the lessons the learning materials (Applets, Worksheets) were very helpful in guiding the group activities and the interactions among students. The lesson units which were well planned and rehearsed by the researcher where tailored for two periods a day for three days. All the lessons were held during the last two periods for each day as a result there was no need extending the time period. Furthermore, whole class discussions were held at the end of each lesson to create a platform for students to discuss and share ideas learnt during the IGS lessons. During these discussions, the researcher asked questions that made the students concentrate on building their conceptual understanding (if they could relate the angles generated by the transversal in the case of parallel line). Students 55 also shared their learning experiences with one another. Groups that had problems manipulating any part of the software or difficulty completing the worksheet brought them out for discussion. Everybody went home fully informed about the concepts learnt for that day. The researcher assumed two reasons for this overwhelming influence of GeoGebra in teaching geometry. The first reason was that most of the students were already exposed to computers, and were conversant with manipulating software like MS Excel and MS word. The students were not scared of the software and therefore were accustomed to working in such environment. The second was that, learning geometric concepts, in a Interactive Geometry Environment supported student-centred learning. The use of Worksheets and Applets were designed and developed more coherently and took into consideration the prior knowledge of the students. In addition, the teaching units for the intervention were intentionally limited that made students took charge of their own learning. 3.7 Data Analysis This study is set out to determine the effect of IGS on students’ understanding of and motivation to learn geometry. The study was a sequential mixed method which employed quasi-experimental as strategy of enquiry. It generated both quantitative and qualitative data. Therefore, Statistical Package for Social Sciences (SPSS) was used for the analysis of the data. The data from both the quantitative (questionnaires and test scores) and qualitative (Interviews) sources were coded and keyed into the SPSS for statistical analysis. For the interviews, meaning coding was used to organize the data into 56 number of respondents holding a particular view and their percentages. According to Charmaz (2005), thorough meaning coding of the material is very important and that codes are: immediate, short and define the action or experience described by the interviewee. Data-driven coding was used, because the researcher sets out without codes but only developed them through readings of the materials. The researcher also employed descriptive data analysis in an attempt to understand, interpret and describe the experiences of the research participants. In specific terms, various statistics such as frequency distribution, measures of central tendency, and correlation coefficients were used to analyse, describe and compare separate sets of data in this study. The Independent-samples t-test statistical procedure at 95% confidence level was used to compare students’ conceptual understanding in the geometry. This is because the researcher was interested in using the sample means as the basis of Comparism between the Control and the Experimental groups. The null hypothesis was that there is no significant difference between the conceptual understanding of students’ in the Pre-test and Post-test. This generated information that enabled the researcher to evaluate the effectiveness of the intervention. The researcher then compared the overall performance between the Experimental and Control groups, using Percentages. This was followed up with a qualitative description of the students working process on how students solved the test items. This generated qualitative information that evaluated the development of students’ conceptual understanding in geometry concepts explored. The researcher also did a within and between group analysis and compare students’ performances thereby eliciting the effects of the IGS on conceptual understanding of geometry. 57 In order to make the process of qualitative analysis easier, the findings from different data sources were tabulated in appropriate Tables. In the analysis, relevant data from the different sources were then cross-referenced and appropriately combined in developing the answers to the research questions. The analysis of the data from the Questionnaire and the Interview served as the basis for answering research questions 2 and 3. To answer research question 1, the analysis of the data from the Pre-test, Post-test and Worksheets, were used. The independent- samples t-test was used to establish the extent of the effect of IGS on students’ conceptual understanding. Before using the t-test, the researcher made sure the following assumptions were satisfied; the data were normally distributed (kolmogorov-Smirnov test), It was measured at interval scale and the scores were independent. In addition, the students’ working process was used to explain the development of their conceptual understanding. 58 CHAPTER FOUR RESULTS AND DISCUSSION 4.0 Overview This chapter focuses on the results of the analyses of the data and discussion of the findings. The data were organized and presented using Tables, Figures, Descriptive and Inferential statistics. The results are presented in this chapter under three main subheadings which reflected the research questions stated in Chapter one, as follows: 1 The Effect of IGS on SHS students’ Conceptual Understanding in Plane geometry 2 The Effect of IGS on SHS students’ Motivation to Learn Plane geometry 3 Ways in which IGS provides Support for Student-centred Learning in a geometry class 4.1 The Effect of IGS on SHS students’ Conceptual Understanding of Plane geometry Results from both the qualitative and quantitative data sources were combined to describe and assign reasons for the development of students’ conceptual understanding in plane geometry. As indicated in Chapter 3, students were made to write a test before and after the intervention. Scores from the test were used to help evaluate the level of students’ conceptual understanding in plane geometry. An interview also provided data on the possible reasons that accounted for the enhanced conceptual understanding in the geometry in students’ working process. These data sources then provided the bases for the evaluation of the differences in students’ conceptual understanding in plane geometry. 59 Table 4.1 summarizes the overall development of students’ conceptual understanding in geometry in both the Experimental and Control groups. The test items were five, questions 1 and 3 were on Polygons and properties of Parallel lines while questions 2, 4 and 5 were on Triangles. Seventy five students from the Control and Experimental groups wrote the test. Responses of students who attempted questions but did not get the total marks allotted per test item were described as partially correct; while the responses that exhibited lack of knowledge about the questions were described as completely wrong. However, the responses of students that demonstrated good knowledge and provided the right responses for the items were described as correct. Table 4. 1: Effect of IGS on students’ Conceptual Understanding in both the Control and Experimental Groups. Pre-test Question 1. Find the angle marked x in the polygon 2. If three sides of a triangle are given as x2 –y2, 2xy and (x2 + y2). Show that the triangle is a rightangle triangle. Correct Partially correct Completely Wrong Correct Partially correct Completely Wrong N (%) N (%) N (%) N (%) N (%) N (%) 69(92%) 2(2.7%) 4(5.3%) 71(94.7%) 2(2.65%) 2(2.65%) 39(52%) 9(12%) 27(36%) 52(69.3%) 8(10.7%) 15(20%) 21(28%) 69(92%) 2(2.7%) 4(5.3%) 6(8%) 3. Calculate the reflex angle in a given polygon. (See appendix ) 48(64%) 4. Calculate the length and area of a rectangle with a diagonal. 5. Given two sides of a right angle triangle, calculate the third side which is the radius of circle? Post-test 32(42.7%) 13(17.3%) 30(40%) 45(60%) 9(12%) 21(28%) 43(57.3%) 9(12%) 23(30.7%) 53(70.7%) 7(9.3%) 15(20%) 60 The results indicates that majority of the students 92% (n=69) got question 1 correct, however 64% (n=48) of the students got question 3 correct. In the students’ working process, they could not identify the appropriate principle of Parallel lines to be used in solving the test items. The working process of student “A” are presented in Figure 4.1. Figure 4. 1: The Working Process of Student A’s question 3 in the Pre-test. The working process of student A is similar to 27 other students who got question 3 wrong in the Pre-test. From Figure 4.1, student A could not correctly identify the two Isosceles triangles in the rhombus. Consequently, the student could not recognize that angles: GTH = THG = GYT = TGY which is a primary property of an Isosceles triangle. This is reflected in the way student A solved the question. Even though this student knew that angles on a straight line add up to 1800 the student still got the answer wrong because the preceding values were wrong. 61 On Triangles, 42.7% (n=32) of the students who wrote the test got question 4 correct, 52% (n=39) got question 2 correct, while 57% (n=43) got question 5 correct. The students were not able to associate the test items to the properties of right angle triangle. Even the students who were able to apply the Pythagoras theorem for question 2, they still could not simplify the equations generated {E.g. (x2-y2) + (2xy) = (x2+y2)}. The working process of student B is presented in Figure 4.2. Figure 4. 2: The Working Process of Student B’s question 2 in the Pre-test This working process of student B is also similar to 36 other students who got the question 2 wrong in the Pre-test. From Figure 4.2, student B could not correctly identify the hypotenuse, adjacent and opposite side of the right angle triangle. This is shown in the way the student substituted the sides of the right triangle into the Pythagoras theorem. Although student B stated the Pythagoras theory correctly he still got the expansion wrong. This is an indication that student had knowledge of Pythagoras theory; however, 62 the student could not correctly identify and substitute the correct sides of the right triangle into the theorem. Also student B had a problem with expanding and simplifying the equation generated. Consequently student B got the solution of the question wrong. Student B later told the researcher that: “I actually did not know which side of the right angle triangle was the hypotenuse and which side was the adjacent. When given numbers, I can easily identify the greatest as the hypotenuse. However, this question was not in numbers, so I could not identify which expression was the greatest that is why I got it wrong in the Pre-test”. Generally, students could not relate what they learnt in the classroom to the test items, due to their low conceptual understanding in plane geometry. In the Post-test, the results showed that 92% (n=69) of students got question 3 correct, as compared to 64% (n= 48) in the Pre-test. Also, 94.7% (n=71) of students got question 1 correct, as compared to 92% (n=69) in the Pre-test. This indicates that there was an increased performance of 32% in question 3 and 2.7% in question 1 from the Pretest. There was a general improvement in the way students answered the questions in the Post-test. Students demonstrated a better understanding of the concepts in the way they presented their solutions in the post-test. Figure 4.3 shows the working process of student A, which is similar to 69 other students who got the question 3 correct in the Post-test. 63 Figure 4. 3: The Working Process of Student A’s question 3 in the Post-test From Figure 4.3 student A was able to identify correctly, the Parallel lines in the diagram. In addition, this student used the properties of Parallel lines very appropriately. This is shown in the way the student linked up the positions of the angles and their descriptions to answer the question. This demonstrated that student A really understood what he was doing. The researcher later spoke to the student and this is what he had to say: “I really did not understand the properties of parallel lines very well before writing the Pre-test. But through the use of the IGS I did not only see or hear these properties, I also had the opportunity to measure the angles to confirm. The IGS lessons made the concept of the properties of parallel lines clearer for me; it was easy to compare angles. Therefore, during the post-test I was able to picture these properties and that helped me in answering the questions”. Furthermore on Triangles, 60% (n=45) of the students got question 4 correct, as compared to 42.7% (n=32) in the Pre-test. Also, 69.3% (n=52) of students got question 2 64 correct, as compared to 52% (n=39) in the Pre-test. While 70.7% (n=53) of students got question 5 correct, as compared to 57.3% (n=43) in the Pre-test. This indicates an increase of 12.8% in question 4, 17.3% in question 2 and 13.4% from the Pre-test scores. Generally, students demonstrated good understanding of the properties of right angle triangles. Students were able to state the principles they used and justify the reason for using them. Figure 4.4 shows the working process of student B which is similar to 60 other students who got the question 2 correct in the Post-test. Figure 4. 4: The Working Process of Student B’s question 2 in the Post-test. From Figure 4.4, student B was able to correctly identify the hypotenuse, adjacent and opposite sides of the right angle triangle. The student then substituted the correct sides of the right triangle into the Pythagoras theorem. More importantly, this student was able to expand and simplify the equation generated. This is an indication that the student had 65 developed some conceptual understanding of the right angle triangle. Student A later told the researcher that: “I now realised that it is not all the time that one would be given numbers to work with. In such cases, like the question given, I had to expand the expressions to verify which expression is the greatest, before substituting it into the formula. The lessons with the IGS have actually made these concepts clearer to me than before”. The development of students’ conceptual understanding of geometry between the Experimental and Control groups demonstrated the extent to which the use of IGS affected students’ learning outcomes. From Table 4.2, 90.6% (n=29) of students got question 1 correct in the Control group, as compared to 97% (n=42) in the Experimental group. Also, 87.5% (n=28) from the Control group got question 3 correct, as compared to 48.8% (n=21) in the Experimental group. The results on Triangles indicated that 53.5% (n=23) of students in the Experimental group got question 2 correct, as compared to 50% (n=16) in the Control group. Also, 53.1% (n=17) of the students in the Control group had question 4 correct, as compared to 34.9% (n=15) in the Experimental group. Furthermore, 53.5% (n=23) in the Experimental group got question 5 correct, as compared to 62.5% (n=20) in the Control group. The Control group indicated a better understanding of the concepts in the Pre-test, when compared with the Experimental group. In the Post-test however, 97.7% (n=42) of students in the Experimental group got question 1 correct, as compared to 90 % (n=29) in the Control group. Also, 90.6% (n=29) of the students got question 3 correct in the Control group, as compared to 93% (n=40) in the Experimental group. On Triangles, 76.7% (n=33) of students in the Experimental group got question 2 correct, as compared to 59% (n=19) in the Control in the group. 66 Also, 62.8% (n=29) students in the Experimental group got question 4 correct, as compared to 59.3% (n=19) of the students in the Control group. In addition, 74.4% (n=32) of students got question 5 in the Experimental group correct, as compared to 65.6% (n=21) in the Control group. The Experimental group demonstrated a better conceptual in the content areas covered in the post-test. Table 4. 2: The Development of Students in Conceptual Understanding of Geometry in the Experimental and Control groups 67 EXPERIMENTAL GROUP CONTROL GROUP Pre-test Pre-test Post-test Post-test Correct Partially correct Completely Wrong Correct Partially correct Completely Wrong Correct Partially correct Completely Wrong Correct Partially correct N (%) N (%) N (%) N (%) N (%) N (%) N (%) N (%) N (%) N (%) N (%) N (%) 40 (93%) 0% 3(7%) 42(97.7%) 1(2.3%) 0% 29(90.6%) 2(6.3%) 1(3.1%) 30(93.8%) 1(3.1%) 1(3.1%) 2. If three sides of a triangle are given as x2 –y2, 2xy and (x2 + y2). Show that the triangle is a right angle triangle. 23(53.5%) 6(14%) 16(32.5%) 33(76.7%) 5(11.65%) 5(11.65%) 16 (50%) 3(9.4%) 13(40.6) 19(59.4%) 3(9.4%) 10(31.2%) 3. Calculate reflex angle in a given polygon. (See appendix) 21(48.8%) 5(11.6%) 18(39.6%) 40(93%) 1(2.3%) 2(4.7%) 28(87.5%) 1(3.1%) 3(9.4%) 29(90.6%) 1(3.1%) 2(6.3%) 4. Calculate the length and area of a rectangle with a diagonal. (See appendix) 15(34.9%) 8(18.6%) 20(46.5%) 29(67.4) 5(11.6%) 11(21%) 17(53.1%) 5(15.6%) 10(31.3%) 19(59.4%) 4(12.5%) 9(28.1%) 5. Given two sides of a right angle triangle, calculate the third side which is the radius of circle? 23(53.5%) 5(11.6%) 15(34.9%) 32(74.4%) 2(4.7%) 9(20.1%) 20(62.5%) 4(12.5%) 8(25%) 21(65.6%) 5(12.5%) 6(18.8%) Question 1. Find the angle marked x in the polygon 68 Completely Wrong Within the Experimental group the results showed an improvement in students’ conceptual understanding of geometry in the Post-test. Table 4.3 compares the Pre-test and Post-test results of the students within the Experimental group. The Minimum score students obtained in the Pre-test was 30%, while the Maximum score was 70%. In the Post-test, the Minimum score was 60%, while the Maximum score was 95%. The mean score of students in the Pre-test was 44.95%, while that of the Post-test was 76.61%, an increase of 31.66%. This is an indication that in the Post-test every student’s performance had increased in the Experimental group. These improvements might be due to the effect of the use of GeoGebra and other factors including teacher factor. Table 4. 3: Means, Standard deviation and Maximum scores for Experimental group Test Pre-test Mean 44.95 Stand Dev 7.077 Maximum 70 Minimum 30 Post test 76.61 12.408 95 60 Within the Control group the results showed a marginal increase in students’ conceptual understanding of geometry in the Post-test. Table 4.4 compares the Pre-test and Post-test results of students within the Control group. The Minimum score students obtained in the Pre-test was 40%, while the Maximum score was 85%. In the Post-test, the Minimum score was 40%, while the Maximum score was 80%. Students mean score in the Pre-test was 56.74%, while that of the Post-test was 58.06%, an increase of 1.32%. This marginal increase might be due to the traditional method alone used and other factors including the teacher factor. 69 Table 4. 4: Means, Standard deviation, Maximum and Minimum Scores for Control group Test Pre-test Post-test Mean 56.74 58.06 Stand Dev 9.248 12.428 Maximum 85 80 Minimum 40 40 As indicated in chapter 3, independent samples t-test inferential analysis was used to find the effect of the use of IGS on students’ conceptual understanding of plane geometry. The results of the independent samples t-test on the participants’ scores in the Pre-test and Post-test are presented in Table 4.5. Table 4. 5 Independent Samples t-test for Experimental and Control groups Test Pre-test Groups Control 3D Experimental 3H Mean 56.74 44.95 Stand Dev 9.248 7.077 t-value 6.252 P-Value 0.114 Post-test Control 3D Experimental 3H 58.06 76.61 12.428 7.212 5.971 0.001 P < 0.05 From Table 4.5, the Experimental group had a mean score 44.95, while that of the Control group was 56.74 in the Pre-test. The results indicated that there was no significant difference in the level of conceptual understanding between the Experimental and the Control groups in the Pre-test at p=0.114>0.05. This means that the Experimental group and the Control group were almost of the level of conceptual understanding of geometry before the start of the intervention. In the Post-test however, the Experimental group had a mean score of 76.61, while the Control was 58.06. The t-test results revealed that there was significant difference in the conceptual understanding of geometry in favour of the 70 Experimental group in Post-test at p= 0.001<0.05. In addition to the test scores, interviews were conducted after the intervention to evaluate the reasons that accounted for the improvement of students’ conceptual understanding in plane geometry. The proportions of students’ and teachers’ views on the effect of the use of IGS and students conceptual understanding of geometry are presented in Table 4.6 Table 4. 6: Students’ and Teachers’ views on the Effect of IGS and Conceptual Understanding of Geometry. Question How does the use of IGS enhance your students' learning outcomes? How does the use of IGS support conceptual understanding of geometry? Teachers' Responses N With better understanding of the concepts, students performed better in the post test Deeper understanding is translated into better Mathematics achievement 2 How does the use IGS enhance your students’ learning outcome 66.7 1 33.3 66.7 2 Students discover geometric concepts with ease Student gets enhanced knowledge of the concepts Question Percentage (%) Students Responses 1 N 33.3 Percentage (%) 5 71.4 Increased my score in the post test Makes me lazy in doing calculations 2 28.6 From Table 4.6, the results show that 71.4% (n=5) of the students were of the view that the use of IGS has deepened their conceptual understanding and enable them improved their performance in the post-test, while 28.6% (n=2) of the students were of a contrary view. They advocated that over reliance on the use of the software can make them lazy, since the software did all the calculations for them. The data from the teachers indicated that 66.7% (n=2) said the use of IGS had enhanced their students’ conceptual 71 understanding of geometry. This confirms students’ assertion that IGS had enhanced their conceptual understanding. In addition, 66.7% (n=2) of teachers were of the view that the use of IGS made the geometry lessons practical. Consequently, some concepts which were difficult to explain during the traditional lessons were made clearer through the use of the software. 4.2 The Effect of IGS on SHS Students’ Motivation to Learn Plane Geometry Results from both the Questionnaire and Interview were combined to describe the effect of IGS on SHS students’ Motivation to learn plane geometry. 4.2.1 The Effect of IGS on Motivating SHS Students’ to Learn Plane Geometry the Student Perspective. Table 4.7 presents the proportion of Students views on the effect of IGS on their motivation. Table 4. 7: The Proportion of Students’ views on the Effect of IGS and their Motivation. Questions Students’ Responses N Percentage (%) In what ways did IGS help you in learning geometry The use of IGS took away boredom during the lessons 4 57.1 3 42.9 In what ways does the IGS motivate you to learn geometry The use of IGS made learning easier and interesting The use of IGS makes learning interesting 3 42.9 The use of IGS encourages us learn from one another The use of IGS does not make one tired during the lessons 1 14.2 3 42.9 72 The results show that 57.1% (n=4) of the students were of the view that the use of IGS helped them to learn geometry by eliminating boredom during the lessons, while 42.9% (n=3) said its usage made learning easier and interesting. Further interrogation revealed that the IGS lessons made learning easier because the software did all the calculations and measurements of the angles for them. The students also exhibited high levels of enthusiasm and interest due to the fascinating nature of the software. One student even remarked that “for the first time, I found Mathematics lessons interesting and free from fear and panic. At the group activities my friends explain some things I don’t understand to me. I was free to ask questions from group members and the teacher”. As indicated in Chapter 3, the students were given a Likert scale questionnaire to rate the extent to which they agreed or disagreed with statements about their motivation during the IGS lessons. The students’ mean level of agreement was computed and presented in Table 4.8 and used as the bases for the analysis. Table 4. 8: Students’ Mean Rating on the Effect of IGS on Motivation to Learn N Question N I I enjoyed the lessons with the use of IGS 43 4.7 0.803 II The IGS helped me a lot to learn the geometrical concepts taught 43 4.5 0.773 VI I am not happy any time I missed the geometry lesson held at the laboratory 43 4.2 1.33 VII IGS helped me to understand geometric concepts when they were taught 43 4.1 1.18 73 Mean level of Agreement SD VIII The use of IGS in the classroom made my learning easier and interesting. 43 4.8 0.820 IV From now on, I want to learn all geometry lessons with computers. 43 4.0 0.683 IX The textbook alone helped me a lot to learn the geometry concepts 43 3.0 0.51 X I felt helpless when asked to explore and study the learning materials presented in the lesson 43 4.1 0.700 III The use of IGS in the geometry lessons makes the lesson boring 43 3.8 0.649 V I hate Mathematics lessons that I take at the computer laboratory. 43 3.5 0.598 From the Table 4.8, six of the items were positively worded (I, II, IV, VI, VII, VIII), while four were negatively worded (III, V, IX, X). The results showed that the highest positive mean agreement level was 4.7 (STD=0.803), which indicates that students enjoyed the lessons with the use of IGS. While the lowest positive mean agreement level was 4.0 (STD=0.683) which suggest that students wanted to learn all their geometry in the laboratory. The levels of agreements were so closed with standard deviations less than1. This implies that majority of the students were of the view that the use of IGS has increased their motivation to learn geometry, since the lowest agreement level on the positively worded items was above the neutral value of 4.0 on the scale of 15. On the negatively worded items, the highest mean agreement level was 4.1 (STD=0.700) which indicated that the students strongly disagreed with the statement that they hated mathematics lessons taken in the computer laboratory. The students also strongly disagreed 3.8 (STD=0.649) with the statement that the use of IGS in the 74 geometry lessons made the lesson boring. This means that students liked the IGS lessons in the laboratory and that the lessons were rather very interesting learning experiences. The students mean rating on their motivation were further analysed and presented as Table 4.8. The results showed that the minimum mean attitude was 2.2, while the maximum mean attitude rated was 3.89. This is an indication that the students were highly motivated and wanted to learn all their geometry with the software. Table 4. 9: Students’ Mean rating on their level of Motivation Descriptive Statistics Mean Attitude Score n Minimum Maximum Statistic Statistic Statistic 43 2.22 3.89 Mean Statistic 3.2351 Std. Deviation Std. Error Statistic .05447 .35716 This showed that the students who did not like the IGS lessons were not strongly against its usage but were not comfortable due to their low experiences with the computer. 4.2.2 The Effect of IGS on Motivating SHS Students’ to Learn Plane Geometry the Perspective of the Teacher. Teachers’ responses on the effect of IGS on students’ motivation to learn plane geometry were analysed and presented in Table 4.10. The results showed that 66.7% (n=2) of the teachers were of the view that IGS was fascinating and interactive. It shows the interactive features of the concepts taught in the laboratory, while 33% (n=1) said its usage made mathematics teaching and learning practical. Consequently, GeoGebra could 75 be adopted as a tool to teach other topics in mathematics including linear transformation and statistics in kanton Senior School. Table 4. 10: Proportion of Teachers’ view on the Effect of IGS on Students’ Motivation to Learn Plane Geometry. Question Give reasons, why you think the use of IGS is important in helping your students to learn geometry In what ways does the use of IGS motivate your students to learn geometry? Teachers Responses N Percentage (%) It is interactive and therefore shows the dynamic features of the concept 2 66.7 Its usage makes Mathematics teaching and learning practical 1 33.3 The use of IGS sustains students' interest to learn geometry 2 66.7 IGS takes away boredom and fatigue in the learning process 1 33.3 The results further indicated, 66.7% (n=2) of the teachers were of the view that the use of IGS sustained students’ interest in the learning process, while 33% (n=1) said it took away boredom and fatigue. As a result, students wanted to learn all their geometry concepts with the use of the computer. 4.3. Ways in which IGS Provided Support for Student-centred Learning in a Geometry Class Results from Teachers and Students were combined to evaluate how IGS supported student-centred learning. 76 4.3.1 Students’ Perspective on Ways in which IGS Supported Student-centred Learning. As indicated in Chapter 3, students were given a Likert scale Questionnaire to rate the frequency at which the teacher observed some activities that indicated Studentcentred lesson. The results are presented in Table 4.11 as number and percentage distribution of students’ view on student-centred learning. Table 4. 11: Percentage Distribution of Students’ views on Student-centred Learning Items How often did you observe the following in the IGS lessons taught in your class? Number and Percentages of students responses N Every Rarely Lesson The teacher made us very active participants in the learning process in the classroom 43 43 (100%) 0 The teacher provided us all opportunities to ask questions 43 43 (100%) 0 Teacher engaged us in a lot of guided group activities 43 31 (72.9%) 12 (27.1%) The teacher provided opportunity for us to interact with each other in the group activity freely and purposively. 43 40 (93%) 3 (7%) The teacher provided opportunity for us to play leading roles in exploring the software. 43 43 (100%) 0 The results showed that all the students 100% (n=43) were of the view that: The use of Worksheets and Applets made them very active in both the small group and whole class discussions. They played lead roles in the software exploration and learned collaboratively in a well guided environment. Students were able to internalize and 77 externalize the concepts in the learning process. This collaborative learning has contributed in deepening students’ conceptual understanding in the geometry concepts explored in class. However, 27% (n=12) of the students were of the view that the teacher engaged them in a lot of guided group activities but rarely. From the questionnaire analysis, the students who ticked “Never and sometimes” were recorded as “0” while students who ticked “every lesson and half of the lesson” were also recorded as “1”. The results are then presented in Table 4.12 as students’ rated means on what student- centred learning lesson is. Table 4. 12: Mean Rating of Students views on Student-centred learning. Student centred learning N Minimum Maximum Mean Std Deviation 43 0 1 0.9767442 0.1525 Form Table 4.12, the results showed that on a scale of 0-1, the students scored a rated mean level of 0.9679 (with 0 been Minimum and 1 Maximum). This indicated that majority of the students are of the view that the use of the IGS in the geometry lessons depicted a typical student-centred lesson. Finally seven students were randomly sampled and interviewed; their responses were coded and presented in Table 4.13, as students’ opinions on student-centred learning from the qualitative data source. 78 Table 4. 13: Proportion of Students’ views on Student-centred learning. Students’ Responses Question In what way does the IGS support studentcentered learning How does the use of IGS support your conceptual understanding of geometry What were some of the challenges you encountered during the IGS lessons How did the worksheets help you in the classroom interaction In summary, how useful is the use of IGS to you as a student? N Percentage (%) The use of IGS encouraged group work 3 42.9 The use of IGS has improve interaction among students It deepens my understanding on the properties of parallel lines 4 57.1 3 42.9 It makes learning of the concepts clearer for me procedures for arriving at solutions are not shown I was not comfortable when moving the computer mouse There was power fluctuation at the computer laboratory. We had free interaction among ourselves 3 42.9 1 14.2 2 28.1 5 71.9 4 57.1 The worksheets made the group work well organized The use of IGS made drawing and measuring of angles easier The use of IGS made learning of geometry interesting and void of boredom 3 42.9 1 14.2 3 42.9 IGS should be integrated in our lessons 3 42.9 From Table 4.13, the results showed that, 57.1% (n=4) of the students were of the view that the use of IGS improved interaction among group members, while 42.9% (n=3) said it encouraged group work. This was because they had to discuss and agree on a point before completing the Worksheets. On whether the worksheet were helpful in learning the geometry concepts, 57.1% (n=4) of the students were of the view that it facilitated free interaction among students and teachers, while 43% (n=3) said it made the group work well organized. This created another opportunity for students to internalize and externalize the concepts through intensive interactions and discussions. All the students 79 were seen grossly involved in the learning process, by either contributing or writing the responses of fellow students. The results indicated that 71.9% (n=5), of the students identified source of power as a major challenge during the IGS lessons, however 28.1% (n=2) said they were not familiar with the use of computer. These students had difficulty moving the computer mouse and clicking the icons on the software. The results of the interview indicated that the students liked the geometry lessons with the computer. They stated that it helped them to interact, discuss and share ideas with each other. 4.3.2 Teachers’ Perspective on Ways in which IGS Supported Student-centred Learning in a Geometry. The results from three teachers’ interview guide were organised and presented as the opinion of teachers on ways IGS supported Student-centred learning in Table 4.14. 80 Table 4. 14: Teachers’ view on Ways in which IGS Supported Student-centred Learning. Question In what ways does the use of IGS support student-centred learning What were some of the challenges your students encountered during the IGS lessons? How did the Worksheets help classroom interaction or otherwise? In what ways, do the IGS lessons depict a typical student centred learning session? In summary how useful is the use of IGS to you and your students? Students’ Responses N Percentage (%) The use of the IGS facilitates students to work in smaller groups all the time The IGS lessons made the student the centre of the learning process Source of electric power was a major challenge 2 66.7 1 33.3 2 66.7 Low IT knowledge of students 1 33.3 The use of IGS facilitated the group activities 2 66.7 The use of IGS enhanced students seriousness towards the work Students worked in active groups 1 33.3 1 34 Students play lead roles and it teaches group dynamics Students interact with each other purposively It makes teaching and learning more effective and meaningful It makes teaching and learning more practical 1 33 1 33 2 66.7 1 33.3 The results indicated that 66.7% (n=2) of the teachers were of the view that students had the opportunity to work in smaller groups all the time, while 33.3% (n=1) said the group activities made the students the centre of the learning process. This made the students the centre of the learning process. They became responsible for their own learning outcomes, while the teacher made notable incursions to facilitate the process. Furthermore 66.7% (n=2) of the teachers said IGS enabled students discover geometry concepts with ease, while 33.3% (n=1) were of the view that the student had an enhanced knowledge of the concepts taught in the laboratory. The students said they were not only 81 told corresponding angles are equal theoretically, but they also saw it. Notwithstanding the immense contribution of the IGS towards the teaching and learning of geometry, there were some challenges that attempted to frustrate the process. Majority (66.7%, n=2) of the teachers sited source of power as the major challenge, while 33.3% (n=1) pointed students’ low experience with the use of computer. Some of the students practically did not have any experience with the computer. They had difficulty manipulating the computer mouse. On whether the Worksheets helped classroom interaction, 66.7% (n=2) of the teachers were of the view that it facilitated group work, while 33.3% (n=1) said it enhanced their level of seriousness during the lessons. As a result the students were seen working in groups throughout the IGS lessons. In a whole 66.7% (n=2) of the teachers commented that teaching with the IGS made teaching and learning more effective, while 33.3% (n=1) said the lessons were practical. 4.4 Discussion of Major Findings. In this research, the aim of the researcher was to explore students’ learning experiences in the IGS learning environment. Therefore, the focus was on how conceptual understanding, motivation and student-centred learning are linked in the teaching and learning of geometry in an interactive geometry environment. The findings indicated that IGS as a learning tool provided new learning experiences in the geometry lessons. These learning experiences included: allowing students to tinker and experiment which increases student’s attention and motivation, while removing instructional constraints such as drawing graphs by hand. Unlike earlier research (Arzarello et, al. 2002; Falcade et, al. 2007; Gawlick, 2002; Laborde, 2001) on the different types of 82 dragging potentials of the IGS tool in geometry learning, the use of the IGS in this research was based on pre-designed Applets. These Applets represented the geometrical concepts which guided the dragging interactions. These guided exploration, reduced the arbitrary dragging and made the students to concentrate on externalizing the geometric concepts. However, this guided exploration did not reduce the efforts to externalize the implicit development of the geometrical concepts. From a cognitive perspective, the findings showed that the use of IGS in teaching and learning facilitated group work and interaction among students in the classroom. This made students to investigate ways the geometry concepts were conceived. The exploration of the concepts using IGS made it possible for students to discuss the ideas embedded in the formation of those concepts, and thereby made them commonly sensible for all students. At the group levels, the ideas based on the visual representations were shared and discussed intensively. This was done both at the small group and whole class discussions. Therefore, most of the factors that challenged the students from internalizing the visual experiences into conceptual understanding were solved due to the intensive nature of the whole class discussions. Thus, all the students left the laboratory with clearer understanding of the concepts explored. This is in consonance with the Social Cognitive Development Theory postulated by Vygotsky. According to Vygotsky, social interaction plays a fundamental role in the development of cognition (Vygotsky, 1978). Furthermore, the results from the Worksheets showed that students carried out practical investigations with the Applets. Students gathered the required responses that characterized the particular geometrical objects explored. They synthesized angles 83 generated and linked them to properties investigated. This was because the students had sufficient time for both small group and whole class discussions which facilitated the internalization of the geometry concepts (Zbiek, et, al. 2007). According to Vygotsky (1978), an essential feature of learning is that which awakens a variety of internal developmental processes that operate when the child interacts with people in his environment and in cooperation with his peers. The external representations of the geometry concepts through the Applets and Worksheets provided a setting for social interactions and discussions. Thus, social interaction is critical as knowledge is constructed in the process (Schunk, 2004). In social interaction, learning takes place in social environment where there are collaborative activities. Through these activities, learners communicate, interact and learn from one another, and consequently, constructing their own world of knowledge (Vygotsky, 1978). This promotes meaningful and effective learning. Both Brunner and Vygotsky in their theory of social interaction stated that learning is shaped and affected by the learner’s background and cultural experiences (Roblyers, 2006). The findings from the Pre-test and Post-test indicated that there was no significant difference between students’ conceptual understanding of plane geometry in the Control and Experimental groups at the Pre-test before the start of the intervention. This showed that both the Control and the Experimental groups were at the same level of conceptual understanding in plane geometry before the intervention. However the low performance of the business class motivated the researcher to choose them as Experimental group. This is in agreement with earlier research by Falcade, Laborde, and Mariotti (2007); Gawlick 84 (2002); Hollebrands (2003); Laborde (2001) and Ruthven, Hennessy and Deaney (2008), who proposed a new learning environment in particular for lesser motivated students to supplement the pencil and paper medium in the teaching of geometry. This entails the application of ICT tools to facilitate the teaching and learning situation in the school. The use of ICT, as suggested by Reynolds (2001), should supplement classroom activity by accessing existing information and knowledge, rather than as an integral part of pedagogical practice. The analysis from the independent-samples t-test showed a significant difference between students conceptual understanding of plane geometry in Experimental and Control groups. From the results obtained, a number of suggestions can be made for improving teaching and learning of geometry in Ghanaian classrooms. The significant differences in geometry achievement of the Experimental group as compared to the Control group indicated that GeoGebra has the potential of enhancing teaching and learning outcomes at the Senior Secondary School level. The results of this study are consistent with the Funkhouser (2002) and Almeqdadi (2005) who reported that the addition of dynamic geometry software in geometric construction has increased their students’ geometry achievement as well as enhanced understanding. This observation can therefore encourage classroom teachers and even curriculum developers on the potential use of GeoGebra in the learning of plane geometry in Ghana. Furthermore, the findings agree with Serhan’s (2004) findings, who examine the effect of Graphing Calculators on students’ conceptual understanding of integral Mathematics. Serhan concluded that students who used graphing calculators during instruction formed a deeper conceptual understanding of the topic studied. The results 85 indicated that students in the Experimental group were better able to interpret numerical representations, understand and explain the connection between the average rate of change and the instantaneous rate of change. This finding also agrees with the findings of Vincent (2005) who used interactive geometry software to explore a series of “linkages” and their underlying geometric structures. Vincent concluded that the software used helped students to achieve an understanding and appreciation for geometric proof. Vincent then noted that, for her students, proof emerged as a valued activity because it offered a verification of the truth of their conjectures and an explanation as to why the linkage works the way it does. Students were then able to extend this value of proof to further geometric proof and investigations. This finding is also in line with Swan, Kratcoski, Schenker, and Van, (2010), who reported that interactive whiteboards in English language arts and Mathematics lessons improved student learning in those areas as measured by students’ scores on state achievement tests. Other researches (Choi-Koh, 2003) on the effect of using Graphing calculator, Excel spreadsheets, and the Geometer Sketchpad on middle and high schools learning of trigonometry. These studies showed that the graphing calculator, Excel spreadsheets, and the Geometer Sketchpad all have a very positive impact on exploring trigonometric functions. However, Scher (2005) noted that the software used led to assessment difficulties for teachers but also gave the opportunity for a meaningful class discussion that may lead to a deeper conceptual understanding. Further interrogations on the effect of the use of GeoGebra in the study, revealed that the Experimental group were business students and did not have the opportunity of studying Elective Mathematics. The average raw score obtained by students in the 86 business class at BECE for which they were placed in Kanton Senior High School by the Computer Selection and School Placement System (CSSPS) was 279 out of 800 marks. This mark was far below the expected average of 400 marks, indicating that the students’ performance was poor. Thus, the high performance of the Experimental group could be attributed to the use of the IGS holding other factors constant. The Control group on the other hand were science students, who in addition to Core Mathematics were also studying Elective Mathematics. The Science students usually have the best of results in Mathematics at the JHS level that qualifies them to pursue science programmes in the school. The average raw scores obtained by the science students at the BECE for which they were placed in Kanton Senior High School by the CSSPS was 307 out 800 marks. Even though, this average mark was also poor it was better than that of the business class. The researcher was therefore not surprised when the science class (Control group) outperformed the business class in the Pre-test, because they were exposed to further Mathematics than the Experimental group. The results also showed that majority of the students motivated to learn geometry through the use of the GeoGebra. They added that it took away boredom and made the lesson interesting, enjoyable and easier. Consequently, it increased the time students spent on learning geometry for the first time in Kanton Senior High School. The findings from teachers also showed that the students enjoyed the lessons, learning became easier and interesting, and for the first time students did not sleep in class during lesson delivery. The use of computer and the software sustained students’ interest in the learning process and consequently, students wanted to learn all their geometry with the computer. The findings of this study are consistent with the Sanders (1998), who reported that the 87 addition of dynamic geometry software in geometric construction has increased her students’ interest in geometry as well as enhancing their understanding. This observation could therefore encourage classroom teachers and even curriculum developers integrate GeoGebra into the learning of geometry. Furthermore, Ruthven and Hennessy (2002) found, through interviews with mathematics teachers in seven schools, that regular access to technology and familiarity with software and hardware made student’s success more likely. The teachers reported that using the IGS led to increased participation and productivity. This provided a new and different learning environment, removing instructional constraints (such as drawing graphs by hand), allowing students to “tinker” and experiment, improving student motivation and engagement, facilitating classroom routines, improving pace and productivity, accentuating mathematical features, increasing student attention, and helping students form and establish ideas. Ruthven and Hennessy’s study supported this claim and concluded that the use of the software led to a variety of benefits for the students, including a deeper conceptual understanding and increased motivation. However, some of the students were not friendly with the use of GeoGebra and therefore disagreed with some of the statements. The students therefore prescribed a blend of the traditional and software based learning to deepen their conceptual understanding. The students pointed out that when the traditional method of teaching is supplemented with computer based teaching, learning outcomes would be better achieved. The findings further showed that majority of students were of the view that in every lesson, the teacher made them very active participants, provided opportunities for 88 asking questions and made them learn from one another in a series of guided group activities. Consequently, the students occupied a central position in the lesson delivery. The teachers confirmed this assertion and stated that the IGS lessons were activity based. The teacher made students play lead roles and interacted with each other purposively in the learning process. The findings concurs with that of Simonsen and Dick (1997), who examined and investigated mathematics teachers’ opinions on the impact of the use of graphing calculators on their classrooms instructions. The results showed that in the Experimental classes where graphing calculators were used regularly led to more student-centered classroom dynamics: increased cooperative and discovery learning; increased student discussion, involvement and enthusiasm. However, teachers identified a need for increased preparation time and a critical need for further professional development and support. Some students also stated that even though the teacher does these things, it was not all the time. In conclusion the teachers commented that teaching with GeoGebra made the lessons practical and teaching and learning more effective. This is in line with Vygotskian view of teaching and learning which aimed at a more effective education. According Lev Vygotsky (1978), a more effective education may be induced by the use of technology, as observed by a number of researchers (Falcade, Laborde & Mariotti 2007; Laborde 2003). 89 CHAPTER FIVE SUMMARY, CONCLUSION AND RECOMMENDATIONS 5.0 Overview This chapter provides the summary of the study and the major findings. It highlights the conclusion of the study and implications for practice. It further outlines some of the limitations and recommendations for future research. 5.1 Summary of Study The study explored the effect of IGS on SHS students’ understanding of and motivation to learn geometry. This study sought to establish the effect of IGS on SHS students’ conceptual understanding of plane geometry. It also looked at the effect of IGS on SHS students’ motivation to learn plane geometry. Finally it investigated ways in which IGS provided support for student-centred learning in a geometry class. The general approach chosen for the study was a sequential explorative mixed method, which employed quasi-experimental design as a strategy of enquiry. The model of quasi-experimental design used was the Pretest-Posttest non-equivalent group strategy. The mixing was therefore done during the data collection, the analysis and presentation of the results. The aim was to use qualitative findings to assist in explaining quantitative findings. The sample used for this research was 75 students from Kanton Senior High School in the Sissala East District Assembly, in the Upper West Region of Ghana. The Experimental class was 43 students in the business class, while the Control group was 32 students in the science class. 90 The results showed that the Experimental group had a mean score of 76.61, while that of the Control was 58.06. The Experimental group demonstrated an enhanced understanding of the concepts investigated. Majority (71.4%) of students stated that the use of IGS has helped enhanced their conceptual understanding and consequently led to an improved performance in their Post-test scores. The findings further revealed that the use of IGS motivated students more in the Experimental group to learn plane geometry. Majority of the students enjoyed the IGS lessons and consequently wanted to take all their geometry lessons in the laboratory (Mean =4.7, SD=0.803). The findings also showed that students mean rating on student-centred learning was 0.976442 (with 0 been minimum and 1 been maximum) which meant that the IGS lessons were activity based and centred on the student. The students had ample time to discuss ideas both at the small group and whole class discussions. Consequently, the use of the IGS enhanced interaction among students and the teacher. The results of the different data sources: the Pre- test and Post-tests, Questionnaires, Interviews and Worksheets were combined to answer the research questions. In particular, each research question was looked at from all relevant data sources. In the case of contradiction between the data sources, the researcher gave more weight to the most objective data sources. Finally, the researcher used qualitative findings to help in assigning reasons to the quantitative findings. 5.2 Summary of Findings The results of the study are summarised and presented under the three sub- heading in line with the research questions. 91 Research question 1: To what extent does the use of IGS affect students’ conceptual understanding of geometry concepts? The findings from the Pre-test and Post-tests showed that students made a significant progress in developing their conceptual understanding as a result of the IGS (GeoGebra) lessons. The students’ working process of the test items indicated that they had developed a better understanding of the geometry concepts. Indeed, students were able to establish the relationship between the angles formed and the technical names of the properties investigated. The findings showed that majority of students (71.4%) had developed a higher conceptual understanding in the geometrical concepts explored. Consequently, there was a significant difference between the results of the Pre-test and Post-test in terms of developed solution methods and scores. Also, the results from the interview with students showed that the Experimental group had developed a better understanding of the concepts taught through the use of the Applets and Worksheets. The interview with the teachers also indicated that the GeoGebra lessons provided students with visual representations that assisted in developing insights into the geometrical concepts explored. The Experimental group had a mean score of 76.61, while that of the Control group was 58.06. The t-test results revealed that there was significant difference in the conceptual understanding of geometry between the Control and the Experimental groups at p=0.001. The findings from all the data sources (Pre-test and Post-test, Interviews and Worksheets) from students and teachers confirmed that the students’ had an enhanced understanding in the geometry concepts explored. This was attributed to the use of the IGS (GeoGebra), the Worksheets and the intensive group discussions during the lessons. 92 However, not all students developed the same level of conceptual understanding in the concepts taught due to individual differences. This lapse was averted during the whole class discussion. Research Question 2: How does the use of IGS motivate SHS students to learn geometry? The findings from the Questionnaire showed that majority of the students enjoyed the IGS lessons (Mean=4.7, SD= 0.803), its usage made learning easier and interesting (Mean=4.8 SD= 0.820). The findings further indicated that the students had positive expressions towards the use of GeoGebra in learning geometry in the interactive geometry software learning environment. Also, findings from the Interview guide with the students showed that the students liked the geometry lessons with IGS because it took away boredom (57.1%, n=4) and affirmed that it made learning easier and interesting (42.9%, n=3). From the discussions and interactions during the lessons, it was noticed that learning with IGS (GeoGebra) raised students’ interest and enthusiasm towards geometry concepts as they explored with the software. Majority of the teachers (66.7%, n=2) also confirmed that the students were highly motivated when working with the GeoGebra. As a result, no students slept in class during mathematics lessons for the first time in Kanton Senior High School. Thus the results from all the data sources indicated that the use of the IGS (GeoGebra) during the geometry lessons has increased and sustained students’ interest more to learn Geometry. It took away boredom and made learning of the geometry concepts easier and free from stress. 93 Research Question 3: In what ways do IGS provide support for student- centred learning in a geometry class? The findings from the worksheets, Questionnaire and Interview indicated a number of ways IGS lesson supported student-centred lessons. The findings from Questionnaires and Interview with the students showed that, the IGS lessons were centred on the student. The students explored the concepts independently, asked questions and learned collaboratively in a series of guided group activities (100%, n=43). The students occupied a central position in the lesson delivery and shared ideas with group members. The findings also showed that the use of the Worksheets and Applets in the IGS lessons engaged students in series of discussions and interactions. Students played lead roles during the discussions and shared ideas before presenting their results for marking (100%, N=43). Furthermore, Findings from the interview with the teachers showed that IGS lessons were activity based (approximately 34%, n=1), students played lead roles (approximately 33%, n=1) and interacted with each other purposively (approximately 33%, n=1). Thus the IGS lessons were student centred. The students used the Worksheets and Applets to investigate the concepts independent of the teachers’ intervention in well coordinated group activities. Consequently, the students played lead roles in these group activities while the teacher facilitated the process. Thus the results from all the data sources indicated the use of the GeoGebra in the geometry lessons supported studentcentred learning activities in a number of ways: The lessons were activity based, students worked in groups, they were engaged in a series of small group and whole class discussions, students learned collaboratively and interacted freely during the lessons delivery. 94 5.3 Conclusion and Implications for Practice The IGS (GeoGebra) learning activities provided an equal support for every student to eventually achieve an enhanced conceptual understanding of the concepts taught. The findings from the Pre-test, Post-test and the Worksheets showed that students’ conceptual understanding had improved. The findings from the Interviews and the Questionnaire revealed that students largely developed conceptual understanding regarding the geometrical concepts, due to the intensive small group and whole class discussions, coupled with the high levels of motivation during lessons. The findings also indicated that majority of the students enjoyed the geometry lessons with IGS (GeoGebra) and thus, they were motivated more to participate actively in the lessons. Nevertheless, the GeoGebra alone may not be the only factor contributing to the emergence of students’ motivation. The emerged motivation might also be linked to other factors such as independent group work, students’ working with Worksheets, and the change of teacher. Furthermore, the findings from the Questionnaire and the Interviews indicated that the IGS lessons supported student-centred lessons in a number of ways. The IGS lessons: were activity based, involved series group work, made students learn collaboratively and provided opportunity for students to interact and discuss with colleagues intensively. The use of the IGS (GeoGebra) with the Worksheets and group work supported student-centred learning activities. The findings of this study showed that the use of the IGS has improved students conceptual understanding in plane geometry. This finding is in line with earlier studies on the effect of the use of IGS in the teaching and learning of geometry (Abdul- Halim & 95 Effandi, 2011; Noraini, 2007; Sanders 1999). These studies indicated that the use of IGS has a number of suggestions for improving teaching and learning of geometry in Ghanaian classrooms. The use of IGS has the potential of helping students understand geometry concepts better, while improving their geometric reasoning. The information generated therefore is available for policy makers, teachers and other stakeholders to help improve students’ geometric reasoning in Ghana, through the use GeoGebra. The findings would be a guide to policy makers on the implementation of the national policy on the ICT integration into the teaching and learning of mathematics. The findings of this study indicated that the use of IGS has motivated students more to learn plane geometry and consequently the students wanted to learn all their geometry with the computer. Its usage has improved classroom interaction and cooperative learning among students in the learning process. This has contributed in deepening students’ conceptual understanding of the concepts explored. This indicates that the use of IGS (GeoGebra) has the ability to improve students’ learning of geometry at the Secondary School level in Ghana. If government wants to improve Ghana’s performance in international examinations like TIMSS, then the use of GeoGebra is imperative. 5.4 Limitations of the Study One school in the Sissala East District Assembly was selected for this study and this has limited the scope of the research. The consequence of this was that, generalization of the research findings was limited. This limitation was mitigated when the computer selection and school placement scheme (CSSPS) posted students from Basic Schools all over the ten regions of Ghana to Kanton Senior High School as 96 students. This has enriched the sample used for the study in terms of students’ cultural and social backgrounds. The sample used therefore represents the characteristics of Ghanaian students in any part of the country who had spent at least two years studying core mathematics in the school. The researcher’s inability to draw from local examples and knowledge in the Ghanaian context limited the contextualization of this research in the Ghanaian setting. This affects a realistic situating of this study in the Ghanaian context. 5.5 Recommendations The IGS (GeoGebra) lessons have helped to improve students’ conceptual understanding of plane geometry in Kanton Senior High School. GeoGebra with its interactive features in a student-centred environment created a platform for students to interact, discuss and learn collaboratively. Based on the findings of the study, the following recommendations are made for the improvement of teaching and learning of mathematics at the Senior High School level. Curriculum developers and policy makers should consider revising the Senior High School Mathematics Syllabus to include the use of IGS (GeoGebra) into teaching and learning of geometry. This will help teachers and other stakeholders improve students’ conceptual understanding of geometry and geometric reasoning in Ghana. The use of Worksheets and Applets helped to improve student-centred learning in an interactive geometry environment. Students were able to internalize and externalize geometric concepts which have deepened their understanding on the concepts studied. 97 The researcher therefore recommends that textbooks and teacher’s handbooks should be revised based on student-centred learning approaches to enhance collaborative learning. It is also recommended that future research studies on the use of IGS could focus on the effect of GeoGebra on students’ achievement and Van Hiele geometric thinking. This will help establish the level Ghanaian students have reached on the Van Hiele geometric reasoning. Finally, the researcher recommends that the study be replicated in many more schools to obtain the general picture of how GeoGebra improves students’ learning of Geometry in Ghana. 98 REFERENCES Affum, O. (2001). Mathematics Crises in our Schools: Causes and Remedies. Mathematics Connection, 2(1), 4-6. Almeqdadi, F. (2007). The effect of using the geometer’s sketchpad (GSP) on Jordanian students’ understanding of geometrical concepts. International Journal for Mathematics and Learning, 23(2), 161-165 Anamuah-Mensah, J. & Mereku, D. K. (2005). Ghanaian Junior Secondary School two students abysmal Mathematics Achievement in TIMSS 2003: A consequence of the Basic school Mathematics. Mathematics Connection , 5(1), 1-11. Anamuah-Mensah, J. Mereku, D. K. & Ghartey-Ampiah, J. (2008). Ghanaian Junior Secondary School students' Achievement in Mathematics and Science: Results from Ghana's participation in the 2007 Trends in International Mathematics and Science Study. Accra: Ministry of Education Youth and Sports. Arzarello, F. Paola, D. & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environment. ZDM, 34(3), 66-72. Asabre-Ameyaw, A. & Mereku, D. K. (2009). Comparative Analysis of performance of eight graders from six African Countries. Mathematics Connection, 8(2), 17-26. Assuah, C. (2010). Use of technology for college Mathematics instruction: African instructors' experiences. Mathematics Connection, 9(5), 41-53. 99 Battista, M. T. (2007). The development of Geometric and Spatial Thinking in Lesta, F. (Ed.), Second hand book of Mathematics of research on Mathematics teaching and learning (pp 843-908). NCTM. Reston, VA: National Council of teachers of Mathematics. Beeland, W. D. (2002). Student engagement, visual learning, and technology: Can interactive whiteboards help? Retrieved October 8, 2007 from http://chiron.valdosta.edu/are/Artmanscrpt/vol1no1/beeland_am.pdf Bransford, J. D. Brown, A. L. & Cocking, R. R. (1999). How people learn: Brain, mind, experience and school. Washington, DC: National Academy Press. Burnard, P. (1999). Carl Rogers and postmodernism: Challenged in nursing and health sciences. Nursing and Health Sciences 1, 241-247. Butzin, S. M. (2001). Using instructional technology in transformed learning environments: An evaluation of Project CHILD. Journal of Research on Technology in Education, 33 (4), 367-73. CEO Forum. (2001). Key building blocks for student achievement in the 21st Century. The CEO Forum School Technology and Readiness Report. Retrieved November 27, 2009 from http://www.ceoforum.org/reports.html Charmaz, K. (2005). Grounded theory in the 21st Century: Application for advancing social justice studies. In N. K. Denzin & Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (3rd ed., pp. 507-535). Thousand Oaks, CA: Sage. Choi-Koh, S. S. (2000). Effect of a graphing calculator on a 10th-grade students’ study of trigonometry. Ninth International Conference on Mathematics Education. 100 Cohen, D. (2005). Pre-calculus: A problems-oriented approach. (6th ed.). Belmont, CA: Thomson Higher Education. Cuthell, J. P. (2005). The impact of interactive whiteboards on teaching, learning, and attainment. Proceedings of SITE 2005 (p. 1353-1355). AACE: Phoenix, Arizona. CRDD, (2009). National Educational Assessment Report on 2009 administration of National Education Assessment primary 3 and Primary 6 English and Mathematics. MOE, Accra. Creswell, W. J. (2003). Research Design: Qualitative, Quantitative and Mixed Methods Approaches (2nd ed). London: Sage Publications. Creswell, J. W. (2009). Research Design: Qualitative, Quantitative and Mixed Methods Approaches, (3rd ed). London: Sage publications. Creswell, J.W. & Clark, V. (2007). Research Design: Qualitative, Quantitative and Mixed Methods Approaches Sage, Thousand Oaks, CA. Drickey, N. (2001). A Comparism of Virtual and Physical Manipulative in teaching Visualisation and Spatial reasoning to middle school Mathematics students. Dissertation Abstracts International, 62(2), 56-78. Driscoll, M. P. (2000). Psychology of Learning for Instruction (2nd Ed). Boston: Allyn & Bacon. Dontwi, I. K. (2001). Research in graphing calculator use: A preliminary report, Mathematics connection, 2 (3) 16-21. Durrheim, K. (1999). Quantitative Measurement: Applied Methods for Social Sciences. Cape Town: University of Cape Town Press. 101 Durrheim, K., & Wassenaar. D. (1999). Putting Design into practice: Writing and evaluating research proposal. In B. T. Durrheim, Research in practice: Applied methods for the social sciences (pp. 57-71). Cape Town: University of Cape Town Press. Falcade, Laborde, & Mariotti, (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317-333. Fitzallen, N. (2005). Integrating ICT into Professional Practice: A case study of four Mathematics Teachers. Tasmania: University of Tasmania. Fredua-Kwarteng, Y. and Ahia, F. (2005). ‘Confronting National Mathematics Phobia in Ghana’ (Part 2) Ghanaweb: Feature Article of Sunday, 11 July 2004 mathsn teachers in Ghana.htm Funkhouser, C. (2002). The effects of computer augmented geometry instruction on student performance and attitudes. Journal of Research on Technology in Education, 35(2), 163175. Galbraith, P. (2006). Students, mathematics, and technology: Assessing the present challenging the future. International Journal of Mathematical Education in Science & Technology, 37(3), 277. Gawlick, T. (2005). Connecting Arguments to Actions Dynamic Geometry as Means for the Attainment of Higher van Hiele Levels. ZDM, 37(5), 361-370. Gawlick, T. (2002). Dynamic Geometry Software in Regular Classroom. ZDM, 34(3), 85-92. 102 Gilmore, C. & Halcomb, G. C. (2004). Technology in the classroom: Investigating the effect on the student-teacher interaction. Usability News 6.2. http://psychology.wichita.edu/newsurl/usabilitynews/62/online_stats.asp. Golafshani, N. (2003). Understanding Reliability and Validity Qualitative Research. Qualitative reports 8(4), 597-607. Hall, I. & Higgins, S. (2005). Primary school students’ perceptions of interactive whiteboards. Journal of Computer Assisted Learning, 1(21), 102-117. Hannula, M. S. (2006). Motivation in Mathematics: Goals reflected in emotions. Educational Studies in Mathematics, 63(1), 165-178. Heafner, T. (2004). Using technology to motivate students to learn social studies. Contemporary Issues in Technology and Teacher Education, 4(1), 42-53. Retrieved March 2, 2009 from http://www.editlib.org/?fuseaction Hershkowitz, R., Ben-zvi, D., Dreyfus, T., Friedlander, A., Hadas, N., & Resnick, T. (2002). Geometry: Concepts and justification. In I. (Ed), Hand book of international research in Mathematics Education (p. 672-678). Mahwah, NJ: Lawrence Erlbaum Associates. Holllebrands, F. K. (2003). High students understandings of geometric transformations in the context of a technological environment. Journal of Mathematical Behaviour, 22(1), 5572. Huitt, W. (2001). Motivation to learn: An overview. Educational Psychology Interactive. Valdosta, GA: Valdosta State University. 103 Joppe, M. (2006). Reliability and Validity of Qualitative and Operational Research Paradigm. University of Education Lahore, Pakistan. Retrieved from, 9 November, 2010 www.Pjsor.com/index.php/pjsor/article/download/59/38 Kember, D. (1997). A re-conceptualization of the research into university academics Conceptions of teaching. Learning and Instruction 7(3), 255-275. Kennewell, S. & Morgan, A. (2003). Student teachers’ experiences and attitudes towards using interactive whiteboards in the teaching and learning of young children. In J. Wright, A. McDougall, J. Murnane, & J. Lowe (Eds.), Young children and learning technologies (pp. 71-76). Sydney: Australian Computer Society. Kleinginna, P. Jr. & Kleinginna A. (1981). A categorized list of motivation definitions, with suggestions for a consensual definition. Motivation and Emotion, (5), 263-291. Kilpatrick, J. Swafford, J. & Findell, (2001). The stands of Mathematics Proficiency. In J. S. Kilpatrict, Adding it up: Helping Children Learn Mathematics (pp. 115-156). Washington, DC: National Academy Press. LeDuff, R. (2004). Enhancing Biology Instruction via Multimedia Presentations. In C. Crawford et al. (Eds.), Proceedings of Society for Information Technology and Teacher Education International Conference 2004 (pp. 4693-4695). AACE: Chesapeake, VA. Retrieved October 8, 2007 from http://www.edcompass.de/NR/rdonlyres/B65DEF10-F8F0-40A3B91AA892BED75292/0/LeDuff_Jan04.pdf Laborde, C. (2001). Integration of technology in the Design of geometry tasks with Cabrigeometry. International Journal of Computers for Mathematical Learning, 6(3), 283-317. 104 Laborde, C. (2003). Technology used as a tool for mediating knowledge in teaching of Mathematics: The case of Cabri-geometry. France: Grenoble. LeCompte, M. D. & Preissle, J. (1993). Ethnography and qualitative design in educational research (2nd Ed.). New York: Academic Press. Lancaster GA, Dodd S & Williamson PR (2004) Design and analysis of pilot studies: recommendations for good practice. Journal of Evaluation in Clinical Practice 10(2): 307-12 Lincoln, Y. S. & Guba, E. G. (1985). Naturalistic inquiry. London: SAGE Publications. Mereku, D. K. (2010). Five Decades of School Mathematics in Ghana. Mathematics Connections 9 (8), 73-86 Mereku, K. D. Seidu A. Mereku, C. W. K. Acquaye, E. & Dowunah-Hammond, C. (2007). Report on the Research on School Management in Ghana. Accra: JICA Ghana Office. Mereku, K. D. (2004). Methods in Ghanaian Primary Mathematics Textbooks and Teachers’ Classroom Practice, Research in Mathematics Education, vol 6. 157-173. McGehee, J. & Griffith, L.K. (2004). Technology enhances student learning across the curriculum. Mathematics Teaching in the Middle School, 9(6), 344-349. McMillan H. J. & S. Schaumacher. (2006). Research in Education: Evidence-Based Inquiry. New York: Pearson Education, Inc. 105 Mehdiyev, R. (2009). Exploring students learning Experiences through Dynamic geometry software in geometry class at a secondary school in Azerbaijan. Netherlands: Amstel Institute of Universiteit van Amsterdam. Miller, D. Glover, D. & Averis, D. (2003). The introduction of interactive whiteboard technology to secondary school Mathematics teachers in training. Paper presented at CERME3: Third Conference of the European Society for Research in Mathematics Education, Bellaria, Italy. Miller, D. Glover, D. & Averis, D. (2004). Motivation: The contribution of interactive whiteboards to teaching and learning in Mathematics. Retrieved October 8, 2007 from http://cerme4.crm.es/Papers%20definitius/9/Miller-Glover-Averis.pdf Miller, D. Glover, D. & Averis, D. (2005). Presentation and pedagogy: The effective use of interactive whiteboards in Mathematics lessons. In D. Hewitt & A. Noyes (Eds.). Proceedings of the Sixth British Congress of Mathematics Education (pp. 105-112). University of Warwick. Mitchelmore, M. (2002). The role of abstraction and generalisation in the development of mathematical knowledge. In Edge, & B. Yeap (Eds.), Proceedings of the Second East Asia Regional conference on Mathematics Education and Ninth Southeast Asian Conference on Mathematics Education, Volume 1 (pp. 157-167). Singapore: National Institute of Education. MOE (2010). Teaching Syllabus for Mathematics. Accra: Ministry of Education. 106 MOE (2006). Ghana Information & Communication Technology in Education Policy. Accra: Ministry of Education Youth and Sports. Morgan, D. (1998). Practical Strategies for combining qualitative and quantitative methods: Application to health. Qualitative Health Research, 8(3), 362-376. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston: VA. CRDD (2009) National Educational Assessment. Report on 2009 administration of National Education Assessment primary 3 and Primary 6 English and Mathematics. Accra: MOE. Noraini I. (2007). The effect of geometer’s sketchpad on the performance of Geometry of Malaysian students’ achievement and Van Hiele Geometric thinking. Malaysian Journal of mathematical science, 1(2), 169-180. Painter, D. D. Whiting, E. & Wolters, B. (2005). The use of an interactive whiteboard in promoting interactive teaching and learning. VSTE Journal, 19(2), 31-40. Reynolds, D. (2001). ICT in Education: The future research and policy agenda. Building an ICT Research Network conference, 12(1), 25-43. Roblyers, M. D. (2006). Integrating educational technology into teaching (4th ed.). Upper saddle River, NJ: Pearson prince Hall. Roblyers, M. D. (2006). Virtually successful: Defeating the dropout problem through online school programmes. Phi: Delta kappan, 88(1), 31-36 107 Rogers, C. R. (1983). The politics of education in freedom to Learn for the 80’s. Ohio: Charles E. Merrill Publishing Company. Roschelle, J. M. Pea, Hoadley, C. M. Gordin, D.N. & Means, B. M. (2000). Changing how and what children learn in school with computer-base technologies. The future of children, 10(2), 76-102. Rosnaini M. Mohd A. & Ismail L. (2009) Development and Evaluation of a CAI Courseware, GReflect’ on Students’ Achievement and Motivation in Learning Mathematics European Journal of Social Sciences, 8 (4), 557-568 Ruthven, K. Hennessy, & Deaney, R. (2008). Construction of Dynamic geometry: A study of the international flexibility of educational software in classroom. Computers and Education , 51(1), 297-317. Scher, D. Steketee, S. Kunkel, P. & Lyublinskaya, I. (2005). Exploring pre-calculus with the Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press. Schunk, D. H. (2004). Learning theories: an education perspective (4th ed.). Upper Saddle River, NJ: Merrill and Pretice Hall Senechal, M. (1990). Shape. In L.A. Steen (Ed.), On the Shoulders of Giants; New Approaches to Numeracy (pp. 139-182). Washington DC: National Research Council. Simonsen, L. M. & Dick, T. P. (1997). Teachers’ perceptions of the impact of graphing calculators in the Mathematics classroom. Journal of Computers in Mathematics and Science Teaching, 16(2-3), 239. 108 Smith, H. J., Higgins, S., Wall, K., & Miller, J. (2005). interactive whiteboards: Boon or bandwagon? A critical review of the literature. Journal of Computer Assisted Learning, 21(1), 91-101. Smith, F. Hardman, F. & Higgins, S. (2006). The impact of interactive whiteboards on teacherpupil interaction in the National Literacy and Numeracy Strategies. British Educational Research Journal, 32(3), 443-457. Stephen J. P. & Tchoshanov, M. A. (2001). The Role of Representation(s) in Developing Mathematical Understanding, Theory into Practice, 40(2), 118-127. Swan, K. Kratcoski, A. Schenker, J. & Van H. (2010). Interactive whiteboards and student achievement. In Thomas, M. and Schmid, E. C. (Eds.), Interactive Whiteboards for Education and Training: Emerging Technologies and Applications. Hershey, PA: IGI Global. Taylor, P. G. (2000). Changing Expectations: Preparing students for Flexible Learning. The International Journal of Academic Development 5(2), 107-115. Vanderstoep, S. W. & Johnson, D. D. (2009). Research Methods for Everyday Life: Blending Qualitative and Quantitative Approaches. San Francisco: Jossey Bass Vincent, J. (2005). Interactive geometry software and mechanical linkages: Scaffolding students’ deductive reasoning. In W. J. Masalski, & P. C. Elliott (Eds.), Technology-supported learning environments: Sixty-seventh yearbook (pp. 93-112). Reston, VA: National Council of Teachers of Mathematics (NCTM). 109 Vygotsky, L. S. (1978). Mind in the Society: The development of higher Psychological process. Cambridge, MA: Harvard University Press. Ward, L. (2003). why aren't our secondary school teachers using computers in their classroom? Educational Research Risks and Dilemmas (pp. 24-67). New Zealand: Auckland publishers. West Africa Examination Council (2003). Senior Secondary School Certificate Examination, Chief Examiners' Report. Accra: West African Examination Council. West Africa Examination Council (2004). Senior Secondary School Certificate Examination, Chief Examiners' Report. Accra: West African Examination Council. West Africa Examination Council (2005). Senior Secondary School Certificate Examination, Chief Examiners' Report. Accra: West African Examination Council. West Africa Examination Council (2006). Senior Secondary School Certificate Examination, Chief Examiners' Report. Accra: West African Examination Council. World Bank (1998), The World Development Report 1998/99. Quoted in Blurton, C. New Directions of ICT-Use in Education. Washington, DC: New Challenges for Tertiary Education. Yidana, Issifu & Amppiah, M. E. (2003). The role of information communication technology (ICT) in national development: the challenges for our society. Mathematics Connections, 3(3), 35-43. Zbiek, M. R., Heid, M. K. Blume, G. W. & Dick, T. (2007). Research on technology in Mathematics education: A perspective of constructs. In F. Lester, Second handbook of 110 research on Mathematics teaching and learning (pp. 1169-1204). Charlotte, NC: Information Age. Zeller, A. R. (1988). Educational research. In A. R. Zeller, Validity. In J.P Kees (Ed) (p. 322330). New York: Pergamon press. Zittle, F. J. (2004). Enhancing Native American Mathematics learning: The use of Smart boardgenerated virtual manipulative for conceptual understanding. Retrieved October 8, 2007 from http://edcompass.smarttech.com/NR/rdonlyres/3E2A063B-6737-400F-BD07 1D239C428729/0/Zittle.pdf 111 APPENDIX I: PRE-TEST AND POST TEST QUESTIONS 112 POST-TEST 113 Table1: APPENDIX II: RESPONSES FROM IN-CLASS ACTIVITIES Responses based on Worksheet 1 and Applet A Group Item 3HB 3HC 3HD 3HE 3HF 3HG 3HH 3H1 Figure 1: X0 AM TA AM TA AM TA AM 67.3 A 90 RA 135.4 O 270 44 A 90 RA 135 O 292.6 84.4 A 90 RA 135 O 274.0 44.8 A 90 RA 135 O 281 61.9 A 90 RA 120.9 O 292.6 26.6 A 90 RA 135 O 270 67.4 A 90 RA 128.4 O 299.6 67.38 A 90 RA 123.7 O 304.3 52.1 A 90 RA 123.7 O 292.6 O 292.6 TA R R R R R R R R R R Figure 2: Y0 Figure 3: Z0 Figure 4: V0 3HA AM: Angle Measure Table 2: Group Figure 1 Figure 2 TA: Type of Angle A: Acute RA: Right Angle O: Obtuse 3HJ 53.2 A 90 RA 134 R: Reflex Responses based on Worksheet 2 and Applet B ISSUE 3HA 3HB 3HC 3HD 3HF 3HG 3HH 3H1 3HJ SOA 90 90 90 90 3E 90 90 90 90 90 90 TN CA CA CA CA CA CA CA CA CA CA SOA 180 180 180 180 180 180 180 180 180 180 TN SA SA SA SA SA SA SA SA SA SA SOA 360 360 360 360 360 360 360 360 360 360 TN AP AP AP AP AP AP AP AP AP AP SOA 360 360 360 360 360 360 360 360 360 360 TN VOA SOA: Sum of Angles SA: Supplementary Angles VOA VOA VOA VOA VOA VOA VOA VOA VOA Figure 3 Figure 4 TN: Technical Name AP: Angles at a point 114 CA: Complementary Angles VOA: Vertically Opposite Angles Table 3: Responses Based on Worksheet 3 and Applet C. Group Marked Angle X0 Y0 W0 Z0 U0 S0 T0 V0 3H A 3H B AD AD 120.26 132.27 59.74 47.73 120.26 132.2 59.74 47.8 120.26 132.01 59.74 47.67 59.74 132.42 120.26 47.65 3H C AD 135 45 135 45 135 45 135 45 3H D AD 139.4 40.6 139.4 40.6 139.4 40.6 139.4 40.6 3H E AD 160 20 160 20 160 20 160 20 3H F AD 135 45 135 45 135 45 135 45 3H G AD 125.54 125.54 54.46 54.46 125.87 54.13 54.13 125.87 3H H AD 137.53 42.47 137.53 42.47 137.53 42.47 137.53 42.47 3H I AD 145.17 145.17 34.83 34.83 145.17 34.83 34.83 145.17 3H J AD 126.87 126.87 53.13 53.13 126.87 53.13 53.13 126.87 AD: Angle in Degree 115 Relationship/Re marks Table 4: CA 3HA W, V, X, U 3HB 132.2 & 132.42 47.8 & 47.73 3HC X=W, Y=X, Z=V, U=T They are equal They are equal Z, S, W, U 132.2 & 132.42, 47.8 & 47.73 U=S, X=V, They are equal They are equal They are equal AA SA Responses based on Worksheet 4 S& V, U & S, T& V, X & Y, X& Z, Y & W, Z& W Add up to 180 W& S, Z &U IA Add up to 180 132.2 & 47.8, 132.27 & 47.73, 132.13 & 47.78 Add up to give 180 132.2 & 47.78 ,132.42 & 47.73 Add up to give 180 Are equal 3HD 139.4 & 139.4, 40.6 & 40.6 Are equal 139.4 & 139.4, 40.6 & 40.6 They are equal 3HE 3HF 3HG 3HH 3H1 3HJ X=U, W=T, Y=S, V=Z X=U, W=T, Y=S, Z=V Y=V, X=V W=T W&S, Y&V W&S, Y&V Are equal Are equal Are equal Are equal Are equal Are equal X=W, U=T, Y=V, Z=S X=S, W=V, W=S, Y=U Z&S, W&U, Z&S, U&Y Z&S, U&Y They are equal They are equal They are equal They are equal They are equal They are equal S+W, Y+Z, U+T, S+V X+Z= 180, W+Y= 180 X&Y, Z&W, U&S, V&T, U&V X&W, Y&S X&W, Y&S Sum up to 180 Sum up to 180 Sum up to 180 Sum up to 180 Sum up to 180 S& W, U &Z Y&S, U&Z Y&S, U&Z Sum up to give 180 Sum up to give 180 Sum up to give 180 20+160 =180, Z+Y=180 Z+U=180 X+S=180 Y+X=180 V+S=180 U+V=180 139.4 + 40.6, 160+20 =180, 139.4 + 40.6 160+20 =180, 20+160 =180 Sum up to 180 X+S=180, U+V=180 Sum up to give 180 Sum up to 180 Sum up to 180 139.4 + 40.6 Y+V, W+S , 139.4 + 40.6 Sum up to give 180 Sum up to give 180 CA: Complementary angles SA: Supplementary angles angles 116 Sum up to give 180 Y+S= 180, U+W= 180 Sum up to give 180 AA: Alternating angles IA: Interior Table 5: Group Responses based on worksheet 5 and Applet D. Item Figure 1: X0 Figure 2: Y0 Figure 3:Z0 Figure 4: V0 3HA 3HB 3HC 3HD 3E 3HF 3HG 3HH 3H1 3HJ 90, 53.13, 36.87 AM 90, 45, 45 90, 45, 45 90, 51.34, 38.66 90, 36.87, 53.13 90, 51.34, 38.66 90, 53.13, 36.87 90, 45, 45 90, 45, 45 90, 45, 45 TT RT RT RT RT RT RT RT RT RT 94.76, 63.43, 21.8 64.44, 58.67, 56.89 67.31, 64.25, 48.44 126.87, 31.33, 21.8 96.9, 49.2, 33.82 133, 33, 14 143.13, AM 135, 26.57, 18.43 23.2, 13.67 126.87, 29.74, 23.39 RT 116.57, 45, 18.43 TT ST ST ST ST ST ST ST ST ST ST AM 67.38, 56.31, 56.31 68.2, 68.2, 43.6 90, 45, 45, 90, 45, 45, 63.43, 63.43, 53.13 90, 45, 45, 63.43, 63.43, 53.13 68.75, 62.5, 58.75 62.25, 58.87, 58.87 63.43, 63.43, 53.13 TT IT IT IT IT IT IT IT IT IT AM 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, IT 56.08, 56.08, 56.08 TT ET ET ET ET ET ET ET ET ET ET AM: Angles measured TT: Type of triangle IT: Isosceles triangle ET: Equilateral triangle RT: Right angle triangle 117 ST: Scalene triangle APPENDIX III: QUESTIONNAIRE The purpose of this questionnaire is to gather data to help evaluate reflections of students on the effects of IGS on students’ motivation and how it supports students’ centred learning. I there for call on your objective responses to facilitate accurate of the evaluation process. The questionnaire is intended to help the researcher to answer the following research questions 1. How does the use of IGS motivate SHS students to learn geometry? MOTIVATION I enjoyed the lessons with the use of IGS The IGS helped me a lot to learn the geometrical concepts taught The use of IGS in the geometry lessons makes the lesson boring From now on, I want to learn all geometry lessons with computers. I hate Mathematics lessons that I take at the computer laboratory. I am not happy any time I missed the geometry lesson held at the 118 Strongly Disagree Disagree Neutral Agree To what extent do you agree with the following statement Strongly Agree 2. In what ways does IGS support student-centred learning in a geometry class? laboratory THE USE OF IGS IGS helped me to understand geometric concepts when they were taught The use of IGS in the classroom made my learning easier and interesting. The textbook alone helped me a lot to learn the geometry concepts I felt helpless when asked to explore and study the learning materials presented in the lesson How often did you observe the following in the IGS lesson taught in your class? Just tick the column that suit your situation Statement Every Half of Some Lesson the lessons lessons STUDENT CENTRED LEARNING The teacher made the students very active participants in the learning process in the classroom The teacher provided all students the opportunities to ask questions Teacher engaged students in a lot of guided group activities Students interacted with each other in the group activity freely and purposively. Students played a lead role in exploring the software on their own, whiles the teacher played a facilitated. 119 Never APPENDIX IV: INTERVIEW TEACHERS AND STUDENTS UNIVERSITY OF EDUCATION, WINNEBA, WINNEBA Department of Mathematics Education, IGS Study INTERVIEW GUIDE FOR TEACHERS Dear Teacher The purpose of this interview is to gather data to help evaluate your reflections on the effects of Interactive Geometric Software (IGS) on the students’ learning. Your objectivity will therefore be greatly appreciated. Thank you. Do you think the use of IGS is important in helping students learn geometry? Yes No Give reasons for your answer ………………………………………………………………………………… 1. Does IGS motivate students to learn geometry? Yes No if no, go to question 3 If yes, in what forms/ways does it motivate students? ………………………………………………………………………………………….. 2. In your opinion do you think IGS enhances the students’ learning outcomes? Yes No If yes, how does it enhance students learning outcomes? …………………………………………………………………………………… If no why does it not enhance students learning outcomes? ……………………………………………………………………………………. 3. As a teacher do you think IGS supports student-centred learning? Yes No If what ways does it support student centred-learning? …………………………………………………………………………………….. If no, why does it not support student centred-learning? ………………………………………………………………………………………… 4. From your perspective, has the IGS been helpful in developing students’ conceptual understanding of geometry? Yes No If yes how does it support conceptual understanding of geometry? ……………………………………………………………………………………… If no, why does it support conceptual understanding of geometry? ................................................................................………………………………… 120 5. Were there any challenges when the geometry lessons were taught using IGS? Yes No If yes, what are some of these challenges you encountered? ……………………………………………………………………………………… 6. Did the Worksheets help classroom interaction among students and between students and teacher? Yes No If yes, how did the Worksheets help classroom interaction? ……………………………………………………………………………………… 7. Was the IGS lesson student centred lessons? In what ways do the IGS lessons depict a typical student centred learning session? 8. In summary how useful is the use of IGS to you and your students? .……………………………………………………………………………………... 121 UNIVERSITY OF EDUCATION, WINNEBA, WINNEBA Department of Mathematics Education, IGS Study INTERVIEW GUIDE FOR STUDENTS Dear Student The purpose of this interview is to gather data to help evaluate your reflections on the effects of Interactive Geometric Software (IGS) on your learning. Your objectivity will therefore be greatly appreciated. Thank you. 1. Do you think the use of IGS has helped you in learning geometry? Yes No In what ways did the IGS help you in learning geometry? ……………………………………………………………………………………… 2. Do you think the IGS has increased your interest and time to learn geometry? Yes No if no, go to question 3 If yes, in what forms/ways does it motivate you? ……………………………………………………………………………………… 3. Do you think the IGS has increased your performance in Mathematics? Yes No If yes, how does it enhance students learning outcomes? ……………………………………………………………………………………… If no why does it not enhance students learning outcomes? ……………………………………………………………………………………… 4. As a student do you think IGS supports student-centred learning? Yes No In what ways does it support student centred-learning? ……………………………………………………………………………………… If no, why does it not support student centred-learning? ………………………………………………………………………………………...... 5. From your perspective, has the IGS been helpful in developing students’ conceptual understanding of geometry? Yes No 122 If yes how does it support conceptual understanding of geometry? ……………………………………………………………………………………… If no, why does it support conceptual understanding of geometry? ................................................................................………………………………… 6. Were there any challenges when the geometry lessons were taught using IGS? Yes No If yes, what are some of these challenges you encountered? …………………………………………………………………………………….... ........... 7. Did the Worksheets help classroom interaction among students and between students and teacher? Yes No If yes, how did the Worksheets help classroom interaction? ……………………………………………………………………………………… 8. In summary how useful is the use of IGS to you and your students? .…………………………………………………………………………………............. 123 APPENDIX V: LESSON PLANS I. LESSON PLAN Subject: Introduction to GeoGebra software Duration of lesson: 90 minutes Target group: SHS 3H Teacher: Bennin Juatie Douri Date: 23rd of September, 2011 ************************************************************************ I. Relevant Previous Knowledge: Students are familiar with basic computer operations. They are able to use mouse and keyboard as inputs and to monitor corresponding outputs on the screen. II. Teaching and learning Materials: 1. GeoGebra 3.4.2.0, installed in all the computers. 2. Projector and Screen. 3. Applets and Worksheets III. Learning Objectives: By the end of the lesson the student should be able to: 1. Recognize the menu and toolbar of the GeoGebra software 2. Identify some basic tools use to draw basic geometric objects IV. Teacher learner activities: The teacher gives a brief explanation about the philosophy of the interactive geometry software and how it is use to learn geometry and other mathematics topics. The teacher then demonstrates to the students how the software works in a geometry classroom. The students are then made to follow the instructions of the teacher and apply them on their 124 desk top computers. The instructions are based on the GeoGebra introductory book taken from the official website of GeoGebra13 for introducing students to GeoGebra the first and second chapters are primarily covered. After getting to know about GeoGebra, the students under the coaching of the teacher are encouraged to draw basic geometric objects. The students are involved in drawing basic geometric objects such as triangle, circle, and polygon. The teacher creates an opportunity for students to explore the software and draw various geometric structures. The students continue working with GeoGebra under the instructions and support of the teacher. The students are then made to explore the software drawing all manner of geometric figures while the teacher comes around to inspect their work. 125 II. LESSON PLAN Subject: Type of angles and their Properties Duration of lesson: 90 minutes Target group: SHS 3H Teacher: Bennin Juatie Douri Date: 25th of September, 2011 ************************************************************************ I. Relevant Previous Knowledge: Students are familiar with the menu and toolbar functions and can draw some basic geometric objects using the GeoGebra. Students can construct angles and parallel lines in the classroom. II. Teaching and Learning Materials: GeoGebra 3.4.2.0 installed in computers, Applets and Worksheets, Projector and Screen. III. Learning Objectives: By the end of the lesson the student should be able to: 1. Construct basic type of angles. 2. Measure angles formed by two straight lines. IV. Teacher Learner Activities: The researcher reviewed students’ previous knowledge on the GeoGebra software. Some students demonstrated on the computer connected to the projector on how to construct triangles and parallel lines to the rest of the class members. The teacher asked students questions on the definition of an angle and the properties of angles. The teacher then introduces the new lesson on type of angles and their properties. 126 The teacher makes each group to launch the GeoGebra programme installed in their computers for the lesson to begin. Each group was given two paper Worksheets prepared as activities for them to work in groups and printed copies of the Applets as a guide. Activity: 1 The teacher gave a brief explanation of what to do with the first paper and the applet 1. Students are to explore Applet 1 and to find out answers needed to complete the table given in Worksheet 1 (see Appendix A). Students are advice to discuss their ideas with each other before completing the Worksheets. The students are to explore and draw four different types of angles as in applet 1 and measure the angles enclosed. In addition the students are to find the technical names of those angles they have drawn using the software. The teacher then goes round to see and inspect the students work and to offer technical advice. Activity 2 In this activity the students are to explore applet 2 (Appendix B), measure the angles and consequently identify the technical names of those angles enclosed. The teacher explains what the students are to do and then goes round to facilitate the students work. The teacher then organizes an end of lesson discussion where students who had serious challenges and problems during the lesson delivery and could not discuss them adequately at the small group level for whole class to brainstorm and find solution to them. 127 III. LESSON PLAN Subject: Properties of parallel lines Duration of lesson: 90 minutes Target group: SHS 3H Teacher: Bennin Juatie Douri Date: 27th of September, 2011 I. Relevant Previous Knowledge: Students are familiar with concept of lines and how to measure angles. Students have also learnt parallel lines in the classroom during the traditional lessons. II. Teaching and Learning Materials: GeoGebra 3.4.2.0 installed in computers, Applets and Worksheets, projector and screen. III. Learning Objectives: By the end of the lesson the student should be able to: 1. Construct basic parallel lines. 2. Identify some basic properties of parallel lines. IV. Activities: The teacher reviewed the students’ previous knowledge on angles and their properties. Some students demonstrated on the computer connected to the projector how to construct triangles and parallel lines to the rest of the class members. The teacher then asked students questions on the definition of parallel lines. The teacher then made each group to launch the GeoGebra programme installed in their computers for the lesson to begin. Each group was given two paper Worksheets prepared as activities for them to work in groups and printed copies of the Applets as a guide. 128 Activity: 1 The teacher gave a brief explanation of what to do with the first paper and the applet 1. The groups were asked to explore Applet 3 and to find answers needed to complete Worksheet 3 (Appendix B). The teacher encouraged students to discuss with each other before completing the worksheet 3. The students are to find the relationship that existed between the angles that are made by the transversal. Based on the numbers they discovered from interacting with Applet 3. The teacher then goes round to inspect the students work and to offer technical advice. Activity 2 In this activity the students are to further explore applet 3 (Appendix B), measure the angles and consequently establish that exist between the angles enclosed in the applet 3. The teacher explains what the students are to do and then goes round to facilitate the students work. At the end of the lesson the teacher initiated whole class discussions were students contribute to common problems with only slight guidance of the teacher. 129 1V. LESSON PLAN Subject: Triangles and their Properties. Duration of lesson: 90 minutes Target group: SHS 3H Teacher: Bennin Juatie Douri Date: 5th October, 2011 ************************************************************************ I. Relevant Previous knowledge: Students are familiar with concepts of the circumference of circle, and the radii of circles both inscribed inside and circumscribed around the given regular polygon. II. Teaching and Learning Materials: GeoGebra 3.4.2.0 installed in computers, Applets and Worksheets. III. Lesson Objectives: By the end of the first lesson the student should be able to: 1. Draw at least two different types of triangles. 2. Measure the angles in a triangle IV. Activities: The teacher reviewed students’ previous knowledge on parallel lines. Some students demonstrated on the computer connected to the projector how to construct and measure angles enclosed by the transversal and the parallel lines. The teacher then introduced the new lesson on triangles and their properties. The teacher asks the students question on the properties of triangles. 130 The teacher then made each group to launch the GeoGebra programme installed in their computers for the lesson to begin. Each group was given two paper Worksheets prepared as activities for them to work in groups and printed copies of the Applets as a guide. Activity 1 The teacher briefly explains what students should do with the applet 4. Students are to explore Applet 4 and find answers that will complete the table given in the worksheet 5. The task in Worksheet 5 demanded that students to explore applet 4 and then draw four different types of triangles as in applet 4 and measured the angles enclosed in the triangles. Activity 2 In this activity students are to discuss extensively among themselves and come out with the technical names of those triangles drawn using the software. The teacher then moves round to make technical inputs while the students take charge of their learning process. At the end of the lesson a whole class discussions is initiated to help students whose problems were not attended to during the lesson delivery. This is to enable the students have common knowledge on the issues discussed in the classroom. 131 APPENDIX VI: APPLETS APPLET: A 132 APPLET: B 133 APPLET: C 134 APPLET: D 135 APPENDIX VII: WORKSHEETS Worksheet 1 Task: Working with the applet 1, fill in the table below with the appropriate data you make from the applet? Figure 1 2 3 4 Type of Angle Angle/range in 90 degrees Worksheet 2 Task 1: Working with the applet 2, fill in the table below with the appropriate data you make from the applet. What can you say about the characteristics of the angles drawn? Figure 1 2 3 Name Sum of Angles 136 4 Worksheet 3 Task 1: Working with the applet 3, fill in the table below with the appropriate data you make from the applet. Measure and record the angles marked on the diagram MARK Xo Yo Wo Zo Uo So To Vo ANGLE IN DEGREE Task 2: Based on the data you have filled in the table, what can you judge regarding the values of the angles recorded? Relationship Angles Remarks Corresponding angles Alternate angles Supplementary angles Interior angles 137 Worksheet 4 Working with the applet 4, fill in the table below with the appropriate data you make from the applet. Measure and record the angles in each diagram. Type of triangle 1 2 Task 3: What is the relationship between the angles? 138 3

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# appendix iv: interview teachers and students