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.
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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.
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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
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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.
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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
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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
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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
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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
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(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
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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).
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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.
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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.
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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
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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
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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.
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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.
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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.
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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?
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3
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appendix iv: interview teachers and students