UNIVERSITY OF EDUCATION, WINNEBA
DEPARTMENT OF MATHEMATICS EDUCATION
FACULTY OF SCIENCE
COMPARISON OF THE USE OF GEOGEBRA AND THE TRADITIONAL
APPROACH TO TEACHING ANGLES, AT BIMBILLA COLLEGE OF
EDUCATION
ANAS SEIDU SALIFU
7100110007
MAY, 2013
UNIVERSITY OF EDUCATION, WINNEBA
DEPARTMENT OF MATHEMATICS EDUCATION
FACULTY OF SCIENCE
COMPARISON OF THE USE OF GEOGEBRA AND THE TRADITIONAL
APPROACH TO TEACHING ANGLES, AT BIMBILLA COLLEGE OF
EDUCATION
ANAS SEIDU SALIFU
7100110007
This Dissertation in the Department of Mathematics Education, Faculty of
Science, submitted to the School of Graduate Studies, University of Education,
Winneba in partial fulfilment of the requirements for award of Master in
Mathematics Education degree
MAY, 2013
DECLARATION
Candidate’s Declaration
I hereby declare that this dissertation is the results of my own original research and that no
part of it has been presented for another degree in this university or elsewhere.
Name of Candidate: Anas Seidu Salifu
Candidate’s Signature………………………
Date……………………………………………
Supervisor’s Declaration
I hereby declare that the preparation and presentation of the dissertation were supervised in
accordance with the guidelines on supervision of dissertation laid down by the University
of Education, Winneba.
Name of Supervisor: DR. Issifu Yidana
Supervisor’s Signature……………….…
Date…………............................…………
3
ABSTRACT
I was interested in conducting action research using geogebra. A free open-source
dynamic software for mathematics teaching and learning that links geometry and algebra
into a single easy-to-use package. Using this software for teaching mathematics at the
College of Education was my attempt to solving poor academic performance in geometry.
The dynamic and interactive medium provides students with the opportunity to share and
discuss the emergent visual phenomena. The research is based on social constructivist view
of learning. I conducted this research study in Bimbilla E. P. College of Education with
seventy-two (72) students. Purposive and convenient sampling procedure was adopted in
selecting the sample, to explore the use of geogebra tool in an active learning arrangement.
I used geogebra as a tool with three (3) research questions relating to the aspect of: (1)
Comparing student’s performance on geogebra approach against traditional approach. (2)
Students attitudes towards geogebra approach. (3) The effects on the student’s ability in
seeing, reasoning and measuring of angles using geogebra.
The intervention was based on teaching Science and General Arts Students for a month.
The collected research data were drawn from Pre-test, Post-test and closed type of
questionnaire. The data were analysed separately and used to answer the research questions
by using SPSS for the achievements test. From the data, it was clear that, the introduction
of geogebra influenced students performance positively in the post test .Also, student’s
attitudes towards geogebra was excellent. Students also improved their skills in measuring,
seeing and reasoning during the intervention. I wish to recommend that, instructional
technology be added to the courses of the Colleges of Education. So that students will
continue to practice and integrate it in the basic levels.
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ACKNOWLEDGEMENT
The printed pages of this dissertation hold far more than the culmination of years of
study. These pages also reflect the relationships with many generous and inspiring people I
have met since beginning my graduate work. The list is long, but I cherish each
contribution to my development as a scholar and teacher.
I would like to express the deepest appreciation to my supervisor, Dr Issifu Yidana
who has the attitude and the substance of a genius: he continually and convincingly
conveyed a spirit of adventure in regard to research and scholarship, and an excitement in
regard to teaching. Without his guidance and persistent help this dissertation would not
have been possible. I would like to thank Dr Johnson Michael Nabie, and Peter Akayuure
of the mathematics department, University of Education, Winneba, for always been ready
to offer advice. In addition, a thank you to late Professor Sandra Turner of Ohio University,
who introduced me to computers literacy, and whose enthusiasm for the “underlying
structures” had lasting effect. Financial support was provided by Mr. Jerry Howarth of
England. I also want to thank him for his fatherly care and motivation throughout my
studies. I also thank my brothers, Sgt Salifu Shirazu 48 Engineers Accra, Cpl Salifu Nasser
of Community two police station, and Salifu Qathafi of university of Ghana, legon for their
brotherly advice. A special feeling of gratitude to my loving parents, W.O.1 (RTD) Salifu
Seidu and Safura Adam whose words of encouragement and push for tenacity ring in my
ears. My sisters Sherifa Salifu and Faiza Salifu have never left my side and are very special.
I dedicate this work and give special thanks to my wife Sawudatu Yahaya of Air force base
Tamale and my wonderful daughters Hifza, Rizka and Shamsia for being there for me
throughout the entire master’s program.
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DEDICATION
I dedicate my dissertation work to my family and many friends.
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TABLE OF CONTENT
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Table of Content
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DECLARATION
Abstract
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Acknowledgement
Dedication
CHAPTER ONE
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Background of the Study
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Statement of the Problem
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Purpose of the Study …
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Objectives of the study
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Expectation of the research
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Significance of the Study
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Delimitations …
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Definition of Terms
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The organizational plan of the study …
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Introduction
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Research Questions
CHAPTER TWO : LITERATURE
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Ghana’s National ICT4AD Policy …
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Brief background on ICTs in education in Ghana
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Research on Technology Integration …
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Overview
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REVIEW
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Introducing New Technology: Calculators and Computers …
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Types of Software Tools Used for Mathematics Education …
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Why integrate ICT in mathematics education?
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Why is GeoGebra different from other Mathematical Software’s? …
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GeoGebra’s User Interface
What is GeoGebra?
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Short History of GeoGebra
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What has GeoGebra got to offer?
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Teaching Mathematics with GeoGebra
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The advantages of using GeoGebra …
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Active Learning
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Seeing …
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Measuring
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Reasoning
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Theoretical Framework
Summary of Literature Review
CHAPTER THREE: METHODOLOGY
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Research Design
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Target Population
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Sampling Procedure …
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Overview
Sample
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Research Instruments …
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Research Design Plan …
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Intervention Process …
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Content Validity
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Data Collection
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Data Analysis …
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Intervention Activities Traditional Approach (Without Computers) …
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Pre-Intervention
Reliability
Intervention Activities Geogebra Approach (With Computers)
CHAPTER FOUR : RESULTS AND DISCUSSION
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Presentation of Pre –Intervention Tables
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Presentation of Post Intervention Tables
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Overview
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Discussion of results …
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Analysis of Results
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATION
Overview of the study
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Summary of key findings
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Recommendation
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Conclusion
Limitations of the study
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Appendix B2: Post Intervention Marking Scheme
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Appendix C: Pre and Post Test Raw Marks …
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Appendix E: Questionnaire and its Results …
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Suggestion Areas for Further Research
References
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Appendix A1: Pre-Intervention Questions
Appendix A2: Pre-Intervention Marking Scheme
Appendix B1: Post Intervention Questions
Appendix D: Intervention Pictures
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Appendix G: List of Figures …
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Appendix J: Sample solution of Post Test… ….
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Appendix F: List of Tables
Appendix H: SPSS Table
Appendix I: Sample Solution of Pre-Test
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CHAPTER 1
INTRODUCTION
1.0 Overview
This chapter deals with background to the study, statement of the problem, purpose
of the study, objectives of the study, research questions, significance of the study,
delimitation and finally the organizational plan of the study.
1.1
Background of the Study
The Teacher Training Colleges of Ghana are now known as Colleges of Education
in transition (since 2005) and are now awarding diploma in Basic Education instead of
teacher’s certificate A. So as a result of that, they are now tertiary institutions since their
bill was passed by the Parliament of Ghana in February 2012. The mathematics content
curriculum has been maintained. However, there has been an introduction of ICT as a
component of teaching mathematics both in the mathematics methods and content course
outlines (Mathematics Syllabus for Diploma in Basic Education, 2005; 2006). The reforms
also introduced special mathematics and science courses among fifteen (15) Science
Colleges, of which Bimbilla College is one. The College is located at the eastern corridor of
Ghana. The district is Nanumba North with Bimbilla its capital city in the Northern Region.
The main purpose of the College is to produce science and mathematics teachers to
cater for the inefficiencies of teaching and learning science and mathematics at the basic
level of the country educational leather. In Ghana, pre-service teachers are admitted directly
into Teacher Colleges of Education based on their performance in the Senior Secondary
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Certificate Examination/ West Africa Senior Secondary Certificate Examination
(SSCE/WASSCE. However, the performance of these students in geometry has been rather
weak or inadequate (The WAEC Chief Examiners report of the SSCE, 1996, 2000, 2001,
2003, 2005 and 2006).
In the Colleges of Education in Ghana, geometry is stated as one of the areas in the
course structure. This is captured in the course outlines of both the content and
methodology. In the content, geometry includes lines, angles, polygons, geometrical
construction, 2D and 3D shapes, circles theorem, geometrical transformation and
coordinate geometry has said that “the inclusion of geometry in both content and
methodology is not only to equip pre-service teachers with subject matter, but more
especially to expose them to more pedagogy on how to teach it effectively at the basic level
of education”. (Institute of Education, UCC, 2005. Acquah, (2008 p.1))
Over the last two decades, many students have tried to draw a perfect circle with a
compass in a Geometry class, only to be frustrated by the results. Depending on the quality
of the compass, or the care of the student, the result may approximate a circle except that it
may appear more like the start of a spiral. Other geometric constructions and activities
suffer similar problems in their accuracy. Cutting geometric shapes as part of an
investigative activity, and reaching unsatisfactory conclusions due to the inexact cutting of
the pieces shows the problem of inaccuracies. These inaccuracies are part of the motivation
for the design of computerized geometrical drawing programs.
Based on this, it is a fact that teaching mathematics is a very challenging thing to do
because mathematics teachers, students, and mathematics content should be running
together in harmony, this means that mathematics teachers teach the subject using an
12
appropriate teaching method for students. When appropriate methods and tools are used in
teaching the subjects, students are able to engage in their learning activities, and the
mathematics content taught is suitable to students’ level of thinking. One of the possible
problems in teaching and learning mathematics is that mathematics teachers, in their
opinion, think that their regular teaching method is appropriate to students without trying to
evaluate whether or not students are really satisfied with their teaching methods. On the
other hand, students do not think that their mathematics teachers use an appropriate
teaching method for teaching them. Meanwhile mathematics is still considered as a difficult
subject to learn by some students. However, they must take it, because mathematics is one
of the compulsory subjects that students must accomplish in their study, at least from
primary school to Colleges of Education levels in Ghana. That is why some students feel
“tortured” every time they meet a mathematics class in their study. Actually, there are some
reasons for this condition, such as students do not see the relevance of mathematics to their
life, which makes it hard for them to understand the subject, mathematics teachers do not
provide suitable and appropriate teaching methods so that students are not engaging in
learning mathematics; there is not enough teaching-learning media to help students become
more enthusiastic and motivated to learn mathematics. I think that to overcome this
condition, mathematics teachers should be more innovative in their teaching methods.
According to observations made by Mereku, et al (2009), teachers are not fully utilizing
these facilities in teaching, especially computers. It is the expectation of the 2007
educational reforms that teachers integrate computers in teaching. Mathematics teachers
also should consider the use of ICT in their teaching, because use of ICT in education can
help students understand and help teachers explain mathematics subjects more effectively.
13
There exists a lot of research (Chrysanthou, 2008; Preiner, 2008; Mulyono, 2010)
investigating the use of ICT in education, and these studies show that students become
more independent in doing their learning activities when they engage in learning through
ICT tools. An example of the use of ICT in education is the use of mathematics software to
teach students. Mathematics software is a kind of application program that has the special
function to help its user understand about the mathematics topics they learn. There are
many kinds of mathematics software, such as Derive 5, Sketchpad, Cinderela, Geogebra,
and Mathematica. Mathematics software nowadays is easy enough to use and to
understand; most are really user friendly, so that users do not need special computer skills
to use them. Some mathematics software is free which means that users can use the
software without needing a license.
One of the free mathematics software programs is Geogebra, a kind of software
called dynamic geometry software (DGS). The definition of dynamic geometry is: "the
theory of construction-like descriptions of function-like objects under parameter changes"
(Kortenkamp, 1999). Meanwhile dynamic geometry software is a computer program by
which a user can construct or create any plane geometrical shape. Therefore dynamic
geometry software is really helpful for teaching and learning geometry, because it has a lot
of tools that can be used to visualize and to construct geometrical shapes in simple ways.
By learning geometry through dynamic geometry software, the researcher hopes that
students will be more excited about learning geometry, and it will make them engage more
in their learning activities.
Teaching and learning geometry traditionally only uses common tools, such as a
chalkboard, a protractor, a ruler, and a compass. This does not necessarily mean that a
14
traditional way of teaching geometry is not appropriate to students. However, to use those
tools is sometimes difficult for students and it takes time to create or construct geometrical
shapes. Therefore some students will lose the time needed to understand geometry, because
the drawing of geometrical figures is so time consuming, and this condition makes students
think that learning geometry is not fun and it is difficult as well. It shows that students’
achievement and opinion are affected by the tools used for teaching.
This condition triggered the researcher to write on teaching geometry through dynamic
geometry software, Geogebra as compared to the traditional methods.
1.2 Statement of the Problem
National ICT policies have reached an established position in both developed and
developing countries. A study funded by the Australian Department of Education, Science
and Training revealed that most national ICT policies focus on the educational sector
(Kearns & Grant, 2002). Education is put forward as the central actor to pursue and attain
the objectives of the ICT policy; other sectors are expected to benefit indirectly from this
approach. Educational ICT policies have been designed in a variety of ways, depending on
the dominant rationales that drive curriculum development.
Although, countries that are at the beginning of using new technology, its future use
in education cannot be underestimated (Carnoy & Rhoten, 2002).
Due to the Ghana
Government policy on curriculum reform, it is necessary for ICT to be integrated and
utilized effectively in schools. Also, exploitation of ICT in teaching improves students
learning and thus develops skills necessary for the competition in knowledge economy and
information society, as explained in the draft copy of Ghana ICT in education policy,
15
Ministry of Education and Sports, (2006). The shift in focus from traditional methods of
teaching to teaching for understanding and investigative learning, computer based
instruction has become ideal tool for this change. It is assumed that ICT brings
revolutionary change in teaching methodologies. The innovation lies not per se in the
introduction and use of ICT, but in its role as a contributor towards a student-centered form
of teaching and learning (Scrimshaw, 2004).
Unfortunately, most Colleges of Education tutors have not been integrating
technology in their teaching. Most tutors use lecture methods when facilitating face-to-face
sessions, especially where study materials are scanty (Bbuye, 2007). The lecture method is
mainly also used by the mathematics tutors at the Colleges of Education in Ghana. In this
era of constructivism, the challenge is for the tutors of mathematics at the Colleges of
Education to incorporate the use of ICT in their teaching. The mathematics course outlines
of Colleges of Education requires teachers to teach students how to use calculators and
computers to solve mathematics problems. Technology, also expect teachers to integrate
ICT in their teaching and learning (CRDD, 2007). Nolan (2008) stated that it is believed
that the use of ICT would enhance teaching and learning and unlikely to be explored in
meaningful ways in school classroom unless there is effective modeling of technology
integration during the teacher education experience.
Integrating technology is a core component of the course outlines for teaching
mathematics in all levels of the colleges of education programmme. The use of computer
based instruction in teaching and learning of mathematics makes certain topics simplified,
easily absorbed and create a picture of the concepts in the mind of students.
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Therefore, the research seeks to compare the use of Geogebra and the traditional
approach to teaching angles: A case study at Bimbilla College of Education using
achievement scores.
1.3 Purpose of the Study
The purpose of this study is to investigate:
(1) Students’ performance in geometry using Geogebra compare to traditional method.
(2) Students’ attitudes towards Geogebra integration in learning geometry.
(3) Active learning approach using Geogebra help students to improve their abilities of
seeing, measuring, and reasoning in learning geometry.
1.4 Objectives of the study
This study was guided by the following objectives:
 Students will acquire some skills for teaching geometry using the geogebra.
 Students will understand and appreciate the social constructivist approach as a
method of teaching.
 Students would also advocate for ICT integration through their headmasters and
headmistresses.
1.5 Expectation of the research
The researcher expected that the researcher’s intervention would make a significant
difference in students’ achievement between those two classes.
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The researcher expected that the intervention would help students to better understand
geometry.
The intervention is aimed at improving students’ abilities in seeing, measuring and
reasoning. The researcher expects to see improvements in students’ ability to recognize
angles and angle patterns, to measure angles, and to reason about angles.
1.6 Research Questions
The following research questions were formulated to guide the study:
1. To what extent does the difference between the performance of General Arts
students and Science students in geometry (angles) when taught using traditional
approach change when active learning approach with Geogebra is used to teach both
groups?
2. How does the use of the active learning approach using Geogebra relate to students’
attitude to learn angles?
3. To what extent does the active learning approach using Geogebra help students to
improve their abilities of seeing, measuring, and reasoning in learning angles?
1.7 Significance of the Study
The results of this study would help to sharpen most students’ analytical skills in
understanding angles. It would promote and sustain students’ interest to learn geometry as
well as motivate slow learners to improve upon their learning.
This will also help address students’ needs as prospective teachers and/or fight the anxiety
of their future students and to instil and improve attitude towards geometry in general. The
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findings will contribute to greater understanding of students’ attitude towards geometry and
enhance the teaching and learning of mathematics.
It will also add to the existing body of knowledge in the teaching and learning of
geometry. Other researchers can use it as reference for further similar study. The research
work will inform, educate and sensitize teacher trainees to develop confidence and greater
interest and cultivate positive attitudes towards the teaching and learning of geometry.
It will guide and facilitate the formulation of new policy and curriculum development in
mathematics for schools and colleges in Ghana
1.8 Delimitations
I would have wished to do the study in all the colleges in Ghana, but owing to
limited time to write and present the thesis and also financial constraints, the study was
limited to only first year students in Bimbilla College of Education Northern Region. Other
teacher trainees from the universities were not involved in this study because the focus of
this study was on those, trainees of Colleges of Education.
Also, there exists a variety of mathematics software for instructing mathematics but
this study focused on the use of Geogebra software for instructing mathematics because of
it user friendliness and effectiveness as teaching and learning software.
1.9 Definition of Terms
TA Test: Traditional Approach Test
GA Test: Geogebra Approach Test
Geogebra: Is the acronym for geometry and algebra. A dynamic mathematics software for
teaching mathematics.
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SSSCE: Senior Secondary Certificate Examination.
WASSCE: West Africa Senior Secondary Certificate Examination
UCC: University of Cape Coast
1.10 The organizational plan of the study
CHAPTER ONE: In this chapter, the researcher discusses the
background
of the
study, the problem that gave rise to the study and the research questions that guide the
research as well as the significance, purpose , objectives, significance, and scope of the
study.
CHAPTER TWO: This chapter focuses on the relevant literature review based on the
subheadings related to the study and theoretical framework. It induces discussions on the
Ghana’s National ICT4AD and ICTs in education in Ghana, research on technology
integration, why integrating ICT in mathematics education, and types of Software Tools
Used for Mathematics Education. The rest were also touched Background of GeoGebra,
why is GeoGebra different from other Mathematical Software’s, GeoGebra’s User
Interface, teaching Mathematics with GeoGebra, and finally the advantages of using
GeoGebra.
CHAPTER THREE: The methodology used in the research process is discussed in this
chapter; this includes discussions on research
design, population, sample, instruments,
data collection procedure, data analysis techniques, reliability, validity and intervention.
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CHPATER FOUR: Data collected from student’s pre test, post test, and questionnaire are
analyzed and presented in this chapter. The results are used to answer the three (3) research
questions.
CHAPTER FIVE : A summary of the research and the findings made are presented
followed by conclusion. Recommendations are made including suggestions for further
investigations and limitations of the study.
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CHAPTER 2
LITERATURE
REVIEW
2.0 Overview
This chapter reviews and discusses issues in literature relating to using geogebra software.
The review has looked briefly at Ghana’s National ICT Policy, ICT education background
in Ghana, research on technology integration and curriculum changes. The review has also
examined calculators and computers, types of mathematics education software, history of
geogebra, why is it different from other software’s and its user interface.
Other areas it looks at included what geogebra got to offer, teaching mathematics with
geogebra, advantages of using geogebra.
Finally, active learning couple with students attitudes, ability of seeing, measuring and
reasoning, theoretical framework and summary of the chapter, were also discussed.
2.1 Ghana’s National ICT4AD Policy
The Government of the Republic of Ghana has committed to pursuing an ICT for
Accelerated Development (ICT4AD) policy (Ministry of Education and Sports, 2006). This
national policy outlines the plans and strategies for the development of Ghana’s
information society and seeks to provide a framework and plan as to how ICTs can be used
to facilitate amongst other objectives the national goal of “transforming Ghana into an
information and knowledge-driven ICT literate nation” (p.22). The national policy outlines
pillars, of which education is highlighted, as both a critical pillar as well as means to socio22
economic development. Towards this end, some key strategies have been identified. These
include:
 promoting the deployment and exploitation of information, knowledge and
technology within the economy and society as key drivers for socio-economic
development;( MOES, 2006, p.9).
 modernizing Ghana’s educational system using ICTs to improve and expand
access to education, training and research resources and facilities, as well as to
improve the quality of education and training and make the educational system
responsive to the needs and requirements of the economy and society with
specific reference to the development of information and knowledge-based
economy and society; and ( MOES, 2006, p.34).
 improving the human resource development capacity and the Research and
Development (R&D) capacity of Ghana to meet the demands and requirements
for developing the nation’s information and knowledge-based economy and
society. ( MOES, 2006, p.33).
As early as 23 years ago, Hawkridge (1990) from Australia discerned four different
rationales that drive policies related to the integration of ICT and the use of computers in
education:
 an economic rationale: the development of ICT skills is necessary to meet the need for
a skilled work force, as learning is related to future jobs and careers;

a social rationale: this builds on the belief that all pupils should know about and be
familiar with computers in order to become responsible and well-informed citizens;

an educational rationale: ICT is seen as a supportive tool to improve teaching and
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learning;
 a catalytic rationale: ICT is expected to accelerate educational innovations.
The government of Ghana’s ICT policy in terms of education may be in the right
direction because the researcher is of the view that, if ICT is employed in mathematics
education, it will have positive impact on academic outcomes. There will also be positive
attitudes towards the study of mathematics in schools and also make the understanding of
abstract concepts in mathematics better. In the Ghana’s ICT4AD policy, it has been stated
that in addition to better performance in traditional measures of academic achievements, a
secondary benefit of ICTs in education is to familiarize new generations with the
technologies that have become integral components of the modern world.
The researcher believes that using ICT in mathematics education can result in improved
teaching and offer the greatest support to learners from disadvantaged backgrounds.
2.2 Brief background on ICTs in education in Ghana
The efforts to introduce ICTs into the education sector by the Ministry of Education
(primarily through the Ghana Education Service (GES)), its development partners and other
private sector agencies cover over ten (10) years. In the early 1990s, ICT was introduced in
all the tertiary institutions as a general course dubbed “computer literacy” in Ghana.
Also, in the late 1990s, some of the senior secondary schools (SSS) now known as
senior high schools (SHSs) in Ghana had private computer laboratories where they
sometimes had lessons in computer technologies.
24
In a study carried out to review and assess the ICT in Education Initiatives in Ghana
(MOES, 2006), initiatives were selected and their impact assessed to see what lessons could
be learnt. Several positive achievements were noted. Two of them are:
 initiatives contributed to a wider number of teachers acquiring ICT skills and
developing strong interests in ICT; schools involved in the initiatives were
motivated to expand the project and / or acquire more ICT equipment; a number of
private-public partners, including Parent Teachers Associations (PTAs) and civil
society collaborated in the efforts and
 lessons learnt from initiatives provided good examples for other schools to
introduce their own ICT programmes (Ghana ICT in Education policy, MOES,
Draft copy 2006, p.10-11).
2.3 Research on Technology Integration
Technology integration into mathematics teaching and learning provides a very
active field of educational research and technology innovations. The great amount of
available literature offers a wide range of theories, methodologies, and interpretations,
which are often related to the potentialities of new technology for mathematics Education.
Although the successful integration of technology into mathematics classrooms is a very
complex process, researchers tend to tackle very specific aspects instead of trying to
understand the process as a whole (Lagrange, Artigue, Laborde, & Trouche, 2003).
Additionally, most of the research on technology integration is conducted in form of
descriptive studies, which contain reports about best practice examples and how they were
implemented into a mathematical learning environment. Researchers often describe what
25
went well in a specific situation and for a single teacher in his/her individual situation
(Mously, Lambdin, & Koc, 2003). Apart from analyzing classroom learning with
technology, many research studies are also conducted in order to find out more about the
‘instrumental genesis’ (Trouche, 2003; Artigue, 2002; Ruthven, 2002; Mariotti,2002;
Haspekian,2005).
In general, research about technology integration into mathematics education is
dominated by studies about the innovative use of new technology tools as well as their
applications in mathematics education (Lagrange et al., 2003). By contrast, hardly any
publications deal with potential difficulties that could occur during the introduction and
integration process of technology into everyday teaching and learning of mathematics, or
with more established uses of technology in teaching practices, which potentially could
help to “gain insights that are better supported by experimentation and reflection”
(Lagrange et al., 2003, p. 256).
2.4 Why integrate ICT in mathematics education?
Oldknow & Taylor (2000) argue that there are at least three reasons for integrating ICT in
mathematics teaching in schools, namely, desirability, inevitability and public policy.
Desirability can be supported in terms of students, teachers and schools; students are
motivated, stimulated and encouraged; teachers improve their efficiency, are less
administrative, allow more time for student-work and gain better records of their students’
progress; schools improve efficiency, educational inclusion and multilingual classrooms. In
addition technology becomes inevitable at the time when conventional alternatives no
longer exist (Oldknow et al., 2000) and when its cost has been reduced to affordable
26
amounts. Researchers showed that ICT integration promotes students interaction,
collaboration and discussion (Agalianos, 2001; Light and Blaye,1989).
As far as public policy in many countries are concerned, there has been an
acceptance of the educational benefits of ICT and thus governments promote its use
wherever and whenever it is possible starting from the first grade of compulsory education.
There are four reasons for incorporating ICT in education: speed and automatic functions,
capacity and range, provisionality and interactivity (Loveless & Dore, 2002). Speed and
automatic functions of ICT allow storing, changing and displaying information, analyzing
and synthesizing information at higher levels leaving students time to think about the
information presented. Capacity and range refers to the ability to access a vast amount of
information that is distributed worldwide (Pachler, 2001). Provisionally enables users to
make changes, try out alternatives, keep trace of their ideas and determine their own path
(Allen, 2007; Loveless, 1995) whereas interactivity can engage students at a number of
levels. Technology is integrated into mathematics teaching and learning in two forms
(Preiner, 2008). First, there are virtual manipulatives which consist of specific interactive
learning environments. In the virtual manipulatives settings students can explore
mathematical concepts without having special computer skills or knowledge about specific
educational software packages. Secondly, there are mathematical software tools that are
appropriate for educational purposes and can be used for a wide variety of mathematical
content topics, thus allowing more flexibility and enabling both teachers and students to
explore mathematical concepts. The environment will promote richer and deeper
interaction than are commonly seen in traditional lessons enriching and facilitating
interaction between all participants (Papert, 1980).
27
2.5 Introducing New Technology: Calculators and Computers
With the production of pocket calculators around 1970, the first ‘technological
revolution’ of mathematics teaching and learning was set off. Though initially very
expensive, costs of pocket calculators dropped during the next six years, making them
affordable for everyone (Wikipedia, 2008a). Thus, their introduction in schools wasn’t
delayed much longer and pocket calculators could be legally used by students in the late
1970s. Expectations concerning changes in pedagogy and mathematical content were very
high (Weigand et al., 2002, p. 4) and the usage of pocket calculators in schools was
expected to:
 increase the importance of experimental and discovery learning
 strengthen modelling and mathematical concepts
 enhance application tasks
 reduce the importance of manual computational skills
 increase the importance of algorithms
Furthermore, the introduction of pocket calculators raised a lot of pedagogical questions
which are very similar to the ones discussed concerning the introduction of computers and
mathematical software nowadays (Weigand et al., 2002, p. 4).
 How can basic objectives of mathematics education be reached more effectively?
 What is the meaning of ‘traditional’ mathematical skills?
 What are we supposed to do with the additional time gained?
 How is using this new technology going to affect weaker students?
Since many schools and teachers were not really prepared for the introduction of this
new tool for teaching and learning mathematics, the full potential of pocket calculators
28
could not be tapped at all in the beginning. Nevertheless, new and innovative ideas were
implemented and the effective usage of pocket calculators increased in schools over the
next decades (Weigand et al., 2002). The use of pocket calculators for teaching and
learning mathematics was controversial and caused considerable discussion about the
potential loss of computational skills among students (Weigand et al., 2002).
Although teaching of several mathematical topics was influenced by this new tool, the
introduction of pocket calculators did not really change objectives, methods, or assessment
in mathematics education (Weigand et al., 2002, Fey & Hirsch, 1992). In general, the
process of introducing new technology in the form of pocket calculators in schools showed
that just providing a new tool along with several best practice examples could not change
mathematics education fundamentally. Instead, teacher education and professional
development needed to be changed as well in order to prepare teachers for this new
methodological tool and teach them how to effectively integrate it into their everyday
teaching. The conclusion drawn from experience was that the full potential of new
technology can only be fully realized if teachers are convinced of its benefits for teaching
and learning mathematics (Weigand et al., 2002).
With a delay of about 10 years, personal computers followed the pocket calculators into
schools. With regard to their expected impact on future everyday life, ‘computer literacy’
became an important keyword, and schools were supposed to prepare students for this new
challenge. The focus is familiarizing students with computers in general while teaching the
basic use often thrust mathematical contents aside.
In the late 1980s pedagogical aspects gained in importance thereby causing a call for
more user-friendly software in order to allow for focusing on content instead of the
29
technology itself (Weigand et al., 2002). Meaningful integration of new technology into
teaching became the general objective, which was supported by the development of the first
dynamic geometry software Cabri Geometry and the computer algebra system Derive
(Weigand et al., 2002 ). Additionally, drill-and-practice programs and computer-assisted
instruction (Kaput, 1992; Kaput & Thompson, 1994), which were the first applications of
computers for mathematics learning (Hurme & J¨arvel¨a, 2005, p. 50), were increasingly
replaced by multimedia learning environments. Thus, the use of technology as a cognitive
tool in order to allow students to construct individual knowledge was fostered (De Corte,
Greer & Verschaffel, 1996).
Again, discussions about effective use of these new tools took place and questions
about selective integration of new technology into teaching and its role for assessment were
raised. However, expectations concerning potential changes of mathematics education in
terms of objectives, contents, and instructional methods were more realistic this time: new
technologies were supposed to be successively integrated into teaching and learning,
supporting an ‘evolution’ instead of causing a ‘revolution’ ( Weigand et al., 2002, p.10).
2.6 Types of Software Tools Used for Mathematics Education
Computer algebra systems, dynamic geometry software, and spreadsheets are the
main types of educational software currently used for mathematics teaching and learning
(Drijvers & Trouche, 2007; Fuglestad, 2005; Leuders et al., 2005). Each of the programs
has its own advantages and is especially useful for treating a certain selection of
mathematical topics or supports certain instructional approaches. Nevertheless, the
boundaries between those types of software become increasingly blurred and features
30
characteristic for one type are often added to another one. Thus, a new type of educational
software, so called dynamic mathematics software, was designed with the purpose to join
the advantages of different types of mathematics software so as to become a versatile tool
for mathematics teaching and learning that can be used for a wider range of mathematical
contents, grade levels, and teaching methods.
Background Information about GeoGebra
2.7 What is GeoGebra?
GeoGebra is dynamic mathematics software (DMS) designed for teaching and
learning mathematics from primary to university level. The software combines the ease of
use of a dynamic geometry software (DGS) with certain features of a computer algebra
system (CAS) and therefore, allows for bridging the gap between the mathematical
disciplines of geometry, algebra, and even calculus (Hohenwarter and Preiner, 2007b)
2.8 Short History of GeoGebra
The development of GeoGebra began in 2001 as Markus Hohenwarter’s Master’s
thesis project at the University of Salzburg, Austria. After studying mathematics education
as well as computer engineering, he started to implement his idea of programming software
that joins dynamic geometry and computer algebra, two math disciplines that other
software packages tend to treat separately. His main goal was to create educational software
that combines the ease of use of dynamic geometry software with the power and features of
a computer algebra system, which could be used by teachers and students from secondary
school up to college level. After publishing a prototype of the software on the Internet in
31
2002, teachers in Austria and Germany started to use GeoGebra for teaching mathematics,
which was, at this point, rather unexpected by the creator, who got a lot of enthusiastic
emails and positive feedback from those teachers (Hohenwarter & Lavicza, 2007).
In 2002, Hohenwarter received the European Academic Software Award EASA in
Ronneby, Sweden, which finally inspired him to go on with the development of GeoGebra
in order to enhance its usability and extend its functionality. Further development of
GeoGebra was funded by a DOC scholarship awarded to Hohenwarter by the Austrian
Academy of Sciences, which also allowed him to earn his PhD in a project that examined
pedagogical applications of GeoGebra in Austrian secondary schools. During the next four
years GeoGebra won several more software and media awards in different European
countries, including Austria, Germany, and France (Hohenwarter, 2005).
Since 2006, GeoGebra’s ongoing development has continued at Florida Atlantic
University, USA, where Hohenwarter works in a teacher training project funded by the
National Science Foundation’s Math and Science Partnership initiative. During the last two
years of close collaboration with a number of middle and high school mathematics teachers,
GeoGebra was enhanced by including a range of important features. This enhanced
functionality enabled the creation of user defined tools and significant simplification in the
steps required for user creation of interactive instructional materials, the so called dynamic
worksheets.
2.9 Why is GeoGebra different from other Mathematical Software’s?
Currently, there are two types of educational software that connect the mathematical
fields of geometry and algebra and are used for mathematics teaching and learning. On the
32
one hand, there is dynamic geometry software (DGS) that allows users to create and
dynamically modify Euclidian constructions. Geometric properties and relations between
objects used within a construction are maintained because manipulating an object also
modifies dependant objects accordingly. Some dynamic geometry programs even provide
basic algebraic features by displaying the equations of lines or conic sections, as well as
other mathematical expressions which usually can not be modified directly by the user. On
the other hand, there are computer algebra systems (CAS) which symbolically perform
algebra, analytic geometry, and calculus. Using equations of geometric objects, a computer
algebra system can decide about their relative position to each other, and display their
graphical representations. Many computer algebra systems are also able to plot explicit and
sometimes even implicit equations. Generally, the geometric representation of objects can’t
be directly modified by the user.
GeoGebra is an attempt to join these two types of software, whereby geometry,
algebra, and calculus are treated as equal partners. The software offers two representations
of every object: the numeric algebraic component shows either coordinates, an explicit or
implicit equation, or an equation in parametric form, while the geometric component
displays the corresponding solution set (Hohenwarter, 2002). In GeoGebra both
representations can be influenced directly by the user. On the one hand, the geometric
representation can be modified by dragging it with the mouse, whereby the algebraic
representation is changed dynamically. On the other hand, the algebraic representation can
be changed using the keyboard causing GeoGebra to automatically adjust the related
geometric representation.
33
This new bidirectional dynamic connection between multiple representation of
mathematical objects opens up a wide range of new application possibilities of dynamic
mathematics software for teaching and learning mathematics while fostering student
understanding of mathematical concepts in a way that was not possible several years ago.
There are no other ways of gaining access to mathematical objects but to produce some
semiotic representations. There is no true understanding in mathematics for students who
do not incorporate into their cognitive architecture the various registers of semiotic
representations used to do mathematics. (Duval, 1999).
2.10 GeoGebra’s User Interface
Since GeoGebra joins dynamic geometry with computer algebra, its user interface
contains additional components that can’t be found in pure dynamic geometry software.
Apart from providing two windows containing the algebraic and graphical representation of
objects, components that enable the user to input objects in both representations as well as a
menu bar are part of the user interface. The main components are graphic window, toolbars,
algebra window, menu bar, and input field bar for entering questions.
34
Figure 1 : GeoGebra’s User Interface
Graphics window: The graphics window is placed on the right hand side of the GeoGebra
window. It contains a drawing pad on which the geometric representations of objects are
displayed. The coordinate axes can be hidden and a coordinate grid can be displayed by the
user. In the graphics window, existing objects can be modified directly by dragging them
with the mouse, while new objects can be created using the dynamic geometry tools
provided in the toolbar.
Toolbar: The toolbar consists of a set of toolboxes in which GeoGebra’s dynamic
geometry tools are organized. Tools can be activated and applied by using the mouse in a
very intuitive way. Both the name of the activated tool as well as the toolbar help, which is
placed right next to the toolbar, give useful information on how to operate the
corresponding tool and, therefore, how to create new objects. In the right corner of the
toolbar the Undo and Redo buttons can be found, which enable the user to undo mistakes
step-by-step.
35
Algebra window: The algebra window is placed on the left hand side of the GeoGebra
window. It contains the numeric and algebraic representations of objects which are
organized into two groups:
• Free objects can be modified directly by the user and do not depend on any other objects.
• Dependant objects are the results of construction processes and depend on ‘parent
objects’. Although they can not be modified directly, changing their parent objects
influences the dependant objects. Additionally, the definition of a dependant object can be
changed at any time.
Additionally, both types of objects can be defined as auxiliary objects, which means that
they can be removed from the algebra window in order to keep the list of objects clearly
arranged.
Algebraic expressions can be changed directly in the algebra window, whereby different
display formats are available (e.g. Cartesian and polar coordinates for points). If not
needed, the algebra window can be hidden using the View menu.
Input field: The input field is placed at the bottom of the GeoGebra window. It permits the
input of algebraic expressions directly by using the keyboard. By this means a wide range
of pre-defined commands are available which can be applied to already existing objects in
order to create new ones.
Menu bar: The menu bar is placed above the toolbar. It provides a wide range of menu
items allowing the user to save, print, and export constructions, as well as to change default
settings of the program, create custom tools, and customize the toolbar.
36
Construction protocol and Navigation bar: Using the View menu, a dynamic
construction protocol can be displayed in an additional window. It allows the user to redo a
construction step-by-step by using the buttons of a navigation bar. This feature is very
useful in terms of finding out how a construction was done or finding and fixing errors
within a construction. The order of construction steps can be changed as long as this does
not violate the relations between dependant objects. Furthermore, additional objects can be
inserted at any position in order to change, extend, or enhance an already existing
construction.
Additionally, the Navigation bar for construction steps can be displayed at the bottom of
the graphics window, allowing repetition of a construction without giving away the
required construction steps ahead of time.
Although GeoGebra’s user interface consists of several components, which can be hidden
on demand, its design is based on the so called KISS principle, known from computer
engineering. This principle expresses the goal of a programmer to ‘keep it short and
simple’, in order to maintain the usability of a software (Hohenwarter, 2006b, p. 109). In
the case of GeoGebra, the developer tries to design the user interface of the software in a
straightforward and clear way, which supports the model of cognitive processes for
learning with multimedia and reduces the cognitive load for the benefit of more successful
learning (Clark & Mayer, 2003).
Multimedia Principle: “Use words and graphics rather than words alone.”(Clark & Mayer,
2003, p. 51). This e-learning principle is implemented in several ways in GeoGebra’s user
37
interface by combining text (in this case numeric and algebraic expressions) with graphical
representations (Hohenwarter & Preiner, 2008).
At first, the software offers two views of each object. The algebraic representation
corresponds to the textual component, whereas the graphical representation adds the visual
component mentioned in this principle.
Secondly, a dynamic construction protocol can be opened and placed next to the graphics
window. It contains the name, definition, command, and algebraic expression for each
object used in the construction and provides a navigation bar to go through the construction
process step-by-step. The current construction step is highlighted within the construction
protocol while the corresponding object appears in the graphics window of GeoGebra.
Thirdly, static and dynamic text can be inserted into the graphics window to emphasize
certain mathematical concepts and relations, show changes in selected algebraic
expressions dynamically, highlight mathematical invariants, or carry out calculations.
Finally, the Multimedia Principle also influences the export possibilities of GeoGebra.On
the one hand, so called dynamic worksheets combine interactive dynamic figures with
explanations and tasks for students. On the other hand, a construction protocol can be
exported for every construction or dynamic figure giving a textual description of all objects
within a table as well as a picture of the actual construction (Hohenwarter, 2006b).
Contiguity Principle: “Place corresponding words and graphics near each other.”
(Clark & Mayer, 2003, p. 67). This e-learning principle is also invested in multiple ways
within the design of GeoGebra’s user interface by placing corresponding words (here:
38
mathematical expressions) and graphics near each other, making it easier to find
corresponding representations of the same object (see (Hohenwarter & Preiner, 2008). At
first, GeoGebra provides pop up text that show the definition of an object when the mouse
is moved over one of its representations. Additionally, pop up text appears when the pointer
hovers over one of the toolbar icons, showing the name of the corresponding tool.
Secondly, labels of objects can either consist of the name, the algebraic value, or both the
name and value of the object. Since the label follows the movements of its object, the
graphical and algebraic representation of the object always stay close to each other.
Thirdly, both representations of an object are displayed in the same color, which can easily
be modified by the user to distinguish between objects of the same type (e.g. two circles).
This makes it easier to find corresponding representations in the algebra window, graphics
window, as well as the dynamic construction protocol.
Fourthly, static and dynamic text can easily be inserted into the graphics window. They can
be placed close to corresponding objects or even attached to them so they follow every
movement dynamically.
Coherence Principle: “Adding interesting material can hurt learning.”(Clark & Mayer,
2003, p. 111).This e-learning principle is also taken into account by avoiding unnecessary
distractions like glaring colors or decorations within GeoGebra’s user interface
(Hohenwarter & Preiner, 2008).
Also, unneeded objects can be hidden in both windows to avoid distracting the students and
help them to focus on the relevant components of a dynamic figure. In the algebra window,
this can be achieved by defining these objects as ‘auxiliary objects’ and hiding them from
39
view, which allows a user to ‘tidy up’ the lists of free and dependant objects. In the
graphics window, the appearance of those objects can be either changed so they don’t
attract attention any more (e.g. dashed lines, lighter color) or the objects can simply be
hidden from view.
2.11 What has GeoGebra got to offer?
As a software package that combines both geometry and algebra, GeoGebra has
much to offer (Hohenwarter & Jones, 2007). Geogebra is specifically designed for
educational purposes and can help students to foster their mathematical learning
(Hohenwarter & Preiner, 2007). Its environment is mathematically rich and due to the fact
that it is interactive it promotes mathematical explorations. It also provides a wide range of
mathematical concepts which are dynamic and thus more accessible to pupils. Geogebra
provides a visual and conceptual feedback to the learner. In addition, it is free, so pupils can
use it not only at school but also at home thus they have the opportunity to do their
homework, practice, revise the lesson and prepare for the next one. It is available in a range
of languages offering a great opportunity to use the software in local languages and in
multicultural classroom environments. The software includes a geometry window, a toolbar
(figure 1), an algebra window, an input field, a menu-bar and construction protocol and a
navigation bar. The construction protocol offers the researcher and the teacher a step-bystep record of the pupils’ computer interaction, which represents an important part of the
pupils’ choices and actions. Thus, it enables them to obtain a relatively precise image of the
strategies used by pupils to solve a given problem. The open source nature of Geogebra has
encouraged a worldwide communication between its users. They can have access to
GeoGebraWiki, a pool of materials that allows everyone to contribute their own creations
40
or take an existing worksheet and produce a customized version. They can also access the
GeoGebra User Forum where they can discuss their questions and ideas (Hohenwarter &
Preiner, 2007).
2.12 Teaching Mathematics with GeoGebra
Skills, pedagogy and curriculum are the three aspects involved in the use of
Geogebra in the classroom. Teachers need to know how it works and how it can be
effectively integrated both within the classroom and within the curriculum. Thus, when
incorporating Geogebra in the classroom these fundamental features should be taken in
mind. Geogebra can be used in many ways in the teaching and learning of mathematics: for
demonstration and visualization since it can provide different representations; as a
construction tool since it has the abilities for constructing shapes; for investigation to
discover mathematics since it can help to create a suitable atmosphere for learning; and for
preparing teaching materials using it as a cooperation, communication and representation
tool (Hohenwarter & Fuchs, 2004). The success of GeoGebra has shown that noncommercial software packages have the potential to influence mathematics teaching and
learning worldwide (Hohenwarter & Lavicza, 2007) without governments having to invest
a tidy sum of money in supplying schools with software.
41
2.13 The advantages of using GeoGebra are:
(a) GeoGebra is more user-friendly. ( b) GeoGebra offers easy-to use interface, multilingual
menus, commands and help compared to a graph calculator; (c) Students can personalize
their own creations through the adaptation of interface (e.g. font size, language, quality of
graphics, color, coordinates, line thickness, line style and other features); (d) GeoGebra is
created to help students gain a better understanding of mathematics by manipulating
variables. This can be done easily by simply dragging “free” objects around the plane of
drawing, or by using sliders. Students can generate changes using a technique of
manipulating free objects, and then they can learn how the dependent objects are affected;
(e) Lecturing should be replaced by a task oriented interactive classroom. The primary role
of teaching is not to lecture, explain, or otherwise attempt to "transfer" mathematical
knowledge, but to create situations for students that will foster their making the necessary
mental constructions. In that sense, GeoGebra provides a good opportunity for cooperative
learning either in small groups, or whole class interactive teaching, or individual/group
student presentations; and (f) GeoGebra stimulates teachers to use and assess technology in
visualization of mathematics; investigations in mathematics; interactive mathematics
classes on site or at a distance (Ljubica Dikovic, 2009).
2.14 Active Learning
In the traditional teaching method, teachers are always being the center of teaching and
learning activities, which means that teachers are active, and students are passive in the
class. In this method teachers give lectures to students and after that teachers give some
examples of what they just taught. Meanwhile, students are only listening to what their
42
teachers explain, and undertaken some exercises after they get some examples of the
exercises.
In my view, such a teaching method makes students become reliant on their teachers, so
that they will not be able to learn how to be critical, innovative, and creative in their
learning activities. Students will only get shallow understanding of what they learned by
such a rote method. In my opinion, to triumph over this problem, mathematics teachers
should modify or even change their traditional teaching method to an active learning
approach.
“Active learning differs from “learning from examples” in that the learning algorithm
assumes at least some control over what part of the input domain it receives information
about” (Atlas, Chon, & Ladner,1990, p.201). This means that a teaching method which
consists of only giving students some examples and then asking them to learn from those
and after that asking students to solve some similar questions by themselves is not an active
learning technique. In an active learning approach, teachers should be more aware of their
students’ actions in learning activities, and teachers should make their students more active,
more engaged, and more critical in class activities. To prepare for active learning activities,
teachers should design a lesson plan in which students must read, write, discuss, or be
engaged in solving problems (Bonwell, 1991). Therefore, in such teaching methods,
teachers are not the centre of the class, but students are the centre of the learning activities.
“Most important, to be actively involved, students must engage in such higher-order
thinking tasks as analysis, synthesis, and evaluation. Within this context, it is proposed that
strategies promoting active learning be defined as instructional activities involving students
in doing and thinking about what they are doing” (Bonwell, et al, 1991). Bonwell suggests
43
some major characteristics associated with active learning strategies: “students are involved
in more than passive learning; students are engaged in activities; there is less emphasis
placed on information transmission and greater emphasis placed on developing student
skills; there is greater emphasis placed on the exploration of attitudes and values; students'
motivation is increased; students can receive immediate feedback from their instructor; and
students are involved in higher order thinking (analysis, synthesis, evaluation).” Therefore I
propose that an active learning approach should be considered as one possible innovation of
teaching methods to be applied in teaching and learning activities.
2.14.1 Seeing
Many educators and researchers in mathematics argue that intuition plays a crucial role in
geometry, and that an intuition process in geometry comes into one’s mind after seeing
shapes of geometrical things. Actually, it is difficult to define what exactly the definition of
intuition in geometry is, but generally it is a skill to ‘see’ geometrical figures even if they
are not drawn on paper. Creating and manipulating such figures in the mind to solve
problems in geometry can be regarded as an intuition skill (Fujita, Jones, and Yamamoto,
2004b). This means that intuition relates to what students see and then think about.
2.14.2 Measuring
Measuring in geometry is one of the important skills in order to establish the size of an
angle, length, area, or volume of geometrical things. Measuring in geometry is frequently
related to using tools such as a ruler, a compass, a protractor, etc. By using such tools
students can measure real geometrical things, and they can investigate whether their
44
intuition about geometrical objects is accurate or not. For example, when students are asked
to investigate whether two given triangles are congruent or not, students can use their
intuition to answer the question. However, to make sure whether the students’
intuitive answer is correct or not, they need to use a ruler and a protractor to measure all
properties of each triangle.
2.14.3 Reasoning
Reasoning in geometry relates to abilities to give logical explanations, argumentations,
verifications, or proofs to arrive at convincing solutions to geometrical problems. Intuition
and measuring skills need to be supported by reasoning skills. This means that reasoning
plays a justification role for what intuition and measurement give as solutions to geometry
problems. Actually, good reasoning will make a solution of a geometry problem more
mathematical and more elegant. Through reasoning skills students can enhance their
understanding about geometry and find it possible to make other theories from what they
have learned and understood.
2.15 Theoretical Framework
As a philosophy of learning, constructivism can be traced at least to the eighteenth century
and the work of the Neapolitan philosopher Giambattista, who held that humans can only
clearly understand what they have themselves constructed (http://www.sedl.org/ scimath/
ncompass /v01n03/2.html).
Many others worked with constructivism, but the pioneers who developed a clear idea of
constructivism as applied to classrooms were Dewey and Piaget.
45
Most constructivists, such as Duffy and Jonassen (1992) stress the need for collaboration
among learners, which is opposite to the traditional approaches where learners are not
considered
as
capable
of
constructing
knowledge.
http:en.wikipedia.org//wiki/
constructivism (learning theory).
This study is based on a social constructivist view of learning: pupils learn mathematics
through active construction of their own knowledge and this can be facilitated in a
computer environment through the interactive process of conjecture, feedback, critical
thinking, discovery and collaboration (Howard et al., 1990). The researcher considers
constructivism to be the socially collective generation and construction of meanings rather
than a meaning-making activity of the individual mind as Crotty (1998) claims. The
researcher does not take constructivism to highlight the unique experience of an individual
that tends to resist the critical spirit (Crotty, 1998); in contrast, the researcher grounded the
research on a social constructivist nature of knowledge in which:
The meanings are negotiated socially and historically. In other words
they are not simply imprinted on individuals but are formed through
interaction with others (hence social constructivism) and through historical
and cultural norms that operate in individuals’ lives (Creswell, 2003, p. 8).
Social constructivism claims that rather than being transmitted, knowledge is created or
constructed by each learner (Leidner & Jarvenpaa, 1995); there is no knowledge
independent of the meaning attributed to experience constructed by the learner (Hein,
1991). According to certain cognitive theories learning does not involve a passive reception
of information; instead, the learning process can be regarded as an active construction of
knowledge in a learner-centered instruction (Kapa, 1999). Constructivism claims that
46
students cannot be given knowledge; students learn best when they discover things, build
their own theories and try them out rather than when they are simply told or instructed.
Vygotsky argues that:
Direct teaching of concepts is impossible and fruitless. A teacher who tries
to do this accomplishes nothing but empty verbalism, a parrot-like repetition
of words by the child, simulating a knowledge of the corresponding concepts
but actually covering up a vacuum (Vygotsky, 1962, p. 83).
By participating in social constructivism activity students have the opportunity not
only to learn mathematical skills and procedures, but also to explain and justify their own
thinking and discuss their observations (Silver, 1996). From a social constructivist
perspective ICT offers teachers a powerful pedagogical tool-kit (O’Neill, 1998). Hoyles
(1991) argues that in mathematics lessons involving computers, learning is achieved
through social interaction for three reasons: the social nature of mathematics; the
collaboration that computer based activities invite; the basis for viewing the computer as
one of the partners of the discourse.
2.15 Summary of Literature Review
 Ghana’s policy for ICT4AD plan & strategies on education was discussed.
 How ICT started in Ghana from SSS to the Colleges of Education. How it initiatives
contributed to teachers acquiring ICT skills.
 How several researches talked of the good aspects of the ICT integration. Also, the
three reasons for integrating ICT in mathematics in schools were discussed. Among
47
them are desirability, inevitability and public policy. The literature also tackled
calculator use and types of software used in teaching mathematics. Examples of
such software’s are Derive, sketchpad, and Geogebra.
 The definition of geogebra and how it started was narrated. Why geogebra is
different from other software and the advantages it has over the others software’s
was enumerated.
 The User interface was discussed supported by the goegebra window well labeled.
The geogebra potentials as well as it using to teach mathematics was explained
emphasizing on it demonstrations/visualization principle.
 Active learning approach, attitudes of students, abilities in seeing, measuring and
reasoning were briefly touched.
 Finally, it explains how constructivist supports active learning approach by
highlighting that: knowledge should be actively constructed by students and
learning is a process in which sense-making of the real world takes a part. How
constructivist
also claim that students cannot be given knowledge; students learn
best when they discover things, build their own theories and try them out rather than
when they are simply told or instructed.
48
CHAPTER 3
METHODOLOGY
3.0 Overview
This study was guided by the following research questions.
4. To what extent does the difference between the performance of General Arts
students and Science students in geometry (angles) when taught using traditional
approach change when active learning approach with Geogebra is used to teach both
groups?
5. How does the use of the active learning approach using Geogebra relate to students’
attitude to learn angles?
6. To what extent does the active learning approach using Geogebra help students to
improve their abilities of seeing, measuring, and reasoning in learning angles?
This chapter covers the following topics: research design, study population, study sample,
instruments, validity and reliability of test items, pre-intervention, intervention, post
intervention, data collection and finally data analysis procedure.
3.1 Research Design
Research designs are procedures for collecting, analyzing, interpreting and reporting data in
research studies. The model for this study is action research since it seeks to find solution to
student’s inability to solve angles problems effectively.
Mills (2003) defined action
research as any systematic inquiry conducted by teachers, administrators, counselors, or
others with
vested interest in the teaching and learning process, for the purpose of
gathering data about how their schools operates or how they teach and how students learn.
49
Furthermore action research is preferred in this context because it deals with an
intervention which is appropriate to a classroom situation in which the researcher carried
out the study.
3.2 Population and Sampling
Identifying the population of a research is important since no research is carried out
in a vacuum. It is, therefore, imperative to know the target population in order to decide on
what sample size to use for the research. Nworgu (2006) classifies population into target
and accessible. The target population is all the members of a specified group to which the
investigation is related, while the accessible population is defined in terms of those
elements in the group within the reach of the researcher.
3.2.1 Target Population
The target population for the research was all teacher trainees of Bimbilla E.P. College of
Education. There are three levels namely level 100, level 200 and level 300, comprising
both Science and General arts programme students. Students in each level were grouped
randomly by the college administration at the start of the academic year, without regarding
various abilities of the students, which were grouped as below average, average and above
average from their entrance interview results. Of these 480 students are studying the
general programme and 270 for the science for all levels. There are 150 female students and
600 male students.
50
3.2.2 Sample
In this research, the students to be investigated were two hundred and fifty-four (254) first
years (level 100) Students of Bimbilla College of Education. There are six (6) parallel
classes at this level. That is four (4) General programme and two (2) Science programme
classes with different number on roll. Thirty-six (36) students each from the two classes is
the sample for the study, making a total of seventy-two (72) students. Representing 28.34%
of the level 100 students. They all came from different regions of the country. The ages of
these students range from 19 to 23 years.
3.3 Sampling Procedure
In this study, non-probability sampling was used. Purposive and convenience sampling
procedures were used because of the action research design that was adopted for this
project. Convenience sampling was also employed because of logistical, financial
constraints and easy accessibility of the students bearing in mind of the Colleges of
Education tight programmes. Intact classes was used for the sample.
3.4 Research Instruments
TA-test, GA-test and questionnaire were the methods used for the collection of data. The
instruments were well designed for easy collection, interpretation, analysis and organization
of the data collected.
51
3.5 Research Design Plan
First Two Weeks
General Art Students
Science Students
TA
TA
PRE -TEST
INTERVENTION
Last Two Weeks
GA
GA
POST -TEST
Figure 2: Research Design Plan
3.6 Pre-Intervention
3.6.1 Phase 1 (First Two Weeks)
Traditional Approach
After the teaching of angles on the traditional approach for both the General Arts Students
and the Science Students for the first two weeks. The researcher wrote numbers from 01
– 36 on pieces of paper and requested students to choose one of them and to copy the
numbers. The chosen numbers were used to identify the participants’ on the pre-test and
post–test instead of participants’ names. TA-test was conducted on angles to access their
knowledge and ability. The TA-test questions were given to students after regular class and
the extra class teaching periods respectively. The researcher asked the students to solve the
questions individually. Each student was given a printed question paper and answer sheets.
Duration for both test were sixty (60) minutes. Both TA-test were made up of ten (10)
questions each. (see Appendix A1). Answers of students to the TA –test (pre-test) items
were marked using a marking scheme the researcher prepared. (See Appendix A2). The
marks were recorded.
52
The general 1A class had their teaching based on the college timetable, that is twice a week,
two (2) hours each for two (2) weeks. Making a total of eight (8) hours. Whiles the science
class had their teaching as an extra class. That was between 3:30 pm to 5:30pm on week
days. They also spent four (4) hours a week, two (2) times a week and two (2) hours per
lesson. Making a total of eight (8) hours.
3.7 Intervention Process
3.7.1
Phase 2 (Last Two Weeks)
Geogebra Approach
After the teaching of angles on the geogebra approach for both the General Arts Students
and the Science Students for the remaining two weeks. The researcher wrote numbers
from 01 – 36 on pieces of paper and requested students to choose one of them and to
copy the numbers. The chosen numbers were used to identify the participants’ on the pretest and post–test instead of participants’ names. The GA-test was conducted; the questions
were given to students after regular class and the extra class teaching periods respectively.
Each student was given a printed question paper and answer sheets. Duration for both test
were sixty (60) minutes. GA-test was made up of ten (10) questions each. (see Appendix
B1). Answers of students on the GA-test items were marked using a marking scheme the
researcher prepared. (See Appendix B2). The marks were recorded.
53
3.8 Validity And Reliability
3.8.1
Content Validity
The quality of a research instrument or a scientific measurement is determined by both its
validity and reliability (Aikenhead, 2005). The procedure by which the content of the test
is judge to be representative of some appropriate domain of content is the validity of
the content. The designed instruments were therefore developed in consultation with my
supervisor and other experts who also provided excellent advice for correction and
amendment to ensure that the instrument was valid. Thus, the items were subjected to
critical examination to ensure that ,they measured the predetermined criteria; objectives
or content of the study.
3.8.2 Reliability
Reliability on the other hand, refers to the consistency of data when multiple measurements
are gathered (Leedy, 1980). A pilot test of the study was conducted on five (5) students
from the school, and using the split – half method, the scores obtained were used to
determine the reliability. The half – test correlated 0.755 giving rise to spearman - Brown
coefficient of 0.860. This correlation coefficient of 0.860 estimates the reliability
of the
full test, an indication that the results of the instruments were sustainably reliable.
3.9 Data Collection
The raw marks of the students were grouped in ranges based on the similar grading system
of University of Cape Coast and University of Education ,Winneba. It caters for mark range
interpretation ,grades, number of students and finally the percentages. The tables with
student’s performance in angles were constructed for both TA-test and GA-test in the first
54
phase and second phase respectively, and can be found in chapter four (4). SPSS was also
employed for the analysis.
Apart from classroom TA-test and GA-test, anonymous questionnaire was developed
and vetted by the researcher’s supervisor. The researcher did a pilot study of the
questionnaires with three (3) his colleagues. The essence was to help make the language and
syntax of the questions less complex in order for the questionnaire to be more clear,
comprehensible, reliable and valid. The aim of the questionnaire pilot study was to simulate the
real thing as closely as possible. The researcher’s colleagues were used as a pilot sample and
setting up the same conditions for administration and response. That was to see roughly how
long it takes to answer the questionnaire and if the questions are clear or need further
explanations (Munn & Drever, 2004). A questionnaire that is anonymous encourages honesty
and it is more economic in terms of time (Cohen et al., 2007). The researcher gave
questionnaires to the students because at this age a questionnaire can be more reliable than
other methods of collecting data. The questionnaires was given to the students after regular
class session; they were brief, easy to understand and reasonably quick to complete.
3.10 Data Analysis
Descriptive statistics, in the form of mark range in percentages, means, and standard
deviation, grades with interpretation, were employed to enhance the discussion. Descriptive
statistics are useful because they make it very easy to compare things (King, Keohane &
Verba 1994). Quantitative analysis collects data that is factual and can be measured and
considered statistically, (Copper & Schindler, 2006). The quantitative data were analyzed
using the SPSS. The SPSS software was chosen for the data analysis because it is
reasonably user friendly and does most of the data analysis one as far as quantitative
55
analysis is concerned. SPSS is also by
far the most common statistical data analysis
software used in educational research (Muijs, 2004). The data entries were done by the
researcher in order to check the accuracy of the data. The responses from the questionnaire
were all stated in the text to support the discussion of the results. Data were analyzed
quantitatively based on the research questions formulated for this study. Quantitative
analysis is used to figure out exactly what happen, or how often things happened. The
questionnaire was developed
using Likert Scale questions as Totally Agree, Agree,
Neutral, Disagree and Total Disagree (see appendix C1), for each question. The student’s
answers were classified as Agree, Neutral, and Disagree with respect to the questionnaire
on the intervention.
3.11 Intervention Activities Traditional Approach (Without Computers)
1ST MEETING:
Objectives: Students would be able to explain, draw lines and angles.
Teachers guides students to explain:
(a) What a line is? (b) A vertex (c) Legs (sides) (d) Interior of an angle (e) Exterior of an
angle
After the brainstorming of the definitions of the above terms: teacher then modifies
their answer by reading out the definition for students to write.
(a) A line is joining two points together.
A
(b) An angle is a shape formed by two lines, diverging from a common point.
56
B
C
A
B
(c) Legs (sides) of an angle is the two lines that make it up. Example is /AB/ and /AC/.
(d) Interior of an angle: is the space between the rays that make up angle and extending
away from the vertex to infinity. Consider the diagram below.
C
A
Φ
B
The interior is the angle Ө.
(e)
Exterior of an angle: is all space on the plane except the interior. Consider the
C
figure below.
γA
(f)
B
A vertex is the meeting point of two or more lines.
Explain the following:
(g)
Parallel lines:…………………………………………………..
(h)
Perpendicular lines……………………………………………..
Topic: Types of Angles
Objectives: Students will able to: Name the types of angles, explain them, and draw
with labels.
57
Activity 2:
Teachers: Ask students to name types of angles they know. Teacher then list them and read
out the definitions on the chalkboard.
Teacher asks students to draw them and label.
(a) Null angles
(c) Right angles
(e) Straight line
(b) Acute angles
(d) Obtuse
(f) Reflex and full.
(g) Adjacent angles
(h) Opposite
(i) Corresponding
(j) Complementary angles
(k) Alternate angles
(l) Supplementary angles
(m) Interior and exterior
2ND MEETING
Objectives: Students will be able to work questions on the previous lesson.
Activity 1: Teacher writes some questions on the board and asks students to solve. The questions
are from the concept of the previous lesson (1st meeting).
1. Find the value of x in the figure below.
Adjacent concept
Expected solution is
30+70+x0=800
0
100+x0=1800
700
x0
30
x=1800-1000
x0=800
2. Find the value of y in the figure below.
Adjacent concept
30
Expected solution is
y
300+y0=900
y0=900-300=600
Activity 3
R
Teacher guides students to solve the following
3.
Find the value of
in the figure below.
58
M
120
a0
O0
500
Q
Vertical Opposite Concepts
Expected solution
a0 +500=1200
a0=1200-500
a0=700
4.
In the diagram below, AC//DF. Angle CBT=400 and angle DET is 1400.
Find the value of the reflex angle y.
Concepts involved are alternate and co-interior
A
CBT= BTS=400
B
C
400
STE+ TED=1800
y
T
STE=180-140=400
0
0
140
E
D
0
y0=360 -40 -40
F
y=2800
5.
In the diagram below, SR//TQ, SRU=800, and RPR=1000. Find PRQ.
U
Concepts involved are adjacent and corresponding angles.
Expected solution
800
S
QPR=1800-1000=800
R
PQR= URS=800
Therefore PRQ = 1800-(QPR+ PQR)
1000
P
T
0
Q
0
PRQ=180-(80 +80 )
PRQ=180-1600=200
6.
Find the size of the angles marked with a letter in the diagram.
Multiple concepts
P
A
70 b
59
c
0
12
E
a
d
500
D
Q
Expected solution
120+a0=70, a=70-12
a=580
Consider triangle ABC. b+580+120+500=1800, b+1800-1200=600
C=58° using alternate angles . Consider triangle ABE, 120+600+ =1800, =1080.
Therefore d=108° vertical opposite
Class Exercise
1. Fin the size of the angle marked a in the diagram below
Expected answer
b 2b
a = 60°
3b
a
2.
560
720
Expected answer
e = 1280
e
60
3. In the diagram below, triangle ABC is an isosceles triangle. Angle ABD is 1080.
A
Find the value of y.
y
Expected answer
y=360
1080
B
D
C
4. In the diagram below, /AB/=/AC/=/CD/. Find the size of BAC.
A
y
Expected answer
y=520
1480
D
C
B
3RD MEETING
Objectives:
Students will be able to apply all the concepts in angles to answer or more
challenging exercise.
Activity 1
Teacher revises with students on the previous concepts e.g isosceles property.
Activity 2
Teacher helps students to solve the following questions.
1.
1400
Find the angle marked w.
Solution: using alternate principle.
w
1100
61
w0=1800-1100,
2.
w0=700
Find x and
D
1000
in the figure below.
Solution: using alternate principle.
E
Consider
GDE
1000+500+X=1800
500
x0=300, =1000
x
G
F
3. Teacher ask students to solve the following, by finding the angles marked below.
R
Q
Expected Solution
800
P
Using alternate angle principles
80+60+x=180
y
x=1800-400
300
600
x
T
x=400
S
y+400+300=1800
y=180-700, = y=1100
E
Expected Solution
From isosceles principle
y=350
A
x
y
110
B
Alternate principles
=35°, Corresponding x=55°
35
55
DCLASS EXERCISE
C
1. Find the angles marked in the following diagram
R
a.
T
Expected solution
840
840+56+x0=1800
x0=400
Q
560 X0
R
62
S
b.
Expected Solution
D=360-312=480
X0+520+480=3600
X
520
0
X0=2600
C
D
3120
4TH MEETING
Objectives: Students will be able to reflex on the whole topic.
Topic: Summary of the topic.
Activity 1 :Teacher guides to brainstorm on the terms. Null, Right angles, Corresponding
angles, etc.
Activity 2 : Answer bordering oral questions from students.
Activity 3: Conducting of the test on traditional approach. Questions can be found on
appendix A1.
3.10 Intervention Activities Geogebra Approach (With Computers) 1st Meeting
Time:2hrs
RP.K: Students are already familiar with Ms Word and Power Point.
Activity 1 Students were guided through the following.
i. How to open to geogebra (ii) Exploring the geogebra Windows and all the geogebra
Tools. Examples Moving tools, Circle tools, Lines tools, construction tools, etc.
63
(iii)
Looking at the functions of some of the tools to be used in the geogebra teaching
approach. (iv) Experimenting some of the tools by constructing points, angles, labeling
them.
Figure 3: Geogebra Tools
(iv) Definitions of the followings were discussed :
1. Straight line 2. Acute angle 3. Obtuse angle 4. Reflex angle 5. Co-interior angles 6.
Alternate angles 7. Corresponding angles.
(v) Students wrote the definitions in their books.
64
2nd Meeting Time : 3hrs
Part 1 : Teacher and Students Activity
Activity 1: In pairs teacher guides students to construct a line segment AB, following the
steps below, using the Grid view as shown below.
Figure 4: intervention activities
Steps :
 Click on New point Icon and bring the mouse to geometry, window and click to get
point A.
 Repeat the same process for point B.
65
 Click on segment between two points.
 Click on point A and drag it to point B.
Part 2: Students Activity in pairs.
Activity 2: using the activity 1, procedure construct an acute angle <DCE, and measure the
angle at C. Without the grid or axes view.
Steps for measuring
 Click on angle icon
 Click three points or two lines
Part 3: Students Activity
Activity 3: Individually, using neither the grid or axes view. Repeat the 2 activity and
proceed to construct an obtuse and reflex angles and measure them.
Part 4: Students Activity in pairs.
Activity 4: Teachers guides students to construct two parallel line and a transversal line.
Using the grid view. Measure all the angles.
Expected window is shown in figure 2 below.
66
Figure 5: intervention activities
Steps :
Construct line segment LM.
Construct another line segment N.O
Construct another line P.Q which is transversal to the parallel lines.
What can you say about the angles.
Which angles are the same?
Which angles add up to 180
Drag point P and write your observation in your book.
Class exercise
67
Without using the grid axes. (1) Construct two parallel lines (2) Construct two vertical
opposite lines and measure their angles.
3rd Meeting : Using the grid view and in pairs.
Activity 1: Using your knowledge in parallel lines construct a “right” parallelogram UVST
and measure all the angles. Again sum the angles in each triangle and comment.
Steps
1. After constructing the parallogram UVST.
2. Construct a line segment SVdividing the parallelogram into two triangles.
3. Drag any of the vertex and comment.
Questions 1. Which angles are alternate and opposite?
Figure 6: intervention activities
Activity 2: With your previous knowledge in parallel lines. Construct a “left”
parallelogram with vertex A1,B,Z and W. Measure all interior angles.
Do the sum of the interior angles sum to 180˚ .if yes state the angles.
68
Drag any vertex and comment.
Steps
After constructing the parallelogram A1,B1,ZW. Join W to B1 and join A1 to z.
Activity 3: Teacher ask students to draw the diagrams below and measured the angles
P
indicated.
A
B
C
x
x
N
G
y
y
D
F
K
E
The expected window is below and named as figure 7.
Figure 7: intervention activities
4th Meeting
O
Activity 1
69
L
R
M
Teacher ask students to open to a document saved on desktop as “Anas”. He again
ask them to measure all the angles and use the drag test and comment on the angles.
The expected window is below.
Figure 8: intervention activities
Activity 2. Teacher does revision with students on some difficult areas of using the
tools of geogebra. Examples: Finding the intersection of two lines, measuring
angles using the correct clicking of points dragging the specific diagram and all
diagrams in a window. Revision was also done on all the windows below.
Figure 9: intervention activities
70
Figure 10: intervention activities
Figure 11: intervention activities
Activity 3: Students wrote the test on geogebra intervention which can be found on
appendix A1 and B1.
71
CHAPTER 4
RESULTS AND DISCUSSION
4.0 Overview
This study was guided by the following research questions.
1. To what extent does the difference between the performance of General Arts
students and Science students in geometry (angles) when taught using traditional
approach change when active learning approach with Geogebra is used to teach both
groups?
2. How does the use of the active learning approach using Geogebra relate to students’
attitude to learn angles?
3. To what extent does the active learning approach using Geogebra help students to
improve their abilities of seeing, measuring, and reasoning in learning angles?
4.1 Presentation of Pre –Intervention Tables
On a score range 0-40, the grading scale used is presented in table 1 analysis the pre
intervention test.
Table 1: Traditional Approach Marks of General Arts Programme Students
Marks
Grade
Interpretation
Number of Students
Percentage
in Range
36-40
A
Excellent
1
2.8 %
33-35
B+
Very Good
2
5.6 %
30-32
B
Good
3
8.3 %
27-29
C+
Average
7
19.4 %
24-26
C
Fair
6
16.7 %
21-23
D+
9
25.0 %
18-20
D
Weak Pass
7
19.4 %
0-17
E
Fail
1
2.8 %
36
100.0
Barely Satisfactorily
Total
From Table 1, six (6) students out of the 36 students representing 16.7% scores lie between
excellent and good. While 29 students representing 80.5 % scores lie between average and
72
weak pass. One student representing 2.8% failed. This results show that majority of the
students performed poorly in the pre intervention test. The marks range from 15 – 40. The
range gave a difference of 25 marks. The average score was 24.75 and standard deviation
5.10. This depicts the performance of General Arts students during the pre-intervention test.
Summary of table 1 is also shown below.
Table 2: Analysis of Traditional Approach Marks of General Arts Programme
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Traditional Approach
36
25.00
15.00
40.00
24.75
5.10
General Arts Students
Table 3: Traditional Approach Marks of Science Programme Students
Marks
Grade
Interpretation
Number of Students
Percentage
Excellent
4
11.1 %
in Range
36-40
A
33-35
B+
Very Good
3
8.3 %
30-32
B
Good
6
16.7 %
27-29
C+
Average
12
33.3 %
24-26
C
Fair
6
16.7 %
21-23
D+
Barely Satisfactorily
1
2.8 %
18-20
D
Weak Pass
3
8.3 %
0-17
E
Fail
1
2.8 %
36
100.0 %
Total
73
From Table 3, thirteen (13) students of the 36 students representing 36.1% scores lie
between excellent and good. While 22 students representing 61.1 % scores lie between
average and weak pass. One student representing 2.8% failed. This results show that
majority of the students performed poorly in the pre intervention test. The range was 22,
that is maximum mark 39, minimum mark 17. The average score was 28.06 and standard
deviation 5.17.
This depicts the performances of science students during pre-intention test. Summary of
table 3 is also shown below.
Table 4: Analysis of Traditional Approach Marks of Science Programme
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Traditional Approach
36
22.00
17.00
39.00
28.06
5.17
Science Students
4.2 Presentation of Post Intervention Tables
Table 5: Geogebra Approach Marks : General Arts Programme Students
Marks
Grade
Interpretation
Number of Students
Percentage
in Range
36-40
A
Excellent
8
22.2 %
33-35
B+
Very Good
4
11.1 %
30-32
B
Good
9
25.0 %
27-29
C+
Average
7
19.4 %
24-26
C
Fair
5
13.9 %
21-23
D+
Barely Satisfactorily
1
2.8 %
18-20
D
Weak Pass
2
5.6 %
0-17
E
Fail
0
0.0 %
36
100.0%
Total
74
From Table 5, twenty-one (21) students representing 58.3% got scores between excellent
and good. While 15 students representing 41.7 % scores lie between average and weak
pass. No student failed. The results show that, majority of the students performed above
average. Minimum mark of 19 and maximum mark of 40, resulting in a range of 21.00. The
mean mark was 30.94 and the standard deviation of 5.89. This gives a clear picture of
general Arts students post intervention test. Summary of table 5 is also shown below.
Table 6: Analysis of Geogebra Approach Marks : General Arts Programme
Students
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Geogebra Approach of
36
21.00
19.00
40.00
30.94
5.89
General Arts Students
Table 7:
Marks
Geogebra Approach Marks: Science Programme Students
Grade
Interpretation
Number of
in Range
Percentage
Students
3640
A
Excellent
9
25.0 %
33-35
B+
Very Good
13
36.1%
30-32
B
Good
6
16.7%
27-29
C+
Average
6
16.7%
24-26
C
Fair
2
5.6%
21-23
D+
Barely Satisfactorily
0
0.0%
18-20
D
Weak Pass
0
0.0%
0-17
E
Fail
0
0.0%
36
100.0
Total
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From Table 7, twenty-eight (28) students representing 77.8% got scores between excellent
and good. While 8 students representing 22.3% scores lie between average and weak pass.
No student failed. The results show that, majority of the students performed
above
average. The mean mark was 33.33 and standard deviation of 4.40. The range is 15, coming
from a maximum mark of 40 and a minimum mark of 25. The total number of students in
that class was 36. Summary of table 7 is also shown below.
Table 8:
Geogebra Approach Marks: Science Programme Students
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Geogebra Approach of
36
15.00
25.00
40.00
33.33
4.40
Science Students
4.3 Analysis of Results
To answer the first research question:
1. To what extent does the difference between the performance of General Arts
students and Science students in geometry (angles) when taught using traditional
approach change when active learning approach with Geogebra is used to teach both
groups?
76
Descriptive statistics with bar charts illustration was used as follows.
Table 9: Analysis of results of pre-intervention of general Arts and science students
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Traditional Approach
36
25.00
15.00
40.00
24.75
17.00
39.00
28.06
5.10
General Arts Students
Traditional Approach
36
22.00
5.17
Science Students
Looking at Table 9 above, the science students had an average mark or mean score of 28.06
and standard deviation of 5.17 which is better then the general Arts score of 24.75 and 5.10
representing mean and standard deviation respectively. Also, after computing their
independent t-test which gave a value 0.008 which is far less than p-value of 0.05. It means
that, there was significant difference in scores of the two classes.
Also, comparing tables 1 and 3.That is the pre test scores of general arts and science
students. From excellent to good, the general arts students got 6 students representing
16.7% while the science students got 13 students representing 36.1 %. Again, between
average to weak pass, the general arts students had 29 students representing 80.5%, whiles
the science students 22 students representing 61.1 %. Both classes had a student failing in
the pre test. Judging from the above percentages the science students performed better than
the general arts students.
77
14
12
NUMBER OF STUDENTS
10
8
GENERAL ARTS STUDENTS
SCIENCE STUDENTS
6
4
2
0
A
36-40
B+
33-35
B
30-32
C+
27-29
C
D+
24-26
21-23
D
18-20
E
0-17
MARK AND GRADE OF STUDENTS
Figure 13: Comparison of Pre-intervention chart of General arts and Science
Students.
Also from the bar chart, the science students performed better than the Art Students. For
instance 4 science students had 4 A’s and compared to general Arts Students of One “A”
representing (1) student. Again, 9 of the science students had grade “Bs” as compared to 5
students of general Arts. Also, 18 of the science students had grade “Cs” as compared to 13
of the general Arts students. Lastly, 4 science students had grades “Ds” as compared to 16
of general Arts. However, both science and general Arts had One “E” each representing a
fail. Based on the mean, standard deviation, and the bar chart comparison. The Science
Students did better than the general Arts Students. This could however be attributed to the
science students requisite knowledge in Mathematics. Science students always have better
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Mathematical understanding than general Arts students, because most of their courses
involve a lot of Mathematical calculations.
Table 10: Analysis of results of Pre Test and Post test of General Arts Students
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Traditional Approach
36
25.00
15.00
40.00
24.75
5.10
General Arts Students
Geogebra Approach of
36
21.00
19.00
40.00
30.94
5.89
General Arts Students
Looking at the table 10 above, Pre-intervention Test had a mean of 24.75 and standard
deviation of 5.10 as compared with the post test mean score of 30.94 and standard deviation
of 5.89.
Also comparing tables 1 and 6, is the pre test and post test scores of general arts students.
For excellent to good, their performance increased from 16.7% to 58.3%. Indicating an
increase in number of students from 6 to 21. Also, for average to weak pass decreased from
80.5% to 41.7%, that is from 29 students to 15 students. The failure rate decreases from 1
student to none. The above percentages show that, the general arts students post test results
was better than pre test. Hence the intervention was good.
79
10
9
8
NUMBER OF STUDENTS
7
6
TRADITIONAL APPROACH
GEOGEBRA APPROACH
5
4
3
2
1
0
A
36-40
B+
33-35
B
30-32
C+
27-29
C
D+
24-26
21-23
D
18-20
E
0-17
MARK AND GRADE OF STUDENTS
Figure 14: Comparison of Pre and post -intervention chart of General Arts Students
Also from the bar chart, the pre test recorded 1 student representing 2.8% obtaining grade
“A” as compared to 8 students representing 22.2%. For grade “Bs”, the pre –test recorded 5
students representing 13.9% as compared to the post test of 36% depicting 13 students.
Analyzing further, 13 students obtained grade “Cs” representing 36.1% for the pre test and
that of post test 33.3% for 12 students. Also, 16 students 44.4% got grade “Ds” for the pre
test and 3 students for 8.4% for the post test. Finally, 1 student representing 2.8% got grade
“E” for the pre test against no student for the same grade in the Post Test.
It can be concluded that based upon the standard deviation, the mean and the bar chart
comparison. There was an improvement in the post test. Because most of the students
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scored around 30 marks in the post test as compared to 24.75 marks in the pre- test. The
paired sample t-test was also used and it gave a value of 0.00 which is far less than 0.05
showing highly significant differences in performance of pre test and post test of the
general arts students.
Table 11: Analysis of Science Students performance in the Pre- Test and Post Test.
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Traditional Approach of
36
22.00
36
15.00
17.00
39.00
28.06
5.17
Science Students
Geogebra Approach of
25.00
40.00
33.33
4.40
Science Students
Looking at table 11 above, Pre-intervention Test had a mean of 28.06 and standard
deviation of 5.17 as compared with the post test mean score of 33.33 and standard deviation
of 4.40.
Also comparing tables 3 and 7, is the pre test and post test scores of science students. For
excellent to good, their performance increased from 36.1% to 77.7%. Indicating, an
increase in number of students from 13 to 28. Also, for average to weak pass decreased
from 61.1% to 22.3%, that is from 22 students to 8 students. The failure rate decreases from
1 student to none. The above percentages show that, the science students improved from
their pre test to post test. Hence the intervention was good.
81
14
12
NUMBER OF STUDENTS
10
8
TRADITIONAL APPROACH
GEOGEBRA APPROACH
6
4
2
0
A
36-40
B+
33-35
B
30-32
C+
27-29
C
D+
24-26
21-23
D
18-20
E
0-17
MARK AND GRADE OF STUDENTS
Figure 15: Comparison of Pre and post -intervention chart of Science Students
Also from the bar chart, the science students performed better in the post test than in the pre
test. For instance 9 students representing 25.0 % had grade “A” in post test as compared to
4 students representing 11.1 % in the pre test. Again, 19 students representing 52.8 % of
had grade “Bs” as compared to 9 students representing 25.0 %. For grade “Cs” 8 students
representing 22.3 % in post test as compared to 18 of the students representing 50.0 %. No
science students got “D” and “E” compared to 4 students representing 11.1 % for the grade
“D” and 1 student getting grade E representing 2.8 %.
Based upon the standard deviation, the mean and the bar chart comparison. There was an
improvement in the post test. Because most of the students scored around 33.33 marks in
the post test as compared to 28.05 marks in the pre- test. Again, computing the paired
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sample t-test gave a value of 0.00 which is far less than p-value of 0.05 indicating that,
there was highly significant difference in their pretest and post test performance.
Table 12: Analysis of Post Test scores for both Science and Arts Students.
Approach
N
Range
Minimum
Maximum
Mean
Std.
Deviation
Geogebra Approach of
36
21.00
36
15.00
19.00
40.00
30.94
5.89
40.00
33.33
4.40
General Arts Students
Geogebra Approach of
25.00
Science Students
Looking at the table 12 above, the science students had an average mark or mean score of
33.33 and standard deviation of 4.40 which is better then the general Arts score of 30.94
and 5.89 representing mean and standard deviation respectively.
Also comparing tables 6 and 7, is the post test of general arts and science students. For
excellent to good, their performance increased from 16.7% in the pre test to 58.3% in the
post test for the general arts students. Whiles the science students’ performance in the pre
test moved from 36.1 % to 77.8 % for the same grade range. For average to weak pass, the
general arts students’ performance decreased from 80.5% in pre test to post test of 41.7%,
whiles the science students own decreased from pre test of 61.1% to post test of 22.3% for
the same grade range. For the failure rate, both classes decreased from 1 student to none.
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14
12
NUMBER OF STUDENTS
10
8
GENERAL ARTS STUDENTS
SCIENCE STUDENTS
6
4
2
0
A
36-40
B+
33-35
B
30-32
C+
27-29
C
D+
24-26
21-23
D
18-20
E
0-17
MARK AND GRADE OF STUDENTS
Figure 16: Comparison of Post -intervention chart of General arts and Science
Students
Also from the bar chart, the science students performed better than the Art Students. For
instance 9 science students representing 25.0 % had grade “A” as compared to general Arts
Students of 8 students representing 22.2% getting grade “A”. Again, 19 science students
representing 52.8 % had grade “Bs” as compared to 36. 1% of General Arts getting grade
“Bs” representing 13 students. For grade “Cs”, 8 science students representing 22.3 % as
compared to 12 of the General Arts students representing 33.3 %. No science student got
“D” and “E” compared to 3 students representing 8.4 % for grade “D” and no student got
grade “E” from the general Arts class.
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Based on the mean, standard deviation marks, and the bar chart comparison, the Science
Students did better than the general Arts Students. This could however be attributed to the
science students requisite knowledge in Mathematics. Because, science students always
have better Mathematical understanding than general Arts students.
Also, after computing their independent t-test value and resulting 0.06 which is more than
p-value of 0.05. It means that there is little to suggest any significant difference in scores of
the two classes’ performance in the post test. Over 60% of the students improved in the
post intervention score individually as compared to the pre intervention score.
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Research Question 2
Tables 13 sought to find out students response on the effects of the use of the active
learning approach using Geogebra on students attitude to learn angles.
I enjoy geometry class with
Number and
1 (1.4%)
72 (100%)
9(12.5%)
0 (0.0%)
63 (87.5%)
72 (100%)
18(25.0%)
2 (2.8%)
52 (72.2%)
72 (100%)
8(11.1 %)
3 (4.2%)
61(84.7%)
72 (100%)
21(29.2%)
1 (1.4%)
50(69.4%)
72 (100%)
23(31.9%)
3 (4.2%)
46(63.9%)
72 (100%)
41(56.9%)
4 (5.6 %)
27(37.5%)
72 (100%)
Percentage
1(1.4%)
Total
70(97.2%)
Disagree
Number (%)
Number (%)
Neutral
1
Agree
Number (%)
NO.
Item
Table13: Students’ attitudes using Geogebra
geogebra
2
I feel upset and confused
when I am learning
geometry using geogebra.
3
Using geogebra is
extremely hard so it takes
the enjoyment of my
learning geometry away.
4
It is extremely hard to
explain my understanding
of geometry by geogebra to
others.
5
I can learn geometry better
without geogebra.
6
Geogebra approach makes
learning boring.
7
Learning through geogebra
requires more time than
traditional approach.
86
8
Geogebra motivates me to
56(77.8%)
4 (5.6 %)
12(16.7%)
72 (100%)
54(75.0%)
8(11.1%)
10(13.9%)
72 (100%)
59(81.9%)
6(8.3%)
7(9.7%)
72 (100%)
72(100%)
0(0.0%)
0(0.0%)
72 (100%)
60(83.3%)
2 (2.8%)
10(13.9%)
72 (100%)
15(20.8%)
2 (2.8%)
55(76.4%)
72 (100%)
47(65.3)
0(0.0%)
25(34.7%)
72 (100%)
18(25.0%)
0(0.0%)
54(75.2%)
72 (100%)
understand the abstract
content through
visualization.
9
The teacher’s use of
geogebra arouses my
curiosity.
10
Exploring angles through
geogebra gives me the
opportunity participate
actively in class.
11
Using geogebra is a new
experience for me and I
appreciate it.
12
Geogebra enables me to
collaborate with my
colleagues, which facilitate
learning of angles.
13
I am always passive,
because learning geometry
through geogebra is
abstract.
14
I am not much stressed
under geogebra as
compared to the traditional
approach
15
I am intimidated to use
geogebra to learn geometry
From Table 13, a total of 70 students representing 97.2% agreed that they enjoy geometry
class with geogebra, again 63 students (87.5%) said they do not feel upset and confused
when learning geometry using geogebra. Also, 72.2% (52 students) disagreed that using
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geogebra to learn geometry is hard and takes their enjoyment away. Out of 72 respondents,
61 (84.7%) dismissed the assertion they find it hard to explain their understanding of
geometry by geogebra to others. In another vain, 50 (69.4%) disagreed that, geometry
learning is better off than geogebra approach. On the issue of boring 46 (63.9%) disagreed
that geogebra makes learning boring. For more time required for learning angles through
geogebra 41 (56.9%) agreed and 27 (37.5%) disagreed that, more time is not needed. From
the 72 students, 56 (77.8%) said geogebra motivates them to understand abstract content
through visualization, also on arousing of curiosity 54 (75%) agreed that, use of geogebra
arouses their curiosity during lessons. Majority of the students representing 59 (89.9%) said
they participate actively in class and all the 72 students representing 100% said learning
angles by geogebra was a new experience. Again with 60 students (83.3%) said they
collaborate with their colleagues. Majority of the students 55 (76.4%) indicated that, they
are not passive learners in geometry lessons using geogebra, also disagreed that they are
always intimidated to use geogebra software to learn geometry that represented 54 (75.2%)
students.
Lastly, 47 (65.3%) said that, they are not stressed in geogebra approach as compared to
traditional approach.
88
Research Question 3
Tables 14 to 16 sought to find out students responses to the research question: To what
extent does the active learning approach using Geogebra help students to improve their
abilities of seeing, measuring, and reasoning in learning angles?
The discussion of research question three (3) has been divided into three (3) parts namely
reasoning, seeing and measuring.
Table 14: Students’ ability of Reasoning
No
Item
Agree
Neutral
Disagree
Number (%)
Number and
Percentage
60 (83.3%)
Number
(%)
5 ( 6.9%)
7 (9.7%)
72 (100%)
51 (70.8%)
10 (13.9%)
11 (15.3%)
72 (100%)
50 (69.4%)
5
7
72 (100%)
Number (%)
1
When I am using geogebra to
Total
learn geometry I do critical
thinking a lot.
2
I reason before drawing angles
with geogebra tools.
3
I analyze a lot, to be able to
(6.9%)
(9.7%)
reconcile with geogebra
concepts.
From Table 14, 60 (83.3%) indicated that, they do a lot of critical thinking when using
geogebra as a tool in learning geometry. Also, 51 (70.8%) agreed to the assertion of
reasoning before drawing angles using the geogebra tools. Finally, 50 (69.4%) students
agreed that they analyze a lot to be able to reconcile with geogebra concepts.
89
Table 15: Students’ ability of Seeing
No
Item
Agree
Neutral
Disagree
Number (%)
Number and
Percentage
60 (83.3%)
Number
(%)
8 ( 11.1%)
4 (5.6%)
72 (100%)
72 (100.0%)
0 (0.0%)
0 (0.0%)
72 (100%)
Number (%)
4
Geogebra makes me picture
Total
angles properties easily.
5
When I drag a point from a
diagram using geogebra, I
observe some patterns.
From Table 15, 60 (83.3%) agreed that geogebra makes them picture angles properties
easily while 72 (100%) also agreed that, when they drag a point from a diagram, they
observe some patterns.
Table 16: Students’ ability of Measuring
No
6
Item
Taking dimensions of a
Agree
Neutral
Disagree
Total
Number (%)
Number (%)
Number and
Percentage
50 (69.4%)
Number
(%)
0 ( 0.0%)
22 (30.6%)
72 (100%)
60 (83.3%)
2 (2.8%)
10 (13.9%)
72 (100%)
triangle under geogebra is
very simple and easy.
7
Geogebra enables me to
determine angles
measurement accurately.
90
From Table 16, 50 (69.4%) student stated that taking dimensions of a triangle under
geogebra is very simple and easy while 60 (83.3%) also agreed that, geogebra enables them
to determine angles measurement accurately.
4.4 Discussion of Results
Research Question 1
The pretest results of general Arts and science students indicated that, the General Arts
reached the score-mean of 24.75 while science students got a mean score of 28.06. It meant
that the science class did better than General Arts class. Difference between means was
3.31. Independent sample t-test was also used to investigate whether the difference were
significant or not. The t-test gave a significance value (2-tailed) of 0.008 which is far less
than 0.05. Therefore there is significant difference between the means of the two classes.
The post tests of the two classes were further analyzed to determine whether the treatment
on both classes yielded improvement in their performances. The mean score of General
Arts students moved from 24.75 to 30.94 an increment of 6.19. Also mean score of the
science class moved from 28.06 to 33.33 a difference of 5.27. The t-test gave a significant
value (2-tailed) of 0.06 which is just greater than 0.05. This means that, the treatment
during the intervention was good because there is no significant difference in their
performances. The analysis goes further to support similar research by (Mulyono, 2010).
The social constructivist approach also helped the students in their intervention because
they collaborated and constructed knowledge together rather than being transmitted to. The
guidance of the teacher gave the students the chance to explain and justify their own
91
thinking and discuss their observation.
Research Question 2
During the lessons students engaged in activities in which they investigated, interacted,
discovered and cooperated with their peers. It is this style that lead to the students
characterizing the lesson as enjoyable and motivating. There was greater collaboration and
task-related interaction when students worked with the software (Wegerif, 1998).
Also, during the lessons students demonstrated collaborative behavior and had the
opportunity to develop their skills of negotiation, observation and interpretation as well as
social skills such as sharing ideas. The environment promoted richer and deeper interaction
than are commonly seen in traditional lessons enriching and facilitating interaction between
all participants (Papert, 1980). The use of computers software like geogebra eliminates
passive learning hence encouraging student centered learning.
The study promoted student autonomy because students were more confident to assert
control of their own learning without the constant need of the teacher. They only sought the
teacher’s help especially in the first two lessons. After that, they were now familiar with the
software hence they were somehow autonomous. Researchers showed that ICT integration
promotes students interaction, collaboration and discussion (Agalianos et al, 2001; Light &
Blaye,1989).
Research Question 3
Students indicated that, they do a lot of critical thinking when trying to use geogebra tools
to draw diagrams and to explain concepts during lessons. From the findings again, students
have appreciated the skill of seeing angle properties through dragging of points which
92
supports the literature on visualization. The students also did mental reasoning which is in
line with (Fujita et al 2004b). Measuring angles clockwise and anti-clockwise is a skill they
have learnt. But some of them at certain stages of the intervention were finding it difficult
in measuring angles. Because, instead of measuring clockwise for acute angles some
students measured anticlockwise for reflex angles, and vice versa.
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CHAPTER 5
SUMMARY, CONCLUSION AND RECOMMENDATION
5.1 Overview of the study
An overview of the research problem, methodology, key findings, conclusions, and specific
limitations of the research are provided in this chapter. Recommendation and suggestions
for future studies are also given.
According to the mathematics syllabus for Diploma in Basic Education (2005, 2006) the
mathematics tutors at the Colleges of Education are required to use ICT in their teaching
and also teach the teacher trainees how to explore ways in which ICT could be used to
enhance teaching and learning of mathematics at the basic level.
The study was guided by the following research questions:
 To what extent does the difference between the performance of General Arts
students and Science students in geometry (angles) when taught using traditional
approach change when active learning approach with Geogebra is used to teach both
groups?
 How does the use of the active learning approach using Geogebra relate to students’
attitude to learn angles?
 To what extent does the active learning approach using Geogebra help students to
improve their abilities of seeing, measuring, and reasoning in learning angles?
94
The study investigated students’ performance on traditional approach and geogebra
approach. Students’ attitude towards geogebra was also investigated. Lastly, it also assessed
student’s abilities on seeing, reasoning and measuring of angles.
The target population for the research was all teacher trainees of Bimbilla College of
Education, who were on the college campus. The population consisted of two hundred and
fifty-four (254) first year students (level 100) of Bimbilla College of Education. Of the
number, seventy-two (72) students were used because of the purposive and convenience
sampling procedure adopted. Action Research design was adapted. Achievement test and
questionnaire were used as the main tools to collect data. The data gathered from the
questionnaire and tests were analyzed quantitatively.
The research design was analyzed as follows:
 Performance of pre-intervention results of General Arts students and Science
students were analyzed using tables and bar charts.
 Performance of pre and post intervention of General Arts students were analyzed
using bar charts and tables.
 Performance of pre and post intervention results of science students were analyzed
using bar charts and tables.
 Performance of post-intervention results of General Arts and Science students were
analyzed using bar charts and tables.
95
5.2 Summary of key findings
Research Question 1: Achievement Test
 The pre-intervention results of general arts programme and science programme
students indicated that, the science students did better than general arts students
taking their mean mark into consideration. It could be found on Table 9 and Figure
13 bar chart. The science student’s performance could be due to their prerequisite
knowledge or foundation in mathematics.
 The pre and post-intervention results of general arts was also analyzed. The general
arts students improved significantly from the pre to post test. The mean was 24.75
for the pre-test and it shot up to 30.94 for the post-test. This means there was an
improvement in their scores individually. Table 10 and Figure 14 supports my
claim.
 The science students also performed creditable from a mean of 28.05 to a mean of
33.33. Examining their individual scores, most students improved individually.
Table 11 and Figure 15 supports my claim on their pre and post intervention results.
 From Table 12 and Figure 16, I can conclude that, the post test scores for both
science and general arts students improved, because their pre-test mean score
moved from 28.05 to 33.33 for science students and from 24.75 to 30.94 for general
arts students. Based on that, both categories of students improved their performance.
Overall conclusion: Though there was sharp rise in performance from the general
arts students as compared to the science students, the science students still did better.
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Questionnaire Findings
Research Question 2
The findings of the questionnaire showed majority of the students liked the lesson with
geogebra. Very few students expressed that the traditional way is good. Also, my
observations during the classroom intervention reveal that the students become motivated
while learning using geogebra and express their delight for more lessons on the software
over a long period of time. This was obvious from their positive attitudes while working
with the software. The student’s behaviour throughout the intervention reflected that they
liked studying geometry with computers. From the student’s discussions and interactions
during the lessons it was noticeable that geogebra based learning arose student’s interest
and enthusiasm towards geometry.
Research Question 3
Based on the questionnaire findings, it was clear that students gradually improved their
abilities of seeing, measuring and reasoning in learning geometry. Even though, students
sometimes still made some mistakes in doing the tasks for each meeting.
Majority of the students said they do a lot of critical thinking when drawing angle and do
mental analysis before they are able to reconcile geogebra concepts. On picturing angle
properties easily and observing angle patterns, most of the students express their
delightfulness about geogebra. Most of the students also admitted that, taking dimensions
and measuring angles are easy, simple and accurate.
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5.3 Conclusion
Based on the findings made in this study, it can be concluded that:
 The students have gained new ICT skills as a result of their participation in the
study using geogebra.
 The students are ready to implement ICT integration into their classrooms when
they go out for their out-segment programme because they have learnt how to be
autonomous and have also developed confidence.
 Students participated actively and collaborated in all the lessons. They were happy
that they gained new knowledge hence they copied the program for continuous
practice in their spare times.
 Students’ attitudes towards geogebra are positive because they participated in all
meetings with eagerness.
 Students’ performance increased because abstract concepts were presented in a
fashionable manner. Therefore, student’s use of geogebra has increased their
performance.
5.4 Recommendation
The following recommendations can be made based on the findings:
 Students had positive attitudes towards geogebra hence ICT workshops should be
organized periodically for mathematics tutors of the colleges of Education so that,
they can be abreast with new software’s like geogebra and sketchpad to be used in
teaching mathematics across the country.
 The University of Cape Coast should collaborate with University of Education,
98
Winneba to organize professional development courses on technology integration.
It is important because the researcher for this study never had the opportunity of
learning fully the geogebra software from an expert but his previous knowledge on
sketchpad and derive helped him to learn it. Such fora will provide collaboration
among teachers to sharing solution to difficult concepts on instructional technology
software.
 Because the students copied the program for continuous practice in their spare
times indicated that they still need more time to learn it fully. Therefore, University
of Cape Coast should add the course instructional technology into the Colleges of
Education courses which should be examinable. So that students can learn it for a
year or two to be perfect with the software.
5.5 Limitations of the study
A major limitation of this study is the consequences of using non-probability sample. Since
this study was not based on a large population, it might have limitations in terms of
generalization of the findings.
The basic restriction of the research is that it is a bi-instrumental case performed by one
researcher over small period of time and under particular circumstances. The results could
have been more reliable and generalized across the Ghanaian student population if more
than one researcher participated. Although, the students had lessons on geogebra, more
time was needed to have adequate training and familiarization with the software which
could lead them to understand all the detail concepts on which geogebra operates.
99
5.6 Suggestion Areas for Further Research
The results for the study showed that, there is a good reason in using geogebra in teaching
mathematics from primary school to university level. Therefore, further research
investigating mathematical ideas, developed through the use of geogebra is necessary.
Accordingly, the present researcher has some suggestions for developers of mathematical
curricula and for future studies, which are summarized in the following points.
 The researcher concentrated on the study of geogebra. A study can be undertaken on
other types such as Geometers sketchpad and mathematica to compare the
effectiveness in student performance of geometry.
 A research can be taken on other topics like circle theorem using geogebra approach
as an intervention.
 A longitudinal research on teachers’ professional development on their use of
geogebra over a period is needed to document teachers experience and challenges.
 Future studies conducted over a longer period of time may produce more reliable
results.
 Future researches should employ multiple instruments, more samples across the
country and finally more than one researcher.
100
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111
APPENDIX A1
PRE
INTERVENTION GEOMETRY ACHIEVEMENT TEST
CODE NUMBER…….
This test is part of an educational research. It aims at examining the geometric
knowledge you have learned from the traditional approach. Any information obtained from
this test will be confidential. Moreover, to feel more comfortable, and write your code.
Please, analyze every question carefully before responding.
DURATION: 1 HOUR
59
i
1)
115
0
j
52 63 q
2)
p
3)
r
m
n
41
K0
66
6)
(7x +11)0
r0 g
(y+26)0
5)
(10y-7)0
4)
(3x -1)°
h
78
8)
7)
(2x+16)
9)
0
(2x+80)
(7x – 25)°
10)
D
1000
(4x +5)°
0
E
R
Q
80
P
0
y
G
500
x
600
x
T
F
112
300
S
113
APPENDIX B1
POST INTERVENTION GEOMETRY ACHIEVEMENT TEST CODE NUMBER…….
Find the angles marked in the following diagrams.
DURATION: 1 HOUR
1)
D
1000
E
R
Q
2)
80
P
r0 g
0
3)
y
78
0
G
50
4)
x
E F
600x
T
5)
300
S
6)
4)
A
55
D
x
y
110
B
520
(7x – 25)°
CX0
h
7
))
(4x +5)°))
)
(2x+16)
°
D
35
C
3120
Open your file on desktop called Geogebra and answer questions 8-10 and save it with
your code
114
(2x+80)0
115
APPENDIX C: RAW MARKS OF STUDENTS. BOTH PRE AND POST TEST
S/NO
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
PRE TEST SCORE
OF TRADITIONAL
APPROACH
GENERAL ARTS
STUDENTS
25
15
20
20
20
25
22
21
24
27
30
35
40
22
28
27
20
21
23
28
25
21
19
20
25
27
29
21
27
20
25
22
23
30
31
33
PRE TEST SCORE OF
TRADITIONAL
APPROACH
SCIENCE STUDENTS
30
25
19
25
30
28
28
35
36
34
29
25
30
36
29
29
27
27
28
27
27
28
18
17
20
24
39
34
30
36
30
22
24
27
31
26
116
S/NO
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
POST TEST SCORE
OF GEOGEBRA
APPROACH
GENERAL ARTS
STUDENTS
25
25
19
35
35
32
32
38
39
37
29
29
35
35
32
31
27
27
27
29
30
25
26
19
25
28
39
40
40
40
40
31
30
30
31
22
POST TEST SCORE
OF GEOGEBRA
APPROACH
SCIENCE
STUDENTS
29
40
40
35
35
28
25
27
29
30
30
34
38
35
35
30
30
35
30
38
35
35
35
28
28
33
25
30
35
35
35
38
37
38
40
40
APPENDIX D
PHOTOGRAPH OF INTERVENTION ACTIVITIES
117
APPENDIX E
QUESTIONNAIRE AND ITS RESULTS
ACTIVITIES ON GEOGEBRA
This questionnaire is part of an educational research in geometry. It is aimed at
understanding your opinion about the suggested activities you went through under
geogebra.
Please, any personal information obtained from this questionnaire will be confidential.
Please, read every item carefully before responding and when you feel hesitated, tick
neutral.
The following are series of statements. They have been set up in a way that permit
you to indicate the extend to which you agree or disagree with the ideas expressed. Please
tick [
] the box that best describes your response as you read the statement.
Part 1 : The effects of the use of the active learning approach using Geogebra on
student’s attitude to learn angles.
118
Students’ attitudes using Geogebra
Agree
Neutral
Disagree
Strongly
Disagree
STATEMENT
Strongly
Agree
NO.
1
I enjoy geometry class with geogebra
65
5
1
1
0
2
I feel upset and confuse when I am learning
0
9
0
60
3
13
5
2
40
12
5
3
3
50
11
geometry using geogebra.
3
Using geogebra is extremely hard so it takes the
enjoyment of my learning geometry away.
4
It is extremely hard to explain my understanding of
geometry by geogebra to others.
5
I can learn geometry better without geogebra.
20
1
1
38
12
6
Geogebra approach makes learning boring.
15
8
3
38
8
7
Learning through geogebra requires more time than
30
11
4
20
7
36
20
4
8
4
9
The teacher’s use of geogebra arouses my curiosity. 32
22
8
5
5
10
Exploring angles through geogebra gives me the
20
39
6
5
2
22
50
0
0
0
30
30
2
5
5
10
5
2
25
30
25
22
0
10
15
10
8
0
32
22
traditional approach.
8
Geogebra motivates me to understand the abstract
content through visualization.
opportunity participate actively in class.
11
Using geogebra is a new experience for me and I
appreciate it.
12
Geogebra enables me to collaborate with my
colleagues, which facilitate learning of angles.
13
I am always passive, because learning geometry
through geogebra is abstract.
14
I am not much stressed under geogebra as
compared to the traditional approach
15
I am intimidated to use geogebra to learn geometry
119
Part 2: To what extend does the active learning approach using geogebra help students to
improve their measuring abilities of seeing, measuring and reasoning in learning angles?
Reasoning, Seeing and Measuring on Geogebra
Neutral
Disagree
Strongly
Disagree
When I am using geogebra to learn geometry I do
Agree
1
STATEMENT
Strongly
Agree
NO.
27
33
5
2
5
critical thinking a lot.
2
I reason before drawing angles with geogebra tools.
30
21
10
5
6
3
I analyze a lot, to be able to reconcile with geogebra
27
33
5
5
2
concepts.
4
Geogebra makes me picture angles properties easily.
30
30
8
2
2
5
When I drag a point from a diagram using geogebra, I
18
54
0
0
0
22
28
0
16
6
21
39
2
5
5
observe some patterns.
6
Taking dimensions of a triangle under geogebra is very
simple and easy.
7
Geogebra enables me to determine angles measurement
accurately.
120
APPENDIX F
LIST OF TABLES
Table 1: Traditional Approach Marks of General Arts Programme
Table 2: Analysis of Traditional Approach Marks of General Arts Programme
Table 3: Traditional Approach Marks of Science Programme
Table 4: Analysis of Traditional Approach Marks of Science Programme
Table 5: Geogebra Approach Marks : General Arts Programme
Table 6: Analysis of Geogebra Approach Marks : General Arts Programme
Table 7:
Geogebra Approach Marks: Science Programme
Table 8:
Geogebra Approach Marks: Science Programme
Table 9: Analysis of results of pre-intervention of general Arts and science students
Table 10: Analysis of results of Pre Test and Post test of General Arts Students
Table 11: Analysis of Science Students performance in the Pre- Test and Post Test.
Table 12: Analysis of Post Test scores for both Science and Arts Students.
Table 13: The effects of the use of the active learning approach using Geogebra on students
attitude to learn angles.
Table 14: Using Geogebra to help students to improve their abilities of seeing.
Table 15: Using Geogebra to help students to improve their abilities of measuring angles.
Table 16: Using Geogebra to help students to improve their abilities of reasoning in
learning angles.
121
APPENDIX G
LIST OF FIGURES
Figure 1 : GeoGebra’s User Interface
Figure 2: Research Design Plan
Figure 3: Geogebra Tools
Figure 4: intervention activities
Figure 5: intervention activities
Figure 6: intervention activities
Figure 7: intervention activities
Figure 8: intervention activities
Figure 9: intervention activities
Figure 10: intervention activities
Figure 11: intervention activities
Figure 12: intervention activities
Figure 13: intervention activities.
122
APPENDIX H
Group Statistics
VAR00003
PRETE PRETEST
ST
POSTTEST
POSTT PRETEST
EST
POSTTEST
N
Mean
Std. Deviation
Std. Error Mean
36
24.7500
5.09552
.84925
36
28.0556
5.17104
.86184
36
30.9444
5.88919
.98153
36
33.3333
4.40130
.73355
Independent Samples Test
Levene's Test for
Equality of
Variances
t-test for Equality of Means
95% Confidence
Interval of the
Difference
PRETEST Equal variances
assumed
Equal variances
not assumed
Mean
Std. Error
Difference Difference Lower
.008
-3.30556
1.20996
-5.71874 -.89237
-2.732 69.985 .008
-3.30556
1.20996
-5.71875 -.89236
-1.950 70
.055
-2.38889
1.22536
-4.83279 .05501
-1.950 64.801 .056
-2.38889
1.22536
-4.83624 .05846
Sig.
t
.030
.864
-2.732 70
Equal variances
not assumed
POSTTEST Equal variances
assumed
Sig. (2tailed)
F
1.953
.167
df
123
Upper
Paired Samples Statistics
Mean
Pair 1
N
Std. Deviation
Std. Error Mean
PRETEST
26.4028
72
5.36199
.63192
POSTTEST
32.1389
72
5.30029
.62465
Paired Samples Test
Paired Differences
95% Confidence
Interval of the
Difference
Mean
Pair
PRETEST -
-
1
POSTTEST
5.73611
Std.
Std. Error
Deviation
Mean
7.48142
.88169
124
Sig. (2Lower
Upper
t
-7.49416
-3.97806 -6.506
df
tailed)
71
.000
125
126
127
128
129
130
131
132
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COMPARISON OF THE USE OF GEOGEBRA AND THE TRADITIONAL APPROACH TO TEACHING ANGLES, AT BIMBILLA COLLEGE OF EDUCATION