exploring how a workshop approach helps mathematics teachers

Exploring How a Workshop Approach Helps

Mathematics Teachers Start to Develop Technological

Pedagogical Content Knowledge

Thesis submitted in partial fulfilment of the requirements for the

degree of Master of Science

Inambao Wakwinji

Supervisor: Andre Heck

Faculty of Science

Universiteit van Amsterdam

The Netherlands

August 2011

Abstract

The use of dynamic mathematics software is said to enhance pupils’ understanding of mathematics.

However, the challenge could be that teachers first need to become knowledgeable about using the

computer teaching and learning environment for this positive effect on pupils’ learning to happen.

The question that arises is the following: What is required for teachers to start using dynamics

mathematics software effectively in their lessons? An immediate answer could be that teachers

have to develop Technological Knowledge and blend it with the Pedagogical Content Knowledge

that they already have.

The aim of my research was to train Zambian mathematics teachers in using GeoGebra and

explore how they started to develop Technological Pedagogical Content Knowledge (TPCK). Ten in-

service mathematics teachers at Mkushi High School participated in my research study. I explored

their development of TPCK through the training course, in which they also developed lesson

materials for use in real lessons. I investigated how teachers thoughts about teaching mathematics

with ICT changed during the training.

Dedicated to:

My wife Mutinta for her ever present love and support. My wonderful children, Chipo Nawa and

Inambao, born at the time of my studies, for their constant encouragement and understanding.

My mother, Delicy Soneka (Deceased), teacher who showed me the world and encouraged me to

go for it.

Acknowledgements

I want to thank God for the opportunity He gave me and for His strength. His words have always

been my source of inspiration in my life. His assurance in Jeremiah 17:5, “Cursed is the one who

trusts in man, who depends on flesh for his strength and whose heart turns away from the LORD,”

gives me hope.

My greatest indebt is to my supervisor, Andre Heck, for his support and encouragement during

the preparation of this work. He has really influenced my thinking about the subject in ways too

numerous to mention. Without his help and guidance this research study review would not have

been a success. In addition, my discussions with Wolter Kaper and Mary Beth Key were really

encouraging. I want to thank them for the patience they had with me in my work and the feedback

I received was really helpful to me.

I would like to thank the Ministry of Education in Zambia especially the Minister Dora Siliya,

for her support to me during my study. I also would like to thank Mr. Hamakumba, the School

manager of Mkushi High School and all his teachers for the support rendered to me during my

research at the school.

I would like also to acknowledge Reshmi Pradeep, Lilia Ekimova and Norbert van Veen for their

encouragements and direction in my work. To you guys I say, God bless you and wish you the best

in your endeavours.

Special thanks to my sons Chipo Nawa, and the special one Inambao III, born while I was doing

my studies, and my wife Mutinta Wakwinji, for their endless support.

Contents

Abstract iii

Acknowledgements vii

List of Figures xiii

List of Tables xvi

1 Introduction 1

1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 My Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theoretical framework 7

2.1 Pedagogical Content Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Technological Pedagogical Content Knowledge . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Evaluation of TPCK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 The Practitioner Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Adapted Practitioner Model for Teacher Training . . . . . . . . . . . . . . . . . . . . 19

3 Research design and methodology 23

3.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Research Setting and Research Methodology . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 The School Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Research Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Research Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

x CONTENTS

3.3.1 The Pre Questionnaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.2 The Post Questionnaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.3 Teacher Interview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.4 Pupil Interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.5 Field Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.6 Summary of the Use of the Research Instruments . . . . . . . . . . . . . . . . 32

3.4 Framework for Analysing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Design of GeoGebra Workshops 35

4.1 The Pilot Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Objectives of the Workshop in the Pilot Study . . . . . . . . . . . . . . . . . 36

4.1.2 Training Session I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.3 Training Session II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.4 Lessons Learnt from the Pilot Study . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.5 The Effects of the Pilot Study . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 The Study in Zambia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Objectives of the First Two Training Sessions . . . . . . . . . . . . . . . . . . 39

4.2.2 Objectives of the Last Five Training Sessions . . . . . . . . . . . . . . . . . . 39

4.2.3 Training Session I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.4 Training Session II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.5 Training Session III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.6 Training Session IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.7 Training Session V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.8 Training Session VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.9 Training Session VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.10 Relationship Between the Themes of the Adapted Practitioner Model and

the Training Sessions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Results and Data Analysis 45

5.1 The Pre and Post Questionnaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.1 The Pre Questionnaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.2 The Post Questionnaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1.3 Fitting Teachers’ Comments to the Adapted Practitioner Model . . . . . . . 58

CONTENTS xi

5.2 Findings and Analysis for Interviews with Teachers and Pupils . . . . . . . . . . . . 60

5.2.1 Transcripts of Interviews with Teachers . . . . . . . . . . . . . . . . . . . . . 60

5.2.2 Fitting Teachers’ Comments to the Adapted Practitioner Model . . . . . . . 65

5.2.3 Findings According to Themes of the Interview Protocol . . . . . . . . . . . . 65

5.2.4 Transcripts of Interviews with Pupils . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Field Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Conclusions and Discussion 77

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1.1 Sub Question 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1.2 Sub Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.1.3 Sub Question 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.1.4 Sub Question 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.1.5 The overarching research question . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3 Limitations of the Study and Recommendations for Future Research . . . . . . . . . 83

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Bibliography 85

Appendices 89

A Pre Questionnaire 89

B Post Questionnaire 91

C Teacher Interview Protocol 95

D pupil Interview Protocol 97

E Themes of the interview protocol 99

xii CONTENTS

List of Figures

2.1 A figure showing components of TPCK framework (sourced from http://tpack.org). 9

3.1 Mkushi High School, where the research was conducted. . . . . . . . . . . . . . . . . 26

3.2 Action Research Cycle adopted from Watts (1985) . . . . . . . . . . . . . . . . . . . 29

4.1 Teachers demonstrating how they were able to develop activities during the pilot

study at Fons Vitae Lyceum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Activity 1 of Training Session I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41



5.1 A picture of teachers engaged in a discussion about activity development. . . . . . . 72

5.2 A picture of teachers engaged in a discussion about activity development. . . . . . . 72

5.3 A picture of teachers engaged an activity without GeoGebra and solved it. . . . . . . 73

5.4 A picture of teachers engaged in converting activity developed without ICT into one

with ICT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 A screen shot of the developed GeoGebra applet. . . . . . . . . . . . . . . . . . . . . 73

5.6 A picture of showing teachers going around assisting pupils during the practical lessons. 74

5.7 Derivative of a quadratic function using slope function . . . . . . . . . . . . . . . . . 75

5.8 Visualizing rotation of a triangle in 1800 clockwise and 1800 anticlockwise . . . . . . 76

xiv LIST OF FIGURES

List of Tables

3.1 Profiles of the teachers at Mkushi High School who participated in my research study. 27

3.2 The timeline of the research study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Data sources used to answer my research sub-questions. . . . . . . . . . . . . . . . . 33

3.4 The seven themes of the Adapted Practitioner Model . . . . . . . . . . . . . . . . . . 33

3.5 Criteria for evaluating the development of TPCK of teachers. . . . . . . . . . . . . . 34

4.1 Structure of the workshop in Zambia. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Classes and topics taught in Training Session VII. . . . . . . . . . . . . . . . . . . . 44

4.3 Themes of the Adapted Practitioner Model addressed in the workshop. . . . . . . . . 44

5.1 Results for question 1: Pupils’ opportunity to use computers in school. . . . . . . . . 46

5.2 Results for question 2: Types of Computer Use in teaching. . . . . . . . . . . . . . . 46

5.3 Results for question 5: How teachers would like to use computers in their lessons. . . 47

5.4 Results for question 6: Expectations of teachers’ of computers use in their lessons. . 48

5.5 Results for question 1: Ever used GeoGebra in a lesson. . . . . . . . . . . . . . . . . 48

5.6 Results for question 2: Teachers’ opinions on the ease for learning to use GeoGebra. 49

5.7 Results for question 2: Teachers’ experiences in learning to use GeoGebra. . . . . . . 49

5.8 Results for question 3: Teachers’ willingness to continue working with GeoGebra. . . 49

5.9 Results for question 3: Choice(s) where to use GeoGebra and reasons thereof. . . . . 50

5.10 Results for question 4.a: Support that teachers need to start using GeoGebra in

teaching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.11 Results for question 4.b: Support that teachers need to start using GeoGebra in

having a whole class discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.12 Results for question 4.c: Support teachers need to start using GeoGebra to let pupils

use it in small groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

xvi LIST OF TABLES

5.13 Results for question 5: How teaching of graphing would be done using GeoGebra. . . 52

5.14 Results for question 6: Comparing learning about functions and graphs using Geo-

Gebra and using paper and pencil and other regular tools. . . . . . . . . . . . . . . . 53

5.15 Results for question 7: Pupils’ understanding of graphs and functions. . . . . . . . . 54

5.16 Results for question 8: Using GeoGebra to do “trial and improvement”. . . . . . . . 54

5.17 Results for question 9: Using GeoGebra to do “trial and improvement”. . . . . . . . 55

5.18 Results for question 10: Opinions of user-friendliness of GeoGebra. . . . . . . . . . . 56

5.19 Results for question 11: Obstacles to be faced if one wanted to use GeoGebra in

teaching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.20 Results for question 12: Plans for working with GeoGebra in teaching. . . . . . . . . 57

5.21 Results for question 13: The most important thing learnt from GeoGebra workshop. 58

5.22 The seven themes of the Adapted Practitioner Model . . . . . . . . . . . . . . . . . . 58

5.23 Adapted Practitioner model’s themes addressed in the post questionnaire by tea-

chers’ responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.24 Adapted Practitioner model’s themes addressed in the teacher interviews. . . . . . . 66

5.25 Results on the impression about the workshop. . . . . . . . . . . . . . . . . . . . . . 67

5.26 Results on the combination of teaching and content with technology. . . . . . . . . . 67

5.27 Results on obstacles to be faced when teachers start to use GeoGebra. . . . . . . . . 68

5.28 Results on GeoGebra’s help in planning and teaching. . . . . . . . . . . . . . . . . . 68

5.29 Results on the development of instructional materials. . . . . . . . . . . . . . . . . . 69

5.30 Results on the change in pedagogy as a teacher. . . . . . . . . . . . . . . . . . . . . . 70

5.31 Results on the effect of the workshop. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Chapter 1

Introduction

“There was a shift, I would say to a large extent, it came from developing lesson ma-

terials that promote pupils’ activities. To me this is how I was able to combine my

knowledge of teaching with the technology. It was not easy to develop activities that

would stimulate pupils to be involved in discoveries but we managed [laughs]. Is that

not good? It was hard at first but with time things develop. Now I know how useful

technology is to teaching.”

This quote from one of the participants of my research study illustrates the subject of my study;

teachers’ development of understanding how technology can enhance mathematics education. In

this introductory chapter, I describe the background information of the study and explain my mo-

tivation for conducting it. I begin with outlining background information leading to the problem(s)

that I identified and wanted to explore and overcome in my research study. In the second section,

I describe a synopsis of teaching of mathematics in a Zambian school and a Dutch school. Finally,

I outline the purpose of my study.

1.1 Statement of the Problem

The continuous rapid spread of technology in recent years has affected the expectations in our daily

lives and in education. This proliferation of Information and Communication Technology (ICT)

has come with changes which have had an impact on the teaching of mathematics, as well as on

the entire educational system worldwide. Grandgenett (2008) argued that use of technology in

the classroom would help to study real world phenomena in ways that were not possible in the

2 Introduction

recent past. He believed that if technology was left out in mathematics education, teachers would

miss opportunities for assisting pupils’ understanding of topics in which development and learning

of concepts are tied to use of computers. Grandgenett used geometry as an example, in which

he thought computers could be used for explaining phenomena: he argued that fractal geometry

uses computer-like algorithms to replicate relative geometric patterns that go beyond the usual

traditional geometry. In his own words, he pointed out that “clouds, coastlines, plants growth,

lightening, and even blood vessels are now routinely modelled with fractals and systematically

investigated with computer technology” (p. 4). He thought that this kind of illustration could help

the pupils to connect mathematics with real life contexts. He expressed his opinion that, although

the mathematics of fractals can be difficult, it has become a mathematics topic which needs to be

included in today’s secondary school mathematics curriculum. Due to the difficulty of learning the

mathematics of fractals when technology is not applied, Grandgenett argued that it was important

to train teachers to develop the knowledge of using computers in teaching, so that they would be

able to make use of it as they teach. However, it is good to keep in mind that Grandgenett never

tried teaching fractals in a classroom, but that he just thought that the topic of fractals would be

very useful if included in the secondary school curriculum. In view of this, I also think that the

use of computers in teaching mathematics will have a far reaching impact in the understanding

of mathematics. This is supported by the National Council of Teachers of Mathematics, NCTM

(2000, p. 24) which pointed out that “technology is essential in teaching and learning mathematics;

it influences the mathematics that is taught and enhances pupils learning” (p. 24). Haciomeroglu,

Bu, Schoen and Hohenwarter (2009) argued that the use of technology in a mathematics classroom

has become extensively acknowledged in educational literature. However, what is fundamental is

how to get teachers started with using technology in their teaching.

Looking at how mathematics is taught in Zambian schools, I come to the conclusion that

teachers are still using traditional methods of teaching, although the ICT policy on education was

launched by the Ministry of Education in 2001. The ICT policy envisaged that ICT shall be used

in schools, especially for teaching in mathematics and sciences. According to Esselaar, Hesselmark

and Silavwe (2002), Zambian teaching of mathematics had at the time of their report not benefited

from this policy. One of the major obstacles had been the lack of training of teachers in using ICT.

Although one of the effects of the policy in ICT had been that many Zambian schools had acquired

computers, they were not used in the teaching and learning because teachers have hardly any skills

in educational use of ICT and have not been trained in learning these abilities. My experience as

1.2 My Motivation 3

teacher and teacher trainer is that this situation still holds (1n 2010). It has kept teaching of many

parts of mathematics difficult, because teachers have for example been deprived of opportunities

for creating interactive and project oriented teaching environments in a mathematics classroom.

In contrast with the Zambian situation, the impression that I got during my stay in the Ne-

therlands is that schools in the Netherlands have been supplied with plenty of ICT equipment and

teachers have had courses to acquire ICT skills. In some schools, such as the International School

of Hilversum, all teachers and pupils have received a personal laptop which they can use both at

school and home. Nevertheless, like in Zambia, the facilities have also not yet been used to their

full potential in mathematics classrooms. In other words, the content knowledge regarding ICT is

present, but many teachers apparently lack knowledge to use ICT effectively in educational settings.

In other words, they lack Technological Pedagogical Content Knowledge (TPCK)1

1.2 My Motivation

The problem identified in the first section was the lack of use of ICT in the teaching of mathematics

albeit the availability of computers and/or laptops in schools. This problem was linked with un-

derdeveloped understanding of educational use of ICT. Noting this problem, I aimed at finding out

how teachers could develop their TPCK of teaching mathematics using technology after providing

them with training, which is coupled with extensive assistance in dynamic mathematics software.

I was also interested in exploring whether the technological pedagogical content knowledge of tea-

chers at a well equipped Dutch secondary school who are not familiar with use of ICT in education

is similar to that of Zambian teachers who have just been introduced to ICT.

For my research study it was necessary to limit it to a particular type of educational techno-

logy. I planned to introduce the use of dynamic mathematics software, in particular GeoGebra, to

Zambian mathematics teachers. Since GeoGebra is open source software and regarded useful in the

teaching and learning processes in many parts of the world, there is an opportunity and, one may

even say, a teacher’s right to have this software environment integrated in the educational system

in Zambia. However, I acknowledged that the introduction of this software in Zambia would not go

without difficulties. For the integration of GeoGebra to be possible, there is a need to look at many

factors that might affect teachers in adopting the use of ICT. This is why I wanted to explore how

1A definition of TPCK, later renamed TPACK, will be given in the next chapter. I will keep using the original

acronym in my study.

4 Introduction

Zambian teachers could develop understanding in the use of GeoGebra in teaching of mathematics.

After training teachers on how to use the software effectively in their mathematics lessons, I wanted

to know how they would blend it with their existing Pedagogical Content Knowledge as they teach

mathematics. I was also curious about the amount of help the teachers needed for being able to

effectively use GeoGebra in their classrooms.

I planned to contrast this with TPCK development of teachers at a Dutch school, in my case

study at Fons Vitae Lyceum, who have access and often use ICT in their teaching. Having been a

teacher and a pupil in a Zambian school, I think Zambian teachers face a lot of difficulties in planning

and executing their teaching. My personal experience as a teacher of mathematics and now a teacher

trainer may serve as an example. I have been teaching mathematics without ICT and the lesson

planning was always teacher-directed. As a result, teaching became very difficult because it deprived

me with the potential of creating learning environments in which pupils have richer opportunities of

constructing mathematical meanings, of exploring and experimenting with mathematical ideas and

expressing these ideas using a wealth of representations. Traditional teaching does not offer pupils

much opportunity to be involved in the learning process, instead they remain passive. Additionally,

in some topics, it is not easy to make learning materials for the pupils.

My personal experience with the use of GeoGebra in a secondary school classroom during

my winter project2 in the Master of Science in Mathematics and Science Education was that the

software could help pupils develop their understanding. So, if pupils’ understanding is facilitated

by the use of dynamic mathematics software, then it makes sense to encourage every teacher and

educator to learn about it (Koehler & Mishra, 2008). The main question is how to help teachers

to develop competencies of using the software in mathematics education. According to Shulman

(1986), teachers posses Pedagogical Content Knowledge which they use in their teaching. How

then can this combined knowledge of content and pedagogy that teachers hold be extended and

enhanced with knowledge about technology and its usibility in education? It may seem obvious

that technology will affect the knowledge that the teachers already have. It is for this reason

that I saw the need for Zambian and Dutch teachers that participated in my research to blend

technology with the knowledge they had in order for them develop learning materials and teach

using GeoGebra. There was also a need to alter the teachers’ altitude towards the use of ICT in

teaching mathematics.

2In winter project, pupils develop materials using ICT and use it on a Dutch school. This is a practical session

that pupils put their learnt theories to practice.

1.3 Summary 5

1.3 Summary

In summary, my research focused on a workshop-based approach for mathematics teachers to

develop Technological Pedagogical Content Knowledge (TPCK) regarding the use of dynamic ma-

thematics software. To this end, I developed two workshops for teachers. First, I developed and

piloted a workshop at Fons Vitae Lyceum, a secondary school in Amsterdam. On the basis of expe-

riences from this pilot workshop and on the basis of my experience as a Zambian teacher trainer, I

adapted and extended the workshop for use in Zambia. My motivation was especially to find ways

to increase the TPCK of Zambian mathematics teachers and to get them started to use computers

and dynamic mathematics software in their teaching practice. My research focused on in-service

teachers with varied experience in teaching mathematics. One of these teachers was a pre-service

teacher. The purpose of my research study was to investigate to what extent the participants, in

my workshops on using ICT in mathematics education, developed TPCK through the training.

6 Introduction

Chapter 2

Theoretical framework

In this research study, I explored how mathematics teachers could start to develop their technolo-

gical pedagogical content knowledge of teaching mathematics after training in the use of dynamic

mathematics software in a workshop. In this chapter, I begin with defining pedagogical content

knowledge, which teachers were expected to possess already or be able to posses. In the second

section, I introduce technological pedagogical content knowledge. Finally, I summarize the contents

of this chapter.

2.1 Pedagogical Content Knowledge

In the process of investigating the knowledge base of teachers, Shulman (1987) coined the term

Pedagogical Content Knowledge and defined it as “a special amalgam of content and pedagogy that

is uniquely the province of teachers, their own special form of professional understanding” (p. 8).

He contended that for effective teaching, teachers need PCK and not only (just) knowledge of the

subject matter or only pedagogical knowledge. He outlined PCK as the ways pedagogy, content, and

knowledge about pupils are intermingled into understanding about how certain topics to be taught

can be represented and adapted to pupils’ characteristics, interest and abilities. Shulman (1987)

described pedagogical content knowledge by outlining its three part framework: (1) pedagogical

knowledge; (2) content knowledge; and (3) pedagogical content knowledge. Under pedagogical

knowledge, he contended that knowing a subject for teaching requires more than knowing its facts

and concepts. This entails that one must also understand the organising principles and structures

and the rules for establishing what is legitimate to do and say in a particular subject. Teachers

should not only understand that something is so, but they must further understand why it is

8 Theoretical framework

so, on what grounds its warrant can be asserted, and under what circumstances the belief in its

justification can be weakened or denied.

One piece of content knowledge, he referred to it as curriculum knowledge, “represented by

the full range of programs designed for the teaching of particular subjects and topics at a given

level, the variety of instructional materials available in relation to those programs, and the set of

characteristics that serve as both the indications and contraindications for the use of particular

curriculum or program materials in particular circumstances” (p. 10). According to Shulman

(1987), curriculum knowledge is simply a grasp of the materials and programmes that serve as

‘tools of the trade’ for teachers (This is the understanding of the structures of the subject matter).

Shulman defined pedagogical content knowledge as the particular form of content knowledge that

embodies the aspect of content most germane to its teachability and includes an understanding

of what makes the learning of specific topics easy or difficult (this includes the conceptions and

preconceptions that pupils of different ages and backgrounds bring with them to the learning of

the most frequently taught topics and lessons).

Angeli and Valanides (2009, p. 155) wrote the following about PCK:

“PCK relates to the transformation of several types of knowledge, includes an unders-

tanding of what makes the learning of specific concepts easy or difficult, and embodies

the aspect of content most germane to its teachability” (p. 155).

According to Jang and Chen (2010), Shulman’s PCK framework has attracted much research and

has been interpreted in different ways. Jang and Chen (2010) noted that PCK influences teachers’

teaching in ways that best engender pupils’ science learning. Mulholland and Wallace (2005) also

argued that the idea of PCK has been interpreted by many researchers to suit the research context.

According to Grossman (1990), some researchers have included the aspect of the knowledge of

curriculum whereas others have excluded it. In experienced teachers, the subject-matter knowledge

and general pedagogical knowledge are blended together due to teaching experiences. Lederman,

Gess-Newsome and Latz (1994) contended that the development of PCK in science teachers is

stimulated by constant use of content knowledge in different teaching environments. As teachers

get rooted in the integrative process in the classrooms practice, PCK is developed and it guides

their actions when dealing with specific subject-matter in the classroom (PCK development never

stops: after pre-service training, teachers develop it further through practising teaching. In-service

training also plays a role in PCK development).

2.2 Technological Pedagogical Content Knowledge 9

2.2 Technological Pedagogical Content Knowledge

Mishra and Koehler (2006) represented Technological Pedagogical Content Knowledge (TPCK) as

a Venn Diagram (see Figure 2.1) consisting of three main subsets and in making reference to the

subsets, they wrote that

“the three, technology, pedagogy and content each have their separate areas, but also

interact with one another in pairs, and also in the centre area which represents the

intersection of knowledge of technology, pedagogy and content” (p. 1029).

The concept of TPCK was derived from the PCK framework of Shulman (1986) who contended

that teachers, for effective education, need to rely on an integrated knowledge base, one that is an

integration of multiple domains of knowledge (knowledge about subject matter, learners, pedagogy,

curriculum and schools) to translate the content in ways that pupils are able to grasp.

Schmidt, Baram, Thomson, et al. (2009) noted that TPCK presents the relationship and com-

plexities between the three components of knowledge, viz., technology, pedagogy and content, as

shown in Figure 2.1.

Figure 2.1: A figure showing components of TPCK framework (sourced from http://tpack.org).

The Venn diagram of the three basic components creates seven intersections as illustrated in Fi-

gure 2.1 namely: (1) Technological Knowledge (TK); (2) Content Knowledge (CK); (3) Pedagogical

Knowledge (PK); (4) Pedagogical Content Knowledge (PCK); (5) Technological Content Know-

ledge (TCK); (6) Technological Pedagogical Knowledge (TPK); and (7) Technological Pedagogical

Content Knowledge (TPCK). I want to remind the reader that my research focuses on in-service


teachers who have already established themselves with CK, PK and PCK. This therefore implies

that my main focus was on the following four components of the framework: TK, TCK, TPK and

TPCK. These four components of TPCK were defined by Schmidt et al. (2009) as follows:

1. Technological Knowledge: This is the knowledge teachers hold about different forms

of technology such as pencil and paper, scientific and graphical calculators, whiteboards,

beamers, and so on.

2. Technological Content Knowledge: This is the knowledge that facilitates the creation of

different representations of specific content. It envisages that when teachers understand and

use the technology appropriately, they would be able to change the way pupils practise, learn

and understand concepts in a specific content area.

3. Technological Pedagogical Knowledge: This refers to the knowledge of how teachers can

use various technologies in teaching, and how it is able to change the way teaching is done.

This also entails better understanding of technology use by teachers.

4. Technological Pedagogical Content Knowledge: This refers to the knowledge required

by teachers for integrating technology into their teaching in any subject area. Teachers

have basic components of knowledge (CK,PK, TK) by teaching content using appropriate

pedagogical methods and technologies (Schmidt et al., 2009; p. 125)

The development of TPCK is at the centre of improving teaching of mathematics and science

with the help of technology. It helps teachers in integrating technology into their teaching and it

enhance the ways of how this knowledge is developed. Schmidt et al. (2009) wrote that

“using TPCK as a framework for measuring teaching knowledge could potentially have

an impact on the type of training and professional development experiences that are

designed for both pre-service and in-service teachers” (p. 125).

This is why there is still a need to design ways for enhancing teachers’ TPCK , after adequate

training has been provided.

Niess (2005) argued that the answer to the question of how to help teachers to develop com-

petencies of using the software in mathematics education, involves guiding the development of

teachers’ knowledge and skills for teaching with technology. Therefore, teachers need programmes

that help them develop a technological pedagogical content knowledge. I concur with this view.

2.2 Technological Pedagogical Content Knowledge 11

The National Council of Teachers of Mathematics, NTCM, (2000 argued that “the effective use of

technology in the mathematics classroom depends on the teacher. [...]. As with any teaching tool,

it can be used well or poorly” (p. 25). Therefore, teachers should use technology to enhance their

pupils’ learning opportunities by selecting or creating mathematical tasks that take advantage of

what technology can do efficiently such as graphing, visualising, and computing.

With the advent of computers in today’s schools, there was indeed a need to extend Shulman’s

original framework of PCK in order to be able to explain the teachers’ cognitive use of the ICT in

the teaching of mathematics. The teachers’ environment becomes more complex when they switch

to computer use as they integrate knowledge of pupils’ thinking and learning, knowledge of the

subject matter, and increasing knowledge of ICT (cf., Harris, 2008). This view is supported by

Smeets and Mooij (2001), who saw the teacher as a major player in the process of integrating ICT

in their lessons. The role teachers play in using ICT determines the kind of learning environment

that is created in the classroom. The use of ICT changes the way teachers teach in their classrooms.

Boylan (2010) argued that while some teachers resist change (from the traditional way of teaching

without ICT to teaching with ICT), once it occurs, teachers tend to appreciate the ICT-supported

environment they create in the classroom for pupils. The effective use of ICT by teachers helps both

pupils and teachers to better understand mathematical relationships that they explore through use

of dynamic mathematics software (Boylan, 2010). This is for example enables by using dynamic

mathematics software in ways that engage the pupils and teachers in the teaching and learning

processes. However, the type of software adopted in the classroom relates to the form of method

or strategy the teacher employs in that particular teaching and learning environment. Software

that is dedicated to skills-based education only has a limited use in terms of exploratory learning

by the pupils. However, software that is both skills-oriented and open-ended allows teachers to

engage their pupils in more individual exploratory forms of learning, thereby allowing their pupils

to construct their own knowledge. Smeets and Mooij (2001), wrote that “teachers who adhere

to traditional transmission approaches, to instruction, prefer skill-based software, whereas most

teachers who support constructivist views of teaching and learning use skill-based as well as open-

ended software” (p. 345). Smeets (2005) pointed out that skills-oriented software endeavours to

develop pupils’ skills by instilling drill and practice tasks, whereas open-ended software could serve

as a tool for aiding pupils develop understanding. In my research, I decided to use GeoGebra, a

software that provides support for both a skills-oriented and open-ended approach. The basis on

which I conducted and planned my research was the following: a workshop followed by teaching


practice using developed lesson materials using GeoGebra. Teachers were free to decide on how

they adopted and used the software, depending on what suited their thinking and situation at a

particular time. But, whatever choices they made, teachers were supposed to develop ICT-inclusive

lessons materials that encouraged learning in which pupils controlled their own learning and could

participate in group work.

2.3 Evaluation of TPCK

Because I wanted to explore to what extent my workshop contributed to the teachers’ development

of TPCK, I was in need of a framework for measuring TPCK. I looked in the research literature for

such a framework, but I could not find one that was appropriate for my purpose. Appropriateness

meant for me that the framework should meet the criteria listed below. The framework should

inform the researcher about the teachers’ TPCK by measuring effects of teacher-designed, ICT-

based instruction in class. In particular, I aimed at measuring the following effects of teachers’ new

instructional approach on pupil performance: The dynamic mathematics software (DMS) and the

DMS-based instruction enable pupils to

• produce and work with accurate graphs;

• study efficiently and effectively, so that they can keep up with the teacher-set pace of instruc-

tion;

• carry out small experiments in a given mathematical problem situation and explore own

mathematical ideas not explicitly taught;

• come to mathematical results by a method of trial and improvement;

• interact with the software and take advantage from the immediate feedback given by the

DMS; and

• enjoy mathematics, get more interested in the subjects taught, and gain confidence in their

mathematical performance.

These are objectives that are hardly met in traditional Zambian instruction.

Instead of a ready for use framework, I found the so-called practitioner model of the use of

computer-based tools and resources to support mathematics teaching and learning, which was

2.4 The Practitioner Model 13

developed by Ruthven and Henessy (2002) and refined into a more compact form by the same

authors in 2003 on the basis of interviews with teachers in mathematics departments. Ruthven,

Deany and Henessy (2008a; 2009) applied their 2003 model to the use of graphing software in

teaching algebraic forms. In these studies they looked at the applicability of the model when

investigating what goes on in a classroom if graphing software is used. Apparently, this model

has hardly been used for exploring teachers’ opinions and educational practice regarding the use

of dynamics mathematics software. However, I considered it useful for my research. In the next

section of this chapter I discuss the practitioner model and elaborate on its use in my research. I

base my exposition on the 2009 study of Ruthven, Deany and Henessy, in which they used it for

exploring educational practice.

2.4 The Practitioner Model

The Practitioner Model of the use of ICT-tools in mathematics education is a model that was

developed by Ruthven, Deaney and Hennessey (2002) out of secondary school teachers’ accounts

of successful use of computer-based tools and resources to support the teaching and learning of

mathematics. These experiences and accounts were elicited through interviewing mathematics

teachers. Out of these accounts and experiences of teachers, a pedagogical model was developed

through organisation of central themes deduced from these accounts. This model, which Ruthven

and Hennessey (2002) called “the Practitioner Model”, had ten themes listed below:

1. Ambience enhanced in changing the general form and feel of classroom activity;

2. Tinkering assisted in helping to correct errors and experiments with possibilities in carrying

out tasks;

3. Routine facilitated in enabling subordinate tasks to be carried out easily, rapidly and

reliably;

4. Features accentuated in providing vivid images and striking effects which highlight pro-

perties and relations;

5. Restraints alleviated in mitigating factors inhibiting pupil participation such as the la-

bouriousness of tasks, the requirement for - and the demands imposed by -pencil-and-paper

presentation, and vulnerability to mistakes being exposed;


6. Motivation improved in generating pupil enjoyment and interest, and building pupil confi-

dence;

7. Attention raised in creating the conditions for pupils to focus on overarching issues;

8. Engagement intensified in securing the commitment, persistence and initiative of pupils

in classroom activity;

9. Activity effected in maintaining the pace and productivity of pupils within classroom ac-

tivity; and

10. Ideas established in supporting the development of pupil understanding and capability

through classroom activity.

The first four themes in the above list depended more directly on exploiting affordance of ICT

and the last six themes depended on the processes stated in the themes themselves (Ruthven and

Hennessey, 2002)

The model was then revised and downsized by Ruthven and Hennessey (2003) to a model with

six themes by merging major themes as listed below.

1. Effecting working processes and improving production. Teachers pointed to ways in

which use of ICT could expedite and —more broadly— facilitate the more routine components

of classroom activity, increasing the productivity of pupils and improving the quality of work

they produced, allowing them to be carried out more quickly and reliably, with greater ease,

and to higher quality.

2. Supporting processes of checking, trialling and refinement. Teachers pointed to

ways in which use of ICT could support various processes of checking, trialling and refine-

ment. There was positive comment on courseware — commercial or teacher-devised — which

presented sequenced items to pupils, testing them at each step to provide “feedback imme-

diately on how they’re doing,” and “giving the kids a chance to check their work, because

they can’t go to the next step unless they get the first question right.” Equally, there was

approval for the use of calculators and spreadsheets to check the results of calculation and

graphing already done ‘by hand’.

3. Enhancing the variety and appeal of classroom activity. Teachers pointed to ways in

which use of ICT could bring variety to classroom activity, and enhance its appeal. There were


many rather general references to activity involving ICT use as being “something different;” as

“mak[ing] a change;” as adding “another dimension;” and — most frequently — as providing

“variety.” Allusions were made to the “novelty value” of work with ICT. There was talk of

how pupils “like a change from the routine of the classroom situation and love to go to the

ICT room. There were suggestions of pupils “enjoy[ing] seeing things done in a different way”

and of “a different teaching and learning style.”

4. Fostering pupil independence and peer support. In a less developed theme, teachers

pointed to ways in which use of ICT could create opportunities for pupils to exercise greater

independence and share their expertise. Comments about pupils willingness to experiment

with ICT have already been reported. Teachers noted how pupils had “shown themselves

more self-sufficient than we are at using things like that,” and how, given the opportunity,

pupils could “go off and do amazing things.” Such independence “gave them [pupils] some

ownership over it and taught them how to do things for themselves, as opposed to just relying

on having a teacher there.”

5. Overcoming pupil difficulties and building assurances. Teachers pointed to ways in

which use of ICT helped to overcome particular difficulties which pupils might encounter

in carrying out schoolwork, so removing the associated disincentives and building pupils

assurance. Computer mediation was seen as alleviating particular difficulties which many

pupils — including those with special needs — experienced in writing, drawing and graphing

by hand. Comments noted how using ICT “[did]n’t involve doing lots of [hand]writing,

which is often something that the lower attainers are unhappy doing,” and overcame pupils’

difficulties in drawing accurate graphs.

6. Focusing on overarching issues and accentuating important features. Teachers poin-

ted to ways in which use of ICT could help to focus the attention of pupils on overarching

issues, and to accentuate important features of situations under consideration. Comments

reported how use of ICT could facilitate or automate subsidiary tasks — typically those invol-

ving routine data handling, calculating and graphing — freeing users to give their attention

to more overarching matters.

Ruthven, Deaney and Hennessey (2003) claimed that each theme points to vital ways of teaching

with ICT. They interviewed teachers and asked their opinions about the use of ICT tools and


resources to reinforce teaching and learning. The model highlighted the contribution of ICT tools

and resources to the themes categorised in the model. In 2009 Ruthven, Deaney and Hennessey

reported about a research that they had done and in which they had observed two teachers teaching

linear forms and quadratic functions. They analysed the lessons, the teachers and their teaching

were analysed by using the themes from the revised 2003 practitioner model. The authors of

the article observed that the contributions of the graphing software to the teaching of algebraic

forms, which the two teachers deployed in their lessons, matched with the following terms in the

practitioner model [the order of the themes was also changed] (p. 295):

1. Effecting working processes and improving production through making it easier to

produce graphs accurately and rapidly, so increasing the efficiency and pace with which related

topics can be covered;

2. Overcoming pupil difficulties and building assurances, through making graphing tasks

more accessible to pupils who have difficulties with organisation and presentation;

3. Supporting processes of checking, trialling and refinement, through enabling les-

son tasks based on trial and improvement, and supporting mathematical speculation and

experimentation within and beyond the lesson agenda;

4. Focusing on overarching issues and accentuating important features, through hel-

ping to bring out the effects of altering particular coefficients or parameters in an equation on

the properties of its graph and through facilitating comparison of gradients and examination

of limiting trends;

5. Enhancing the variety and appeal of classroom activity, through reducing ‘labourious’

written work, increasing the immediacy and interactivity of classroom tasks and helping to

create new forms of playful challenge within lessons;

6. Fostering pupil independence and peer exchange, through providing support for ex-

ploration by pupils and consequent sharing of discoveries, including software techniques and

mathematical ideas within and beyond the lesson agenda.

The model of using graphing software to teach algebra consisted of six themes that were considered

useful in evaluating the use of ICT in the teaching of mathematics. Ruthven, Deaney and Hen-

nessy’s (2009) review of how teachers thought about and implemented use of graphing software in


mathematics education offers a constructive outline to the practitioner model which was specifically

developed out of teachers’ experiences in teaching mathematics using graphical software; graphical

calculators in particular. Ruthven, Deaney and Hennessey (2009) argued that “the themes from the

model provide a useful organising framework for synthesising the thinking reported by the teachers

in association with each lesson” (p. 289).

Lagrange and Monaghan (2010) explained that in their quest to understand the teachers’ prac-

tices when using technology, they made use of various models. They found Ruthven and Hennessy’s

(2002) model useful in their study. They argued that this model helped them to understand how

teachers could connect potentialities of technology to their pedagogical needs. Lagrange and Mo-

naghan, (2010) wrote that “the observation of two teachers [whom they were working with in their

study] using dynamic geometry showed what happens when the connection does not work: the tea-

cher tries to re-establish the connection by becoming a technical assistance” (p. 1610). Lagrange

and Monaghan (2010) found out that the model provided useful patterns of giving an account of

teachers’ pedagogical reasons for the classroom use of dynamic geometry.

In my research, I adopted the practitioner model for evaluation of the teachers’ use of GeoGebra

in the lesson planning, the development of teaching and learning materials, and in the teaching in

the classroom. Since Lagrange and Monaghan (2010) used the model in their research on dynamic

geometry and found it useful and appropriate, I thought adopting it in my own work could be of

great help. However, I adjusted the Ruthven, Deaney and Hennessy’s (2009) model to revert it in

one aspect to its original model of 2002, which had motivation as a theme on its own. Leaving the

other themes intact, I made the Ruthven, Deaney and Hennessy (2009) model more aligned to the

use of dynamic geometry, so that in general it more is suitable for my study. In addition to the

six themes of Ruthven, Deaney and Hennessey’s (2009) model, I included the aspect of motivation

in the use of dynamic mathematic software through the provision of feedback to both pupils and

teachers. I included motivation to the model because I believed that motivation influences the

learning processes of pupils. Dweck (1986) argued that “motivational processes influence a child’s

acquisition, transfer, and use of knowledge and skill, [...].” (p. 1040). I thought that if pupils were

motivated to using dynamic mathematics software, they could develop confidence and start en-

joying their lessons thereby enhancing understanding. Hershkovitz and Nachmias (2009) perceived

motivation as a factor that explains individual differences in intensity and direction of behaviour.

Hershkovitz and Nachmias (2009) defined motivation as “an internal state or condition that serves

to activate or energise behaviour and give it direction.” (p. 200). Motivation therefore, has an


impact on the way pupils learn and adapt to using dynamic mathematics software.

On the other hand, teachers could as well be motivated if the use of the software they adopted

enhances effective teaching. Torrance and Pryor (1998) argued that it was in the best interest

of the teacher to foster learning goals in the pupils. For a person to be actively engaged in the

educational endeavour, he or she must value learning, achievement, and accomplishment even with

respect to topics and activities he or she does not find interesting. Deci, Vallerand and Pelletier

(1991) argued that value of a topic comes from internalization and integration. They wrote the

following:

“in terms of education, it has become ever more apparent that self-determination, in

the forms of intrinsic motivation and autonomous internalization, leads to the types of

outcomes that are beneficial both to individuals and to society” (p. 342).

By reintroduction of motivation in this model, I wanted to focus on learning goals in the teaching

and learning processes. Dweck (1986) argued that children with learning goals usually use obstacles

as a signal for increasing their efforts and vary their strategies which often end up in improved

performance. She believed that “the more children focus on learning or progress, the greater

the likelihood of maintaining effective strategies” (p. 1042). Deci et al. (1991) pointed out that,

if teachers provided support for competence such as positive feedback, there would be enhanced

motivation in general. But it could enhance intrinsic motivation and integrated internalization

only if it was administered in ways that are autonomy supportive. The effects of supports for

competence and relatedness on motivation and internalization have been studied in many studies.

In such studies, positive feedback has generally been found to increase intrinsic motivation because

it enhances perceived competence although studies have shown that this enhancement occurs only

when the feedback is accompanied by support for autonomy (Ryan, 1982).

One of the ways of enhancing motivation was through developing lessons that would provide

useful feedback to the pupils. In the process, the pupils would be encouraged to develop independent

learning. This could be one of the attributes that every teacher would want his or her pupils to

develop in their learning. My adapted practitioner model was expected to be used in ways that

would help teachers [those who attended my training] develop expertise in the use of dynamic

mathematics software through interacting with the researcher and among themselves. I called my

model ‘the adapted practitioner model’. For every theme of this model presented in Section 2.5, I

give justification for the adaption.

2.5 Adapted Practitioner Model for Teacher Training 19

2.5 Adapted Practitioner Model for Teacher Training

In this section, I outline the adapted practitioner model that I adopted for my teacher training.

Each theme of the original model has been adjusted so that it is aligns to the use of dynamic

mathematics software. After every theme, I give my reasons for the adaptation of the theme.

1. Effecting working processes and improving production: The use of dynamic mathema-

tics software should foster effectiveness in the teachers’ planning process as well as improving

their teaching. As Zambian and Dutch teachers incorporated the software in their teaching

and learning, my interest was whether there was improved accuracy in the way the produced

lessons and teaching materials [for instance the use of graphing and presentation, for both

pupils and teachers] resulted in effective lessons, thereby increasing the pace at which topics

are covered. Reasons for Adaptation: The adjustment was done in order to capture teachers’

planning, accuracy in graphing and presentation, as well as to capture the productivity of

pupils, the quality improvement of their work and the ease with which they used the dynamic

mathematical software. I explored how teachers incorporated technology in their PCK as

they planned to make their lessons effective and I investigated whether there was an increase

in the pace of coverage of topics.

2. Overcoming pupil difficulties and building assurances: The dynamic mathematics

software should afford teachers with algebraic and graphical usability and let them find the

software user-friendly and useful. When teachers are experienced with the software and realise

its potentiality in simplifying mathematical concepts (for example equations and graphs), they

could devise ways of overcoming their pupils’ difficulties in learning mathematics. This could

encourage the teachers to adopt dynamic mathematics software for their use in teaching of

mathematics. In doing so, they would help their pupils in averting difficulties in graphing,

calculations through easy access to tasks and presentations, and develop full confidence in

mathematics (pupils that formerly were not self-confident and were struggling in mathematics

should now appreciate the subject). Many pupils face difficulties in writing, drawing and

graphing by hand. These problems can be alleviated through use of dynamic mathematics

software, resulting in accurate drawing and graphing.

Reasons for Adaptation: The change was to allow teachers to get used to the software and see

how it afforded them with algebraic and graphical skills so that they were able to help pupils

overcome difficulties and develop assurance in pupils, as they use the software in their tasks.


Here I was interested in seeing teachers getting used to the use of the dynamic mathematical

software in their preparations and teaching processes. The researcher wanted teachers to use

the software effectively in order to avert graphing difficulties that many pupils have.

3. Supporting processes of checking, trialling and refinement: The lesson materials

developed by the teachers should afford pupils the opportunity to use dynamic mathematics

software, through trial and improvement so that pupils are able to end up with a refined

understanding of the problem at hand. I wanted to explore how these teachers developed

materials using dynamic mathematics software that would be motivating pupils. Developed

learning materials should be able lead pupils in playing with the software so that they develop

their own understanding through the use of speculative and experimentation tasks thereby

allowing them to do tasks beyond the classroom tasks.

Reasons for Adaptation: The change was made so that I could focus on the development

of lesson materials by teachers using the dynamic mathematical software. The developed

materials should be able to help pupils to develop their own understanding through working

with the dynamic mathematics software. The learning materials made by teachers should

foster inquiry based learning in which pupils are engaged in independent learning, in which

they control their own learning process.

4. Focusing on overarching issues and accentuating important features: Teachers

should be able to develop activities that could help pupils focus on overarching issues, and to

give emphasis on vital concepts. Since the software enables playing around with parameters

in the equations in the algebra window pupils must develop the ability to observe and analyse

the effect it has on the geometric figure in the geometric window. Teachers make use of these

features that the software provides as they prepare or plan their lessons, and help pupils to

develop understanding on the relationship between the equations and their graphs. The un-

derstanding of geometry by the pupils depends on the effective use of dynamic mathematics

software by teachers. Effective use of dynamic mathematics software must foster subsidiary

tasks such as those involving routine data handling, calculating and graphing.

Reasons for Adaptation: The change was made to help me capture how teachers make use of

GeoGebra in connecting algebra and geometry by making sure that they planned their lessons

in ways that effectively affords pupils develop understanding of properties of equations and

their graphs through manipulation of parameters in the algebra window and see how it affects

2.5 Adapted Practitioner Model for Teacher Training 21

the corresponding graphs. The emphasis was to explore how the teachers made use of features

of the software in their TPCK.

5. Enhancing the variety and appeal of classroom activity: The use of dynamic mathe-

matics software in the classroom lessons should enhance the creation of a learning environment

that brings about variety to classroom activity, thereby giving pupils a different taste of lear-

ning as opposed to routine one. With the acquired knowledge, teachers should be able to

create lesson activities that could enhance appeal in pupils, resulting in pupils to value their

learning process. Working with dynamic mathematics software should make classroom tasks

unique each time to the extent that challenge tasks are included as it helps motivate pupils

in doing the task with fun.

Reasons for Adaptation: The change was made in order to see how teachers made use of

dynamic mathematical software in creating a learning environment that promotes variety

and appeal of classroom activity. Teachers should allow pupils being involved in interactivity

through the creation of a variety of tasks which motivate them [the pupils]. I explored how

teachers created this learning environment for making sure that there was variety, appeal and

interactivity in the way the classroom tasks were done as well as promotion of a variation in

classroom activities or tasks.

6. Fostering pupil independence and peer exchange: Teachers should be able to develop

the skills of using dynamic mathematics software in ways that would foster pupils developing

independent exploration. Teachers can develop lessons that require the pupils to be involved

in exploring beyond what they have been provided with in class, thereby fostering critical

thinking in pupils. I wished to explore how these teachers blended the use of dynamic mathe-

matics software in developing lessons that encouraged discovery lessons through independent

work.

Reasons for Adaptation: I changed this so as to monitor how teachers were able to develop

lesson activities that called for pupils’ independent work, in which they are allowed to make

discoveries as they do their tasks. Here I explored how well the software was used by teachers

in creating a learning environment in which pupils worked with minimum direct help from

the teacher.


7. Motivating teachers and pupils through provision of feedback, generating pupil

enjoyment and interest: Learning and using dynamic mathematics software, combined

with appropriate feedback, should encourage teachers to use dynamic mathematics software

with their own pupils. The applets that teachers developed must avail feedback to the pupils

so that they are motivated in the lessons. Activities that generate enjoyment and interest

help pupils’ understanding and builds their interest. When pupils use GeoGebra applets, that

in a sense gives feedback to the teacher: Pupils can send GeoGebra file to the teacher. This

could help the teacher to focus on activities with which pupils have difficulties.

Reasons for inclusion: The theme about motivation was re-included in the model so that I

could find out how the training and individual guidance in the use of dynamic mathematics

software encouraged teachers to adopt its use in in teaching mathematics. I was also inter-

ested in exploring how pupils developed understanding through using dynamic mathematics

software and providing feedback to the teacher, so that their problems/difficulties could be

understood or taken care of by the teacher.

May I remind the reader that, in this research study, I wanted to investigate to what extent the

participants in my workshops on using ICT in mathematics education developed TPCK through

the training. Since the use of technology in teaching and learning has to be blended with the

teachers’ PCK, I wished to explore how useful the workshop was to this end. In addition, I

wanted to compare the experiences and understanding the needs of teachers in the use of dynamic

mathematics software of Zambian teachers with those of Dutch teachers.

Chapter 3

Research design and methodology

In this chapter, I outline the research questions, the research setting, the research instruments, and

the methodology that I used in data collection and data analysis.

3.1 Research Questions

The purpose of this study was to explore whether my workshop helped teachers who already have

developed TPCK and change their teaching accordingly? This study will answer the following main

research question:

To what extent could participants of my training course in using ICT in

mathematics education start developing TPCK through the workshop?

By “training course”, I mean the training in using GeoGebra in mathematics education that I gave

to mathematics teachers through the workshop. TPCK is the technological pedagogical content

knowledge which has to do with how teachers combine knowledge about technology with knowledge

of how to teach a curriculum course to their pupils. ICT is simply all the uses of digital technology

that exist to help individuals collect, process and use information. In my case, I considered the use

of technology to help teachers teach mathematics. I used the adapted practitioner model, explained

in Section 2.5, to assess the development of TPCK of the teachers who participated in my research

study.

The following questions are research sub-questions that helped me to answer the above main

research question of my study.

24 Research design and methodology

Sub question 1. What obstacles do teachers encounter when they start using dynamic

mathematics software?

Sub question 2. What assistance do teachers need to overcome obstacles?

Sub question 3. Does collaboration of less experienced teachers with each other

during the development of teaching materials and lesson planning help them in

developing their TPCK?

Sub question 4. Does the teachers’ personal experience in the classroom teaching

using the materials they developed during the training course contribute to their

development of TPCK?

On the one hand, sub questions 1 and 2, have to do with identifying and overcoming the problems

that teachers may encounter when they start using GeoGebra in their teaching. On the other hand,

I wanted to find remedies teachers needed in order to avert the teachers’ problems. The interest

was more focused on the Zambian teachers who used ICT in mathematics teaching for the first

time.

My expectations of the obstacles that teachers could face (research question 1) were;

• lack of time to prepare the learning and teaching materials,

• lack of time to work on the pupils’ activities;

• lack of ICT facilities to enable teachers start the use of GeoGebra in teaching;

My expectations of the assistance that teachers could need (research question 2 ) were;

• enough time to get to know how to use the software in their teaching;

• enough help in terms of planning and development of lesson material;

• enough training in terms of workshops, which must be conducted continuously, to help tea-

chers in getting started to using GeoGebra in teaching and to help them become proficient

users.

My expectations on the collaboration of teachers (research sub question 3) were as follows:

• When teachers discuss on the development of teaching and learning materials, they are able

to think of ways of helping their pupils get involved in activities.

3.2 Research Setting and Research Methodology 25

• Working in groups helps teachers to develop critical thinking about what they want to develop

in their activities which could be useful for pupils’ learning.

• There is a risk to working in groups, namely, that it is a waste of time as teachers just end

up arguing. Some structure of collaboration must first be agreed upon amongst the teachers.

Regarding research sub question 4, my workshop not just about training teachers to use Geo-

Gebra and how to develop lesson materials, but also included training sessions in which teachers

were offered to opportunity to have a personal experience in the classroom teaching using the lesson

materials they developed. It is something that I wanted to investigate in my study. In my opinion,

this made my research quite special.

Please take notice of the fact that the teachers referred to in the research questions are in fact

the Zambian teachers that I trained and supported.

3.2 Research Setting and Research Methodology

To be able to answer the research questions, the study was conducted in two high schools, one in

the Netherlands and one in Zambia, with mathematics teachers as participants. In Zambia, I used

pre and post questionnaires to investigate how teachers developed their TPCK and how motivated

they were in utilizing the knowledge. In the Netherlands, only the post questionnaire was used to

assess the teachers’ developed TPCK. This was set up in this way because I used the study in the

Netherlands as a pilot study to help me develop a comprehensive Zambian study. My initial plans

of conducting the research at a pedagogical institute in Kabwe, in which I planned to conduct seven

training sessions in two months (one session per week) never materialised. This was because the

computer laboratory that I was supposed to use was in use for other educational activities. Instead,

the workshop was held every day for seven consecutive working days at a high school in another

city. This helped both teachers and me to work continuously without disruption. Teachers were

first introduced to GeoGebra in the first workshop and trained how to use it. The pre and the post

were administered to teachers before and after the whole set of training sessions, respectively. The

unexpected change of research setting was made possible by the help of the Ministry of Education.

In the next subsection, I outline the school setting. Furtheron, I describe the research methods

and analysis framework that I used in my research.


3.2.1 The School Setting

Mkushi High School (See Figure 3.1), where the study was conducted, is in a rural district called

Mkushi. The district is in the Central Province of Zambia, but it is actually in the northern part

of the country. According to data from The Zambian Central Statistics Office, CSO (2003), the

district had a population of 107,438 persons in 2000. The school is a core education boarding

institution with a pupil population of about 2000 pupils from grade 10 to grade 12. Each class has

about sixty pupils.

Figure 3.1: Mkushi High School, where the research was conducted.

Supported by the Ministry of Education, I applied for and received consent to conduct this

study at Mkushi High School. I also received permission to recruit mathematics teachers, and two

grade 12 classes from the school for participation in my study. Ten teachers, profiled in Table 3.1,

participated in the training course. Half of them had 10 or more years of teaching experience. Of

these ten teachers, one teacher was female. The two grade 12 classes engaged in the study were

used for the purpose of teachers experiencing actual teaching with their developed ICT-inclusive

materials. The school has ICT facilities (about 40 computers in the computer lab, but with only

15 working well or in use).

3.2 Research Setting and Research Methodology 27

S/N Teacher Subject Combination Experience (yrs) Use of ICT

1 A maths/chemistry 10 no knowledge

2 B maths 12 Word processing

3 C maths/history 4 Word processing and email

4 D maths/chemistry 10 Word processing and email

5 E maths 12 Word processing

6 F maths 4 Word processing and email

7 G maths 4 Word processing and Facebook

8 H maths 1 Word processing

9 I maths 17 Word processing

10 J maths on teaching practice no knowledge

Table 3.1: Profiles of the teachers at Mkushi High School who participated in my research study.

In the Netherlands, the pilot study was conducted at Fons Vitae Lyceum, a secondary school in

Amsterdam offering the HAVO and VWO examination programme. The school has a population

of 950 pupils. Five mathematics teachers from this school participated in the study. The school is

well equipped with ICT facilities and the teachers recruited had different levels of ICT expertise.

Two short workshops (1 hour each) were conducted for the teachers in which they were introduced

to the use of GeoGebra and were assisted by the researcher in their practice of using ICT.

The Timeline of the Research and Details

My research study was divided into three main stages (see Table 3.2): (1) the pilot study in the

Netherlands; (2) training sessions in Zambia; and (3) teaching practice in Zambia.

3.2.2 Research Method

This section describes the procedures for collecting and analysing data, and the measures taken

to ensure both reliability and validity. In this research, I used in fact a number of methods for

the collection and analysis of data. I undertook a kind of action research because it involved

teachers and myself as researcher and teacher trainer trying to implement the ICT policy of the

Zambian Ministry of Education on a small, local scale. The reasons for undertaking such a study

was specifically for my intention carrying out a research that would inform and change the practice

in the teaching of mathematics in Zambian schools. In addition, the study was carried out within

the context of the teachers’ environment, as described by Watts (1985).


Research Stage Period Activities

Pilot study October, 2010 1. Review of literature for background knowledge

2. Training of teaching in GeoGebra.

3. Interviews and assistance of teachers develop materials.

4. Discussion with teachers and their reflections.

5. Review and revision of workshop materials used in

pilot study and adaptation for use with Zambian teachers.

Training sessions February, 17-26, 2011 1. Search for government permission to use Mkushi

High School and teachers for my research.

2. Introduction of GeoGebra use to teachers.

3. Training sessions on 8 consecutive working days.

4. Observation of teachers developing lesson materials

and demonstration.

5. Assistance of teachers.

Practice teaching February, 26, 2011 1. Observation of teachers teaching pupils using

the developed materials from the workshop.

2. Assistance of teachers in helping pupils in their lessons.

3. Conclusion of two lessons of teaching practive.

Table 3.2: The timeline of the research study.

Because I have been a mathematics teacher in the past and currently work as a teacher trainer,

I learned what teachers are able to influence and what could be changed to produce results in

the teaching of mathematics. The research provided me the opportunity to work with teachers in

a school setting and learn to share ideas. In my research, I followed four sequences of the basic

themes of action research: (1) empowerment of teachers; (2) collaboration through participation in

the workshops or training and discussions; (3) acquisition of knowledge; and (4) social change or

change in the way teachers teach and perceive mathematics (Oja & Smulyan, 1989).

I followed a routine proposed by Watts (1985), that is guided by movement through five phases

of inquiry (see Figure 3.2): (1) identification of a problem; (2) data collection and organisation;

(3) data interpretation; (4) action based on the data; and (5) reflection.

My research involved interacting with the participants in the process of using GeoGebra through

the process of helping them and finding out how they thought about the use of ICT as they

went about developing lesson materials. This principle of collaborative resource was identified by

Ferrance (2000) as a key element of action research. According to Ferrance (2000), action research

is a procedure by which participants evaluate their own educational practices. This is in line with

3.3 Research Instruments 29

Figure 3.2: Action Research Cycle adopted from Watts (1985)

.

how Mills (2003) wrote about action research, he described it as any systematic inquiry executed

by teachers with the interest in the teaching and learning process so that data is collected.

I am a teacher trainer who investigated my support programme for secondary school teachers. I

was inspired by Thomas, Krantz, and Ramirez (1986), who believed that action research is aimed at

contributing to both practical concerns of people in immediate problematic situation and furthering

the goals of social science. In other words, I used action research to investigate my our teacher

training. In the next section, I outline methods used for data collection and data analysis.

3.3 Research Instruments

I used the following five research instruments for data collection in my research: (1) pre ques-

tionnaire; (2) post questionnaire; (3) teacher interview; (4) pupil interview; and (5) field notes. I

describe them in more detail in the following subsections.

3.3.1 The Pre Questionnaire

The purpose of the pre questionnaire was to establish the teachers’ background knowledge in ICT

and how they used it in teaching. The pre questionnaire (Appendix A) was designed in such way

that the researcher would receive a teacher’s account of his or her vision on ICT use in teaching

mathematics. The teachers’ knowledge about and classroom use of ICT in teaching mathematics

was explored in the pre questionnaire. The questionnaire consisted of 6 questions of which the


first 4 were closed questions, requiring teachers to mark or circle their options. The last two

questions were open-ended, requiring teachers comments. Filling in the pre questionnaire was

expected to take 20 minutes. I used the pre questionnaire to assess teachers’ initial knowledge of

ICT, information which I needed for answering the main research question: To what extent could

participants of my training course in ICT in mathematics education develop TPCK through the

workshop? My assumption was that teachers possessed and continued developing PCK as they

were already teaching for some time after graduating form the pedagogical training college. The

questionnaire concnered four themes, each serving a specific purpose:

1. The computers use by teachers and pupils: The purpose of the questions under this theme

was to help the researcher to assess teachers’ and pupils’ knowledge of ICT.

2. The computer use during lessons: The purpose of the questions in this theme was to find out

the type of ICT teachers use in their lessons. The assumptions were that the teachers used

ICT as a presentation tool, to help pupils discover mathematical concepts, for educational

games, and for accessing information from the internet.

3. How teachers wanted to use computers in their lessons: The purpose of the question was to

explore what and how teachers wanted to use computers in teaching mathematics. In other

words, I wanted to explore what teachers felt was the role of computers in teaching.

4. Teachers’ expectations for using ICT in teaching: The purpose of the question in this theme

was to explore what teachers’ expectations were after undergoing the training in the workshop.

3.3.2 The Post Questionnaire

The post questionnaire was developed to assess teachers’ development of TPCK after the training

course. This questionnaire was developed with the training course in mind so that it could help

analyse the effectiveness of my training programme on the teachers’ development of ICT skills.

The first four questions were designed to emphasise learning to use GeoGebra. Two questions were

designed to emphasise use of GeoGebra as a teacher. Four questions concerned possible benefits

of GeoGebra for pupils, and the last four questions emphasised on the workshop and afterwards.

The results in the questionnaire would give an indication of the teachers’ knowledge and use of

ICT in teaching and learning. This would be compared with the results of the pre questionnaire

on the same subject. The post questionnaire was developed for the purpose of answering the

3.3 Research Instruments 31

main research question. It was divided into the four parts (see Appendix B): (1) learning to use

geoGebra; (2) teacher’s use of GeoGebra; (3) pupils’ benefits of using GeoGebra; and (4) teachers’

development of TPCK during and after the workshop.

3.3.3 Teacher Interview

At the end of the training course, four out of ten teachers were interviewed using a semi-structured

protocol (Appendix C). The teacher interviews were designed to get feedback from the teachers

about: (1) their experience of the workshop; (2) the teaching practice of conducting a mathema-

tics lesson using dynamic mathematics software; (3) the collaboration with fellow teachers in the

process of learning to use dynamic mathematics software in their teaching and in the process of

discussing their classroom observations and their thoughts about the use of ICT in mathematics

education. All interviews were audio- and video-recorded. The data collected from the interviews

were analysed to determine how teachers developed their TPCK. These interviews were used to

answer sub questions 2, and 4.

In summary, the interviews were intended to explore how teachers felt about the whole training

course in the workshop. The interview protocol for teachers was divided into seven themes [not

the themes of the adopted practitioner model] that helped me fit the interview protocol into the

adopted practitioner model discussed in Section 2.5 which I used to answer the main research

question in section 3.1. However, interview for each teacher was open-ended resulting in differences

in questions but the questions for all teachers were fell in seven themes of the interview protocol.

3.3.4 Pupil Interviews

Two pupils were interviewed, one from each class in which practice teaching was conducted. The

semi-structured interviews using the protocol in Appendix D were used to get the pupils’ reactions

to the lessons after being introduced to using dynamic mathematics software. I also wanted to

learn about the pupils’ impressions of the lessons and the use of ICT in learning mathematics. I

wanted to support my impression of the teaching practise. These interviews helped me to answer

the research sub question 4.

Pupils were interviewed to explore how they felt about the lesson they had using GeoGebra. I

interviewed pupils who volunteered to be interviewed; it was not based on their performance in the

lesson.


3.3.5 Field Notes

I used both observation and participation notes as input for answering the main research question.

No teacher accepted to be an observer, although everyone wanted to be actively involved in the

training course. I made notes (in my diary) on what I observed that teachers were doing and what

I was involved in myself, since I was an active participant in the course.

Observation notes were also made when I looked again at the video-recordings of each training

session to carefully review what I could have missed during the actual session. As researcher, I

focused particularly on how teachers used the training materials in their teaching practice and how

they worked in groups to develop teaching and learning materials.

3.3.6 Summary of the Use of the Research Instruments

In summary, I collected and analysed research data via the following instruments:

• the Pre questionnaire;

• the Post questionnaire;

• interviews with six teachers after the workshop (audio and video-recorded);

• interviews with two pupils after the lesson (audio and video-recorded);

• field notes obtained from observing teachers develop teaching and learning materials (audio

and video-recorded). Two kind of field notes can be distinguished: (1) workshop observation

notes; and (2) participatory observation notes. The observation notes concerned teachers

involved in teaching. All teachers of a group were involved in jointly teaching the class

and went around assisting and guiding pupils in their activities. They used the developed

teaching and learning materials developed during the training course. Both lessons were audio

and video-recorded. The participatory observation notes concerned my collaboration with

teachers, my assistance in their activities, and my help to pupils in the lessons. Everything

was audio- and video-recorded.

Table 3.3 lists the research instruments that were used to answer particular research sub-questions.

3.4 Framework for Analysing Data 33

Research Instrument Research Question

1 2 3 4

Pre questionnaire 4

Post questionnaire 4 4 4 4

Interviews with teachers 4 4 4 4

Interviews with pupils 4

Field notes 4 4 4 4

Table 3.3: Data sources used to answer my research sub-questions.

3.4 Framework for Analysing Data

The analysis of qualitative data collected through the research instruments outlined in Section 3.3

was done to describe teachers’ development of TPCK in my training course through the workshop.

I used the Adapted Practitioner Model discussed in Section 2.5 for analysing the pre and post

questionnaires, observation and participation notes, and interviews of teachers and pupils. The

themes of this model are listed in Table 3.4

Number Theme

1 Effecting working processes and improving production

2 Overcoming pupil difficulties and building assurances

3 Supporting processes of checking, trialling and refinement

4 Focusing on overarching issues and accentuating important features

5 Enhancing the variety and appeal of classroom activity

6 Fostering pupil independence and peer exchange

7 Motivating teachers and pupils through provision of feedback, generating pupil enjoyment

and interest

Table 3.4: The seven themes of the Adapted Practitioner Model

In order to study the change in the way teachers used dynamic mathematics software and the

development of TPCK of the teachers, I analysed the pre-and post questionnaires. Observation and

participation notes were analysed to demonstrate how teachers blended technology in their PCK

when developing lesson materials and as they taught.

The data from interviews of teachers were analysed to gather teachers’ reflections about the

training programme [especially on the use of ICT in teaching mathematics] and pupils’ learning in

the lesson, and find out how TPCK developed in teachers through working with dynamic mathema-

tics software. Data from interviews of teachers were also analysed to assess on teachers’ planning


and development of teaching and learning materials. Interviews of pupils provided data on pupils’

reflections on lessons and their impressions.

In Table 3.5, I list the criteria used for assessing and evaluating the development of TPCK

using the themes in the adapted practitioner model.

Evaluation Criterion Addressed Theme Instrument Addressed Research Question

Produce and work 1, 2 Observation 4*

with accurate graphs Interviews

Post Questionnaire

Study efficiently and effectively 1,7 Observation 3*

to keep up pace of instruction Interviews

Post Questionnaire

Carry out small experiments and 3, 5, 6 Interviews 3

get own mathematics ideas Post Questionnaire 4

not explicitly taught

Come to mathematical results 3 Observation 4

by trial and improvement Participation

Interact with software and take 1, 3, 4, 7 Interviews 4

advantage of feedback by DMS Post Questionnaire

Observation

Enjoy mathematics and get 3, 5, 7 Interviews 4

more interest in mathematics Observation

and gain confidence

Table 3.5: Criteria for evaluating the development of TPCK of teachers.

1 Effecting working processes and improving production;

2 Overcoming pupil difficulties and building assurances;

3 Supporting processes of checking, trialling and refinement;

4 Focusing on overarching issues and accentuating important features;

5 Enhancing the variety and appeal of classroom activity;

6 Fostering pupil independence and peer exchange;

7 Motivating teachers and pupils through provision of feedback, generating pupil enjoyment and interest.

* This analysis framework was mainly designed for answering research sub-questions 3, and 4.

I examined how pupils reacted to the lessons using GeoGebra. I observed pupils while they

were involved in learning processes. Their behaviour and reaction was analysed. I also observed

their group discussions while they did activities in the computer laboratory.

I analysed the whole dataset collected through audio and video recordings, pre and post ques-

tionnaires, field notes, and interviews with pupils and teachers.

Chapter 4

Design of GeoGebra Workshops

I designed materials to be used for teacher training in the use of dynamic mathematics software.

These materials were meant to introduce teachers to working with GeoGebra. The training mate-

rials were designed in ways that introduced teachers to GeoGebra and helped them develop skills

in using it in developing own instructional materials for their teaching. The materials I developed

were meant for a workshop for Zambian mathematics teachers consisting of seven training session.

Table 4.1 lists the components of the programme of my training sessions and the activities that

Zambian teachers did.

Parts of the Training Action

Introduction to GeoGebra Researcher introduced GeoGebra to teachers through presentation.

Activity sessions and practice Participants did several activities (including installation of GeoGebra).

Material development Participants developed in teams instructional materials,

which they later practised in a classroom with pupils.

Teaching Developed materials were utilized for teaching by the participants.

Table 4.1: Structure of the workshop in Zambia.

In order to plan for the study in Zambia, pilot workshops were carried out at Fons Vitae

Lyceum and the International School of Hilversum in the Netherlands. They helped me develop

training materials suitable for Zambian teachers. The workshop at Fons Vitae Lyceum also included

activities that focused on concepts of algebra and calculus in addition to geometry (cf., Preiner,

2008). The workshop at the International School of Hilversum only concerned ICT-supported

teaching of geometry.

The Fons Vitae Lyceum workshop was especially useful for my research because the teachers

36 Design of GeoGebra Workshops

who participated in it had different knowledge levels of ICT, which depicted the same scenario

expected in Zambia. Although, the research could have been conducted in the Netherlands, I

wanted the study to be done in Zambia because I want to explore how it could facilitate change in

the teaching of secondary school mathematics in Zambia.

In summary, the research was conducted in two phases, namely, a pilot study in the Netherlands

and the main study in Zambia. I describe both parts in the next sections. These descriptions are

rather close to what actually happened the workshops because substantial parts of the programme

were (re-)designed after the workshops had already started.

4.1 The Pilot Study

I used the pilot study to develop materials and get practical experience with them before adapting

activities for use with Zambian teachers. This study took place within the period of February-

December 2010 at Fons Vitae Lyceum and the International School of Hilversum. Each workshop

consisted of two sessions of two hours.

4.1.1 Objectives of the Workshop in the Pilot Study

My goals for running the workshop were to,

• adopt activities suitable for secondary school teaching and create own activities;

• learn about the difficulties teachers and I (as researcher/workshop leader) would encounter

during the workshop in Zambia;

• develop strategies for helping teachers in the workshop;

• develop a method of measuring/evaluating materials made by teachers; and

• develop and test the questionnaires to be used in the research study.

4.1.2 Training Session I

Teachers were introduced to GeoGebra and were given an opportunity to come up with ideas for

applets that would be used in their own lessons. Worked out activities and problems were given to

teachers to practise working with the software. The other activities were used for discussions with

teachers.

4.1 The Pilot Study 37

4.1.3 Training Session II

During the second session, I observed pupils working with GeoGebra during a national contest for

solving mathematics problems in the pilot study. It is relevant to mention here that the second

session of the pilot study coincided with the national mathematics contest at the school. Pupils

were using the software for the first time in this contest. I was interested to explore how pupils

reacted to the use of GeoGebra. I also had discussions with teachers about the use of the software

in teaching mathematics. Some pupils and one teacher were interviewed afterwards. At the end of

the workshop, the post questionnaire was administered to each teacher.

Figure 4.1: Teachers demonstrating how they were able to develop activities during the pilot study

at Fons Vitae Lyceum.

4.1.4 Lessons Learnt from the Pilot Study

The pilot study was useful because I learnt a lot of things from it. This study affected my study

in Zambia because I had to make changes to the training materials and how the training was to be

handled. Listed below is what I learnt from the study and how it helped me make changes to the

Zambian study:

• Training sessions should be done in ways that allow teachers to see the relevance of the

dynamic mathematics software to teaching and learning of mathematics.


• Teachers should be given enough time that allows them to teach during the practical teaching.

This is necessary because pupils need to settle down in the lessons and get used to the software

before they can get on with the intended activities.

• Teachers should learn and practise computer skills. Once this is done, computer skills help

teachers get started with the dynamic mathematics software. Basic knowledge of computers

is a pre-requisite for participation in the research study.

• Teachers should be given all the assistance they require. Working with teachers is one way of

knowing the kind of assistance they need.

• Teachers should be given enough activities that would allow them to practise in using dynamic

mathematics software. Only after practising will they be able to develop the understanding

of using the dynamic mathematics software. This gives them a lead on how to get started

with developing their own instructional materials.

4.1.5 The Effects of the Pilot Study

The lessons learnt from the pilot study helped me in my research to make the following changes:

• Extension of the presentation of the researcher to allow more practical examples that are

relevant to teaching and learning of mathematics.

• Extension of time for practical teaching from 45 minutes to 1 hour.

• Development of the pre questionnaire.

• Modification of the post questionnaire.

• Increase of the number of activities for teachers’ practice.

• Modification of training materials with goal of making them more interactive (Teachers be-

coming more interactive among themselves and also with the researcher).

• Provision of time for teacher to teach the other participants (including the researcher).

• Extension of the research study in the directions of developing materials that could be used

to introduce Zambian teachers to computers (This was done because I assumed that some

Zambian teachers would have no computer skills necessary to participate).

4.2 The Study in Zambia 39

The changes outlined above were done in order to make the training more suitable for Zambian

mathematics teachers.

4.2 The Study in Zambia

In the Zambian workshop, seven training sessions were held, each lasting about 4 hours. Each

session started with new knowledge or revision of the previous day’s tasks, or with discussions.

The sole purpose of the workshop was to help teachers start developing ICT skills or TPCK about

the use of GeoGebra, so that they could work in the end on their own. After each session, teachers

were given a handout1 adopted from Judith and Markus Hohenwarter, containing activities on

which teacher could practise in order to develop knowledge of using GeoGebra effectively and

getting familiar with toolbar and icons. The activities were meant for teachers to work through

at home for practice purposes. In these section I give a detailed description of the workshop at

Mkushi High School in Zambia.

4.2.1 Objectives of the First Two Training Sessions

The training sessions were introducing GeoGebra and were designed to achieve the following ob-

jectives: After the training sessions, participants are able to

• acquaint themselves independently with the use of GeoGebra;

• get help and support in the use of GeoGebra via the help system and internet;

• compare paper-and-pencil constructions with dynamic mathematics software constructions;

• develop additional skills on making constructions in GeoGebra; and

• learn more about the use of GeoGebra through (teaching) practice.

4.2.2 Objectives of the Last Five Training Sessions

In the last five training sessions, teachers were trained and involved in using GeoGebra in mathema-

tics teaching. These sessions were designed to achieve the following objectives: After the sessions,

participants are able to

1Handouts accessed from http://foresthillsggb.pbworks.com/w/page/13562767/FrontPage


• design instructional materials that foster discovery, mathematical experimentation and pupil-

centred learning;

• collect information about the design and development of instructional materials with GeoGe-

bra;

• design ICT-supported lessons and teach mathematics using GeoGebra;

• use GeoGebra in mathematics lessons in their classroom; and

• develop more skills in GeoGebra use in mathematics education.

4.2.3 Training Session I

In Training Session I, consisting of three activities, I introduced teachers to GeoGebra through a

10 minutes presentation. In this presentation, teachers received general information about GeoGe-

bra. I provided information about how GeoGebra has been and still is developed. Teachers were

given examples how GeoGebra has the potential of meeting pupils’ needs for guidance in learning

mathematics, provision of learners’ support and tools for learning about functions, and meeting

teachers’ needs and tools for instruction and active engagement of pupils in learning. I provided

teachers with examples of lesson activities using GeoGebra.

Activity I: Addition and Subtraction Through Visualisation

The subject of the activity was teaching addition and subtraction of negative numbers to pupils at

primary school level (grade 4; aged 9). The aim of the activity was to demonstrate teachers how

to help pupils understand addition and subtraction of negative numbers through visualisation and

interactivity with sliders. Sliders a and b in the GeoGebra applet shown in Figure 4.2 represent

numbers being added or subtracted. The movement of the sliders a and b results in the operation

of either addition or subtraction which is depicted by the movement of ”Bunny”, the rabbit.

Activity II: Transformation Through Representation

The subject of the activity was reflection of an object in the line x = 0 by using either a matrix

or by changing the vector direction along the x-axis. The aim of the activity was to make teachers

reflect on how they taught the topic in their grade 11 class and how ICT could help in doing so

(see Figure 4.3).


Figure 4.2: Activity 1 of Training Session I


Activity III: Angles Through Experimentation

The subject of the activity was measuring angles using a digital protractor that was uploaded into

GeoGebra window. The aim was to demonstrate to teachers how to help pupils develop accuracy

in measurements. Measurements were done by moving the ends of the protractor so that the

hypotenuse got in line with one side of an angle (see Figure 4.4).

Hereafter, teachers worked on activities given in first Handout 2. The handout was used to let

teachers familiarise themselves with the use GeoGebra. This also provided teachers an opportunity

to practise their newly gain knowledge in preparation of the next session the following day.

2Handout accessed from http://foresthillsggb.pbworks.com/w/page/13562767/FrontPage



4.2.4 Training Session II

The session focused on letting teachers develop their own lesson materials using GeoGebra. I let

teachers think of topics that were either difficult for them to teach, or topics that pupils had

problems with in terms of understanding. Teachers worked in groups of five persons for developing

instructional materials at the time they needed the abilities. Teachers were taught some of the

skills required for developing lesson materials. Listed below are some of the skills that teachers

practised:

• Transferring screen shots of windows applets to be used in the lesson materials, created as

Word documents.

• Customising toolbars. The GeoGebra window can be used as a lesson activity for pupils.

For this purpose, teachers can customize the toolbar in order to limit the freedom of pupils

when they do their activities or in order to avoid distraction by tools not needed for solving

a problem.

• Making a macro (own tool or icon) to simplify constructions: Teachers can create their own

tool for drawing or constructing a GeoGebra object. This helps them to save time on drawing.

4.2.5 Training Session III

In session III, teachers were engaged in preparing lesson activities to be used in a demonstration

teaching (teaching the other participants including the researcher) in the next training session.


This micro-teaching was used to allow participants to evaluate the lessons, so that improvements

could be made. I went round providing assistance to teachers who needed help. Teachers were

involved in a discussion as they engaged in developing activities. Teachers first developed activities

that required working on paper-and-pencil before resorting to using GeoGebra. Each group looked

at a different topic in mathematics, upon which they decided beforehand.

4.2.6 Training Session IV

In this session, two groups were given an hour each to present their work to the other participants of

the workshop. The presentations took place in the same computer laboratory which we were using

for our workshop. Each group presented their work for an hour and used 30 minutes for questions

and suggestions from the audience for improving its activity. In general, the presentations were

about how teachers thought about the activities and how they were able to develop them thereafter

with GeoGebra.

4.2.7 Training Session V

Teachers engaged in a discussion about selecting topics on which to develop instructional materials

for their teaching practice. Topics were selected: One group chose ‘Area of triangles’ and the other

group went for ‘Locus of points’.

4.2.8 Training Session VI

In the sixth session, the groups continued working on their activities by making improvements and

changes to the activities, arising from suggestions made to them in the previous session. On this

day, they made final preparations for the teaching practice on the following day. It was during this

time that printing and photocopying of materials were done and I was assisting, participating and

observing the teachers during their preparatory work.

4.2.9 Training Session VII

In this session, the teacher teams taught their developed activities in class (See Table 4.2)


Teacher Team Topic Grade

12G 12F

A Area of triangles 60

B Locus of points 60

Table 4.2: Classes and topics taught in Training Session VII.

4.2.10 Relationship Between the Themes of the Adapted Practitioner Model

and the Training Sessions

Table 4.3 shows the themes of the adapted practitioner model addressed in the training sessions in

my workshop.

Theme Training Session

Session I Session II Session III Session IV Session V Session VI Session VII

Theme 1 4 4 4

Theme 2 4 4 4

Theme 3 4 4 4 4

Theme 4 4 4 4 4

Theme 5 4 4 4 4 4 4 4

Theme 6 4 4 4

Theme 7 4 4 4 4

Table 4.3: Themes of the Adapted Practitioner Model addressed in the workshop.

1 Effecting working processes and improving production;

2 Overcoming pupil difficulties and building assurances;

3 Supporting processes of checking, trialling and refinement;

4 Focusing on overarching issues and accentuating important features;

5 Enhancing the variety and appeal of classroom activity;

6 Fostering pupil independence and peer exchange;

7 Motivating teachers and pupils through provision of feedback, generating pupil enjoyment and interest.

Chapter 5

Results and Data Analysis

In this chapter, I present results and findings of the research study emanating from the pre and

post questionnaires, the interviews with teachers and pupils, and form the field notes.

5.1 The Pre and Post Questionnaires

The pre and post questionnaires contained questions that were grouped in accordance with the

themes distinguished in Subsections. 3.3.1 and 3.3.2. For every question theme and actually for

every question, questions and results are tabulated. The main findings are briefly introduced before

the tables.

5.1.1 The Pre Questionnaire

Teachers filled out the pre questionnaire form (Appendix A), so that I could explore their experience

in using ICT. The question themes were:

1. The computers use by teachers and pupils;

2. The computer use during lessons;

3. How teachers wanted to use computers in their lessons;

4. Teachers’ expectations for using ICT in teaching.

1. The computer use by teachers and pupils

The first two questions of the questionnaire were about the use of computers by teachers and pupils.

The results show that all of the teachers agreed that their pupils have a (limited) opportunity to

46 Results and Data Analysis

use computers at the school (see Table 5.1). All teachers reported that they never used a computer

during their teaching (see Table 5.2)

Question 1 Number of Teachers

Never Sometimes Often

Do your pupils have an opportunity to use computer at school? 2 8 0

Table 5.1: Results for question 1: Pupils’ opportunity to use computers in school.

Question 2 Number of Answers

Yes No

Have you ever used computers in your teaching? 0 10

Table 5.2: Results for question 2: Types of Computer Use in teaching.

2. Type of computer use during lessons

The four items in the third question concerned how teachers used computers in their lessons.

The fourth question addressed the type of software potentially used. Because no teacher used a

computer during mathematics teaching in the classroom, question 3 and 4, which addressed this

theme, remained unanswered.

3. How teachers wanted to use computers in their lessons

In the fifth question, teachers were asked how they would like to use computers in their lessons.

The results were that all teachers felt that the use of computers would be useful for teaching

mathematics (see Table 5.3).

5.1 The Pre and Post Questionnaires 47

Teacher Question 5 How would you like to use computers in your lessons?

A Computers act as teaching and learning aids to both learners and facilitator,

especially with this modern world.

B I would like to use computers when explaining mathematical concepts,

i.e., in development stage of the lesson.

C Using lessons in the computer lab once a week by having teacher group meetings

to discuss how to improve the performance of maths in schools with experts on ICT.

D Each pupil will have a computer with relevant software used

E Create an environment and questions/situation that will enable learners to explore

freely [..] to acquire knowledge. Its through this approach that learners’ ability will be enhanced.

F When explaining concepts in various topics which may appear abstract to pupils

G In developing a lesson. Reason is that when developing a lesson under a

topic mensuration pupils [..] see how polygons are constructed.

H With the help of ICT, I would be able to teach Geometry and Algebra since it is

much easier to use a self learning device like computer.

I Teaching pupils in small groups. Pupils discover concepts from the computer.

J Presentation of diagrammatic illustrations. Solving mathematical problems.

Table 5.3: Results for question 5: How teachers would like to use computers in their lessons.

4. Teachers expectations for using ICT in teaching

In the sixth question, teachers were asked what were their expectation for using ICT in their

classroom lessons. The results were that all teachers expressed high expectations about the use of

ICT in teaching mathematics (see Table 5.4).

In all the teachers’ expectations, what came out prominently was that teachers thought that

ICT makes their teaching easier and more effective.

5.1.2 The Post Questionnaire

The post questionnaire was administered at the end of the training course for the purpose of

exploring how teachers started developing TPCK during the workshop. Eight out of ten teachers

filled out the post questionnaire form form (Appendix B), so that I could explore their experience

their development of TPCK in the workshop. The questions were categorized into the following

themes:


Teacher Question 6 What are your expectations for using ICT in your teaching?

A Learners will find it easier to workout problem in maths. It is a tool and offers guide

to both learners and facilitator.

B Use of ICT will enable me deliver mathematical concepts more effectively

in class as it [..] me to drift from teacher centred lessons to more child-centred lessons.

A more practical approach to teaching and learning.

C To improve in my delivery of knowledge using ICT. To understand and improve my teaching

D Teaching will become easier and interesting.

E It will make teaching easier as I will mainly play the role of a facilitator and motivator.

F To teach effectively and have pupils centred lessons.

G ICT programme will break the barrier that is between the learner and mathematics.

topic mensuration pupils [..] see how polygons are constructed.

H To acquire the necessary skills of using ICT. I expect to get exposure

to computer and I expect to become conversant with the use of computer in teaching.

I To learn how different mathematical concepts can be taught to children using computers.

J Get conversant with the software. Learn new concepts in ICT.

Table 5.4: Results for question 6: Expectations of teachers’ of computers use in their lessons.

1. learning to use geoGebra;

2. teacher’s use of GeoGebra;

3. pupils’ benefits of using GeoGebra;

4. teachers’ development of TPCK during and after the workshop.

1. Learning to use GeoGebra

The results showed that all the teachers were new to using GeoGebra in their lessons. (see

Table 5.5). For this reason, the follow-up question in question 1 can be ignored.


Yes No

Have you ever used GeoGebra in your lesson? 0 8

Table 5.5: Results for question 1: Ever used GeoGebra in a lesson.

Table 5.6 shows the opinions of the teachers about the ease of learning to use of Geogebra.

Teachers explained that they found the identification of icons in the toolbar to use for specific tasks


difficult. With time, every teacher seemed to have familiarised with the icons and the toolbar (see

Table 5.7).


Very easy Hard at first but became easier Still hard

What is your opinion about the ease for 4 4 0

you of learning to use GeoGebra?

Table 5.6: Results for question 2: Teachers’ opinions on the ease for learning to use GeoGebra.

Teacher Please give reasons for your response to Question 2?

A It was hard but it is very easy to use GeoGebra to teach now that I have learnt

B It was difficult because I could not understand the software but I have found GeoGebra

beneficial to teaching of mathematical concepts

C [left blank].

D Identification of icons of GeoGebra is sometimes difficult, but as one gets used, it

becomes simple

E A computer is very user friendly and as such one can learn even by own discovery

provided the GeoGebra software is installed in the computer

F It enables to do a text in a short time and improves the effectiveness.

H It was easy. GeoGebra is a self learning software explicitly if you are familiar with the

use of computers.

I Hard in the sense that one needs to get used to the icons and operation of a computer.

Table 5.7: Results for question 2: Teachers’ experiences in learning to use GeoGebra.

Table 5.8 shows the willingness of the teachers to continue working with Geogebra. Reasons

are listed in Table 5.9


Yes No

Would you like to continue working with GeoGebra? 8 0

Table 5.8: Results for question 3: Teachers’ willingness to continue working with GeoGebra.


Follow-up question to Question 3 Choice of where to use GeoGebra

Teachers Please explain your choice(s) where to use GeoGebra

A Because I have learnt and it is very beneficial a teacher work group

so I would like to educate other teachers about GeoGebra.

B The will enable us teachers to share our experiences a teacher work group

C So as to share ideas amongst teachers. in a teacher work group

D To consolidate what we have learnt. in a teacher work group

E Will enable me to consolidate my skills so far acquired another workshop similar

and interact with fellow mathematics teachers and [..] to do to this one; a teacher work group

more application or use of GeoGebra in teaching mathematics.

F To effectively explain the concepts to pupils who have other

challenges in class.

H To help other teachers learn the software. For instance, a teacher workshop

science teachers.

I Provision of programme to individuals makes people to other

be experts in the area.

Table 5.9: Results for question 3: Choice(s) where to use GeoGebra and reasons thereof.

2. Teacher’s use of GeoGebra

For question 4.a, the results were that all teachers wanted support in terms of facilities that would

help them start using GeoGebra to show their classes something (see Table 5.10).

Teacher Question 4.a: What support would you need to start GeoGebra in teaching?:

To show the whole class something.

A A computer to help him/her teach the lesson to pupils.

B [Left blank]

C To have more computers, if possible a laptop for the teacher

D Computer, beamer and software. Pupils will see what needs to be done

E I need a beamer, a laptop and all the accessories.

F The school have enough computers to accommodate learners for effective support as the teacher.

Also facilities.

H Introduce them to the software and the importance of using it.

I Provision of computers, enough for individual pupils to have access.

Table 5.10: Results for question 4.a: Support that teachers need to start using GeoGebra in teaching.


Teachers responses to question 4.b were somewhat similar to the previous question. Teachers

felt that in order to start using GeoGebra for a whole class discussion, they needed facilities that

would help pupils engage in a class discussion (see Table 5.11).

Teacher Question 4.b: What support would you need to start GeoGebra in showing the whole class something:

To have a whole class discussion.

A The teacher should have a projector to be used in discussion as pupils.

would be able to see by themselves on the screen

B [Left blank]

C Support in terms of computers.

D Computers and GeoGebra software. Pupils should be seeing what they need to do.

E A well furnished ICT laboratory with not less that 40 computers.

F Learners will share ideas and move at the same pace, hence having few

cases of learners facing difficulties.

I Materials to have guiding questions/instructions e.g. typing and printing the materials.

Table 5.11: Results for question 4.b: Support that teachers need to start using GeoGebra in having

a whole class discussion.

Apart from the support of ICT facilities teachers wanted, the results were that teachers wanted

to start using GeoGebra to let pupils use it in small groups so that teachers would give activities

to pupils to work on their own (see Table 5.12).

Teacher Question 4.c: What support would you need to start GeoGebra in teaching?:

To let pupils use it in small groups.

A The pupils must have computers in each of small groups

to be used during their discussion.

B [Left blank]

C Give some activity with some guidlines then give a question.

D Computers and GeoGebra software. Pupils will be exposed to the latest methods

used in teaching.

E At lest 12 computers and connections.

F So that they share ideas in order to consolidate the content.

H Let them use the software so that they discover the use of the software by themselves.

I production of learning materials. Time for plaining how the groups would progress.

Table 5.12: Results for question 4.c: Support teachers need to start using GeoGebra to let pupils

use it in small groups.


In question 5, I explored how teachers would like to use GeoGebra in teaching graphing. The

results were that many teachers would use GeoGebra to help pupils consolidate their understanding

as they compare the handwritten graphs to those produced by GeoGebra. This is why the majority

of teachers chose teaching of graphing in handwritten format first, followed by using GeoGebra (see

Table 5.13).

Question 5 How would use like to use GeoGebra in teaching graphing? Choice

Teachers Reasons for the choice(s)

A Because GeoGebra is easier to use and it is accurate. with GeoGebra alone.

so I would like to educate other teachers about GeoGebra.

B [Left blank] left blank

C Use GeoGebra and handwritten and comparison them. handwritten followed.

by GeoGebra graphs

D Use GeoGebra and handwritten and comparison them. handwritten followed.

by GeoGebra graphs.

E It will enable my learners make concrete handwritten followed

the theory as they marry it with practice on computer. GeoGebra graphs.

F Learners should first have skills to compute the use GeoGebra. by GeoGebra graphs.

GeoGebra would enable learners to consolidate concept very well.

H Use GeoGebra and handwritten and comparison them. handwritten followed

by GeoGebra graphs.

I Certain concepts need explanations first handwritten followed

by GeoGebra graphs.

Table 5.13: Results for question 5: How teaching of graphing would be done using GeoGebra.

3. Pupils’ benefits of using GeoGebra

In this theme of the questionnaire, I explored what teachers thought about possible benefits of

GeoGebra for pupils. In question 6, I explored for instance how teachers compared learning about

functions and their graphs using GeoGebra and using paper-and-pencil and other regular tools.

The results were that all teachers preferred using GeoGebra when teaching about functions and

their graphs (see Table 5.14).

In question 7, I explored teachers’ opinions about pupils’ understanding of graphs and their

functions when using GeoGebra. I wanted to know whether teachers felt that GeoGebra helps pupils

make connections between graphs and functions. The results were that all teachers saw GeoGebra

as a software that could help pupils understand functions and graphs through visualisation (see

Table 5.15).


Teacher Question 6: How would you compare learning about functions and their graphs

using GeoGebra and using paper and pencil and other regular tools?

A It is a bit difficult without GeoGebra and time consuming.

B In learning about functions and their graphs without GeoGebra pupils’ abilities to make

their own discoveries is restricted. Introduction of GeoGebra make it easy for them

to make connections.

C Using GeoGebra is fast and accurate than the others.

D It is easier for pupils to learn than when they are using paper and pencil.

E the use of GeoGebra increases on the pupils’ retention capacity

as they actual see a great deal of what the hand-on activity brings about.

F Time consuming on paper and pencil. It takes an average learner to observe the concepts.

H Using paper and pencil is more abstract, it diverts pupils’ attention away

from reality. But the use of GeoGebra show how the graphs vary in size, shape, and any

other relevant information. GeoGebra also shows the relationships of graphs and the equations.

I Computer literacy, Reduced sourcing of teaching and learning materials.

Table 5.14: Results for question 6: Comparing learning about functions and graphs using GeoGebra

and using paper and pencil and other regular tools.

In question 8, I explored teachers’ thoughts on whether using trial and improvement could help

pupils reach a satisfactory solution to a mathematical problem. The results were that all teachers

felt that using GeoGebra could help pupils arrive at a satisfactory solutions to the problem using

trial and improvement, as long as pupils were guided on how to do it (see Table 5.16).

Through question 9, I explored what teachers thought about GeoGebra in helping pupils who are

weak or unsure to learn about graphs and their functions. The results were that all teachers believed

in GeoGebra being able to help and motivate weaker and unsure pupils to understand graphs and

their functions. Table 5.17 gives summaries of teachers’ comments. For further illustration: Listed

below are two unsummarized comments (from teachers B and E) about the ability of GeoGebra in

helping weaker and unsure pupils:

Mr B: Most pupils do not enjoy learning about graphs and their functions because of limited aids.

The use of GeoGebra will enable pupils to learn extensively make their own personal discoveries.

For example, using GeoGebra pupils will discover the relationship between y = x2 + 2x + 1 and

y = −x2 + 2x + 1 i.e. one is a reflection of the other.

Mr E: Yes. It will motivate them to achieve more as computer based learning is more practical than

paper and pencil. e.g. they (pupils) may find more problems to graph y = x2 on paper but following

procedure correctly, GeoGebra will enable them achieve with less difficulties.


Teacher Question 7: Pupils understanding of graphs and functions: What is your opinion about using

GeoGebra to help pupils make connections between graphs and their functions?

A Pupils discover themselves how a linear question changes to a quadratic

function and it becomes very interesting.

B GeoGebra make it easy for pupils to connect between graphs and their functions.

For example, a pupil can connect from a simple linear function to a quadratic

function just use of GeoGebra.

C It is beneficial and helps pupils to understand as it is visual.

D Pupils will easily find it easier to learn mathematics, hence understanding of mathematical

concepts will be improved.

E Pupils could easily see the connection as it is quick to draw and, compared with drawing of

computer, give connections and roots as they actual see a great deal of what the hand-on

activity brings about.

F Easy and fast. Enables the teacher to explain concepts and learners see and consolidate

Many concepts can be taught in the shortest possible time.

H There is a direct relationship between graphs and functions. GeoGebra gives the right

graph for the function in less time. Their understanding is enhanced and draw easy conclusions.

I It is the best way. For the solutions are always accurate.

Table 5.15: Results for question 7: Pupils’ understanding of graphs and functions.

Teacher Question 8: Using GeoGebra to apply “trial and improvement”, do you think pupils, using the

method of trial and improvement, can arrive at a satisfactory solution to a mathematical problem?

A Yes pupils may arrive at a satisfactory solution

B Yes, through investigation, a pupil will make his/her own discoveries.graph a pupil discovers

what happens to the y-intercept as the linear graph is moved away from the y-axis. For instance,

by sliding a linear function just use of GeoGebra.

C Yes they can.

D They can as long as they know the basic operations of a computer. concepts will be improved.

E They can arrive at a satisfactory solution to a mathematical problem. Help them correct errors [..]

F yes

H With a little guidance it is very possible.

I Yes, as long as pupils are given enough time to explore the computer.

Table 5.16: Results for question 8: Using GeoGebra to do “trial and improvement”.


Teacher Question 9: Do you think GeoGebra can help pupils who are weak or unsure to learn

about graphs and their functions?

A GeoGebra would help the weak learners as it is very accurate provided

instructions are proper.

B [..] use of GeoGebra will enable pupils to learn extensively, making own discoveries, [..]

C Yes it can.

D Weak pupils can be helped because GeoGebra is motivating and additive.

E Yes. It will motivate them to achieve more as computer based learning is more practical and [..].

F The software can be beneficial to learners who are weak. It is the effective

software to help weak pupils..

H Yes. Because it shows the direct relationship of the nature of the functions and the graph.

I The use of computers also motivates pupils. As the achieve some tasks the get

interested consequently improve their skill acquisition.

Table 5.17: Results for question 9: Using GeoGebra to do “trial and improvement”.

4. Teachers’ development of TPCK during and after the workshop

After the training course, I explored teachers’ thoughts about the workshop. I wanted to know

whether the course had an impact on teachers, and therefore I wanted to find out what teachers

could do afterwards regarding the use of GeoGebra in their teaching. There were four questions

under this theme.

For question 10, the results were that all teachers found or thought that GeoGebra was user-

friendly as it was easy to use after getting used to the software (see Table 5.18). Listed below are

four unsummarised comments from teachers B, E and H about user-friendlyness of GeoGebra;

Mr B: GeoGebra is so pupil-friendly in that it is practical in nature and enables pupils to learn on

their own and the teacher just facilitates the learning process. I enjoyed this approach to teaching

because it is more pupil-centred.

Mr E: It is surely user friendly- One needs to be introduced and thereafter would easily discover

more through practice.

Mr H: GeoGebra is so pupil-friendly in that it is practical in nature and enables pupils to learn on

their own and the teacher just facilitates the learning process. I enjoyed this approach to teaching

because it is more pupil-centred.

Mr H: GeoGebra is more friendly, user-friendly that is. At first I was sceptical about how this could

work, but finally my scepticism is over.


Teacher Question 10: What is your opinion about the user-friendliness of GeoGebra

now at the end of the workshop series?

A I would suggest that GeoGebra is introduced in teaching of mathematics in school.

B GeoGebra is user friendly [..] and practical in nature which [..] learners [..]

C It is very user friendly.

D GeoGebra is very friendly.

E It is user friendly and one [..] to easily discover more through practice and [..].

F It is user-friendly and enables easy understanding of concepts.

H GeoGebra is more friendly. At first i was sceptical but now its fine.

I Pupils motivated to work with GeoGebra.

Table 5.18: Results for question 10: Opinions of user-friendliness of GeoGebra.

For question 11, the results were that the major obstacles teachers imangined that they could

face were: (1) lack of enough computers in school; (2) lack of computer skills in both teachers and

pupils; and (3) lack of other ICT facilities such as internet, which could make using GeoGebra in

their teaching difficult (see Table 5.19). Some of the full teachers’ comments are listed below;

Mr B: Limited number of computers in school. The number of computers in school is limited to

cater for the number of pupils taking mathematics. The whole school takes the subject. Computer

literacy: There are a few pupils who have knowledge about computer manipulation. This makes

teaching using the computer a difficult undertaking.

Mr E: Our school hasn’t got enough computers as such most pupils will have no hand-on activities.

Most pupils have no knowledge on how to use a computer .i.e they are few pupils who are computer

literate. Not all teachers are computer literate. And finally, access to internet is limited to one

computer in the school as such adequate research is very difficult to achieve.

All these comments point to the fact that lack of ICT facilities and lack of computer literacy could

make using GeoGebra in teaching difficult for teachers.

For question 12, results were that all teachers planned to make use of GeoGebra in their

teaching (see summarized comments in Table 5.20). Interesting results are listed below in their

unsummarised form about teachers’ plans for working with GeoGebra in their teaching;

Mr E: To orient pupils on the use of GeoGebra, through small groups. And also request the school

administration to procure more computers and accessories as well as increasing access to internet.


Teacher Question 11: What sort of obstacles do you think you would face if you wanted to

use GeoGebra in your teaching?

A You need to be acquainted with and understand icons. Drawing of polygons.

B Limited computers in school and computer literacy.

C Non availability of computers.

D Lack of computers, if any adequate. Pupils’ lack of knowledge on the use of computer.

E lack of computers, computer literacy in both teachers and pupils and [..].

F The ratio of computer to pupil is high. Too much use of computers may affect

pupils knowledge as everything is done by the computer. Software to help weak pupils.

H Lack of computers and computer literacy.

I Over enrolment in classes.

Table 5.19: Results for question 11: Obstacles to be faced if one wanted to use GeoGebra in teaching.

Mr F: Use the software when having a remedial work with pupils who may have difficulties to

consolidate concepts while in classroom.

All these comments indicated that the teachers who participated in my research study planned to

use GeoGebra in their teaching for various reasons.

Teacher Question 12: What are your plans (ideas) for working with GeoGebra after the

workshop Training?

A Let it be implemented as it help to teach if the lesson material is well prepared.

B My plans are to group the pupils and carry out lessons on GeoGebra on regular intervals

C Use it to explain some mathematical concepts and facts.

D To frequently teach pupils using GeoGebra.

E Orient pupils on the use of GeoGebra, [..] and ask for more computer accessories

F Use the software when having remedial work [..]. software to help weak pupils..

H Once in a while I will want to be taking my class to GeoGebra to consolidate their

understanding by checking their work and proving their accuracy

I To once in a while develop GeoGebra lessons.

Table 5.20: Results for question 12: Plans for working with GeoGebra in teaching.

For question 13, results were that teachers perceived the use of ICT in teaching mathematics

as the most important aspect and noted that GeoGebra was helpful in making teaching effective

and helping pupils understand mathematical concepts (see Table 5.21).


Teacher Question 13: What do you think has been the most important thing that you have gotten

from GeoGebra workshop?

A Finding the solutions to the equations has been very helpful. Teachers became computer literate.

It arises the interest of learners.

B I have learnt the importance of using ICT in the teaching of mathematics, as well as the relevance

of ICT in teaching and learning.

C The use of GeoGebra to answer mathematics in my lesson.

D Improved way of teaching pupils in mathematics.

E The use of GeoGebra in the learning and teaching of mathematics is very vital.

F The software improves understanding. software to help weak pupils [..]

H The use of GeoGebra in teaching functions and graphs is more effective and easier,

because the software help you to discover things on your own.

I The GeoGebra concept itself is important.Reduced work for teachers..

The child-centredness of GeoGebra lessons very important.

Table 5.21: Results for question 13: The most important thing learnt from GeoGebra workshop.

5.1.3 Fitting Teachers’ Comments to the Adapted Practitioner Model

The responses of each teacher to post questionnaire’ questions are fitted in the adapted practitioner

model framework. This was done to help the researcher explore teachers’ TPCK development during

the GeoGebra training, which in the end helped the researcher to answer the main research question

of this study (see Table 5.23). For the readers’ convenience, the themes of the Adapted Practitioner

model are repeated in Table 5.22.

Number Theme

1 Effecting working processes and improving production

2 Overcoming pupil difficulties and building assurances

3 Supporting processes of checking, trialling and refinement

4 Focusing on overarching issues and accentuating important features

5 Enhancing the variety and appeal of classroom activity

6 Fostering pupil independence and peer exchange

7 Motivating teachers and pupils through provision of feedback, generating pupil enjoyment

and interest

Table 5.22: The seven themes of the Adapted Practitioner Model


Teacher Themes of the Adapted Practitioner Model addressed by the post questionnaire

Theme 1 Theme 2 Theme 3 Theme 4 Theme 5 Theme 6 Theme 7

A Ra 6 R 9 R 8 R 7 R 13

R 10

R 12

B R 13 R 10 R 7 R 6 R 9 R 8

R 8 R 7 R 10 R 9

R 8 R 10

C R 4c R 7 R 7 R 7 R 4c

R 6 A 9 R 8

R 10

D R 13 R 7 R 8 R 4b R 4c R 9

R 9

R 10

E R 7 R 8 R 7 R 5 R 6

R 8 R 6

R 9 R 9

R 10 R 12

F R 2 R 3 R 8 R 7 R 4b

R 3 R 4b

R 4 R 6

R 6 R 9

R 7 R 10

R 12

R 13

H R 5 R 5 R 8 R 6 R 6 R 4c

R 7 R 9 R 7 R 13

R 12 R 10 R 9

R 12 R 13

I R 4c R 8 R 7 R 9 R 13 R 9

R 7 R 10

R 13 R 9

aR is the response to the specified question in the questionnaire.

Table 5.23: Adapted Practitioner model’s themes addressed in the post questionnaire by teachers’

responses.


5.2 Findings and Analysis for Interviews with Teachers and Pupils

Interviews were conducted at the end of the training programme to explore how teachers felt

about the whole training course. Four out of ten teachers and two out of one-hundred-and-twenty

pupils were interviewed [They volunteered for the interviews]. In the interviews outlined below, the

characters R, T and P stand for researcher, teacher and pupil, respectively. (See Appendix C and

Appendix D for the planned structure for the semi-structured interviews).

5.2.1 Transcripts of Interviews with Teachers

Interview with Teacher A

R1. What is your general impression about the workshop?

T1. When we began, I faced difficulties, especially that I had no knowledge about computers. [..] now I

am fine. It is more interesting to use GeoGebra in teaching than teach in the traditional way. This is

really educative and effective. For me I loved it.

R2. What was your initial perception, especially when you were introduced to use of Geo-

Gebra at the beginning?

T2. At the beginning I thought it was not possible to teach mathematics using ICT but at the end of the

first session, I saw and imagined things happening.

R3. Is your perception about teaching now different to the one you had before?

T3. There I would say, there is a big change in me. Now I have experienced a new approach in [..] use

GeoGebra to let pupils control their own learning. It makes me comfortable when preparing teaching

materials.. [....].

R4. From now onwards, how do you look at Teacher A’s teaching of mathematics?

T4. I will adopt the new approach in which pupils and I, will be using GeoGebra in defining my teaching

and learning.

R5. How do you think your pupils will be helped in developing a clear understanding of

mathematics through using GeoGebra?

T5. The experience we had during teaching practice showed that it was easier for pupils to solve and

understand problems using GeoGebra than when we use the traditional way. My understanding now

is GeoGebra will help pupils understand mathematics concepts that are abstract through observation

and doing.

5.2 Findings and Analysis for Interviews with Teachers and Pupils 61

R6. Do you suggest that the best way to teach mathematics at [..] is to let pupils learn on

their own?

T6. Yes, exactly [laughs], this to me is what I have come to adopt from this training. It is good [laughs]

what is better that this?

R7. What do you think this kind of teaching will do to you and your pupils?

T7. This will help improve results in mathematics and my teaching approaches [laughs]. This will even

clear the belief in pupils that mathematics is difficult. You saw here .[..] when we taught pupils, they

were saying mathematics is simple. [laughs]. This changed pupils’ perception about mathematics.

R8. You are showing a shift in your teaching from your talk. What do you say about that?

T8. On change, there is something I have learnt during this training. When it comes to understanding

concepts, in traditional way, it is the teacher’s show to explain all parts to pupils . [....] how can a

pupil learn like that and how effective can a teacher be in? The approach of letting pupils develop and

discover things for themselves is a good way of teaching since pupils develop critical thinking. This is

what ought to be supported. pupils are [..]

R9. Where do you stand after this workshop?

T9. [...] who will devote his time in teaching mathematics using GeoGebra. A teacher who has moved

from ICT ignorant teacher to an expert use [...] did I know I will ever have this chance? wow! I love

this innovation, so these things are there?

R10. What are some of the obstacles that you will face in starting to use GeoGebra?

T10. Lack of equipment and low levels of ICT knowledge in both teachers and pupils.

Interview with Teacher E


T1. [..] before the programme I was sceptical and with time I was expectant because I was not sure what

I expected. [...] setting Zambia in particular, our teaching is traditional and use of ICT has not taken

root. but you [..] change is often resisted. I realise now that there was something missing in my

teaching. What I learnt was worth learning and I have learnt a lot and what remains is to put it in

practice.

R2. Looking at how you have been developing materials in the workshop, which way of

teaching do you think is the best?


T2. Having practised teaching with the materials we developed, I saw that teaching was made easy and it

helped me to evaluate which way is better. Use of GeoGebra allows more pupil-centred teaching than

the traditional one. For this I think GeoGebra is better as it brings concrete understanding to pupils.

[..] I head applause from pupils when they got things right [...].

R3. How did you manage to combine your teaching and content with technology?

T3. There was a shift, I would say to a large extent, it came from developing lesson materials that promote

pupils’ activities. To me this is how I was able to combine my knowledge of teaching with the

technology. It was not easy to develop activites that would stimulate pupils to be involved in discoveries

but we managed [laughs] is that not good? It was hard at first but with time things develop. Now I

know how useful technology is to teaching.

R4. With a few facilities avilable here, Is your department thinking about using GeoGebra

interm of Continuous Professional Development (CPD)?

T4. I feel as a department, we will get back to thinking of individual topics then see how we could first

orient ourselves as staff before it is brought to pupils. The use of GeoGebra will not end here, this is

just the beginning [..].

R5. You talked about resistance to change among your teachers. Are there those that you

have seen giving in to change after this workshop?

T5. I would say, all teachers attended the workshop. Teachers are happy and have welcomed the use of

GeoGebra in teaching and are saying it has come to stay. I hope to see them more in future activities.

R6. What are some of the obstacles that you may face as you start using GeoGebra?

T6. At the inception stage, the major obstacle is that we are all still learner as such we expect to move at

a slow pace and that start slow in using GeoGebra in our teaching. It will take a little of some time

for us to be experts and the other obstacle will be the lack of enough room in the computer lab since

our classes are over enrolled-60 pupils is not easy to teach.

R7. What is the way forward with the use of GeoGebra?

T7. I will be using GeoGebra in my lessons and I think I hope to encourage other teachers to do the same.

R8. At a personal level, how does the use of GeoGebra help you in planning your teaching?

T8. It is a positive effect on my planning. It will help me plan in such a way that pupils will have hand-on

activities that could help them understand mathematics. It will be easy for me to develop activities

that would engage pupils in independent learning thereby making development and discoveries in

mathematics.


Interview with Teacher F

R1. : What is your general impression about the workshop?

T1. This workshop has really helped me as a teacher. This software is actually good for teachers and

pupils in understanding mathematics. It enables a pupils to consolidate their understanding. It is

purely pupil-centred. When it comes to teachers, we shared ideas and together could see how this

facilite help us to effectivly teach and I would love the department to adopt it.

R2. With regards teaching and material preparations,how do you compare the use of Geo-

Gebra and the traditional way of teaching?

T2. Actually it is a blessing to have less talking as I will do less talking and let pupils to do the learning.

The abstract part of teaching will be reduced. When teaching functions and graphs [..] pupils will be

able to understand the relationship [..] makes the teachers’ work on development of teaching materials

easily and [..] ends up developing activities that encourage independent learning. [..].


T3. Interest part of this software is addictive. Nevertheless, there is [..] to use both traditional and

GeoGebra. The obstacle will be to let pupils develop mathematics skills using the traditional way

before getting to GeoGebra. [..] pupils must must have skills and then letting them to use GeoGebra

will make them understand more. Starting teaching with this software without having pre-requisite

of plotting, it would have a problem to transfer the knowledge which is a big problem.

R4. How do you look at your pedagogy as a teacher?

T4. One of the things that I have observed is that the contact with pupils is big. This I have seen

that pupil-teacher contact is good. It helps teachers to approach individual pupils and monitor their

progress in the process, and it has helped me a lot in that I look at it as a good teaching skill. [..]

The shift in my teaching is, I will make it a habit to involve pupils in practical tasks that help them

develop their own understanding. To me, this is a very good virtual as a teacher that will help give

feedback to both my pupils and I.

R5. Are you adopting the use of GeoGebra in teaching?

T5. The use of GeoGebra in teaching is nice. The first time you introduced it to us and demonstrated

some of its uses, I was convinced that this is what I was looking for to enhance my teaching. The

adoption of use of GeoGebra will entail adapting the best method of teaching in my class [laughs] [..]

i will endeavour to use it in my teaching.


R6. Since you are saying you will change your teaching, what are some of the benefits will

you get in your teaching when you adopt use of GeGebra?

T6. I will now lessen on talking in class and let pupils be involved in the learning process. This will make

my pupils busy and develop mathematical skills easily which is difficult to achieve in a traditional way.

When pupils discover things on their own, understanding is more permanent that when I talk it to

them?

Interview with Teacher H


T1. The workshop was very successful as we learnt a lot of things. I liked the way to use ICT in teaching

mathematics. It is new to me but I have like it so much that I will make sure that I practice so much.

R2. How has the workshop changed the way you teach mathematics?

T2. Use of GeoGebra has made me have a second thought on how I should teach. In my teaching, I use

text books and do a lot of talking in front of pupils which now I see deprives pupils the freedom to

learn on their own. Use of GeoGebra helps pupils see reality [..] functions and graphs, pupils are able

to see the relationship and each time the change in values of one its effect is seen in the other. Is that

not wonderful? [laughs]

R3. How does GeoGebra help you in your teaching?

T3. It has helped me in a lot of things. There are things I did not know, but the workshop made it possible

for me to known.[..] use of GeoGebra has helped me find out some concepts which were missing in my

teaching. This will end up in enhancing my teaching. Even the pupils themselves, once in a while will

be coming to the computer lab to consolidate what we were learning and teaching in class through

using GeoGebra.

R4. How has your teaching been affected by the workshop?

T4. From today onwards, I will blend ICT in my teaching. This means that I will develop a teaching

strategy in which I will let pupils develop learning themselves and will only help by guiding them. To

me, the workshop has had a big effect on my view of teaching. I will make sure I prepare my lesson

materials in ways that will encourage discoveries in pupils. I feel this will help my pupils to create

proper understanding. My appeal to the administration is that, computer laboratory must be made

accessible to teachers and pupils.

R5. What is your impression of the practical teaching you had?


T5. I must say that at first when you introduced this workshop, I was sceptical and doubted the success of

it but after practising and doing activities, I was shock I could even develop learning materials which

we used on pupils during our practical session. Pupils were really happy and I could see how they were

doing things and made me see how they were thinking through their activities which is not possible to

see when teaching using traditional way. I have really develop interest. My teaching now has changed

and I will make sure I use GeoGebra in my teaching.

R6. How much help do you need to be able to start using GeoGebra?

T6. As at now we need to activities and also development of materials has become very easy and what

requires on instruction to the pupils and in so doing, pupils develop their own understanding. This

makes teaching real and not cosmic in that pupils have control of their learning process. I was happy

with pupils I considered slow learners because they were the first to complete that task and understood

the lesson.


T7. Obstacles will mainly be over enrolment in classes and literacy levels on ICT use.

5.2.2 Fitting Teachers’ Comments to the Adapted Practitioner Model

The responses of each teacher to the interviews were fitted to the Adapted Practitioner Model of

ICT use in mathematics teaching and learning. This was done to help me explore teachers’ TPCK

development during the GeoGebra training (see Table 5.24).

5.2.3 Findings According to Themes of the Interview Protocol

I reorganised the teachers’ responses from the interviews into the seven themes of the interview

protocol (Appendix E) that not only helped to structure the interviews but also helped to analyse

them. The themes were:

1. The overall impressions of the workshop;

2. blending technology with pedagogy and content in teaching;

3. obstacles and challenges faced when teachers start to use GeoGebra;

4. use of GeoGebra helps in planning and teaching;

5. development of instructional materials;


Teacher Themes of the Adapted Practitioner Model addressed by the post questionnaire

Theme 1 Theme 2 Theme 3 Theme 4 Theme 5 Theme 6 Theme 7

A Ta 1 T 1 T 5 T 3 T 2

T 3 T 5 T 6 T 7

T 4 T 7 T 8 T 8

T 7 T 9

T 8

E T 3 T 2 T 2 T 2 T 1

T 8 T 3 T 8 T 3

T 5

F T 1 T 1 T 2 T 2 T 2 T 1 T 1

T 4 T 2 T 4 T 2 T 4

T 5

H T 3 T 3 T 2 T 2 T 2 T 2 T 1

T 4 T 4 T 3 T 4 T 2

T 5 T 5 T 5 T 6 T 3

T 7 T 6 T 5

T 7

aT is the response to a given question in the questionnaire.

Table 5.24: Adapted Practitioner model’s themes addressed in the teacher interviews.

6. change in perception about teaching; and

7. effect of the workshop on the teachers.

The results in Table 5.25 illustrate that the workshop was beneficial to teachers and they found

the use of GeoGebra motivating.

Results in Table 5.26 illustrate that all teachers thought that their teaching shifted from teacher-

centred to pupil-centred teaching when they combined technology with teaching and content. Tea-

chers also thought pupils and teachers could easily get feedback from teaching and learning once

technology is used.


Teacher Theme 1: The overall impressions of the workshop

A When we began, I faced difficulties, especially that I had no knowledge about computers. [..]

now I a fine. It is more interesting to use GeoGebra in teaching than traditional way.

This is really educative and effective. For me I loved it.

E [..] before the programme I was sceptical and with time I was expectant because I was not sure

what I expected. [...] setting Zambia in particular, our teaching is traditional and use of ICT has not taken

root. but you [..] change is often resisted. I realize now that there was something missing in my teaching.

What I learnt was worth learning and I have learnt a lot and what remains is to put it in practice.

F This workshop has really helped me as a teacher. This software is actually good for teachers and

pupils in understanding mathematics. It enables a pupils to their understanding. It is purely pupil-centred.

When it comes to teachers, we shared ideas and consolidate together could see how this facility help us to

effectively teach and I would love the department to adopt it.

H The workshop was very successful as we learnt a lot of things. I liked the way to use ICT in

teaching mathematics. It is new to me but I have like it so much that I will make sure that I practice so much.

Table 5.25: Results on the impression about the workshop.

Teacher Theme 2: Blending technology with pedagogy and content in teaching

A There I would say, there is a big change in me. Now I have experienced a new approach

in [..] use GeoGebra to let pupils control their own learning. It makes me comfortable when preparing

teaching materials.. [....].

E [There was a shift, I would say to a large extent, it came from developing lesson materials

that promote pupils’ activities. To me this is how I was able to combine my knowledge of teaching with the

technology. It was not easy to develop activites that would stimulate pupils to be involved in discoveries but

we managed [laughs] is that not good? It was hard at first but with time things develop. Now I know how

useful technology is to teaching.

F One of the things that I have observed is that the contact with pupils is big. This I have seen that

pupil-teacher contact is good. It helps teachers to approach individual pupils and monitor their progress in the

process, and it has helped me a lot in that I look at it as a good teaching skill. [..] The shift in my teaching

is, I will make it a habit to involve pupils in practical tasks that help them develop their own understanding.

To me, this is a very good virtual as a teacher that will help give feedback to both my pupils and I.

H Use of GeoGebra has made me have a second thought on how I should teach. In my teaching, I use

text books and do a lot of talking in front of pupils which now I see deprives pupils the freedom to learn on

their own. Use of GeoGebra helps pupils see reality [..] functions and graphs, pupils are able to see the

relationship and each time the change in values of one its effect is seen in the other. Is that not wonderful?

[laughs]

Table 5.26: Results on the combination of teaching and content with technology.


Results in Table 5.27 illustrate that the over-enrolment, ICT knowledge, and access to compu-

ters were the major obstacles for teachers to start using GeoGebra in their teaching.

Teacher Theme 3: Obstacles and challenges faced when teachers start to use GeoGebra

A Lack of equipment and low levels of ICT knowledge in both teachers and pupils.

E At the inception stage, the major obstacle is that we are all still learner as such we expect

to move at a slow pace and that start slow in using GeoGebra in our teaching. It will take a little

of some time for us to be experts and the other obstacle will be the lack of enough room in

the computer lab classes are over enrolled - 60 pupils is not easy to teach.

F Interest part of this software is addictive. Nevertheless, there is [..] to use both traditional and

GeoGebra. The obstacle will be to let pupils develop mathematics skills using the traditional way

before getting to GeoGebra. [..] pupils must must have skills and then letting them to use

GeoGebra will make them understand more. Starting teaching with this software without having

pre-requisite of plotting, it would have a problem to transfer the knowledge which is a big problem.

H Obstacles will mainly be over enrolment in classes and literacy levels on ICT use.

Table 5.27: Results on obstacles to be faced when teachers start to use GeoGebra.

Results in Table 5.28 showed that teachers perceived use of GeoGebra as a software to help their

teaching and planning in ways that could enhance pupils’ independent learning and understanding.

Teacher Theme 4: Use of GeoGebra helps in planning and teaching

A This will help improve results in mathematics and my teaching approaches. [laughs]. This will even

clear the belief in pupils that mathematics is difficult. You saw here .[..] when we taught pupils,

they were saying mathematics is simple. [laughs]. This changed pupils’ perception about mathematics.

E It is a positive effect on my planning. It will help me plan in such a way that pupils will have

hand-on activities that could help them understand mathematics. It will be easy for me to develop

activitiesthat would engage pupils in independent learning thereby making development

discoveries in mathematics.

F The use of GeoGebra in teaching is nice. The first time you introduced it to us and demonstrated

some of its uses, I was convinced that this is what I was looking for to enhance my teaching.

The adoption of use ofGeoGebra will entail adapting the best method of teaching

in my class [laughs] I will endeavour to use it in my teaching.

H It has helped me in a lot of things. There are things I did not know but the workshop made it

possible for me to known.[..] use of GeoGebra has helped me find out some concepts which were

missing in my teaching. This will end up enhancing my teaching. Even the pupils themselves,

once in a while will becoming to the computer lab to consolidate what we learning and

teach in class through using GeoGebra.

Table 5.28: Results on GeoGebra’s help in planning and teaching.


Results in Table 5.29 showed that teachers found the use of GeoGebra helpful in the development

of instructional materials. They also thought that GeoGebra helps teachers develop materials that

foster independent and creative learning.

Teacher Theme 5: Development of instructional materials

A On change, there is something I have learnt during this training. When it comes to understanding

concepts, in traditional way, it is the teacher’s show to explain all parts to pupils. [....] how can a pupil

learn like that and how effective can a teacher be in? The approach of letting pupils develop and discover

things for themselves is a good way of teaching since pupils develop critical thinking. This is what ought to

be supported. pupils are [..]

E Having practised teaching with the materials we developed, I saw that teaching was made easy and it

helped me to evaluate which way is better. Use of GeoGebra allows more pupil-centred teaching than the

traditional one. For this I think GeoGebra is better as it brings concrete understanding to pupils. [..] I

head applause from pupils when they got things right [...].

F Actually it is a blessing to have less talking as I will do less talking and let pupils to do the

learning. The abstract part of teaching will be reduced. When teaching functions and graphs [..] pupils will be

able to understand the relationship [..] makes the teachers’ work on development of teaching materials easily

and[..] ends up developing activities that encourage independent learning. [..].

H I must say that at first when you introduced this workshop, I was sceptical and doubted the success of it

but after practising and doing activities, I was shock I could even develop learning materials which we

used on pupils during our practical session. Pupils were really happy and I could see how they were

doing things and made me see how they were thinking through their activities which is not possible

to see when teaching using traditional way. I have really develop interest. My teaching now has changed

and I will make sure I use GeoGebra in my teaching.

Table 5.29: Results on the development of instructional materials.

Results in Table 5.30 illustrate that all teachers thought that their perception of teaching had

changed. Instead of teaching in a traditional way, the use of GeoGebra had brought meaning to

their teaching.

Results in Table 5.31 illustrate that all teachers felt the impact of the workshop on their

teaching. Teachers thought that use of GeoGebra helps pupils understand mathematics and also

that teachers and pupils were motivated. They hoped to start using GeoGebra in their teaching

from that day onwards.


Teacher Theme 6: Change in perception about teaching

A I will adopt the new approach in which pupils and I, will be using GeoGebra in defining my teaching

and learning.

E I will be using GeoGebra in my lessons and I think I hope to encourage other teachers to do the same.

F I will now lessen on talking in class and let pupils be involved in the learning process. This will

make my pupils busy and develop mathematical skills easily which is difficult to achieve in a traditional

way. When pupils discover things on their own, understanding is more permanent that when I talk it

to them?

H I must say that at first when you introduced this workshop, I was sceptical and doubted the success

of it but after practising and doing activities, I was shock I could even develop learning materials

which we used with pupils during our practical session. Pupils were really happy and I could see

how they were doing things and made me see how they were thinking through their activities

which is not possible to see when teaching using traditional way. I have really develop interest.

My teaching now has changed and I will make sure I use GeoGebra in my teaching.

Table 5.30: Results on the change in pedagogy as a teacher.

Teacher Theme 7: Eeffect of the workshop on the teachers.

A The experience we had during practical teaching showed that it was easier for pupils to solve and

understand problems using GeoGebra than when we use the traditional way. My understanding now is

GeoGebra will help pupils understand mathematics concepts that are abstract through

observation and doing.

E I would say, all teachers attended the workshop. Teachers are happy and have welcomed the use of

GeoGebra in teaching and are saying it has come to stay. I hope to see them more in future activities.

F The use of GeoGebra in teaching is nice. The first time you introduced it to us and demonstrated some

of its uses, I was convinced that this is what I was looking for to enhance my teaching. The

adoption of use of GeoGebra will entail adapting the best method of teaching in my class [laughs] [..]

I will endeavour to use it in my teaching.

H From today onwards, I will blend ICT in my teaching. This means that I will develop a teaching

strategy in which I will let pupils develop learning themselves and will only help by guiding them.

To me, the workshop has had a big effect on my view of teaching. I will make sure I prepare my lesson

materials in ways that will encourage discoveries in pupils. I feel this will help my pupils to

create proper understanding. My appeal to the administration is that, computer laboratory

must be made accessible to teachers and pupils.

Table 5.31: Results on the effect of the workshop.


5.2.4 Transcripts of Interviews with Pupils

Interview with Pupil X

R1. How was your lesson?

P1. It was interesting. I thought for a moment that mathematics can be this simple. I loved the lesson

and I could see what was happening to the graph.

R2. What was different from the lessons you usually have?

P2. This lesson was nice because I now understood the topic on locus of points which in class I could

hardly understand. Why can’t my teacher be using computers? Did he know this?

R3. How do you want your teacher to teach you?

P3. I want him to be allowing us here in the lab to learn using computers.

R4. Do you have any suggestions to your teachers about your learning process?

P4. Yes, I do. I think learning mathematics using computers is good and helps us understand. We need

to learn like this.

R5. Do you have anything to say about using GeoGebra in your learning?

P5. GeoGebra is good. All my friends who hated mathematics now have loved mathematics because of

GeoGebra and I hope our teacher can adopt this in our lessons.

Interview with Pupil Y

R1. How was your lesson?

P1. The lesson was good and I saw what was happening.

R2. What was different from the lessons you usually have?

P2. Learning like this helps you to understand on your own instead of memorizing, you see what to do.

R3. How do you want your teacher to teach you?

P3. I want the teacher to teach us mathematics like the way he and his friends did today.

R4. Do you have any suggestions to your teachers about your learning process?

P4. It is good to learn using GeoGebra but it is important to check on us because some pupils start playing

computer games instead of doing tasks.


R5. Do you have anything to say about using GeoGebra in your learning?

P5. It is good. Are you going back with GeoGebra?

Results from the two interviews with pupils, X and Y, illustrate that pupils liked the way they

learnt mathematics and that working with ICT helped them understand mathematical concepts

of the particular topic (lesson). They were also motivated in using GeoGebra and wished their

teachers continued using the software in their teaching.

5.3 Field Notes

I observed teachers engaged in preparing and developing instructional materials and activities. I

went around assisting the groups that required my help. Teachers were involved into discussions

as they were developing activities. I got also involved in these discussion (see Picture 5.1).

Figure 5.1: A picture of teachers engaged in a discussion about activity development.

Figure 5.2: A picture of teachers engaged in a discussion about activity development.

5.3 Field Notes 73

I noticed that teachers, while developing activities, first did it without using GeoGebra and once

they were done, they discussed how they could blend technology into their activity. For instance,

teachers developed an activity on kinematics without using GeoGebra and solved the problem

(Figure 5.3). Thereafter they discussed how to use GeoGebra (Figure 5.4) and they created a

Geogebra applet (Figure 5.5).

Figure 5.3: A picture of teachers engaged an ac-

tivity without GeoGebra and solved it.

Figure 5.4: A picture of teachers engaged in

converting activity developed without ICT into

one with ICT.

Figure 5.5: A screen shot of the developed GeoGebra applet.


When teachers were engaged in the teaching practice with their develop material, I could see

that they were actively involved in providing assistance to pupils. They were able to have face to

face contact with each group of pupils working behind a computer, thereby monitoring each pupil

while walking around in the computer lab (see Figure 5.6).

Figure 5.6: A picture of showing teachers going around assisting pupils during the practical lessons.

Figure 5.7 and Figure 5.8 are illustrative examples of pupil activities in the teaching practice

sessions.

After the teachers got used to the software, I observed them developing activities using Geo-

Gebra, without first doing it without GeoGebra. This showed how teachers were able to combine

technology with PCK.

I observed that working in teams helped teachers, especially the ones that hardly had computer

skills. Interestingly, even with no initial computer training, Teacher A, one of the teachers with

no computer skills, was able to present to other participants what their team had developed.

Looking at how he was able to use the computer and GeoGebra in particular, I concluded that

he had developed computer skills and TPCK (He was one of the teacher who volunteered to be

interviewed).

5.3 Field Notes 75

Figure 5.7: Derivative of a quadratic function using slope function


Figure 5.8: Visualizing rotation of a triangle in 1800 clockwise and 1800 anticlockwise

Chapter 6

Conclusions and Discussion

In this chapter, I conclude and discuss my research study: I answer the research question, describe

the limitations of my study, and make recommendations for future research.

6.1 Conclusions

Research Question

“To what extent could participants of my training course in using ICT in mathematics education

start developing TPCK through the workshop?”

To be able to answer the over-arching research question, four sub questions were developed. In

the following subsections I answer each sub question based on the results and findings presented in

Chapter 5.

6.1.1 Sub Question 1

What obstacles do teachers encounter when they start using dynamic mathematics software?

The most common obstacles that teachers encountered when they started using dynamic mathe-

matics software were:

- In the initial stage, teachers had to get used to GeoGebra, e.g., to understand what every icon

meant and how it worked, and the teachers needed time for getting acquainted and feeling

comfortable with the software.

78 Conclusions and Discussion

- Many teachers and pupils had little or no ICT knowledge. For those that had no knowledge,

they had two things to know, use of computers and also use of dynamic mathematics software

(DMS).

- Over enrolments in classrooms made it difficult to develop activities for pupils to work on.

This meant that working in groups could make some pupils not participant in the group

discussions since the groups were crowded.

- The limited number of computers in the school made access to the computers difficult. The

school had only 12 working computers and that meant that some pupils had not access to

computers.

- The school had only one computer with Internet connection and besides of this, Internet was

down most of the time. This made use of internet (online) resources hardly accessible to

teachers.

All the above obstacles are findings from interviews with teachers, post questionnaire and field

notes.


What assistance do teachers need to overcome obstacles?

Teachers indicated that they needed more practice in the use of GeoGebra so that they could get

started using the software in their teaching. They thought that this could be realized through

holding in future more workshops of the same kind as the one reported on in this thesis. Teacher

E, for example, suggested turning Continuous Professional Development (CPD) weekly meetings

into use of GeoGebra practice sessions so that teachers could become expert users of the software

before fully engaging in teaching using GeoGebra.

Teachers indicated also that they needed teachers and pupils trained in ICT use (training how to

use computers), so that it could be easier for teachers to teach pupils mathematics using GeoGebra

without facing difficulties in computer use. They also stated that the school needed to purchase

more computers or repair the ones that were not functioning, so that they could be used to cater

for the large numbers of pupils in classes.

As I went around providing assistance to teachers and pupils, I saw that teachers required more

time to prepare their instructional materials and when I enquired from them through discussions,

6.1 Conclusions 79

they indicated to me that over-enrolment makes them overworked, thereby having no time to

prepare for tnext class. They indicated to me that having enough computers would be useful for

them in their teaching.


Does collaboration of less experienced teachers with each other during the development of teaching

materials and lesson planning help them in developing their TPCK?

Teachers indicated that collaborating with their colleagues helped them so much in knowing how

to use GeoGebra and how to develop instructional materials for teaching. Teacher A, who used

computers for the first time in his life, indicated that he developed ICT knowledge through working

with more knowledgeable colleagues. Interestingly, Teacher A was chosen by his teachers’ team to

present the developed materials to the other participants of the workshop during the third training

session: He was confident and presented the team’s work with enthusiasm. From his presentation

one could not tell that the teacher had just mastered ICT skills through his collaboration with

colleagues. In his interview, this teacher indicated that his colleagues contributed to his knowledge

and skills regarding use of computers and GeoGebra, simply by allowing him to participate in the

team.

The findings from interviews with teachers and pupils, the post questionnaire and the field notes

about practical teaching experience indicated that ICT supported effective team work, because

dynamics mathematics software helped the teachers provide structure, direction and support to

their pupils. This made management of group work easier for teachers. Teachers also indicated

that pupils working in small groups on a computer activity worked together for longer time. When

the pupils did computer activities, they discussed ideas, listened to each other and built on each

other’s experience and knowledge. Often the process of explaining to peers what they think helped

pupils understand the mathematics and to learn it more effectively. In my research I found that

this kind of teaching seemed to motivate teachers and the teachers stated that they could use group

work in their teaching as it helped pupils to develop skills in using dynamic mathematics software

in learning mathematics.


Does the teachers’ personal experience in the classroom teaching using the materials they developed


during the training course contribute to their development of TPCK?

Teachers stated that teaching practice using their own developed instructional materials gave

them a practical experience to teaching using dynamic mathematics software. All teachers felt that

teaching using GeoGebra was a worthwhile experience in their teaching experience.

Observing and taking part in the practical lessons, I saw how enthusiastic teachers and pupils

were. I had rarely experienced this in my own teaching at high school. Development of instructional

materials plus teaching practice motivated teachers in the sense that they could actually see what

it means to teach mathematics using ICT and see how pupils better understand mathematics in

this way of instruction. All teachers indicated that the use of dynamic mathematics software in

teaching mathematics was useful because it provided opportunities for interactions among pupils

and between teachers and pupils. Teachers indicated that practical teaching provided them with

feedback on the instructional materials they developed using GeoGebra. They stated that feedback

from the activities helped pupils to learn in a range of different ways that helped teachers understand

the usefulness of dynamic mathematics software in teaching.

Pupils also indicated that use of dynamic mathematics software in learning was very useful for

them as it made them understand the concepts more easily than in traditional instruction. Inter-

estingly, one of the interviewed pupils stated in his interview that the use of dynamic mathematics

software resurrected his interest in mathematics, while he had before given up understanding of

mathematics. The pupils’ experience in the ICT-supported lesson was that the introduction of ICT

in the lesson gave them a different taste of mathematics and that they liked lessons to continue in

this format.

These findings made me look upon my research study as a special experience. After training

teachers on how to use dynamic mathematics software and giving them the opportunity of a teaching

practice, helped me also to have a personal feel of how ICT could be used in teaching. In other

words, the research study also contributed to my personal development of TPCK. It were not only

teachers who were participating in my research study who got motivated for and realised that use

of ICT could improve mathematics teaching, but I also became even more enthusiastic during the

research. I was very happy with the comments of Teacher E:

“There was a shift, I would say to a large extent, it came from developing lesson ma-

terials that promote pupils’ activities. To me this is how I was able to combine my

knowledge of teaching with the technology. It was not easy to develop activities that

6.1 Conclusions 81

would stimulate pupils to be involved in discoveries but we managed [laughs]. Is that

not good? It was hard at first but with time things develop. Now I know how useful

technology is to teaching.”

This clearly showed how motivated teachers were during teaching practice.

6.1.5 The overarching research question

To what extent could participants of my training course in using ICT in mathematics education

start developing TPCK through the workshop?

The findings and conclusions discussed in the previous subsections indicate that teachers who

participated in the training course started developed Technological Pedagogical Content Know-

ledge in stages. One of the most important stages in the teachers’ development of TPCK was the

part of the workshop in which the teachers created instructional materials that were more oriented

towards a constructivist approach, in the sense that instructional materials were created with the

purpose of engaging pupils in their learning process via ICT support and having pupils collaborate

with peers.

All teachers indicated that it was difficult at first to blend technology in their activities. Brains-

torming in teacher teams helped them to develop instructional materials that suit small group tea-

ching and learning as well as independent learning. The interviewed teachers felt that development

of instructional materials and teaching helped them see how to blend technology in to their teaching

and content knowledge.

By participating in developing instructional materials, teachers were confronted with building

a technological artifact based on the particular requirements of the subject matter to be taught,

the instructional goals to be achieved. They think about what is possible with the technology in

the subject matter. The findings and conclusions of research study indicated that teachers found

blending of technology with their teaching and content (Pedagogical Content Knowledge - PCK)

challenging and motivating.

All teachers stated that their teaching had been affected in a positive way such that they felt

now that there was something missing in their previous way of teaching. This they noticed after

having had an experience of including technology to their PCK. They indicated that they will

have to change the way they taught mathematics. Whether they really make this change is at the

moment difficult for me to conclude.


In addition, pupils’ interest and motivation in the lessons indicated the fact that teachers had

developed TPCK in their teaching. Both pupils interviewed recognised the impact the lessons had

on them. Their positive reaction to the lesson exhibited just how well received the lessons were to

the pupils.

6.2 Discussion

There is evidence that teachers’ ways of teaching changes when technology is blended with teachers’

PCK (Boylan, 2010). From interviews with teachers, it was confirmed that the use of the dynamic

mathematics software affected all the four components of TPCK framework: TK, TCK, TPK and

TPCK (discussed in Section 2.2). The interviews with teachers, the post questionnaire and the field

notes indicated that teachers needed to be motivated to be able to adopt the use of the dynamic

mathematics software effectively.

In order to use the dynamic mathematics software effectively, teachers needed to develop TPCK.

As teachers began to use the dynamic mathematics software, concerns became more tense in the

area of managing classroom activities. Results in Chapter 5, demonstrated that there was significant

change, from the pre to post questionnaire, interviews with teachers and pupils, and the field notes,

on teachers’ ways of teaching using the dynamic mathematics software. The results suggested that

teachers’ development of TPCK had a potential to change teachers’ beliefs about teaching after

participation in the workshop. This is similar to findings of Boylan (2010), who discussed a single

teacher’s account of change of teaching style.

It is possible that the training sessions and the personal experience in teaching practice invol-

vement using the dynamic mathematics software enabled teachers to gain a clear understanding of

how they could implement pupil-centred technology use in a classroom setting. I viewed this as

the first step towards teachers’ development of TPCK and change in their future practice as well

as their beliefs about effective technology use.

Teachers acknowledged that having personal experience via practice teaching with their ins-

tructional materials created during the training sessions was meaningful, because it allowed them

to get feedback from both the pupils and the lessons. The practical sessions made teachers realise

how vital technology use in teaching of mathematics was. In addition, pupils’ positive feedback

and reaction to the lessons was clear indication how teachers’ TPCK development had impacted

them in terms of understanding.

6.3 Limitations of the Study and Recommendations for Future Research 83

However, for teachers to start to develop TPCK, they need more support. This support ranges

from individual support to departmental support (all teacher in the department).

In Zambia, technology use in schools is not intensified as is the case with schools in the Nether-

lands. Nonetheless, with fewer or available computer facilities in schools, technology use in teaching

mathematics could make a big difference at a local level (teacher’s classroom). With positive results

from my study, individual teachers could resort to technology use in their own teaching. This does

not require the curriculum change but just allowing individual teachers using dynamic mathema-

tics software in teaching. However, it is important to mention here that, I cannot be sure whether

the teachers at Mkushi High School will resort to technology use in near future, even though they

indicated this in the post questionnaire and in the interviews.

6.3 Limitations of the Study and Recommendations for Future

Research

One limitation of my research study was the short period of time in which I conducted the study,

which also means that I was not able to to get results that could be easily generalised. In addition,

lack of enough computer facilities in the school affected the practice teaching that was part of my

workshop. Therefore not all pupils were able to interact with computers to the extent that I had

in mind, although this was also partly because of over-enrolment in the classes. However, findings

of this study could provide insight in how technology could be deployed by Zambian mathematics

teachers, as well as by teachers of other subjects.

A longitudinal study would provide the opportunity to investigate the effects of teachers’ de-

veloped TPCK on their pupils’ performance, as well as on teachers’ teaching profession. Further

research might also draw attention on the need for integrated approaches to investigating the role

of technological on teachers’ development of PCK through workshops.

In order to set a high quality standard for professional development events with GeoGebra,

follow-up application workshops as well as additional documentation, like online-courses, a Geo-

Gebra introductory book, and instructional materials about GeoGebra and its integration into

mathematics classrooms should also be developed in the future. Further study is needed to re-

search how teachers’ development of TPCK can be measured at every stage of its development.

This could help teachers and researchers determine how much more support teachers could require

in their work.

References 85

References

Akkoc, H., Yesildere, S., & Ozmantar, F. (2007). Prospective mathematics teachers’ Pedagogical

Content Knowledge of definite integral: The problem of limit process. In D. Kuchemann

(Ed.), Proceedings of the British Society for Research into Learning Mathematics, 27 (3).

Angeli, C., & Valanides, N. (2009). Epistemological and methodological issues for the conceptua-

lization, development, and assessment of ICT-TPCK: Advances in technological pedagogical

content knowledge (TPCK). Computers and Education, 52 (1), 154-168.

Ball, D. L. (2010). Content knowledge for teaching: What makes it special? Journal of Teacher

Education, 59 (5), 389-409.

Boylan, M. (2010). It’s getting me thinking and I’m an old cynic: exploring the relational dynamics

of mathematics teacher change. Journal of Mathematics Teacher Education, (13)(5), 383-395.

Bu, L., & Haciomeroglu, E. S. (2010). GeoGebra in mathematics teacher education: The case of

quadratic relations MSOR Connections, 10 (1) 6-9.

Central Statistical Office (2003). Summary Report for the 2000 Census of Population and Housing.

www.zamstats.gov.zm Retrieved on 14/04/2011.

Deci, E. L., Vallerand, R. J., Pelletier, L. G., & Ryan, R. M. (1991). Motivation and education:

The self-determination perspective. Educational Psychologist, 26 (3&4), 325-346.

Dikovic, L. (2009). Applications GeoGebra into teaching some topics of mathematics at the college

level. Computer Science and Information Systems, 6 (2), 191-203.

Doering, A., Veletsianos, G., & Miller, C. (2009). Using the Technological, Pedagogical, and Content

Knowledge framework to design online learning environments and professional development.

J. Educational Computing Research, 41 (3): 319-346.

Dweck, C. S. (1986). Motivational processes affecting learning. American Psychologist, 41 (10),

1040-1048.

Esselaar, P., Hesselmark, O., & Silavwe, E. (2002). A Country ICT Survey for Zambia. Stockholm,

Sweden: SIDA Retrieved on September 24, 2010, from www.sida.se

Ferrance, E. (2000). Action Research. Providence, RI: Northeast and Islands Regional

Educational Laboratory, Brown University. Retrieved on September 24, 2010, from

www.lab.brown.edu/pubs/themes ed/act research.pdf

Grandgenett, N. F. (2008). Perhaps a matter of imagination: TPCK in mathematics education.

In M. J. Koehler & P. Misra (Eds.), Handbook of Technological Pedagogical Content Know-

86 References

ledge for Teaching Technology (pp. 145-165). New York: American Association of Colleges of

Teacher Education and Routledge.

Grossman, P. L. (1990). The Making of a Teacher: Teacher Knowledge and Teacher Education.

New York, NY: Teacher College Press, Columbia University.,

Haciomeroglu, E. S., Bu, L., Schoen, R. C., & Hohenwarter, M. (2009). Learning to develop ma-

thematics lessons with GeoGebra. MSOR Connections, 9 (2) 24-26.

Harris, J. (2008). TPCK in in-service education Assisting experienced teachers’ Planned improvi-

sations. In M. J. Koehler & P. Misra (Eds.), Handbook of Technological Pedagogical Content

Knowledge for Teaching Technology (pp. 251-271). New York: American Association of Col-

leges of Teacher Education and Routledge.

Hennessey, S., Ruthven, K. and Brindley, S. (2005). Teacher perspectives on integrating ICT into

subject teaching: Commitment, constraints, caution and change. Journal of Curriculum Stu-

dies, 37 (2), 155-192.

Hershkovitz, A., & Nachmias, R. (2009). Learning about online learning processes and pupils’

motivation through web usage mining. Interdisciplinary Journal of E-Learning and Learning

Objects, 5, 197-214.

Jang, S-J. and Chen, K-C (2010). From PCK to TPACK: Developing a transformative model for

pre-service science teachers. Journal for Science Education and Technology, 19 (6), 553-564.

Koehler, M. J., Mishra,P. (2008). Introducing TPCK. In M. J. Koehler & P. Misra (Eds.), Handbook

of Technological Pedagogical Content Knowledge for Teaching Technology (pp. 3-29). New

York, NY: American Association of Colleges of Teacher Education and Routledge.

Lagrange, J. B., & Monaghan, J. (2010). On the adoption of a model to interpret teachers’ use of

technology in mathematics lessons. In Proceedings of CERME6 (pp. 1605-16154). Retrieved

on November 1, 2010, from www.inrp.fr/editions/cerme6

Lederman, N. G., Gess-Newsome, J., & Latz, M. S. (1994). The nature and development of pre-

service science teachers’ conceptions of subject matter and pedagogy. Journal of Research in

Science Teaching, 31 (2), 129-146.

Mills, G. (2003). Action research: A guide for the teacher researcher (2nd ed.). Upper Saddle

River, NJ: Merrill/Prentice Hall.

Mishra, P., & Koehler, M. J. (2006). Technological Pedagogical Content Knowledge: A Framework

for Teacher Knowledge. Teacher College Record, 108 (6), 1017-1054.

Mulholland, J., & Wallace, J. (2005). Growing the tree of teachers knowledge: Ten years of learning

References 87

to teach elementary science. Journal of Research in Science Teaching, 42 (7) 767-790.

National Council of Teachers of Mathematics, NCTM (2000). Principles and Standards for School

Mathematics. Retrieved September 20, 2010, from www.nctm.org/standards/

Niederhauser, D. S., & Stoddart, T. (2001). Teachers’ instructional perspectives and use of educa-

tional software. Teaching and Teacher Education, 17 (1), 15-31.

Niess, M. L. (2005). Preparing teachers to teachers to teach science and mathematics with techno-

logy: Developing a technology pedagogical content knowledge. Teaching and Teacher Educa-

tion, 21 (5), 509-523.

Oja, S. N., & Smulyan, L. (1989). Collaborative Action Research: A Developmental Approach. New

York, NY: Falmer Press.

Ozmantar, M. F., Akkoc, H., Bingolbali, E., Demir, S. & Ergene, B.. (2010). Pre-service Mathe-

matics Teachers’ Use of Multiple Representations in Technology-Rich Environments. Eurasia

Journal of Mathematics, science & Technology Education, 6 (1), 19-36.

Preiner, J. (2008). Introducting Dynamic Mathematics Software to Mathematics Teachers: the case

of GeoGebra. Doctorol thesis, University of Salzburg.

Ruthven, K. (2007). Teachers, technologies and the structures of schooling. In Proceedings of

CERME5. Retrieved on November 1, 2010, from

http://ermeweb.free.fr/CERME 5/Plenaries/Plenary3.pdf

Ruthven, K. ,Hennessey, S., & Deaney, R. (2005). Incorporating Internet resources into classroom

practice: pedagogical perspectives and strategies of secondary-school subject teachers. Com-

puters and Education, 44 (1), 1-34.

Ruthven, K., & Hennessey, S. (2002). A practitioner model of the use of computer-based tools and

resources to support mathematics teaching and learning. Educational Studies in Mathematics,

49 (1), 47-88.

Ruthven, K., & Hennessey, S. (2003). Successful ICT use in secondary mathematics-a teacher

perspective. Micromath, 19 (2), 20-24.

Ruthven, K., Deaney, R., & and Hennessey, S. (2008a). Practitioner use of graphing software to

teach about algebra forms. In M. Joubert (Ed.), Proceedings of the British Society for Research

into Learning Mathematics, 28 (3), 108-113.

Ruthven, K.,Deaney, R. and and Hennessey, S. (2008b). Construction of dynamic geometry of: A

study of interpretative flexibility of educational software in classroom practice Computers and

Education, 51 (1), 297-317.

88 References

Ruthven, K., Deaney, R., & Hennessey, S. (2009). Using graphing software to teach about algebraic

form: a study of technology-supported practice in secondary-school mathematics. Educational

Studies in Mathematics, 71 (3), 279-297.

Ryan, R. M. (1982). Control and information in the intrapersonal sphere: An extension of cognitive

evaluation theory. Journal of Personality and Social Psychology, 43 (3) 450-461.

Schmidt, D. A., Baran, E., Thompson, A. D., Mishra, P., Koehler, M. J. and Shin, T. S. (2009).

Technological Pedagogical Content Knowledge (TPACK): The development and validation

of an assessment instrument for preservice teachers. Journal of Research on Technology in

Education, 42 (2), 123-149.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Resear-

cher, 15 (2), 4-14.

Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational

Review, 57 (1), 1-22.

Smeets, E. (2005). Does ICT contribute to powerful learning environments in primary education?

Computer & Education, 44(3), 343-355.

Smeets, E., & Mooij, T. (2001). Pupil-centred learning, ICT, and teacher behaviour: Observations

and educational practice. British Journal of Educational Technology, 4 (32), 403-417.

Smeets, E., & Mooij, T. (2000). Modelling and supporting ICT implementation in secondary

schools. Computer and Education, 36, 265-281.

Smeets, E., & Mooij, T. (2006). Design, Development and Implementation of Inclusive Education.

European Educational Research Journal, 5 (2), 94-109.

Thomas G., Krantz, J., & Ramirez, R. (1986). Action based modes of inquiry and the host-

researcher relationship, Consultation 5.3 (Fall 1986), 161.

Torrance, H., & Pryor, J. (1998). Investigating Formative Assessment. Philadelphia, PA: Open

University Press

Twining, P. (2002). ICT in Schools Estimating the level of investment. Report No. meD8 02.01,

retrieved on October 29, 2010, from www.meD8.info/docs/meD8 02-01.pdf

Unwin, A. (2007). Technological Pedagogical Content Knowledge (TPCK), a conceptual framework

for an increasingly technology driven higher education? Bulgarian Journal of Science and

Education Policy, 1 (1), 231-247.

Watts, H. (1985). When teachers are researchers, teaching improves. Journal of Staff Development,

6 (2), 118-127

Appendix A

Pre Questionnaire

Mkushi High School Workshop

Introduction to GeoGebra

In my research project, I am investigating the development of teachers’ technological pedagogical

content knowledge, that is, what teachers need to know in order to become comfortable using

technology with pupils. Your responses to this questionnaire will help me determine this. Also I

take your impressions and opinions as advice to me and my colleagues on how we can learn to help

Zambian mathematics teachers more effectively.

I hope that you will also be so kind to give a short explanation of your reasons for each of your

answers.

Please do be assured that I will keep your personal information absolutely confidential¿ I only

ask for your name and contact details in case I want to discuss something with you personally.

I am very grateful to you for taking time to answer my questions. Thank you very much for

your help. [Wakwinji]

Personal Data

Name:.......................

School Name:.............

Teaching experience:......yrs

Instructions:

1. Do your pupils have an opportunity to use computer at school? (please circle)

Never//Sometimes//Often

90 Pre Questionnaire

2. Have you ever used computers in your teaching? (please circle) yes//No. If yes,

in how how many lessons per week? (please circle) None// 1-3// 4 or more.

If no, you may proceed to question 5.

3. How do you use a computer during your lesson? You may choose as many as possible. (mark

with X for your choice)

As a presentation tool with a beamer

To help my pupils discover mathematical concepts

For educational games

For accessing information from the internet

If other, please indicate...........................................................................................

4. What type of technology do you use for teaching? You may choose as many as possible.

(mark with X for your choice)

Presentation software (powerpoint)

Word processing software

Spreadsheet

Geometer’s Sketchpad

Cabri

Graphical calculater

If other, please indicate...........................................................................................

5. How would you like to use computers in your lessons? Please give reasons.

.................................................................................

.................................................................................

6. In these workshops, you are going to learn how to use mathematical software in your teaching.

In general, what are your expectations for using ICT in your teaching?

.................................................................................

.................................................................................

Appendix B

Post Questionnaire

Mkushi High School Workshop

Introduction to GeoGebra

In my research project, I am investigating the development of teachers’ technological pedagogical

content knowledge, that is, what teachers need to know in order to become comfortable using

technology with pupils. Your responses to this questionnaire will help me determine this. Also I

take your impressions and opinions as advice to me and my colleagues on how we can learn to help

Zambian mathematics teachers more effectively.

I hope that you will also be so kind to give a short explanation of your reasons for each of your

answers.

Please do be assured that I will keep your personal information absolutely confidential¿ I only

ask for your name and contact details in case I want to discuss something with you personally.

I am very grateful to you for taking time to answer my questions. Thank you very much for

your help. [Wakwinji]

Personal Data

Name:.......................

School Name:.............

Teaching experience:......years

How may I contact you? (telephone number, email address, other)

Feel free to continue any of your answers on the reverse side of a page.

92 Post Questionnaire

Learning to use GeoGebra

1. Have you ever used GeoGebra in your lessons? Yes No (please circle)

If so, please mention about how often you use it by circling one of the options given below:

Always// Frequently// Regularly // Occasionally// Rarely.

2. What is your opinion about the ease for you of learning to use GeoGebra?

(very easy// hard at first but became easier// still hard)? Please explain

..........................................................................................

...........................................................................................

3. Would you like to continue working with GeoGebra (choose as many as appropriate):

Yes No (please circle)

If so,

- in another workshop similar to this one?

- in a teacher work group in your school?

- other? (please state)

Please explain your choice(s)

...........................................................................................

............................................................................................

............................................................................................

Use of GeoGebra as a teacher

4. What support would you need to start using GeoGebra in your teaching? (please explain or

give examples.)

(a) To show the whole class something?

...........................................................................................

............................................................................................

............................................................................................

(b) To have a whole class discussion?

...........................................................................................

............................................................................................

............................................................................................

93

(c) To let pupils use it in small groups (Describe briefly.)

...........................................................................................

............................................................................................

............................................................................................

5. How would you like to use GeoGebra in teaching the subject of graphing:

- with handwritten graphs first, followed by using GeoGebra?

- with GeoGebra alone?

- without GeoGebra?

Please explain your choices.

...........................................................................................

............................................................................................

............................................................................................

Possible benefits of GeoGebra for pupils

6. How would you compare learning about functions and their graphs using GeoGebra and

using paper and pencil and other regular tools (without GeoGebra)?

...........................................................................................

............................................................................................

............................................................................................

7. Pupils understanding of graphs and functions: What is your opinion about using GeoGebra to

help pupils make connections between graphs and their functions? (please give your reasons.)

...........................................................................................

............................................................................................

............................................................................................

8. Using GeoGebra to do “trial and improvement”:

Do you think pupils, using trial and improvement, can arrive at a satisfactory solution to a

mathematical problem (either on their own or with a friend)?

...........................................................................................

............................................................................................

............................................................................................

94 Post Questionnaire

9. Do you think GeoGebra can help pupils who are weak or unsure to learn about graphs and

their function? Please be as specific as you can (you can always give an example)

...........................................................................................

............................................................................................

............................................................................................

About this workshop and afterwards

10. What is your opinion about the user-friendliness of GeoGebra now at the end of the workshop

series?

...........................................................................................

............................................................................................

............................................................................................

11. What sort of obstacles do you think you would face if you wanted to use GeoGebra in your

teaching? Please list and explain.

...........................................................................................

............................................................................................

............................................................................................

12. What are your plan (ideas?) for working with GeoGebra after the workshop training?

...........................................................................................

............................................................................................

............................................................................................

13. What do you think ha been the most important thing that you have gotten from the GeoGebra

workshop?

...........................................................................................

............................................................................................

Appendix C

Teacher Interview Protocol

A list of questions that acted as interview guide for the researcher when interviewing teachers.

1. What is your general impression of the workshop?

2. Is GeoGebra easy to use and helpful in solving problems?

3. What was your initial perception, especially when you were informed of of the training pro-

gramme?

4. Is your perception about teaching now different to the one your had before?

5. From today onwards, what do you think of GeoGebra use in teaching mathematics?

6. What benefits will your pupils get from using GeoGebra in terms of development of their

understanding?

7. Why do think GeoGebra is easy to use?

8. What teaching method do you suggest is the best way to teach mathematics?

9. What do you think this kind of teaching will do to you and your pupils?

10. What are some of the challenges that you faced personally when you began using GeoGebra?

11. What kind of assistance do you need in order to use GeoGebra in your teaching?

12. Where do you stand in your teaching after this workshop?

13. You resisted change at the beginning of the training course, how did you look at everything

at the start?

96 Teacher Interview Protocol

14. How do you want ICT use initiated in school?

15. You prepared lesson materials in groups and later used them in practice, what was the reaction

of the pupils about the lesson?

16. Do you have any suggestion to the whole training programme or on the use of GeoGebra in

teaching mathematics?

17. How did you manage to combine your teaching and content with technology?

18. With a few facilities avilable here, Is your department thinking about using GeoGebra interm

of Continuous Professional Development (CPD)?

19. At a personal level, how does the use of GeoGebra help you in planning your teaching?

20. How do you look at your pedagogy as a teacher?

21. Are you adopting the use of GeoGebra in teaching?

22. How does GeoGebra help you in your teaching?

23. How has your teaching been affected by the workshop?

Appendix D

pupil Interview Protocol

A list of questions that acted as interview guide for the researcher when interviewing pupils.

1. How was your lesson?

2. What was different from the lessons you usual have?

3. How do you want your teacher to teach you?

4. Do you have any suggestions to your teachers about your learning process?

5. Do you have anything to say about using GeoGebra in your learning?

98 pupil Interview Protocol

Appendix E

Themes of the interview protocol

The seven themes used in analysing teacher interview protocol. There were drawn from teacher

interview protocol.

1. The overall impressions of the workshop;

2. blending technology with pedagogy and content in teaching;

3. obstacles and challenges faced when teachers start to use GeoGebra;

4. use of GeoGebra helps in planning and teaching;

5. development of instructional materials;

6. change in perception about teaching; and

7. effect of the workshop on the teachers.

exploring how a workshop approach helps mathematics teachers

Documents