Linking Geometry and Algebra

Linking Geometry and Algebra:

A multiple-case study of Upper-Secondary

mathematics teachers’ conceptions and practices of

GeoGebra in England and Taiwan

Yu-Wen Allison Lu

Thesis submitted for the degree of

Master of Philosophy in Educational Research

Faculty of Education

University of Cambridge

Supervisor: Dr. Paul Andrews

July 2008

Abstract

The idea of the integration of dynamic geometry and computer algebra and the

implementation of open-source software in mathematics teaching underpins new

approaches to studying teachers’ conceptions and technological artefacts in use. This

study opens by reviewing the evolving design of dynamic geometry and computer

algebra, teachers’ conceptions and pioneering uses of GeoGebra, and early sketches

of GeoGebra mainstream use in teaching practices.

This study has investigated English and Taiwanese upper-secondary teachers’

conceptions and practices regarding GeoGebra. It has more specifically sought to gain

an understanding of the teachers’ conceptions of technology and their pedagogies

incorporating dynamic manipulation with GeoGebra into mathematical discourse.

Moreover, the impact of teachers’ conceptions of GeoGebra with respect to their

practices has been explored. In order to answer the research questions, a multiple-case

study has been followed, involving two English and two Taiwanese teachers. For data

triangulation purposes, various methods have been employed, such as documentation,

expert interviews with observation of the teachers using the software, and informative

interviews with the GeoGebra creator and an advanced user.

According to the results of this study, some teachers tended to perceive GeoGebra as

not merely a tool but rather an environment for teaching and learning mathematics.

They viewed GeoGebra as serving the purpose of supporting pupils learning, and

performing the functions of visualising and conceptualising their mathematical

understandings. The study also found that the teachers employed a wide variety of

strategies to integrate GeoGebra into their teaching practices, such as preparation for

teaching materials, presentation of mathematical content and concepts, classroom

activities for interaction with pupils and investigation of mathematics. Their practices

regarding GeoGebra integration have many weaknesses, but there has been evidence

of some good examples of GeoGebra teaching being applied. The findings also

suggest that teachers’ teaching practices are considerably influenced by their

conceptions of GeoGebra in relation to mathematical knowledge and their cultural

traditions.

Acknowledgements

First and foremost, I would like to thank my supervisor Dr. Paul Andrews. He has

been a truly inspiring mentor and has offered the most invaluable support over this

past year.

I extend my appreciation to all those individuals whom I interviewed for this project,

and to those who offered suggestions at all stages of this thesis. I am particularly

grateful to Dr. Markus Hohenwarter, the designer of GeoGebra, for his invaluable

information in the interview and help during the data collection.

My sincere gratitude is also given to Cambridge Overseas Trust and St Edmund’s

College who supported my study.

Many thanks to my greatest friend Rebecca Day who helped me enormously through

difficulties I encountered.

Last, but not least, my fondest regards are given to my beloved family; especially my

parents and brother who have always given me the greatest love, care and support.

CONTENTS

CHAPTER ONE Introduction …………………………………………………........... 1

1.1 Research Context ………………………………………………..……………… 1

1.2 Rationale and Significance of the Study ……………………...…………….... 3

1.3 Outline……………………………………………………………………………4

CHAPTER TWO: Literature Review…………………………….……….…………… 6

2.1 Introduction…………………………………………………………………... 6

2.2 The Role of ICT in Mathematics Education ……………………………………. 6

2.2.1 Technology Integration in Education……………….…………………….. 6

2.2.2 An Overview of Technology Use in Mathematics Teaching …………….. 8

2.2.3 Cross-Cultural Studies on Technology and Mathematics Teaching ………. 9

2.2.4 England and Taiwan: Two Opposite Systems? ………………………….. 11

2.2.5 Teachers’ Beliefs, Conceptions and Practices ……………………….…... 12

2.3 Teaching algebra and geometry with Technology………………………………13

2.3.1 Geometry +Algebra=GeoGebra? .................................…………………… 16

2.4 Statement of Research Questions………………………………………………18

CHAPTER THREE: Research Methodology…………...……………………………. 19

3.1 Introduction………………………………………………………………….. 19

3.2 Theoretical Framework………………………………………………………. 19

3.3 Epistemology and Theoretical Perspective………………….………………... 20

3.4 Methodology…………….…………………………………………………….. 22

3.4.1 Selection of Appropriate Research Approach………………………….. 22

3.4.2 Case Study Research Approach………………………………………….. 23

3.5 Methods……..………….…………………………………………………….. 25

3.5.1 Data Collection………………………………….……………………….. 25

3.5.2 Interviews……….……………………………….……………………….. 26

3.5.3 Research Settings and Participants….…………….…………………….. 28

3.5.4 Data Analysis…….……………………………….……………………….. 29

3.5.5 Research Considerations.…………………….....………………………..31

CHAPTER FOUR: Data Analysis…………………………..………………………. 33

4.1 Introduction…………………………………………………………………. 33

4.2 The Cases……………………………………………………………………... 33

4.2.1 Jay…………………………….………………………………………….. 34

4.2.2 Li.…………………………….…………………………………………….38

4.2.3 Richard……………………….……………………………………………. 43

4.2.4 Tyler………………………….……………………………………………. 47

4.3 Informative Interviews…………………………………………………………..51

4.3.1 Interview with GeoGebra creator………………………………………….. 51

4.3.1 Interview with an advanced user………………………………………….. 52

4.4 Cross-Case Analysis…………………………………………………………… 53

4.4.1 Emerging Categories……….……………………………………………… 53

4.4.2 Educational Tool…………….……………………………………………. 54

4.4.3 Teacher Transition……………….……………………………………….. 56

4.4.4 Mathematical Scope……………………………………………………… 58

4.4.5 Infrastructural Change……………………………………………………. 59

4.5 Cross-Cultural Exploration…………………………………………………… 61

4.6 Summary……………………………………………………………………… 62

CHAPTER FIVE: Discussion and Concluding Remarks……………..…………….. 63

5.1 Introduction………………………………………………………………….. 63

5.2 Findings………………….…………………………………………………... 63

5.3 Discussion.....……………………………………………………..................... 65

5.4 Reflections and Limitations……………………………………..................... 67

5.5 Implications and Recommendations for Further Research………..................... 68

5.6 Concluding Thoughts……………………………………………..................... 71

REFERENCES…………………………………………………………………….. 72

APPENDICES ………………………………………………………………... 83

APPENDIX I: Interview Transcripts……………………………….………………… 83

Appendix II: A Matrix for Generating Theme-Based Assertions …..……………….111

Appendix III: Research Timeline …………………………………………………… 112

Appendix IV: Informed Consent- England ………………………………………….113

Appendix V: Informed Consent- Taiwan ………………………………………….114

Appendix VI: Sample Interview Protocol ………………………………………… 115

Appendix VII: Jay’s Example of Geometrical Constructions ……………………… 116

Appendix VIII: Li’s Examples of Proofs of Theorems and Problem-solving ……… 118

Appendix IX: Li’s Revision Worksheet ……………………..…………………… 119

Appendix X: Tyler’s Example of Geometrical Constructions ……………………… 121

Appendix XI: Summary of Emerging Themes……………….…………………… 123

LIST OF FIGURES AND TABLES

Tables:

Table 3.1: Data collection for the Intended Study………………………………… 26

Table 3.2: The themes for within-case analysis…………………………………….. 30

Figures:

Figure 4.1: One example of Jay’s geometrical construct with GeoGebra……………37

Figure 4.2: One example of Li’s exponential function constructs with

GeoGebra…………………………………………………………………… 42

Figure 4.3: One example of school mathematics website on the topic:

transformations……………………………………………………………… 45

Figure 4.4: One example of school mathematics website on the topic: angles in the

same segment……………………………………………………………….. 46

Figure 4.5: One example of linking algebra and geometry with GeoGebra by

Richard …………………………………………………………………… 46

Figure 4.6: One example of Tyler’s use of transformation activity on

‘enlargement’………………………………………………………………... 50

Figure 5.1: The general schema of teachers’ conceptions and practices integrating

GeoGebra…………………………………………………………………… 65

LIST OF ABBREVIATIONS AND GLOSSARY OF TERMS

AST: Advanced Skills Teacher

Cabri: Cabri-Geometry

CAS: Computer Algebra System

DGS: Dynamic Geometry Software

GSP: Geometer’s Sketchpad

ICT: Information and Communication Technology

IGI: International GeoGebra Institute

IT: Information and Technology

NCETM: The National Centre for Excellence in the Teaching of Mathematic

OECD: Organisation of Economic Cooperation and Development

OFSTED: Office for Standards in Education

PGCE: Postgraduate Certificate in Education

PISA: Program for International Student Assessment

TIMSS: Third International Maths and Science Study

‘GeoGebra has changed the way I teach’ Peter

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CHAPTER 1

Introduction

‘Algebra is concerned with manipulation in time, and geometry is concerned

with space. These are two orthogonal, aspects of the world, and they represent

two different points of view in mathematics.’ (Michael Atiyah)

1.1 Research Context

Algebra and geometry are two core strands of mathematics curricula throughout the

world and are considered the „two formal pillars‟ of mathematics (Atiyah, 2001). It is

therefore not surprising that they have been specifically targeted by the field of

technology (Sangwin, 2007). Many researchers consider mathematics education as

one of the earlier education fields to introduce technology as an assistant tool in

classrooms (Papert, 1980; Hoyles and Sutherland, 1989; Noss and Hoyles, 1996).

The dynamic and symbolic nature of computer environments can provoke

students to generalise and formalise and make links between their intuitive

notions of mathematics and the more formal aspects of mathematical knowledge.

(Godwin and Sutherland, 2004:131-132)

The major application of technology in mathematics education is the integration of

mathematical software in teaching practices. In respect of geometry, the most widely

used computer applications, known as Dynamic Geometry Software (DGS) and

include, Cabri-géomètre and Geometer‟s Sketchpad (GSP), etc. One important feature

of DGS is the drag mode, encouraging interactions between teachers, students and

mathematics (Jones, 2000). The drag mode can be used to explore and visualise

geometrical properties by dragging objects and transforming figures in ways beyond

the scope of traditional paper-and-pencil geometry (Laborde, 2001; Ruthven, 2005).

DGS also has options to visualise the paths of objects as they move. For algebra, the

most widely used applications are known as Computer Algebra Systems (CAS) and

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include programmes such as Mathematica, Maple and Derive. Some graphical

visualisation and symbolic representations of algebraic expressions are implemented

in CAS (Kokol-Vojic, 2003). CAS software processes algebraic variables, equations

and functions and provides immediate computations (Harris, 2000; Lavicza, 2007).

These provide opportunities for investigation and checking as feedback can be given

promptly which assists the learning of algebraic topics.

Using the metaphor of the two „formal pillars‟ of mathematics, geometry and algebra

are afforded prominent positions especially at the secondary level (Hohenwarter and

Jones, 2007). However, the connection between geometry and algebra, namely „the

beam connecting the two pillars‟, is apparently missing, as evident that in some

countries geometry and algebra are entirely separate in their curricula (ibid).

Since CAS and DGS are two completely different mathematical constructs, the

„beam‟ is weakly constructed within current mathematical software. Historically,

CAS programmes have mainly provided algebraic and numerical computations while

DGS have provided graphical and dynamic demonstrations. Hohenwarter and Jones

(2007) point out that „forms of CAS have begun to include graphing capabilities in

order to help to visualise mathematics; likewise, DGS have begun to include elements

of algebraic symbolisation in order to be useful for a wider range of mathematical

problems‟(p. 127) . In recent years, the need to integrate CAS and DGS has become

apparent as Schumann and Green (2000: 337) claim that „[t]here is a need for further

software development to provide a single package combining the desired features [of

DGS and CAS]‟. The recently published software GeoGebra by Markus Hohenwater

(2004) explicitly links the two (as evidenced by the name Geometry and alGebra).

This integration aims to provide unprecedented opportunities for mathematics

education (Sangwin, 2007; Hohenwarter et al., 2007). GeoGebra affords a

bidirectional combination of geometry and algebra that differs from earlier software

forms. The bidirectional combination means that, for instance, by typing in an

equation in the algebra window, the graph of the equation will be shown in the

dynamic and graphic window. Similarly, by dragging the graph, the equation changes

accordingly (Hohenwarter and Fuchs, 2004). A closer connection between the

visualisation capabilities of CAS and the dynamic changeability of DGS is therefore

offered by GeoGebra (ibid).

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1.2 Rationale and Significance of the study

GeoGebra has the potential to clearly demonstrate to students the close connection

between geometry and algebra and is becoming a recognised part of mathematical

knowledge (Jones and Edwards, 2006; Hohenwarter and Jones, 2007). Edwards and

Jones (2006: 30) believe a significant feature of GeoGebra is „its activities which

require high-level thinking and enable pupils to engage with the potential that

technology brings, such as learning through feedback, seeing patterns, making

connections, working with dynamic images, etc‟.

In addition, teachers and students can download and use GeoGebra at home as it is

open-source1 software. This outperforms commercial software such as GSP, Cabri-

géomètre or Autograph, which offer similar affordances (Lu 2007; Hohenwarter and

Preiner, 2007). There is therefore a growing belief among international mathematics

educators that GeoGebra has the potential to transform mathematics education

(Sangwin, 2007; Jones and Edwards, 2006). It must not be forgotten, however, that

teachers play a vital role in the enhancement of learning as they are the gateway to

larger cultures of knowledge, and no amount of technology will replace teachers in

this respect (Sutherland et al., 2004).

There is evidence of GeoGebra being used extensively around the globe; it has been

translated into forty languages and has been used by approximately a hundred

thousand teachers worldwide (Hohenwarter and Lavicza, 2007). However, systematic

enquiries into the effectiveness of GeoGebra in teaching practices are limited.

Consequently, this study aims to provide one of the first rigorous accounts of this

potentially liberating software and how it can support or enhance mathematics

teaching.

1 Cross-platform open source tools and collaborative software provides educators opportunities to join

an online community and overcome technological and financial barriers. All materials in this

environment are subject to a Creative Commons license that allows everyone to make customized

works for non-commercial purposes (Hohenwarter and Preiner, 2007).

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There is a noticeable demand for the pedagogical development of technology

implementation in the teaching of geometry and algebra (Ruthven, 1990; 2002; 2005;

2008; Artigue, 2002, Sutherland et al., 2004; Hennessy, et al., 2005; Laborde, 2007).

My rationale behind carrying out this inquiry into GeoGebra is not only due to its

being open-source software with freely available support and online materials (Suzuki,

2006), but also due to its unique capacity to integrate geometry and algebra. The

significance of this research is not only the investigation of how GeoGebra usage can

be incorporated into the teaching of either geometry or algebra alone, but more

importantly, how the teaching of geometry and algebra can be linked using GeoGebra,

thus contributing to a better understanding for students of their interrelationships.

Studies such as this one will contribute to knowledge of GeoGebra-mediated teaching

and the future pedagogical development.

Recent research has indicated that culture influences the ways that teachers behave

and inter-culture differences appears to be stronger than intra-culture differences

(Schmidt et al., 1996; Givvin et al., 2005; Andrews, 2007). In particular, comparing

eastern and western traditions with their respective Confucian and Socratic

underpinnings can be enlightening as there are great differences in teacher beliefs and

practices (Leung, 1995; Tweed and Lehman, 2002; Andrews, 2007). There is little

comparative research of technology use in mathematics education, especially between

Eastern Asian and Western countries (Graf. and Leung, 2001). Consequently, seeing

how culture influences technology-mediated mathematics teaching in England and

Taiwan is a pertinent issue. The comparisons between the two countries will help

obtain a sense of the uniformities and dissimilarities of GeoGebra use. In so doing, I

decided to study the transformative potential of this software and its multiple uses as

well as providing further recommendations for its improvement in teaching practices.

1.3 Outline

Chapter 2 presents a review of theoretical and research literature illuminating the use

of technology in mathematics education. It also highlights teachers‟ conceptions and

practices of new technologies for their teaching of geometry and algebra. Chapter 3

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states the philosophical stance and methodology employed to conduct the study. It

also gives an account of the rationale for research design and the methods for data

collection and analysis undertaken on the issue of Taiwanese and English teachers‟

ways of using GeoGebra. Presentation and analysis of the multiple-case study are

reported in Chapter 4. Results and findings of the study followed by discussion,

reflection and implications for further research are presented in the final chapter.

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CHAPTER 2

Literature Review

2.1. Introduction

In this chapter, I firstly review the general role of technology in education and its

contribution to mathematics education, followed by an exploration into the ways in

which mathematical software is used to support mathematics teaching. Secondly, I

give an account of the relationships between teachers‟ conceptions and practices in

relation to their software usage. Thirdly, the decision for a cross-cultural approach is

described by looking at comparative mathematics education. In the final part, I focus

on the research questions extracted from the paradigm used.

2.2 The Role of Technology in Mathematics Education

2.2.1 Technology Integration in Education

There has been an increasing awareness that interactions between humans and

technologies can facilitate effective teaching and learning (e.g. Hennessy et al., 2005;

Arcavi, 2003). During the 1990s, Information Technology (IT) was a term reserved

for computers and other electronic data handling and storage devices used to provide

speedy automatic functions, capacity and range (Monaghan, 1993; Andrews, 1996).

More recently, the word „communication‟ was incorporated to acknowledge the

increase in interaction between people and technology; this is widely known as

Information and Communication Technology (ICT) and is a term extensively used in

the UK. Kennewell (2004: 4) explains that „the term ICT covers all aspects of

computers, networks (including the internet) and certain other devices with

information storage and processing capacity, such as calculators, mobile phones and

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automate control devices‟. I, therefore, use the term ICT to refer to new technologies

with an emphasis on communication.

In the context of education, ICT integrates teaching and learning as a complete

activity, which has a number of features. Kennewell (2004) points out that some key

features that ICT can offer in this respect are speed and automatic functions, capacity,

range and interactivity. Deaney et al. (2006: 465) identify teachers‟ ‘practical theory’

concerning the contribution of ICT to education as:

Broadening classroom resources and references;

Enhancing working processes and produces;

Mediating subject thinking and learning;

Fostering more independent pupil activity;

Improving pupil motivation towards lessons.

The „practical theory’ could be seen as a starting point for the development of explicit

models of ICT into different subject teaching and learning. Nevertheless, after

decades attempting to incorporate technology in education, it is still problematic

(Cuban et al., 2001). A pervading notion suggests that ICT alone cannot enhance

learning (Godwin and Sutherland, 2004; Noss and Hoyles, 2003). In light of this issue,

the constraints preventing some teachers from using ICT to help students learn has

been examined (Sutherland et al., 2004). Four barriers to implementing ICT in the

classroom are pinpointed as (Steen, 1988; Brown, 2001):

Accessibility of computers: teachers and students need ready and regular

access to computers;

Openness of programmes: they must be easily available and the emphasis is

on learning the mathematics rather than the programme;

Curriculum scope: teachers‟ pedagogical practices through using ICT along

with the curriculum need to be accomplished more effectively;

ICT competence: individual teachers‟ level of confidence and skills in ICT.

Consequently, it has been suggested that further areas for development in terms of the

contribution that ICT lends to education include: improvements in pedagogical

8

development and teacher training of ICT competence (Ofsted, 2004; Johnston-Wilder

and Pimm, 2005; Webb and Cox, 2004; Gülbahar, 2007).

2.2.2 An Overview of Technology Use in Mathematics Teaching

With the introduction of ICT to mathematics education, one question to consider is

whether mathematics education changes when ICT is introduced? Hershkovitz

and Schwartz (1999) research the differences between ICT-integrated environment

and paper-and-pencil environment and suggest that the paper-and-pencil environment

is relatively passive in supporting learning. Current studies have found that there are

changes in terms of active engagement with the implementation of ICT into

mathematics education as ICT holds higher efficiency in mathematics manipulation

and communication as well as interactivity between teachers, students and

mathematics (Hershkovitz, et al, 2002).

Nevertheless, the paper-and-pencil environment has simplicity and convenience that

cannot be ousted from classroom practices. It can be argued that inappropriate uses of

ICT may potentially block teaching and learning processes in problem-solving and

justifying, or perhaps create cognitive obstacles in understanding (Yerushalmy, 2005;

Arzarello, 2005). Bramald et al. (2000) concur and warn against underestimating the

effect of the personal relationship existing between teacher and student. They cite it as

an important factor in successful educational development. Since ICT and paper-and-

pencil environments both have advantages and disadvantages, it is not necessary to

separate them but to combine them. Considering the integration of both ICT and

paper-and-pencil can be beneficial; the implementation of ICT into mathematics

education has been the main direction of current research in the field of mathematics

education and ICT (Ruthven et al. 2008; Sutherland et al., 2004; Becker, 2001; Cuban

et al., 2001).

Despite official encouragement and enormous investment across the developed

world, the global movement to integrate digital technologies into school

mathematics has had limited impact on mainstream classrooms (Ruthven et al.,

2008:1)

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Since the implementation of ICT in classroom practices has been slow, recent studies

shift their attention to the role of the teacher as a mediator for appropriate integration

of ICT into teaching practices (Becta, 2004; Ruthven et al., 2008; Sutherland et al.,

2004). Teachers‟ pedagogical knowledge in the use of ICT to bolster students‟

learning requires them to tackle potential problems (Ofsted, 2004). Possible

misunderstandings may arise from multiple representations within the software, or

improper use of ICT to investigate mathematical ideas (Deaney et al., 2006). In effect,

there is a constant demand for teachers‟ pedagogical development of embedding ICT

into everyday classroom practice to occur (Godwin and Sutherland, 2004; Hennessy

et al., 2005; Kendal and Stacey, 2001). Consequently, one of my main focuses is

researching teachers‟ instructional practices incorporating technology.

2.2.3 Cross-Cultural Studies on Technology and Mathematics Teaching

Since there is evidence that each educational system has a different approach to

mathematics education (Schmidt et al., 1996; Stigler and Hiebert, 1999), one crucial

question to address is „what precisely is meant by mathematics education?‟ in

different cultural contexts. Would it be possible to address multiple definitions of

mathematics education by comparing and contrasting different cultural traditions and

approaches to mathematics?

To all intents and purposes, cross-cultural studies usually refer to comparative studies

(Kaiser, 1999a). Osborn (2004: 265) argues that „comparative approaches which

combine careful measurement with up-close, deep understanding of real-world

contexts, can be a very powerful mix‟. The most crucial reason for conducting such

cross-cultural research is that it may contribute towards improving the approaches to

mathematics teaching, thereby providing a better understanding of the context of

different environments (Conway and Sloane, 2005). The contribution of cross-cultural

studies is processed as „a means of conveying in a powerful and compelling form

significant applied and theoretical insights across a range of disciplines and

professional fields’ (ibid: 13).

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There are large-scale quantitative studies such as TIMSS2 and PISA

3 and small-scale

qualitative studies, for example, Andrews and Sayers‟ (2004) comparative research in

five European countries. These studies highlight both similarities and differences

between mathematics education in different cultural contexts in depth and in breadth.

Large scale surveys are limited, however, by the fact that they often compare

students‟ academic achievements without taking cultural and social factors into

consideration (Prais, 2007). Quantitative studies such as TIMSS have also been

reproached for their uncritical evaluation and for promoting globalisation over

curricular and cultural diversity (Andrews, 2007). In contrast, small qualitative studies

acknowledge cultural differences without attempts for generalisation. Particularly,

when comparing East Asian and Western traditions with their respective Confucian

and Socratic underpinnings, there is a significant difference between what are

classically designed with the educational traditions (Leung, 1995; Kaiser et al., 2005;

Tweed and Lehman, 2002). In particular, Kaiser et al. (2005) proposed a framework

analysing East Asian and West European cultural traditions in mathematics education.

The framework is listed as follows:

Understanding of mathematical theory- scientific knowledge versus

pragmatic understanding

Organisation by subject structure versus spiral-type curriculum

Introduction of new mathematical concepts and methods

The position and function of proofs

Focus on justifications or rules versus work with examples

The role of precise language

The role of real-world examples

Teaching and learning styles

The framework by Kaiser et al. is adapted partially in terms of teaching styles as I

undertake a small-scale qualitative study in countries that exemplify East and West

with a focus on teachers‟ perspective and their use of technology in mathematics

teaching. The Eastern country chosen is Taiwan since it is viewed as „the one most

often cited admiringly by educators in the West for the level of its students‟

2 TIMSS- Third International Maths and Science Study

3 PISA- Program for International Student Assessment

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educational achievements (Broadfoot, et al., 2000: 13)‟ and a high mathematics

performing country in international comparative studies such as TIMSS and PISA

(Mullis, 2003; OECD, 2004; 2007). The Western country chosen for the study is

England due to its contrasting educational system (Broadfoot et al., 2000).

2.2.4 England and Taiwan- two opposite systems?

Taiwan and England are at two ends of a value and beliefs continuum, as Taiwan is

influenced by the Confucian-heritage culture of learning (Wong, 2004), in contrast

with English Socratic tradition. Taiwanese core educational values are centred on

Collectivism, based on a teacher-centred classroom culture, whereas in England,

Individualism seems to predominate around a student-centred ideal (Osborn et al.,

2000; Hofstede, 1986). Jacques (1996) compares Taiwanese and English educational

systems pointing out that the Taiwanese educational system has: „a commitment to all

children succeeding which means that, unlike Britain, there is no trailing edge of

failure (p.1)‟. Despite Taiwanese students‟ high level of apparent achievement in

response to a particular teaching style or curriculum emphasis, I would argue that the

influence of a rigorous exam-oriented training and rote-learning culture should also be

taken into account.

It appears that the awareness and perceptions of using technology in mathematics

education varies from nation to nation as education is influenced by the social

environment, economic state and historical background as well as by cultural

traditions (Graf. and Leung, 2001). Technology has been integrated in different ways

internationally, often due to the availability of financial funding. Taiwan is one of the

„Asian Tiger‟ economies (Jacques, 1996). Provided that both countries in my study

have sufficient finances for computer technology, investigating the ways in which it is

used differently can be significant.

The use of technology in Taiwan is relatively scarce as it has not been developed and

researched to as great a degree as in England. Taiwanese research shows little

awareness of technology practicability in secondary schools (Hung and Hsu, 2007),

except that, GSP is the most widely used mathematical software for secondary school

mathematics (Yen, 2003). Compared with England, the use of technology in Taiwan

12

seems underdeveloped in terms of professional training, curriculum and educational

policy in teaching practices.

The notion of technology usage as a provider of information in the teaching of

mathematics has become evident in a significant number of countries cross the world.

After surveying the literature of cultural studies, I contend that not enough

comparative research in technology and mathematics education has been done despite

a few examples of research (e.g. Kyriakidou et al., 1999; Graf. and Leung, 2001).

Consequently, there is also a lack of theoretical frameworks for cross-cultural studies

on teachers‟ uses of technology in Eastern and Western cultural contexts.

2.2.5 Teachers’ beliefs, conceptions and practices

To investigate the ways in which teachers use technology, it would be necessary to

study how their views, attitudes, beliefs, or conceptions influence their practices as

research suggests that teachers‟ conceptions are crucial factors with regard to their

practices (Thompson, 1992). Teachers‟ behaviour and choices of technologies can be

seen related to their attitudes, conceptions and beliefs; accordingly, this would

influence their technology integration into instructional practices. Therefore, an

understanding of teachers‟ conceptions in relation to their practices is necessary in

order to allow a reflective „transfer‟ of effective measures from one system to another

(Kaiser, 1999b; Andrews, 2007). There have been studies examining the relationships

between teachers‟ beliefs of mathematics and their instructional practices which

indicate that they are related in a complex way (Thompson, 1984; Ernest, 1989).

There are factors, such as cultural underpinnings of practices and sources of cultural

and educational traditions that influence mathematics teaching, teachers‟ beliefs,

conceptions and practices. However, the complexity of terms in relation to beliefs,

such as conceptions, attitudes and views are difficult to delve into, therefore,

researchers have organised beliefs into belief systems which can be called

conceptions (Nespor, 1987; Ernest, 1989). Thompson (1992) defines conceptions as

„conscious or subconscious beliefs, concepts, meaning, rules, mental images and

preferences (p. 132)‟. I adopt the term „conceptions‟ providing clarity of the

13

inclusiveness of terms. In the following, I examine literature in relation to teachers‟

instructional practices with technology.

2.3 Teaching algebra and geometry with Technology

Providing immediate graphing and calculation to benefit mathematics teaching and

learning are some of the merits of technology (Papert, 1996; Hoyles and Jones, 1998;

Noss and Hoyles, 1996). Wright (2005) asserts that ICT, particularly mathematical

software, helps to provide better visual and dynamic representations of abstract ideas

and the links between symbols, variables and graphs. Consequently, I have chosen to

investigate the ways in which teachers gain from using mathematical software in their

teaching practices. In the following sections, I discuss mathematical software used for

teaching geometry and algebra, followed by an account of open-source software and a

combination system of geometry and algebra. I then position my research and identify

research questions.

Generally, there are several types of software used in the teaching of mathematics:

Computer Algebra System (CAS), Dynamic Geometry Software (DGS) such as GSP,

Cabri-géomètre, and open source software- Java Applets, GeoGebra, etc. (Laborde,

2001; 2003; 2007; Strässer, 2001; Kokol-Voljc, 2003). Each form is generally

associated with particular aspects of mathematical teaching and learning. For example,

CAS is often used for teaching algebraic topics, whilst DGS programmes are used for

geometrical topics. CAS focuses on manipulation of expressions and DGS

concentrates on relationships between points, lines circles and so on (Schneider,

2007). However, such distinctions are not always clear with considerable overlap due

to the duality of mathematics in terms of geometry and algebra. Schumann and Green

(2000) state that „graphical, numerical and algebraic should not be considered

separate, but rather as constituting a holistic comprehensive computer-aided

approach‟ (p.324). The awareness of integrating graphical, numerical and algebraic

representations has become noticeable in recent years.

14

Pederson (2004) claims that „geometry is a skill of the eyes and the hands as well as

the minds‟. There are more visual and dynamic areas in geometry than in algebra.

Since mathematical software offers great visualisation capability and dynamic

changeability for teaching, it is well placed to support this important element. The

properties of DGS and the ways in which it supports learning are demonstrated in the

following.

Dynamic geometrical constructions, visualisation for motions of objectives by

dragging and investigation under various angles are some affordances of DGS

(Laborde, 1998; Osta, 1998; Olivero, 2003; Healy and Hoyles, 2001). Laborde (1998)

emphasises the properties of DGS on a „real‟ model for the theoretical field of

Euclidean geometry where it is possible to handle theories in a physical sense. The

feedback of diagrams resulting from the use of geometrical primitives is also a vital

component of DGS. There are a myriad of opportunities offered by DGS. These are:

the direct interaction with the tools provided by the system that allows construction,

manipulation and exploration of figures and discovery of the relationships between

multiple representations. Also, the essential features of DGS are efficiency in

mathematics manipulation and communication for learning. Furthermore, the efficient

coupling of visual representation with other forms of representations and interactivity

between students and mathematics can enhance learning (Healy and Hoyles, 2001).

DGS is not only for teacher demonstrations but also for students‟ interactive learning.

Potentially, some mathematical software programmes offer algebraic and numerical

computations and symbolic representation providing the linkage between multiple

representations. This sort of mathematical software, known as Computer Algebra

System (CAS), include: Derive, Mathematica, Maple, etc (Fey et al., 1995; Ruthven,

2002; Kendal et al., 2005). CASs can work with strings of symbols enabling students

to concentrate on developing their conceptual understanding of mathematics (Keller

and Russell, 1997; Shaw, 1997; Cuoco, 2002). Ruthven (2008a:1) argues that „when it

comes to mainstream use of these technologies, the uptake of dynamic geometry has

been wider and more longstanding…similar studies are currently lacking for computer

algebra.‟ This can be explained by Artigue (2002:1): „professional mathematicians

and engineers know that these sophisticated new tools do not become immediately

efficient mathematical instruments for the user: their complexity does not make it

15

easy to master, and fully benefit from, their potential‟. Yerushalmy (2004) mentions

the transformation of the content of algebra curricula by technology as this new

digital culture may shift teachers‟ pedagogical strategy. However, „the transitions

between fundamental concepts and operations remained the difficult and non-trivial

parts (Yerushalmy, 2004: 19)‟ and still needs further research. Ruthven (2008a: 1)

researches the specific examples of computer algebra and dynamic geometry, and

highlights „three important dimensions- interpretative flexibility, instrumental

evolution and institutional adoption-of the incorporation of new technologies into

educational practices‟. The interpretative flexibility of technologies refers to „varied

conceptions of technology‟s functionalities and modalities of use (ibid)‟. The

instrumental evolution of scenarios is categoried as four types: a convenient parallel

to paper-and-pencil, invariant properties through visual salience under dragging, new

types of solution to familiar problems and posing novel forms of problem. The

institutional adoption means that the official curriculum should show explicit

recognition and provide the instrumental genesis of manual tools. These three

dimensions cover the major issues concerning the incorporation of new technologies

into mathematics teaching.

Although research into current technology use of computer algebra and dynamic

geometry in teaching practices separate each sphere into distinct areas for study; I

argue against this separation as there are areas overlapping algebra and geometry such

as functions and graphs (Dubinsky and Harel, 1992). Examining both together has

great educational implications and the connections between the two should not be

ignored (Edwards and Jones, 2006). However, there is a gap in the literature dealing

with this linkage between both fields and the use of technology. Despite an awareness

of the need for a combination of DGS and CAS (Hohenwarter and Fush, 2004),

software designers struggle to combine them as there are completely different

constructs in software design. GeoGebra could be seen as pioneering software,

although whether or not it is successful in linking DGS and CAS still needs research

as the supporting evidence is limited at present.

One problem is that most mathematical software in mainstream use is commercial,

which means the availability of software is subject to the school or student‟s finances.

Therefore, some teachers or students who cannot afford to buy commercial software

16

search for free, open-source software for their own purposes. The wide-spread usage

of open-source software is beyond researchers‟ awareness since Hippel and Krogh

(2003) explain that „the phenomenon of open-source software development shows

that users programme to solve their own as well as shared technical problems, and

freely reveal their innovations without appropriating private returns from selling the

software‟. There is positive potentiality and improvement offered by encouraging a

collaborative community of open-source software users and voluntary software

developers.

2.3.1 Geometry + Algebra= GeoGebra?

Hohenwarter (2004) developed GeoGebra with the intention of supporting secondary

mathematics teaching by bridging students‟ understanding of the connection between

geometry and algebra. GeoGebra is a multi-platform dynamic mathematical software

with its window divided into two parts (Fig. 2.1, Hohenwarter, 2006) - „Algebra

window‟ (left side) and „Geometry and Graphics window‟ (right side).

Figure 2.1: GeoGebra window- Algebra window and Geometry and Graphic window

Algebra Window

Geometry and Graphics Window

17

On the one hand, GeoGebra is a dynamic geometry system, much like any other,

which works with points, vectors, segments, lines, and conic sections. On the other

hand, equations and coordinates can be entered directly into the grid at the bottom of

the window (Fig. 2.1). It provides a bidirectional combination and a closer connection

between visualisation capabilities of CAS and dynamic changeability of DGS.

Although most research attention on GeoGebra pertains to the teaching of geometry,

GeoGebra has great potential in the teaching of algebra which lies mainly in functions

and graphs. Functions can be defined algebraically and then changed dynamically

afterwards (Sangwin, 2007). For example, by entering the equation y=x2 the

corresponding graph can be seen directly. The visualisation of two windows provides

a connection between algebraic and geometric representations. It also works the other

way around, by dragging the line or curve of the graph to change the equation. The

change in the equation can be seen on the algebraic window. This encourages the

investigation of the connection between variables in the equations and graphs in a

bidirectional experimental way (Hohenwarter and Preiner, 2007). This is particularly

significant as it connects the crucial parts of multiple representations of mathematics,

which are numerical, algebraic, geometrical and graphical; far beyond the reach of

other DGS and CAS.

GeoGebra being open-source software may face criticism as it may be thought that

free software lacks quality control compared to commercial software. Acknowledging

that it would be insufficient to only provide free software without proper training and

collegial support, the International GeoGebra Institute (IGI)4, therefore, is organised

for supporting the collaboration between teachers and researchers and provides

professional development for teachers (Hohenwarter and Lavicza, 2007). Since it is a

non-profit organisation, funding has been sought mainly from Europe and the U.S.

(Hohenwarter et al., 2008). Teachers need a support system and professional

development to improve their skills in teaching mathematics using GeoGebra

(Hohenwarter and Preiner, 2007). With this guidance and support from IGI,

GeoGebra enhances teachers‟ willingness to integrate this new technology into their

teaching practices. Despite its important ramifications, there has been little research

4 IGI is a virtual, not-for-profit organization which has established the following three goals: training

and support, development and sharing, and research and collaboration.

18

into this area. It is hoped that this cross-cultural study will contribute to the IGI

development of GeoGebra implementation in mathematic teaching in terms of

pedagogical strategies and innovative ways of using GeoGebra in classroom practices.

Nevertheless, one might ask the question: „does GeoGebra offer sufficient linkage

between geometry and algebra?; does it provide both functionalities of DGS and

CAS?‟ I, therefore, aim to explore whether GeoGebra offers linkage between

geometry and algebra in teachers‟ practices.

2.4. Statement of Research Questions

A cross-cultural study between Taiwan and England will help obtain a sense of the

commanalities and discrepencies of teachers‟ conceptions and practices in relation to

GeoGebra use. I have chosen to research at the upper-secondary level (students aged

15-18) as this level is less researched but is a crucial step for bridging students‟

secondary mathematics learning and higher education. Therefore, the overarching

research questions are:

What are the upper-secondary mathematics teachers‟ conceptions of

technology in relation to GeoGebra in England and Taiwan?

In what manner is GeoGebra used for the teaching of geometry and algebra by

Taiwanese and English teachers?

How are the teachers‟ conceptions of technology and GeoGebra related to

their teaching practices in both countries?

19

Chapter 3

Research Methodology

3.1 Introduction

The previous chapters were concerned with literature examining the use of technology

in mathematics education; particular attention was paid to the teachers‟ perspective of

implementing mathematical software into classrooms. Also highlighted in the

literature was the apparent need for research on open-source software and possible

assistance of a cross-cultural approach. Integrating the literature review and research

questions, I outline my decisions with respect to my research design to address these

issues. The research design aims to investigate how the use of open-source software

supports the teaching associated with the links between geometry and algebra. In this

chapter, I introduce the theoretical framework used for research design, epistemology,

theoretical perspective, methodology and methods followed by a discussion of

research considerations.

3.2 Theoretical Framework

The research process often follows a certain path, beginning with the problem to be

solved or an issue, which then becomes the core concern of the study. This is

followed by identifying research questions, reviewing the literature, choosing research

methods, developing the research design, collecting data, analysing the data,

interpreting the results, formulating conclusions and identifying implications (Robson,

2002; Cohen et al., 2007; Creswell, 2007).

20

Theoretical frameworks for methodological rationale can be used to assist researchers

in highlighting and structuring the range of approaches, and the methods to

investigate their research questions. Crotty (1998) proposes a general framework

containing four elements: epistemology, theoretical perspective, methodology and

methods. Epistemology deals with „how education researchers can know the reality

they wish to describe… the belief they have about the nature of that reality‟ (Scott and

Morrison, 2005: 84). The theoretical perspective here means „the philosophical stance

lying behind the methodology‟ (Crotty, 1998:66). Methodology holds the assumption

that a researcher conceptualises the research process in a certain way (Creswell, 2007)

and detailed procedures of data collection, analysis and writing, are called methods

(Creswell, 2003). This framework allows careful development of research processes.

These elements help researchers answer how their philosophical underpinnings,

epistemology and theoretical perspectives relate to methodology and methods.

3.3 Epistemology and Theoretical Perspective

Epistemology helps researchers make sense of research information transforms it into

data detailing how that analysis might be patterned, reasoned, and compiled and

shows the belief they have about the nature of the reality they describe (Willis, 2007;

Creswell, 2007; Scott and Morrison, 2005). In the following, I provide a brief

description of the philosophical theories and discuss the justifications of my

theoretical perspective.

Crotty (1998) points out „the great divide‟ between objectivist research and

constructionist/subjectivist research. Acknowledging there is a debate against this

divide (Howe, 2003), my discussion of epistemology still targets these two strands as

they are representative of the mainstream research. On the one hand, positivism

claims that knowledge exists whether we are conscious of it or not (Crotty, 1998;

Cohen et al., 2007). Post-positivism has a commitment to objectivity but is

approached by recognising that reality or knowledge can only be known imperfectly,

and researchers‟ biases yield limitations in the production of knowledge (Philips and

Burbules, 2000; Robson, 2002). My research focuses on the usage of mathematical

21

software by teachers to help construct meanings for students. Reflecting on my

research questions, I inquire as to how technology is used by teachers in their teaching

practices; therefore, I do not take the position of assuming that knowledge is

independent from social construction. The positivist and post-positivist

epistemologies are not applicable to my research as it is my belief that knowledge

exists through human interactions with social environment and technologies.

On the other hand, constructionist and subjectivist epistemologies acknowledge

different sets of beliefs. For instance, constructionism is the belief that knowledge is

constructed by people and we come to „know‟ through our interactions with others;

each one of us constructs his or her own knowledge based on a unique set of

experiences with the world (Bassey, 1999). Social constructionism, more specifically,

is the understanding of the world and each other as socially constructed through our

interaction with the environment (Crotty, 1998; Robson, 2002; Creswell, 2007).

Interpretivism is a quest to generate understanding of the subjective world of human

experiences. In symbolic interactionism, people ‘act’ on the basis of the meanings and

understandings that they develop through group actions and interactions (Blumer,

1969; Crotty, 1998; Cohen, 2007).

Due to the research focus being the subjective nature of various perceptions of

teachers‟ use of technology; I have not attempted to claim to be an objectivist. I see

knowledge as constructed within interactions that people have with other people or

social environments. In my view, meaning and understanding of knowledge are

created, constructed and negotiated rather than told, given or shown. Thus, humans

construct meaning in engaging with the world through their interpretation of it. In

taking a constructionist epistemology, I adopt an interpretivist theoretical perspective

with a view of symbolic interactionism as I look at teachers‟ accounts of mathematics

teaching through technology, as teachers „act‟ on their understandings and beliefs

about the use of technology in classroom practices.

I agree with the ontological assumption that reality is subjective and multiple as

evidenced by participants‟ responses in the study (Creswell, 2007). However, I do not

intend to find evidence of multiple realities on multiple quotes from individuals to

present different perspectives. Ontological questions such as: „what is the nature of

22

mathematics‟ and „what is the pedagogy of mathematics‟ are not the focal point of my

research, but rather the epistemological questions about „how technology can help to

construct an understanding of mathematics‟ and „how GeoGebra can be used

interactively to scaffold the construction of mathematics knowledge‟. In an attempt to

discover the answers to my research questions, I have tried to minimise the „distance‟

or „objective separateness‟ between those being researched and myself (Guba and

Lincoln, 1988: 94).

3.4 Methodology

3.4.1 Selection of Appropriate Research Approach

Before discussing the methodology, I revisit my research questions in the light of the

decision concerning theoretical perspective. The questions informed by the literature

are: „What are the upper-secondary mathematics teachers‟ conceptions of technology

and GeoGebra in England and Taiwan? ; in what manner is GeoGebra used for the

teaching of geometry and algebra by Taiwanese and English teachers? how are the

teachers‟ conceptions of technology and GeoGebra related to their teaching practices

in both countries?‟

In order to answer these questions and select an appropriate research approach, I

begin with an investigation of three approaches to research suggested by Creswell

(2003), namely qualitative, quantitative and mixed methods. Quantitative research and

statistical data mainly provide the knowledge about what is happening, rather than

why or how (McKnight, et al., 2000). Mixed methods research considers the

knowledge for both what is happening and why or how and could be considered for a

large-scale project as both breadth and depth of a research topic can be examined.

These approaches are not applicable to the study as my inquiry of knowledge is based

on ‘how’ questions in a small scale approach. My proposed research lends itself to the

qualitative approach, as it tends to focus on a single concept or phenomenon, bringing

personal values into the study, studying the context or settings of participants and

validating the accuracy of findings (Creswell, 2003).

23

Yin (2003) states that case studies are the preferred strategy when how or why

questions are being posed, when the investigator has little control over events, and

when the focus is on a contemporary phenomenon with some real-life context.

Amongst all qualitative research traditions, a case study would fit best with my

methodology in relation to the ‘how’ research questions I have proposed.

Consequently, I have chosen to use case study research, as I do not intend to study

historical or phenomenological perspectives of the research topic.

3.4.2 Case Study Research Approach

Case studies examine the particularity and complexity of a bounded system, single

case or multiple cases over time (Stake, 1995; Bassey, 1999). The method involves

„the detailed, in-depth data collection and recording of data about a case or cases,

involving multiple sources of information rich in context‟ (Creswell, 2007: 73). Stake

(2006) categorises two types of case studies: single-case and multiple-case studies. A

single case study can be seen as a single scrutiny bounded by time and activity that

necessitates the collection of detailed information (Merriam, 1998). Multiple-case

studies are special efforts to examine something having a number of cases, parts or

members when four to fifteen cases are involved (Stake, 2006) as a larger number of

cases might require different methodology to tackle. They aim to answer specific

questions, involve an empirical investigation of a particular contemporary

phenomenon and seek a range of multiple sources of evidence (Robson, 2002). The

evidence has to be abstracted and collated to get the best possible answers to the

research questions (Stark and Torrance, 2000). My study examines teachers‟

mathematical software uses in Taiwan and England, and thus involves multiple-cases

rather than a single case. Multiple-case studies help obtain valuable information from

different cases between countries and therefore they are used as my main research

strategy.

As the rapid growth of technology use in secondary schools is a relatively modern

phenomenon (Hennessy et al., 2003; Laborde, 2003; Deaney, et al. 2006), there is

little research into the use of open-source software to guide this investigation. In view

of my research assumption that GeoGebra may be useful in upper-secondary

24

mathematics teaching, and by exploring these case studies, I can have a deeper

understanding of how GeoGebra is adopted by teachers. Thereby I comprehend the

applications for possible usage and affordances using the case study approach. Yin

(2003) writes that case studies can be exploratory, descriptive or explanatory. The

object of this study is framed into the exploratory model though elements of

explanatory and descriptive models are in use inside this context. In order to

investigate GeoGebra, which is a new tool, I emphasise that my multiple case studies

are not simply aimed at describing or explaining how GeoGebra can be used, but

rather exploring its potentialities in supporting mathematics teaching. Since there is

little research into GeoGebra usage to date, this study is exploratory (Marshall and

Rossman, 2006; Creswell, 2007).

In brief, exploratory and multiple-case studies are my chosen methodology as the

research focuses on this particular mathematical software, requiring specific teachers

who utilise GeoGebra to teach upper-secondary level mathematics. Comparing and

contrasting cases of teachers with interest in using GeoGebra from Taiwan and

England provide a comprehensive understanding of how GeoGebra can be used in

two very different cultural traditions, pedagogies and curricula.

I define mathematics teaching with the use of GeoGebra in Taiwan and England as

the two main units of analysis. These have embedded cases of teachers who use this

software. Moreover, within the units, four cases of English and Taiwanese teachers

are studied to obtain evidence of their views on GeoGebra teaching practices.

Studying teachers‟ use of technology in two countries invokes a particular

methodological response asking „what is comparable?‟ In order to compare,

conceptual, linguistic, measurement and sampling equivalence must be ensured

(Warwick and Osherson, 1973, cited Osborn, 2004). To achieve the comparability

between cases and units, pre-determined themes: teacher background, views on

technology and GeoGebra, software comparisons and ways of using GeoGebra have

been set for research design and data collection which are illustrated as follows.

25

3.5 Methods

3.5.1 Data Collection

The emerging issues of my research are teacher conceptions of mathematics teaching

and learning in relation to GeoGebra and its usage in teaching geometry and algebra. I

also consider teachers‟ conceptions of strengths and weaknesses of GeoGebra for

upper-secondary mathematics teaching both in Taiwan and in England. To delve into

my research questions comprehensively, I explore the reasons for, and ways of,

utilising GeoGebra. Following a qualitative research approach and multiple-case

study methodology, the data was collected mainly through interviews with teachers.

In seeking to gain insights into professional perceptions of the role of teachers for

integrating GeoGebra in practice, I was mindful of the opportunity for informative

interviews with GeoGebra creator and an advance user.

To gain a more in depth picture for each case, I espoused „data triangulation‟ (Denzin

and Lincoln, 2008) for different resources and information with respect to using

GeoGebra and teaching with the use of GeoGebra. This serves as a means by which

the weaknesses of one data collection technique could be compensated for by the

strengths of another technique. External information about technology facilities and

technological issues might also be investigated through observations, documentation,

informal conversations and e-mail communications with the teachers. The methods I

intend to use for data collection in my case studies are fully documented in Table 3.1.

26

Table 3.1: Data collection for the Intended Study

Data collection Participants Techniques Data Aim of data collection

Documentation Teachers GeoGebra-

related

websites,

teaching

materials.

Print-outs Teachers and

researchers‟ views in

respect to GeoGebra and

its teaching as well as

technical problems

Informal

communications

Teachers Emails and

GeoGebra-

related

websites

Field-notes To achieve feedback and

confirmation from the

teachers

Formal

interviews

Teachers Video

recording

Transcripts To provide insightful

and targeted evidence

directly on the case

study topic

Observations of

teachers‟

mathematics

constructs with

GeoGebra

Teachers Note-taking Field notes To obtain a more holistic

sense of the ways

teachers utilise the

software

Informative

Interviews

GeoGebra

creator and

advanced

users

Video

recording

Transcripts To obtain in-depth

evidence of practical

usage of the software

from international

perspectives and

disciplines

3.5.2 Interviews

Interviews have a „central importance in social research because of the power of

language to illuminate meaning‟ (Legard et al., 2003: 139) and can provide access to

the meanings people attribute to their experiences and social worlds. Therefore,

27

interviews are the focus of my research and the major method for data collection.

However, interviews can take different forms for various purposes. This next section

discusses these in relation to my research questions.

There are a number of interview types with different terms used. For instance, Denzin

and Lincoln (2008) categorise interviews as structured and unstructured interviewing,

group interviewing, creative interviewing, post-modern interviewing, gendered

interviewing and electronic interviewing. Bogdan and Biklen (1992) add semi-

structured interviews. I offer a summery of these types of interviews with their

strengths and weaknesses before explaining why I dismiss them as inappropriate.

Unstructured interviews have questions that emerge from the immediate context and

are asked in the natural course of discussion with no predetermination of question

topics or wording (Cohen et al., 2007; Patton, 1980). They allow for the salience and

relevance of questions but are less systematic. However, the interview flexibility in

sequencing and wording questions can result in substantially different responses, thus

reducing the comparability of responses. Structured interviews use predetermined

questions and fixed response categories. Data analysis of this sort of interview data is

simple but may be perceived as impersonal, irrelevant and mechanistic (Cohen et al.,

2007). I have chosen not to use these approaches, as they do not fit the nature of my

exploratory and multiple-case study. I also require the data not only to be comparable

between cases but also exploratory which means there is space for interviewees to

express their thinking without any influence by directive interview questions.

Consequently, semi-structured interviews can be useful for my research as wording of

opening questions can be determined in advance (Patton, 1980). As the interviews

were conducted in Taiwan and England, translation of wording might create problems

in respect of reliability. To increase the comparability of responses, the use of exact

wording in interview questions can reduce this concern. Pre-determined set questions

also reduce interview effects and bias and facilitate organization and analysis of the

data (Cohen et al., 2007).

28

There are some strategies for achieving depth of interviews through the use of content

mining questions (Legard et al., 2003: 150). These include amplificatory probes5,

exploratory probes6, explanatory probes

7 and clarificatory probes

8 (Legard et al.,

2003). These interviewing strategies were employed to encourage participants to

elaborate on their thoughts. To ensure the interview data was being collected

appropriately with these strategies, I conducted pilot interviews with two teachers

prior to the formal interviews, to allow time for amendment of interview questions

and personal reflections. In the pilot study, I discovered that I could not gather useful

information when the teachers were not particularly skilful or experienced with

GeoGebra. I then decided to find teachers who have at least six months of experience

teaching with GeoGebra.

To get empirical experiences of GeoGebra usage from teachers, interviews assist in

grasping teachers‟ points of view and personal accounts through talking about the

software. One may argue that the teachers might not show their authentic experiences

with the software through talking alone. As an aid to communication during the

interviews, a laptop with GeoGebra was prepared for the teachers. They were invited

to demonstrate their thoughts and ideas about GeoGebra that came up in the

discussion. This observational method was an interview aid and helped me understand

the ways in which they use the software and its related teaching tasks or activities.

3.5.3 Research Settings and Participants

Before introducing the criteria for selection of participants, I give a brief description

of the Taiwanese Education system. In Taiwan, formal schooling starts at the age of

six and includes two six-year phases: elementary and high school. High schools

include junior high schools (students in the 12-15 age range) and senior high schools

(upper-secondary equivalent, students aged 15-18). In England, the upper-secondary

level often refers to post-16 education. To ensure the age equivalence, I decided to

5 Amplificatory probes are used to encourage interviewees to elaborate further by questions such as

„can you tell me a little more about …?‟ (Legard et al. 2003: 150) 6 Exploratory probes help to explore the views and feelings that underlie descriptions of behaviour,

events or experience and show the meaning that experiences hold for interviewees (ibid). 7 Explanatory probes are repeatedly sought for reasons by asking „why?‟ (Legard et al. 2003: 151).

8 Clarificatory probes explore issues in depth, which require a high degree of precision and clarity

(ibid).

29

choose teachers who teach in the 15-18 age range as the upper-secondary level in both

countries.

The data collection took place in one grammar school and one Village College in

England and senior high schools in Taiwan. A total of four cases were involved in this

study. GeoGebra is newly-published open-source software with online resources and

networking on the GeoGebra User Forum. These GeoGebra related websites,

conferences, workshops and seminars provided a great opportunity to gain access to

the participants in both Taiwan and England. My focused participants were teachers

who are enthusiastic about and skilled in the use of GeoGebra in teaching, as they

have experience and knowledge of GeoGebra‟s applications and limits.

3.5.4 Data Analysis

The use of multiple sources of evidence, with their strengths and weaknesses, is what

characterises a case study (Yin, 2003; Stake, 2006). A complete set of data was

collected from four school visits. All of the interviews were audio and video-recorded,

lasted for approximately an hour each and took place in classrooms using either a

laptop or a computer connected to an interactive whiteboard. Through observations

during the interviews the teachers demonstrated ways they utilised the software. The

interview data were collated and summarised for each of the four case studies. The

interview data was later transcribed (Full transcripts in Appendix I) according to the

predetermined themes for each case analysis.

To ensure the comparability, the framework proposed by Kaiser et al. (2005) was

partially adapted with a focus on teachers‟ perspective of the use of technology. I pre-

determined the themes (Table 3.2) as a framework for the within-case analysis (Stake,

2006; Miles and Huberman, 1994). The decision for pre-determined themes is

because it is vital to explore and describe individual cases before comparisons

between cases, especially cases from two different countries. After within-case

analysis, similarities and differences between cases were noted through cross-case

analysis (Appendix II, Stake, 2006). I used a mixture of a priori analysis (top-down)

in relation to my research questions and inductive analysis (bottom-up) that allowed

new categories to emerge from the cross-case analysis (Dey, 1993). The transcriptions

30

of these interviews were subsequently highlighted by themes, which helped several

categories to emerge from the data.

Table 3.2: The themes for within-case analysis (adapted from Stake, 2006, P. 43)

Key Themes Sub Themes Examples of Interview Questions

Theme 1:

Participant

Background

Teaching experience,

acceptance of and

participation with

technology

Student age and

achievement

How many years have you been teaching?

What grade or year of students have you

taught/ do you teach?

Theme 2:

Conceptions of

GeoGebra

Teacher conceptions of

technology and

GeoGebra, the teaching

in relation to geometry

and algebra

What to you think about GeoGebra?

Do you think it provides linkage between

geometry and algebra?

Theme 3:

Software

Evaluation

Strengths and

weaknesses of

GeoGebra,

comparisons with other

softwares

What advantages and disadvantages do

you think that GeoGebra has?

Theme 4:

GeoGebra Usage

Ways of using

GeoGebra and its

materials and websites

Reasons for the chosen

mathematical topics

when teaching with

GeoGebra

How do you use GeoGebra?

For which topics do you use GeoGebra to

teach?

Why and in what ways do you teach

them?

Do you use GeoGebra in bridging

geometry and algebra?

During the cross-case analysis, new categories emerged from the data according to the

themes using the constant comparison method (Glaser and Strauss, 1967). This

method has four distinct stages (Lincoln and Guba, 1985: 339):

1. comparing incidents applicable to each category;

31

2. integrating categories and their properties;

3. delimiting the theory, and

4. writing the theory.

The reason for my choice of constant comparative method is that „the qualitative

analyst‟s effort at uncovering patterns, themes, and categories is a creative process

that requires making carefully considered judgements about what is really significant

and meaningful in the data‟ (Patton, 1990: 406). This is an important aspect where I

tended to display my data in an organised and compressed way that allowed me to

make verifiable findings (Denzin and Lincoln, 2008). A sequence of procedures was

involved, firstly, according to Creswell (2003), I read through the interview

transcripts several times to get an overall sense of the data. Secondly, I connected

relationships in the process of comparing and contrasting the data. After categorising

data by the themes, some categories were developed until the data was exhausted.

Thus, the final findings were discovered after delimitation.

3.5.5 Research Considerations

Some weaknesses of case studies are that they are „not easily open to cross-checking,

hence they may be selective, biased, personal and subjective‟, and „prone to problems

of observers‟ bias, despite attempts made to address reflexivity‟ (Cohen et al., 2007:

256). I was aware of these research weaknesses and tried to be objective both while

conducting the case studies and analysing the data. It could be argued that we cannot

avoid subjectivity. This is because even in quantitative studies that claim to be

objective, the data chosen and the procedures used to analyse the data go through a

human filter and thus rely on a certain level of subjectivity (McKnight et al., 2000).

If this study was not constrained by a time limit (Appendix III), it could be addressed

by a large-scale quantitative approach or by using mixed methods research. The

research questions could be elaborated in other ways, for instance, „to what extent

does upper-secondary teachers‟ use of GeoGebra highlight the relationship between

algebra and geometry?‟. Therefore, pragmatic views with a mixed-methods approach

could be applied for further or larger scale study.

32

Ethics

Letters of consent were sent to the teachers prior to the interviews (Appendices IV

and V). This was to give assurance that they remain anonymous in any written reports

arising from the study. The contents in the interviews are treated in the strictest

confidence.

Triangulation, Validity and Reliability

The theoretical perspective of the researcher and the nature of reliability and validity

are relative (Maxwell, 2002). Validity refers to the appropriateness, meaningfulness,

correctness, and usefulness of any inferences a researcher draws based on data

obtained through the use of an instrument (Fraenkel and Wallen, 2003). In the

qualitative research, the validity of interviews is dependent upon: depth, honesty,

extent of triangulation and objectivity of the researcher (Cohen, et al., 2007).

Reliability stands for „the extent to which research findings can be replicated‟

(Merriam, 1998:205).

To ensure greatest validity and reliability, I took the following actions. I used

methodological triangulation (including interviews with observation and

documentation) to strengthen the validity (Yin, 2003; Cohen et al., 2007). Since there

were a variety of instruments for data collection: video-recording, audio-recording,

field-notes and observations - this enhanced the validity of the findings. When

interviewing the participants, I video-recorded all the conversations, allowing my

contribution to be identified and enabling more careful analysis of the participants‟

answers to be carried out. This also reduced the danger of data distortion due to

selective memory, thereby improving the reliability of the study. As similar wording

of open-ended questions (Appendix VI) was used for every participant, reliability of

the interviews could also be heightened.

33

CHAPTER 4

Data Analysis

‘This is different. This is maths by interacting;

This is maths by trying things out, by conjecturing, by having a go’

Tyler

4.1 Introduction

This chapter discusses a summary of each teacher‟s conceptions and practices of

technology and mathematics teaching in relation to the use of GeoGebra. It is

presented in accordance with the pre-determined themes in the methodology section,

that is: the teachers‟ background, their conceptions of technologies and views on

GeoGebra compared to other mathematical software and their methods of using

GeoGebra. In order to report the cases in a systemic way, I follow Thompson‟s (1984)

framework for the data analysis to discuss each case study as they pertain to the four

themes.

This chapter is structured by the presentation of a mixture of a priori analysis (top-

down) and inductive analysis (bottom-up) of the data. Firstly, within-case analysis of

the four individual cases by the pre-determined themes is demonstrated. Secondly, I

report two informative interviews with the software creator and one advanced user.

Finally, emerged categories from cross-case analysis and cross-cultural exploration

are discussed.

4.2 The Cases

In an attempt to validate the collected data, I volunteered to join the GeoGebra

translation team in Taiwan and have worked as a research assistant on the NCETM9

GeoGebra project. Through commonality of background along with my five-year

mathematics teaching experience, the Taiwanese teachers- Jay and Li were able to

9 The National Centre for Excellence in the Teaching of Mathematics where the GeoGebra project is

funded in the U.K.

34

talk openly, with a common understanding of the mathematical content and

educational system. My involvement with GeoGebra workshops, conferences and

seminars allowed me the opportunity to meet English teachers who use GeoGebra.

Since Richard and Tyler have acted, respectively, as a software developer and as a

GeoGebra trainer, they were enthusiastic, cooperative and willing to share their

thoughts and practices.

4.2.1 Jay

Background

Jay has been teaching mathematics for twelve years in two senior high schools in

Taiwan (students aged 15 -18) and has also worked as a system analysis engineer (SA)

in the field of IT for two years in the US. Being a SA engineer helped him perform

actively in translating software and develop advanced skills in using mathematical

software, such as GSP and GeoGebra. His mathematical knowledge was enriched

during his undergraduate study, when he majored in mathematics.

Views on the use of technology and GeoGebra in mathematics teaching

Jay‟s views about the incorporation of technology into teaching practices are

generally more negative than positive. He inferred that both students and teachers

viewed computers as a tool for entertainment rather than a learning or teaching tool.

He described this phenomenon in relation to technology in Taiwan:

Nowadays, there is a reactionary phenomenon in Taiwan; students will notice the

software. However, they might think that they do not even have enough time to

learn mathematics, so how can they spend time learning and investigating the

software? We consider that there is an improvement through software use, but

they might think that it is going backwards.

Jay‟s remarks about this situation also applied to the teachers:

People take the computers as a tool for entertainment rather than as a tool for

research. Consequently, teachers are no exception… to be honest; most teachers

are not used to this software because using a computer itself is a challenge. At

present, it is still like this for most teachers.

35

Furthermore, he noted that the human brain does the thinking, believing that a

computer:

…has its own limitation. It cannot do the logical and deductive thinking for you.

We need human brains to do that job. Some theories are a process of deduction.

The software itself helps to strengthen the fact that these are right; however, it

does not explain why they are right.

When he talked about students of higher abilities, he also devalued technology and

expressed that:

Mathematically talented students might not need to use this kind of software to

help them learn. Sometimes, doing mathematics is a matter of logical thinking

and deduction.

On the contrary, he held positive attitudes only with regard to GeoGebra. He claimed

GeoGebra to be a convenient tool, which can be used for demonstrations, checking

and visualisation as well as research. He states that: „I would use this software for

drawing graphs or even calculation as it has command list functions for calculations.

They are very convenient…I consider it convenient tool.‟ He mentioned that

GeoGebra provides powerful capabilities that other software packages cannot offer:

„It is actually very good, especially when you want to do addition and subtraction in

the grid coordinate system.‟ He added that GeoGebra links algebra and geometry: „as

you might know, its name is a combination of geometry and algebra. Therefore, I

think it has been done perfectly well regarding this part.‟ His views on GeoGebra are

revealed in the following statements:

(a) It is difficult to display and demonstrate on the blackboard. This is what

makes GeoGebra stand out and it is really useful.

(b) It is easier for students to come to an understanding of the described situation

when we drag one point to see how it affects the motion of another point.

Otherwise, it is difficult to relate concepts to images...it makes life so much

easier having this software.

36

(c) Mathematics is alive, but when it is written in textbooks - it becomes dead.

Consequently, I use it to make mathematics come to life. It is ‘resurrection’

software!

In general, Jay was discouraged by the current educational environment regarding

technology and both students‟ and teachers‟ attitudes toward mathematical software in

Taiwan. He also asserted that support from mathematical software was limited as

human brains do the logical deduction. However, he emphasised that GeoGebra

provides quality functionalities that encouraged his use of this software in his teaching

practice.

GeoGebra evaluation in relation to other mathematical software

GeoGebra has been the most successful software among similar software

packages so far. It is very impressive because it has the capability of algebra. I

really must use it at certain times.

Jay highlighted GeoGebra‟s distinguished features and made comparisons between

GSP and GeoGebra. He argues that:

GeoGebra has buttons that you can basically do the same thing apart from the

conic section. GSP is not good at that, as it does not have this function. In fact, it

does not have much about circles, only a few of them. On top of that, there are

parts like tangent lines, etc. that GSP does not have. GeoGebra, in contrast, is

very good at them, which make it very convenient for users. For instance, if you

use GSP to make tangent lines, it is very difficult as you need to calculate it

yourself at times.

However, he pointed out two weaknesses of GeoGebra: - the lack of the animation

button and iteration capability. Apart from these two parts, he thought GeoGebra

provides much better capabilities than GSP for mathematics teaching: „The algebra

window and command line, especially the command line and the bottom part with

equation input. These are where GSP cannot even compare. It does not even have

these.‟ He mentioned that his school purchased Cabri but he dismissed it due to its

lack of command line and algebraic window.

37

Ways of Using of technology and GeoGebra in mathematics teaching

There are different modes where Jay specifically used GeoGebra. He stated:

I mainly use GeoGebra as a tool for teaching and researching… I use it as a

checking tool…to test and verify thinking, or sometimes, when it is

inconvenient to draw graphs on the blackboard, I use it as a demonstration tool to

emphasise their impression.

He mostly used GeoGebra to teach Cartesian coordinate systems. Occasionally, Jay

used GeoGebra for preparation, investigation or classroom practice. He said:

I bring my laptop to the classroom whenever I need to. For example, to

demonstrate conic sections, it is very inconvenient to draw by hand.

Consequently, I use the software; enter equations, the graph shows up. You don‟t

need to draw for a long time and students think that is very cool.

He demonstrated examples of his strategies in operating GeoGebra with one of them

about the trace (Fig.4.1 and Appendix VII for step-by-step constructions).

Figure 4.1: One example of Jay‟s geometrical construct with GeoGebra

38

He illustrated that:

We often see this kind of exercise in textbooks: one point moving on the edge of

a circle, if we take the perpendicular bisector of the segment to find the

intersection of another segment. We ask students what the trace of that

intersection is. Very often, students are dumbfounded after this long description.

If you draw the graph, they can visualise it so they can feel it... It really moves

along with a trace of an ellipse. At the least you can use it as a demonstration

tool.

His skilfulness at geometrical constructions and algebraic calculations are apparent,

however, he only used GeoGebra for presentational purposes which are strongly

bounded to textbooks. This limited inclusion of the software does not engage students

to its full potential.

The salient categories emerged from the data are listed as follows:

Tool use Graphing, calculations, visualisation, demonstration, dragging,

checking, test and verify, teaching and research

Mathematics topics Cartesian coordinate systems, both algebra and geometry

Teaching style Textbook-oriented

Infrastructure Laptop demonstration in the classroom

4.2.2 Li

Background

Li has thirteen years of teaching experience at the upper-secondary level (Year 10-12

equivalence) in Taiwan. Since his first degree was in applied mathematics, he gained

an interest in IT during his undergraduate study. He was enthusiastic about new

technologies and volunteered to translate the Traditional Chinese version of

GeoGebra. Moreover, he had been creative in using different software packages, free

software in particular, and trying to use a combination of different open-source

software to make teaching materials. He has written some journal articles comparing

new, free software packages detailing how they might be incorporated into

mathematics teaching for Taiwanese teachers. He maintains the school mathematics

website which includes GeoGebra related teaching and problem-solving materials. In

addition, he proposed and conducted GeoGebra training courses and workshops in

39

senior high schools in Taipei. He had also set up his website and uploaded his up-to-

date GeoGebra materials and step-by-step tutorial materials for students or teachers.

Views on technology and GeoGebra

Li had a similar opinion to Jay on students‟ and teachers‟ attitudes towards the use of

computers. He said:

I do not think the use of computers raised students‟ motivation, because since

they were young, they perceived computers as a tool for entertainment. When

they discovered you can actually use computers for mathematics, they think it is

interesting but it still can not motivate them to learn mathematics with computers.

He also added that, „Generally, people are afraid of using computers in teaching

mathematics because it feels different from using Microsoft Office on the computer‟

Consistent with Jay‟s comment he also mentioned students‟ passive attitudes about

technology in learning:

…when you are demonstrating mathematics on the computer to students, some

of them at the back of the classroom might fall asleep if they are not interested.

They do feel that using computers is interesting but if you want them to use or

design with it, it is impossible. Using computers to them is for entertainment,

such as surfing the Internet, chatting, and playing computer games. Now some

students in my class can be called - “kidnapped by the computer”. They are

addicted to it.

Despite knowing how students and teachers feel about the software, he feels proud of

his achievements in developing and translating GeoGebra not to mention creating

related Traditional Chinese websites and GeoGebra teaching materials. His

enthusiasm for using GeoGebra when teaching mathematics was bountiful. He

insisted that: „you can use GeoGebra to teach almost all topics. It is brilliant!’

GeoGebra evaluation

Li had published one journal article about the comparisons between GSP and

GeoGebra. Four areas of differences between them were found: price, speed, Java10

10

One programming language- Java technologies are made available most as free software under

General Public Liscense.

40

and Latex11

support and international cooperation. He believed that because

GeoGebra is free, its potential to have a great impact on mathematics education could

be all-pervading. In terms of speed, since GeoGebra has algebraic capabilities, some

graphs made by GSP could take more than ten steps, whereas in GeoGebra equations

are simply keyed in and the enter key pressed to form them. Furthermore, since

GeoGebra was written in Java language, it inherited the advantages of Java in terms

of multi-language, multi-platform and the support for Latex language. In his view, it

could be a great choice for mathematicians to discuss mathematical problems over the

GeoGebra websites such as user forums. This would serve as an international

community within which everyone could benefit from support of others.

By contrast, GSP is scarcely comparable to GeoGebra in these areas. However, he

highlighted one weakness of GeoGebra: „In the Grade II (Year 11 equivalent), there

are topics like vectors in the space. It is trickier when it is in the 3D.’ Therefore, he

chose to use SketchUp for the three dimensional topics instead.

Ways of Using of technology and GeoGebra

Li has used GeoGebra for one and a half years trying many different ways of using it.

He is positive that exploiting GeoGebra can change students‟ attitude towards

mathematics learning. Some of his designed teaching materials and tutoring examples

of using GeoGebra in solving examination problems were displayed on the websites.

He also encouraged students to use the websites for reference and discussion. His

ideal teaching environment would incorporate technology and GeoGebra, he said:

I would bring them [the students] to the IT room and introduce them to the

GeoGebra website…I would also use projectors and computers in the

classroom. I would show them how to use it and tell them about my website… I

mainly want students to use the website for reference and hope they will go

home and visit it. Then they can make the connection between graphs and the

contents in textbooks.

When talking about the content and topics, he stated:

11

A programming language for writing professional papers

41

I use GeoGebra to teach more on geometrical topics and some algebraic

calculation and graphs. Actually both, it really depends as these two parts have

very close relationships… As for our curriculum, it doesn‟t separate geometry

and algebra in a clear and detailed way. When teaching functions we link the

concepts to graphs because we have to follow the curriculum. In the textbooks,

they always link geometry and algebra together.

This could account for his view on GeoGebra linking geometry and algebra as

Taiwanese curriculum does not necessarily separate them.

Li provided a number of strategies for exploiting GeoGebra and put a significant

stress on examination exercises and problem-solving as well as proofs of theorems

(Appendix VIII). He occasionally took students to the IT room where they went

through revision for examinations. One of these lessons was observed. He pre-

prepared a worksheet (Appendix IX) for students to investigate graphs of linear

function, quadratic functions, trigonometry, exponential functions, and logarithmic

function. He mediated GeoGebra for demonstration first, and then guided students to

interact with it and investigate properties of the function family.

One example that he created with GeoGebra where he used the slider to show the

changes of graphs of exponential functions is shown (Fig. 4.2). This example

indicates the changes in the graphs in relation to the base number a of y= a 2

. His

scenario was initially a presentation followed by a pre-prepared sheet guiding students

to investigate by typing in different equations or moving sliders to observe the

changes in the graphs. It could be argued that he orchestrated interaction between the

students, mathematics and software; however, this didactic way of supporting

students‟ revision provides less freedom for students to explore themselves.

The salient categories emerged from the data are listed as follows:

Tool use Graphing, calculations, demonstration, problem-solving,

revision, investigation, and interaction

Mathematics topics Geometrical topics and algebraic calculations

Teaching style Curriculum-based, textbook-oriented and exam-driven, self-

developed teaching materials and website with GeoGebra

Infrastructure Home, IT room or computer and projector in classroom

42

(1) The graph of y= a 2

when a <1, (a is at the left side of the slider)

(2) The graph of y= a 2

when a =1 (the point on the slider was moved to the right)

(3) The graph of y= a 2

when a >1 (the point moved to the right)

Figure 4.2: One example of Li‟s exponential function constructs with GeoGebra

43

4.2.3 Richard

Background

Richard has taught secondary and A-level mathematics for twelve years in England.

He is skilled in computer programming and is in charge of the school mathematics

website where a combination of GeoGebra, Yacas12

and JavaScript13

are used for

developing online mathematics materials and tests. He designed a piece of DGS and

used it to teach before starting to use GeoGebra. Previously, he was working as a

software developer and cooperated with the NCETM GeoGebra project.

Views on technology and GeoGebra

Richard has an ambivalent view of GeoGebra. He expressed that he was not

convinced that GeoGebra links geometry and algebra but then stated that: „it does the

connection between algebra and geometry much better than other programmes -

anywhere you can enter a number you can also enter a formula‟.

He asserted that GeoGebra had changed the way he taught as he had been taking

students to IT rooms more often and some students liked the revision with GeoGebra

as it sped up some processes of preparation for examinations and for accuracy. He

added that some students, however, preferred printed-out sheets with longer questions

as in examinations they had to use paper-and-pencil.

Since Richard had a personal interest in computer programming and he utilised a

combination of GeoGebra and JavaScript to create online materials, he stated: „I think

because of the interface of JavaScript, you can display anything you want. You can do

anything you want to. You really CAN do anything!‟ He stressed „the fact that you

can animate any variable by turning it into a slider is a very powerful feature‟.

12

One open-source software which is viewed as a CAS 13

A programming language that controls a software application

44

Nevertheless, he also pointed out that GeoGebra is a tool that „like any other tool can

be used badly or well‟.

GeoGebra evaluation

Richard summarised several features which he particularly appreciated about

GeoGebra and compared it to GSP:

The fact that it is free… that students can use it at home, the fact that it is Java

that you can use in any platform and with JavaScript you can control and put it in

the webpage. For me, that is tremendously useful…It is specialised, not many

people can write with JavaScript… Geometer‟s Sketchpad, I don‟t like how it

looks, I don‟t like its interface. So GeoGebra looks nice and the interface is easy

to use.

He picked out that GeoGebra was good for teaching gradients of a curve, both for

the concept and the proof. However, he pointed out one weakness of GeoGebra was

that the fractions could not be typed in.

Ways of Using of technology and GeoGebra

With respect to pedagogical practices, Richard discussed two ways in which he used

GeoGebra to teach. Firstly, he used it for demonstration in the classroom due to the

fact that „it does the questions quickly. It is quite easy if I want to demonstrate on the

board‟. His way of using GeoGebra in the classroom followed an orderly sequence of

using paper-and-pencil first, and then demonstrating graphs using GeoGebra. He

taught topics with linkage to graphs such as transformations in a different order as he

explained: „possibly because it takes a long time to draw the graphs.

Transformations, this is what I might get student to the IT room first in the future‟.

Richard‟s second way of using GeoGebra was taking students to the IT room to work

on activities or revision. However, he attempted this less frequently than in

classrooms as he believed that students should learn in a paper-and-pencil

environment initially as: „in the exam they‟ve got to use paper-and-pencil. I think if I

do everything on the computer. They‟re probably not gonna do well. They‟ll get

bored. I do it with Year 11 for one lesson every fortnight. I think that‟s been about

45

right.‟ He described the ways in which IT lessons were carried out: „I take the class

into the IT room … I tend to do two activities in a lesson. I set up a combination

system. The good students you get them to move on to different activities.‟

A few examples of online lessons are presented in the following figures. He set up

tests (Fig. 4.3) for pupils on his school website in which he used GeoGebra and Java.

The example shown in the figure 4.3 is used for testing students‟ understanding of

transformations of equations and their graphs.

Figure 4.3: One example of school mathematics website on the topic: transformations

The example in the figure 4.4 is a designed activity on Richard‟s school website for

students to drag the points interactively and discover that angles in the same segment

within a circle are the same.

46

Figure 4.4: One example of school mathematics website on the topic: angles in the

same segment

Richard demonstrated his idea of linking algebra and geometry with GeoGebra

(Fig.4.5). He plotted several points by using GeoGebra and the input sequence [(n,

4n-2), n, 1,100] followed by entering the equation y=4x-2 to show the link between

the algebraic and graphic representations.

Figure 4.5: One example of linking algebra and geometry with GeoGebra by Richard

47

The salient categories emerged from the data are listed as follows:

Tool use Graphing, calculations, demonstration, revision, student

activities, investigation with the slider

Mathematics topics Mainly geometrical topics, gradients of a curve and

transformations

Teaching style Activity-based, a combination system of paper-and-pencil and

computer environments

Infrastructure Home, IT room or computer and projector in classroom

4.2.4 Tyler

Background

Tyler has taught mathematics to 11-16 year olds in a college for twelve years. He has

spent three days a week teaching at the school and one day a week teaching secondary

trainee mathematics teachers for PGCE14

in a university. He has also acted as an

AST15

supporting schools and as a part-time school consultant, cooperated with the

NCETM GeoGebra project and hosted a GeoGebra training workshop at his college.

Views on technology and GeoGebra

Tyler‟s utterances reflected a view of GeoGebra as an environment for exploring

dynamic geometry rather than algebra. He viewed GeoGebra as a replacement to

Cabri, which he used before GeoGebra. However, he mentioned that his experience

with GeoGebra was approximately half a year, which meant that there were areas of

using GeoGebra that were under-explored and underdeveloped, such as using

GeoGebra in teaching algebra. He stated his expectation for GeoGebra development:

„It‟s still very new. But it‟s really exciting so far. It‟s going to be really, really

exciting to see how it develops and how we can develop using it.‟

14

Postgraduate Certificate in Education 15

Advanced Skills Teacher

48

Some criticisms about current usage of technology in schools were brought up in

terms of the IT rooms and school websites. He described his intention to change the

way his pupils work from being passive to actively involve in learning through

software. Getting pupils to work with mathematics in the IT room was difficult in his

experience. Moreover, he did not expect that students would not undertake much

thinking in the IT room. In addition, some school mathematics websites have

mathematics tests for pupils to log on to at home with their personal passwords which,

in his view, allowed no room for discussion and interaction. He pointed out that

GeoGebra is interactive and intuitive so he could set up diagrams and activities for

students to interact with easily: ‘This is different. This is maths by interacting; this

is maths by trying things out, by conjecturing, by having a go.’

He emphasised that GeoGebra could not only be used as a presentation tool by

teachers but also as an investigation tool for pupils.

GeoGebra evaluation

Tyler spoke positively about the features of GeoGebra in terms of changeability of

the font size, projection capability and the slider in which he considered GeoGebra

outperformed other DGS packages such as Cabri and GSP. He used Autograph in the

way he preferred to have pupils conjecture on topics related to trigonometry such as

sine waves. However, he specified that the bidirectional capability of GeoGebra in

linking algebra and graphs which can also be used by pupils to investigate at home

was superior to other software.

Ways of Using of technology and GeoGebra

An enthusiasm for GeoGebra was apparent in Tyler‟s strategies of using GeoGebra in

mathematics teaching. He systematically summarised three different ways in which he

considered GeoGebra could be used for teaching:

Demonstration

He thought potentially teachers could potentially use GeoGebra as a presentation tool

where there is only interaction between teachers and GeoGebra: „One way is where I

49

demonstrate, so with me at the board, using it as a teaching tool using it to

demonstrate to the class.‟

Interaction

The second way was setting up some particular parts of mathematics for pupils to

work on and find out as a whole class activity in advance. At this stage pupils interact

with GeoGebra within the whole class. „I set up particular GeoGebra files for them to

look at, to explore, to make changes to, and then for them to make hypotheses of

what might be happening.‟

He offered one example of pre-prepared files on the topic of „transformation‟ (Fig.4.6,

Appendix X) as a whole class teaching activity. He demonstrated how he would use

this activity while teaching „enlargement‟:

We can spend a proper amount of time talking about what happens if I move this

to the left. And only at the very end of that discussion, do we then actually do

it… then wonderfully pupils want to know: can you make it a decimal? That‟s

how they call it, what happens if I make this point to the centre? Can you make it

negative? What happens if that‟s a zero? There are very nice things you can do

with this.

From this example, his ways of questioning to provoke students‟ thinking along with

the designed activity revealed that the whole class activity worked under teacher

demonstration and interaction.

(1) Enlargement with a=2.4 (a>1), the transformed triangle on the left became

bigger

50

(2) Enlargement with a=0.3 (a<1), the transformed triangle on the right became

smaller

(3) Enlargement with a=- 0.7 (a<0), the transformed triangle on the right became

smaller and inverted

Figure 4.6: One example of Tyler‟s use of transformation activity on „enlargement‟

Investigation

The third way of using GeoGebra was conceived to be an ideal state where pupils

investigate their mathematical ideas with GeoGebra by making conjectures and

testing them out. He exemplified his experience of using GeoGebra this way:

51

It‟s in the IT room that children use GeoGebra for themselves. There they work

in pairs, they discuss what they are doing, and they are encouraged to have ideas

and test those ideas out … they started to create things. Some of them started

with a blank sheet and they wanted to us, maybe, the reflection… and they

wanted to do reflections to make their own pictures and interact.

Overall, Tyler was reflective and explorative about different practices with GeoGebra,

and eager to find out possible areas where GeoGebra could be useful in mathematics

teaching. He also drew a distinction between „knowing how’ to use it and „getting

used to’ using it in relation with GeoGebra. This inferred that he acknowledged the

differences between using GeoGebra and teaching with the use of GeoGebra.

The salient categories emerged from the data are listed as follows:

Tool use Demonstration, interaction, investigation, exploration, testing

hypothesis, creation, projection capability and the slider

Mathematics topics Mainly geometrical topics

Teaching style A whole-class teaching activity

Infrastructure Home, IT room or computer and projector in classroom

4.3 Informative interviews

4.3.1 Interview with the GeoGebra creator

Markus created GeoGebra during his Master‟s study in Computer Science and

Mathematics Education and finished the first prototype in March 2002. He received

the European Academic Software Award 2002 while he was teaching in a high school

in Austria. His PhD was funded for GeoGebra development and he now works as a

visiting professor in Florida. When talking about how GeoGebra could be used by

teachers, he said:

I think there is a huge variety in how teachers use it. This depends a lot on the

teachers‟ background. Both their background in mathematics, mathematics

content knowledge and of course also how much they have been doing in

technology before.

52

Based on his experiences training mathematics teachers to use GeoGebra, he

concluded there are four different stages that teachers possibly go through from

learning to use GeoGebra to teaching mathematics with GeoGebra:

Stage 1: teachers have to get comfortable with the software alone at home, using

the software to create nice pictures for tests;

Stage 2: teachers use GeoGebra as a presentation tool;

Stage 3: teachers do construction on the fly. They use GeoGebra to visualise

what has been discussed…And then the way to get students interact a little bit

more would be to let them present. So students do some kind of exercise and

GeoGebra can be used as a checking tool. Then students walk out to the

teacher‟s computer. They type in what they think is the answer and show it to the

class and we compare different answers. GeoGebra can just be used to present by

students as well;

Stage 4: teachers can ask much more open questions. Students can play with

GeoGebra to come up with conjectures. So not just checking the conjecture but

also developing the conjectures. That is what I really want to see.

During the discussion about whether teachers would use GeoGebra to teach geometry

or algebra, he clarified: „Basically, lower grades can use it for geometrical

constructions and higher grades for families of functions with the sliders and basic

calculus like derivatives.‟

When talking about what had been missing in the status quo, he mentioned that there

was not enough training and support for teachers new to technology as well as limited

research on the impact of GeoGebra for teaching and learning of mathematics.

4.3.2 Interview with an advanced user

Peter has been teaching mathematics in a university and masters course for higher

ability students (Year 10-12) for eight years in England. He taught with GeoGebra at

university and upper-secondary levels. He emphasised that: „the simplicity and ease of

use’ and ‘the ability to go between algebra and graphics and again contribute to

factor’, make GeoGebra an effective tool.

Peter used GeoGebra mostly for showing diagrams and suggested that:

53

You can pre-prepare things and then they look great, but I think it is useful to go

through the construction, I think it is crucial to go through the construction step-

by-step in front of them so that they can understand.

Since he warmed to GeoGebra he found it useful and effective for illustrating

graphical part as he could show proper graphs briefly and quickly when students have

questions. „I would say GeoGebra has changed the way I teach. I am incredibly into

this. GeoGebra certainly makes things, some things, easier and that has benefits‟

4.4 Cross-Case Analysis

4.4.1 Emerging Categories

Some extracted findings from each case were collected in the within-case analysis. By

following the constant comparative method (Glaser and Strauss, 1967), several

categories emerged from the data when comparing incidents applicable to each

category. The classification involved subdividing the data as well as assigning the

data into as many categories as possible that fitted an existing category. For example,

the category of teachers‟ conceptions and uses of GeoGebra as an „educational tool‟

emerged quickly from comparisons of the teachers‟ responses to the ways in which

they viewed and used GeoGebra as a tool for a variety of purposes.

Categories appeared when comparing the interview data across the cases. In relation

to environments within which teachers use GeoGebra, infrastructural change of IT

facilities and settings seemed to be one of the major concerns. With regard to

teachers’ behavioural change, two aspects, teachers‟ mathematical and IT

background and the transition that they experienced through using GeoGebra, were

scrutinised. The third category is the way they viewed GeoGebra as an educational

tool. The fourth main category- mathematical topics had been targeted for different

levels of mathematics. Out of those categories, some sub-categories emerged, which

will be discussed in the following analysis.

After splitting categories into sub-categories, I followed Dey‟s (1993: 139) strategy

for splicing categories: „when we splice categories, we join them by interweaving the

54

different strands in our analysis‟. This is for the purpose of integrating categories and

their properties. Following the sequence of splitting, splicing categories and linking

the data, a framework for analysing cross-cases was then identified. In the final stage,

there are four main categories (Appendix XI) in relation to the use of GeoGebra

integrated:

(a) the ways in which teachers view and use GeoGebra as an educational tool

(b) the transition that teachers experience when they go through different stages

from learning GeoGebra to teaching with the use of GeoGebra

(c) the mathematical topics that teachers choose for teaching aided by GeoGebra

(d) the infrastructural change of technology environment under which teachers

work in relation to their practices of GeoGebra

These four dimensions are used to examine the differences and similarities among

these four cases in the following.

4.4.2 Educational Tool

The case studies show that, besides differences in teachers‟ views on and methods of

using GeoGebra, they all referred to GeoGebra as an educational tool. Two

possibilities of GeoGebra as an educational tool are that teachers might view it as a

tool or use it as a tool in their classroom practices. As a consequence, this dimension

overlaps two themes- views on and uses of GeoGebra. Applying comparative analysis

cross the cases and themes, GeoGebra can be identified as an educational tool for:

research and analysis;

immediate feedback and reflective checking;

creating teaching materials and online materials;

demonstration, presentation and visualisation;

problem-solving, computation and calculation;

classroom activities, tasks- investigation, experimentation and conjecture;

geometrical proof of theorems;

revision for examinations;

55

Jay viewed GeoGebra as a tool for research, checking, calculation, teaching and

demonstration and used it mainly for presentation in the classroom. He mentioned that

GeoGebra was a „resurrection‟ tool that activated and visualised some mathematical

concepts in textbooks. He also stressed his position of viewing GeoGebra merely as a

tool which was useful and convenient. After one year of using GeoGebra, he had not

changed the way he viewed it as an additional tool for speeding up teaching processes.

He did not give students guidance to learn or to engage with GeoGebra. Jay‟s limited

ways of using GeoGebra could be the result of his conceptions that its effectiveness

was low and that not many teachers would use it or students find it a useful tool.

Li considered GeoGebra as a tool for a broader range of affordances, such as making

teaching materials, editing online tutoring worksheets for problem-solving,

conjectures, geometrical proof of theorems, students‟ reference after school and

revision for examinations. This is likely to reflect his high level of enthusiasm and

confidence in GeoGebra. Moreover, his extensive production of GeoGebra

applications could be inferred from his profound mathematics content knowledge.

However, a lack of pedagogy in teaching with GeoGebra seemed apparent. During

observation of a lesson in the IT room, he used GeoGebra as a revision tool. Students

followed his pre-prepared worksheet step-by-step to observe how graphs change when

different functions were typed in. It seemed that students simply acted according to

the required task and did not engage in actively thinking about the task. Therefore,

this is understandable that students might unlikely to be inspired or motivated by

learning through GeoGebra. This view of missing appropriate pedagogy was also

indicated in his aspiration to raise students‟ motivation to learn by using GeoGebra-

he uttered: „I hope to use GeoGebra to move students‟ hearts and grasp them back‟.

Although Li‟s self belief that his design work with GeoGebra might persuade students

to engage more fully with mathematics, the unappreciative reaction of his students to

his efforts indicate otherwise. This has prompted him to improve the situation.

Richard regarded GeoGebra as a tool for a variety of practices, even for different

subject areas such as physics. He asserted that „you really can do anything’ with

GeoGebra, such as designing tests or tasks on school websites. Nevertheless, his main

use for it was as a presentation device in the classroom and a tool for revision for

examinations in the IT room. His enjoyment of mathematics was derived from

56

combining different software packages for producing online tests for students to

practice at home. One limitation of his use of GeoGebra stemmed from the fact that

most of the material he designed only required „yes‟ or „no‟ answer. Additional

explanation or help was not offered if students answered questions incorrectly. His

intention was to help students learn through these online tests, IT room activities,

classroom tasks and demonstration. Arguably, these activities might assist students

with procedural understandings rather than conceptual ones.

Tyler did not appear to consider GeoGebra as a tool but rather as an environment for

exploring mathematics. However, he stated that he would use GeoGebra as a

presentation tool in the classrooms but preferred students to use it as a tool for

working on tasks, investigation and testing conjectures. He was aware students simply

observing teachers present work with the software hinders their interactive

participation and is different from doing the work themselves. Therefore, he claimed

that GeoGebra is most useful when students actually experiment and investigate with

it. He viewed GeoGebra as an educational tool, not only for teachers but also for

students.

Comparing the four teachers‟ behaviours with GeoGebra, Richard and Li approached

GeoGebra in a similar fashion although they are from different countries. They both

had a combination system of working with GeoGebra, creating their own teaching

materials and websites as well as providing revision section for students‟ examination

preparation. However, Jay and Tyler both approached GeoGebra differently. Jay was

more demonstration-oriented which indicated that his teaching practice was consistent

with his conceptions of GeoGebra being software for visualising mathematics. Tyler‟s

practice was student interaction-based which might be in relation to his conceptions of

GeoGebra being interactive.

4.4.3 Teacher Transition

Teachers might experience changes in their manipulation of GeoGebra providing

more time and exploration. According to the interview with Markus, he thought that

57

teachers seem to go through phases and changes from starting to learn GeoGebra to

teaching utilising GeoGebra. These four stages are:

(a) Preparation- teachers begin with basic constructs, such as making triangles,

circles and graphs of equations. They create diagrams for preparation of

arranged lessons and generate printed worksheets or test sheets.

(b) Presentation- teachers start using GeoGebra in the classroom for demonstration,

either displaying pre-prepared files or constructing graphs step-by-step in front

of students.

(c) Interaction- teachers design whole class activities and encourage interactions

between students and GeoGebra.

(d) Investigation- teachers ask open questions and students work in pairs to

investigate their mathematical ideas, conjectures with GeoGebra.

Given this framework for examining teacher transition, I determined that Jay was the

only one who stayed at the presentation stage; Li and Richard seemed to move on to

the interaction stage whilst Tyler had proceeded to the investigation stage as a result

of his personal expertise as an AST. However, I could argue that these teachers are

not teachers who are new to using technology: some might have experiences using

other software in the past, particularly similar DGS packages. Therefore, they did not

necessarily need to go through the first stage. For instance, Jay had experience using

GSP during the past twenty years and he started GeoGebra straight into the second

stage without changing for years. His perception of the uses of GeoGebra was limited

and possibly so were his intentions of exploring different uses of mathematical

software. Consequently, there are probably teachers who stay at one stage, never

moving forward.

The data suggested that teachers can be categorised into three types: unskilled

teachers who have never used technology in teaching, technology-skilled teachers and

GeoGebra advanced skills teachers. Some teachers who are not used to technology

can download GeoGebra online materials or worksheets for their classroom practices.

They could be at the pre-stage phase where they might simply want to use it for

demonstrations and are unwilling to learn more advanced mechanisms of the software.

Teachers who are skilful using technology are possibly the ones who progress from

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stage to stage. Advanced skills teachers use GeoGebra across all stages as a network.

They change their plans adapting to different topics or student abilities and employ

GeoGebra for preparation of lessons to encourage interaction with students,

preparation for presentation on particular topics or preparation of activities for student

investigation. Given more time and experiences of teaching with GeoGebra, a

combination of all stages is exploited.

4.4.4 Mathematical Scope

The mathematical topics that GeoGebra supports in terms of my research focus can be

categorised as algebraic topics, geometric topics or both algebraic and geometric

related topics. The differences and commonalities of the four teachers‟ choices of

mathematical topics using GeoGebra are discussed.

Jay taught with GeoGebra mainly on topics related to coordinate systems, which is a

possible subcategory of both algebra and geometry. Li listed all topics16

related to a

wide range of mathematics areas apart from 3D topics which can also be set to both

algebra and geometry. Both Taiwanese teachers viewed algebra and geometry as two

sides of a coin that should not be separate. Li pointed out that there were no particular

separation in the curriculum, therefore, they both sometimes taught algebra and

geometry at the same time which seems to be the reason why they used GeoGebra to

teach both algebraic and geometric related topics.

In contrast, Tyler and Richard used GeoGebra mainly for geometric topics possibly in

consequence of their perception that GeoGebra is a DGS. They expressed that they

would not want to use GeoGebra for all topics as there are certain topics that are not

appropriate for incorporating technology. Therefore, they preferred to find out what

topics GeoGebra is appropriate for then use it for those specific topics. For example,

Tyler had shorter period of time exploiting GeoGebra and was interested in exploring

16

Cartesian coordinate systems, linear equations, graphs of polynomial functions, quadratic functions

followed by higher order functions, exponential functions, logarithmic functions and trigonometry,

circles and balls- the equations of circles and conic sections- parabolas, ellipses and hyperbolas.

Furthermore, there are topics related to pre-calculus such as differentiation and integration involving

rectangles, upper sum and lower sum and tangent and inequality and symmetric graphs.

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GeoGebra for topics related to algebra. According to the interviews, both Richard and

Tyler chose to use different software for algebraic topics as they did not seem to be

convinced by the algebraic capability of GeoGebra. The difference between

Taiwanese and English teachers‟ choices of topics might be due to discrepancies in

the structure of mathematics curricula and their perceptions of GeoGebra.

According to the teacher evaluation of GeoGebra and other software, most of them

regarded GeoGebra as a replacement to GSP and Cabri. However, GeoGebra could

not work with particular topics such as 3D topics for Taiwanese teachers. Jay chose to

use Archimedes 3D17

whilst Li chose SketchUp for the 3D related topics. Richard

designed online materials with Yacas and JavaScript for algebraic topics whilst Tyler

used Autograph for teaching topics related to functions. When the weaknesses of

GeoGebra capability were discovered all teachers were proficient in embracing other

software packages for their chosen topics.

4.4.5 Infrastructural Change

The infrastructure of the educational environment is closely related to the ways

GeoGebra can be used. Since GeoGebra is open-source software, one advantage

offered by it is that both teachers and students have options to use it at school and at

home. Teachers can use GeoGebra at home for either research or preparation for

mathematics teaching materials whilst students can do coursework with it at home.

According to the interviews, some of the teachers encouraged students to go on

GeoGebra-related websites to practice mathematics exercises at home. Most teachers

used GeoGebra to demonstrate mathematical objects or visualise mathematics in their

classrooms using a laptop or a computer connected to a projector. Some of them

brought students to the IT room and a few prepared laptops in the classroom for

students to investigate. There are therefore three different environments that teachers

used GeoGebra – at home for research or preparation, in classrooms for

demonstration or student interaction and in IT rooms for activities, revision or student

investigation.

17

One three dimensional DGS which is commercial

60

Jay and Li conceived of the infrastructure of GeoGebra usage for conventional

presentation of mathematical work in classrooms. Before teaching, both of them used

GeoGebra at home, however, Jay would use it for research and making teaching

portfolios whereas Li used it for making GeoGebra worksheets, online tutorial

materials and examination sheets as well as teaching preparation. During teaching,

Jay used GeoGebra solely for demonstration in classrooms where he brought a laptop

and showed graph works to the class whenever needed. Li would not use GeoGebra

in classrooms but in IT rooms for demonstration and revision.

Richard and Tyler worked with GeoGebra in different ways. Richard used it at home

for designing mathematical tasks and tests on the school website, whereas Tyler set up

activities for presentation at home and student investigation in IT rooms. Richard

mainly used it for demonstration in classrooms and revision in IT rooms. Tyler used it

for activities in classrooms and tasks for students to investigate in IT rooms where

they work in pairs, making conjectures and testing their mathematical ideas out.

Comparing these four cases, the English teachers taught both in classrooms and IT

rooms whilst the Taiwanese teachers chose one environment instead of switching

between classrooms and IT rooms.

Most teachers expressed that there was a certain degree of difficulty in approaching

appropriate IT facilities as the time spent on setting up laptops and projectors or

getting students in IT rooms and logging on to the computers could take up to 20

minutes in one lesson. In addition, there were distractions when computers were

available as students occasionally attempted to check emails, surf the web, or listen to

music. These factors could contribute to their frustrations towards implementing

GeoGebra.

Compared with English teachers, Taiwanese teachers held more negative attitudes

with respect to infrastructure of technology and therefore it influenced their ways of

using GeoGebra. This is not only because it is more time-consuming but also due to

students‟ passive response to technology. For example, Jay stated that, „I don‟t dare to

say that it enhances students‟ motivation in learning. In fact, I don‟t even put the idea

in my head. Because learning in the field of mathematics, is considered a hard subject,

61

students need very strong motivation if they are willing to learn or they are interested

in logical thinking.‟

4.5 Cross-Cultural Exploration

There are several areas with respect to the use of GeoGebra in Taiwan which are

different from England. However, ascertaining the commonalities and differences of

the use of GeoGebra between Taiwan and England is not particularly easy as cultural

influence is a complex issue. In addition, the presentation of four cases cannot offer a

broad understanding or generalisation of what is happening in both countries. What

this study offers is an exploration into teachers‟ commonalities and discrepancies in

using GeoGebra in England as compared to their Taiwanese counterparts according to

their personal characteristics, conceptions and practices.

By adopting Kaiser et al. (2005)‟s framework for analysing mathematics education in

Eastern and Western traditions, teachers‟ conceptions of mathematics and their

practices in relation with GeoGebra and cultural influences are chosen for cross-

cultural comparison. In an attempt to identify „what is universal‟ and „what is context

bound‟ (Osborn et al., 2000), this study would help understand the role played by

cultural context and the ways in which teachers use GeoGebra with different forms of

pedagogy as Taiwan and England have contrasting values.

Responding to „what is context bound‟, there are three aspects generated from the data

that could be seen as significantly different between the cultures in England and

Taiwan. Firstly, teachers‟ attitudes towards technology in both countries varied. The

participating Taiwanese teachers held negative conceptions of technology use for

teaching practices, whereas the English teachers were positive about it not only

because they were confident and comfortable about using technology but also because

students seemed to have a higher level of acceptance. Secondly, the Taiwanese

teachers experienced greater difficulties pertaining to infrastructure as the classroom

settings were not particularly designed for technology use in Taiwan whilst the

English classroom settings implemented interactive whiteboards and projectors which

offered convenience for teachers.

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Finally, in terms of pedagogy, the Taiwanese teachers tended to follow a curriculum

based teaching strategy and mostly related GeoGebra exercises to textbooks; therefore,

GeoGebra was used specifically for assistance of visualisation of textbooks examples.

Again, the English teachers appeared to be more creative and flexible in choosing

their teaching methods. As the Taiwanese educational system has an examination-

driven culture, there are several areas being used extensively such as problem solving

for university entrance examinations and proof of theorems as well as revision for

examination preparation. In contrast with Taiwan, the English educational system has

a focus on individual learning, therefore, there seemed to be an emphasis on students‟

individual investigation and interaction with GeoGebra. Identifying „what is

universal‟ cross cases, one noticeable commonality is that all teachers conceived

GeoGebra as a useful tool for mathematics teaching practices.

4.6 Summary

Teachers‟ practical elaboration of GeoGebra can be seen as interrelated within the

four dimensions. The infrastructure of technology has a great impact on the ways in

which teachers regard GeoGebra as an educational tool since if technology facilities

are not available or advanced, it would definitely influence the way teachers use the

software. Given technology provision, teachers‟ mathematical content knowledge and

conceptions may affect their mathematical scope utilising GeoGebra. Certainly,

provided there is sufficient support for the use of GeoGebra, teachers might start

experiencing changes in their behaviour with GeoGebra. This teacher transition will

move them from beginners to advanced users of GeoGebra as well as help them

develop their pedagogical practices in teaching practices.

In spite of these common dimensions between Taiwan and England, there are

substantial discrepancies in technological artefacts and adaptation of curricular

resources which underpin English and Taiwanese teachers‟ decisions and practices

with GeoGebra applications. These significant differences could be explained by the

two opposing Eastern and Western cultural traditions.

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CHAPTER 5

Discussion and Concluding Remarks

5.1 Introduction

In this chapter, I summarise the findings and elaborate on their contribution to current

thinking. I discuss firstly the findings in relation to the research questions and

literature. Secondly, I offer my reflections on my role as a researcher and thirdly, I

consider the implications of the study in connection with the reviewed literature.

Finally, I suggest areas for further research and introduce my proposed future study.

5.2 Findings

I begin this study with, in essence, three research questions. Firstly, I investigated

teachers‟ conceptions regarding technology and GeoGebra in Taiwan and England.

Secondly, I set out to see the manners in which GeoGebra is used for the teaching of

algebra and geometry. Thirdly, I intended to understand whether or not their

conceptions are related to their practices with GeoGebra in both countries.

The purpose of this study is neither to draw generalisations nor to criticise or rank the

teachers but rather to explore the relationship between their conceptions and practices

regarding GeoGebra in order to make suggestions for improvement. Analysing the

data thematically across the case studies revealed four salient dimensions in relation

to GeoGebra-assisted teaching: educational tools, teacher transition, mathematical

scope and infrastructural change.

The findings are introduced in the following, which indicate that understanding the

linkage between teachers‟ conceptions and practices is crucial. Firstly, the teachers‟

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conceptions of GeoGebra seemed to be strongly rooted in their conceptions of the

effectiveness and infrastructure of technology. The English teachers imbued a more

positive attitude towards technology than their Taiwanese counterparts. However,

teachers in both countries expressed favourable opinions regarding GeoGebra‟s

agreeable contribution to their teaching.

Secondly, GeoGebra was commonly used as a tool for visualisation, demonstration

and interaction of mathematical topics, whereas for algebraic topics it was rarely

utilised in England. It appeared that the English teachers associated GeoGebra

primarily with geometric topics. Conversely, Taiwanese teachers worked with

GeoGebra on both geometric and algebraic topics as they did not consider algebra and

geometry to be necessarily separate; possibly as a result of the structure of Taiwanese

curriculum and textbook-oriented culture.

Thirdly, there were three different environments where teachers engaged with

GeoGebra: - preparation of teaching materials at home, presentation and interaction in

classrooms and activities for pupil investigation in IT rooms. Teacher transitions

evolved from and were influenced by the infrastructure as they moved from

preparation to presentation, incorporating interaction with pupils and finally

encouraging investigation.

In effect, GeoGebra can be implemented in upper-secondary mathematics teaching as

a network of preparation, presentation, interaction and investigation whereby teachers

mediate their practices with flexibility. Based on the findings above, I present the

general schema of this thesis (Fig. 5.1). Arguably, there is a conceptual change in

accordance with infrastructural change when technology is introduced in

mathematics teaching. Teachers are the first to encounter this re-conceptualisation of

pedagogical practices. They not only experience changes in their conceptions but also

modification of their practices when they experience the transition. This transition

would possibly alter teachers‟ choices of the mathematical scope and their uses of

GeoGebra as an educational tool in light of their new pedagogical practices.

65

Figure 5.1: The general schema of teachers‟ conceptions and practices integrating

GeoGebra

5.3 Discussion

A number of studies have highlighted significant policy-rooted pressure on and

academic support for English teachers‟ integration of technology into classroom

practice (e.g. Sutherland et al., 2004; Hennessy et al., 2005; Hayes 2007). It is

possible that this may explain why Richard and Tyler appeared positive about

technology. They seemed confident and skilful exploiting GeoGebra and this might

due to their enthusiasm for it. In Taiwan, on the other hand, educational policies

expect little by technology use with the consequence that, perhaps, study teachers

often felt unsupported in terms of infrastructure and pedagogical support. Inevitably,

organisational and pedagogical challenges of technology integration are clearly major

issues in both countries. Ruthven‟s (2008b) exposition regarding the limited success

of the existing policies and provisions of technology is supported by my findings. His

proposed three dimensions of the incorporation of new technologies into mathematics

educational practices in terms of interpretative flexibility18

, instrumental evolution19

18

Varied conceptions of technology‟s functionalities and modalities of use, discussed in Chapter 2.3.

19 The instrumental evolution of scenarios is categories as four types: a convenient parallel to paper-

and-pencil, invariant properties through visual salience under dragging, new types of solution to

familiar problems and posing novel forms of problem.

Teacher transition

Infrastructural change

Educational tool

Mathematical scope

Implementation

of GeoGebra

+

New pedagogical

practices

Re-conceptualisation

Conceptions

Practices

66

and institutional adoption20

are recognised in this study. In respect of interpretive

flexibility and instrumental evolution, I spotted the evolution of teachers‟ material

design and their propagation as a finished product and appropriation as a practical tool

from teacher transition. Evidently, Taiwanese mathematics curriculum needs further

institutional adoption of new technologies.

However, I argue that the „practical theory’21

proposed by Deaney et al. (2006) is too

idealistic in its claims as I perceived limited evidence in this study that support the

contribution of technology in terms of improving pupil motivation towards lessons.

Prominently, the Taiwanese teachers were against the idea that technology could

improve students‟ motivation and this may be due to the cultural differences in their

conceptions of technology in mathematics teaching.

According to the interviews, teachers in both countries valued the bidirectional

capability of GeoGebra as a key feature. Bidirectional interaction not only includes

the drag mode but also the inverse way of changeability in the algebraic window,

which is an improvement over DGS. DGS has been used for supporting the

development of geometrical concepts and Euclidean geometry in particular. This is

characterised by the drag mode- a dynamic modelling of the traditional paper-and-

pencil environment which allows interaction and becomes a progressively more

central salient feature in the design of DGS (Kokol-voljc, 2003; Olivero and Robutti,

2007; Ruthven et al., 2008). It is evident that GeoGebra was valued as an advanced

DGS.

In contrast, the development of CAS capabilities in GeoGebra has limited

achievement which is not only acknowledged by Hohenwarter but is also supported

by this study. The data analysis indicates that some teachers did not appear to be

aware of or intend to include CAS in their teaching practices. Owing to different

contents and levels of curricula, CAS can be seen as being more widely utilised at the

university level (Lavicza, 2007). Therefore, I argue that the implementation of CAS

20

The institutional adoption means that the official curriculum should show explicit recognition and

provide the instrumental genesis of manual tools. 21

Disscussed in Chapter 2.2.1, page 7.

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has been premature at the upper-secondary level as well as the integration of

GeoGebra in algebraic topics.

Consistent with the findings of Thompson (1984), this study ascertained that teacher

conceptions appear to play a significant role in affecting instructional decisions and

behaviour, although conceptions and practices are related in a complex way. For

instance, the Taiwanese teachers‟ conceptions were related to their practices in a

contradictory manner as they were negative about general technology integration but

enthusiastic about using GeoGebra in mathematics teaching, while the English

teachers‟ conceptions and practices were linked in a straightforward manner.

Uniting the use of technology in mathematics teaching, many factors appear to

interact with teachers‟ conceptions, decisions and behaviour. These could be, for

example, their choices of mathematical software and pedagogical issues linking

mathematical content knowledge and technology implementation. In accordance with

the study of Almas and Krumsvik (2008), for teachers who do not feel comfortable

with changes in classrooms, their teaching practices stay the same. This reflects one

of the reasons why the Taiwanese teacher Jay did not change his practice in years.

Furthermore, Almas and Krumsvik (2008) suggested that teachers are more likely to

develop a digital pedagogical content knowledge in technology-rich classrooms. This

can be evident by the English teachers‟ willingness for pursuing pedagogical

development as they worked under better technology-assisted environments.

5.4 Reflections and Limitations

My experience of being a mathematics teacher helped me understand the cultural

context, educational system and curriculum structure in Taiwan. I felt confident and

comfortable speaking in my first language to the Taiwanese teachers who had similar

interests in the field of mathematics and technology. Fortunately, I have been

involved in the GeoGebra NCETM and IGI projects, workshops, seminars and

conferences, gaining insight into the workings of GeoGebra in England. Although I

interviewed the teachers with English as a second language, I would argue that I had

accumulated a degree of knowledge about mathematical practices during my previous

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postgraduate study in mathematics education in England, and therefore, was not

wholly an outsider. However, the roles I had experienced in Taiwan and England

were different and could have been differentiated from the role of a researcher. These

involvements might have biased the way I conducted the study, therefore, I have

striven to maintain the validity and reliability by data triangulation.

The study‟s limitations stemmed from the exploratory nature of my research- there are

difficulties in determining teachers‟ practices using GeoGebra through interviews.

Interviews with teachers might not reveal a great detail about the existing relationship

between teaching and GeoGebra. Due to limitation of resources and time, the study

could not be as in-depth as hoped for. More time spent in conversation with

informants and increased observation of teaching would have been advantageous as it

would have allowed me to identify discrepancies between their conceptions and

practices more clearly. Language barriers and preconceptions of foreign researchers,

and social expectations could have influenced in probing and prompting during the

interviews. In addition, the Taiwanese teachers were given longer notice, thus, more

time for preparation might have affected the quality of the data. Consequently, the

data collected in England was less detailed compared to that collated in Taiwan.

Moreover, there is a limitation to my study as GeoGebra has only recently been

published. Most teachers in both Taiwan and England were at the stage of learning

and exploring rather than having fully implemented GeoGebra into their classroom

practices. Any research has the potential for follow-up studies, where it would be

constructive to look in depth at teachers‟ teaching practices in relation to students‟

mathematics learning processes using GeoGebra.

5.6 Implications and Recommendations for Further Research

The research findings helped me gain a better understanding of teachers‟ pioneering

use and extract the potentialities of GeoGebra elaboration in practice. I recommend

three areas that need further research- teacher pedagogical development with

GeoGebra, evaluation into GeoGebra integration in teaching practices and cultural

implications.

69

Becoming advanced users of GeoGebra or role models of students‟ use of GeoGebra‟

has become the next challenge of teachers‟ practices. According to the findings,

teachers encountered difficulties in delivering mathematical knowledge through

GeoGebra. Thus, pedagogical issues of GeoGebra in teaching practices have become

apparent. Teaching practices of presentational skills such as step-by-step

constructions and ready-made demonstrations are also in need of examination. In

addition, there is also a demand for the development of pedagogical support in terms

of linking affordances of CAS and DGS.

The pedagogy of GeoGebra should not be limited for presentation, as using it beyond

demonstration contributes to exploring challenges and potentialities of GeoGebra

implementation. To mediate GeoGebra in an interactive environment, it is important

for students to follow the mathematics step-by-step processes slowly; learning to

engage and then achieve results, working alongside GeoGebra. Consequently, a

further recommended study investigating students‟ coursework in making conjectures

and testing whilst proving their finding, would also be favourable.

Another area of investigation is whether the educational infrastructure, school

academic objectives and individual student mathematics level of attainment and

achievement influence how teachers deliver their mathematics lessons using

GeoGebra. Does teachers‟ personal preference of using GeoGebra for

experimentation along with their mathematics content knowledge influence their

teaching behaviours? What factors contribute to teachers‟ creativity in presenting and

interacting mathematical ideas with GeoGebra? Do the educational implications of

GeoGebra being open-source software with the nature of internationally-shared

materials significantly contribute to factors?

Several features and functional tools of GeoGebra, such as hiding and then revealing

strategies, as well as the bidirectional interaction of the drag mode are incorporated as

a way of evaluating teachers‟ effectiveness using the programme. Using a certain

criteria, teachers‟ abilities to utilise GeoGebra‟s capabilities can be monitored.

Hohenwarter and Preinder (2007) have developed a handbook containing professional

knowledge in using GeoGebra. The next stage is to ensure these guidelines are

70

followed by teachers resulting in sufficient implementation of GeoGebra into

classroom practices. With more direction regarding successful teacher practices with

GeoGebra, curriculum-focused mathematical topics could then be incorporated.

The Taiwanese educational system has an examination-driven culture which impacts

on teachers‟ mediation of GeoGebra. Despite the highest student mathematics

performances internationally, are Taiwanese teachers applying their excellent

mathematics content knowledge to using GeoGebra to teach? The effectiveness of

Taiwanese teachers‟ extensive applications of revision, problem-solving and

geometrical proofs to enhance learning needs examination.

Moreover, learning from different cultural contexts is useful for pedagogical

improvement. It would be significant to compare the different ways Taiwanese and

English teachers benefit from GeoGebra and learn from each other. For example, the

study has indicated that the English teachers were likely to be more flexible and

creative with their engagement exploiting GeoGebra and the Taiwanese teachers were

skilful in implementing GeoGebra into examination and textbook to improve

students‟ mathematical achievement. More research on cultural exchange of

GeoGebra implementation could benefit pedagogical development.

Suggestions for my proposed PhD research

Within the recommended areas for further research, I would like to extend my future

work on the cross-cultural investigation of pedagogical application of GeoGebra in

relation to teaching practices. To research further, I propose to study how GeoGebra

support upper-secondary teachers‟ teaching practices and explore the underlying

mathematics in depth and how these representations enhance students‟ learning.

Answering these questions and achieving the stated objectives will contribute to the

growing development of GeoGebra usage in designing geometrical and algebraic

pedagogy that promotes mathematics learning.

71

5.6 Concluding thoughts

The journey exploring teachers‟ conceptions in relation to their use of GeoGebra has

been stimulating. I have come to acknowledge that GeoGebra could be used more

than as merely a tool, in line with Markus‟s expectation, Peter‟s suggestion and

Tyler‟s utterance- it can be an environment where teachers and students collaborate

for the creation of complete pieces of mathematical work. Implementing GeoGebra in

classroom practices effectively will result in a plethora of mathematical ideas,

thoughts, conjectures and investigation between teachers and students.

Despite the potentiality of GeoGebra, teachers have not fully discovered its capability

to link geometry and algebra but acknowledged that it offers pervading possibility in

teaching practices. As Markus Hohenwarter puts it, „GeoGebra is free software

because I believe education should be free. This philosophy makes it easy to convince

teachers to give this tool a try, even if they haven‟t used technology in their

classrooms before‟.

With the widespread idea of using open-source software, there is evidence showing

that GeoGebra is widely used across the world. However, research into its mainstream

use is still limited. I would like to conclude by highlighting the importance of

instrumental dimensions and the underlying mathematics within the use of GeoGebra

and the crucial role of interface features, the underlying mathematics and the

pedagogical possibilities of open-source software integration into teaching practices.

72

REFERENCES:

Almas, A. G, and Krumsvik, R. (2008) Teaching in Technology-Rich Classrooms:

is there a gap between teachers‟ intentions and ICT practices?, Research in

Comparative and International Education, 3 (2), 103-121.

Alvesson, M. (2002) Postmodernism and Social Research, Chapter 6,

(Buckingham, Open University Press).

Andrews, P. (1996) The impact of first computer encounters on mathematics

teachers‟ computer competence, Journal of Information Technology and Teacher

Education, 5(3), 301-316.

Andrews, P. (2007) Negotiating meaning in cross-national studies of mathematics

teaching: kissing frogs to find princes, Comparative Education, 43(4), 489-509.

Andrews, P. and Sayers, J. (2004) International comparisons of mathematics

teaching: searching for consensus in describing opportunities for learning, British

Educational Research Association Annual Conference, University of Manchester.

Artigue M. (2002) Learning Mathematics in a CAS environment: the genesis of a

reflection about instrumentation and the dialectics between technical and

conceptual work. International Journal of Computers for Mathematical Learning,

7(3), 245-274.

Arzarello, F. (2005) Technology and mathematics in the classroom: lights and

shadows, CIEAEM 57 Oral presentations in Working Groups 23, 23-27.

Atiyah, M. (2001) Mathematics in the 20th Century: geometry versus algebra,

Mathematics Today, 37(2), 46-53.

Bassey, M. (1999) Case Study Research in Educational Settings (Open University

Press).

Becker, H. J. (2001) How are teachers using computers in instructions? Meetings

of the American Educational Research Association: National Survey Report,

Irvine, CA: Center for Research on Information Technology and Organizations,

University of California Irvine.

Becker, J. P., Sawada, T., and Shimizu, Y. (1999) Some findings of the Us-Japan

cross-cultural research on students‟ problem-solving behaviors, in: Kaiser, G.,

Luna, E. and Huntley, I. (Eds) International Comparisons in Mathematics

Education, 121–139 (London, Falmer Press).

Blumer, H. (1969) Symbolic Interactionism: Perspective and Method, (Berkeley:

University of California Press).

Bogdan, R. G. and Biklen, S. K. (1992) Qualitative Research for Education, 2nd

edition (Boston, MA: Allyn and Bacon).

73

Bramald, R., Miller, J. and Higgins, S. (2000) ICT, Mathematics and effective

teaching, Mathematics Education Review, 12, 1-13.

Bransford, J., Brown, A. and Cocking, R. (1999) How people learn (Washington,

DC: National Academy Press).

British Education Communication Technology Agency (Becta) (2004) Using

Web-based Resources in Secondary Mathematics (Coventry: British Educational

Communications and Technology Agency).

Broadfoot, P. (2000) Comparative education for the 21st century: retrospect and

prospect, Comparative Education, 36 (3), 357-372.

Brown, G., Cadman, K., Cain, D., Clark-Jeavons, A., Fentem, R., Foster, A.,

Jones, K., Oldknow, A., Taylor, R. and Wright, D. (2005) ICT and mathematics: a

guide to learning and teaching mathematics 11-19. Leicester, UK, Mathematical

Association, 80, http://eprints.soton.ac.uk/41376/.

Brown, R. (2001) Computer Algebra Systems and the challenge of assessment.

The International Journal of Computer Algebra in Mathematics Education, 8(4),

295-308.

Cohen, L., Manion, L. and Morrison, K. (2007) Research Methods in Education,

6th

edition, (London: Routledge).

Conway, P. F. and Sloane, F. C. (2005) International Trends in Post-Primary

Mathematics Education, National Council for Curriculum and Assessment

(NCCA).

Creswell, J. W. (2003) Research Design – Qualitative, Quantitative, and Mixed

Methods Approaches, 2nd

edition (Sage Publications Inc., Thousand Oaks).

Creswell, J.W. (2007) Qualitative Inquiry and Research Design: Choosing among

five traditions, 2nd

edition (Thousand Oaks, London, New Delhi, Ca: Sage).

Crotty, M. (1998) The Foundations of Social Research: Meaning and Perspective

in the Research Process, (London: Sage).

Cuban, L., Kirkpatrick, H., Peck, C. (2001) High access and low use of

technologies in high school classrooms: Explaining the apparent paradox.

American Educational Research Journal, 38(4), 813-834.

Cuoco A. (2002) Thoughts on reading Artigue‟s learning mathematics in a CAS

environment, International Journal of Computers for Mathematical Learning,

7(3), 293-299.

Deaney, R., Ruthven, K., and Hennessy, S. (2006) Teachers‟ developing „practical

theories‟ of the contribution of information and communication technologies to

subject teaching and learning: an analysis of cases from English secondary

schools, British Educational Research Journal, 32(3), 459–480.

74

Denzin, N. K. and Lincoln, Y. S. (2008) Collecting and Interpreting Qualitative

Materials, 3rd

edition (Los Angeles, London, New Delhi, Singapore: Sage).

Dey, I. (1993) Qualitative Data Analysis- A user-friendly guide for social sciences,

(Routldge: New York).

Drijvers, P. Herwaarden, O. (2001) Instrumentation of ICT Tools: The Case of

Algebra in a Computer Algebra Environment. International Journal of Computer

Algebra in Mathematics Education, 7 (4), 255-275.

Dubinsky, E. and Harel, G. (1992) The nature of the process conception of

function. In Harel, G and Dubinsky, E., The concept of function aspects of

epistemology and pedagogy (Washington, D.C.: Mathematical Association of

America), 85–106.

Edwards, J. A. and Jones, K. (2006) Linking geometry and algebra with

GeoGebra, Mathematics Teaching, incorporating MicroMath, 194, 28-30.

Ernest, P. (1989) The knowledge, beliefs and attitudes of the mathematics teacher:

a model. Journal of Education for Teaching, 15(1), 13–33.

Fey, J. T., Heid, M. K., Good, R. A., Sheets, C., Blume, G. W., and Zbiek, R. M.

(1995) Concepts in algebra: A technological approach. Dedham, (MA, Janson

Publications, Inc).

Gillham, B. (2000) Case Study Research Methods (London, Continuum).

Givvin, K., Hiebert, J., Jacobs, J., Hollingworth, H. and Gallimore, R. (2005) Are

there national patterns of teaching? Evidence from the TIMSS 1999 video study,

Comparative Education Review, 49(3), 311–343.

Glaser, B and Strauss, A. (1967) The discovery of grounded theory, (Chicago,

Aldine).

Godwin, S. and Sutherland, R. (2004) Whole class technology for learning

mathematics: the case of functions and graphs, Education, Communication and

Information, 4 (1), 131-152.

Graf, K. and Leung, K. (2001) Mathematics education in different cultural

traditions: a comparative study of East Asia and the West, ICMI Comparative

Study Discussion Document.

Guba, E. G., and Lincoln, Y. S. (1988) Do inquiry paradigms imply inquiry

methodology? In D. M. Fetterman (Ed.), Qualitative approaches to evaluation in

education, 89-115. (New York: Praeger).

Gülbahar, Y. (2007) Technology planning: A roadmap to successful technology

integration in schools, Computers and Education, 49 (2007) 943–956

Haapasalo, L. (2007) Adapting mathematics education to the needs of ICT, The

Electronic Journal of Mathematics and Technology, 1 (1).

75

Haapasalo, L. and Silfverberg, H. (2007) Technology enriched mathematics

education. To appear in E. Pehkonen, M. Ahtee, J. Lavonen (Eds.) How Finns

Learn Mathematics and Science. (Sense Publishing, The Netherlands).

Harris, G. A. (2000) The use of a Computer Algebra System in capstone

mathematics courses for undergraduate mathematics majors. The International

Journal of Computer Algebra in Mathematics Education, 7 (1), 33-62.

Healy, L. and Hoyles, C. (2001) Software tools for geometrical problem solving:

Potentials and pitfalls. International Journal of Computers for Mathematical

Learning, 6, 3.

Heinze, A., Cheng, Y.-H. and Yang, K.-L (2007) Students' performance in

reasoning and proof in Taiwan and Germany: Results, paradoxes and open

questions, ZDM, Springer Berlin / Heidelberg.

Hennessy, S., Deaney, R. and Ruthven, K. (2003) Pedagogic Strategies for Using

ICT to Support Subject Teaching and Learning: An Analysis Across 15 Case

Studies (Faculty of Education, University of Cambridge).

Hennessy, S., Ruthven, K., and Brindley, S. (2005) Teacher perspectives on

integrating ICT into subject teaching: commitment, constraints, caution, and

change. Journal of Curriculum Studies, 37(2), 155-192.

Hershkovitz, R. and Schwarz, B.B. (1999) Reflective processes in a technology

based mathematics classroom. Cognition and Instruction, 17, 665–91.

Hershkovitz, R., Dreyfus, T., Ben-Zvi, D., Freidlander, A., Hadas, N., Resnick, T.,

Tabach, M. and Schwarz, B. (2002) Mathematics curriculum development for

computerized environments: A designer-researcher-teacher-learner-activity. In

English, L. (Eds), Handbook of International Research in Mathematics Education,

(Mahwah, New Jersey and London: Lawrence Erlbaum Associates).

Hewitt, D. (2005) Thinking Numerically, in Johnston-Wilder and Pimm (Ed.)

Teaching Secondary Mathematics with ICT (London: The Open University Press).

Hino, K., Kaiser, G. and Knipping, C. (2002) Comparing Teaching Mathematics

in Eastern and Western traditions – looking at France, Germany, England and

Japan. In: Pre-conference Proceedings of the ICMI Comparative Study

Conference, 17-22.

Hofstede, G. (1986) Cultural differences in teaching and learning, International

Journal of Intercultural Relations, 10, 301–320.

Hohenwarter, M. (2004) Bidirectional Dynamic Geometry and Algebra with

GeoGebra, http://www.geogebra.at

Hohenwarter, M. (2006) Dynamic investigation of functions using GeoGebra,

Proceedings of Dresden International Symposium on Technology and its

Integration into Mathematics Education, Dresden, Germany: DES-TIME

76

Hohenwarter, M. and Fuchs, K. (2004) Combination of Dynamic Geometry,

Algebra and Calculus in the Software System GeoGebra, in Computer Algebra

Systems and Dynamic Geometry Systems in Mathematics Teaching Conference,

P´ecs, Hungary

Hohenwarter, M. and Jones, K. (2007) BSRLM Geometry Working Group: Ways

of linking geometry and algebra: the case of GeoGebra, in D. Küchemann (Ed.)

Proceedings of the British Society for Research into Learning Mathematics, 27 (3),

126-131.

Hohenwarter, M. and Lavicza, Z. (2007) Mathematics Teacher Development with

ICT: Towards an International GeoGebra Institute, in D. Küchemann (Ed.)

Proceedings of the British Society for Research into Learning Mathematics, 27 (3),

49-54.

Hohenwarter, M. and Preiner, J. (2007) Design Guidelines for Dynamic

Mathematics Worksheets, the Proceedings of the CADGME Conference

Hohenwarter, M., Preiner, J, and Yi, Taeil. (2007) Incorporating GeoGebra into

teaching mathematics at the college level. (pdf) Proceedings of the International

Conference for Technology in Collegiate Mathematics 2007, Boston, USA:

ICTCM

Hohenwarter, M., Jarvis, D. and Lavicza, Z. (2008) Report of the First Meeting of

the International GeoGebra Institute, May 2008, Faculty of Education, University

of Cambridge.

Howe, K. (2003). Closing methodological divides: Towards democratic

educational research, (Dordrecht, Netherlands: Kluwer).

Hoyles C. and Jones K. (1998) Proof in dynamic geometry contexts In:

Perspectives on the Teaching of Geometry for the 21st century-An ICMI study, C.

Mammana and V. Villani (eds.) (chap. 4, section II, 121-128) (Dordrecht: Kluwer

Academic Publisher).

Hung, Y.-W., and Hsu, Y.-S. (2007) Examining teachers' CBT use in the

classroom: a study in secondary schools in Taiwan, Educational Technology and

Society, 10 (3), 233-246.

International Commission on Mathematical Instruction (ICMI) (2001): ICMI

Comparative Study: Mathematics education in different cultural traditions: A

comparative study of East Asia and the West, Discussion Document. Zentralblatt

Didaktik der Mathematik, 33(4), 113 -122

Jacques, M. (1996) Taiwan: a case study, Presentation to Home Lessons from

Abroad: learning from international experience in education. Neil Stewart

Associates Conference, Hammersmith, London.

John P. D, and Sutherland, R. (2004) Teaching and Learning with ICT: new

technology, new pedagogy? Faculty of Education, University of Plymouth, UK

77

Johnston-Wilder and Pimm (2005) Teaching secondary mathematics with ICT,

(London: The Open University Press).

Jones, K. (2000) Providing a Foundation for Deductive Reasoning: Students'

Interpretations when Using Dynamic Geometry Software and Their Evolving

Mathematical Explanations, Educational Studies in Mathematics, 44 (1-3), 55–85.

Kaiser, G. (1999a) „International comparisons in mathematics education under the

perspective of comparative education‟ In G. Kaiser, E. Luna and I. Huntley (eds),

International Comparisons in Mathematics Education, (London: Falmer), 3–17.

Kaiser, G. (1999b) Comparative studies on teaching mathematics in England and

Germany In G. Kaiser, E. Luna and I. Huntley (eds), International Comparisons

in Mathematics Education, (London: Falmer),140–150

Kaiser, G.; Hino, K.; Knipping, C. (2005) Proposal for a framework to analyse

mathematics education in Eastern and Western traditions – looking at England,

France, Germany and Japan. To appear in: K.D. Graf and F. Leung (Eds.), ICMI

Comparative Study on Mathematics education in different cultural traditions: a

comparative study of East Asia and the West. Dordrecht: Kluwer Academics.

Kendal, M., Stacey, K., and Pierce, R. (2005) The influence of a computer algebra

environment on teachers‟ practice. In Guin, D., Ruthven, K., and Trouche, L. op.

cit., 83-112.

Kennewell, S. (2004) Meeting the standards in using ICT for secondary

teaching: a guide to the ITT NC, (London: Routledgfalmer).

Kokol-Voljc, V. (2003) Early Development of Mathematical Concepts Using

Dynamic Geometry, The 8th Asian Technology Conference in Mathematics

(ATCM 2003), Taipei, Taiwan, 202-209.

Kyriakidou, K., Chrisostomou, C. and Banks, F. (1999) Primary Teachers‟

Attitude to the Use of ICT: a comparative study between Cyprus and the UK. The

European Conference on Educational Research, Lahti, Finland, 22-25

Laborde C. (2003) The design of curriculum with technology: lessons from

projects based on dynamic geometry environments, the CAME Symposium, Reims.

Laborde, C. (1998) Relationship between the spatial and theoretical in geometry:

The role of computer dynamic representations in problem solving. In J. D. Tinsley

and D. C. Johnson (Eds.), Information and communications technologies in school

mathematics, 183–195(London: Chapman and Hall).

Laborde, C. (2001) Integration of technology in the design of geometry tasks with

Cabri-géomètre, International Journal of Computers for Mathematical Learning,

6, 283-317.

Laborde, C. (2007) The design of task talking full advantage of dynamic geometry:

what kinds of knowledge does it require from teachers? Proceedings of the 1st

Central- and Eastern European Conference on Computer Algebra- and Dynamic

Geometry Systems in Mathematics Education, Pecs, Hungary.

78

Lavicza, Z. (2007) Exploring the current and future roles of Computer Algebra

Systems in teaching mathematics at the university level – A work in progress,

MSOR Connections, 7 (1), 14-16.

Legard, R., Keegan, J. and Ward, K. (2003) Qualitative Research Practice: A

Guide for Social Science Students, Chapter 6: In-depth Interviews, (London: Sage).

Leung, F. K. S. (1995) The Mathematics Classroom in Beijing, Hong Kong and

London, Educational Studies in Mathematics, 29 (4), 297-325.

Leung, F. K. S. (2001) In search of an East Asian identity in mathematics

education, Educational Studies in Mathematics, 47, 35-51.

Lincoln, Y., and Guba, E. (1985) Naturalistic Inquiry, (Beverly Hills, Sage).

Lu, Y. W. (2007) Understanding Mathematical Representations and

Transformations of Functions and their Graphs with the Use of ICT, Proceedings

of the 31st Conference of the International Group for the Psychology of

Mathematics Education, Seoul, South Korea ,July 2007, 1, 1-262.

Marshall, C. and Rossman, G. B. (2006) Designing qualitative research, 4th

edition (Thousand Oaks, CA: Sage).

McKnight, C., Magid, A., Murphy, T. J., and McKnight, M. (2000) Mathematics

Education Research: A Guide for the Research Mathematician, (Rhode Island:

AMS).

McLead, D. B. (1993) Research on Effect in Mathematics Education: A

Reconceptualisation, In D. A. Grouws (Ed.) Handbook of Research on

Mathematics Teaching and Learning, (Macmillan Publishing Co., London), 575-

596.

Merriam, S. B. (1988) Case Study Research in Education: A Qualitative Approach.

San Fransisco: Jossey-Bass. Inc.

Merriam, S. B. (1998) Qualitative Research and Case Study in Education: A

Qualitative Approach, (San Fransisco: Jossey-Bass. Inc).

Miles, M. B. and Huberman A. M. (1984). Drawing Valid Meaning from

Qualitative Data: Toward a Shared Craft, Educational Researcher, 13(5), 20-30.

Miles, M. B., and Huberman, A. M. (1994). Qualitative Data Analysis, (Thousand

Oaks: Sage).

Monaghan, J. (1993) IT in mathematics initial teacher training-factors influencing

school experience, Journal of Computer Assisted Learning, 9(3), 149-160

Monaghan, J. (2005) Thinking algebraically, in Johnston-Wilder and Pimm (Ed.)

Teaching Secondary Mathematics with ICT, (London: The Open University Press).

Mullis, I. V. S., Martin, M. O., Gonzalez, E. J.,and Chrostowski, S.

J.(2004).TIMSS 2003 International Mathematics Report: Findings From IEA‟s

79

Trends in International Mathematics and Science Study at the Fourth and Eighth

Grades. TIMSS and PIRLS International Study Centre, Lynch School of

Education, Boston College

National Council of Teachers of Mathematics (2000) Principles and standards of

school mathematics, (NCTM Publications).

NCTM (2000) Principles and Standards for School Mathematics. Reston, VA:

The National Council of Teachers of Mathematics.

Nespor, J. (1987) The role of beliefs in the practice of teaching, Journal of

Curriculum Studies, 19(4), 317–328.

Noss, R. and Hoyles, C. (1996) Windows on mathematics meanings：learning

cultures and computers, (Netherlands：Kluwer Academic Publishers).

Noss, R. and Hoyles, C. (2003) What can digital technologies take from and bring

to research in mathematics education? In Bishop, A. J. Clements, M. A., Keitel, C.,

Kilpatrick J. and Leung, F. K. S (Eds.), Second International Handbook of

Mathematics Education, (Dordrecht, The Netherlands: Kluwer), 1, 323-350.

Office for Standards in Education (Ofsted) (2004) 2004 Report: ICT in schools-

the Impact of Government Initiatives. Secondary Mathematics, (London: Office

for Standards in Education).

Olivero, F. (2003) Cabri as a shared workspace within the proving process.

Proceedings of the 28th Conference of the International Group for the Psychology

of Mathematics Education, 3, 429 – 436.

Olivero, F. and Robutti, O. (2007) Measuring in dynamic geometry environments

as a tool for conjecturing and proving, International Journal of Computer

Mathematics Learning, 12, 135-156.

Organisation of Economic Cooperation and Development (2004). Learning for

tomorrow’s world -First results from PISA 2003. Paris: OECD.

Osborn, M. (2004) New methodologies for comparative research? Establishing

„constants‟ and „contexts‟ in educational experience, Oxford Review of Education,

30(2), 265–285.

Osborn, M., Broadfoot, P., Planel, C. and Sharpe, K. (2000) Promoting Quality in

Learning. Does England Have the Answer? Findings from the QUEST project

(London, Cassell).

Papert, S. (1996) An Exploration in the Space of Mathematics Education,

International Journal of Computers for Mathematical Learning, 1(1), 95-123.

Patton, M. Q. (1980) Qualitative Evaluation Methods (Berverly Hills, CA: Sage).

Patton, M. Q. (1990) Qualitative evaluation and research methods, 2nd

edition

(Newbury Park, CA: Sage).

80

Pederson, J. (1983) „Why We Still Need to Teach Geometry.‟ In Proceedings of

the Fourth International Congress on Mathematical Education (Boston:

Birkhauser Boston), 158-159.

Philips, D.C. and Burbules. N. C. (2000) Postpositivism and Educational

Research, (Lanham and Boulder: Rowman and Littlefield Publishers).

Prais, S. (2007) Two recent (2003) International surveys of schooling attainments

in mathematics: England's problems, Oxford Review of Education, 33 (1), 33-46.

Robson, C. (2002) Real World Research, 2nd edition (Oxford: Blackwell).

Ruthven K. (2002) Instrumenting mathematical activity: reflections on key studies

of the educational use of computer algebra systems, International Journal of

Computers for Mathematical Learning, 7, 275-291.

Ruthven, K. (1990) The influence of graphic calculator use on translation from

graphic to symbolic forms. Educational Studies in Mathematics, 21 (5), 431-450.

Ruthven, K. (2005) Expanding Current Practice in Using Dynamic Geometry to

Teach about Angle Properties, Micromath, 21(2), 26-30.

Ruthven, K. (2008a) The interpretative flexibility, instrumental evolution and

institutional adoption of mathematical software in educational practice: the

examples of computer algebra and dynamic geometry, the Annual Meeting of the

American Educational Research Association, March 2008, New York.

Ruthven, K. (2008b) The Didactical Challenge of Technology Integration in

School Mathematics, the Proceedings of ELEM08, in press.

Ruthven, K. and Hennessy, 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), 1-148.

Ruthven, K., Hennessy, S. and Deaney, R. (2008) Constructions of dynamic

geometry: A study of the interpretive flexibility of educational software in

classroom practice, Computer and Education, 51, 297-317.

Ruthven, K., Hennessy, S., and Brindley, S. (2004) Teacher representation of the

successful use of computer-based tools and resources in secondary-school English,

mathematics and science. Teaching and Teacher Education, 20, 259-275.

Sangwin, C. (2007) A brief review of GeoGebra: dynamic mathematics, MSOR

Connections, 7(2), 36-38.

Schmidt, W. H., Jorde, D. and Cogan, L. S. et al. (1996) Characterizing

pedagogical flow: an investigation of mathematics and science teaching in six

countries (Dordrecht, Kluwer).

Schneider, E. (2007) CAS a didactical challenge, Proceedings of the 1st Central-

and Eastern European Conference on Computer Algebra- and Dynamic Geometry

Systems in Mathematics Education, Pecs, Hungary.

81

Schumann, H. and Green, D. (2000) „New protocols for solving geometric

calculation problems incorporating dynamic geometry and computer algebra

software‟ International Journal of Mathematical Education in Science and

Technology, 31(3), 319–339.

Schwartz, J. L. (1999) Can technology help us make mathematics curriculum

intellectually stimulating and socially responsible? International Journal of

Computers for Mathematical Learning, 4, 99-119.

Scott, D. and Morrison, M. (2005) Key Issues in Educational Research, (London:

Continuum Press).

Stake, R. E. (2006) Multiple Case Study Analysis, New York: Guilford.

Stake, R.E. (1995) The Art of Case Study Research, Thousand Oaks, Ca.: Sage

Stark, S. and Torrance, H. (2000) Case Study In B. Somekh and C. Lewin (eds.).

33-40 Research methods in the social sciences, (London: Sage).

Steen, L. A. (1988). The science of patterns. Science, 240, 611-616.

Stigler, J. W. and Hiebert, J. (1999) The teaching gap: best ideas from the world's

teachers for improving education in the classroom, (New York: The Free Press).

Strässer, R. (2001) Cabri -geometre: Does Dynamic Geometry Software (DGS)

Change Geometry and its Teaching and Learning? International Journal of

Computers for Mathematical Learning, 6(3), 319-333.

Sutherland, R., Armstrong, V., Barnes, S., Brawn, R., Breeze, N., Gall, M.,

Matthewman, S., Olivero, F. Taylor, A., Triggs, P., Wishart, J. and John, P. (2004)

Transforming teaching and learning: embedding ICT into everyday classroom

practices, Journal of Computer Assisted Learning, 20, 413–425.

Suzuki, J. (2006) The Open Source Revolution. In: Focus, 26(6)

Thompson, A. G. (1984) The relationship of teachers‟ conceptions of mathematics

andmathematics teaching instructional practice. Educational Studies in

Mathematics, 15(2), 105–127.

Thompson, A. G. (1992) Teachers‟ beliefs and conceptions: a synthesis of the

research. In D. A. Grouws (ed.), Handbook of Research on Mathematics Learning

and Teaching (New York: Macmillan), 127–146.

Tweed, R. and Lehman, D. (2002) Learning considered within a cultural context:

Confucian and Socratic approaches, American Psychologist, 57(2), 89–99.

Van Hippel, E. and Von Krogh, G. (2003) Open Source Software and the „Private-

Collective‟ Innovation Model: Issues for Organization Science Organization

Science, Sloan School of Management, Massachusetts Institute of Technology, 14

(2), 209-223.

82

Warwick, D. and Osherson, S. (Ed.) (1973) Comparative research methods: an

overview, (Englewood Cliffs, NJ, Prentice Hall).

Webb, M and Cox, M (2004) A Review of Pedagogy Related to Information and

Communications Technology, Technology, Pedagogy and Education, 13(3), 235-

286.

Willis, J. W. (2007) Foundations of Qualitative Research: Interpretive and

Critical Approaches. (London: Sage).

Wong, N. Y., Han, J., and Lee P. Y. (2004) The mathematics curriculum:

Towards globalization or westernization? In L. Fan, N. Y. Wong, J. Cai, and S.

Li (Eds.), How Chinese learn mathematics: Perspectives from insiders, Singapore:

World Scientific, 27-70.

Wright, D. (2005) Graphical calculators’ in Teaching Secondary Mathematics

with ICT, in Johnston-Wilder and Pimm (ed). (London: The Open University

Press).

Yen, Y. L. (2003) A study of dynamic mathematical exploring in GSP, the 8th

Asian Technology Conference in Mathematics, Chung Hua University, Hsin-Chu,

Taiwan, 144-152

Yerushalmy, M. (2004) Does technology transform the content of algebra

curricula? An analysis of critical transitions for learning and teaching, ICME 10.

Yerushalmy, M. (2005) Function of interactive visual representations in

interactive mathematical textbooks. International Journal of Computers for

Mathematical Learning, 10 (3), 217-249.

Yin, R. K. (2003) Case Study Research: Design and Methods, 3rd edition

(London: Sage).

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APPENDICES:

Appendix I:

Interview Transcripts

Jay

Interviewer (I): 請問你大概教書幾年?

Jay (J):十多年了 正確來講應該十二年吧 其中有幾年跑掉

I: 去哪?

J: 去念碩士 跑去電腦公司工作

I: 所以你會跟資訊…

J: 有很大的關係

I: 你在電腦公司做什麼?

J: 做 SA 所謂的系統工程師 不過 後來轉去做翻譯部門 去美國 IBM 做像當時

Windows 95 的中文化諸如此類的 像中文化的工作 我有去支援

I: 是否跟參與 GeoGebra 翻譯有關?

J: 當然 一些工作經驗跟這個可以直接可以套用得上 不然有時候我們在翻譯東西

的時候不知道介面在哪裡 我們有這個經驗之後會比較知道我們翻譯是什麼東西

會比較精準 有些人在翻譯的時候不知道這個東西出現是在某個地方 那當然照字

面上翻譯事 OK 的 但套到那個地方就不是那麼合用

I: 像翻譯 WIKI 若知識不足是否不容易上手?

J: 入門要有一點時間 我也是摸了一兩個月 那個 WIKI 的部分 但是我是先玩

wikipedia 上面自己玩一玩 反正他有 sam BOX 可以變來變去 一兩個月就上手 還

好是可以直接編 線上就看 看到成果

I: 可以先完練習版的

J: 習慣就還好 雖然還是有些是手打的 不是按個按鈕就出來 比較不是那麼方便

但還是可以用啦!

I: 這裡你的學生是高中幾年級…?

J: 每年都會輪 今年是二年級 明年是三年級 每年都會輪 不像建中是固定

I: 所以你會去跨國中嗎?

J: 我們是分開的 除非是過去支援 聘書是分開的 國中老師上高中教也要經過考

試 目前還在協調便於老師的方式

I: 學生大部分的程度和軟體的接受度?

J: 我們學校大概以台北公立學校算是中後面 以學生數學程度不能算很高 但考得

上公立也不算低。 那軟體的接受度 以校內整個環境來講 也不是那麼方便 說可

以隨時操作 但我上課有時候會帶筆電去 那講到比方說...現在在上圓錐曲線 你們

用手畫半天很麻煩 所以常用這個 輸入方程式圖形就出來了 不用手畫半天 所以

學生會覺得很炫 跟他們說上那個網站 他們也會去 回家也是會用 所以漸漸得有

一些學生知道

I: 教室設備?

J: 投螢幕是有啦 但是因為投影機要自己帶 現行情況下 帶投影機去 東西架設起

來 開機 接好所有的線 十幾分鐘過了 下課還要再弄一次 所以沒有人會這麼做...

也不是沒有人這麼做 是會這麼做 只是很少啦! 現在是有規劃二年級每個教室安

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裝投影機 但是電腦還是要自己帶 這會比較便利 但是還是有些小缺憾 因為學生

只能看 不能實際操作

I: 在英國有在教室上課 還有 ICT 課 學生去電腦教室 有一個 package 可以練習

這樣的配套措施 是否國內有出版社附的整個做好的光碟 可以讓學生練習 或是

要回家使用?

J: 對 對 對 我知道的有些軟體在網路上 類似 web conference 就是老師在線上 學

生也可以在線上 可以同時操作 老師可以講 在筆電上可以直接操作 學生可以看

到 現在是有這種系統 只是沒看到有人這樣做而已

I: 凡事都有人先開始做 那之後再...?

J: 這個系統如果架設得好的話 反正學生在家 老師在家裡 也彼此可以玩 只是說

現在環境也許不是每一個人家裡都有電腦 所以那是理想狀況 那也許學校的環境

可以這樣架設...

I: 還是有時候 也許學生回去的話 他們也不一定會真的自己去玩 所以有環境讓

他們操作比較好?

J: 我們頂多是在專科教室或電腦教室裡面可以這樣做 但實際操作的時候 會發現

老師一廂情願的成分頗大

I: 這我可以理解

J: 對 你想要她們做什麼 他們可不這樣想 這是主要的問題 因為教學現場 你老師

都覺得自己主導一切 事實上不然 其實是學生主導一切 他們主導自己的路 我們

有時候可以介紹學生說有這種東西 至於他們會不會去用 有沒有興趣再去研究

那是另一回事。 現在台灣有一種反動的現象 他會看到這種東西 但是他會想說

我數學都學不完了 我還弄這個 哪有時間再去學那些東西 我們認為這是進步的

但他們認為是退步的 我不用花這麼多時間在這上面 有一部分的人是這樣想的

I: 現在這個軟體 你大概怎麼使用?

J: 一般來講 我們通常在介紹到有關座標系 方程式 方面的東西 為了方便起見 我

會拿這個軟體來畫圖有時甚至於作計算 因為他本身其實的命令列的計算 也是蠻

方便 主要是一種方便的工具啦…嗯… 那當然有時候有些數學內容是動態性的東

西 像比方說現在有些討論在軌跡的部分 在課本上軌跡是靜態的描述 學生很難

去理解 那就用這個 來展現給他們看 我們再拉一個點的時候 另一個點會怎麼動

這樣他就會感覺到比較能夠融入所描述的情景 不然很難理解

I: 那你主要就是教學的時候使用

J: 對

I: 請問是否使用學習單或是做好的活動來用 還是直接畫給他們看?

J: 我都沒有做那個 只要是把他拿來當做教學或研究的工具 當然那現在不只是那

個 我還有翻譯另外兩套的 3D 軟體 也希望可以用得上 像比方說最近的圓錐曲線

用圓錐去切 學生怎麼想也想不出來 他切出來會是那個東西 所以想說把我在翻

譯的 3D 軟體 給他們看 放在螢幕上可以轉來轉去 也許會更容易切入一些觀念

所以我的定位很簡單 就是把他當工具使用

I: 那 3D 的軟體是什麼?

J: 目前我翻譯的 還有像我們學校友買一套是 Cabri 3D

I: Cabri-Geometre?

J: 對

I: 另外還有我再翻譯的德國人做的 Calques 3D 不對 這是英國人做的 另一個才是

德國人做的 Archimedes 3D

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J: 你怎麼知道這些軟體的?

I: 用 google 搜尋

J: 這個不錯 他操作的方式有點類似 GeoGebra 其實很像 因為它本身有(工區?)

不過這不是免費軟體 因為他功能還不錯 我主動跟作者提說要翻譯這個軟體 他

有給我一些 feedback 目前在翻譯中 我們數學有很多是動態的 空間的 黑板在那

邊畫其實不是太好的辦法 有工具當然就要拿出來用

I: 你說你在教學生軌跡的部分是怎麼用?

J: 如果舉一個例子好了 我看要用什麼例子比較好 比方說橢圓好了 橢圓本身就

是一個軌跡了 假設隨便畫一個

J: 像我們橢圓的題目 我們參考書上常有這種題目: 一個點在圓上面跑 然後拉這

種線段作中垂線 跟另外的線段取交點 問學生這個點的軌跡是什麼? 當然剛碰到

的學生就傻眼了 光聽這些敘述學生就已經暈掉了 實際上話給她們看的話 會比

較有感覺 但推論出來軌跡是橢圓還是有一段小路啦 不過至少它們可以看得出來

它會這樣子跑 真的會繞一個橢圓的軌跡在跑 當然這種東西在黑板上是很難表現

的 這個就是它比較好用的地方 至少這個工具可以作用來展示 所為軌跡 已前沒

有這種軟體的時候 很難去體會什麼是軌跡 你只看到一個點阿 哪有軌跡 也很難

體會什麼是動點 有這個的話 就會方便很多 不過僅止於主動學習的同學 它們會

想要去發掘這種現象 那不主動學習的同學 當然還是沒有用

I: 所以你先會介紹學生有這個軟體 然後這是免費的 可以上網下載嗎?

J: 對 對 所以她們會回家 按照網址 去直接點 web start 的那個地方 就可以啟動 對

於會去使用的同學 多了一個工具 就好像多了一台腳踏車一樣 才可以跑快一點

所以我覺得說數學本身也是一種技藝啦 工欲善其事 必先利其器 對…

I: 那學生看到之後 知道這些軌跡 要怎麼讓學生以後遇到類似的題目 可以推想

到軌跡?

J: 當這些東西她們如果真正要用的時候 其實還是要叫她們釣魚拉 不能餵魚給她

吃 就像老師們也不直接將教學檔案下載 就直接使用 其實還是要知道內部的ㄧ

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些機制是怎麼作的 所以學生也是應該如此 也不能只看老師弄一弄 然後就出來

如果是一個這樣的學生 以後不會有什麼發展 至於如果要去作一些研究的話 她

當然要知道怎麼去作某些圖 有些圖不是... 你說要作中垂線 要什麼哪裡跟哪裡等

長 這是觀念上的 但是你要用工具去畫 又是另外一回事了 就像說 你如果要去作

一個 比方說要作一條線 一個圓 要作另一個圓和這兩個東西相切 我們當然如果

在紙上畫圖的話 隨便畫一畫 看起來就相切了 但是要用這軟體作精確的相切的

話 其實要有一些數學底子的 不然你還做不出來

I: 若一些數學底子好的學生又比較主動想要使用這些軟體 可是她們可能也是需

要去學 你會怎麼建議讓她們學習使用這個軟體?

J: 那個數學底子 還是在... 嗯 不見得要靠這種軟體 有時候還是要靠邏輯的推演

數學沒辦法避免的是邏輯推理要很強 那不是靠軟體可以導出來 那是要獨力訓練

那就像學生已經是一隻老虎了 可以讓她如虎添翼 就這樣而已 這種軟體沒有辦

法說學生是一隻小貓 還可以幫她變成老虎 這是沒辦法的

I: 所以你平常是把它當成工具輔助 加強的時候 像軌跡 要呈現時會用一些動畫

的小工具 可以用?

J: 當然 當然沒錯 東西工具是好用是一回事 但是它總是有它的侷限 它不能代替

邏輯推理能力的思考 那個部分是人腦必須要作的事情 對 所以還是回有個界線

在 我想我們在課堂上如果要幫得上學生的忙 也主要是在這個部分

I: 你平常如何訓練學生邏輯推理的部分 如何加強?

J: 以我的模式… 其實每個老師的模式不一樣 我的模式是互動式的 有時候我會

主動去問 比方說今天作到一個計算用到了畢氏定理 我會問為什麼畢氏定理是對

的 但是很令人訝異的是 十個有九個高中生答不出來 證不出來 不知為什麼 就是

認為它是對的 所以在某個程度底下我們台灣的教育訓練方式 台灣這樣子教 某

些方式已經僵化了 把一些數學的東西視為理所當然 這是很危險的ㄧ件事 但是

一樣可以學到高中高三畢業 一樣可以 這就很納悶… 昨天有一個學生 我們在討

論問題他問老師這是一定要用所謂的餘弦定理嗎? 不用它也可以 用別的方法 但

餘弦定理 相當於是整個座標系統的 你如果不知道她你的半壁江山就毀了 這麼

重要的定理你不知道 那麼你還可以學到高三結束就很奇特… 畢氏定理和餘旋定

理是相當於等同重要的ㄧ種東西 不曉得怎麼活下去

I: 那些定理 你會用什麼方式去呈現? 你會想要用軟體還是呈現還是先讓學生先

了解?

J: 有些定理是因為是一種…推理的過程 那軟體本身它可以去來強化一個事實就

是說某個東西是對的 可是它沒辦法說明為什麼是對的 就像比方說 昨天我也用

這軟體展示 那個„雙曲線上的點到兩個漸近線的距離的乘積是固定的‟ 這個用軟

體當然很容易展現 就畫一畫然後去拉雙曲線上的點 兩線段雖然短長都在變 但

乘積乘起來卻是不變的 這很神奇 對… 但是這軟體沒辦法告訴你為什麼是這樣

所以推理的部分還是人腦的工作 對…

I: 所以你會譬如說先推理給學生看說 有這個定理 然後再用這個軟體去呈現視覺

的部分?

J: 如果它們想要看的話 我的上課的方式是比較特別 是扣鳴法 不扣則不鳴 小扣

則小鳴 大扣則大鳴 所以如果你不問的話 我也不會說什麼 你如果自己不覺得這

是個問題的話 那我認為你的大腦沒在動 對不對? 我可能會丟給他們一個問題 讓

他們去想 那如果還是不動 就責不付已

I: 若有問的話 刺激他們去想?

87

J: 若學生有回饋的話 我會去繼續講 那如果學生在某個地方停住了 不再往前 那

我也停住了 但是很多學生會只希望知道答案是什麼 怎麼算? 用什麼公式最快 那

如果要的話 我也會去提供它公式 但也僅止於此 因為這是屬於比較沒有發展性

的學生 那個多說無益

I: 我常想到說有些老師可能自己比較不會操作 那會不會有些老師去下載一些已

經做好的學習單 就是那種已經做好直接撥放之類的?

J: 那比較不是我的模式

I: 所以這裡的老師是否會使用或偏向已經做好的 直接撥放?

J: 你是說我們學校的老師喔?

I: 對

J: 嗯…我們學校老師會用的不多

I: 在台灣會用的老師呢?

J: 通常是主動 就是找到它的 然後才會用嘛 原則上我覺得目前還不是很多吧我

想 但是它的用法跟我們已經大部分老師知道 GSP 有用過 GSP 雖然不是很熟 但

至少有用過 所以對這個入門來講還是 比較簡單一點 但是話說回來 大部分的老

師還是不是很習慣用這個 因為用電腦本身是一種挑戰 對目前大部分的老師來講

還是如此

I: 你對於使用這個軟體和使用軟體來教學有什麼不同? 讓一個老師學這個軟體大

概多久時間?

J: 若一個老師有心的話 這個軟體基本的操作 我想頂多花個兩三個小時應該就可

以有一些基本的概念了 當然如果說你要去熟練它的話 就像騎腳踏車一樣 你會

騎 和你騎得很厲害 那個是兩回事 對… 所以你要熟練它 我覺得至少要花兩三個

月的時間 也許就是兩三天就要摸一次 怎麼去操作它 那有些難一點的作圖的東

西 那你可能要去深入研究有某些幾何的理論 你才畫得出來 不然有些圖你知道

但你不見得畫不出來 那個部分 深入的話就看個人造化 但是如果是基礎一些作

圖 作作三角形 作作正多邊形 那些 花不了兩三個小時

I: 所以 你的焦點主要是放在幾何還是代數?

J: 通常我們放比較多的部分是放在幾何的呈現上面 因為畢竟它來講 作代數的運

算是可以啦 它本身就有那個功能 不過沒有必要 主要是沒有必要 因為代數運算

我們沒有它也一樣 因為代數運算是紙筆可以取代的東西 或者是甚至於其他軟體

可以取代的東西 對…

I: 所以你把它當成算是一個幾何軟體?

J: 主要是...不過它有一個與眾不同的是 它可以輸入方程式 這是它與眾不同的

I: 對...所以可以教函數, 像三角函數?

J: 對..甚至說方程式本身可以是動態的 我們可以設一個橢圓方程式 可以去動態

的改變短軸，長軸 或半長 之類的

I: 你是用 slide?

J: 對... 就是說 比方說 現在隨便作個... 先設一些變數好了 m=3, n=4...比方說 然後

隨便弄個那個橢圓... x 除以 m 平方加上... (輸入橢圓方程式) 這樣應該沒問題...

這樣就出來就是一個橢圓 那當然這個 slide 設 m 跟 n 這個可以讓它出來 所謂動

態就是你可以去拉它 對不對? 就是任何數字都可以當作一種參數或變數在操弄

這是它無可取代的地方 對... 你像一般 GSP 它就沒有這種東西 它沒辦法對一個

方程式作操弄 但他(GeoGebra)可以

I: 那它也是有辦法用來提供一些代數的部分

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J: 當然可以

I: 可是僅止於函數，就像方程式那些?

J: 如果它可以搭配 因為像剛剛那個變數 還是反映在圖形上面 通常我會用它 還

是多多少少都跟圖形是相關的部分 我才會去用它 如果說單只是計算 我大概不

會用它 還是會有 所以你應該看到 它的幾何區這麼大

I: 對 大部分還是...

J: 那一塊才是它要展現的東西 雖然它的命令列真得非常好用

I: 那有些功能像顯示或隱藏 你會常常用到嗎?

J: 當然會阿 像我們有時候會畫一大堆線 一堆圖 那事實上不希望它那麼亂 就會

把一些東西隱藏起來 它現在有一個功能還不錯 就是顯示...顯示隱藏...我記得它

有一個東西 這個東西是可以作的... 就是說這個東西是必較新的東西...按一下...

隨便按一下 再回頭去選壓 把東西藏起來...比方說這個圓 也把它藏起來...我隨便

亂按一些東西...然後就套用 ok ... 當然這個名字是可以改的 它這個打勾是 你可

以把它切換掉

I: 喔就是 可以隨時可以要隱藏 或是..

J: 對對對 剛剛應該把這個箭頭 所以這個就可以切換 這個還算方便 這算是新的

功能 而且它還可以拿來作計算可以作所謂的邏輯運算 還蠻方便

I: 那還有沒有其他 就是你覺得比較新一點的或是其他軟體不太一樣的功能?

J: 當然它有蠻多是蠻好的功能 比方說 像這邊的這個移動位置區 也是其他軟體

所沒有的 然後甚至於它可以放大縮小... 這個右邊...好...不管... 像這種功能就非

常好 GSP 完全沒有這種東西 如果你要畫一個很小的東西 GSP 辦不到 你要點半

天 連兩個點靠很近 它就會黏在一起 那個沒辦法作 這就是它的優點 ok 那當然以

插入圖形 這是也是它的優點 GSP 可以插入圖形 但就是放正的 它可以作所謂的

矩陣式的變形 它可以放正的或是變成歪的 甚至於旋轉 什麼的 它都可以作所以

要展示的話這個就是很大的優點

I: 你剛剛提到就是主要在座標系的部分很好用 那有沒有用到不是座標系的部分?

J: 比較少 因為像之前...前幾天我在幫另一個出版社作有關機率統計的教材 我就

發現 它是指定用 GSP 來作 那我就發現 這太困難了 因為它完全...因為那個不是

他擅長的 如果你要作機率統計圖 也是可以 但是 不是擅長的東西 我們變成說要

花十倍的力氣達到一點點的效果 所以通常我就認為這不是適當的東西 所以我們

在某些課題的時候 就會找另外的工具來用

I: 那有適合教機率統計的軟體?

J: 那當然很多阿 只是說大部分都是要錢的 比方說像 EXCEL, SPSS, SARS 之類

的東西 不過當然有些免費線上的所謂的 online worksheet 可以作一些圖表 也是

ok 的

I: EXCEL 不是每個電腦都有嗎?

J: 那還是要花錢 那是 Microsoft office 理面的ㄧ個部分 雖然有 open office 它也有

類似的東西 不過一般人不熟 不過還好 現在大部分都有 就是 online 的東西 像

google documents 理面有線上的文件 其實可以直接點 還算蠻方便的 總之 機率統

計的來講 我覺得用它就不是很適合

I: 那哪些主題 你會使用 GeoGebra?

J: 通常像... 比方說...如果以高中數學來講 我就甚至於空間座標我都不會拿它來

用 就用直接用 3D 的軟體 因為空間來講 它當然可以表現 但是它要花很多倍的

力氣去作空間的表現 就像以前有很多老師用 GSP 作 3D 的 作出來很漂亮 但你

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要看背後花了多少力氣再做那個東西 所以通常我會花在跟平面座標有關的ㄧ些

課題 或者是比方說微積分什麼的 但是當然還是要跟平面座標系有關的 我們才

會去用它 所以這個領域裡面的東西才會去用 其實說穿了 它理面就是處理平面

座標嘛 目前我蠻期望是它有矩陣處理的能力 但目前看起來好像沒有 就是它這

邊的會有一些所謂的 command 通常會有 這些東西通常是在作圖用的 或是作向

量 什麼的 但是要如果要作一些平面的變換的時候 平面變換的時候就...

I: 你是說像 transformation 的部分?

J: 對 transformation 對 有時候我們在作 linear transformation 的時候 那麼說當然

說如果有矩陣能力的話 會比較方便 不過也許是我沒有注意到 我沒有發現到這

樣的東西

I: 好像是你要自己作變換參數 讓它感覺上可以在轉 有在變大變小 或者是...

J: 對 那要自己作 它沒有內建的東西 不過當然還是可以自己作 而且 它理面的加

減乘除的計算 還是可以處理向量什麼的 其實也還不錯 點作標也可以拿來加加

減減的 也都還不錯 就像那個計算功能 比方說 這 a b c 這三點 如果說要作三點

重心的話 傳統要畫中線 連來連去 這個比較好的是 還可以用來計算 也還蠻方便

a+b+c 這樣子 圖形就出來了 當然 也許因為圖形太大 看不到 縮小一下 就可以看

出來了 ( 喔 我知道了 ... 應該是因為這個 把它關掉了 喔 這是個小問題 它只能在

這邊切換而已 對 所以這樣子的話 這個重心...是 abc 的重心...它可以直接用計算

的方式)

I: 所以你只要點就可以 就有重心?

J: 這也是另一個軟體 GSP 作不到的事情 它一定要用線畫 所以說我們知道公式

的話 就可以直接在命令列 直接輸入 那 當然那就需要一些背景知識 這個東西以

學生來講 不見得會這樣子用 那老師的背景知識如果多一點就可以直接這樣作

甚至很多東西直接在命令列裡輸入 不過 那恐怕就是比較熟悉的老師才有辦法這

樣作 不然一般來講 大概很少會用到這邊

I: 就你剛剛提到一些 GSP 和一其他 3D 的軟體 比較來講 你會覺得 有什麼不同?

J: 各有擅長啦 那當然 如果說以這個和 GSP 來講 這個的優點是遠超出 GSP 的

GSP 大概就是目前看起來只有兩個比較好的地方 可能就是它可以作按鈕 所以按

下去 可以自己跑 另一個就可能是有關... 可能就是 iteration 的地方 可以作蝶代的

東西 可以作比方說 fractle 比方說就是 碎形的東西 GSP 在這個地方還算是表現

得不錯 目前是沒看到 GeoGebra 有這方面的功能 不過據說他們將來會作 目前功

能上就這兩個小部分 比它稍微弱一點 但其他強太多了 其它真的是 那 GSP 沒得

比

I: 就像哪邊是它的優點的部分?

J: 就是這邊(Algebra window) 和這邊阿(Command Line) 尤其是底下這個

(Equation input) 這是 GSP 根本沒得比的 它根本沒有 對不對? 所以我們在用...

I: GSP 有軌跡嗎?

J: 有 當然 它上面的那些按鈕基本上都有 除了那個圓錐曲線 GSP 處理能力比較

差之外 GSP 沒有這種東西 事實上他的圓的部分也沒有那麼多 只有其中幾個 然

後這邊其他的像 GSP 都沒有這種切線極線這種東西 這個都沒有 這是 GeoGebra

的強項 非常方便 不然用 GSP 作切線 其實是很麻煩的 有時候甚至要自己算 對...

I: 所以他就是多了這些部分 還有圓...?

J: 按鈕大部分是類似的

I: GSP 有這個嗎?

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J: 沒有 沒有完全沒有 其他就只有這個部分

I: 那你說你們學校買的 Cabri 呢?

J: Cabri 的話 他是比較像這個 不過 Cabri 我的印象...但我很少用他 我的印象是沒

有 command line 他沒有命令列那個地方 所以我覺得總之他表現比人家好 又不

用錢 所以不用它實在是沒道理

I: 那你覺得... 他一開始軟體的訴求是 他希望提供幾何的部分又有連結到代數的

部分 你覺得有作到連結的部分嗎?

J: 當然啦 他會有幾何區和代數區嘛 那個 command line 就是代數的部分 也是它

主要的部分 這邊事實上者是給你看而已 代數區是這個地方 所以他的名字就叫

GeoGebra 就是 Geometry 和 Algebra 結合起來的 所以我覺得在這個地方做得蠻

厲害的 這是同一類的軟體裡面 目前成功的把代數和幾何兩個領域結合起來的

這是蠻難得的 也就是因為它有代數的功能 所以有些時候還非用它不可呢

I: 那你覺得學生有意識到在你使用這個軟體的時候 幾何和代數的連結嗎? 有加

強觀念的連結嗎?

J: 幾何和代數 我想這個連結的部分 是在你使用這個軟體的時候獨立的一種經驗

但是如果說要幫助他們代數的處理能力 我想這不是這個軟體會幫得上忙的 反而

是你自己本身的數學知識要夠強才能用這個地方的代數 反而是這樣 而不是電腦

去幫助你 不是

I: 那她可以用來解決比如說像解方程式或是必須用代數軟體才可以?

J: 解方程式大概是有啦 但那個東西也僅止於展示嘛 像有類似 solve 之類的 喔 沒

有 是用 root 的命令 當然可以解出他的根 他會把算出來的東西列在代數視窗 對

不過問題是他只是 show 出來 他不會告訴你為什麼 對不對? 所以關於背後的那

個推理 我還是一貫的立場 推理的部分是人腦本身要去發展的 他只能展示結果

給你看 不能告訴你為什麼

I: 所以你會用到這些軟體 都是幾何的軟體?

J: 就是拿來去驗證我們的東西 或是有些不方便畫在黑板給學生看 用這個來給學

生看 加強他們的印象 有些東西是活的 寫在書本上就是死的 我們就用這個軟體

讓他活過來 所以這是一個 那個 是一種復活的軟體...

I: 還有沒有需要補充的部分或提供的意見?

J: 關於這個東西 我覺得如果真正的要讓它變成一種我們全民的習慣 我覺得還有

一場硬仗要打 因為 電腦本身的使用 在台灣是很奇怪的

I: 奇怪的 怎麼說?

J: 對 電腦本身是一種有點類似娛樂工具 不是研究工具 所以包含老師也是一樣

老師拿到它通常也是當成娛樂的工具 上上網 看看 dvd...

I: 可是你如果把它應用在教學上 你覺得是否可以提升學生學習的動機 注意力增

加 或是有什麼影響?

J: 如果說靠他學生學習動機增加 這個我不敢講 事實上我不敢奢望 因為數學這

個領域來講是一種硬學科 要學的話要必須有很強的動機 或者是你本身對推理感

興趣 你寧願去打那三國無雙也不會打這個東西 或者上網打網上電動也不會打這

個 所以我想很多時候 都是老師比較一廂情願

I: 你覺得有辦法改變嗎?

J: 你說改變老師的一廂情願嗎? 還是學生的態度?

I: 其實很多老師也不是那麼一廂情願 但是就是 有兩個難的部分 其實有很多老

師不是那麼情願使用軟體...這個部分有沒有...

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J: 你說老師不情願使用軟體 那是沒辦法改變的 因為從 GSP 時代就是如此 到現

在也沒改善 因為使用工具是一種習慣 你如果不習慣敲榔頭 不回拿起他來敲 對

不對? 這種東西也是一樣

I: 那有些新老師或可能剛畢業出來教書的老師 會不會比較容易接受使用軟體?

J: 我只能說比較不害怕開機 這件事情 至於說會不會使用新的軟體 會不會比較

容易接受 這個我還是保留 因為 我們會去接觸一些新的東西都是主動去發掘 像

我們也不是學校學來的 不是 都是主動去發掘 後來發現誰在用這種東西 就會去

用 至於說沒有在用的老師 就算他知道了也不會去用 這是蠻大的分截線 主動和

被動之間 這是一條分水嶺 學生和老師都是一樣的 當然我們可以去透過研習 或

是說大學裡面可以去推廣這個東西 不過 我覺得這還是有很長的ㄧ條路要走 因

為 我開始用 GSP 已經是二十年前了

I: 對 我剛剛在想問你說 你所謂 GSP 時代是 之前可能沒有很多很多不同的軟體

發展出來 那時後大部分的老師大概只知道一個 GSP 但現在時代可能比較不一樣

可能很多老師會接觸到各種多元化的軟體 那在的時代 到底是?

J: 這個其實跟以前差不了多少 會用 GSP 的老師還是那幾個

I: 他們會想要去學一些新的嗎? 比如說像 GeoGebra 或是?

J: 如果是會使用 GSP 的老師 也就是說平常就有用在用這個東西在教學 或者是

作研究用 那他當然會去找別的工具 那如果平常就沒有在用這個 再告訴他另外

一個 GeoGebra 他只會更頭痛而已 這是一種自然現象 對不對? 那當然新老師來

講 我覺得他們還是有比較好的機會去接觸 畢竟年紀輕吧 我想越來越多人會用

它是一種趨勢 不過我不覺得有太多人用 因為我想我們那個年代 我們那個數學

系 教授會開機就了不起了 更別說用什麼軟體 現在當然好一點 那如果說以後每

個老師都會用 我是不抱這種希望 因為真正教學現場 他們也可以不用啊 所以何

苦再學一套 又沒有加錢 加薪 這是一種非常基本的東西 因為老師本身要有很大

的動力 才會去學這種東西 人說穿了就是這樣 除非你有很大的動力 不然你為什

麼要學習呢? 就算當老師的人也是如此 老師也是人

I: 那學生呢? 學生你平常會提供他學習 然後如果說他有興趣 他會去學 你會教他

還是?

J:那當然了 不只是學生 只要任何人有興趣 我都會非常開心跟他討論 畢竟要找

到同好不容易 不管是學生或是老師 甚至於校外人士 都無所謂 不過我覺得說要

同好 我才比較會提得起興趣去講 如果說今天 比方說辦研習 其實 說真的 我是興

趣缺缺 因為你也知道 通常我們都是 辦完研習 聽完就回家了就再見 對... 所以我

寧可有同好的彼此討論 辦研習 可有可無

I: 那現在有一個國際 GeoGebra 協會 因為這個協會...

J: 對... 那是什麼東西?

I: 他是因為很多老師不知道該怎麼用 也不會用 也沒有很多補充教材或是資源可

以提供 這個協會希望說要知道有興趣用的老師 到底想要什麼 想要得到什麼樣

的幫助 他們可以提供任何的幫助 那你會想要什麼?

J: 我先問一下你 這是它們官方所舉辦的?

I: 這個是因為整個軟體是免費的 沒有任何的營利 所以 他只是說 架設維護這個

軟體 然後他成立各個不同的國家的使用 他現在三十六個語言

J:所以你是中文的負責人

I: 繁體中文?

J: 繁體中文 對 繁體中文 對

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I: 所以有這個協會 有很多國家都有設立 幫助這些老師可以去認識 然後使用它

所以說今天如果有這樣的協會來幫助 你會希望得到什麼樣的幫助?

J: 這個是我不會考慮的問題 當然如果說這樣的ㄧ個團體 有需要我來協助的話

要我協助什麼 我會作

I: 我知道你可能跟一般的老師不太一樣 那一般的老師的話?

J: 我想最多的 大概就是研習課程吧 因為畢竟如果對這套軟體有興趣 就是想知

道怎麼用它吧 所以有研習課程的話 比較好上手 這是肯定的 而且要對這套軟體

上手 說真的不用超過兩三個小時 所以 當然除了研習課程之外 最好是背後要有

一群支持的諮詢老師 就算是離開了研習課 我們還是有地方可以得到諮詢 我想

這樣應該就夠了吧 畢竟他也不是高等微積分

I: 其實有人用來教大學數學

J: 這是當然的 因為它背後有很多蠻不錯的東西 就像微積分 他事實上可以弄一

些所謂的黎曼積分的東西來展現 所以用它是理所當然的

I:台灣若有這個協會 可能要麻煩你幫忙

J: 這個沒問題阿 只要是使得上力的地方 當然是沒有問題阿 對... 其實我們台灣

學習的動力是比較低落的 這個是整體環境的問題 我們整體環境是在追求分數

聯考成績 再來呢就沒了 考上大學就沒事了 所以當然老師也是一樣 我只要送你

上大學就好 我這麼累幹嘛 研究什麼軟體阿 所以我想我們整個教育的心態是不

一樣的 我們覺得學習 就是要考上好的大學 又不是要作什麼高深研究 所以通常

一個人是考上大學之後就結束了

I: 不過 也許他之後出去工作 又會想要回來學習

J: 那是工作之後 也許覺得知識不足 那又另當別論了 不過通常是當職業學生的

時候 不會體會到自己的不足 通常都是如此 像當老師來講 一方面因為太穩定了

不用學 月新照樣會來 這是主要的問題

I: 謝謝你今天的幫忙

J: 你也是很辛苦 還要到處跑來跑去訪問

I: 不會 我也學到很多

Li

Interviewer (I): 想先了解一下 你教書多久?

Li (L):我教民國 84 年 1995 年到 2008 是 13 年

I: 你之前有資訊科技的背景嗎?

L: 我大學是念應用數學 研究所念土木工程

I: 為什麼?

L: 因為大學畢業不知道念什麼 看同學都有念研究所 所以就考一個 可是沒什麼

興趣 還是比較喜歡算數學

I: 所以土木工程有幫助你加強資訊方面的技巧嗎 還是本來就喜歡?

L: 其實本來大學對電腦蠻有興趣的 喜歡摸電腦 不過那時候軟體沒有那麼發達

那時候著重理論 資料結構 作業系統 都念一些根本的東西 研究所不知道在幹什

麼得 沒有興趣 只是念個學位而已

I: 是先教書才去念還是念完?

L: 念完 大學畢業當兵 當完念研究所 研究所畢業教書

I: 你大概教幾年級?

93

L: 目前是教高一和高三

I: 學生呢?

L: 還好 算好教 學生就土土的 被考試壓得死死的 沒辦法阿 要念書阿 被動的啦

學習蠻被動的

I: 之前有提到 你好像想用…

L: 我有帶她們到電腦教室 實際教他們畫圖 可是她們對這個不怎麼有興趣 全班

大概四十幾個 有興趣的不到十個

I: 那有興趣的會來問你嗎?

L: 有 課堂… 就是因為我有寫網站寫部落格 後來我發現真正去上部落格的 沒有

幾個 不到一半

I: 但他們比較稍微對數學有興趣的 還是會去看 然後去研究?

L: 因為我教的高三 是文組班 文組班對數學比較逃避的 上數學是被動的 有什麼

就聽 然後因為電腦教室可以上網路 趁我不注意就看圖片看一些…

I: 可是她不是有切換功能?

L: 對我切換完 我示範 就 阿…就長嘆一聲 我說切換回去了 就很高興 我就盯他

們 畫圖阿

I: 所以你大概都 是用什麼 畫什麼?

L: 網路上有 因為我有寫部落格 從我學校的網站進去

I: 這是新的嗎? 好像一陣子就有新的

L: 因為我覺得部落格蠻方便的 學生可以上網去照著操作

I: 學生上去看就知道

L: 這是最近上的課 上課的資料打成一個 就照著這個打 方程式圖形 照著打就出

來了 像這個底數不一樣 二分之一 三分之一 底數大小他的圖形 還有底數二和二

分之一 這樣對稱的

I: 這些都是你設計的?

L: 對 這邊是用拉桿 拉桿拉動以後 看圖形變化 拉桿主要是變動底數 y=a 的 x 次

方 y=a 的 x 次方 拉動上面的拉桿 黑色的 對 按一下 然後拉 底數大於一和小於一

方向不一樣

I: 所以學生就可以這樣看 那你都是上課的時候示範還是讓學生回家上網看?

L: 我就上課 會帶到電腦教室 跟他們說有這個網站 讓他們看 告訴他們有部落格

點這個呢 會看到這個東西 y= logax 底數是 a 這同樣也是用拉桿 拉這個拉桿

I: 像你自己設計是花多久時間?

L: 這個 就是說 上課在我教到這個單元的時候 感覺到這個很重要 同學對於底數

a 圖形的變化 感覺沒這麼強 就設計 設計這個很快啦 不用半小時

I: 有些這個底子 作得很快 還有什麼可以看一下?

L: 在往下看 這些題目都不錯 像 Lambert 是 94 年數甲的ㄧ題考題 這題就是 那年

很多人都錯 因為這要用 Lambert 的定理 圖畫出來 才知道他的答案 因為他題目

說跟三條切線相切 會有幾條拋物線? 然後他有一個定理說 有四條切線的話 才可

以決定唯一的拋物線 只三條切線的話 那拋物線就是無窮多 無窮多的圖形畫出

來就是 Lambert 定理 就是說三條切線 倆倆有個交點 倆倆有個交點 那就有三個

交點 你可以畫一個圓 那個圓 就是說拋物線的交點 就在外接圓上面

(部落格:大多數的人看到這一題直覺的反應就是開口向上的拋物線,至於斜方向

的拋物線要如何畫出來呢?這就要用到 Lambert 定理.

以下的所連到的網頁是用 GeoGebra 畫圖,然後轉成 Java 網頁的, 平常看習慣書本

94

的數學算式,圖形,現在換成由電腦螢幕來顯示,可是別有一番風味的.

http://twww.jhsh.tpc.edu.tw/~jojoba26/Geogebra/Lambert/Lambert3.html,

http://learn.jhsh.tpc.edu.tw/~smath/Book4/9401/p2.html

可利用 Wink 擷取下來，輸出為網頁動畫，讓學生也可在家利用網路觀看,並按

步驟繪圖。)

I: 喔所以就可以拉出他的軌跡

L: 嗯 就可以畫軌跡 然後 你可以改變焦點位置 上面的 F 改 F 焦點的話 焦點的位

置就變了 然後再拉動下面的 P 點 看拋物線的軌跡 像這個 一開始碰到題目也是

不知道怎麼辦 那是有一個建中的老師 (沈朋裕) 看再下面用的資料 再看上一頁

那個 Lambert 的定理 之後 交點 F 然後切線 四點共圓 所以要把圓先畫出來

I: 這應該不好畫吧 要設計很久嗎?

L: 不用阿 就是你畫出來 轉一下 下個指令 把圖轉出來就好 這畫圖還蠻簡單的

一般對於電腦有點恐懼 因為電腦裡面文書作業打 word 跟這個是不一樣的感受

像這個圖是建中老師 沈老師 他首先在龍騰發表一篇文章 就是那個定裡 那題用

Lambert 定理 可是它是書面的 我覺得書面上 看定理證明和電腦畫出來 感受不一

樣 電腦化的話 圖形可以改變顏色 角可以把它標出來 像拉動 H I 改變一下切線

那切線改變 相對角度位置不一樣 可以看旁邊的這個(algebra window) GeoGebra

多方面應用 可以在課堂上秀給學生看 帶去電腦教室 或且說 一些定理證明過程

想跟人家分享 而且 GeoGebra 旁邊是支援 Latex 的 有一些符號呢就是說 有一些

平常在 html 格式打不出來 用 Latex 可以秀出來 所以蠻好的

I: 所以你在設計這個的時候 你都是用 GeoGebra 先把東西弄好 然後?

L: 對 GeoGebra 打得出來 你就可以轉成網頁

I: 其它的呢? 你有用其它輔助軟體嗎?

L: 沒有沒有 這完全就是 GeoGebra

I: 那文字的部分呢?

L: 文字也是 GeoGebra 因為 GeoGebra 它有 一個可以加入文字 所以他蠻好的阿

像它就擺脫 html 的束縛 因為 html... 就數學呢... 數學家一直很想就是在網路上

打數學方程式 可是網路格式被很多大廠商之間彼此互壟嘛 不相容 所以數學的

東西很難寫上去 可是 GeoGebra 因為是用 Java 寫的 Java 是另外一個平台 ru.4

可以把 Latex 弄上去 它是解決數學符號 這就是它偉大的地方

I: 所以這些都是直接打上去就好 不用特別的...?

L: 就完全都是 GeoGebra 的東西

I: 所以這都是你設計的 還有?

L: 嗯 設計得蠻有成就感

I: 很棒耶

L: 這都是考題啦 就是說這幾年的指考題 喔 然後 GeoGebra 還有一個就是說 它

可以把橢圓旋轉 上面有一個 它可以旋轉橢圓的題目 這邊是 拉動紅色的橢圓旋

轉 http://learn.jhsh.tpc.edu.tw/~smath/Book4/9402/9402.html

這一題就是說兩個橢圓 如果是斜方向交 交出來四邊形是什麼樣的四邊形?

I: 喔 那就可以看出來

L: 對 就看出來了 所以這一題就是考你圖形的概念強不強

I: 所以你大概會怎麼用 GeoGebra 呢? 就是你會用來教學 或是當比如說 學生的

參考資料或輔助工具?

95

L: 主要應該是參考資料 希望他們課後上網看一看 然後對圖形跟課本內容所教

的有個連結

I: 那課堂上呢?

L: 會 用單槍投影機 電腦

I: 那會不會就是比較不方便呢?

L: 裝上去要一段時間 大概要五分鐘吧 不過有一個缺點 你在秀給學生看 後面有

些沒興趣就睡著了 他們覺得很新奇 要她們自己去用 去設計 不可能 然後看一看

覺得很新奇 但 要真的打動他們的心 恐怕沒幾個 對電腦 對他們來說 主要是娛樂

上網 聊天 打電動玩具 我們班上有幾個學生 可以說被電腦綁架了 整個心都沉迷

下去了

I: 可是 這個也是跟電腦有關 會不會使用可以幫助增加學習動機 注意力?

L: 沒有 因為他們從小開始 就是他們覺得電腦覺得是一個娛樂的東西 他們發現

到電腦可以弄數學 覺得很新奇 但動機還是不夠說 讓他們用這個來學數學

I: 你大概多久? 會很常秀給學生看嗎?

L: 偶爾 大概一個月一次 主要還是課本為主 這是輔助而已

I: 你會請學生去看部落格嗎?

L: 對 希望用這個慢慢打動他們的心 把他們的心抓回來

I: 像我剛剛看到部落個上面 有一些就是步驟 讓他們去… 你會用這個來讓他們

學習?

L: 喔 對 看這個步驟

I: 你要教他們嗎還是?

L: 對 就是說 他們只要有心學 這邊就是步驟動畫 他只要看一遍這個就知道了 好

像真正有人秀給他一樣

I: 你是用什麼軟體?

L: 對對對 這都免費軟體 不用錢 這叫 wink

I: 這個會作很久嗎?

L: 這大概要一個小時 一題要一個小時

I: 所以他就是把動作記下來?

L: 對 他有很多模式 可以選擇 我選擇是說當使用者在按下滑鼠或是動 keyboard

的時候把畫面抓下來 那從這個畫面到那個畫面 滑鼠會移動 好像感覺說真正有

人在旁邊操作給他看一樣 這就是網路教學的ㄧ種

I: 所以你作這個是為了學生設計嗎?他就

L: 對 不一定學生 只要想要學的 像其他學校老師有興趣也可以上網去學阿 因為

我有到建中學科中心去貼這個公告 說有這個網站 要學可以上去

I: 所以你覺得一般老師要大概多久學會 GeoGebra?

L: 這個要學多久? 其實 如果一天花一個小時 大概一個禮拜就會了 這蠻簡單的啦

其實他簡單就圓規和直尺 就一般的尺規作圖 不過他這邊還有一些精密作圖工具

座標 像點的變換啦 內分點這些

I: 你覺得學會用這個軟體和用他來教學有什麼不同?

L: 教學的話 因為教學和學生有互動 要懂得他們的心在想什麼 他們這時候是不

是真正在聽 還是說… 學生的心比較難抓啦 自己要學這個比較簡單

I: 所以學生如果有心的話 你也會教一下?

L: 對阿 但學生沒有 但其他學校有些老師蠻有興趣的 下個月我會去麗山高中辦

研習

96

I: 所以大概哪些主題你會用到 GeoGebra?

L: 像高一…嗯…哪些主題..高一高二幾乎都用的到 像高一上有座標平面 平面上

直線方程式也可以用 高一上還有多項函數圖形也可以用 一開始教二次函數 再

教高次的函數 高一下的話就指數函數 對數函數 三角函數 幾乎都可以用 所以起

碼每個月用一次 高二上是向量 平面向量 那空間向量就比較麻煩

I: 那個地方可能就借要用 3D?

L: 對 就要用 3D 軟體 然後高二上接著就是圓和球 圓的方程式又可以用 高二下

的話 就是圓錐曲線 拋物線 橢圓 雙曲線 都可以用

I: 那高三?

L: 高三的話 向自然組 高三上是平面旋轉 矩陣嘛 平面旋轉可以用 自然組下學期

是微積分 微分 積分啦 什麼切線啦 什麼長方形的上和下和啦 可以用

I: 社會組呢?

L: 社會組的話 高三社會組上學期 那是複習啦 什麼不等式 都可以用

I: 那機率統計呢?

L: 那比較少 某些特殊題型可以用 然後高三下數乙的話 有對稱圖形

I: 那幾乎都可以用

L: 對阿 幾乎都可以用 太好了

I: 那你大多用來教幾何還是代數?

L: 幾何 比較多

I: 那代數的部分?

L: 代數有些算一算 然後畫圖 都有啦 這不一定

I: 所以課程比較沒有分代數或幾何?

L: 兩個都有 其實兩個都關係蠻密切的

I: 你會分開教還是幾何部分代數部分都有?

L: 都有 沒有分得說很仔細 就是這方程式有這圖型喔 這函數有這圖形 因為還是

配合課本內容 課本指數函數有圖形 底數怎樣圖形就怎樣 就是這樣子 對阿 課本

都是把幾何跟代數結合在一起

I: 對你來講 當初學這個很久嗎?

L: 因為我之前有學過 GSP 學一陣子再學 GeoGebra 就學很快

I: 你覺得他們有什麼不一樣嗎?

L: 像 GSP 就比較強在幾何方面 GeoGebra 比他厲害就是它跟代數有一個結合 可

以在下面輸入代數 GSP 代數方面不夠的話 所以很多都是畫的 畫出來 然後

GeoGebra 可以先計算 計算好後 他有那個點 就比較容易畫 可是 GSP 不一樣 他

就是 他那個點就是把他畫出來 把代數意義 就根據代數意義一把他畫出來 所以

GSP 畫圖的動作就是蠻複雜的 你可以去看師大有一位教授 那個陳創義教授 他

就寫得很多 GSP 的講義 你會發現說 他很多圖形他都是用畫的 是畫的沒有錯 可

是 GeoGebra 他是用算的 用算的話 直接把打方程式就好 不像說 GSP 要一直畫

一直畫 你去看他的一個教學網站 會發現他那個滑鼠動好快 原來他都是在畫東

西 可是 GeoGebra 就是說你可以把方程式打出來只要打一個方程式 你按 enter 那

個點就出來了 然後 GSP 要一直畫才畫出來 簡單一個動作 GeoGebra 就打一個方

程式 可是 GSP 要畫十幾個步驟才畫出來 那個東西很麻煩 那個動作一個畫錯整

個就畫錯了

I: 就是比較快速一點?

97

L: GeoGebra 比較好 而且讓外人來看 那個動作就是一個方程式 GSP 是十幾個動

作 你看動作也不知道方程式…

I: 對 他沒有跟方程式作連結

L: 對對對對

I: 那他有什麼優缺點呢? 他蠻多優點 有沒有什麼缺點呢?

L: 我在龍騰的網站有投稿一篇 我有列一個表格 他的優缺點 有蠻多文章蠻想寫

得 就是有關很多數學軟體 其實國內蠻缺乏的

(GeoGebra 和 GSP 的比較

提到數學的繪圖教學軟體，大家第一個想到的便是 GSP(Geometry Sketch

Pad)，因此筆者特地在此將兩種軟體比較一下（如附表），並針對其中數項發

表個人的看法。 首先是價格，這可是很重要的！教學軟體要能夠普級，影響很

大，筆者在電腦教室以學校所購買的 GSP 教導班上同學們繪製圓錐曲線的圖

形，一堂課短短的 50 分鐘，學生們均意猶未盡，紛紛詢問何處可下載試用版軟

體，當我告訴他們網路上沒有試用版可供下載○註，要回家用只有自行購買，學

生們均十分失望，無法想像擁抱數學竟如此痛苦，要花費 1600 元，才能享受繪

圖樂趣。如此一來，教師若在學校以 GSP 教授數學繪圖，無異成為推銷員。若

是以 GeoGebra 來進行教學就沒這個困擾。由於 GeoGebra 是以 Java 程式語言撰

寫，Java 的優缺點，如支援多國語言、跨平台、執行速度慢，GeoGebra 也都

有，不過在比較舊的電腦才感覺出速度慢的困擾，以筆者的 Acer 512TE 筆記型

電腦為例（7 年前的機型），硬體配備： CPU：Pentium III 680HZ 256MB

memory，第一次執行 GeoGebra 時，以為是當機，費時 15 秒才見繪圖畫面出

來，在繪圖過程中若按 Delete 鍵要刪除某物件，或是按 Ctrl-Z 要回復前一步

驟，都要停個 1 秒多才會顯示結果，還好其他的繪圖操作一切順暢，當然若使

用的電腦機型不要那麼舊，如 Pentium IV 512MB memory 等級以上的電腦，就

感覺不出上述的情形。 GeoGebra 以 Java 撰寫的最大好處是將圖形儲存成網頁

時，可以 100％轉換為網頁畫面，GeoGebra 還支援 Latex 語法，可在畫面上顯

示根號、次方及分數，這都是 GSP 所望塵莫及的，因此數學家在網路上要討論

數學問題時，用 GeoGebra 來繪圖是一個很好的選擇。

國內引進 GSP 較早，使用 GSP 的師生較多，相對之下 GeoGebra 似乎乏人問

津，這可以由 Google 蒐尋相關的中文網頁看出來，但若看看世界各地的使用情

形，以 Google 輸入 Geometry sketchpad 及 GeoGebra 分別所找到的網頁數

387000 及 362000 來看，兩者在這世界的使用人口差不多，但以 GeoGebra 的支

援多國語言特性，使用 GeoGebra 可稱的上是與世界接軌。

GeoGebra 號稱是高中數學的教學軟體，當然有許多和高中數學直接相關的指

令，如輸入 2 2 1x xy y ，可畫出來斜方向的橢圓，右下方有指令視窗，以滑

鼠下拉可看到一大串的指令，如極限、單位向量、切線…等，另外還有許多指

令有待更正以符合我國現行高中教材，如：單位垂直向量（單位法向量）、第

一軸線（長軸）、第二軸線（短軸）…等，不過瑕不掩瑜，這些小問題是可以

克服的。若對 GeoGebra 的使用有任何問題，可上 GeoGebra 的討論區和世界各

地的使用者請教解決方法，也可對 GeoGebra 的未來發展提出建議，是要多加些

什麼功能或指令，程式設計者 Markus Hohenwarter 還會親自回答。學習數學繪

圖軟體並不難，主要是利用滑鼠來點選圓規或直尺工具來畫圖，因此操作 GSP

和操作 GeoGebra 其實大同小異，若是有操作過 GSP 的經驗，學習 GeoGebra 應

很快就可熟悉，此外筆者針對高中課程常見的數學題目圖形以 GeoGebra 繪製，

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並用 Wink（哦!又是一個好用的 freeware）錄下畫面，讓初學者看一眼也知道如

何來使用 GeoGebra，有興趣的人歡迎來錦和高中數學網學習。)

I: 有哪些其他軟體你想要研究?

L: 有一些是用得蠻有心得 有 愛麗絲 (Cinderella) 是美國的科學基金會 他所…

他是一個博士班的博士論文 他是講好像電動玩具 可以讓學生了解 3D 座標的東

西 但我還沒有時間研究

I: 你覺得這些軟體中 哪一個軟體最好用?

L: 就 GeoGebra 一直都在用 GeoGebra

I: 所以你大部分都用這個軟體? 有沒有說你覺得 GeoGebra 用不上 會去找其他軟

體?

L: 最主要是立體的 像四面體 角錐圓錐啦 他是可以畫 可是要用平面去模擬 那就

比較複雜 是可以畫 只是麻煩 要是平面就沒問題

I: 所以 我剛剛看到的 你是已經作好的網路學習單…

L: 喔… 那是 WINK 我在畫圖的時候先畫一遍 畫熟了以後 然後第二遍就同時開

啟 WINK 程式 他就把我的步驟 畫面擷取下來

I: 你上課是用那些已經作好的去撥放還是說當場示範?

L: 我是當場示範一遍 因為你要停下來 然後問說有沒有問題 或者是說強調這個

步驟重點是在什麼 畫圖是用到 譬如說中垂線的什麼特性 我會停下來講解

I: 所以你是當場…

L: 對 當場我示範一遍 接下來同學自己操作自己操作 那有不懂的 忘記的步驟再

去看 wink 程式

I: 所以你是讓他們參考

L: 嗯 是讓他們參考的 大部分是看是我示範一遍 他們自己有個印象 可是 實際細

節還是要看 wink 程式

I: 哪你有沒有接觸到其他老師是用已經作好的直接撥放還是?

L: 這個我不知道耶 對對 因為我網站有公開 他們可能…但是不知道他們怎麼用

還蠻多老師來看的

I: 有沒有什麼我們剛剛沒有談到 想要補充說明? 對於軟體改善?

L: 網路上有人建議看能不能寫 3D 好像不前沒有這個構想 Markus 有一個學生

可是寫一寫跑掉了能有 3D 最好

I: 那老師在使用的時候 有沒有一些建議?

L: 其實老師 有一些是是希望說畫出來的圖 把圖剪下來 存檔啦 可以弄在 word

做成講義 考卷使用

I: 應該是可以吧?

L: 是可以 大家都只是摸索 要剪下來 存成 jpeg 檔 可是有些老師不熟

I: 可以 print screen?

L: 可是要用小畫家 功能蠻差的 可能要用一些抓圖軟體 可是一些老師對電腦可

能不熟 他又不太會

I: 所以要可以存成

L: 因為用的ㄧ些蠻有心得 所以想說寫一些文章 之前有寫一些 但沒寫完 有關一

些繪圖軟體

I: 還有什麼可以再補充?

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L: 我希望有其他老師可以一起研究 有問題有人可以一起討論 不要每次都找外

國人 像我寫得網站也是有老師 寫信來問我問題 不過有好幾個 像有一個是南投

的 問說有沒有研討會 可是太遠了

I: 你有沒有辦過研討會?

L: 林口一場 之前台中大甲高中 有邀請我去辦研討會 但是我覺得好遠

I: 可能需要多一點人力?

L: 嗯 我下個月去麗山 像台北縣的話 就有一個縣政府教育局推廣的資訊教育 數

位學校 我們學校有些老師有參加推廣 這是小學一年級到高中 我們學校有加入

我是負責我們學校的數學的成員之一 我是負責 GeoGebra 的 然後 他的意思就是

說學校就各自研究 對教學和資訊化做一個結合 各個學校就負責一些東西 我覺

得大部分是小學在弄 國中稍微多一點 高中就更少

I: 你學得這個可以推廣到高中數學老師真的可以常常使用在平時教學 我知道有

很常一段路要走 你覺得大概要什麼?

L: 其實就是老師要先會啦 其實我覺得老師可能在當初大學授的教育都是書本上

的 就算有電腦 也是計算機概論而已 就是說有些當營養學分 或者是說必修不得

不修 不是真正對電腦有興趣的 但真正有興趣的… 所以老師要先知道軟體怎麼

操作 然後介紹給學生

I: 那就以比較像你這樣子 對於使用軟體很有技巧的老師 在學校上課有沒有困難

影體的部分呢?

L: 硬體的部分 我覺得我們學校都還蠻夠的 像我們學校還有電子互動白板

I: 可能學校硬體有改進的話 對教師使用會比較容易?

L: 對對對

I: 所以你有使用過?

L: 還沒 要辦校內研習使用 像電子白板有軟體模擬的下載程式 用 wii 可以改裝

I: 你覺得他的潛力呢? 譬如說十年後 會不會很普遍還是說?

L: 會阿 應該會普遍

I: 之前那個時代是 GSP 大部分老師會用 可能就是 GSP 那你覺得有沒有可能以

後?

L: GSP 就倒閉啦

I: 可是現在有些學校大部分是買?

L: 對阿會買 Cabri 沒辦法阿 因為 3D 的阿 但 GSP 就沒人會買 你還要花錢 又…

但 Cabri 蠻貴的 校園版要五萬

I: 那學校會願意買嗎? 還是要看學校?

L: 我們學校會願意 可是目前沒有錢啦 那買個人版 九章不賣學校個人版的 因為

他知道 賣個人版就變成全校版 個人買就遍全校 我們有去買 他說不賣

I: 那不就被封鎖了 沒有人使用?

L: 沒辦法阿 如果賣個人版就變成全校版了 那就權益受損了 在學校就賣校園版

I: 那如果說就是 不知道如果你有在上課使用可以過來看一下?

L: 可以阿 下禮拜四第八堂

Richard Interviewer (I): How many years have you been teaching?

Richard (R): Since 1996, 12 years

I: When did you know about GeoGebra?

R: Probably a year ago, not very long.

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I: Have you been using it in teaching?

R: umm…about a year… I started doing more intensively last year and I started doing

some development.

I: Are you the software developer or? I am wondering if you can do programming.

R: It‟s just my hobby really. I use a combination, mainly GeoGebra and everything

outside is Java Script.

I: Did you do your first degree in mathematics or?

R: Just mathematics. Just I mean… here just says…this just changes the points are at

different place and…enlargement just different transformations.

I: Does it do enlargement automatically?

R: It randomly does enlargements, reflection, rotation…

I: Do students have to try to get the answer or? Does it have the button shows that are

how you do it?

R: It could do. It could be quite of an enhancement.

I: But it is more interactive?

R: Yeah… None of them got any teaching on it. It‟s for them that they can practice.

I: It is the school website so they can visit, play and practice?

R: I use it two ways; I take the classes into the IT room which probably 50 minutes

lessons. It takes about 10 minutes for pupils to log on, so that two loses about 20

minutes. So I tend to do two activities in a lesson.

I: What kind of activities do you use?

R: I sort of set combination system. The good students you got them to move on to

different activities.

I: Do you use GeoGebra more to demonstrate or?

R: I might use this to demonstrate because it does the questions quickly. It quite easy

if I want to demonstrate on the board transformations... I‟ve got something for this…

I: I am just wondering that it is a task, is it asked the concept of transformation?

R: I probably use that to teach them and probably get them go on the board.

I: So what is your way of teaching mostly? Using in a variety of ways?

R: Probably a variety of ways. I am using mostly on the whiteboard using the pen.

I: On the whiteboard or on the screen (smart board)?

R: Just on the whiteboard. As for algebra I use a lot of writing on the board.

I: Do you use GeoGebra in teaching algebra?

R: Yes I will show you in a minute. So we do classes one or two on sequences, the

connection between… (Using GeoGebra and input sequence [(n, 4n-2),n,1,100] and

enter the equation y=4X-2)

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I: Do you use mostly in the geometry part or?

R: Yes. Mostly for the moment, I am always looking for new things to try out but it‟s

quite hard to put things to link these two…It‟s always quite difficult to…

I: Do you think GeoGebra links geometry and algebra?

R: Not convinced yet.

I: Why?

R: I don‟t know. I can‟t find things I can try at the moment…

I: How about functions?

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R: mmm… these are more related to algebra and geometry. I don‟t normally use the

algebra window…

I: Do you teach all different years? all of them?

R: Yes.

I: How do you differentiate the way you teach to different levels?

R: We‟ve got to have sets…a few are good at maths…bottom sets and top ones apart

from Year 7.

I: What kind of topics of A-level do you use GeoGebra in teaching maths with

GeoGebra?

R: Probably with the graph work. So this is all module C which I do in the year 13.

Just very simple… you predict the function… The curve…

I: So when you teach you use already made ones for them to try out?

R: Yes mostly probably more.

I: Do you construct image steps in front of them or?

R: Sometimes, but yeah…probably a mixture. Sometimes I use pre-made ones,

sometimes, when teaching gradient, I‟ll probably show them how to use GeoGebra to

make little triangle, show them tangent or things like that.

I: So do they try as well?

R: Probably just in this room. (Classroom)

I: Have you ever taken them to the IT room?

R: No no… I haven‟t tried that yet. Sometimes it‟s difficult to arrange the IT rooms.

Probably get some laptops. I encourage them to use it at home but not sure…it‟s hard

to tell how…

I: Do you have some activities used which are more related to Geometry? or your

favourite one?

R: I will show you …again I am not sure how good this is as I was just learning

GeoGebra really…These are the GCSE, sometimes I have done some revision

lessons…

I: Do students know about GeoGebra?

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R: Some do. So I just sort of get students to recognise this. I try and get students

recognise all sorts of different triangles.

I: What do you think normal teachers think about GeoGebra?

R:

I: Did you prepare all materials?

R: yes just me

I: If a teacher wants to learn GeoGebra, how long does it take?

R: Probably about a month.

I: This year since you started teaching with GeoGebra, did it change the way you

teach?

R: Yes definitely. I go to the IT room a lot more. Probably it‟s not just GeoGebra, it‟s

more a combination of GeoGebra and Yacas. It‟s another CAS.

I: How did you know about it?

R: I knew it quite a while. I looked it a couple of years ago…I emailed the author and

asked a few java functions to do some interaction stuff and he did that. So I have been

using Yacas to check students‟ answers. Every time I do to make it better, so this one

I can stand at the back of the room to see what all students are doing and help them

out… ask them to go to the front which is quite good.

I: What do you think students think about this computer aid?

R: I think they like it. I asked Year 8 yesterday for what is the best sort of revision for

them. And a few of them thought using the computer stuff is better. Most of them

thought the actually printed out sheets are better. Some of them said they speed for the

exam for the accuracy. So you don‟t need to do the same thing again and again to be

able to understand it. But still, they don‟t prefer this, they still prefer the written work,

longer questions as well.

I: They use it as a checking tool or?

R: They just stuck on when they want to revise a certain topic they just do.

I: How do they know?

R: Just tell them right or wrong, that‟s what it does. They get the score. But I haven‟t

got yet…it would be nice to get the score…picked up…that‟ll be really cool.

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I: So generally their attitudes towards ICT or software are positive?

R: Yes I think so, I had my bottom set Year 11. They finished last Friday and we had

the last lesson for the exam in the IT room. I got some work with them. Last lesson

was surprising.

I: So the revision?

R: Yes I gave them some revision. They were pretty well motivated.

I: Do you think they first start learning, they basically learn use paper-and-pencil

environment, latter they use computer to review or?

R: Yes that‟s how I do it. I prefer to use paper-and-pencil first. Possibly I‟ll say

transformations cause it takes a long to draw the graphs. Transformations, this is what

I might get student to the computer room first in the future. But still, in the exam

they‟ve got to use paper-and-pencil. I think if I do everything on the computer.

They‟ll probably not gonna do well. They‟ll get bored. I do it with Year 11 for one

lesson every fortnight. I think that‟s been about right.

I: In the normal classroom you use the projector?

R: Yes I use quite that a lot.

I: Is it every lesson or sometimes when you need to?

R: Yeah when I need to really.

I: Do you use it quite often or once a week?

R: Depends on what topics I am doing, probably once a day on average. Probably

more the start of the year, because the first term we do a lot more on teaching and the

rest are a lot of revision.

I: Which Year you use more on the computer?

R: mmmmmmm…probably…the thing I found it hard to teach was transformations…

I think this topic is probably Year 8. This is a Grammar school so we do things a bit

earlier. But it‟s gonna be they do SATS in Year 9. So there are simple transformations

like reflections, rotations, enlargements yah…, so UK Year 9, everyone went on to

Year 9, simple ones. Then they do…For GCSE, for Year 11 they‟ll probably do

enlargement with matrix. That‟s the topic I find most useful for cause that I don‟t

think I could be quicker just before.

I: Students here are more able?

R: they are selective so should be quite able.

I: Do you have any favourite functions or activities in GeoGebra?

R: I should have thought about it before you came, but I will find something. Ohh..

this is a great one.

I: So you take GeoGebra as what kind of software? For geometry or algebra?

R: mmm…depends if you can use the algebra, graphs of algebra? It‟s nice as it does

everything.

I: Is anything you think you can not teach with GeoGebra?

R: It can‟t do fractions at the moment

I: Yeah I think Markus does…

R: I suppose you can do… I suppose you can‟t type in…You can make worksheet

with fractions but you can‟t type in half quarter or stuff like that. I think because the

interface of that Java scripts. You can display anything you want. You can do

anything you want to. You really CAN do anything.

I: So you can teach with it in any topics?

R: If you get the right idea, you can do anything. You can do any subject you can do

Physics… if you have the idea and time to do it.

I: What do you think the strengths of it if you compare to other software?

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R: Strengths…the fact that it is free… that students can use it at home. The fact that it

is Java that you can use in any platform and with Java script you can control and put it

in the webpage for me that‟s tremendously useful…It is very specialised use not many

people can write with Java Scripts. Just the fact that it looks nice. Geometer‟s

Sketchpad, I don‟t like how it looks I don‟t like its interface. So GeoGebra looks nice

and the interface is easy to use.

I: Did you use Sketchpad before?

R: I got one of myself called Cabriolet. It just does Geometry… First of all I use this a

little bit.

I: Do you compare?

R: So what Autograph can do it, Autograph does graphs like hexagons better.

I: Do they have 3D?

R: yes it‟s got 3D. But in the syllabus here there is not much 3D here. They do a little

bit in GCSE, in Year 13 they use Pythagoras in a cuboids.

I: How did you know all these software?

R: Just search and people telling you. One of my colleagues told me about GeoGebra.

He went to a course and the course tutor told him this is good and he came back and

mentioned it to me. When I tried to use I realised it is better than I thought.

I: what do you think about GeoGebra as a piece of mathematical software and an

educational tool?

R: It does the connection between algebra and geometry much better than other

programs - anywhere you can enter a number you can also enter a formula eg you can

enter (4,5) but also expressions like (3 a,2-b) so it's not actually necessary to have the

algebra window open for students to see the connection. Also the fact that you can

animate any variable by turning it into a slider is a very powerful feature. As an

educational tool: like any other tool it can be used badly or well.

I: what benefits do you think GeoGebra brings to your teaching of geometry and

algebra at the A-level?

R: I've some files to help show the proofs for the formulae for

Sin(A+B) etc. It's very good for teaching iteration (cobweb file attached). It's very

good for teaching gradient of a curve (both the concept and the proof)

I: Could I possibly also have the GeoGebra files (check box and rolling polygon) you

showed me?

These are the files:

http://www.geogebra.org/en/upload/files/piman/Non_Standard_Wheels.zip and this is

the thread with explanations:

http://www.geogebra.org/forum/viewtopic.php?f=2&t=3249&start=75

Tyler Interviewer (I): How do you view GeoGebra as a piece of mathematical software?

Where did you do your degree and how you have come to be a teacher?

Tyler (T): I did a Maths degree in Oxford and I then did a PGCE in KCL. I have been

in Comberton Village College for 12 years. I was a cross teacher first and I stand head

of maths for four years. I also teach one day a week to teach PGCE for the faculty. I

have worked as an AST, supporting teachers of maths, sometimes in primary school

and secondary schools.

I: Sounds very busy

T: mmm yes it is.

I: When did you start using GeoGebra?

T: Earlier this year, about 6 months ago.

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I: Do you use in teaching in upper level?

T: I‟ve used it with 10 year olds from our primary school through to Year 11.

I: How do you use GeoGebra when you teach?

T: Ok… I‟ve used it in three different ways or I can consider there have been three

different ways it can be used, I‟ve used along three of these ways. One way is where I

demonstrate, so to me at the board, using it as a teaching tool using it to demonstrate

to the class, using it for the class to react, it seems that I am doing for them to be able

to say. So I used like that with some pupils. Certainly, I‟ve set up in advance at some

particular things for them to work out, to find out, but that‟s a whole class activity.

I‟ve used it where I‟ve set things up for them to pupils to interact with. So with Year

11 when we studied circle theorems, so things like the angle of semi-circle is always

90 degrees, the angle at the centre is twice as the side, I set up particular GeoGebra

files for them to look at, to explore, to make changes to, and then for them to make

hypothesis of what might be happening.

I: So that‟s in the classroom?

T: No it‟s in the IT room that‟s children using that for themselves. There they work in

pairs, they discuss what they are doing, and they are encouraged to have ideas and test

those ideas out.

I: You prepare some pre-made files?

T: That‟s it, so that‟s me preparing things for them to interact with. So I‟ve done the

work preparing some activity I‟ve don the work and prepare the files so pupils just

interacting with things what I‟ve made. And again, it‟s for them to find out specific

links to things about particular parts of maths, particular areas of maths. We also then,

as parts of that, discover the ideas of proof and GeoGebra wasn’t providing proof.

The particular angles were the same, or particular angles added up to 180 degrees or

particular angle were twice other angles. But we were then able to use those diagrams,

the diagrams that GeoGebra produced, in fact, some of the children to screen shots,

for them to look at geometrical proofs.

I: So many pupils just investigate these ideas?

T: Yes that‟s right

I: Did you ask them to make conjectures to test and produce a piece of mathematics

work or?

T: Yes and for something we called alternate circle theorem, when if you‟ve got

tangent to the circle (making the constructs with GeoGebra) so this I didn‟t tell them

what‟s going on… and ask them to find out what happened, and try to explain, so they

conjectured it and I set up the diagram for them to find out what‟s going on.

I: So you didn‟t tell them and ask them to investigate?

T: That‟s right, so they can, first of all, see what‟s happening, so they could answer

questions like, if I move this point, which angles will be bigger, which angels will get

smaller which angles will stay the same, if I move this point around the circle to the

right, which angles will get bigger, so they will say, well… this angle at A get bigger,

these angles at C get smaller, these angles stay the same…. So they will have a real

feeling for as I move these, these angles getting bigger, this angle‟s getting smaller,

this angel‟s getting bigger as to what‟s going on. And we discuss very often if we

move a point pass the other point then things change. If you take this point here, then

it ended up two angles are 90 degrees.

So they will be able to see how it works, what things going on, so they will be able to

see properly and dynamically, … So a conjecture if we don‟t have the tangent

involved. We know this angle as we move this point, as

107

Well… also… so maybe there is a link…so they both can be increasing in the same

way. They were just using it for …

For example if they do this… so they know the tangent form 90 degrees…so they

know in this particular diagram…so …Can we prove that? As a way into it, in one

hour session, it‟s perfect.

I: How often do you use it in the IT room?

T: Up to now, we have a week lessons of circle theorem. I‟ve used it in a one hour

section with 10 year olds, Year 6. It‟s been it so far. One of the things I‟ve like to do

is to explore what sort of subject areas GeoGebra can do and to see how pupils might

be able to learn to use themselves.

I: You mentioned three days?

T: Yes I think of it at the moment, there are three different ways, using it to

demonstrate something. If I use it to the whole class, if I was to say what happens if I

move this point, that‟s me demonstrating something, children having ideas of using

this as a dynamic programme. But one projective version, the pupils might not have

ideas. The second one is for them to work with a computer with a partner, interacting

this with their partner. The third way then is with Year 6 where they starting to learn

to some tools, they started to create things. Some of them started with a blank sheet

and they wanted to use like the reflection… and they wanted to do reflections to make

their own pictures and interact which is lovely. We are doing things like…

I: How do you guide them without using GeoGebra as a recipe?

T: With these pupils

T: They can create pictures they loved. They really enjoy it. I like this.

I: So you mainly use it for teaching geometry or?

T: So far yes,

So for example, I am quite intrigued by the idea of using this with transformations, it

uses for rotations, uses A dashed, B dashed to show the reflections, so show

transformations,…which is lovely…which it will be a very interesting way of

working or a very interesting way to use, there will be, it may be…that some pupils

can understand the rotation. So yes we have to wait and see.

I: How do you view GeoGebra as a piece of mathematical software?

T: Because where I‟ve come from, I used Cabri in the past so I started by seeing it as

a replacement to Cabri… so as dynamic geometry… the…yeah…so that‟s so far…

that‟s how I‟ve used it … so it will be interesting to see what are the ways of using….

appropriate ways of using it for algebra as well.

I: So comparing to Cabri how do you think about GeoGebra and why you think

GeoGebra can replace it?

T: Comparing to Cabri, I think there are a number of advantages over Cabri. One is

that the axes are very much clearer that you can change axes much much more easily

and much much more regularly. The fact that font size changes every font size, I think

it‟s brilliant. And it changes font size of the icon size at the top that it change the font

size for everything I think it is stunning. The fact the axes and grid are so easy to use.

It‟s brilliant. The fact that the properties and changing colours is so easy to use, I

think this has… and it projects far better than Cabri does. So that‟s why… huge

advantage and changing the properties on Cabri is frustrating and annoying. It‟s far

far superior in terms of projection. It also now can be used to put pictures at the

background…but to put pictures can be rotated and reflected. One of the things that I

am really looking forward to do with some pupils is reflecting and rotating and

enlarging pictures. So being able to ask them to take a photograph and we could then

use as part of these. That‟s what Cabri can‟t, as far as I‟ve known, do yet. Yes you can

108

use it at the background but it‟s only in Cabri II. Cabri is also a commercial package.

We‟ve got Cabri II at school. Cabri II+ is different. Cabri II+ will run Cabri II files

but they are not backward and compatible. So my Cabri II won‟t run Cabri II+ files

which I think is naughty cause that means it‟s a different version, and it ought to be

called version III. But to update to upgrade from version II to version II + you have to

pay it…again it‟s a different version, and that‟s expensive. For pupils to be able

download it to use it at home. It would be absolutely ideal.

I: So the fact it is free, will you ask pupils can use it for homework or course work at

home in the future?

T: Ideally yes. We have… One of the things that worries me mathematics teaching in

general is that we have subscription at school and in the same way that lots of school

do to a website that pupils can access at home. They have their own passwords for it,

this is my maths. I have a personal password for it. They can get on the website and

they can answer, it shows them particular areas of maths and they can answer

questions. The problem is that it really is very very constraint and the technology is on

the stage that it will take yes no questions and it will mark answers to questions. But it

won‟t actually allow pupils to explore things properly which means that pupils seeing

maths as it‟s something where you listen and read the instruction and then you do

results. This (GeoGerba) is very different. This is maths by interacting. This is

mathematics by trying things out, by conjecturing, by having a go…

I: How about algebra? What software do you use to teach algebra?

T: I‟ve used in fact with the same Year 11 group I used Autograph with them. The

reason I‟ve used Autograph. I wanted it to start with the sine wave and I wanted to

start the sine curve they be able to write it like that. The way Autograph does…is

using gradients. I could have defined f(x) as sine. But I wanted to start with y equals f

(x).So then pupils can explore y=sin(x). How does that transform? What happens with

4 sin(x) and sin (4x) the lovely thing about the trick of functions, you can tell the

differences between vertical and horizontal stretch and squash. You can‟t tell it‟s been

squashed it out. And again children would use Autograph to conjecture.

I: Do you think GeoGebra can do this?

T: I think it can. I would want it to be able to make y=sin(x). So if there is a way of

making it happen on GeoGebra, I think it will be ideal.

What happens if y=x cubed. So they will be conjecturing and testing those conjectures.

I: But when you type in y=f(x) does it work with Autograph?

T: You will have to type y=f(x). If I use y= sin(x) …I am assuming that I can… does

pi works here as pi? So…yeah we can do it like that but it happens to draw to use this.

So now if I do… that will work but that‟s for…to the left by 90… But like I said I

want it to work that way. It does the work around here but it didn‟t do what exactly I

wanted to do. So I used Autograph.

I: So compare GeoGebra to Autograph you still think you will?

T: Well only because…The only thing that Autograph still has over…well…the

solution is to put in the degrees so perfect… so if I do y= cos(x)…

degrees…yeah…try y=sin x degrees take away 5… so it worked. If you put degrees in

it then works… I didn‟t see that happen but I do now.

I: Will you use GeoGebra to teach algebra in the future?

T: It makes sense to me because of things you can do with it, you can‟t do on

Autograph. So if I drag this left and right we can see the changes happening (on the

algebraic window) so yeah…that‟s a big deal. The pupils again can it with Autograph

but they can also do it with GeoGebra at home.

I: Have you used it for upper level?

109

T: Only with Year 11. Year 11 is the highest we‟ve got. That‟s the circle theorems.

I: What topics do you think you can use GeoGebra?

T: I think semi-circle theorems, transformations, some transformations. I wouldn‟t

personally want to use areas of circles or conferences of circle. That‟s what I want

pupils to do in practical ways. I think these would be more powerful than something

on GeoGebra. So that‟s why I wouldn‟t use it for that, whereas the circle theorems,

where the alternatives is either been taught they all are like that or having to draw lots

of diagrams and measure lots of things. That‟s sensible as it is not accurate when the

things you‟ve made are not accurate. So similarly with trigonometry graphs, it might

at some point be useful to introduce the ideas of a sine graph by using a circle by

doing some measuring. But if you want to do transformation for the graphs, it doesn‟t

seem worth to re-plotting all the points every single time. You want to use the power

so let GeoGebra to do that. Similarly with reflections and rotations, there is one stage

having a mirror and see how it is reflected, it‟s great. Having a piece of paper you can

fold, it‟s great, it‟s important and it‟s great but after a while if you are wanting to do

multiple transformations, it may well be that GeoGebra is more effective.

I: So you use a mixture of different ways?

T: Yeah… what I wouldn‟t want to do is to get to a stage where you are using

GeoGebra for everything, whether sensible or not. Because it is clearly there are times

when interacting with physical materials, it‟s going to be more useful.

I: Do you think GeoGebra links Geometry and algebra?

T: That‟s particularly obvious for the drawing graphs and that‟s one of the reasons

why I like to use this for drawing graphs…it‟s that it with pupils I‟d talk about

different representations, talk about how y= 3x+2 could be a way of describing a

sequence. It could be algebra it could be about …it could be formula it could be

equations of lines or graphs it could be different ways of presenting the same idea…

and they might mean very different things, where you take a taxi journey, you‟ve got

something to do with the cost about the taxi journey… you can‟t do all the fractions

with penny…. You have 20 pence, 20 pence equivalence. Nowadays so there is some

kind of integer issue going on whereas the graph straight lines every point of line is on

there. So then, this is a nice way to talk about the link between the algebra and picture

which again is another thing that GeoGebra still has over Cabri or Geometer‟s

Sketchpad that you see specifically some of those ...

I: Is there any thing or functions you particularly like about GeoGebra?

T: I love concepts I love the idea you can set up diagrams and interact with it. I love

the slider.

I: Do you use it?

T: I do, some of them I showed on Saturday…yes I‟ll probably…. (drawing the graph

with GeoGebra) so yes for pupils to see what happens… to be able to see what

happens if it‟s minus

I: Do you create lots of GeoGebra worksheets yourself?

T: Yes a few like this.

I: What‟s your favourite?

T: This one. Because partly not for the part of GeoGebra sheet but to say…what will

happen if I take that enlargement to the left… And as I mentioned on Saturday, if I

move this to the left, almost everybody thinks this will get bigger. It‟s really

interesting to unpick why… and I think the reason is that lots of people when they

first introducing enlargement as a torch or an overhead projector where if you have

torch if you move the torch away, the size changes. That‟s a fundamental

misunderstanding of how an enlargement works. Because with a torch it‟s been

110

projected to a wall or a screen but that wall or screen is not moving whereas here

that‟s not how it works. So the idea we can talk about what will happen without even

having to do it. So we can show: what happens if I move this point downwards? The

other point moves three times as fast, shows that they are linked, then we can spend

proper amount of time talking about what happens if I move this to the left. And only

at the very end of that discussion, do we then actually do it. So yeah… when we have

a file a few down to c then it does stay the same. Then they want to know things like:

What happens if I make this one, then wonderfully pupils want to know: can you

make it a decimal? That‟s how they call it, what happens if I make this point to the

centre? Can you make it negative? What happens if that‟s a zero? Then again there

are very nice things you can do with this.

I: Is there anything you‟d like to add about your views on GeoGebra?

T: Only that it‟s still very new. But it‟s really exciting so far. It’s going to be really

really exciting to see how it develops and how we can develop using it.

I: Anything you think could be improved?

T: Well I didn‟t know how to do the trig graphs so y=sin(x)… I couldn‟t get how to

make it work in degrees but now I can get it to work in degrees. So we‟ve sorted. So

no…not… nothing I can think of at present… I am sure there will be things that crop

up. But I have got enough things to be thinking about that I can use it for. That‟s

intuitive. But I do make a difference between… I would draw a distinction between

knowing how to use it these tools and still getting used to using it as tools. You‟ve

seen that I make mistakes and I am still learning. I wanted to do something and I‟ve

gone to the wrong menu. But it‟s nice enough it works in a way I found it‟s intuitive

so if I want to find one, I go on to the next one until I find what I am wanting. So

that‟s not a big deal. But making a different between the tools and knowing how to

use GeoGebra tools…and the ways of using it, so which topics do you use? So sort of

pedagogical ways of using it.

I: As you mentioned tools? What tool do you take it?

T: As an environment for exploring dynamic geometry.

111

Appendix II: A Matrix for Generating Theme-Based Assertions for Case

Finding Rated Important (adopted from Stake 2006, P. 51)

Themes

Teacher

Background

Views on

GeoGebra

Software

Evaluation

Ways of using

GeoGebra

Jay 1 2 3 4

Finding I 12 years

teaching

experience

Positive about

GeoGebra

GSP and GeoGebra Home-

Research,

Teaching

portfolio

Finding II Translator Negative about

ICT in general

SketchUp and

GeoGebra

Classroom-

Demonstration

Finding III SA/ IT Negative about

student attitude

towards

technology

Li 1 2 3 4

Finding I 13 years

teaching

experience

Positive about

GeoGebra

GSP and GeoGebra Home-

Research/

teaching

material

preparation

Finding II School

website

designer

Less positive

about

technology in

general

Cabri and

GeoGebra

IT room-

Demonstration,

Revision

Finding III GeoGebra

translator/

trainer

Negative about

student attitude

towards

technology

Richard 1 2 3 4

Finding I 12 years

teaching

experience

Balanced views

on GeoGebra

GSP and GeoGebra Home-

research/

teaching

material

preparation

Finding II Software

developer

Balanced views

on technology

Cabri and

GeoGebra

Classroom-

Demonstration

Finding III School

website

designer

Positive about

student attitude

towards

technology

Yacas and

GeoGebra

IT room-

Activities

Revision

Tyler 1 2 3 4

Finding I 12 years

teaching

experience

Balanced views

on GeoGebra

Cabri and

GeoGebra

Home-

research/

teaching

material

112

preparation

Finding II GeoGebra

teacher

trainer

Balanced views

on technology

Autograph and

GeoGebra

Classroom-

Demonstration

Student

interaction

Finding III AST/ school

mathematics

consultant/

PGCE tutor

Positive about

student attitude

towards

technology

GSP and GeoGebra IT room-

Activities/

student

investigation

Appendix III:

Research Timeline

Task March

2008

April

2008

May

2008

June - July

2008

Preliminary classroom naturalistic

observation

from to

Documentation- website pages, publications from to

Semi-structured and in-depth interviews from to

Teacher observations from to

Analysing interviews from to

Analysing supplementary data from to

Writing up from to

113

Appendix IV:

Informed Consent- England

St. Edmund‟s College

Cambridge CB3 0BN

Dear teachers,

I am investigating the educational experiences and views of pre-university

mathematics teachers on ICT (particularly the use of GeoGebra) in the teaching of

mathematics cross-culturally. Therefore, I hope to interview teachers form Taiwan

and England.

For the purpose of data preservation, the interview will be video-recorded and will

take approximately one hour. The data and results of these interviews will be used for

academic purposes only.

For ethical reasons, the names of participants will be anonymous. Participants have

the rights to opt out from the research as well as the rights to review materials in

respect of the interviews both in written and taped form. All involvement in the

interview is voluntary.

It is my greatest wish that this study will facilitate the improvement of and

contribution to educational development in mathematics teaching with the use of ICT.

I do sincerely hope you will be willing to be involved and thank you very much in

advance for participating in this study.

With kindest regards and many thanks,

Yours sincerely,

Yu-Wen Allison Lu

114

Appendix V:

Informed Consent- Taiwan

劍橋大學教育學院

聖艾德蒙學院

親愛的老師：

因目前致力於研究 GeoGebra 於數學教學的教育經驗及高中數學老師的看

法，我希望訪談兩位台灣及兩位英國高中數學教師。

為了研究內容的整理及資料保存，希望以錄音錄影的方式進行約一小時的訪

談。 研究資料及結果僅限於劍橋大學研究的學術目的及使用。

參與教師的姓名將以匿名方式於研究論文中呈現。教師有權利暫停此訪談及參

考訪談的錄音及錄影內容。

此研究對於結合中西方資訊溝通科技應用於數學教育及教師專業發展極

有貢獻價值。我竭誠地希望您能參與並感謝您對此研究的配合。專此

敬頌

教祺

盧郁汶 敬上

民國九十七年三月十九日

115

Appendix VI: Sample Interview Protocol

Background

How long have you been teaching mathematics?

Do you use any kind of technology applications for your teaching?

Probe: How do you use these technologies? How did you become familiar

with GeoGebra?

Actual use of GeoGebra

How would you describe your experiences with GeoGebra?

What kinds of changes would you want in ICT/ GeoGebra when you would

consider using ICT in your teaching?

Views on GeoGebra

What do you think about the use of technology in your teaching practices?

Do you think GeoGebra is useful for your teaching?

Follow up

Is there anything that we did not mention before and you would like to add?

Thank you very much for your time!

Would you be interested in receiving a summary about findings of the study?

116

Appendix VII: Jay’s Example of Geometrical Constructions

117

118

Appendix VIII:

Li’s Examples of Proofs of Theorems and Problem-solving

119

Appendix IX:

Li’s Revision Worksheet:

一.GeoGebra 輸入操作注意事項

1. 次方符號 ^ : 按 Shift 6

2. 絕對值符號 | : 按 Shift

3. 按 CTRL-Z 可取消前一步驟

4. 若要設定滑桿, 先按滑桿 按鈕, 設定名稱及最大最小後, 按套用, 然後記得

按一下 ESC, 變回滑鼠選取狀態

二．輸入函數或方程式看看圖形的變化

1. 直線方程式

(1). 斜截式 y = mx+k , m: 斜率 k: Y 截距 (2). 截距式 1b

y

a

x,

a, b 分別為 X 截距及 Y 截距

設定兩滑桿, 名稱分別為 m, k

輸入 y= mx+k

拉動滑桿看看圖形的變化

2. 配合絕對值 | |, abslute value

120

y=|x| 輸入 y=abs(x)

y=|x|-1 輸入 y=abs(x)-1

y=|x-1| 輸入 y=abs(x-1)

y=|x-1|-1 輸入 y=abs(x-1)-1

3. 二次函數 y=ax2+bx+c

y=2x2+x+1 輸入 y=2x^2+x+1 加上絕對值, 看看圖形會轉彎

○1 y=x2+|x-2| 輸入 y=x^2+abs(x-2)

因 x-2=0, x=2 圖形在 x=2 處轉彎

○2 y=|x2+x|-2 輸入 y=abs(x^2+x)-2

因 x2+x =0, x= -1, 0 圖形在 x= -1, 0 處轉彎

4. 高次多項式函數

y=x4-x

3-9x

2+2x+12 輸入 y=x^4-x^3-9x^2+2x+12

試找出 f(x)=x4-x

3-9x

2+2x+12, 所有實根的

位置, 分別在那兩個相鄰整數之間

x -3 -2 -1 0 1 2 3 4

f(x)

5. 指數函數 y = ax

按滑桿 按鈕, 設定名稱: a

及最大: 3 最小: 0 後按套用,

然後記得按一下 ESC

輸入 y=a^x

6. 對數函數 y = logax

按滑桿 按鈕, 設定名稱: a

及最大: 3 最小: 0 後按套用,

然後記得按一下 ESC

輸入 y=ln(x)/ln(a) (換底公式)

7. 三角函數 (以 y=sin x 為例) y = a sin( bx+c ) + d

a: 振幅 b: 週期 c, d 左右上下平移

輸入 y= sin(x) 輸入 y=2sin(x) 振幅 2 倍 輸入 y=sin(2x) 週期變一半

輸入 y=sin(x+2) 圖形左移 2 單位 輸入 y=sin(x)+2 圖形上移 2 單位

-------------------------------------------------------

輸入 y=sin(x+pi/2) 左移 2 單位, 變成…..

自行更改函數圖形顏色, 看一看這許多曲線之間的差異

★三角函數的疊合 y=Asin(x)+Bsin(x)

輸入 y=sin(x)+cos(x) 看看和 y=sin(x+45°) y=1.41*sin(x +45°) 的關係

★ 其他的三角函數

輸入 y=tan(x) 輸入 x=pi/2 ( 漸近線 ) 輸入 x= -pi/2 輸入 x= -3pi/2

注意 y=tan(x) 的週期為 π

★下列那一個正切函數值最大?

121

(1) tan 7π

3 (2) tan

5π

4 (3) tan

-4π

5 (4) tan

-11π

6 (5) tan

-20π

7

解: 輸入 a= 7pi/3 輸入 a= 7pi/3 輸入 b= 5pi/4 輸入 c= - 4pi/5

輸入 d= -11pi/6 輸入 e= - 20pi/7 輸入 (a, tan(a) ) 輸入 (b, tan(b) )

輸入 (c, tan(c) ) 輸入 (d, tan(d) ) 輸入 (e, tan(e) ) 看看這 5 個點的高度

★ GeoGebra 並未定義 cot(x), sec(x), csc(x)

要用倒數關係 輸入 y= 1/tan(x) 輸入 y= 1/cos(x) 輸入 y= 1/sin(x)

8. 反三角函數

定義域 值域

輸入 y= asin(x)

輸入 y= acos(x)

輸入 y= atan(x)

Appendix X:

Tyler’s Example of Geometrical Constructions

122

123

Appendix XI: Summary of Emerging Themes

1) Conceptions of Mathematics- Mathematics Content Knowledge

2) Conceptions of the use of technology- Conceptions of GeoGebra Conceptions

of mathematics teaching and learning in relation to conceptions of GeoGebra

Teacher transition- teaching practices

Benefit for students‟ future studies and work

GeoGebra as a educational tool

3) Infrastructure Change- Technology related themes

Critiques of hardware

Critiques of software

Encouraging advancement of technology

4) Practices with GeoGebra-

Teachers‟ mathematical content knowledge

GeoGebra affordances

Mathematical Scope- algebra-related, geometry-related or both

algebraic and geometrical topics

5) Cultural issues

Student-centred interactive pedagogy/ Teacher-centred didactic

Curriculum-focused

Textbook-oriented

Exam-driven