Forum MIPA Vol 13 2010

(1)

ISSN

=

1410-1262

Volume

13

No.

2

Juli

ZoI

0

UpayaMeningkatkan Hasil BelajarMahasiswaMelalui PenerapanModel Pembelajarun ThinkPair and SharePadaMataKuliah KimiaDasar 1 (A. Rachman

lbrahim)

Learning

Geometry

using Dynamic

Geometry Software

(DGS)

in Active

Learning Approach (Budi

Mulyono)

;.t

Peningkatan Kemampuan Mahasiswa dalam Membuktikan

Melalui

StrategiAbduktif-Deduktif padaMata

Kuliah

StrukturAljabar

Di

Program Studi Pendidikan Matematika FKIP-Unsri (Cecil

Hiltrimartin

& Yusuf

Hartono)

Pengaruh Bioakumulasi

Merkuri

pada Pertumbuhan Eceng

Gondok

_fEichornia

crassipes (Martius) Solms.l

(Ermayanti)

Upaya Meningkatkan Keaktifan dan Hasil Belajar Kimia Siswa Kelas X,

MAN

Sakatiga Indralaya Melalui Model P embelaj ar an Inquiry T erbimbing (Penelitian Tindakan Kelas)

(Fatihayani)

Pendidikan Lingkungan bagi

Masyarakat

sebagai

Mitigasi

Dampak Perubahan

Iklim

Melalui

Upaya Penyimpanan Karbon pada Kawasan Hij au

(Hitda

Zulkifl i)

Produk Transgenik Hikmah atau Bencana

(Laihat)

Pengembangan Bahan

Ajar

Mata

Kuliah

Pendahuluan

Fisika

Inti di

Program Studi Pendidikan Fisika FKIP Unsri

(Murniati)

Sintesis dan Penentuan Struktur Senyawa Kompleks

Ni(Ii)

dengan Ligan

Dipiridin

dan Turunannya (M. Hadeti

L.)

Pembelaj aran Perubahan Konseptual : Pilihan Penulisan Skrip si Mahasi swa

(Syuhendri)

ffi

Majalah

FORUMMIn

llmiah

Jurusan

Universitas

Sriwiaya

(2)

cd+vott

jO lorolSO

LEARNING

GEOMETRY USING

DYNAMIC

GEOMETRY SOFTWARE (DGS)

IN

ACTIVE LEARNING

APPROACH

Budi

MulYono t-/

Universitas Sriwijaya, Jln. Raya Palembang-Prabumulih KM 32 Indralaya

e-mail : [email protected]

Abstract: Nowadays the use of ICT in teaching-leaming activities becomes a trend in education

field. Therefore all people involving in teaching-leaming process should update their knowledge in technology especiaily ubititi"r

in

using ICT

to

improve the quality

of

students' achievement.

Mathematics teachers are expected to be more creative and innovative in designing lesson activities

which are oriented to an aitive leaming approach. DGS can be used as a tool to create lesson

activities which support to higger students

-ore

"ngag"d and active in their learning activities. DGS can help

to

visualize geometrical shapes. As we know that geometry

is

a topic which needs visualization. By creating lesson activitiis in an active leaming approach in which DGS is embedded

will help to improve and to increase students' understanding in leaming geometry.

Abstrak Saat ini penggunaan media ICT dalam proses belajar mengajar sudah menjadi salah satu

trend dalam dunia penaiaikan. Oleh karena

itu

semua pihak yang terlibat dalam proses belajar

mengajar sudah seharusnya mengikuti informasi kemajuan teknologi khususnya kemampuan dalam

p"nglunuun ICT untuk meningkatkan kualitas hasil belajar siswa. Guru matematika saat ini pun

aituntut untuk lebih kreatif dan inovatif dalam mendesain aktifitas belajar mengajar dengan

berorientasi pada peningkatan keaktifan siswa dalam proses tersebut. Salah satu caranya adalah

mendesain umintu. beiajar yang menggunakan DGS. Geometri merupakan salah satu topik

matematika yang memerlukan vizualisasi dimana hal tersebut dapat terbantu dengan menggunakan DGS.

Keywords: the use of ICT, Geometry, DGS, Active learning, Lesson activities, Students' achievement

eometry

is a

branch

of

mathematics

studying shapes

and

configurations- In learning geometry there are some

skills

that students

should acquire such

as

intuition,

measuring, and reasoning skills. Students should

have

all

those abilities after

they

learned geometry well. One of the geometry topics is the

angle

concept,

which

is

foundational

for learning geometry. There are some stages in which children understand the concepts of angle

which

are from

concrete

to

abstract

(Mitchelmore

and

White,

2004).

Geometry topics are sometimes related to visualization

of

concepts and definitions

of

geometry objects.

For example, a line can be created by connecting

two

points. To visualize this concept, a picture

of

a line should be drawn to make the concepts

more real

to

sfudents. Many kinds

of

tools can

be used to visualize geometry concepts. One

of

the

tools

is

dynamic geometry software. By using such software, students

will

easily be able

to draw and manipulate a geometrical picture. In teaching and learning activities, especially in

teaching

mathematics, mathematics teachers

82

F)RUM M:PA vol. L3 No. 2 Edisi Juli 201"0

should

not be the

center

of

the

class, and

students should be more active and independent

in

their

learning activities.

In my

opinion, an

active learning approach which combines with using DGS

will

help students learn mathematics

much better and also

will

make them active and

critical in learning activities.

I

tried to find some

literatures that support my opinion.

Aim and research question

The aim of this literature review (LR) is to find

out that teaching geometry (the angle concept) is

appropriate through an active learning approach

using DGS.

The

question

of

this

LR is:

What

kind teaching methods

is

appropriate

to

use

teaching geometry (the angle concept)?

Methodology

To

answer the research questiofl,

I

used some

literatures

which

supported

the

idea

that

an

active

learning

approach

using DGS

is

appropriate to use in teaching geometry. To find

of

in

(3)

Budi Mulyono

the

literafures,

I

used

search

engine

"scholar.google.com"

by

Uping

some search

terms into

it,

such as: teaching geometry using

ICT, active learning, concept angle in geometry,

and use of ICT in education.

Skills in

learning geometry

Many educators and researchers in mathematics

argue

that intuition

plays

a

crucial

role

in

geometry,

and that

an

infuition

process in

geometry comes

into

one's

mind

after seeing

shapes

of

geometrical

things. Actually,

it

is

difficult to define what exactly the definition

of

intuition in geometry is, but generally it is a skill to 'see' geometrical figures even

if

they are not

drawn on paper. creating and manipulating such

figures

in

the

mind

to

solve

problems in

geometry can be regarded as an intuition skill

(Fuj

ita,

Jones, and Yamamoto, 2004

b).

This means that intuition relates to what students see

and then

think

about.

P.

Treutlein ( I 9l I

;

in

Fuj

ita,

Jones,

and

Yamamoto,

2004

_b)

considered

intuition

as an

essential

skill

in

geometry as well as in everyd ay life, and argued

that training

sfudents"imagination'

through

geometry was

very

important.

An

interesting

example

of

Treutlein's tasks

for

students is

when he asked students to make new figures in

their

mind

by

manipulating

two

(given) triangles. (see _figureI ₎

Figure I

Students were asked

to

make as many

combination figures as they could

by

mentally manipulating the

first two

triangles. The more

often

students

use

their

'imagination'

in

geometry,

the

higher

the

possibility that they

improve their intuition skill in geometry. To be a

successful problem solver in geometry, a sfudent

must practice and exercise

a

skill,

which

is

called'geometrical

intuition',

in

creating and

manipulating geometrical figures

in

the mind,

perceiving

geometrical properties,

relating

images

to

concepts and theorems

in

geometry,

and deciding where and how to start showing a

qiven problem

in

geometry (Fuj ita, Jones, and

Yamamoto, 2004 a).

Measuring

in

geometry

is

one

of

the

important skills in order to determine the size

of

an angle, length

)

area, or volume of geometrical

/55N; 141-0-L262

Leorning Geometry Using Dynomic Geometry ...

things. Measuring in geometry is mostly related

to

using tools such

as

a

ruler,

a

compass, a

protractor, etc. By using such tools students can

measure real geometrical things, and they can

investigate

whether

their

intuition

about

geometrical objects

is

accurate

or

not.

For example, when sfudents are asked to investigate

whether

two

given triangles are congruent or not, students can use their infuition to answer the

question. However,

to

make sure whether the

sfudents'

infuitive

answer

is

correct

or

not, sfudents need to use a ruler and a protractor to measure all properties of each triangle.

Reasoning

in

geometry

relates

to

abilities

to give

logical

explanations, argumentations, _{verifications, or proofs to arrive} at convincing solutions to geometrical problems.

Intuition and

measuring

skills

need

to

be

supported

by

reasoning

skills.

It

means that reasoning plays

a

justification

role

for

_what

intuition and measurement give as solutions to

geometry problems.

Actually,

good reasoning

will

make

a

solution

of

a

geometry problem

more mathematical and more elegant. Through reasoning

skills

students

can

enhance their understanding

about

geometry

and

find

it possible to make other theories from what they have learned and understood.

Geometry

deals

with

mental entities

(geometrical figures) which possess conceptual

and

figural

characters

(Fischbein,

1993).

concepts

and

images

are

considered two basically distinct categories

of

mental entities.

Pieron (1957;

in

Fischbein, 1993) defines a

concept

as

a

symbolic representation (almost always verbal) used

in

the process

of

abstract

thinking and possessing

a

general significance corresponding

to

an

ensemble

of

concrete representations with regard to what they have in common. Meanwhile,

an

image

is a

sensorial representation

of

an object or phenomenon. For example, an angle

is

an abstract ideal concept,

but

it

also possesses figural properties. Actually, the absolute perfection

of

a

geometrical angle

cannot be found in reality, even though we can

find many different contexts of angle.

How children

think

and learn about

concepts

of

angle

Mathematical objects may best be described as

abstract-apart, since mathematics is essentially a

self-contained system,

but

on the

other hand,

fundamental mathematical

ideas

are

closely related

to

the real world

and

their

learning

.! ./i .t: ,j:

,,' i

-/i

)i ./? +...__-j

i.i

\i

.ii_\:

i! t

ew figures

(4)

Learning Geometry Using Dynamic Geometry ...

involves empirical concepts (Mitchelmore and

White,

2004).

In

everyday

life

we

can

see

situations around

us

as many kinds

of

angle

contexts, such as the intersection between two streets, inclination

or

slope, corners

of

a table, an end point of a pen, etc. That is why the angle

concept

is

special because

it

can appear

in

so

many different contexts. Henderson

(in

Lehrer,

2003)

suggests

three

conceptions

of

angle,

which are (1) angle as movement, (2) angle as a

geometric shape, and

(3)

angle as

a

measure.

The

angle

concept urs movement

can

be

contextual

in

rotation

or

sweep,

the

angle

concept as a geometric shape can be contextual

in

a

delineation

of

space

by two

intersecting

lines, and the angle concept as a measure can be

contextual

in

a perspective that coordinates the

first two.

Children

find

it

difficult

to

learn the angle concept because of the multifaceted nature

of

angle (Mitchelmore

&

White, 2000), and to

acquire a general concept ofangle, students need

to see the similarities between the various angle

contexts

and

identifr their

essential common

features (Mitchelmore

&

White,

2004).

In understanding about concepts

of

angle, children

pass through some developmental stages, during which "children progressively recognize deeper

and deeper similarities between

their

physical angle experiences and classiff them

firstly

into

specific

situations,

then

into

more

general

contexts, and finally into abstract domains", and

during which,

from the

classification

at

each

stage

of

development,

an

angle

concept is abstracted (Mitchlemore

&

White, 2000).

Active

Learning

In

the traditional teaching method, teachers are

always being the center of teaching and learning activities, which means that teachers are active,

and students are passive

in

the

class.

In

this method teachers give lectures

to

students and

after that teachers give some examples of what they

just

taught. Meanwhile, students are only listening

to

what their teachers explained, and

doing

some

exercises

after

they

got

some

examples

of

the exercises.

In

my opinion, such

teaching method makes

students

become dependent on their teachers, so that they

will

not be able

to

learn how

to

be critical, innovative,

and creative in their learning activities. Students

will

only

get rote understanding

of

what they

learned

by

such

a

method.

In

my

opinion, to overcome

this

problem, mathematics teachers

84

FORUM MlPAVol. 73 No. 2 Edisi Juli 2070

Budi Mulyono

should

modiff or

even change their traditional

teaching method to an active learning approach.

"Active

learning differs from "learning

from

examples"

in

that

the

learning algorithm

assumes at least some control over what part

of

the input domain

it

receives information about" (Atlas,

L.,

Chon, D.,

&

Ladner,

R.

1994). This

means that a teaching method which consists

of

only

giving students some examples and then

asking them

to

learn from those and after that

asking sfudents

to

solve some similar questions

by

themselves

is

not an

active

learning technique.

In

an

active

learning

approach, teachers should be more aware of their students'

actions

in

learning activities,

and

teachers

should make

their

students more active, more engaged, and more critical in class activities. To prepare

for

active learning activities, teachers

should design

a

lesson plan

in

which students

must

read,

write,

discuss,

or

be

engaged in

solving problems (Bonwell,

C.

Charles, 1991).

Therefore in such teaching methods, teachers are

not the center

of

the class, but students are the

centre of learning activities. "Most important, to be actively involved, students must engage in

such higher-order

thinking

tasks

as

analysis, synthesis, and evaluation. Within this context,

it

is

proposed

that

strategies promoting active

learning

be

defined

as

instructional activities involving students

in

doing and thinking about

what they are

doing"

(Bonwell,

C.

Charles,

l99l).

Based

on

Bonwell's opinion, there are

some of the major characteristics associated with active learning strategies: "students are involved

in

more than

passive learning; students are

engaged

in

activities;

there

is

less emphasis

placed on information transmission and greater

emphasis placed

on

developing student skills;

there

is

greater

emphasis

placed

on

the

exploration

of

attitudes

and

values; students

motivation

is

increased; students

can

receive

immediate feedback

from their

instructor; and

students are involved

in

higher order thinking (analysis, synthesis, evaluation)."

After

all,

I

propose that an active learning approach should

be

considered

as one

possible innovation

of

teaching methods

to

be applied in teaching and

learning activities.

Using

ICT in

teaching

and learning

activities

The use

of ICT

in

education

is

an issue which nowadays becomes popular

to

be implemented

in teaching and learning activities.

"If

designed

(5)

Budi Mulyono

and

implemented

properly,

ICT-supported education can promote

the

acquisition

of

the

knowledge and skills that

will

empower students

for

lifelong

learning"(Tinio,

V.L.,

2002). By

using ICT in education, teaching methods can be

shifted from a teacher-centered pedagogy to one

that is a student-centered. ICT use

in

education

can

support

active

learning,

collaborative learning, creative learning, integrative learning, and evaluative learning approaches.

o

"Active

learning: ICT-enhanced learning

mobili

zes tools for

examination, calculation and analysis information, thus

providing a platform

for

student inquiry,

analysis

and

construction

of

new

information. Learners therefore learn as

they do and, whenever appropriate, work

on

real-life

problems in-depth, making learning less abstract and more relevant to

learner's

life

situation. In this way, and in contrast

to

memo rization-based

or

rote

learning, ICT-enhanced learning promotes

increased

learner

engagement. ICT-enhanced learning

is

also

_{Just-in-time'}

learning

in

which

learners

can

choose

what to learn when they need to learn it.

o

Collaborative

learning:

ICT-supported

learning

encourages

interaction

and

cooperation among students, teachers, and

experts regardless

of

where

they

are.

Apart from

modeling

real-world

interactions, ICT-supported

learning provides learners the opportunity to work

with

people

from

different

cultures,

thereby

helping

to

enhance learner's

teaming and communicative skills as well

Dynamic Geometry Software

(DGS)

Teaching mathematics in a regular or traditional way without using

ICT

does not mean that the

/55N; 1410-L262

Learning Geometry Using Dynomic Geometry ...

as their

global

awareness.

It

models

learning done throughout

the

learner's

lifetime

by

expanding the learning space

to

include not just peers but also mentors and experts from different fields.

o

Creative learning: ICT-supported learning

promotes

the

manipulation

of

existing information

and

creation

of

real-world products rather than the regurgitation

of

received information.

o

Integrative

learning:

ICT-enhanced

learning promotes a thematic, integrative

approach

to

teaching and learning. This

approach eliminates

the

artificial

separation

between

the

different

disciplines

and

between

theory

and

practice that characterizes the traditional

classroom approach.

o

Evaluating

learning:

ICT-enhanced

learning

is

student-directed

and

diagnostic. Unlike static,text-or

o

print-based educational technologies, ICT-enhanced learning recognizes

that

there

are many different learning pathways and

many different articulations of knowledge.

ICTs allow

learners

to

explore

and

discover rather

than merely listen

and

remember." (Tinio, V .L.,2002)

Table below

will

show some compar-isons between a traditional teaching method and

a teaching method which is using ICT. (Tinio, v.L. ,2002)

method is not appropriate for students. However,

teaching

mathematics

through

using

ICT

nowadays has become a familiar trend in many countries. Using ICT in education is a method to help teachers and students to interact in a better

Aspect Less (Traditional teaching method) More (Teaching method usins ICT)

Active o Activities prescribed by teacher

.

Whole class instruction

.

Little variation in activities

o Pace determined by the programme

o

Activities determined by learners

o

Small groups

.

Many different activities

o

Pace determined by learners

Collaborative o Individual

o Homogenous groups

o Everyone for himlherself

Working in teams Heterogeneous groups

Supporting each other

o o o

Creative

.

Reproductive learning

o Apply know solutions to problems

o

Productive learning

o

Find new solutions to problems Evaluative O

o

Teacher-directed Summative

Student-directed

Diagnostic o

o

(6)

Learning Geometry Using Dynamic Geometry

"'

way in teaching-learning activities (Jhurree,

V',

2005).

One

example

of

the

use

of

ICT

in education

is

using dynamic geometry software

to

teach mathematics

to

students, especially

geometry.

Dynamic

geometry software

is

a

certain type of software which is predominantly used

for

the consffuction and analysis

of

tasks

and problems

in

elementary geometry (Straber, Bielefeld, and Lulea, 2002)- With this software a

user can construct, create, and manipulate all

kinds

of

geometrical shapes.

Using DGS

in learning activities

will

be helpful

to

students,

because

it

provides them

with

access

to

the

world

of

geometrical theorems,

which

is

mediated

by

features

of the

software

environment, certainly

in

the

vital

early

and

intermediate stages of using the software (Jones,

2000).

"The ability

of

a

student

to

obtain,

analyze, measure

and

compare

the

many

instances

of

a

mathematical proposition

in

a

dynamic geometry environment gives him/her opportunities

to

make conjectures and

test

a

proposition. These roles

for

dynamic geometry

roft**"

are widely acknowledged as having the

potential

to

enrich the teaching

of

geometr;r" (Guven, B., 2008).

There are many kinds

of

DGS such as

Cabri, Cinderela, Geogebra, etc. Some are free

software,

which

means users

do not

need a

license

to

use

the

software;

one

of

these is

Geogebra.

This

software can

be

downloaded free

of

charge on the official site

of

Geogebra

which is

http://www.geogebra.orglcmsl. Geogebra provides many tools

in

which users

can

interactively create

and

manipulate geometrical shapes to find geometrical theories.

This software does not require special skills in

computer programming to use it. To learn about

angle concepts, Geogebra also provides many tools

to

students

to

enable them

to

understand

angle

concepts. Students

can

do

many

experiments by creating, drawing, constructing, and manipulating any angles they desire. Since

geometry always relates

to

shapes

and

configurations, visualization is very important in

helping

students

to

learn

geometry.

By providing visualizations

of

geometrical shapes

children

will

easily see, and after this they can

use

their

intuition,

measuring,

and

reasoning

skills to respond to every question related to the

shapes. Since Geogebra provides such good tools

for

visualization

of

geometrical objects'

I

suggest

that

mathematics teachers should use

Geogebra as the DGS in teaching

geometry-86

FORUM M\PAVol. 73 No.2 Edisi luli 2070

Budi MulYono

Teaching geometry

using

GDS

As

we know, nowadays teaching and learning

through

DGS has

been

known

among

mathematics teachers, and also there had been a

lot of

research which conducted

to

investigate

about learning geometry through DGS- One

of

them

is

the research conducted

by

Sang Sook

Choi-Koh (a professor of mathematics education

from Korea), which investigated the geometric

learning

of a

secondary

school

during instruction, on the basis of the van Hiele model,

with

dynamic

geometry software

as

a

tool

(Choi-Koh, S.S.,

1999).

In

his

research, he

examined how changes in the students' learning to the van Hiele levels

of

geometric thought for the geometric topics of right triangles, isosceles

triangles,

and

equilateral triangles.

The

participant

of

his research was

a

student

of

a

iecondary school. However, the student whom

he chose had not taken geometry but had taken a

computer course

or

had had experience with computer

at

home,

which

means

that

his

experimental students did not yet have learning-geometry experience, but had computer skills' In

his

research,

he

investigated

a

student (Fred) through

four

learning stages,

which

were: 1'

Intuitive

learning stage,

2.

Analytical learning

stage,

3.

Inductive learning

stage,

and

4.

Deductive

learning stage.

During

his

investigation,

he

saw that Fred was

really enthusiastic doing the given task, and he also found that Fred properly performed the task, and

also Fred did the task

in

a much simpler way

than

he

expected.

He

also found that

the

visualization

by

dynamic computer software

helped

Fred

made some

conjectures about

relationships between triangles- The results

of

his

research about

using

dynamic geometry

software show that learning using DGS helps

motivate

students

to

learn

with

more

enthusiasm, which could lead students reaching better achievements in their study.

Another

research

of

using DGS

in teaching geometry

is

Guven's research (2008) which shows that how using dynamic geometry

software

can

provide

an

opportunity

to

link

between empirical and deductive reasoning, and

how such software can utilized

to

gain insight into a deductive argument.

Gonclusion

After all,

I

can say that using ICT in education is

one

of

innovation ways

to

make teaching and

learning much better. In this way, teaching and

(7)

Budi Mulyono

learning activities

will

be

shifted

from

a

traditional way

to

one

that

is

innovative way using

ICT.

In

the

innovative

way

using ICT, students

will

have

more

opportunities

to

be

active and

to

explore their talent. Meanwhile a

teacher

is

more passive and

just

gives some

guidance and help when students need

to

face

problems.

In

teaching mathematics, teachers

should design lesson plans which make students

more active and allow students

to

explore their talent

in

mathematics. As we know that one

of

mathematics

topics taught

to

students

is

geometry

which

needs some visual ization to make sfudents easy to understand some concepts

of

geometry. The concept

of

angle

is

one

of

geometry topics which students

find

it

difficult to understand because

of

its multifaceted nature

of

angle.

By

using DGS, students

will

have an

opporfunity

to

experience themselves

to overcome the problem, because such software

will

provide students

a

link

between empirical

and deductive reasoning

to

learn the concept

of

angle.

Therefore,

I

suggest

that

an

active

learning approach using DGS is an appropriate

way to teach the concept of angle to students.

References

Atlas, L., Chon, D,,

&

Ladner, R. ( 1994). Improving generalization

with

active learning. Machine

Learning, I5:201-221 .

Bielefeld, Lulea,

&

Straber, R. (2002). Research on

dynamic geometry software

(DGS)

an

introduction. ZDM,Vol. 34 (3).

Bonwell, Charles C.,

&

Eison, James

A.

(1991).

Active

learning: creating excitement

in

the

classroom. ENC Digest. ED340272 l99l -09-00.

Choi-Koh, S.S. (1999).

A

student learning of geometry

using

computer. The

Journal of

Educational Research, Vol. 92 (Itto.S)

Fischbein, E. ( 1993). The theory of figural concepts.

Educational Studies in Mathematics, Vol. 24, It{o.

2, 139-162.

Fujita,

T.,

Jones,

K.,

&

Yamamoto, S. (2004 a).

Geometrical intuition and the learning and the

teaching of geometry. Paper Presented to Topic

Study Group

29

at

the

I0'h

lfiernational

Congress

on

Mathematical

Education.

Copenhagen, Denmark.

Fujita, T., Jones, K,, & Yamamoto, S. (2004 b). The

role of intuition in geometry education: learning

from the teaching practice

in

the early 20'fi

century. Paper Presented to Topic Study Group

29

at

the I0'h

Internationit Congress on

Mathematical Education. Copenhagen, Denmark.

Learning Geometry Using Dynamic Geometry ...

Guven, B. (2008). Using dynamic geometry software to gain insight into a proof. _{International Journal}

of

Computers

for

Mathematical Learning, 13:

251-262.

Jones,

K.

(2000). Providing

a

foundation for

deductive reasoning: sfudents' interpretations when using dynamic geometry software and their

evolving mathematical explanation s. Educational

Studies in Mathematics, 44,55-85.

Juhrree,

V.

(2005). Technology integration in education in developing countries: Guidelines to

policy makers. International Education Journal, 6(4), 467-483.

Lehrer,

R.

(2003). Developing understanding of measurement.

A

Research Companion to

Principles and Standard _forschool Mathematics,

179-t92.

Mitchelmore,

M,C., &

White,

p.

(2000).

Development

of

angle concepts by progressive

abstraction

and

generalization

.

Educational

Studies in Mathematics, 4

I,

209-2:35.

Mitchelmore, M.C.,

&

White, p. (2004). Abstraction

in

mathematics

and

mathematics learning,

Proceedings

of

the

2B'h Conference

of

the

International Group

for

the

Psychology

of

Mathematics Education, Vol. 3, 329-339.

Tinio, V.L. (2002). ICT in education. E-Primers on

the

Application

of

Information

and

Communication Technologies

(ICTs)

to

Development. UNDP's regional project, the

Asia-Pacific Development Information Programme.

www.eprimers. org or www. apdip.net

Bibliography

HennesSy, S. & Ruthven, K. (2002). A practitioner

model of the use of computer-based tools and

resources to support mathematics teaching and

learnin g. Educational Studies in Mathematics,

49, 47-gg.

Kortenk?-p, U. (1999). Foundation of Dynamic

Geometry. A Dissertation Submitted to the Swrss

Federal Institute of Technology Zurich _forthe

Degree of Doctor of Technical Sciences. Diss.

ETH N0 I 3403.

Mitchelmore, M.C., Prescott, A.

&

White, P. (2002).

Student difficulties in abstracting angle concepts

from physical activities with concrete materials.

Proceedings

of

the 2 5'h Annual Conference

of

Mathematics Education Research Group

of

Australia, Auckland, 583-591 .

Straesser, R. (2001). Cabri-Geometre: Does dynamic

geometric software (DGS) change geometry and its teaching and learning? International Journal

of Computers _forMathematical Learning, 6,

319-333.