Forum MIPA Vol 13 2010
ISSN
ffi
=
1410-1262
FORUMMIn
Majalah llmiah Jurusan PMIPA FKIP
Universitas Sriwiaya
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 StrategiAbduktifDeduktif 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
Pendidikan Fisika FKIP Unsri (Murniati)
Inti di Program Studi
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)
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
and active in their learning activities. DGS
activities which support to higger students -ore
"ngag"d
that geometry is a topic which needs
we
know
As
can help to visualize geometrical shapes.
approach in which DGS is embedded
leaming
an
active
in
activitiis
visualization. By creating lesson
geometry.
leaming
in
understanding
students'
increase
to
will help to improve and
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
Abstrak
Saat
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
is foundational for
learning geometry. There are some stages in
which children understand the concepts of angle
angle concept, which
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 of
teaching methods is appropriate to use in
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
/55N: L410-1262
Budi Mulyono
the
Leorning Geometry Using Dynomic Geometry ...
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
for
Treutlein's tasks
students is
when he asked students to make new figures in
their mind by manipulating two (given)
triangles. (see figure I )
.!
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-/i
,,'
i
)i
./?
+...__-j
i.i
\i
\:
.ii
i!
t
Figure I
ew fig ures
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
things. Measuring in geometry is mostly related
compass, a
protractor, etc. By using such tools students can
measure real geometrical things, and they can
to using tools such as a ruler, a
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
FORUM MIPA Vol. L3 No. 2 Edisi Juli
2010
83
Learning Geometry Using Dynamic Geometry ...
Budi Mulyono
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
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
dependent on their teachers, so that they
84
FORUM MlPAVol. 73 No. 2 Edisi Juli 2070
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;
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
there
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
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Budi Mulyono
Learning Geometry Using Dynomic Geometry ...
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
o
and
construction
of
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
o
products rather than the regurgitation of
received information.
o Integrative learning:
with people from different cultures,
thereby helping to enhance learner's
ICT-enhanced
learning promotes a thematic, integrative
approach to teaching and learning. This
approach eliminates the artificial
separation between the different
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. ICTenhanced learning is also Just-in-time'
learning in which learners can choose
what to learn when they need to learn it.
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
and experts from different fields.
Creative learning: ICT-supported learning
promotes the manipulation of existing
information and creation of real-world
disciplines and between theory
and
practice that characterizes the traditional
classroom approach.
o Evaluating learning: ICT-enhanced
learning is student-directed and
o
diagnostic. Unlike static,text-or
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)
will show some comparisons between a traditional teaching method and
Table below
a teaching method which is using ICT. (Tinio,
v.L. ,2002)
teaming and communicative skills as well
Aspect
Active
Less (Traditional teaching method)
o Activities prescribed by teacher
.
.
Whole class instruction
Little variation in activities
o Pace determined by the programme
Collaborative
Creative
Evaluative
o Individual
o Homogenous groups
o Everyone for himlherself
.
Reproductive learning
o Apply know solutions to problems
O
o
Teacher-directed
Summative
More (Teaching method usins ICT)
o
o
.
o
o
o
o
o
o
Activities determined by learners
Small groups
Many different activities
Pace determined by learners
Working in teams
Heterogeneous groups
Supporting each other
Productive learning
Find new solutions to problems
o
Student-directed
o
Diagnostic
method is not appropriate for students. However,
Dynamic Geometry Software (DGS)
teaching mathematics through using ICT
Teaching mathematics in a regular or traditional
way without using ICT does not mean that the
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
/55N; 1410-L262
FORUM MIPA Vol. 73 No. 2 Edisi Juli
20L0
85
Learning Geometry Using Dynamic Geometry
Budi MulYono
"'
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
(Jones,
intermediate stages of using the software
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
to learn geometry.
By
shapes
geometrical
providing visualizations of
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-
helping students
86
FORUM M\PAVol. 73 No.2 Edisi luli 2070
Teaching geometry using GDS
As we know, nowadays teaching and learning
through DGS has been known
among
been a
had
there
also
and
teachers,
mathematics
investigate
to
conducted
lot of research which
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 learninggeometry 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
/55N; 7470-1262
Budi Mulyono
Learning Geometry Using Dynamic Geometry ...
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
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 for school Mathematics,
179-t92.
Mitchelmore,
M,C., & White, p.
Development
of angle
concepts
(2000).
by progressive
abstraction and generalization . Educational
Studies in Mathematics, 4 I, 209-2:35.
active
& White, p. (2004). Abstraction
mathematics and mathematics learning,
Proceedings of the 2B'h Conference of the
International Group for the Psychology of
learning approach using DGS is an appropriate
way to teach the concept of angle to students.
Tinio, V.L. (2002). ICT in education. E-Primers on
angle. Therefore,
I
suggest that
an
in
Mathematics Education, Vol. 3, 329-339.
the Application of
Information
Communication Technologies
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Fischbein, E. ( 1993). The theory of figural concepts.
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/SSN; 14L0-1262
FORUM MIPA Vol.
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No. 2 Edisi Juli
2070
87
ffi
=
1410-1262
FORUMMIn
Majalah llmiah Jurusan PMIPA FKIP
Universitas Sriwiaya
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 StrategiAbduktifDeduktif 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
Pendidikan Fisika FKIP Unsri (Murniati)
Inti di Program Studi
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)
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
and active in their learning activities. DGS
activities which support to higger students -ore
"ngag"d
that geometry is a topic which needs
we
know
As
can help to visualize geometrical shapes.
approach in which DGS is embedded
leaming
an
active
in
activitiis
visualization. By creating lesson
geometry.
leaming
in
understanding
students'
increase
to
will help to improve and
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
Abstrak
Saat
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
is foundational for
learning geometry. There are some stages in
which children understand the concepts of angle
angle concept, which
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 of
teaching methods is appropriate to use in
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
/55N: L410-1262
Budi Mulyono
the
Leorning Geometry Using Dynomic Geometry ...
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
for
Treutlein's tasks
students is
when he asked students to make new figures in
their mind by manipulating two (given)
triangles. (see figure I )
.!
./i
.t:
,j:
-/i
,,'
i
)i
./?
+...__-j
i.i
\i
\:
.ii
i!
t
Figure I
ew fig ures
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
things. Measuring in geometry is mostly related
compass, a
protractor, etc. By using such tools students can
measure real geometrical things, and they can
to using tools such as a ruler, a
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
FORUM MIPA Vol. L3 No. 2 Edisi Juli
2010
83
Learning Geometry Using Dynamic Geometry ...
Budi Mulyono
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
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
dependent on their teachers, so that they
84
FORUM MlPAVol. 73 No. 2 Edisi Juli 2070
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;
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
there
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
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Learning Geometry Using Dynomic Geometry ...
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
o
and
construction
of
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
o
products rather than the regurgitation of
received information.
o Integrative learning:
with people from different cultures,
thereby helping to enhance learner's
ICT-enhanced
learning promotes a thematic, integrative
approach to teaching and learning. This
approach eliminates the artificial
separation between the different
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. ICTenhanced learning is also Just-in-time'
learning in which learners can choose
what to learn when they need to learn it.
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
and experts from different fields.
Creative learning: ICT-supported learning
promotes the manipulation of existing
information and creation of real-world
disciplines and between theory
and
practice that characterizes the traditional
classroom approach.
o Evaluating learning: ICT-enhanced
learning is student-directed and
o
diagnostic. Unlike static,text-or
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)
will show some comparisons between a traditional teaching method and
Table below
a teaching method which is using ICT. (Tinio,
v.L. ,2002)
teaming and communicative skills as well
Aspect
Active
Less (Traditional teaching method)
o Activities prescribed by teacher
.
.
Whole class instruction
Little variation in activities
o Pace determined by the programme
Collaborative
Creative
Evaluative
o Individual
o Homogenous groups
o Everyone for himlherself
.
Reproductive learning
o Apply know solutions to problems
O
o
Teacher-directed
Summative
More (Teaching method usins ICT)
o
o
.
o
o
o
o
o
o
Activities determined by learners
Small groups
Many different activities
Pace determined by learners
Working in teams
Heterogeneous groups
Supporting each other
Productive learning
Find new solutions to problems
o
Student-directed
o
Diagnostic
method is not appropriate for students. However,
Dynamic Geometry Software (DGS)
teaching mathematics through using ICT
Teaching mathematics in a regular or traditional
way without using ICT does not mean that the
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
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FORUM MIPA Vol. 73 No. 2 Edisi Juli
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Learning Geometry Using Dynamic Geometry
Budi MulYono
"'
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
(Jones,
intermediate stages of using the software
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
to learn geometry.
By
shapes
geometrical
providing visualizations of
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-
helping students
86
FORUM M\PAVol. 73 No.2 Edisi luli 2070
Teaching geometry using GDS
As we know, nowadays teaching and learning
through DGS has been known
among
been a
had
there
also
and
teachers,
mathematics
investigate
to
conducted
lot of research which
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 learninggeometry 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
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Learning Geometry Using Dynamic Geometry ...
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
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 for school Mathematics,
179-t92.
Mitchelmore,
M,C., & White, p.
Development
of angle
concepts
(2000).
by progressive
abstraction and generalization . Educational
Studies in Mathematics, 4 I, 209-2:35.
active
& White, p. (2004). Abstraction
mathematics and mathematics learning,
Proceedings of the 2B'h Conference of the
International Group for the Psychology of
learning approach using DGS is an appropriate
way to teach the concept of angle to students.
Tinio, V.L. (2002). ICT in education. E-Primers on
angle. Therefore,
I
suggest that
an
in
Mathematics Education, Vol. 3, 329-339.
the Application of
Information
Communication Technologies
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