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93THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
Jurnal Teknologi, 49(E) Dis. 2008: 93–107
© Universiti Teknologi Malaysia
THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
TO DEVELOP GEOMETRIC THINKING
ABDUL HALIM ABDULLAH1 & MOHINI MOHAMED2
Abstract: The purpose of this research is to measure the level of students’ geometrical thinking
after using a newly developed module based on Form Two topic on Circles using an Interactive
Geometry software (IGS). The research comprises of two phases which are the development phase
and the implementation phase. In the development phase, a learning module using KDE Interactive
Geometry (KIG) software was developed. The questions in the module were developed based on
the levels as proposed in the Van Hiele Model which are visualization, analysis and informal
deduction. In the implementation phase, the developed module was then used as a treatment on
three Form Two students involving one-group of pre-test post-test research design. The students
were chosen randomly and they were given a pre and post test before and after the learning
sessions of circles using IGS. While using the module, the students’ conversations with the researcher
were recorded. The transcript was then analysed to determine the types of critical and creative
thinking utilised by the students. The critical and creative thinking indicators as suggested by the
Curriculum Development Centre were referred. The students’ work completed in the module,
together with their work saved in hard disk were analysed for the purpose of identifying the
students’ geometrical thinking levels. The results showed that the students were able to utilise
various types of critical and creative thinking while solving the questions in the module. In addition,
the students were also able to achieve the third level of geometrical thinking as proposed in van
Hiele Model. The post test results indicated that the students’ geometrical thinking showed clear
improvement after using the module.
Keywords: Geometrical thinking; critical and creative thinking; interactive geometry software
Abstrak: Kajian ini bertujuan untuk mengukur tahap pemikiran geometri pelajar selepas
menggunakan modul pembelajaran bulatan tingkatan dua menggunakan sebuah perisian geometri
interaktif. Kajian ini melibatkan dua fasa, iaitu fasa pembangunan dan fasa perlaksanaan. Dalam
fasa pembangunan, sebuah modul pembelajaran topik bulatan tingkatan dua menggunakan perisian
KDE Interactive Geometry (KIG) dibangunkan. Soalan-soalan di dalam modul dibina berdasarkan
kepada tiga tahap pemikiran geometri sebagaimana yang dicadangkan dalam Model Van Hiele,
iaitu visualisasi, analisis dan deduksi tidak formal. Dalam fasa perlaksanaan, modul tersebut
kemudiannya dijadikan rawatan dan digunakan oleh tiga orang pelajar tingkatan dua yang terlibat
dalam kajian eksperimen berbentuk praujian-pasca ujian satu kumpulan ini. Pelajar yang terlibat
dalam kajian ini dipilih secara rawak dan mereka diberikan ujian pra dan ujian pos sebelum dan
selepas rawatan diberikan. Semasa menggunakan modul tersebut, perbualan pengkaji dengan
pelajar dirakamkan dan kemudian ditranskripsikan untuk mengenal pasti jenis-jenis kemahiran
berfikir kritis dan kreatif (KBKK) yang telah digunakan. Petunjuk KBKK yang dicadangkan oleh
1&2Department of Science and Mathematics Education, Faculty of Education, Universiti Teknologi
Malaysia, 81310 UTM Skudai, Johor Bahru, Malaysia
Email: p-halim@utm.my, p-mohini@utm.my
ABDUL HALIM ABDULLAH & MOHINI MOHAMED94
Pusat Perkembangan Kurikulum dirujuk bagi tujuan pengenal pastian KBKK. Hasil kerja pelajar
di dalam modul dan di dalam cakera keras juga telah dianalisis bagi mengenal pasti tahap pemikiran
geometri mereka. Hasil dapatan menunjukkan pelbagai KBKK telah digunakan pelajar semasa
mereka menyelesaikan soalan yang terdapat di dalam modul. Di samping itu, pelajar juga dapat
mencapai tahap tiga pemikiran geometri dalam Model Van Hiele. Keputusan ujian pos turut
menunjukkan peningkatan tahap pemikiran geometri pelajar setelah mereka menggunakan modul
yang telah dibangunkan.
Kata kunci : Pemikiran geometri; kemahiran berfikir kritis dan kreatif; perisian geometri interaktif
1.0 INTRODUCTION
Geometry is one of the most important components of the school mathematics
curriculum. In Malaysia, the syllabus for Mathematics under the Kurikulum Bersepadu
Sekolah Menengah (KBSM) highlights and includes geometry. One of the main
reason is to provide the platform for students to expand their visualisation skills and
to enhance their knowledge and understanding in the area. This will enable them to
utilise geometrical structures and theorems ( Jones, 2002). According to Curriculum
Development Centre (2000), shapes and space are vital components in secondary
level Mathematics. The knowledge and skill in this field and its connection to other
topics are useful in our daily lives. By increasing the level of understanding of students
in this field, students can solve geometric problems effectively. Besides that, students
can also increase their visual thinking and appreciate the aesthetic values of shapes
and space. Principles and Standards for School Mathematics documented by National
Council of Teachers of Mathematics (NCTM) accentuated that through the study of
geometry, students will learn about geometric shapes and structures and analyze
their characteristics and relationships. Furthermore Sherard (1981) stated seven reasons
on why geometry is supposed to be taught in school and the reasons are (1) geometry
is an important aid for communication, (2) geometry has important applications to
real life problems, (3) geometry has important applications to topics in basic
mathematics such as arithmetical, algebraic and statistical concepts, (4) geometry
gives valuable preparation for courses in higher mathematics and the sciences and
for a variety of careers requiring mathematical skills, (5) geometry provides
opportunities for developing spatial perception, (6) geometry can serve as a vehicle
for stimulating and exercising general thinking skills and problem solving abilities,
and (7) there are cultural and aesthetic values to be derived from the study of
geometry. Hence, the techniques in learning geometry concepts should be
implemented in an effective way. Instead of using the traditional approach which is
concentrating more on memorization of geometrical concepts, students’ geometrical
learning experience should be changed in a more meaningful way.
In the new millennium era, a lot of interactive geometrical software are being
introduced. Geometer’s Sketchpad (GSP) was one of the early kind. The other
examples of such software are Cabri, Thales, Cinderella and Dr Geo ( Jones, 2001).
KDE Interactive Geometry (KIG) is another model of interactive geometrical software
95THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
with some similar capabilities to those of GSP (Zaleha, 2007). KIG is an open source
geometrical software and runs on LINUX operating system. Computer-based learning
environment especially the use of dynamic geometry software has been shown to
have some benefits for learning geometry (Knerr, 1982; NCTM, 2000; Norhayati,
2005). There are a few attributes of dynamic geometrical software that can make
students’ geometrical learning more meaningful. One attribute is the ability to specify
the geometrical relationships between objects created on the computer screen, such
as points, lines and circles. A second attribute is the ability to explore graphically the
implications of the geometrical relationships established in constructing a figure.
Within the computer environment the geometrical objects created on the screen can
be manipulated, moved and reshaped interactively with the use of the mouse
(Christou, 2005). In addition, the tools, definitions, exploration techniques and visual
representations associated with dynamic geometry contribute to a learning
environment fundamentally removed from its straightedge-and-compass counterpart
(Laborde, 1998).
2.0 PROBLEM BACKGROUND
Geometry is not only an important field in the mathematics curriculum in schools, it
is also an important element in Mathematics itself (Atiyah, 2001). Nonetheless, many
parties are of the opinion that the learning of geometry in a classroom environment
today is a traditional approach and not student centric (Noraini, 2005; Olkun, 2005).
Based on the evaluation conducted by National Assessment of Educational Progress
Right Toolbar
Window
Menu Bar
Left Toolbar
Figure 1 An interface of KIG
ABDUL HALIM ABDULLAH & MOHINI MOHAMED96
(NAEP ), students fail to grasp the basic concepts of geometry and fail to master the
skill of geometry problem solving (Kouba et al. 1990). The factors that lead to this
weak achievement is the students’ habit and practice that only focus on identifying
and naming geometric shapes and learning symbols for basic geometry concepts
(Carpenter et al. 1980 & Flanders, 1987). This is also agreed by Battista (2002) who
stated that the traditional method in learning mathematical topics related to geometry
only gives focus in memorising the definition of various shapes and its characteristics.
Students should be learning geometry by understanding shapes, movements and its
relations with the surrounding space (Clements & Battista, 1986). The early experience
of students in learning geometry should be focused on informal learning of shapes
and its physical characteristics. This knowledge is then made the first objective to
expand their knowledge in the surrounding space. The next experience should
involve students analysing and collating concepts and relations of shapes in geometry.
Due to the fact that the geometry curriculum in secondary schools only focus on
identification of shapes and usage of geometric themes (Kouba et al. 2000), the
opportunity to solve geometric problems are limited. This will impact the students’
chance in expanding their spatial thinking, and this is a basic skill required in geometry.
Students also have limited opportunity to analyse and conceptualise the ideas of
geometry. In addition to that, according to National Advisory Committee of
Mathematical Education (NACOME, 1995), the contents of geometry in text books
are insufficient and the task of teachers are to transfer what is already available in
these text books (Noor Azlan Ahmad Zanzali, 1987). Hence, after learning geometry
in secondary schools, a study revealed that some university students do not have
solid background in geometry to take up further courses in geometry (Hoffer, 1991).
They also face difficulties in learning other mathematical ideas related to geometry
like vectors, transformation, coordinates and trigonometry (Fey et al. 1994). Besides
that, the current syllabus for geometry does not focus on the relation in geometry
understanding. Too much emphasis is given in naming and identifying geometric
symbols formally. Exploring of space, analysis, synthesis and solving geometry
problems were not emphasised.
The result of the study from NAEP revealed that most students in every level
regard their role is geometry classes as passive. They feel that their chance to interact
with their peers on geometry, participating in geometry activities and working on
geometry shapes are very limited (Carpenter et al. 1980). This matter has also been
highlighted by David Wagner (2004) who stated that students feel that their roles in
mathematics and notably geometry classes are passive. They spend most time listening
to geometric facts, observing the teacher solve problems on the blackboard while
solving questions in the text books were done on their own. The study by Kouba
et al. (1998) showed that 80 percent of grade 7 students think that studying geometry
is based on rules while 50 percent studying geometry by memorising. This result is
contradicting the proposal by the National Council of Teachers of Mathematics.
97THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
The proposal states that mathematics teachers should provide the opportunity for
students to actively participate in the learning of geometry besides giving focus on
experimental activities, exploration and communication on geometric problem solving
in a conducive environment (NCTM, 1980). This proposal is inclined with the belief
that learning geometry should be student oriented, encourages the constructive process
whereby students build and modify their knowledge and are responsible in referring,
filtering and expanding their knowledge in this field (Hatfield, 1999). As an alternative
to replace the traditional method in delivering and presenting topics of geometry,
students themselves need to build their own concepts and meaningful evaluating
skills to enable them to analyse and solve related problems (Battista, 2001). However,
topics in geometry are always neglected in secondary schools due to several factors
(Olkun, 2005). These include the lack of concrete resources, computer softwares
and lack of expertise in the skills of computer handling. Hence, mathematics teacher
should think and practice the best way on how to enhance students geometrical
learning.
3.0 VAN HIELE MODEL
Van Hiele Model is a model concerning the learning of geometry. It is in hierarchical
sequence of five levels of thinking which are visualization, analysis, informal deduction,
formal deduction and rigor. At level 1 which is visualisation, geometric figures are
recognized by their shape as a whole. For example, students can identify both squares
and rectangle but they think that squares are not rectangles. At level 2 which is
analysis, students can discern some characteristics of figures. At level 3 which is
informal deduction, students can see interrelationships between figures and derive
relationship among figures. At level 4 which is formal deduction, students understand
the significance of deduction and the role of postulates, theorems and proof. They
can write proofs with understanding. At level 5 which is rigor,
students understand
the formal aspects of deduction, such as establishing and comparing mathematical
systems. They also can understand the use of indirect proof and proof by contra
positive, and can understand non-Euclidean systems. Although there are five levels
of thoughts in the van Hiele model, this research will focus on the first three levels
which are visualisation, analysis and informal deduction. This is based on the study
by Knight, K.C. (2006) which stated that the National Council of Teachers of
Mathematics (NCTM) expected that students who completed Grade 8 (Form 2) are
able to attain the third level in the van Hiele model.
4.0 PURPOSE OF THE STUDY
This study has two main objectives:
(1) To develop a module for learning circles with an IGS
ABDUL HALIM ABDULLAH & MOHINI MOHAMED98
(2) To identify the types of critical and creative thinking the van Hiele level of
understanding of geometry among students who have used the developed
module
5.0 THEORETICAL FRAMEWORK
Cognitive approach looks at students as active knowledge explorers and focus on
the cognitive processes. Learning styles such as discovery learning, inquiry and so
on have been recommended to be implemented in the classroom because these
learning styles can encourage the use of students’ minds (Battista, 2002). In geometry,
students’ thinking in two-dimensional geometry could be best explained using the
Van Hiele theory. Crowley (1987) stated that there are a few characteristics of Van
Hiele theory. One of them is the development of geometric ideas progresses through
a hierarchy of levels. Students first learn to recognize whole, shapes and then to
analyze the relevant properties of shape. Later they can see relationships between
shapes and make simple deductions. Teachers should provide their students with
appropriate experiences and the opportunities to discuss them. Besides that, teachers
can assess their students’ levels of thought and provide instruction at those levels.
The teacher should provide experiences organized according to the phases of learning
to develop each successive level of understanding. Research done by Suguna (2005)
proved that the use of various types of critical and creative thinking while learning
geometry with interactive geometry software can help students achieve the third
level in the Van Hiele theory.
Figure 2 Theoretical framework
Creative and Critical Thinking (KBKK)
Level 3
Informal
Deducation
Level 1
Visualisation
Level 2
Analysis
99THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
6.0 THE DEVELOPMENT OF THE MODULE AND THE CIRCLE
TEST
A module on learning circles with IGS was developed by the researcher. The
content in the module was based on a topic of circles in form two mathematics
syllabus. There are four activities in the module representing each subtopic of circles.
Practice questions were developed for each activity and the questions were developed
based on first three levels in Van Hiele Model. The levels are visualisation, analysis
and informal deduction. Table 1 shows the content summary of the developed
module.
Table 2 displays the sample activity and practice questions that cover the three
Van Hiele levels.
Table 1 Summary of the module
Activities Subtopics Questions Tested van Hiele level
Informal deduction
1 Parts of a Circle 1 Analysis
Visualisation
Informal deduction
2 Analysis
Visualisation
Informal deduction
2 Concept of Circumference 3 Analysis
Visualisation
Informal deduction
3 Arc of a Circle 4 Analysis
Visualisation
Informal deduction
4 Area of a Circle and Sector 5 Analysis
of a Circle Visualisation
Table 2 Sample activity in the module
Level Question
In this module, draw a circle with
Level 1 - visualisation (i) a minor sector, OKLM
(ii) a major sector, OKJM
(iii) a chord, MN
By using KIG, construct a circle with a diameter 12 cm, labeled as ST and
a chord, labeled as TU.
Level 2 - analysis What is the difference between diameter and chord of a circle?
ABDUL HALIM ABDULLAH & MOHINI MOHAMED100
Diameter Chord
(i) Construct another diameter of a circle, labeled as XY. Measure its
length (To do this, right click at diameter XY → “Set Label” → Length)
(ii) Construct another diameter of a circle, labeled as UV.
Level 3 - informal (iii) Measure its length (To do this, right click at diameter UV → “Set
deduction Label” → Length)
Save your exercise in desktop. Name the file as Exercise 2. Complete the
table below
Diameter Length (cm) Conclusion
ST 12
XY
UV
A set of test items on form two circles topic based on Van Hiele was developed.
The test consists of six questions where two questions represent the first three levels
in Van Hiele Model which are visualisation, analysis and informal deduction. The
test was then used as a pre and post test in this study. Both module and the test were
given to three mathematics teacher for the purpose of validity.
7.0 METHODOLOGY
One group pre-test-posttest research design was adopted in this study. The
experimental research was designed to determine the effect of independent variable
towards dependent variable which is students’ geometrical thinking levels. Pre and
post-test were given to the students before and after the treatment. Basically, the
research design can be illustrated as follow:
Table 3 Research design
Pre-test Treatment Post-test
C1OC
2
Table 2 (Continued)
101THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
Three form two students named as A, B and C were chosen randomly to participate
in the study. They were given a pre-test which was the test designed based on Van
Hiele before they learnt the form two circles topic using the developed module.
While using the module, the samples’ conversations with the researcher were recorded.
The transcript was then analysed to determine the types of creative and critical
thinking that could be picked up from their conversations. The critical and creative
thinking indicators as suggested by the Curriculum Development Centre were
referred. Then the researcher analysed the students’ work completed in the module,
together with their work saved in hard disk for the purpose of identifying the students’
geometrical thinking levels. Finally the students were given a post-test to determine
the effectiveness of the module to increase students’ geometrical thinking levels.
8.0 DATA ANALYSIS
The performances of the three students in the pre-test are displayed in Table 4.
Table 4 Pre-test results
Sample Level 1 Level 2 Level 3
Question Question Question Question Question Question
123456
A33XXXX
B3XXXXX
C33XXXX
3“ = achieved
X = did not achieve
After the pre-test was given, the students used the module in learning the topic of
Circles using the KIG. Question 1 asked the students to construct a circle with
center O and radius OP with length 6cm. Then on the same circle, the students were
asked to construct a minor sector OPFG and a major sector, OPHG. Finally they
had to make conclusion about the lengths of radius in a circle. After the transcript of
students’ conversations with the researcher were analysed, the following types of
critical and creative thinking were identified. Some parts of dialogue are given below
where R represents the researcher and A, B and C represent students.
R : how to draw minor sector?
A : it is small right?
R : yes
A : it does not matter to draw it anywhere right?
ABDUL HALIM ABDULLAH & MOHINI MOHAMED102
R : yes. okay, now label it as?
A : OPFG
R : where to put H to construct a major sector?
A : it is big, right?
From the above transcript, the researcher indicated that student A can distinguish
a major and minor sector before she can construct them by using KIG (refer Figure
2). Refering to the critical and creative thinking indicators as suggested by the
Curriculum Development Centre, the students had the ability to utilize the “compare
and differentiate” type of critical and creative thinking if they can find, state, list and
arrange the similarity and difference of geometrical object. Then, by using KIG, the
students made conclusion about the lengths of radius in a circle. Referring to the
indicators of KBKK from the Curriculum Development Centre, students have the
ability to utilise “making inference” type of creative and critical thinking if they can
make a conclusion based on a set of information. Figure 3 shows the conclusion
made by student B as taken from her written work in the module. She used KIG to
find several length in one circle before she can make a conclusion about the length
of radius in a circle. Student B concluded that “In a circle, all the radius have the same
length.”
After all discussion sessions with the researcher for Activity and Question 1 were
analysed, the researcher found that the students were able to utilise various types of
Figure 3 Minor and major sector constructed with KIG
Figure 4 Conclusion made by sample B
Radius Length (cm) Conclusion
OP 6
OQ 6
OR 6
In a circle al the radius have
the same length
103THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
critical and creative thinking while solving the questions in the module. The thinking
that are involved are classifying, comparing and differentiating, analysing, inference
making and mental illustrating. Furthermore, the students’ completed work for
Question 1 in the module, together with their work saved in hard disk were analysed
to identify the students’ geometrical thinking levels based on Van Hiele Model. The
Table 5 shows the result for Question1.
Table 5 Van Hiele thinking level for question 1
Sample Van Hiele thinking level
Level 1 Level 2 Level 3
A333
B333
C333
3 = achieved
X = did not achieve
Question 2 asked the students to make a conclusion about the length of diameters
in a circle. The question also asked the samples to find the difference between a
diameter and a chord in a circle. All the activities were done by using KIG. After the
transcript of students’ conversations with the researcher were analysed, the following
types of critical and creative thinking were identified.
C : diameter 12cm?
R : how to draw it…
C : 6,6 (refer to radius)
From the above transcript, student C knew that a diameter is twice of radius.
Cochrane, C. (1996) explained that if the students can link facts, ideas and notions
and generate new data from information collected, they had the ability to utilize
critical thinking skills.
C : chord
R : label as..
C : TU
R : where to put TU?
C : here..
R : from point T to…
C : it does not pass through center right?
ABDUL HALIM ABDULLAH & MOHINI MOHAMED104
Figure 5 The differences between a diameter and a chord in a circle made by sample C
Figure 6 Length of diameters of a circle constructed by sample A
Diameter Chord
Have to cross the centre Don’t have to cross the centre
Radius X2 . Diameter Depends
From the above transcript, student C knows that a chord does not pass through
a center of a circle. Newman, D. R. (1996) stated that the most important points to
determine students had utilized any critical thinking if they can find new problem-
related information, new ideas for discussion, new solutions to problem and welcoming
new ideas.
R : add another diameter
C : label as…
R : UV
C : from here to there
R : yes. As long as it passes through..
C : center of a circle
From the above transcript, student C knew that a diameter passes through a
center of a circle.
So from the three transcripts above, it is apparent that student C had utilized
classifying and analysing types of creative and critical thinking.
Then the samples find the lengths of a few diameters of a circle by using KIG.
Based on the information about the lengths of diameters in a circle, sample A made
the following conclusion. “therefore, the length of a diameter in a circle are all the
same.”
105THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
After all students’ conversations with the researcher for activity and question 2
were analysed, the researcher found that the students were able to utilise various
types of critical and creative thinking such as classify, mentally illustrate, rearrange,
synthesis and integrate while solving the questions in the module. Furthermore, the
students’ completed work for question 2 in the module, together with their work
saved in hard disk were analysed to identify the samples’ geometrical thinking levels.
The Table 6 shows the results for question 2.
In this study, the students had gone through five questions in three days before
the post-test was given to them. The post-test results shown in Table 7.
Table 6 Van Hiele thinking level for question 2
Sample Van Hiele thinking level
Level 1 Level 2 Level 3
A333
B333
C333
3 = achieved
X = did not achieve
Table 7 Post-test results
Sample Level 1 Level 2 Level 3
Question Question Question Question Question Question
123456
A333 33X
B333 333
C333 33X
3 = achieved
X = did not achieve
9.0 DISCUSSION AND CONCLUSION
In general, the study revealed that by engaging in solving problems with IGS has
successfully enabled students to attain a higher level of geometrical understanding.
The study also showed that there is a significant difference in the level of geometry
understanding before and after learning with IGS. In the pre test, all three students
ABDUL HALIM ABDULLAH & MOHINI MOHAMED106
only achieved the lowest level in the Van Hiele model which is visualisation. After
using the module for learning circles with IGS, they attained the third level of the
model which is informal deduction. This result is similar with studies done by Corley
(1990), Yusof (1990), Fitzsimmons (1995), Mac Clendon (1990) and Olkun (2005)
who developed learning modules for geometry with IGS that involves active
participation of students. The activities with IGS modules developed by the researchers
are used as treatment in group experiments. In the end, students participated in the
research achieved the third level in the Van Hiele model. When activities based on
the Van Hiele with IGS is given in this research, the usage of various creative and
critical thinking skills by students have enabled them to move from one lower level
to the next higher level in the Van Hiele model. Activities for all levels are required
because learning geometry must be supported by suitable activities for each level in
the model (Noraini, 2007). For visualisation questions, most students use their thinking
skill to obtain a mental image. For example, to build a circle with centre at O and
radius OP, students must move the cursor to functions in the menu bar to draw the
right diagram. This is similar to the study by Suguna (2005) that determined that
students use their thinking skills to obtain a mental picture in order to direct the
turtle to draw the polygon. For analytical questions, students compare and differentiate,
analyse, arrange sequence and make relations. Students analyse the parts of a circle,
compare and differentiate the parts before attaining the next level in the Van Hiele
model. A sample question in this module where students are required to analyse the
area of sectors for ½ circle, ¼ circle and 1/8 circle. They compare and differentiate
the areas and sector angles to determine the formula for area of circle and area of
sectors. This is similar to the study by Suguna (2005) that determined that student
use their analytical skill to modify information while drawing polygons to a smaller
size in order to understand the concept. The pair of student draw the circle by
analising the turtle movement from a bigger angle to a smaller angle and longer
sides to shorter sides.
For questions under informal deduction, the students have employed the creative
and critical skills such as inventing, sequencing and making inference. This is in line
with the study by Noraini (2007) whereby learning geometry is challenging and
require the involvement of higher level thinking skills. An example of question of
this nature required students to determine the relation between the length of diameter
and circumference. To attain this level, students must make a few circles with varying
diameters. By using the KIG software, they then analyse the diameter and
circumference of the respective circles. Based on the information obtained, they
finally make an inference on the relation between the length of diameter and
circumference.
In summary, various critical and creative thinking skills were used by students
when solving problems with IGS in this module. This will help them achieve the
suggested thinking levels in the Van Hiele module.
107THE USE OF INTERACTIVE GEOMETRY SOFTWARE (IGS)
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