abstraction processes in learning geomet
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
TSG
ABSTRACTION PROCESSES IN LEARNING GEOMETRY USING
GSP
Jozua Sabandar
Indonesia University of
Education
f4121da_n@yeahoo.com jsabandar@yahoo.com
Farida Nurhasanah
Sebelas Maret University
Yaya S. Kusumah
Indonesia University of
Education
yayaskusumah@yahoo.com
Abstraction is a fundamental process in learning mathematics. So far, there are still few
researchers and professionals discussed about how this process occurs in mathematics
classroom, especially in geometry. In Indonesia, 50% of the content of Junior High
School Mathematics Curriculum consists of geometry. In line with the development of
information technology, dynamic geometry software emerges; one of them is
Geometers’ Sketchpad (GSP). This software was created to assist students in learning
geometry by visualizing and manipulating geometry objects. Unfortunately, this
software is still rarely used in Indonesia. The concern of this research is to investigate
how the abstraction process in learning geometry using GSP takes place. The objective
of this research is to figure out students’ abstraction process in learning geometry
concepts using GSP and students’ abstraction process during solving geometry
problems in a classroom. This study is a qualitative research and it was conducted in a
public junior high school in Indonesia. The subjects of this research are seventh grade
students. The data were collected through observation, test, and interviews and the data
were analysed using analytical induction and constant comparative techniques. The
results of this research are that both types of the students’ abstraction process during
construction of geometry concepts and solving geometry problems falls into the
category of empirical abstraction. In addition, students’ ability in the aspect of
representation of mathematical ideas into symbols or mathematics language as a part
of students’ abstraction is the most dominant aspect that appears when students solved
geometry problems.
Keywords: empirical abstraction, theoretical abstraction, geometers’ sketchpad, van
Hiele’s Model.
INTRODUCTION
Mathematics should be learned by students from primary school until university level in
almost every country in the world. Why should they learn mathematics? The reason is
that mathematics is useful and can help them to live in this complicated world when
they must solve problems in their daily life. Mathematics is related to the concept of
time, distance, trading, and many more. Unfortunately, learning mathematics is a
complicated process, because the objects of mathematics are abstract. Therefore the
concepts of mathematics could not just be transferred into students’ mind just like a
piece of information. It should be learned through a kind of processes.
The process of learning abstract things can be considered as an abstraction process.
In learning mathematics, abstractionis a process that is related to the emerging of
mathematics concepts.This means that this process is very important in learning
mathematics. Ferrari(2003) also stated that abstraction is a fundamental process in
learning mathematics. So, abstraction process is very significant in creating effective
1
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
mathematics education classroom in order to achieve learning objectives of
mathematics education (Goodson & Espy, 2005)
School mathematicsconsists of some strandsamong others: numbers, algebra,
analysis, geometry. All these components have their own portion in each level of
school. In Indonesia, since 2006, mathematics education curriculum contains a large
portion of geometry in secondary school level.
As it is known that geometry consist of many abstract things in which we need to
learnusing deductive reasoning. There are many students around the world who have
trouble in learning geometry (Laborde etal, 2006) and this also happens in Indonesia
(Nurhasanah, 2004).
Mitchelmore and Tall (2007) mentioned that abstraction has significant role in learning
geometry related to the formation of triangle and quadrilateral concepts, when students
learn the shape of a triangle or a quadrilateral. They identifythe shapes by observing
the similarities, doing classification based on the characteristics of the objects, finding
the embodied properties of the concepts, and constructing a concept of each shape.
Another perspective of learning geometry has been studied by Dina van HieleGeldof
and Pierre Marrie van Hiele. They proposed teaching model for geometry, based on
their theory of geometry teaching model which consists of five phases:iinquiry, directed
orientation, explication, free orientation, and integration. This model was designed in
order to enhance the ability to think geometry. At the time when this theory was
emerged, the tools for learning geometry were still rare. But now, there are many
interactive tools for learning geometry, for istance, Geometers’ Sketchpad, Cabri, Logo,
etc., which can help students in learning geometry. This kind of interactive tools is
usually called as Geometry Dynamic Software (GDS).
Geometers’ Sketchpad (GSP) is one of GDS that is widely used for helping students in
learning geometry concepts through a series of construction. This software is
intentionally designed to help students in learning Euclidean Geometry. GSP gives
opportunity to students and also teachers to intuitively or inductively explore the
possible relationships between geometry figures in two dimensions and their
characteristics intuitively or inductively through a series of geometry dynamic
construction.
Some of the studies related to the use of GSPwere conducted by Choi-Koh (2000) and
Olkunetal(2002). The result of the studiesindicate that GSP can createpotential
situation in the classroom in order to build and develop thinking process in learning
geometry that can lead to students’ understanding into abstract concepts in geometry.
As stated before that learning abstract conceptswill always gothrough abstraction
process. Therefore, abstraction processalso significant role in the process of learning
geometry. Related to the emerging of GDS, it is also interesting to investigate how the
abstraction process occurs in the classroom where GSPis used as a tool in learning
geometry and solving geometry problems.
Abstraction Process and van Hiele’s Model of Teaching Geometry
There are two main theories in the abstraction processes: an empirical abstraction and
theoretical abstraction (Mitchermoreand White, 2007). The concept of empirical
abstraction was derived from Skemp’s conception (1986). Based on his conception,
abstraction starts from similarity recognition then it is followed by embodiment of the
similarity in a new mental object. The result of this processis called a concept. Since
the process started from a series activity on experiences, this process is called as an
empirical abstraction.
2
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
On the other hand, the concept of theoretical abstraction was originated from Soviet
Psychologists, Vygotsky and Davydov. In essence, theoretical abstraction consists of
the creation of concepts to fit into some theory.
Based on the understanding of both theoriesof empirical abstraction and theoretical
abstraction, an indication of abstraction in learning process can be identified from the
following aspects:
1.
2.
3.
4.
5.
Identify the characteristics of the objects through direct experience
Identify the characteristics of manipulated or imagined objects
Making generalization
Representing mathematical objects into symbols or mathematical language
Creating relationships between processes or concepts to form a new
understanding
6. Applying the concepts into appropriate context
7. Manipulation abstract mathematical concepts
8. Idealization or removing material properties from an object
Related to van Hiele’s model of teaching geometry, the abstraction potential activities
that can emerge in some levels of teaching can be seen on Table 1 below:
Table 1:The Relationships between van Hiele’s Model of Teaching Geometry and the
Aspects of Abstraction Process
Steps of Teaching
Geometry
1. Inquiry/ Information
2. Directed Orientation
3. Explication
Abstraction Aspects that could be
Involved in every Phase
Identify the characteristics of the objects
through direct experience
Empirical
abstraction
Identify the characteristics of manipulated or
imagined objects
Empirical
Abstraction
Representing mathematical objects into
symbols or mathematical language
Creating relationships between processes or
concepts to form a new understanding
Type of
Abstraction
Theoretical
Abstraction
Idealization or removing material properties
from an object
4. Free Orientation
Applying the concepts into appropriate
context
Theoretical
Abstraction
Making generalization
5. Integration
Manipulation of abstract mathematical
concepts
Creating relationships between processes or
concepts to form a new understanding
Theoretical
Abstraction
Based on Serow (2008), geometry instruction using van Hiele’smodel of teaching and
assisted by GDS is an effective instructional design. In line with Serow, Olkunet al
(2002) said that the application of GDS can create a potential environment for students
in order to do many kinds of investigations to gain construction experiencesregarding
geometry shapes. Geometers’ Sketchpad as one of the GDS can be a good choice to
3
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
assist van Hiele model of teaching geometry, in particular for seconddimensional
geometry. GSP not only dynamic and has many menus to construct many geometry
objects using geometry concepts but it is also a user friendly tool.
Previous studies show that GSP can be an effective toolfor creating a potential
situation in order to establish a good interaction between a teacher and students in
such a way that the teacher give students times to investigate the conjectures (ChoiKoh, 2000).
RESEARCH METHOD
This paper reports on the abstraction process of two students in learning geometry
using GSP in the topic of triangle and solving geometry problems related to the topic of
triangle. Based on the aims of this research to identify the abstraction process of
student in learning geometry and solving geometry problems, this research can be
considered as qualitative research (Creswell, 2008).
Related to the aims of the research, the school involved in this research should have
good computer laboratory and the students must be familiar with GSP. The research
was conducted in one of the international standar schools (SBI) in Cimahidistrict of
WestJavaProvince. The research involves 7th grade students in SBI consisting of 26
students, but only 6 students who are assigned as the subject of this research. These
students were selected based on their achievement, performance, their communication
skills, and the result of the test.
The data were collected using observation, test, and interview. The observations were
conducted in the class during learning process of triangle, assisted by GSP. During this
process all students’ activitieswere observed and videotaped. In this class, the teacher
used van Hiele model of teaching using GSP. The observation was held by using a
camera, particularly when the students solved problemson triangles. The test was
constructed in order to stimulate abstraction during problem solving process. Then, the
data from observation and test were analysed to determine the subject of this research
and also to determine which students should be interviewed.
The data from observation, test, and interviews were analysed using analytic induction
techniques and constant comparison (Alwasilah, 2003). The data were classified into
some categories, and then verification measure between the categories is taken.
Based on the defined categories, a posteriori act arose from the data gathered, while
maintaining the focus of the study and the theoretical framework.
RESULT
The data from observation, test, and interview were analyzed based on the aspects of
abstraction which emerge during the learning process and solving problems about
triangles. Students learned the conceptsof triangles such as: definition of triangle, types
of triangle, interior angles, exterior angles, relation between interior and exterior angles,
area and perimeter of triangle, altitude, median, bisector, and perpendicular bisector.
Abstraction Process in Learning Geometry Using van Hiele’s Model with GSP
The abstraction process was observed during instructional process, then the emerged
of abstraction aspects were recorded and analyzed based on each aspect. The result
from observation and interview about the aspects of abstractions that occur during
learning process using van Hiele’s model with GSP indicates that only six from eight
aspects that emerge. Two aspects of abstraction i.e., “aspect of applying the concepts
into appropriate context” and “manipulating abstract mathematical concepts”did not
appear during learning process. The aspect of applying the concepts into appropriate
context not appear, it is because this aspect could not be observed in classroom.
Based on the lesson activity, students continue their learning activities at home as a
4
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
home work. For another aspect which did not appear was influenced by levels of
students’ thinking.
Based on the observation during introduction triangle concept, abstraction aspects at
most, could be detected in the phase of directed orientation. The aspects of abstraction
process that are prominent in introducing triangle concept during learning process are:
Identifying the characteristics of the objects through direct experience; identifying the
characteristics of manipulated or imagined objects; and representing mathematical
objects into symbol or mathematical language. These three aspects occurred at in
same phase of the van Hiele model of teaching, in the directed orientation phase. The
use of GSP which is blended with lesson activity triggered such kind of situation.
If we refer to van Hiele’s model of teaching theory, the aspect of representing
mathematical objects into symbol or mathematical language should appear in the
phase of explication through teacher’s explanation. But using GSP this aspect could
also appear before the explication phase because students unintentionally learn the
geometry symbols as displayed by the GSP. This is become one of the significant
result from this research.
Generally, the abstraction process that occurred in learning geometry using van Hiele’s
model with GSP can be viewed in Figure 1.
Phase of van Hiele’s
Model of Teaching
Abstraction Aspects
E
Creating relationship between
processes or concepts to form a
new understanding
CONCEPT
Integration
M
P
I
R
Identify the characteristics of the
objects through direct experience
Identify the characteristics of
manipulated or imagined objects
Making generalization
Representing mathematical objects
into
symbol
or
mathematical
language
Creating
relationship
between
processes or concepts to form a
new understanding
I
C
ACTIVITIES
WITH GSP
Explication
Free orientation
Directed orientation
A
L
A
B
S
T
Identify the characteristics of
the objects through direct
experience
Idealization or removing
material properties from an
object
R
CONTEXTS
Inquiry Through
some experiences
with GSP
A
C
T
I
O
N
Students’ Prior Knowledge
Figure 1. Flowchart of Students’ Abstraction Process in Learning Triangle Using GSP
5
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
The abstraction process starts from a context but also depends on the students’ prior
knowledge.Then through the series of activities that were designed in lesson activity
using GSP students experience many kind of abstraction aspect in order to
comprehend a new concept. Geometers’ Sketchpad has significant role in three
phases of van Hiele model. The activities were designed for directed orientation, free
orientation, and explication included instructions activities using GSP. Related to the
inquiry phase, GSP also provide many features for students to identify the
characteristics of objects through direct experience such as measurement activity and
object manipulation. This process is still aligned to the theory of van Hiele’s model of
teaching.
Related to the theory of empirical and theoreticalabstraction that has been mentioned
before, the abstraction process that occurscan be considered as empirical abstraction.
Abstraction Process in Solving Geometry Problems
See the
Figure
Start
Read the
Problem
Collecting Relevance
Information
Observed the
Figure
Analyzed
the Figure
Representing the
Information into
Figure
Predicting the Size
of Angles and the
Length of Sides
Does the
Information
enough?
Yes
No
Choose a
Triangle
Doing Direct
Measurement
Comparing The
Triangle with Right
Triangle
Determine Type of
the Triangle based
on its Shape
Identify the
Characteristics
Yes
Are there any
special
characteristics on
its sides?
No
Making Connection
between the
concepts that
enable
Find the
Characteristics of
Angles
Determine the
Type of the
Triangle
Did I Already found all
six types of Triangles?
Classify the
Triangle
No
Yes
Finish
Figure 2. Students’ Abstraction in Solving Problem Number 1
6
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
Abstraction processes in solving geometry problems were identified using test and
interviews. The test was designed based on the aspects of abstraction. The test
consists of five geometry problems related to the concept of triangle and the content of
all testitems need some abstraction aspects to be held. The results of test were
clarified through interview.
If we refer to Figure 2, aspects of abstraction that are clearly identified are aspects of
identifying characteristic object through direct experience. It can be seen when
students tried to use direct measurement from the picture given; aspect of making
connection between object or concepts were interesting to be analyzed, students tried
to use concept of the size of right angle in a triangle as a reference to determine the
size of angles in a triangle without doing measurement. Another aspect was also
identified in the process of representing mathematical objects into symbols or
mathematical language. This aspect emerged when students converted the information
from the given problem into symbols that attached on the figure and also when
students wrote their answer, they used symbols like ∆ for triangle, // for parallel line,
and for an angle.
DISCUSSION
The new concepts associated with triangle formed by abstraction process can be
categorized into conceptual-embodied and proceptual-symbolic (Gray dan Tall, 2001).
conceptual-embodied was formed when students built a triangle concept based on
perception and reflection to the similarities characteristics of geometry shapes which
were constructed using GSP. Furthermore, the proceptual-symbolic concepts was
formed when students built a concept through awareness to the similarities
characteristics in action and making concept symbolization into something that can be
conceived. This process is called as empirical abstraction process.
The GSP has significant role in the process of forming concept of manifold proceptual
symbolic. GSP can be an effective tool in creating potential situation so that the
students can be more effectiveand efficient in their construction process, which
includes the process of identification of the objects’ characteristics or relationship
between concepts in learning geometry for junior high school students.
Geometry is also potential in helping students to solve their problems in accordance to
the needs and the characteristic of each student. It can be seen from the variation of
students’ answers that learning geometry using GSP is also strengthened by studied of
Serow (2008).
From the eight aspects, the aspect of manipulating abstracts objects did not emerge.
This situation is certainly related to the level of development of students' ability in
learning geometry. This aspect is equivalent to the level of rigor based on van Hiele’s
level theory of learning geometry. This aspectdid not appear, it is still considered
normal for student at this age level. Students at this age are mostly still at the level of
relational as described by Matsuo (1993) and Currie Pegg (1998) in Guiterrez and
Boero (2006).
CONCLUSION
The conclusion of this research are; firstly, that the students’ abstraction process in
learning geometry using van Hiele’s model of teaching with GSP belongs to empirical
abstraction which occurred during the process of concept formation; secondly,the
students’ abstraction process in solving geometry problems belongs to empirical
abstraction that emphasis on the aspect of representing mathematical objects into
symbol or mathematical language
7
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
Based on the conclusion of this research there are some suggestions for further
studies especially, further study about abstraction in geometry using other models of
teaching geometry, and using other GDS. This research cannot explore the relationship
between the abstraction process and the students’ achievement. This study, however,
can give much information to educators and practitioners about what type of
abstraction process that better occur in specific condition related to the situation.
REFERENCES
Alwasilah, C. (2003). Pokoknya Kualitatif. Bandung: PT. Kiblat Buku Utama.
Choi-koh, S.(2000). The Activities Based on van Hiele Model Using Computer as a
Tool.Journal of the Korea Society of Mathematics education series D: research in
Mathematics Educationvol. 4, No. 2. November 2000, 63-67. Seoul.
Creswell, J.W. (2008). Educational Research: Planning, Conducting and Evaluating
Quantitative and Qualitative Research. Upper Saddle Creek, NJ: Pearson
Education.
Ferrari, P. (2003). Abstraction in mathematics. Philosophical Transactions of the Royal
Society of London. Series B: Biological Sciences, 358(1435), pp. 1225–1230.
Goodson, T & Espy. (2005). Why Reflective Abstraction Remains RelevantIn
Mathematics Education Research. Proceeding of 27th Conference of the
International Group for the Psychology of Mathematics Education (Vol. 2, pp. 1 - 7).
Bergen, Norway: PME.
Gray, E. & Tall, D. (2001.) Relationships between embodied objects and symbolic
procepts:an explanatory theory of success and failure in mathematics. In HeuvelPanhuizen, M.(Ed.) Proceedings of the 25th conference of the international group
for the psychology of mathematics education. Utrecht,Netherland: Freudenthal
Institute.
Mitchelmore, M & White, P. (2007). Abstraction in Mathematics Learning. Mathematics
Education Research Journal. (Vol 19(2). pp 1-9). Retrieved from:
http://www.merga.net.au/documents/MERJ_19_2_editorial.pdf.
Nurhasanah, F. (2004). Proses Berpikir Siswa Sekolah Menengah Tingkat Pertama
Dalam Belajar Geometri Pada Pokok Bahasan Jajargenjang, Belahketupat, LayangLayang Dan Trapesium[Unpublied A Thesis]. Sebelas Maret University: Surakarta.
Olkun, S et al. (2002). Geometric explorations with Dynamic Geometry Applications
based on van Hiele Levels. International Journal for Mathematics teaching and
Learning.(pp 1-12).Retrieved from: http://www.ex.ac.uk/cimt/ijmtl/ijmenu.htm.
Serow, P. (2008). Investigating a Phase Approach to using Technology as a Teaching
Tool.Retrieved from: http://www.merga.net.au/dokuments/ RP532008.pdf.
Copyright © 2013 grant a
non-exclusive license to the organisers of the EARCOME6, Center for Research in
Mathematics Education, to publish this document in the Conference Proceedings. Any
other usage is prohibited without the consent or permission of the author(s).
8
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
9
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
TSG
ABSTRACTION PROCESSES IN LEARNING GEOMETRY USING
GSP
Jozua Sabandar
Indonesia University of
Education
f4121da_n@yeahoo.com jsabandar@yahoo.com
Farida Nurhasanah
Sebelas Maret University
Yaya S. Kusumah
Indonesia University of
Education
yayaskusumah@yahoo.com
Abstraction is a fundamental process in learning mathematics. So far, there are still few
researchers and professionals discussed about how this process occurs in mathematics
classroom, especially in geometry. In Indonesia, 50% of the content of Junior High
School Mathematics Curriculum consists of geometry. In line with the development of
information technology, dynamic geometry software emerges; one of them is
Geometers’ Sketchpad (GSP). This software was created to assist students in learning
geometry by visualizing and manipulating geometry objects. Unfortunately, this
software is still rarely used in Indonesia. The concern of this research is to investigate
how the abstraction process in learning geometry using GSP takes place. The objective
of this research is to figure out students’ abstraction process in learning geometry
concepts using GSP and students’ abstraction process during solving geometry
problems in a classroom. This study is a qualitative research and it was conducted in a
public junior high school in Indonesia. The subjects of this research are seventh grade
students. The data were collected through observation, test, and interviews and the data
were analysed using analytical induction and constant comparative techniques. The
results of this research are that both types of the students’ abstraction process during
construction of geometry concepts and solving geometry problems falls into the
category of empirical abstraction. In addition, students’ ability in the aspect of
representation of mathematical ideas into symbols or mathematics language as a part
of students’ abstraction is the most dominant aspect that appears when students solved
geometry problems.
Keywords: empirical abstraction, theoretical abstraction, geometers’ sketchpad, van
Hiele’s Model.
INTRODUCTION
Mathematics should be learned by students from primary school until university level in
almost every country in the world. Why should they learn mathematics? The reason is
that mathematics is useful and can help them to live in this complicated world when
they must solve problems in their daily life. Mathematics is related to the concept of
time, distance, trading, and many more. Unfortunately, learning mathematics is a
complicated process, because the objects of mathematics are abstract. Therefore the
concepts of mathematics could not just be transferred into students’ mind just like a
piece of information. It should be learned through a kind of processes.
The process of learning abstract things can be considered as an abstraction process.
In learning mathematics, abstractionis a process that is related to the emerging of
mathematics concepts.This means that this process is very important in learning
mathematics. Ferrari(2003) also stated that abstraction is a fundamental process in
learning mathematics. So, abstraction process is very significant in creating effective
1
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
mathematics education classroom in order to achieve learning objectives of
mathematics education (Goodson & Espy, 2005)
School mathematicsconsists of some strandsamong others: numbers, algebra,
analysis, geometry. All these components have their own portion in each level of
school. In Indonesia, since 2006, mathematics education curriculum contains a large
portion of geometry in secondary school level.
As it is known that geometry consist of many abstract things in which we need to
learnusing deductive reasoning. There are many students around the world who have
trouble in learning geometry (Laborde etal, 2006) and this also happens in Indonesia
(Nurhasanah, 2004).
Mitchelmore and Tall (2007) mentioned that abstraction has significant role in learning
geometry related to the formation of triangle and quadrilateral concepts, when students
learn the shape of a triangle or a quadrilateral. They identifythe shapes by observing
the similarities, doing classification based on the characteristics of the objects, finding
the embodied properties of the concepts, and constructing a concept of each shape.
Another perspective of learning geometry has been studied by Dina van HieleGeldof
and Pierre Marrie van Hiele. They proposed teaching model for geometry, based on
their theory of geometry teaching model which consists of five phases:iinquiry, directed
orientation, explication, free orientation, and integration. This model was designed in
order to enhance the ability to think geometry. At the time when this theory was
emerged, the tools for learning geometry were still rare. But now, there are many
interactive tools for learning geometry, for istance, Geometers’ Sketchpad, Cabri, Logo,
etc., which can help students in learning geometry. This kind of interactive tools is
usually called as Geometry Dynamic Software (GDS).
Geometers’ Sketchpad (GSP) is one of GDS that is widely used for helping students in
learning geometry concepts through a series of construction. This software is
intentionally designed to help students in learning Euclidean Geometry. GSP gives
opportunity to students and also teachers to intuitively or inductively explore the
possible relationships between geometry figures in two dimensions and their
characteristics intuitively or inductively through a series of geometry dynamic
construction.
Some of the studies related to the use of GSPwere conducted by Choi-Koh (2000) and
Olkunetal(2002). The result of the studiesindicate that GSP can createpotential
situation in the classroom in order to build and develop thinking process in learning
geometry that can lead to students’ understanding into abstract concepts in geometry.
As stated before that learning abstract conceptswill always gothrough abstraction
process. Therefore, abstraction processalso significant role in the process of learning
geometry. Related to the emerging of GDS, it is also interesting to investigate how the
abstraction process occurs in the classroom where GSPis used as a tool in learning
geometry and solving geometry problems.
Abstraction Process and van Hiele’s Model of Teaching Geometry
There are two main theories in the abstraction processes: an empirical abstraction and
theoretical abstraction (Mitchermoreand White, 2007). The concept of empirical
abstraction was derived from Skemp’s conception (1986). Based on his conception,
abstraction starts from similarity recognition then it is followed by embodiment of the
similarity in a new mental object. The result of this processis called a concept. Since
the process started from a series activity on experiences, this process is called as an
empirical abstraction.
2
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
On the other hand, the concept of theoretical abstraction was originated from Soviet
Psychologists, Vygotsky and Davydov. In essence, theoretical abstraction consists of
the creation of concepts to fit into some theory.
Based on the understanding of both theoriesof empirical abstraction and theoretical
abstraction, an indication of abstraction in learning process can be identified from the
following aspects:
1.
2.
3.
4.
5.
Identify the characteristics of the objects through direct experience
Identify the characteristics of manipulated or imagined objects
Making generalization
Representing mathematical objects into symbols or mathematical language
Creating relationships between processes or concepts to form a new
understanding
6. Applying the concepts into appropriate context
7. Manipulation abstract mathematical concepts
8. Idealization or removing material properties from an object
Related to van Hiele’s model of teaching geometry, the abstraction potential activities
that can emerge in some levels of teaching can be seen on Table 1 below:
Table 1:The Relationships between van Hiele’s Model of Teaching Geometry and the
Aspects of Abstraction Process
Steps of Teaching
Geometry
1. Inquiry/ Information
2. Directed Orientation
3. Explication
Abstraction Aspects that could be
Involved in every Phase
Identify the characteristics of the objects
through direct experience
Empirical
abstraction
Identify the characteristics of manipulated or
imagined objects
Empirical
Abstraction
Representing mathematical objects into
symbols or mathematical language
Creating relationships between processes or
concepts to form a new understanding
Type of
Abstraction
Theoretical
Abstraction
Idealization or removing material properties
from an object
4. Free Orientation
Applying the concepts into appropriate
context
Theoretical
Abstraction
Making generalization
5. Integration
Manipulation of abstract mathematical
concepts
Creating relationships between processes or
concepts to form a new understanding
Theoretical
Abstraction
Based on Serow (2008), geometry instruction using van Hiele’smodel of teaching and
assisted by GDS is an effective instructional design. In line with Serow, Olkunet al
(2002) said that the application of GDS can create a potential environment for students
in order to do many kinds of investigations to gain construction experiencesregarding
geometry shapes. Geometers’ Sketchpad as one of the GDS can be a good choice to
3
th
6 East Asia Regional Conference on Mathematics Education (EARCOME6)
17-22 March 2013, Phuket, Thailand
assist van Hiele model of teaching geometry, in particular for seconddimensional
geometry. GSP not only dynamic and has many menus to construct many geometry
objects using geometry concepts but it is also a user friendly tool.
Previous studies show that GSP can be an effective toolfor creating a potential
situation in order to establish a good interaction between a teacher and students in
such a way that the teacher give students times to investigate the conjectures (ChoiKoh, 2000).
RESEARCH METHOD
This paper reports on the abstraction process of two students in learning geometry
using GSP in the topic of triangle and solving geometry problems related to the topic of
triangle. Based on the aims of this research to identify the abstraction process of
student in learning geometry and solving geometry problems, this research can be
considered as qualitative research (Creswell, 2008).
Related to the aims of the research, the school involved in this research should have
good computer laboratory and the students must be familiar with GSP. The research
was conducted in one of the international standar schools (SBI) in Cimahidistrict of
WestJavaProvince. The research involves 7th grade students in SBI consisting of 26
students, but only 6 students who are assigned as the subject of this research. These
students were selected based on their achievement, performance, their communication
skills, and the result of the test.
The data were collected using observation, test, and interview. The observations were
conducted in the class during learning process of triangle, assisted by GSP. During this
process all students’ activitieswere observed and videotaped. In this class, the teacher
used van Hiele model of teaching using GSP. The observation was held by using a
camera, particularly when the students solved problemson triangles. The test was
constructed in order to stimulate abstraction during problem solving process. Then, the
data from observation and test were analysed to determine the subject of this research
and also to determine which students should be interviewed.
The data from observation, test, and interviews were analysed using analytic induction
techniques and constant comparison (Alwasilah, 2003). The data were classified into
some categories, and then verification measure between the categories is taken.
Based on the defined categories, a posteriori act arose from the data gathered, while
maintaining the focus of the study and the theoretical framework.
RESULT
The data from observation, test, and interview were analyzed based on the aspects of
abstraction which emerge during the learning process and solving problems about
triangles. Students learned the conceptsof triangles such as: definition of triangle, types
of triangle, interior angles, exterior angles, relation between interior and exterior angles,
area and perimeter of triangle, altitude, median, bisector, and perpendicular bisector.
Abstraction Process in Learning Geometry Using van Hiele’s Model with GSP
The abstraction process was observed during instructional process, then the emerged
of abstraction aspects were recorded and analyzed based on each aspect. The result
from observation and interview about the aspects of abstractions that occur during
learning process using van Hiele’s model with GSP indicates that only six from eight
aspects that emerge. Two aspects of abstraction i.e., “aspect of applying the concepts
into appropriate context” and “manipulating abstract mathematical concepts”did not
appear during learning process. The aspect of applying the concepts into appropriate
context not appear, it is because this aspect could not be observed in classroom.
Based on the lesson activity, students continue their learning activities at home as a
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home work. For another aspect which did not appear was influenced by levels of
students’ thinking.
Based on the observation during introduction triangle concept, abstraction aspects at
most, could be detected in the phase of directed orientation. The aspects of abstraction
process that are prominent in introducing triangle concept during learning process are:
Identifying the characteristics of the objects through direct experience; identifying the
characteristics of manipulated or imagined objects; and representing mathematical
objects into symbol or mathematical language. These three aspects occurred at in
same phase of the van Hiele model of teaching, in the directed orientation phase. The
use of GSP which is blended with lesson activity triggered such kind of situation.
If we refer to van Hiele’s model of teaching theory, the aspect of representing
mathematical objects into symbol or mathematical language should appear in the
phase of explication through teacher’s explanation. But using GSP this aspect could
also appear before the explication phase because students unintentionally learn the
geometry symbols as displayed by the GSP. This is become one of the significant
result from this research.
Generally, the abstraction process that occurred in learning geometry using van Hiele’s
model with GSP can be viewed in Figure 1.
Phase of van Hiele’s
Model of Teaching
Abstraction Aspects
E
Creating relationship between
processes or concepts to form a
new understanding
CONCEPT
Integration
M
P
I
R
Identify the characteristics of the
objects through direct experience
Identify the characteristics of
manipulated or imagined objects
Making generalization
Representing mathematical objects
into
symbol
or
mathematical
language
Creating
relationship
between
processes or concepts to form a
new understanding
I
C
ACTIVITIES
WITH GSP
Explication
Free orientation
Directed orientation
A
L
A
B
S
T
Identify the characteristics of
the objects through direct
experience
Idealization or removing
material properties from an
object
R
CONTEXTS
Inquiry Through
some experiences
with GSP
A
C
T
I
O
N
Students’ Prior Knowledge
Figure 1. Flowchart of Students’ Abstraction Process in Learning Triangle Using GSP
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The abstraction process starts from a context but also depends on the students’ prior
knowledge.Then through the series of activities that were designed in lesson activity
using GSP students experience many kind of abstraction aspect in order to
comprehend a new concept. Geometers’ Sketchpad has significant role in three
phases of van Hiele model. The activities were designed for directed orientation, free
orientation, and explication included instructions activities using GSP. Related to the
inquiry phase, GSP also provide many features for students to identify the
characteristics of objects through direct experience such as measurement activity and
object manipulation. This process is still aligned to the theory of van Hiele’s model of
teaching.
Related to the theory of empirical and theoreticalabstraction that has been mentioned
before, the abstraction process that occurscan be considered as empirical abstraction.
Abstraction Process in Solving Geometry Problems
See the
Figure
Start
Read the
Problem
Collecting Relevance
Information
Observed the
Figure
Analyzed
the Figure
Representing the
Information into
Figure
Predicting the Size
of Angles and the
Length of Sides
Does the
Information
enough?
Yes
No
Choose a
Triangle
Doing Direct
Measurement
Comparing The
Triangle with Right
Triangle
Determine Type of
the Triangle based
on its Shape
Identify the
Characteristics
Yes
Are there any
special
characteristics on
its sides?
No
Making Connection
between the
concepts that
enable
Find the
Characteristics of
Angles
Determine the
Type of the
Triangle
Did I Already found all
six types of Triangles?
Classify the
Triangle
No
Yes
Finish
Figure 2. Students’ Abstraction in Solving Problem Number 1
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Abstraction processes in solving geometry problems were identified using test and
interviews. The test was designed based on the aspects of abstraction. The test
consists of five geometry problems related to the concept of triangle and the content of
all testitems need some abstraction aspects to be held. The results of test were
clarified through interview.
If we refer to Figure 2, aspects of abstraction that are clearly identified are aspects of
identifying characteristic object through direct experience. It can be seen when
students tried to use direct measurement from the picture given; aspect of making
connection between object or concepts were interesting to be analyzed, students tried
to use concept of the size of right angle in a triangle as a reference to determine the
size of angles in a triangle without doing measurement. Another aspect was also
identified in the process of representing mathematical objects into symbols or
mathematical language. This aspect emerged when students converted the information
from the given problem into symbols that attached on the figure and also when
students wrote their answer, they used symbols like ∆ for triangle, // for parallel line,
and for an angle.
DISCUSSION
The new concepts associated with triangle formed by abstraction process can be
categorized into conceptual-embodied and proceptual-symbolic (Gray dan Tall, 2001).
conceptual-embodied was formed when students built a triangle concept based on
perception and reflection to the similarities characteristics of geometry shapes which
were constructed using GSP. Furthermore, the proceptual-symbolic concepts was
formed when students built a concept through awareness to the similarities
characteristics in action and making concept symbolization into something that can be
conceived. This process is called as empirical abstraction process.
The GSP has significant role in the process of forming concept of manifold proceptual
symbolic. GSP can be an effective tool in creating potential situation so that the
students can be more effectiveand efficient in their construction process, which
includes the process of identification of the objects’ characteristics or relationship
between concepts in learning geometry for junior high school students.
Geometry is also potential in helping students to solve their problems in accordance to
the needs and the characteristic of each student. It can be seen from the variation of
students’ answers that learning geometry using GSP is also strengthened by studied of
Serow (2008).
From the eight aspects, the aspect of manipulating abstracts objects did not emerge.
This situation is certainly related to the level of development of students' ability in
learning geometry. This aspect is equivalent to the level of rigor based on van Hiele’s
level theory of learning geometry. This aspectdid not appear, it is still considered
normal for student at this age level. Students at this age are mostly still at the level of
relational as described by Matsuo (1993) and Currie Pegg (1998) in Guiterrez and
Boero (2006).
CONCLUSION
The conclusion of this research are; firstly, that the students’ abstraction process in
learning geometry using van Hiele’s model of teaching with GSP belongs to empirical
abstraction which occurred during the process of concept formation; secondly,the
students’ abstraction process in solving geometry problems belongs to empirical
abstraction that emphasis on the aspect of representing mathematical objects into
symbol or mathematical language
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Based on the conclusion of this research there are some suggestions for further
studies especially, further study about abstraction in geometry using other models of
teaching geometry, and using other GDS. This research cannot explore the relationship
between the abstraction process and the students’ achievement. This study, however,
can give much information to educators and practitioners about what type of
abstraction process that better occur in specific condition related to the situation.
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Creswell, J.W. (2008). Educational Research: Planning, Conducting and Evaluating
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Ferrari, P. (2003). Abstraction in mathematics. Philosophical Transactions of the Royal
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