Abstraction Processes in Learning Geomet


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  ABSTRACTION PROCESSES IN LEARNING GEOMETRY USING GSP Farida Nurhasanah Jozua Sabandar Yaya S. Kusumah Sebelas Maret University Indonesia University of Indonesia University of

  Education Education f4121da_n@yeahoo.com jsabandar@yahoo.com 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.


  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 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.

  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.

  Steps of Teaching Geometry Abstraction Aspects that could be Involved in every Phase Type of Abstraction

  Theoretical Abstraction

  5. Integration Manipulation of abstract mathematical concepts Creating relationships between processes or concepts to form a new understanding

  Theoretical Abstraction

  4. Free Orientation Idealization or removing material properties from an object Applying the concepts into appropriate context Making generalization

  Theoretical Abstraction

  3. Explication Representing mathematical objects into symbols or mathematical language Creating relationships between processes or concepts to form a new understanding

  Empirical Abstraction

  2. Directed Orientation Identify the characteristics of manipulated or imagined objects

  Empirical abstraction

  1. Inquiry/ Information Identify the characteristics of the objects through direct experience

  Aspects of Abstraction Process

  Based on the understanding of both theoriesof empirical abstraction and theoretical

  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

  7. Manipulation abstract mathematical concepts

  6. Applying the concepts into appropriate context

  5. Creating relationships between processes or concepts to form a new understanding

  4. Representing mathematical objects into symbols or mathematical language

  3. Making generalization

  2. Identify the characteristics of manipulated or imagined objects

  1. Identify the characteristics of the objects through direct experience

  following aspects:


abstraction, an indication of abstraction in learning process can be identified from the

  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 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 (Choi- Koh, 2000).


  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 th WestJavaProvince. The research involves 7 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.


  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 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 Abstraction Aspects Model of Teaching E

   Creating relationship between  Integration M processes or concepts to form a

  CONCEPT new understanding


  I R

  I  Identify the characteristics of the objects through direct experience

  C  Identify the characteristics of manipulated or imagined objects

   Explication A  Making generalization  Free orientation


   Representing mathematical objects  Directed orientation


  into symbol or mathematical language A

   Creating relationship between processes or concepts to form a B new understanding

  S T R  Identify the characteristics of

   Inquiry Through the objects through direct A some experiences experience


  C  Idealization or removing material properties from an

  T object

  I O


Students’ Prior Knowledge

  Figure 1. Flowchart of Students’ Abstraction Process in Learning Triangle Using GSP 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 Read the Start

  Collecting Relevance Figure Problem Information Observed the

  Figure Does the Analyzed Representing the

  Yes Information the Figure Information into enough?

  Figure Predicting the Size No Choose a of Angles and the Triangle

  Length of Sides Doing Direct Making Connection Measurement between the concepts that

  Comparing The Identify the enable Triangle with Right Characteristics

  Triangle Determine Type of the Triangle based on its Shape

  Are there any Yes No Find the special

  Characteristics of characteristics on Angles its sides?

  Determine the Type of the Triangle

  No Did I Already found all Classify the six types of Triangles?

  Triangle Yes Finish

  Figure 2. Students’ Abstraction in Solving Problem Number 1 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.


  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).


  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

  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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