Realistic Mathematics Education

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Math Instructional Model & Strategy

1st GROUP
JALIL SETIAWAN JAMAL NOOR AZIZAH ARINI ANSAR WIDURI ANANDA PUTRI

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PREFACE

Praise were offered the presence of Allah Almighty, who has been delegated His guidance and aid, so this paper can be solved by not a hitch. The purpose of this paper was written to fulfill the task. Fully aware that this paper can not be resolved without assistance, guidance, and support from various parties. Therefore in this opportunity authors express appreciation and gratitude to the lecturer of this subject, who has given us like a forward of this material. Hopefully, participation and guidance they can add insight and knowledge. However, in this paper is still far from realizing, so with all humility the writer expects criticism and constructive suggestions for the perfection of writing in the foreseeable come. Finally, the authors hope this paper can be useful and be supporting the development of educational science, particularly mathematics education

Makassar, 10th of May 2011

Writer

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CONTENTS

TITTLE .................................................................................................................i PREFACE ............................................................................................................ii CONTENTS ........................................................................................................iii CHAPTER I INTRODUCTION A. Background .............................................................................................1 B. Problem Statement ................................................................................2 C. Objective of Paper ..................................................................................2 CHAPTER II REALISTIC MATHEMATICS EDUCATION A. Definition of realistic mathematics education ......................................4 B. Realistic mathematics education’s key principles ..................................6 C. Characteristic of rme's teaching and learning principles ........................ 17 D. The steps of realistic mathematics education ........................................20 E. Advantages and disadvantages of realistic mathematics education .....23 F. Theories that support the realistic mathematics education .................23 CHAPTER III CLOSING A. Conclusion ..............................................................................................32 B. Suggestion...............................................................................................32 BIBLIOGRAPHY .................................................................................................34

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CHAPTER I INTRODUCTION
B. Background
Mathematics is one of the basic sciences, which increasingly felt interkasinya with other scientific fields such as economics and technology. The role of mathematics in this interaction lies in the structure of science and equipment used. One characteristic of mathematics is an abstract object that has this can cause many students have difficulty in math. Student mathematics achievement, both nationally and internationally has not been encouraging. In math students are not yet meaningful learning, so that student understanding of the concept is very weak. "According to Jenning and Dunne (1999) says that, most students have difficulty in applying mathematics to real life situations." This is causing the difficulty of mathematics for students is due in less meaningful mathematics learning, and teachers in learning in the classroom does not associate with scheme which has been owned by the students and the students lack the opportunity to rediscover the ideas of mathematics. Linking real-life experiences, children with mathematical ideas in learning in the classroom is very important for learning mathematics meaningful. According to Van de Henvel-Panhuizen (2000), when children learn mathematics separate from their everyday experience, then the child will quickly forget and can not apply mathematics. One of the math-oriented learning matematisasi everyday experience and apply mathematics in everyday life is a realistic mathematics learning.

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Realistic Mathematics Education (RME) is a theory of teaching and
learning in mathematics education. Realistic mathematics first introduced and developed in the Netherlands in 1970 by the Freudenthal Institute. These lessons relate and involve the surrounding environment, the real experience of having experienced students in daily life, and make mathematics as a student activity. With the RME approach, students do not have to be brought into the real world, but problems associated with real situations that exist in the minds of students. So students are encouraged to think how to solve problems that may or often experienced by students in their daily life. Here in this paper will elaborate further on Realistic Mathematics Education or Realistic Mathematic Eduacation (RME) C. Problem Statement Base to the background before, can be made some problem statement like: 1. What the definition of realistic mathematics education? 2. What the realistic mathematics education’s key principles? 3. What the characteristic of rme's teaching and learning principles? 4. What the steps of realistic mathematics education? 5. What the advantages and disadvantages of realistic mathematics education? 6. What theories that support the realistic mathematics education?

D. Objective of Paper
Base to the problem statement before, can be made the objective of paper like:
1. To know the definition of realistic mathematics education. 2. To know the realistic mathematics education’s key principles.

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3. To know the characteristic of rme's teaching and learning principles. 4. To know the steps of realistic mathematics education. 5. To know the advantages and disadvantages of realistic mathematics

education.
6. To know theories that support the realistic mathematics education.

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CHAPTER II REALISTIC MATHEMATICS EDUCATION
As a reaction to the New Math or Mathematics Modern, the Wiskobas project in the Netherlands developed the instructional theory called 'Realistic Mathematics Education (RME)' (see Freudenthal, 1973, 1991; van den HeuvelPanhuizen, 1996; Gravemeijer, 1994, 1997; Klein, 1998; Streefland, 1991, 1991a; Treffers, 1987, 1991). The label 'realistic' is taken from a classification by Treffers (1987) that discerns four approaches in mathematics education: mechanistic, structuralistic, empiristic and realistic (these approaches will be discussed in section 3.3.1). Later on, based on Freudenthal's interpretation of mathematics as a human activity (Freudenthal, 1973), a realistic approach to mathematics education became known as Realistic Mathematics Education (RME). To give more insight into this theory the following section outlines some notions of RME.

A. Definition of Realistic Mathematics Education Understanding realistic approach by Sofyan, (2007: 28) "an educational approach that tries to place education at the basic essentials of education itself." According Sudarman Benu, (2000: 405) "realistic approach is an approach that uses real-world problem situation or a concept as a starting point in learning mathematics." Learning math is equivalent Realistic Realistic Mathematics Education (RME), an approach to learning mathematics developed in the Netherlands Freudenthal. Gravemeijer (1994: 82)

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“Realistic

mathematics

education

is

rooted

in

Freudenthal's

interpretation of mathematics as an activity.”
Gravemeijer expression above shows that learning mathematics Freudenthal realistic on the view that states of mathematics as an activity. Furthermore Gravemeijer (1994: 82) explains that which can be classified as such activities include problem-solving activities, search and organize subject matter. According to Freudenthal's activities called mathematization with the concept of realistic mathematics above Gravemeijer (1994: 91) states

“Mathematics is viewed as an activity, a way of working. Learning mathematics means doing mathematics, of which solving everyday life problem is an essential part”
Gravemeijer explained that with regard mathematics as an activity then studied mathematics means working with math and solving problems of everyday life is an important part of learning. Another concept of learning mathematics put forward realistic Treffers (in Fauzan, 2002: 33-34) in the following statement

“The key idea of RME Is That Children Should Be given the opportunity to reinvent mathematics under the guidance of an adult (teacher). In addition, the formal mathematical knowledge can be developed from children's informal knowledge.”
In the expression above Treffers explains the key ideas of realistic mathematics learning which emphasizes the need for opportunities for students to reinvent mathematics by an adult (teacher). Also mentioned that the formal mathematical knowledge can be developed (re-discovered), based on 8

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informal knowledge that students possessed. These statements mentioned above describes a way of looking at learning matamatika placed as a process for students to find their own informal knowledge-based mathematical knowledge he had. In this view mathematics is presented not as good "so" that can be moved by the teacher into the minds of students. From some of the above opinions can also be said that the RME or Realistic approach is a learning approach that uses everyday problems as a source of inspiration in the formation of concepts and apply these concepts or can be considered a learning mathematics based on real things or real for students and refers to the social constructivist. The purpose of Realistic Mathematics Education as follows (Kuiper & kouver, 1993): 1. Making math more interesting, relevant and meaningful, not too formal and not too abstract. 2. Consider the student's ability level. 3. Emphasize learning mathematics "learning by doing". 4. Facilitate completion of math problems without using a standard solution. 5. Using context as a starting point of learning mathematics.

B.

Realistic Mathematics Education’s Key Principles
According to Gravemeijer (1994, 1997) there are three key heuristic

principles of RME for instructional design (see also Gravemeijer, Cobb, Bowers, and Whitenack, 2000) namely guided reinvention through progressive mathematization, didactical phenomenology, and self developed models or 9

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emergent models. These principles are discussed consecutively in more detail in the following sections. 1. Guided reinvention through progressive mathematization According to de Lange (1987), in RME the real world problem is explored in the first place intuitively, with the view to mathematizing it. This means organizing and structuring the problem, trying to identify the mathematical aspects of the problem, to discover regularities. This initial exploration with a strong intuitive component should lead to the development, discovery or (re) invention of mathematical concepts. These criteria lead to the first key principle of RME for instructional design that is 'guided reinvention through progressive mathematizing'. In the guided reinvention principle, the students should be given the opportunity to experience a process similar to that by which

mathematics was invented (Gravemeijer 1994, 1999). With regard to this principle, a learning route has to be mapped out (by a developer or instructional designer) that allow the students to find the intended mathematics by themselves. When designing the learning route

(Gravemeijer (1994) calls this conjectured learning trajectory), the developer/designer starts with a thought experiment, imagining a route by which he or she could have arrived at a solution him-or herself. Gravemeijer (1994) says that the conjectured learning trajectory should be emphasized on the nature of the learning process rather than on inventing mathematics concepts/results. It means we have to give students the opportunity to gain knowledge so that it becomes their own private knowledge, knowledge for which they themselves are responsible. This implies that in the teaching learning process students should be given the opportunity to build their own

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mathematical knowledge on the basis of such a learning process. According to Gravemeijer (1994, 1997) there are two things that can be used to realize the reinvention principle. Firstly, from knowledge of the history of mathematics we can learn how certain knowledge developed. This may help the developer/instructional designer to lay out the intermediate steps, by which the intended mathematics could be reinvented. It means that students can learn from the work of mathematicians. Secondly, by giving a contextual problem that has various informal solution procedures, continued by mathematizing similar solution procedures, will also create the opportunity for the reinvention process. To do so the developer/instructional designers need to find contextual problems that allow for a wide variety of solution procedures, especially those which considered together already indicate a possible learning route through a process of progressive mathematization. Gravemeijer (1999) sees the reinvention principle as long-term learning process in which the reinvention process evolves as one of gradual changes. The intermediate stages always have to be viewed in a long-term perspective, not as goals in themselves, and the focus has to be given on guided exploration. To realize this view, the developer/instructional designers

need to design a sequence of appropriate contextual problems. What we mostly find in traditional mathematics instruction is in the contrary to this view. Here the learning path is structured in separate learning steps, in which each step can be mastered independently. To understand the guided reinvention principle better, let us see the differences between the realistic approach and information processing regarding reinvention process. According to Gravemejer (1994) the

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information processing approach views mathematics as a ready-made system with general applicability, and mathematics instruction as

breaking up formal mathematics knowledge into learning procedures and then learning to apply them. On the other hand, the realistic approach is emphasized on mathematizing. Mathematics is viewed as human activity and learning mathematics means doing mathematics in which solving the everyday problems is an essential part. The different view of the two approaches is essentially reflected in the mathematical learning processes as shown in the next models in solving a contextual problem.

Source: Gravemeijer, 1994. application of formal mathematics realistic problem-solving Mathematical learning process in the information processing and realistic approaches. The first model describes the process of solving a contextual problem by using the formal mathematical knowledge. In the first step, the problem is translated to a mathematical problem (mathematical terms), then the mathematical problem is solved by using the relevant mathematical means. At the end, the mathematical solution is translated back into the original context. Gravemeijer criticizes thismodel because there is reducing

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information in the process of solving the problem. Transformation of a contextual problem into a mathematical problem causes a reduction of information because many aspects of the original problem will have been obliterated. When the mathematical solution is translated back into the original context, it involves an interpretation. On the other side, the aspects that were obliterated should be taken into account again. What frequently happens is that the suggestion obtained from mathematical solution does not really fit the original problem. Moreover, solving the problems by using this model is due to recognizing problem types and establishing standard routines. In the second model, solving the problem also passes through three stages: describing the contextual problem more formally, solving the problem on this level, and then translating the solution back into the context. But because in the realistic approach mathematics is taught based on human activity, it makes that the three activities have a very different meaning than those in the first model. Gravemeijer describes the advantages of solving the problem by using this approach as follow:  The problem is the actual aim rather than the use of a mathematical tool;  Solving the problem is done in an informal way rather than applying a standard procedure;  The problem is described in a way that allow pupils to come to grips with it;  By schematizing and identifying the central relations in the problem situation, pupils will understand the problem better;  The description we provide can be sketchy and using self-invented

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symbol (it needs not be presented in commonly accepted mathematical language);  The description also simplifies the problem by describing relations and distinguishing matters of major and minor importance;  Translation and interpretation of the solution are easier because the symbol are meaningful. So far we can see that 'mathematizing' is a very important activity in RME. This activity (Gravemeijer, 1994). Formalizing includes modeling, symbolizing, schematizing and defining, and generalizing is to understand in a reflective sense. By mainly involves generalizing and formalizing

solving the contextual problems in realistic approach students learn to mathematize contextual problems. This process is called mathematization (Treffers, 1987, 1991a). As the process of mathematization is very important to develop knowledge from children's thinking (Freudenthal, 1968; Resnick, Bill & Lesgold, 1992; Trffers, 1991a), it is necessary to start the process by mathematizing those contextual problems that come from children's everyday-life reality. By doing that, children have the opportunity to solve the contextual problems using informal language (Treffers (1987, 1991a) calls this process as horizontal mathematization). In the long term, after the students have experienced similar processes (through simplifying and formalizing), the informal language will be developed into a more formal or standardized language. At the end of these processes the students will come to an algorithm. The process of mathematization of mathematical matter is called vertical mathematization (Treffers, 1987, 1991a). Freudenthal (1991) makes the distinction between horizontal and vertical mathematization: 14

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"Horizontal mathematization leads from the world of life to the world of symbols. In the world of life one lives, acts (and suffers); in other one symbols are shaped, reshaped, and this is manipulated, mechanically,

comprehendingly,

reflectingly:

vertical mathematization. The

world of life is what is experienced as reality (in the sense I used the word before), as is symbol world with regard to abstraction". De Lange (1987) distinguishes between horizontal and vertical mathematization in more detail based on type of activities. The activities in horizontal mathematization involve identifying the specific mathematics in a general context; schematizing; formulating and visualizing a problem in different ways; discovering relations; discovering regularities; recognizing isomorphic aspects in different problems; transferring a real world problem to a mathematical problem; and transferring a real world problem to a known mathematical model. Meanwhile, in vertical mathematization the activities include representing a relation in a formula; proving regularities; refining and adjusting models; using different models; combining and integrating models; formulating a new mathematical concepts; generalizing The process of horizontal and vertical mathematization is described in Horizontal mathematization takes place when pupils describe contextual problems using their informal strategies in order to solve them. If the informal strategies lead the pupils to solve the problems using mathematical language or to find an algorithm, then this process of movement shows a vertical mathematization.

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Source: Gravemeijer, 1994 Horizontal Mathematization ( . . . . . ) Vertical Mathematization (------>) Due to this learning process, if the students can (re) construct the formal mathematical knowledge, it means they do reinvention process. Gravemeijer (1994) schematizes this process in the next figure.

Reinvention process Although in figure above the reinvention process is presented sing a one way arrow, in reality it is a repeated process. In other words, before

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reinventing the formal mathematical knowledge, pupils experience the processes of describing and solving the contextual problems that have similar procedure solutions. In these processes the pupils develop their informal strategies into mathematical language or algorithm. The four approaches in mathematics education mentioned in section 3.1 are classified by Treffersr (1987) using criteria of horizontal and vertical mathematization. In the realistic approach, horizontal and vertical mathematizations are used to construct the long-term learning process. Here the students will start with contextual problems, idiosyncratic, informal knowledge and strategies. They then have to construct formal

mathematics by mathematizing the contextual problems (horizontally) and by mathematizing solution procedures (vertically). The mechanistic approach is the opposite of the realistic approach because it lacks both the horizontal and vertical mathematization. The structuralistic approach only emphasizes on vertical mathematization, while the empiristic approach focuses on horizontal mathematization. These conditions can be summarized as follows.

The sign '+' means much attention paid to that kind of mathematization, and the sign '-' means little or no attention at all (see De Lange, 1987). 2. Didactical Phenomenology In contrast to the anti-didactic inversion (see section 3.2),

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Freudenthal (1983) advocated the didactical phenomenology. This implies that in learning mathematics we have to start from phenomena that are meaningful for the student, that beg to be organized and that stimulate learning processes. In didactical phenomenology, situations where a given mathematical topic is applied are to be investigated for two reasons (Gravemeijer, 1994, 1999). Firstly, to reveal the kind of applications that have to be anticipated in instruction. Secondly, to consider their suitability as points of impact for a process of progressive mathematization. According to Gravemeijer (1994, 1999), the goal of a

henomenological investigation is to find problem situations for which situation-specific approaches can be generalized, and to find situations that can evoke paradigmatic solution procedures that can be taken as the basis for vertical mathematization. This goal is derived from the fact that mathematics is historically evolved from solving practical problems. In mathematics instruction we can realize this goal by finding the contextual problems that lead to this evolving process. An implication of the didactical phenomenology principle is that the developer/ instructional designer has to provide students with contextual problems taken from phenomena that are real and meaningful for them. But sometimes mathematics educators misunderstand the label 'real' or 'realistic' in RME. They interpret it as referring to a 'really' real objects or situations in the surroundings. Considering this, it is important to notice the next statement from Gravemeijer (1999). 'The use of the label 'realistic' refers to a foundation of mathematical knowledge in situations that are experientially real to the students. Context problems in RME do not necessarily have to deal with authentic every-day life

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situations. What is central, is that the context in which a problem is situated is experientially real to students in that they can immediately act intelligently within this context. Of course the goal is that eventually mathematics itself can constitute experientially real context for the students.' 3. Self-developed models The third key principle for instructional design in RME is selfdeveloped models or emergent models (Gravemeijer 1994, 1999). This principle plays an important role in bridging the gap between informal knowledge and formal knowledge. It implies that we have to give the opportunity to the students to use and develop their own models when they are solving the problems. At the beginning the students will develop a model which is familiar to them. After the process of generalizing and formalizing, the model gradually becomes an entity on its own. Gravemeijer (1994) calls this process a transition from model-of to model-for. After the transition, the model may be used as a model for mathematical reasoning (Gravemeijer, 1994, 1999; Treffers, 1991a). To give a clearer meaning of models, Gravemeijer (1999) differentiates between embodiment and models. He says that embodiment are presented as pre-existing models in product-oriented mathematics education, while models emerge from the activities of the students themselves in realistic mathematics education. Related to this, Gravemeijer suggests that the primary aim of the use of models should not be regarded as something to illustrate mathematics from an expert point of view, but that they should support students in constructing mathematics starting from their own perspective. The next figure illustrates the use of models in three different approaches in mathematics education.

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Source: Gravemeijer, 1994 The process of using models in three different approaches At the beginning of this section the term emergent models was introduced. This term is used by Gravemeijer (1999) to indicate the character of the development of model-of to model-for. An RME model emerges from the informal solutions of the students when they solve the contextual problem. Firstly, the model is used to support informal strategies that correspond with situation-specific solution strategies. After the students experience similar solution procedures, the choice of a strategy is no longer dependent on its relation with the problem situation, but is much influenced by mathematical characteristics of the problem. Here the role of model begins to change because it gets a more general character. Finally the model becomes an entity on its own after a process of reification takes place. Gravemeijer (1999) argues that at this stage the model becomes more important as a base for mathematical reasoning than as a way to represent a contextual problem.

C. Characteristic of Realistic Mathematics Education
The previous section has discussed the important principles in RME for instructional design. Suppose that we have designed curriculum material based on the RME theory, now comes the questions: how should

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the teaching learning process using this curriculum material be conducted; how should teachers present the curriculum in the classrooms; and how are students supposed to learn from the curriculum material? Related to these questions, Treffers (1991a) proposes five learning and teaching principles namely constructing and concretizing, levels and models, reflection and

special assignments, social context and interaction, and structuring and interweaving (Note: in each pair, the learning principle is indicated first). These teaching and learning principles are parallel to five tenets mentioned by de Lange (1987): (1) the use of real-life contexts; (2) the use of use models; (3) student's free production; (4) interaction; (5) intertwining. The following parts discuss the RME's learning and teaching principles one by one. 1. Constructing and concretizing The first learning principle of RME is that learning mathematics is a constructive activity, something which contradicts the idea of learning as absorbing knowledge which is presented or transmitted (Treffers, 1991a). On the teaching idea, the instruction should start with a concrete orientation basis. In other words, the instruction has to be emphasized on a phenomenological exploration (Gravemeijer, 1994). From phenomena that need to be organized as a starting point, teachers can stimulate students to manipulate these means of organizing. 2. Levels and models In this principle, the learning of a mathematical concept or skill is viewed as a process which is often stretched out over the long term and which moves at various levels of abstraction (from informal to formal and from the intuitive level to the level of subject-matter systematics) (Treffers, 1991a). Now how to help bridge the gap between these various levels? Using the

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term bridging by vertical instruments. Gravemeijer (1994) advocates that a broad attention has to be given to visual models, model situations, and schemata that arise from problem solving activities because it will help students to move through these various levels. 3. Reflection and special assignments The third learning principle in RME is related to the raising of the level of the learning process. According to Graveimeijer (1994) and Treffers (1991a) the raising process is promoted through reflection, therefore serious attention has to be paid to a student's own constructions and productions. On the teaching principle: the students must constantly have the opportunity and be stimulated at important junctures in the course, to reflect on learning strands that have already been encountered and to anticipate what lies ahead (Treffers, 1991a). To realize this principle we have to provide students with special assignments, for example the conflict problems, those that can stimulate students' free productions. 4. Social context and interaction The fourth learning principle is related to the importance of social context, as Treffers (1991a) says that learning is not a solo activity but it occurs in a society and is directed and stimulated by the socio-cultural context. By working in-groups for example, students have the opportunity for the exchange of ideas and arguments so that they can learn from others. This principle implies that mathematics education should by nature be interactive. It means interactivity that includes explicit negotiation, intervention,

discussion, cooperation and evaluation become very essential elements in a constructive learning process (Gravemeijer (1994) 5. Structuring and Interweaving

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The last learning principle is connected to the first principle. According to Treffers (1991a) learning mathematics doe not consist of absorbing a collection of unrelated knowledge and skill elements, but is the construction of knowledge and skills to a structured entity. In addition, Gravemeijer (1994) says that the holistic approach, which incorporates applications, implies that learning strands can not be dealt with as separate entities; instead, an intertwining of learning strands is exploited in problem solving. These statements bring us to the teaching principle: the learning strands in mathematics must be intertwined with each other.

D. The Steps Of Realistic Mathematics Education
Reviewing interactive characteristics in realistic mathematics above seems necessary in a learning design that is able to build interaction between students and students, students with teachers, or students with their environment. In this case, Asikin (2001: 3) believes the need for teachers to give students the opportunity to communicate his ideas through individual presentations, group work, group discussions, and class discussion. Negotiation and evaluation with fellow students and teachers are also learning an important factor in this constructive learning. The implications of the social aspect is quite high in these students' learning activities, the teacher needs to determine appropriate teaching methods and in accordance with those needs. One method of teaching that can meet those objectives is to include discussions on student learning. Activity is seen to encourage discussion and interaction between members of the class launched. According to Kemp (1994: 169) discussion is a form of face-to-face teaching is the most commonly used for exchanging information, thoughts and opinions. More than that in a discussion of the learning process that goes not only activities that are considering the sheer information, but also allows 23

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the process of thinking in the analysis, synthesis and evaluation. Further discussions have also determined the form which would be to consider the condition of the existing class. Because learning in the framework of the research was conducted in a classroom, which generally consists of 40 to 44 students with placement of students that are difficult to form a large group discussion, the interaction among the students raised through small group discussions in pairs, in addition to class discussions. Basing on the condition of class as the above description as well as some of the characteristics and principles of realistic mathematics learning, the learning steps undertaken in this study consisted of: Step 1. In this step the teacher presents to students a contextual

problem. Next the teacher asks the students to understand the problem first. Characteristics of realistic mathematics that appears in this step is to use context. Use of context seen in the presentation of contextual issues as the starting point of student learning activities. Step 2. Describes the contextual problem. This step was taken when students have difficulty understanding the contextual issues. In this step the teacher provides assistance by giving the necessary instructions or questions that can lead students to understand the problem. Characteristics of realistic mathematics that appears in this step is interactive, ie the interaction between teachers and students and between students with a student. While the principle has guided Reinvention least arise when teachers try to give direction to the students in understanding the problem. Step 3. Solving contextual problems. At this stage students are encouraged to solve the problem individually, based contextual ability to use the clues provided. Students have the freedom

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to use his own way. In the process of solving problems, students actually lured or directed to think discover or construct knowledge for themselves. At this stage it is possible for teachers to provide the necessary assistance (scaffolding) to students who really need help. At this stage, the two principles of realistic mathematics that can be raised is Reinvention and progressive mathematizing guided and self-developed models. While the characteristics that can be raised is the use of models. In solving problems students have the freedom to build models of the problem. Step 4. Compare and discuss answers. At this stage the teacher first asks students to compare and discuss answers with their partner. This discussion is a vehicle for a pair of students discuss their answers. From these discussions are expected to appear to answer agreed upon by both students. Next the teacher asks students to compare and discuss answers in class discussions held. At this stage the teacher pointing or giving students the opportunity to partner to bring the answers he has to face the class and encourage other students to examine and respond to the answers that appear in front of the class. Characteristics of learning mathematics that appears realistic at this stage are interactive and use student contributions. Interactions can occur between students and between teachers and students are also students. In this discussion, student contributions useful in solving problems. Step 5. Conclude From the results of class discussions teachers lead students to draw conclusions about problem solving, concepts, procedures or principles that have built together. At this stage of learning the characteristics of realistic mathematics that emerges is interactive and use student contributions.

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E. Advantages and Disadvantages Realistic Mathematics Education
Some of the advantages of learning metematika realistic among others: 1. Lessons to be quite enjoyable for students and the atmosphere did not seem tense. 2. Material can be understood by most students. 3. Props are the objects that are around, so easily obtained. 4. Teachers are challenged to learn the material. 5. Teachers become more creative to make props. 6. Students have a high enough intelligence seem more intelligent. Some weaknesses of metematika realistic learning include: 1. Difficult to apply in a large class (40-45 people). 2. It takes a long time to understand the subject matter. 3. Students who have the intelligence was needed more time to be able to understand the subject matter.

F. Relevant Learning Theory with Realistic Mathematics Education
As described in the previous section, realistic mathematics developed with reference to and inspired by the constructivist philosophy. Meanwhile, according Soedjadi (1999: 156) constructivism in the field of study can be viewed as one approach that was developed in line with the theory of cognitive psychology. The essence of constructivism in the field of learning is that students have a major role in constructing knowledge that are meaningful to him. While the teachers position themselves more as facilitators of learning. Some cognitive learning theory which is deemed relevant to the realistic approach to learning mathematics is the theory of Piaget, Vygotsky theory,

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theory theory Ausubel andBruner. 1. Piaget's Theory (in Ibrahim, 1999:16) Holds that, children have the potential to develop intellectual. Intellectual development of their departure from curiosity and to understand the world around him. Understanding and appreciation about the world around it will encourage them to build their minds about the world view in his brain. The view that a mental structure is called a schema or schemata (plural). Suparno (1997: 30) described the scheme as a network of concepts or categories. By using the scheme, a person can process and identify a stimulus that it receives so that he can put it in the category / concept accordingly. Piaget states that the basic principle of development is the ongoing adaptation of one's knowledge of one's thoughts into the reality around him. This adaptation process is inseparable from the existence of such schemes owned and involves assimilation, accommodation and equiliberation in mind (Suparno, 1997: 31). Assimilation is the cognitive process by which one can integrate perception, concept or new experiences into the scheme has. Through assimilation, the scheme evolved but not changed person. Thus the development of the scheme a person means the enrichment of one's perception and knowledge of the surrounding world. Therefore, assimilation can be viewed as a process by which individuals to adapt and organize themselves into the environment so that change their undrstanding. Cognitive process of assimilation is not always possible someone. This happens if new stimuli are not received in accordance with the scheme has. If this happens, it will be the process of accommodation. Through a process of accommodation, one's thoughts will form a new scheme that fits with that stimulus or modify the existing scheme so that it matches with that stimulus (Suparno, 1997: 32). In developing his knowledge, the process of assimilation and 27

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accommodation continue to take place in a person. Both occurred not stand alone. Both these processes take place in the balance that is set mechanically. The process of setting this balance is called equilibrium (Suparno, 1997: 32). However, in accepting a new experience can happen in such a state of imbalance to occur between assimilation and accommodation. This situation is referred to as dissequilibrium. This imbalance arises when there is a discrepancy between the current experience with new experiences that lead to accommodation. If there is an imbalance then someone also managed to find a balance between assimilation and accommodation. According Dahar (1991: 182) a person who is able to regain its balance will be at a higher intellectual level than before. It can be concluded that Piaget's theories with the reality or knowledge not as an object that was already finished and available to humans, but he must be obtained through construction activities by man himself through the process of adapting his mind into the reality around him. Furthermore, Piaget (in Atkinson, 1999: 96) explains that in his intellectual development stages of a child is already involved in the process of thinking and pondering life logically. The thought process took place in accordance with the level of child development. For optimal intellectual development of children take place then they need to be motivated and facilitated to develop theories that explain the world around him (Ibrahim, 1999: 19). Related to this effort Piaget (in Ibrahim, 1999:18) argues that a good education is education that involves the child to experiment on their own, in the sense: a. Trying everything to see what happens. b. Manipulate signs and symbols c. Ask questions d. Finding the answers yourself

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e.Verify what he had discovered at some point with what he found at the other f. Comparing the findings with the findings of others. Realistic mathematics learning is one approach that is consistent with Piaget's view of the above. Realistic mathematics developed based on constructivist philosophy, looking at knowledge in mathematics is not as something that is already finished and ready to be given to students, but as a result of students who are studying construction. Therefore, in realistic mathematics student learning is central to the learning process itself, while teachers act more as facilitators. The implication of this view is a must for teachers to facilitate and encourage students to engage actively in the learning process. Students should be encouraged to construct knowledge for themselves. For this purpose the students should have greater freedom in expressing his thoughts in solving the problems it faces. To realize the situation and conditions to learn that so then in managing the learning teachers need to consider several views of Piaget. Among them are teachers should encourage students to dare to try various possible ways to understand and solve problems. In this activity students construct knowledge by realized by providing contextual problems. Contextual problems were designed so that it allows students to construct knowledge independently. 2. Vygotsky Theory Matthews and O'Loughlin (in Suparno, 1997: 41) argues that Piaget's theory was developed with greater emphasis on personal aspects. This theory is considered too subjective and less social, so that community and environmental factors less attention in the process of a child's intellectual development. Unlike Piaget, Vygotsky (in Ibrahim, 1999: 18) argues that the

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process of knowledge creation and development of children is inseparable from social interaction factors. Through interaction with friends and the environment, a child helped his intellectual development. Vygotsky's view about the importance of social interaction in the intellectual development of children looked out of four key ideas that build a theory. a. Emphasis on the social nature The first key idea is to explain Vygotsky views about the importance of social interaction in the learning process of children. Vygotsky (in Nur, 1999: 3) argued that children learn through interaction with adults or peers. In such a learning process, a child who is learning not only convey your understanding of a problem to himself, but he also can deliver it to other people around him. Cooperative learning are interwoven by social intraksi participants learn to give the benefit of learning outcomes that are open to all students and other students' thinking process open to other students. b. Areas nearby development (zone of proximal development). Vygotsky explained that there are two levels of intellectual development, namely the level of actual development and potential development level. At the level of actual development of a person is able to learn or solve problems by using the capabilities that exist in him at that time. While the level of potential development is the level of one's intellectual development achieved with the help of someone more capable. The level of potential development is located above the level of actual development of a person. Changes in the level of actual development leading to the potential level of development does not occur directly and automatically. The changes that take place through a process of learning that occurs in the region nearest the development. Regional development is located just above the nearest actual development of a person. According to Slavin (1994: 49) a child who 30

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works in areas closest developments involved in the tasks themselves are not capable of completion. It requires the presence of people who are better able to help. By doing a series of learning tasks in the area nearest the development of a child is expected to reach a certain proficiency level in next time. Thus the learning process in the region nearest the development can be viewed as a process of transition or a transition from the level of actual development to the level of potential development. c. Cognitive Apprenticeship The key idea is a combination of two key ideas of the first, namely the social nature and development of nearby areas. According to Vygotsky, in the process of cognitive apprenticeship a student to gradually achieve expertise in interaction with an expert, adults or their peers with more knowledge (Nur, 1999: 5). Implementation of this idea is the formation of heterogeneous cooperative learning groups so that students are more intelligent may help students who are less good at completing tasks. d. Scaffolding Scaffolding (the stairs) is a principle that refers to the assistance provided by adults or peers who are competent. In the learning process for help was given to students in the form of a large amount of support in the early stages of learning. Further aid was getting less and eventually none at all so the kids take over full responsibility for what is done after he was able to do so (Slavin, 1997: 48).

The key idea is to explain the views of Vygotsky about the need for complex tasks, difficult and realistic to students. Through problem solving in the task he was receiving, a student is expected to find the basic skills useful for him. Thus learning that takes place more emphasis on top-down teaching models (Nur,

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1999: 5). From this, in contrast to traditional bottom-up model, where basic skills are given in stages to achieve a more complex skills. The implications that arise on the view Vygotsky in education of children is the need for an encouragement to students to interact with people around him who have better knowledge that can provide assistance in intellectual development. More broadly than that, the constructivist emphasis for educators pay attention to the existence of the school situation, community and friends around someone who can affect the intellectual development of his students (Cobb in Suparno, 1997: 96). One of the characteristics of realistic mathematics learning is the discovery of concepts and problem solving as a result of false ideas of the students. The contribution of these ideas could be realized through the learning process in which there is interaction between students and students, between students and teachers or between students with their environment. Thus, in addition to any mental activity that was personal, in realistic mathematics teachers need to encourage the emergence of social interaction between members of the class in the process of construct knowledge. Through these social interactions are more capable students the opportunity to convey the understanding that it has on other students who are weaker. This allows for students who are weaker are obtained from actual development to increase potential growth for students who are better able to help. While on the other hand teachers have a role in helping students who have difficulty with giving directions, instructions, warnings and encouragement. Thus it appears that the learning process in line with realistic math that Vygotsky theories emphasize the importance of social interaction in the intellectual development of children. In this case, the social interaction between members of the classes is realized through a phase of discussing and negotiating the settlement of 32

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problems at the group level or grade level. In the discussion group or the class teacher should encourage the spirit of sharing and respect the views of other parties. While the interaction that can be built by teachers with students is to provide assistance as needed without having to limit the range of students to express her ideas. 3. Ausubel Theory Ausubel, Noval and Hanesian classify study of two types of learning memorization and meaningful learning (Suparno, 1997: 53). According to Nur (1999: 38) refers to the memorization learning to memorize facts or relationships, eg multiplication tables and symbols of chemical atoms. Meanwhile, according to Ausubel learning meaningful to say if the information to be studied are prepared in accordance with students' cognitive structure so that these students relate new information to the cognitive structure that has (Hudojo, 1988: 61). According Parreren through meaningful learning one's own concept of the structure had been developed. In addition, the concepts are connected to one another in a meaningful delivery rules are useful in solving the problem (Winkel, 1991: 57). This view is in line with the opinion that says that the knowledge learned significantly will allow to apply to the broader situation in real life (Nur, 1999: 34). Contrary to the explanation above, if the knowledge that should be taught in a meaningful but are taught by rote will produce inert knowledge. Inert knowledge is real knowledge can be applied to more general situations, but in fact only applicable in special situations (Nur, 1999: 38). Students who simply memorizing a concept without really understand it is a form of verbal victim (Winkel, 1991: 58). One of the characteristics of realistic mathematics learning is the use of

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context. The use of realistic context for learning mathematics means that the everyday environment or students' existing knowledge can be used as part of learning materials for students. What is going on around students and student knowledge is a valuable material to serve as the contextual issues that became the starting point of students' thinking activity. The problem is thus more meaningful to students because it is still within range of knowledge that has been owned by previous students. Therefore, to solve contextual problems a student should be able to relate the knowledge that has been held with the problem. Thus a student will successfully solve the problem if he has enough contextual knowledge related to the problem. In addition, students also must be able to apply knowledge that has been held to resolve these contextual issues. Thus the presentation of contextual problems for students in mathematics realistic in line with the theory of meaningful learning Ausubel. 4. Bruner Theory Bruner (in Hudojo, 1988: 56) argues that learning mathematics is to learn about the concepts and structures and to find relationships between the concepts and structures are. According to Bruner understanding of a concept and its structure makes the material more memorable and can be understood more comprehensively. Similar to what was raised as Piaget, Bruner argues that there are three stages of mental development through which students in the learning process. But the third stage of thinking by

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CHAPTER III CLOSING

A. Conclusion
Based on the description above, then the conclusion can be delivered more of the following. Realistic Mathematics (MR) is a school mathematics performed by placing the realities and experiences of students as a starting point of learning. MR use of realistic learning problems as a starting base of learning, and through the horizontal-vertical matematisasi students are expected to find and reconstruct the concepts of mathematics or formal mathematical knowledge. Furthermore, students are given the opportunity to apply mathematical concepts to solve everyday problems or issues in other fields. In other words, learning-oriented MR matematisasi everyday experience (mathematize of

Everyday experience) and apply mathematics in everyday life (everydaying mathematics), so that students learn the significance (understanding).
MR student-centered learning, while teachers only as a facilitator and motivator, so it requires a different paradigm of how students learn, how teachers teach, and what is learned by students with mathematics learning paradigm so far. Therefore, changes in teacher perceptions of teaching needs to be done if you want to implement realistic mathematics learning.

B. Suggestion
To make this paper we (writer) have arduousness to collect the literatures about this material because in library there are just few books about

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that. And we also have limitation to find literature in another place or out of library campus or we just can see in the internet. So that, we suggest to library administration to more maximize their collection books especially in subject strategies and models of learning. Actually in the making of this paper, we still have confusing about the materials because there are many term we don’t know what its mean. So for the lecture, please explain more about this material.

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BIBLIOGRAPHY
Freudental, H. 1973. Mathematics as an Educational Task. Dordrecht: Reidel Publising. Gravemeijer, K. 1994. Developing Realistic Mathematics Education. Utrecht: Freudental Institute. Fauzan, Muhammad. 2002. Application Realistic Mathematics Education

(RME) In Teaching Geometry In Indonesian Primary School. Surabaya.
Hergenhahn, B.R. and Matthew H. Olson. 2008. Theories of Learning. Perason Education.

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