RME learning model was first introduced and developed in the Netherlands since 1970 by the Institute Freudenthal and showed good results, based on the results of the Third International Mathematics and Science Study (TIMSS) in 2000. According to Freudenthal, the principal activities are carried out in the RME covers.
a. Find problems or questions contextual (looking for problems).
b. Solving problems (problem solving).
c. Organize teaching materials (organizing a subject matter).
It can be realities that need to be organized and also mathematically mathematical ideas that need to be organized in a broader context. Organizing activities is called mathematization.
An expert argues as follows.
In RME students learn mematematic contextual issues. In other words, students identify that the contextual matter should be transferred to the matter in the form of mathematics to understand further, through penskemaan, formulation, and pemvisualisasian. This is a horizontal mathematization process. Furthermore, the vertical mathematization, the students completed the mathematical form of contextual problems using concepts, mathematical operations and procedures that apply and understand students (Armanto, 2001: 43).
So the next step horizontal mathematization departing from the real world issues drawn into the world of symbols. While the vertical mathematization is the process of implementing solutions to problems in the form of mathematical symbols corresponding mathematical procedures. RME Freudenthal refers to the opinion that "Mathematics should be associated with reality and mathematics is a human activity". This means that mathematics should be close to the child and relevant to real life everyday. (In the Ministry of National Education, 1994: 21) says that "Mathematics as a human activity means that man should be given the opportunity to rediscover the ideas and concepts of mathematics with the guidance of an adult". This work is done through the exploration of a variety of situations and problems "realistic". Realistic in this case meant not refer to reality but on something that can be imagined by the students (in the Ministry of National Education, 2000: 34). The principle can be inspired by the rediscovery of procedures solving information, while the discovery process again using the concept of mathematization.
Two types of mathematization formulated (in the Ministry of National Education, 1991: 25), ie horizontal and vertical mathematization. Examples of horizontal mathematization is the identification, formulation, and pemvisualisasian problems in different ways, and transforming the real world problem to a mathematical problem. Examples of vertical mathematization is a representation of the relationships in the formula, repair and adjustment of mathematical models, the use of different models, and generalizing. Both of these types of mathematization balanced attention, because both have the same value as this mathematization (in MONE, 2000: 17).
Based on the horizontal and vertical mathematization, approaches in mathematics education can be divided into four types: a mechanistic approach, empiristik, strukturalistik, and realistic (in the Ministry of National Education, 2005: 95).
1. Mechanistic Approach
Mechanistic approach is the traditional approach and is based on what is known from his own experience (starting from the simple to the more complex). In this approach the man regarded as the machine. Mathematization type is not used.
2. Approach empiristik
Empiristik approach is an approach in which mathematical concepts are not taught, and students are expected to find through horizontal mathematization.
3. Approach strukturalistik
Strukturalistik approach is an approach that uses formal systems, such as teaching summation long way should be preceded by the value of the place, so that a concept is achieved through vertical mathematization.
4. Realistic Approach
Realistic approach is an approach that uses realistic problems as a starting base of learning. Through horizontal and vertical mathematization activities students are expected to find and construct mathematical concepts.
RME has the characteristics of learning using real-world context, models, production and construction students, interactive, and relatedness (interteinment) (the Ministry of National Education, 1991: 35).
a. Using Context "Real World"
In the picture above shows two mathematization process in the form of a cycle in which the "real world" not only as a source of mathematization, but also as a place to re-apply mathematics.
In Figure Concepts of Mathematics (De Lange, 1987: 15) argues that, "In RME, learning begins with contextual issues (" real world "), thus enabling them to use previous experience directly".
Liquefaction process (core) of the corresponding concept of the real situation is expressed by De Lange (1987: 18) as a conceptual mathematization. Through abstraction and formalization of the students will develop a more complete concept. Then students can apply mathematical concepts to a new field of the real world (Applied mathematization). Therefore, in order to bridge the mathematical concepts to students' everyday experiences to note mathematization everyday experience (mathematization of everyday experience) and the application of mathematics in everyday (in Freudenthal, 2000: 12).
b. Using Models (mathematization)
The term models with regard to the model situation and mathematization models developed by the students themselves (self-developed models). The role of self-developed models are a bridge for students of the real situation to the abstract or from informal to formal mathematical mathematics. This means that students create their own models in solving the problem. The first is a model of a situation that is close to the real world of students. Generalization and formalization of the model will be transformed into a model of the problem. Through mathematical models of reasoning would be shifted to a model similar-far problem. In the end, there will be a formal mathematical model.
c. Using the Production and Construction
Stretland (in MONE, 1991: 45) emphasizes that "the making of" Free Production "students are encouraged to reflect on what they think is an important part of the learning process". Informal strategies of students who form a contextual problem-solving procedure is a source of inspiration in the development of further learning is to construct a formal mathematical knowledge.
d. Using Interaction
The interaction between students and teachers is fundamental in RME. Explicitly forms of interaction in the form of negotiation, explanation, justification, agree, disagree, question or reflectors are used to achieve formal forms of informal forms of students.
e. Using Linkage
In RME integration unti-unit mathematics is essential. If we ignore the learning linkages with other fields, it will affect the solution of the problem. In applied mathematics, usually required a more complex knowledge, and not just arithmetic, algebra, or geometry but also other fields.
Realistic Mathematics (MR) which is intended in this case is the mathematical school conducted by placing the realities and experiences of students as a starting point for learning. Problems realistically be used as a source of the emergence of mathematical concepts or formal mathematical knowledge, learning-oriented class MR characteristics RME, so that students have the opportunity to rediscover the concepts of mathematical or formal mathematical knowledge, then students are given the opportunity to apply conceptual mathematical concepts to solve everyday problems or problems in other areas. This is very different learning mathematics learning during these tend to be oriented to provide information and use mathematical ready to solve the problems. Because realistic mathematics using realistic problems as a starting base of learning the problem situation needs to be put completely contextual or appropriate experience of students, so that students can solve problems with informal ways through horizontal mathematization. Informal ways shown by students as an inspiration for the formation of concepts or aspects of mathematics maths through vertical. Through the process of horizontal-vertical mathematization students are expected to understand or discover mathematical concepts (formal mathematical knowledge).
According to the constructivist view of learning mathematics is to give students the chance to construct concepts or principles of mathematics with its own capabilities through internalizing the teacher in this case acts as a facilitator.
According to Davis (in the Ministry of National Education, 1996: 42),
Constructivist view of learning oriented mathematics:
1. Knowledge is built in the mind through the process of assimilation or accommodation.
2. In mathematics workmanship, every step students are exposed to the problems.
3. New information must be connected with the experience of the world through a logical framework that transport, organize, and interpret their experiences.
4. The learning center is how students think, not what they say or write.
1.1.1 The principle - the principle of Indonesian Realistic Mathematics Education (PMRI)
In line with the concept of origin, PMRI developed of the three basic principles that started RME, which is guided reinvention and progressive mathematization, didactial phenomenology, as well as self-developed models (2009: 2). The principle according to Van den Heuvel RME-Panhuizen in Supinah (2009: 75) is as follows.
a. The principle activity
Namely mathematics is a human activity. Learners must be active both mentally and physically in the learning of mathematics. According Freudental, because the idea of mathematization process associated with the view that mathematics is a human activity, then the best way to learn is by doing the math to work.
b. The principle of reality
That lesson should begin with realistic problems or can be imagined by the students. The main objective is to enable students to apply mathematics. Thus the ultimate goal is to make students able to use mathematics to what they understand to solve the problems encountered. The reality principle is not only developed in the late stages of a learning process, but is seen as a resource for learning math.
c. Principle tiered
This means that in learning mathematics students through the various levels of understanding, that of being able to find a solution to the problem of realistic contextual or informally, through schematization gain knowledge about basic things until able to find the solution of a mathematical problem formally.
d. Principle braid
This means that various aspects or topics in mathematics should not be viewed and studied as separate parts, but intertwined with each other so that students can see the relationship between the materials is better.
e. The principle of interaction
Namely mathematics is seen as a social activity. Students need and should be given the opportunity to express its strategy in resolving a problem to others to be addressed, and listening to what other people are found and the strategy found it and respond to it.
f. Principles guidance
Ie students must be given the opportunity to find (reinvention) guided math knowledge.
1.1.2 Quality Assurance Standards PMRI
To complement the characteristics of RME, the development team PMRI in Quality Assurance Conference held in Yogyakarta on 17-18 April 2009 agreed to set some standards of quality assurance PMRI. Established standards including standards include teacher PMRI, PMRI learning standards, and standards of teaching materials PMRI. The standard can be used and referred to the teachers of mathematics. Here is a question that is related to the standard math teacher.
a. Teacher Standards PMRI
1. Teachers have adequate knowledge and skills of mathematics and PMRI and can apply it in the study of mathematics to create a conducive learning environment.
2. The teacher facilitates students in thinking, discussing and negotiating to encourage initiative and creativity of students.
3. Teachers assist and encourage students to dare to express ideas and finding strategies according to their own problem solving.
4. Teachers manage classroom so as to encourage students to work together and discussed in the context of construction of student knowledge.
5. Teachers with students extract (summarize) the facts, concepts, and principles of mathematics through the process of reflection and confirmation.
b. Standards of Learning According PMRI
1. Learning to meet the demands of the achievement of competency standards in the curriculum.
2. Learning begins with a realistic problem that motivated and helped students learn mathematics.
3. Learning provides students with opportunities to explore a given problem and discuss teachers so that students can learn from each other in order to constructing knowledge.
4. Learning to associate various mathematical concepts to make learning more meaningful and form a complete knowledge.
5. Learning ends with a reflection and confirmation to extract facts, concepts, and principles of mathematics that have been studied and followed by exercises to strengthen comprehension.
c. Standard Teaching Material PMRI
1. Teaching materials are prepared in accordance with the applicable curriculum.
2. Teaching materials using realistic problems to motivate students and help students learn math.
3. Instructional materials containing various mathematical concepts are interrelated so that students gain meaningful mathematical knowledge and intact.
4. Teaching materials containing enrichment materials that accommodate differences in the way and thinking ability of students.
5. The teaching materials are formulated or presented so as to encourage or motivate students to think critically, creatively, innovatively and interact in learning.
1.1.3 Reflection and Assessment in Learning PMRI
In each study, reflection is a main thing to give an overview of the learning process that had gone before. Reflection is an activity with intensive listening back to the learning process, among other subject matter, experience, ideas, suggestions, or a spontaneous reaction in order to understand and grasp the deeper meaning. Thus, it will be able to reveal what has been and is being done. What does it correspond to what people think? With the reflection teachers can determine the learning progress made. Results of reflection can be a picture for teachers in taking action in future activities. The importance of reflection declared Supinah (2009: 78) as follows.
1. For the teacher
Get information about what students are learning and how students learn. In addition, teachers can make improvements in planning and learning on the occasion of the next or future.
2. For students
Improving students' mathematical thinking ability, and also it as well as did the teacher.
About the things that need to be in reflection according Arvold, Turner, and Cooney in Supinah (2009: 79) recommends students to give answers or responses to the questions below.
1. What did I learn today?
2. The difficulty is that I learned today?
3. Which part of mathematics that I like?
4. On the math section where was I having trouble?
Of the teacher, in doing good reflection if it can include methods of teaching, pedagogy, completion exciting and rewarding for him as well as how to manage a good learning atmosphere in the classroom. In RME, assessment not only on the final result, but also on the learning process itself. Ideally, during the learning activities, assessment process was carried out. There are many things that can be used as a means to carry out the assessment. Among them, the ability of students to solve problems by using different strategies, student interaction, discussion during the learning process.
The objective of the assessment to provide an overview of information about the learning process has been implemented and can be also as a tool to aid decision-making process.
De Lange (1987) in Zulkardi (2002: 35) "formulated five guiding principles of assessment or assessment in RME", as follows.
1. The main purpose of testing is to improve the teaching and learning process.
2. The method of assessment should be able to make it easier for students to demonstrate what they know rather than what you do not know.
3. The assessment should operationalize all goals of mathematics education.
4. Quality assessment of mathematics is not determined by the ease of access to objective assessment.
5. The assessment tool should be practical, suitable to the practice of public schools.
In RME, processes and products have an important effect in the assessment so expect a good assessment conducted during the process of interaction as well as their results.
There are several techniques that can be used assessment. Suryanto (2010) provide some alternatives that can be used as a means of assessment, ie.
1. The final results of the students, can be a journal, videos, demonstrations, magazine wall, art, and the results of the construction of mathematical models.
2. The student portfolio is a collection of student work produced by the students. May include images, reports, results of the analysis of a problem, or the process of resolving a problem.
3. Resolution on solving problems or responses to open-ended questions as outlined in the article.
4. The ability to investigate issues related to other subject areas such as general science, social sciences, or the completion of math problems themselves.
5. The student's response to a case, situation, and open issues given teacher.
6. Assessment of the performance of students in both groups or individuals to solve problems.
7. Direct observation of the student in his quest solve a given problem teachers.
8. Interviews were conducted to determine the depth of student understanding of the issues presented.
9. Asking questions can provide opportunities for students to think that the teacher is able to gather information on student understanding.
10. Students are given the opportunity to assess their own ability in learning, adapted to the development that they develop.
2.1.4 Design of Realistic Mathematics Education
Design realistic mathematics learning, as follows.
a. The Purpose
Lange (1995) states that there are three levels in mathematics education objectives are:
(1) Lower level
(2) Middle level
(3) Higher order level
Realistic mathematics learning must include all levels of objectives. Low level is more focused on conceptual and procedural knowledge. Middle and upper level is more focused on problem-solving abilities, argue, communicate, and the formation of critical attitudes of students.
b. Material
Lange (1996) asserts that "Content is the association of real-life activities that are very specific, knowledge and strategies used in the context of the situation". Various contextual matter incorporated in the curriculum at the start from the beginning.
c. Activity
The role of teachers in realistic mathematics learning in the classroom (Lange, 1996; Gravemeijer, 1994) is as facilitator, regulator, translator, and evaluators as the basis of a mathematical process, can be generally described steps the teacher's role as a basic process in mathematics realistic as follows.
1. Give contextual questions on student learning-related topics as a starting point.
2. At the time of the interaction, give students instruction for example, to describe a table on the board, mixing students individually or in small groups in which if a teacher needs help.
3. Provide opportunities for students to compare their answers with his answer in class discussions. The discussion aims to guide students in translating a matter of interpretation of contextual and conclude a more efficient solution than some of the answers are varied.
4. Allow students to find solutions on their own. This means that students are free to make a statement with their ability level, to build experience in knowledge, and played a short answer on the steps they are doing.
5. Give other matter in the same context.
6. In other respects, the role of students in realistic mathematics should work individually or in groups, they should have more confidence in themselves, and they responded with a free production or contribution.
d. Evaluation
Lange (1995) formulated the five principles in the evaluation that can be used as a reference in making the evaluation in realistic mathematics learning, namely:
1. The basic purpose of the test is to improve the quality of learning and teaching. This means that the evaluation should be able to measure the students for learning, not just the provision of information about learning outcomes in the form of value.
2. The method of assessment should be designed in such a way that allows students to describe what they know not reveal what they do not know. It can be held with having an open question, or have a different answer strategy.
3. The test should involve all the goals of mathematics education, low-level thinking processes, medium, and high.
4. The evaluation tool should be practical, so that construction can be prepared test different formats according to the needs and achievement of objectives to be disclosed.
2.1.5 Advantages RME
According Suwarsono (2001: 5) there are some strengths or advantages of realistic mathematics learning, ie.
1. Learning realistic mathematics give students a clear understanding of the relevance of mathematics to everyday life and usability in general for humans.
2. Learning realistic mathematics provide a clear understanding to the students that mathematics is a field of study that is constructed and developed by students not only by so-called experts in the field.
3. realistic mathematics learning gives students a clear understanding that the way of solving a problem or issue should not be single and not necessarily equal to one with the other. Everyone can find or use their own way, so long as that person seriously work on the problems or the problems. Furthermore, by comparing the way the completion of one another by way of settlement, would be gained most appropriate way of settlement, in accordance with the objectives of the problem solving process.
4. Learning realistic mathematics provide a clear understanding to the students that in the study of mathematics, the learning process is the main thing and people have to undergo the process and trying to find their own mathematical concepts with the help of others who know better (eg teachers). Without the willingness to undergo the process yourself, meaningful learning can not be achieved.
1.1.6 Weaknesses RME
The existence of certain requirements in order to excess RME may arise which raises its own difficulties in applying it. These difficulties, ie.
1. It is not easy to change the fundamental outlook on many things, for example regarding the students, teachers and the role of contextual questions or problems, while the change was a condition for the implementation of RME.
2. Search contextual issues that meet the conditions required in realistic mathematics learning is not always easy for each subject studied mathematics students, the more so because such questions must be resolved by a variety of ways.
3. It is not easy for teachers to encourage students to be able to find various ways to solve problems or solve problems.
4. It is not easy for teachers to provide assistance to students in order to perform rediscovery of concepts or principles learned math.
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