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Sabtu, 26 Maret 2011

Realistic Mathematics Education

In the study of mathematics during this, the real world only to be a place to apply the concept. Students having trouble in math class. As a result, students are less appreciate or understand math concepts, and students had difficulty applying mathematics in everyday life.
One of the math-oriented learning experience everyday  (mathematize of Everyday experience) and apply mathematics in everyday life is a learning Realistic Mathematics (MR).

Characteristics of RME is to use the context of "real world", models, production and construction students, interactive, and linkages (intertwinment). Related to this, this paper aims to describe theoretically realistic mathematics learning, learning the implementation of MR, and MR links between learning with understanding. Realistic Mathematics Education provides the opportunity for students to rediscover and reconstruct the concepts of mathematics, so students have a strong understanding of mathematical concepts. Thus, learning Realistic Mathematics will have a very high contribution to the understanding of students.
Keywords: mathematics realistic, real world, the reconstruction of mathematical concepts, models, interactive.


1. Preliminary

One characteristic of mathematics is to have objects that are abstract. This abstract nature causes many students have difficulty in math. Student mathematics achievement, both nationally and internationally has not been encouraging. Third International Mathematics and Science Study (TIMSS) reported that the average mathematics score of students' level 8 (junior II level), Indonesia is far below the average mathematics scores of international students and is at rank 34 out of 38 countries (TIMSS, 1999). The low student mathematics achievement of students caused by factors having problems comprehensively or partially in mathematics.

In addition, meaningful mathematics learning yet, so that student understanding of the concept is very lemah.Jenning and Dunne (1999) says that, most students have difficulty in applying mathematics to real life situations. Another thing that causes the difficulty of mathematics for students is that learning math is less meaningful. Teacher in classroom learning does not associate with the scheme which has been owned by the students and the students lack the opportunity to rediscover and construct their own mathematical ideas. Linking real-life experiences of children with mathematical ideas in learning in the classroom is necessary for meaningful learning (Soedjadi, 2000; Price, 1996; Zamroni, 2000).

According to Van de Henvel-Panhuizen (2000), when children learn mathematics apart from their daily experience, the child will quickly forget and can not apply math Based on the above opinion, the learning of mathematics in the classroom emphasis on the linkages between the concepts of mathematics with experience children daily. In addition, the need to apply the mathematical concepts that have been re-possessed child in everyday life or in other fields are essential.

One of the math-oriented learning experience everyday (mathematize of Everyday experience) and apply mathematics in everyday life is a learning Realistic Mathematics (MR).

MR learning first developed and implemented in the Netherlands and is considered very successful to develop students' understanding.

This paper aims to describe theoretically realistic mathematics learning, learning the implementation of MR, and MR links between learning with understanding.

2. Theory Study

2.1 Realistic Mathematics Education (RME)

Realistic Mathematics Education (RME) is a theory of teaching and learning in mathematics education. RME theory was first introduced and developed in the Netherlands in 1970 by the Freudenthal Institute. This theory refers to Freudenthal opinion which says that mathematics should be associated with reality and mathematics is a human activity. This means that math should be close to children and relevant to real life everyday. Mathematics as a human activity means that people must be given the opportunity to rediscover the ideas and concepts of mathematics with the guidance of adults (Gravemeijer, 1994). This work is done through exploring a variety of situations and problems "realistic". Realistic in this case meant not refer to reality but to something that can be imagined by the students (Slettenhaar, 2000). The principle of rediscovery can be inspired by the informal resolution procedures, while the process of rediscovery of the concept.

Two types of  formulated by Treffers (1991), namely  horizontal and vertical.

Examples of horizontal  is the identification, formulation, and penvisualisasi problem in different ways, and pentransformasian real world problems into mathematical problems.

Examples of vertical  is a representation of the relations in the formula, repair and adjustment of mathematical models, the use of different models, and penggeneralisasian. Both types of these balanced attention, since both has the same value (Van den Heuvel-Panhuizen, 2000).

Based on the horizontal and vertical, approaches in mathematics education can be divided into four types, namely mechanistic, emperistik, strukturalistik, and realistic.

Mechanistic approach is the traditional approach and is based on what is known from his own experience (starting from simple to more complex). In this approach the man regarded as the engine. Both types of  not used.

Emperistik approach is an approach where math concepts are not taught, and expected students to discover through horizontal.

Strukturalistik approach is an approach that uses formal system, such as teaching the sum of how long should be preceded by the value of the place, so that one concept is achieved through vertical.

Realistic approach is a realistic approach that uses problems as a starting base of learning. Through horizontal and vertical activity is expected students to discover and construct mathematical concepts.

2.2 Characteristics of RME

Characteristics of RME is to use: the context of "real world", models, production and construction students, interactive, and linkages (intertwinment) (Treffers, 1991; Van den Heuvel-Panhuizen, 1998).

2.2.1 Using Context "Real World"

The following figure shows two processes that form a cycle in which "real world" not only as a source, but also as a place to apply the math again. Figure 1 Concept (De Lange, 1987) In RME, the learning begins with contextual problems ("real world"), allowing them to use previous experience directly. Penyarian process (core) of the appropriate concept of the real situation is stated by De Lange (1987) as a conceptual. Through abstraction and formalization of the students will develop a more complete concept. Then, students can apply mathematical concepts to new areas of the real world (applied mathematization). Therefore, to bridge the mathematical concepts with everyday experiences children need to be considered matematisi everyday experience (mathematization of Everyday experience) and the application of math in everyday (Cinzia Bonotto, 2000)

2.2.2 Using models

The term model associated with the model situation and the mathematical model developed by the students themselves (self-developed models). The role of self-developed models is a bridge for students from the real situation to situation or from abstract mathematics informal to formal mathematics. This means that students create their own models in solving problems. The first is a model situation that is close to the real world of students. Generalization and formalization of the model will change to the model-of the problem. Through mathematical model-of reasoning will be shifted into the model-for a similar problem. In the end, will be a formal mathematical model.

2.2.3 Using the Production and Construction

Streefland (1991) stressed that by making "free production" students are encouraged to reflect on what they consider important part in 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.

2.2.4 Using the Interactive

Antarsiswa interaction with teachers is fundamental in the RME. Explicitly, the forms of interaction in the form of negotiation, explanation, justification, agree, disagree, question or reflection is used to achieve formal forms of informal forms of student.

2.2.5 Using the linkage (Intertwinment)

In RME integration of the units of mathematics is essential. If we ignore them in the learning relationship with other fields, it will affect the problem-solving. In applied mathematics, usually required a more complex knowledge, and not just arithmetic, algebra, or geometry but also in other fields.

3. Discussion

3.1 Realistic Mathematics (MR)

Realistic Mathematics (MR) which is meant in this case is that school mathematics performed by placing the realities and experiences of students as a starting point of learning. Realistic problems are used as a source of the emergence of mathematical concepts or knowledge of formal mathematics. MR in the classroom learning-oriented characteristics of RME, so students have the opportunity to rediscover 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.

Learning is very different from learning mathematics during this that tends to give information-oriented and ready to use mathematics to solve problems.

Because of realistic mathematics using realistic problems as a starting base of learning the problem situation really needs to be put contextual or in accordance with the student experience, so that students can solve problems with informal ways through horizontal. Informal ways shown by the students used as inspiration mathematical aspects of concept formation or enhanced through vertical. Through a process of horizontal-vertical expected students to understand or discover mathematical concepts (formal mathematical knowledge.)

3.2 Lessons Realistic Mathematics (MR)

According to the view of Constructivist Learning constructivist view of mathematics is to provide opportunities for students to construct mathematical konsep-konsep/prinsip-prinsip own abilities through a process of internalization. Teachers in this role as facilitator.

According to Davis (1996), constructivist view of mathematics learning-oriented:

(1) the knowledge constructed in the mind through a process of assimilation or accommodation,

(2) of the work of mathematics, each step students are confronted with what,

(3) new information should be associated with the experience of the world through a logical framework to transform, organize, and interpret their experiences, and

(4) a center of learning is how students think, not what they say or write.

This constructivist criticized by Vygotsky, which states that the student to construct a concept needs to consider the social environment. Constructivism is by Vygotsky called social constructivism (Taylor, 1993; Wilson, Teslow and Taylor, 1993; Atwel, Bleicher & Cooper, 1998).

There are two important concepts in the theory of Vygotsky (Slavin, 1997), the Zone of Proximal Development (ZPD) and scaffolding.

Zone of Proximal Development (ZPD) is the distance between actual developmental level is defined as the ability to independently problem solving and the level of potential development is defined as the ability of problem solving under adult guidance or in collaboration with more capable peers.

Scaffolding is the provision of assistance to students during the early stages of learning, and reduce support and provide the opportunity to take over greater responsibility after he or she can do it (Slavin, 1997). Scaffolding is the help given to students to learn and solve problems. Such assistance may include guidance, encouragement, warnings, outlined the problem into solving steps, providing examples, and other measures that allow students to learn independently.

The approach refers to social constructivism (philosophy of social constructivism) is called a social constructivist approach. Philosophy of social constructivist view of mathematics is not absolute truth and identify mathematics as a result of solving problems and filing problem (problem posing) by man (Ernest, 1991). In learning mathematics, Cobb, Yackel and Wood (1992) call with constructivism social (socio-constructivism). Students interact with teachers, with other students and based on informal experience students develop strategies to respond to a given problem. Characteristics of socio constructivist approach is well suited to the characteristics of RME.

The concept of ZPD and Scaffolding in social constructivist approach, in a study called the rediscovery of MR guidance (guided Reinvention). According Graevenmeijer (1994) although the two approaches have similarities but the two approaches be developed separately.

The differences are both socio-constructivist approach to learning is the approach of a general nature, while learning the MR is a special approach that is only in mathematics.

3.3 How Learning Implementation MR?

To give an idea about the implementation of the MR study, the following is an example of learning fractions in elementary school (SD). Fractions in Elementary interpreted as part of the whole. This interpretation refers to the division of units into the part of the same size. In this case as a framework for students is an area, length, and volume models. Part of the whole also can be interpreted on the idea pempartisian a set of discrete objects.

In the study, before students enter the formal system, students first brought to the "situation" informal. For example, learning fractions can be preceded by division into equal parts (eg the division of the cake) so there is no stepping informal knowledge of children with math concepts (formal mathematical knowledge.)

Once students understand the division into equal parts, newly introduced term fractions. This is very different from conventional learning (not MR) in which students since the beginning of term fed with fractions and some types of fractions.

Thus, learning begins with the phenomenon of MR, then the student with the help of teachers are given the opportunity to rediscover and construct their own concept. After that, applied to everyday problems or in other areas (see figure 02).


Figure 2 The discovery and construction of concept
(Adapted from Van Reeuwijk, 1995)

MR 3.4 Linkages between Learning with Understanding

If we look at the teachers in teaching mathematics ever uttered the word "how, what do you understand?" Students are usually answered quickly understand or have. Students often complain as follows, "Sir ... at the time in my class understand the explanation you, but when I got home I forgot", or "Sir ... at the time in my class understand the example that you gave, but I can not solve problems exercise "What are experienced by students in the illustration above shows that students do not understand or do not have the conceptual knowledge. Students who understand the concept or have conceptual knowledge to rediscover the concept that they forget.

Mitzel (1982) says that, student learning outcomes are directly influenced by the experience of students and internal factors. Students' learning experiences are influenced by the performance of teachers. When students in meaningful learning or place links between new information and the network representation, the students will gain an understanding. Develop an understanding of mathematics teaching purposes. Because without understanding one can not apply the procedures, concepts, or processes.

In other words, mathematics is understandable if the mental representation is part of a network representation (Hiebert and Carpenter, 1992). Generally, since the children of people already familiar with mathematical ideas. Through his experiences in daily life they develop ideas that are more complex, for example about numbers, patterns, shapes, data, size, etc.. Children prior to school to learn math ideas naturally. This shows that students come to school not with your head "blank" that is ready to be filled with anything.

Learning in school will be more meaningful if teachers relate to what has been known to the child. Student understanding of mathematical ideas can be built through the school, if they are actively linked with their knowledge.

Hanna and Yackel (NCTM, 2000) says that learning with understanding can be enhanced through classroom interaction. Conversation classes and social interaction can be used to introduce the linkages between ideas and organize knowledge back.

MR Learning provides the opportunity for students to rediscover and to construct mathematical concepts based on the realistic problems that are given by the teacher. Realistic situation in the problem allows students to use informal ways to resolve the problem. Informal ways students who are student production plays an important role in the rediscovery and construction of the concept. This means that the information given to students has been associated with the scheme (network representation) of the children. Through classroom interaction scheme child relationship will be stronger so that student understanding of concepts that they themselves become strong construction. Thus, learning the MR will have a very high contribution to the understanding of students.

4. Conclusion and Suggestions

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 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 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. In accordance with the above conclusion, it is recommended:

(1) to an expert or a lover of mathematics education to research-oriented research on learning in order to obtain global MR MR theory of learning appropriate to the social culture in Indonesia and

(2) to teachers of mathematics to try to implement the MR is gradually learning, for example, began by providing realistic problems to motivate students to express opinions.

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