# Theory of RME

Realistic Mathematics Education (RME) is a domain specific-instruction theory for mathematics education (e.g., Treffers, 1987; De Lange, 1987; Streefland, 1991; Gravemeijer, 1994a; Van den Heuvel-Panhuizen, 1996; Heuvel-Panhuizen, 2003). It is a Dutch answer to reform the teaching of mathematics. According to Van den Heuvel-Panhuizen (1996), although still under development, and not yet entirely implemented in the classroom practice, the reform of mathematics education begun at that time has left its mark upon today’s primary school mathematics education. More than three-quarters of the Dutch primary schools now use a mathematics textbook that was inspired to a greater or lesser degree by this reform movement.

The theory is based on the idea of Hans Freudenthal (1977)  that mathematics must be connected to reality, stay close to children and should be relevant to society (Van Den Heuvel-Panhuizen, 2003). Hence, the use of realistic contexts become the determining characteristic of this approach. The contexts are not necessarily situations where mathematics is applied to real world problems, because the most important thing is that they allow students to take ownership of the mathematics. Puzzles, fictitious, some kinds of fairy tales and even formal mathematics can all provide suitable contexts, as long as they are real in the students’ mind.

Later on, according to Treffers (1978, 1987), mathematizing is distinguished into two types which are ‘horizontal’ and ‘vertical’ mathematizing. The two types have totally different characteristic. In horizontal mathematizing, the students come up with mathematical tools to help organize and solve a problem located in a real-life situation. On the other hand, vertical mathematizing is the process of a variety of reorganizations and operations within the mathematical system itself. In other words, Freudenthal (1991) explained that horizontal mathematizing involves going from the world of life into the world of symbols, while vertical mathematizing means moving within the world of symbols. Finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries is implicit in vertical mathematizing. Freudenthal emphasized, however, that the differences between these two worlds are far from clear cut. In addition, in his eyes, the two forms of mathematizing were of equal value and he stressed the fact that both activities could take place on all levels of mathematical activity. In other words, even on the level of counting activities, for example, both forms may occur. The concept of horizontal and vertical mathematizing is one of the salient features of the RME teaching methods. It contains, in fact, all of the important aspects of the RME educational theory.

Treffers classifies mathematics education into four types with regard to horizontal and vertical mathematization. These classifications are described clearly by Zulkardi (2010) after Freudenthal (1991):

• Mechanistic approach, or ‘traditional approach’, is based on drill-practice and patterns, which treat the person like a computer or a machine (mechanic). It means the activities of students in this approach are based on memorizing a pattern or an algorithm. The errors will be occurred if the students are faced with other problems that are different from the one they have memorized. In this approach, both horizontal and vertical mathematization are not used.
• Empiristic approach, the world is a reality, in which students are provided with materials from their living world. This means students are faced with the situations in which they have to do horizontal mathematization activities. However, they are not prompted to the extended situation in order to come up with a formula or a model. Treffers (1991) pointed out that this approach, in general, it is one that is not taught .
• Structuralist approach, or ‘New Math approach’ that is based on set theory, flowchart and games that are kinds of horizontal mathematization but they are stated from an ‘ad hoc’ created world, which had nothing in common with the learner’s living world.
• Realistic approach, a real-world situation or a context problem is taken as the starting point of learning mathematics. And then it is explored by horizontal mathematization activities. This means students organize the problem, try to identify the mathematical aspects of the problem, and discover regularities and relations. Then, by using vertical mathematization students develop mathematical concepts.

There are five tenets for realistic mathematics education defined by Treffers (1987) in Zulkardi (2010) that are described in the following ways:

1. Phenomenological exploration. As the first instructional activity, a concrete context is used as the base of mathematical activity. The mathematical activity is not started from a formal level but from a situation that is experientially real for students.
2. Using models and symbols for progressive mathematizing. The second tenet of RME is bridging from a concrete level to a more formal level by using models and symbols. Students’ informal knowledge as the result of experience-based activities needs to be developed into formal knowledge.
3. Using students’ own construction. The freedom for students to use their own strategies could direct to the emergence of various solutions that can be used to develop the next learning process. The students’ strategies in each activity were discussed in the following class discussion to support students’ acquisition of the basic concepts of the topic which is learned.
4. Interactivity. The learning process of students is not merely an individual process, but it is also a social process. The learning process of students can be shortened when students communicate their works and thoughts in the social interaction emerging in the classroom.
5. Intertwinement. Intertwinement suggests integrating various mathematics topics in one activity. Pupil should develop an integrated view of mathematics as well as the flexibility to connect to different sub-domain and/or to other  disciplines.

According to Gravemeijer (1994, 1997) which is mentioned by Fauzan (2002) 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 emergent models..

1. Guided reinvention through progressive mathematization. This implies that in the teaching learning process students should be given the opportunity to build their own mathematical knowledge on the basis of such a learning process.
2. 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.
3. Self developed models or emergent models. 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).

REFERENCES:

Fauzan, A. 2002. Applying Realistic Mathematics Education (RME) in Teaching Geometry in Indonesian Primary Schools. Thesis University of Twente, Enschede.

Van den Heuvel-Panhuizen, M. (1996). Assesment and Realistic Mathematics Education. Utrecht: CD-b Press / Freudenthal Institute, Utrecht University. New Theory: Realistic Mathematics Education.

Van den Heuvel-Panhuinzen, M. 2003. The Didactical Use of Models in Realistic Mathematics Education: An Example from A Longitudinal Trajectory on Percentage. Educational Studies in Mathematics 54: 9-35.

Zulkardi. 2010. How to Design Mathematics Lesson Based on The Realistic Approach.