Understanding of RME is one of the sub titles of the Prof. Zulkardi’s thesis that titled *Developing A Learning Environment On Realistic Mathematics Education For Indonesian Student Teachers. *This thesis was written in 2002

Prof. Zulkardi explains that the philosophy and characteristics of RME consist of views on what mathematics is, how pupils learn mathematics, and how mathematics should be taught. It is strongly influenced by Hans Freudenthal’s concept of mathematics as a human activity that means the students aren’t passive recipients of ready-made mathematics, but they should be guided to discover and reinvent mathematics by doing it themselves.

According to Van Hiele, that was quoted by Prof. Zulkardi, there are 3 levels in learning mathematics. First, students can manipulate the known characteristics of a pattern that is familiar to them. Second, students learn to manipulate the interrelatedness of the characteristics. Third, students start manipulating the intrinsic characteristics of relations. Based on that levels, RME starts from the first level. It is different with the traditional instruction that is inclined to start from second or third level. In order to start from the first level, RME starts from a meaningful contextual problem. Furthermore, by guided reinvention through progressive mathematization, students are guided didactically to progress efficiently from one level to another level of thinking.

Five tenets (characteristics) of RME such as :

**1. ****The use of context in phenomenological exploration**

RME uses real problem and contextual situation as the starting point of learning. The phenomena by which mathematics concepts appear in reality should be the source of concept formation. The process of conceptual and applied mathematization according de Lange is showed below.

Figure 1. Conceptual and applied mathematization

**2. ****The use of models or bridging by vertical instruments**

Four levels of models that are developed by the students in RME according to Gravemeijer are described below.

Figure 2. Levels of models in RME

**3. ****The use of pupils’ own creations and contributions**

Students create concrete things and free productions, such as write an essay, do an experiment, collect data and draw conclutions, design exercises or test for other students.

**4. ****The interactive character of the teaching process or interactivity**

Interaction between students and students and teachers is very important. Students are engaged in explaining, justifying, agreeing and disagreeing, questioning alternatives and reflecting.

**5. ****The intertwining of various mathematics strands or units**

In RME, the integration of mathematical strands or units that is often called the holistic approach is very important. An intertwining of learning strands is exploited in solving real life problems.

These characteristics can be used as a study guideline both in the process of adapting RME curriculum materials to the Indonesian context and in the process of pre-service training for student teachers in teacher education.

Three levels of construction that was developed by Streefland in realistic mathematics lesson materials are

**1. ****The classroom level**

At this level, instructional activities are designed based on all the characteristics of RME. How all the characteristics of RME are pictured in a model for designing RME curriculum materials is showed below.

Figure 3. A model for designing RME curriculum materials

**2. ****The course level/instructional sequence**

After the materials from the classroom levels were tried out and revised, students are expanded to other contents and contexts in order to develop the instructional sequence of that topic.

**3. ****The theoretical level**

A theory in the form of a local theory for a specific area of learning is constructed, revised, and tested again during additional cyclic developments.

RME exemplary lesson materials refer to learner materials and teacher guides that are used as a learning trajectory for teacher and consist of content materials, learner and teacher activities, and assessment.

RME materials are related to real-life activities and uses contextual problems that should be appropriate for the goals of the particular mathematics topic. Three levels of goals in mathematics education that were characterized by De Lange are lower level, middle level, and higher-order level. While the goal of the traditional program were classified as lower level that are based on formula skills, simple algorithms and definitions, RME goal includes middle and higher level goal (reasoning skills, communication and the development of a critical attitude).

While the RME teacher in the classroom as a facilitator, an organizer, a guide, and an evaluator, the students work individually or in a group.

De Lange formulated 5 guiding principles of assessment in RME, such as

- The primary purpose of testing is to improve learning and teaching
- Methods of assessment should enable the pupils to demonstrate what they know rather than what they don’t know
- Assessment should be operationalize all of the goals of mathematics education
- The quality of mathematics assessment is not determined by its accessibility to objective scoring.
- The assessment tools should be practical

Assessment can be conducted in the classroom using strategies both formative and summative.

Based on that explanation, we can conclude that to apply realistic teaching and learning in the class, we have to refer to all of the RME characteristics.