This article will present about key principles of two approaches in teaching and learning mathematics, namely Realistic Mathematics Education (RME) and the approach developed by National Council of Teacher of Mathenatics (the NCTM approach).

There are three key principles of Realistic Mathematics Education that can be clearly seen as heuristics for instructional design as follows.

- Guided reinvention and progressive mathematizing

The teacher should give chance to the students to do experience process to invent mathematics by using historical mathematics or informal solution procedure.

2. Didactical phenomenology

Situation where a given mathematical topic is applied are to be investigated for two reasons: to allow the kind of application that must be anticipated in instruction be seen and also to consider their suitability as point of impact for a process of progressive mathematization.

3. Self developed models

The students develop model to process information by themselves to bridge the gap between informal knowledge and formal mathematics.

Moreover, there are six key principles of NCTM approach to mathematics education as follows:

1. The equity principle

Excellence in mathematics education requires equity, high expectation and support for all students.

2. The curriculum principle

Curriculum is more than collection of activities, it has to be coherent and focused on important mathematics.

3. The teaching principle

Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

4. The learning principle

Students have to learn mathematics with understanding by actively construct their knowledge through do experience

5. The assessment principle

Assessment should support learning process

6. The technology principle

Technology can enhance students ability in learning mathematics, it is essential property in learning mathematics.

References:

Gravemeijer, Koeno. 1994. *Developing Realistic Mathematics Education.* Utrecht: Freudenthal Institute.* *

Suwarsono. 2008. *Realistic Mathematics Education and NCTM Approach to Mathematics Education: Similarities and Differences. *Yogyakarta: Widya Dharma.

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