The learning process starts from contextual problems. Using activities in the horizontal mathematization, for instance, the student gains an informal or a formal mathematical model. By implementing activities such as solving, comparing and discussing, the student deals with vertical mathematization and ends up with the mathematical solution. Then, the student interprets the solution as well as the strategy which was used to another contextual problem. Finally, after the student has used the mathematical knowledge. Treffers classifies mathematics These classifications are described clearly by Freudenthal (1991):

*mechanistic*, 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*, 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.

Type |
HorizontalMathematization |
Verticalmathematization |

Mechanistic | – | – |

Empiristic | + | – |

Structuralist | – | + |

Realistic | + | + |

*Table 1: Four types of mathematics education (Freudenthal,1991)*

The realistic approach is different with the other approaches in the term of mathematizing since it covers both important things so that students can study mathematics meaningfully.

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