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Freudenthal stressed the idea of mathematics as a human activity. Education should give students the “guided” opportunity to “re-invent” mathematics by doing it. This means that in mathematics education, the focal point should not be on mathematics as a closed system but on the activity, on the process of mathematization (Freudenthal, 1968). Later on, Treffers (1978, 1987) formulated the idea of two types of mathematization explicitly in an educational context and distinguished “horizontal” and “vertical” mathematization. In broad terms, these two types can be understood as follows.
 
In horizontal mathematization, the students come up with mathematical tools, which can help to organize and solve a problem located in a real-life situation.
 
Vertical mathematization is the process of reorganization within the mathematical system itself, like, for instance, finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries.
 
Or, in other words (Freudenthal 1991): “horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols.”
 
The transition from horizontal to vertical mathematizing, and facility in vertical mathematzing, are core goals of RME.
Core goals of RME
More information:
Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), pp. 9-36.
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Conferences
(c) Freudenthal Institute US & Freudenthal Institute for Science and Mathematics Education 2013
FIUS: mathematics education, development and research