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.

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.

(c) Freudenthal Institute US & Freudenthal Institute for Science and Mathematics Education 2013

FIUS: mathematics education, development and research