The development of what is now known as RME started around 1970 in Netherlands. Freudenthal and his colleagues at the Freudenthal Institute laid the foundations. The present form of RME has been mostly determined by Freudenthal’s view on mathematics. He felt mathematics must be connected to reality, stay close to children’s experience and be relevant to society, in order to be of human value. Instead of seeing mathematics as a subject to be transmitted, Freudenthal stressed on the idea of mathematics as a human activity. Mathematics lessons 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.
Later on, Treffers explicitly formulated the idea of two types of mathematization in an educational context; he distinguished between ‘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 organize and solve a problem, set in a real life situation. Vertical mathematization is the process of reorganization within the mathematical system itself, for instance, finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries. Thus horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols. Although this distinction seems to be free from ambiguity, Freudenthal stated that it does not mean that the difference between these two worlds is clear-cut. He also stressed that these two forms of mathematization are of equal value. Furthermore, one must keep in mind that mathematization can occur at different levels of understanding.
Misunderstanding of ‘realistic’
Despite this clear statement about horizontal and vertical mathematization, RME became known as ‘real-world mathematics education’. This was especially true outside the Netherlands, but the same interpretation can also be found within the Netherlands. It must be acknowledged that the name ‘Realistic Mathematics Education’ is somewhat confusing in this respect. The reason, however, why the Dutch reform of mathematics education was called ‘realistic’ is not just because of its connection with the real world, but is related to the emphasis that RME puts on offering the students problem situations which they can imagine. The Dutch translation of ‘to imagine’ is ‘zich REALISEren.’ It is this emphasis on making something real in your mind that gave RME its name. For the problems presented to the students, this means that the context can be one from the real world but this is not always necessary. The fantasy world of fairy tales and even the formal world of mathematics can provide suitable contexts for a problem, as long as they are real in the student’s mind.
