### RME, History and founding principles

This text is based on the NORMA-lecture, by Marja van den Heuvel-Panhuizen, held in Kristiansand, Norway on 5-9 June 1998

The development of what is now known as RME started almost thirty years ago. The foundations for it were laid by Freudenthal and his colleagues at the former IOWO, which is the oldest predecessor of the Freudenthal Institute. The actual impulse for the reform movement was the inception, in 1968, of the Wiskobas project, initiated by Wijdeveld and Goffree. The present form of RME is mostly determined by Freudenthal’s (1977) view about mathematics. According to him, mathematics must be connected to reality, stay close to children and be relevant to society, in order to be of human value. Instead of seeing mathematics as subject matter that has to be transmitted, 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.

In short, one could say — quoting 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.” Although this distinction seems to be free from ambiguity, it does not mean, as Freudenthal said, that the difference between these two worlds is clear cut. Freudenthal also stressed that these two forms of mathematization are of equal value. Furthermore one must keep in mind that mathematization can occur on different levels of understanding.

### Misunderstanding of “realistic”

Despite of this overt statement about horizontal and vertical mathematization, RME became known as “real-world mathematics education.” This was especially the case outside The Netherlands, but the same interpretation can also be found in our own country. It must be admitted, 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 the 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 the verb “to imagine” is “zich REALISEren.” It is this emphasis on making something real in your mind, that gave RME its name. For the problems to be presented to the students this means that the context can be a real-world context but this is not always necessary. The fantasy world of fairy tales and even the formal world of mathematics can be very suitable contexts for a problem, as long as they are real in the student’s mind.

### The realistic approach versus the mechanistic approach

The use of context problems is very significant in RME. This is in contrast with the traditional, mechanistic approach to mathematics education, which contains mostly bare, “naked” problems. If context problems are used in the mechanistic approach, they are mostly used to conclude the learning process. The context problems function only as a field of application. By solving context problems the students can apply what was learned earlier in the bare situation.

In RME this is different; Context problems function also as a source for the learning process. In other words, in RME, contexts problems and real-life situations are used both to constitute and to apply mathematical concepts.

While working on context problems the students can develop mathematical tools and understanding. First, they develop strategies closely connected to the context. Later on, certain aspects of the context situation can become more general which means that the context can get more or less the character of a model and as such can give support for solving other but related problems. Eventually, the models give the students access to more formal mathematical knowledge.

In order to fulfil the bridging function between the informal and the formal level, models have to shift from a “model of” to a “model for.” Talking about this shift is not possible without thinking about our colleague Leen Streefland, who died in April 1998. It was he who in 1985* detected this crucial mechanism in the growth of understanding. His death means a great loss for the world of mathematics education.

Another notable difference between RME and the traditional approach to mathematics education is the rejection of the mechanistic, procedure-focused way of teaching in which the learning content is split up in meaningless small parts and where the students are offered fixed solving procedures to be trained by exercises, often to be done individually. RME, on the contrary, has a more complex and meaningful conceptualization of learning. The students, instead of being the receivers of ready-made mathematics, are considered as active participants in the teaching-learning process, in which they develop mathematical tools and insights. In this respect RME has a lot in common with socio-constructivist based mathematics education. Another similarity between the two approaches to mathematics education is that crucial for the RME teaching methods is that students are also offered opportunities to share their experiences with others.

In summary, RME can be described by means of the following five characteristics (Treffers, 1987):

- The use of contexts.
- The use of models.
- The use of students’ own productions and constructions.
- The interactive character of the teaching process.
- The intertwinement of various learning strands.

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