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

Since the early days of RME much development work connected to developmental research has been carried out. If anything is to be learned from the Dutch history of the reform of mathematics education, it is that such a reform takes time. This sounds like a superfluous statement, but it is not. Again and again, too optimistic thoughts are heard about educational innovations. The following statement indicates how we think about this: The development of RME is thirty years old now, and we still consider it as "work under construction."

That we see it in this way, however, has not only to do with the fact
that until now the struggle against the mechanistic approach to mathematics
education has not been completely conquered— especially in classroom practice
much work still has to be done in this respect. More determining for the
continuing development of RME is its own character. It is inherent to RME,
with its founding idea of mathematics as a human activity, that it can
never be considered a fixed and finished theory of mathematics education.

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.

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.

This concludes a brief overview of the characteristics of RME.

Click for more about:

- progress in understanding (in primary school)
- RME and assessment
- trends and (current) issues in RME
- a selection of relevant references

* Streefland (1985). Later on, this idea of a shift in models became a significant element in RME thinking about progress in students’ understanding of mathematics (see Streefland, 1991; Treffers, 1991; Gravemeijer, 1994; Van den Heuvel-Panhuizen, 1995).