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

Realistic Mathematics Education, or RME, is the Dutch answer to the world-wide felt need to reform the teaching and learning of mathematics. The roots of the Dutch reform movement go back to the early seventies when the first ideas for RME were conceptualized. It was a reaction to both the American "New Math" movement that was likely to flood our country in those days, and to the then prevailing Dutch approach to mathematics education, which often is labeled as "mechanistic mathematics education."

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."

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.

**RME in brief**

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.

**"Progress" issues to be dealt with**

This self-renewing feature of RME
explains why it is work in progress. But, there are at least two more aspects.
One significant characteristic of RME, is the focus on the growth of the
students’ knowledge and understanding of mathematics. RME continually works
toward the progress of students. In this process, models which originate from
context situations and which function as bridges to higher levels of
understanding play a key role. Finally, considering the TIMSS results, it seems
that RME really can elicit progress in achievements.

**RME, History and founding principles**

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.

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.

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).

You can see this article from http://www.fisme.science.uu.nl/en/rme/

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