Realistic Mathematics Education
work in progress
This text is based on the NORMA-lecture, by Marja van den Heuvel-Panhuizen,
held in Kristiansand, Norway on 5-9 June 1998
RME in brief
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
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
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.
In horizontal mathematization, the students come up with mathematical
tools which can help to organize and solve a problem located in a real-life
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
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
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.
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* 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).