## Marja van den Heuvel-Panhuizen
## Realistic Mathematics Education as work in progress
#### Summary
*This lecture addresses several “progress” issues related to the Dutch approach to mathematics education, called “Realistic Mathematics Education” (RME). The most important of these issues is the way in which RME facilitates the progress of children’s understanding in mathematics. This topic forms the heart of the lecture. Attention is paid to both the micro-didactic and the macro-didactic perspective of the students’ growth. Progress in achievements, as the result of this learning, is the next progress issue to be dealt with. Finally, the spotlights are turned towards the developments within RME itself. The general focus in the lecture is on primary school mathematics education.* ## 1 Introduction #### RME in brief Realistic Mathematics Education, or RME, is the Dutch answer to the need, felt worldwide, to reform the teaching of mathematics. The roots of the Dutch reform movement go back to the beginning of the seventies, when the first ideas for RME were conceptualized. It was a reaction to both the American “New Math” movement, which was likely to flood our country in those days, and the then prevailing Dutch approach to mathematics education, which often is labeled as “mechanistic mathematics education.” Since the early days of RME much design work connected to developmental research (or design 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. It looks a superfluous statement, but it is not. Again and again, too optimistic thoughts are heard about educational innovations. Our experience is that reforms in education take time. 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 conquered completely — 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. Inherent to RME, with its founding idea of mathematics as a human activity, is that it can never be considered a fixed and finished theory of mathematics education. ^ This self-renewing feature of RME was an important reason for choosing “work in progress” as the title for this lecture. But there were more reasons for this choice. The title also refers to another significant characteristic of RME, namely its focusing on the growth of the children’s knowledge a nd understanding of mathematics. The way in which RME continually works on the progress of children is the first progress issue to be dealt with in this lecture. This progress work is distinguished in two levels of working on the mathematical development. Attention is paid to both the micro-didactic perspective and the macro-didactic perspective of the students’ growth. The micro-didactic perspective clarifies how within the context of one or two lessons shifts in comprehension and abilities can happen. In this process, models which originate from context situations and which function as bridges to higher levels of understanding have a key role. The macro-didactic perspective deals with the progress in understanding over a longer period of time. The focus here is on learning-teaching trajectories – including the attainment targets to be reached at the end of primary school and the landmarks along the route – that serve as a longitudinal framework for teaching mathematics. The coherence between the various levels of mathematical understanding that is made apparent in this trajectory description plays a key role in stimulating students’ growth. A following progress issue has to do with the students’ achievements in mathematics. The question is whether RME brought Dutch primary students to the top level of mathematics achievements. Although the TIMSS results and results from other comparative studies are suggesting this, there are also arguments against. Finally, the lecture deals with the progress in the RME approach to mathematics education. Although this approach is already some thirty years old it is still “under construction.” ^ #### History and founding principles As I said already, almost thirty years ago the development of what is now known as RME started. Freudenthal and his colleagues laid the foundations for it at the former IOWO, which is the earliest 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 has been 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 — and here I am 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 (ibid.) said, that the difference between these two worlds is clear-cut. Freudenthal (ibid.) 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. ^ 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. ^ In any way, the use of context problems is very significant in RME. This is in contrast with the traditional, mechanistic approach to mathematics education, where programs mostly only contain problems with bare numbers. If in the mechanistic approach context problems are used, 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 format. In RME this is different. Here, context problems function also as a source for the learning process. In other words, in RME, context 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 give support for solving other but related problems. Eventually, the models give the students access to more formal mathematical knowledge. In order to fulfill 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 1998. It was he who in 1985 detected this crucial mechanism in the growth of understanding.^{1} 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 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.
This concludes a brief overview of the characteristics of RME. Now, I will continue with the issue of progress in understanding and will start with the micro-didactic perspective. |