**History and basic philosophy**

# The development of what is now known as RME started around 1970 in Netherlands. Freudenthal and his colleagues at the Freudenthal Institute laid the foundations. The present form of RME has been mostly determined by Freudenthal’s view on mathematics. He felt mathematics must be connected to reality, stay close to children’s experience and be relevant to society, in order to be of human value. *Instead of seeing mathematics as a subject to be transmitted, Freudenthal stressed on the idea of mathematics as a human activity*. Mathematics lessons 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.*

# Later on, Treffers explicitly formulated the idea of two types of mathematization in an educational context; he distinguished between ‘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 organize and solve a problem, set in a real life situation. Vertical mathematization is the process of reorganization within the mathematical system itself, for instance, finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries. Thus *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, Freudenthal stated that it does not mean that the difference between these two worlds is clear-cut. He also stressed that these two forms of mathematization are of equal value. Furthermore, one must keep in mind that mathematization can occur at different levels of understanding.

** ****Misunderstanding of ‘realistic’**

Despite this clear statement about horizontal and vertical mathematization, RME became known as ‘real-world mathematics education’. This was especially true outside the Netherlands, but the same interpretation can also be found within the Netherlands. It must be acknowledged that 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 because of its 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 ‘to imagine’ is ‘zich REALISEren.’ It is this emphasis on making something real in your mind that gave RME its name. For the problems presented to the students, *this means that the context can be one from the real world but this is not always necessary*. The fantasy world of fairy tales and even the formal world of mathematics can provide suitable contexts for a problem, *as long as they are real in the student’s mind*.

**How arithmetic is taught in primary school**

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RME reflects a certain view on mathematics as a subject, on how children learn mathematics and on how mathematics should be taught. These views can be characterized by the following six principles. Some of them originate more from the point of view of *learning* and some are more closely connected to the *teaching* perspective. The list below is a mix of principles each reflecting a part of the identity of RME.

**1. ****Activity principle**

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The idea of mathematization clearly refers to the concept of mathematics as an activity, which according to Freudenthal can best be learned by doing. *The students, instead of being receivers of ready-made mathematics, are treated as active participants in the educational process*, in which they develop all sorts of mathematical tools and insights by themselves. According to Freudenthal, using scientifically structured curricula, in which students are confronted with ready-made mathematics, is an ‘anti-didactic inversion.’ It is based on the false assumption that the results of mathematical thinking, placed in a subject-matter framework, can be transferred directly to the students. The activity principle means that students are confronted with problem situations in which, for instance, they can produce fractions and gradually develop an algorithmic way of multiplication and division, based on an informal way of working. In relation to this principle, *‘own productions’* play an important role in RME.

**2. ****Reality principle**

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As in most approaches to mathematics education, RME aims at enabling students to apply mathematics. The overall goal of mathematics education is that students must be able to use their mathematical understanding and tools to solve problems. This implies that they must learn *‘mathematics so as to be useful’*. In RME, however, this reality principle is not only recognizable at the end of the learning process in the area of application, reality is also conceived as a source for learning mathematics. Just as mathematics arose from the mathematization of reality, so must learning mathematics also originate in mathematizing reality. Even in the early years of RME it was emphasized that if children learn mathematics in an isolated fashion, divorced from their experiences, it will quickly be forgotten and the children will not be able to apply it. Rather than beginning with certain abstractions or definitions to be applied later, one must start with rich contexts demanding mathematical organization or, in other words, *contexts that can be mathematized*. Thus, while working on context problems, the students can develop mathematical tools and understanding.

**3. ****Level principle**

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Learning mathematics means that students pass through various levels of understanding: *from the ability to invent informal context-related solutions, to the creation of various levels of short cuts and schematisations, to the acquisition of insight into the underlying principles and the discernment of even broader relationships*. The condition for arriving at the next level is the ability to reflect on the activities conducted. This reflection can be elicited by interaction. Models serve as an important device for bridging this gap between informal, context-related mathematics and more formal mathematics. First, the students develop strategies closely connected to the context. Later on, certain aspects of the context situation can become more general, which means that the context more or less acquires 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 formal levels, models have to shift from a ‘*model of’ a particular situation* to a *‘model for’ all kinds of other, but equivalent, situations*. An important requirement for having models functioning in this way is that they are rooted in concrete situations and that they are also flexible enough to be useful in higher levels of mathematical activities. This means that the models will provide the students with a foothold during the process of vertical mathematization, without obstructing the path back to the source.

The strength of the level principle is that it guides growth in mathematical understanding and that it gives the curriculum a longitudinal coherence. This long-term perspective is characteristic of RME. There is a strong focus on the relation between what has been learnt earlier and what will be learnt later. A powerful example of such a ‘longitudinal’ model is the number line. It begins in first grade as (a) a beaded necklace on which the students can practice all kinds of counting activities. In higher grades, this chain of beads successively becomes (b) an empty number line for supporting additions and subtractions, (c) a double number line for supporting problems on ratios, and (d) a fraction/percentage bar for supporting working with fractions and percentages.

**4. ****Intertwinement principle**

It is also characteristic of RME that mathematics, as a school subject, is not split into distinctive learning strands. From a deeper mathematical perspective, the chapters within mathematics cannot be separated. Moreover, solving rich context problems often means that you have to apply a broad range of mathematical tools and understandings. For instance, if children have to estimate the size of the flag, this estimation involves not only measurements but also ratio and geometry.

The strength of the intertwinement principle is that it renders coherence to the curriculum. This principle involves not only the mutual relationship between the different chapters of mathematics but can also be found in the different parts of one chapter. In the number strand, for instance, topics like number sense, mental arithmetic, estimation and algorithms are closely related.

**5. Interaction principle**

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Within RME, *the learning of mathematics is considered a social activity*. Education should offer students opportunities to share their strategies and inventions with one another. By listening to what others find out and discussing these findings, the students can get ideas for improving their strategies. Moreover, the interaction can evoke reflection, which enables the students to reach a higher level of understanding. The significance of the interaction principle implies that *whole-class teaching* plays an important role in the RME approach to mathematics education. However, this does not mean that the whole class is proceeding collectively and that every student is following the same track and is reaching the same level of development at the same moment. On the contrary, within RME, *children are considered as individuals*, each following an individual learning path. This view on learning often results in pleas *for splitting up classes into small groups of students*, each following its own learning trajectory. In RME, however, there is a strong preference for keeping *the class together as a unit of organization* and for adapting the education to the different ability levels of the students instead. This can be done by means of providing the students with problems, which can be solved at different levels of understanding.

**6. Guidance principle**

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One of Freudenthal’s key principles for mathematics education is that it should give students* a ‘guided’ opportunity to ‘re-invent’ mathematics*. This implies that, in RME, both the teachers and the educational programs have a crucial role in how students acquire knowledge. They steer the learning process, but not in a fixed way by demonstrating what the students have to learn. This would be in conflict with the activity principle and would lead to pseudo-understanding. Instead, the students need room to construct mathematical insights and tools by themselves. In order to reach this desired state, the teachers have to provide the students with a learning environment in which the process of construction can emerge. One requirement is that *teachers must be able to foresee where and how they can anticipate the students’ understandings and skills that are just coming into view in the distance*. Educational programs should contain scenarios, which have the potential to work as a lever in shifting students’ understanding. It is important for these scenarios that they always hold the perspective of the long-term teaching/learning trajectory based on the desired goals. Without this perspective, it is not possible to guide the students’ learning. The guidance principle leads to the curriculum ideas of RME.

*Edited version of: ‘Mathematics education in the Netherlands: A guided tour’, by Marja van den Heuvel-Panhuizen*