Realistic Mathematics Education (RME)

Realistic Mathematics Education (RME) is a domain-specific instruction theory for mathematics education (e.g., Treffers, 1987; De Lange, 1987; Streefland, 1991, Gravemeijer, 1994a; Van den Heuvel-Panhuizen, 1996). This theory is the Dutch answer to the need, felt worldwide, to reform the teaching of mathematics. The roots of RME go back to the early 1970s when Freudenthal and his colleagues laid the foundations for it , the earliest predecessor of the Freudenthal Institute. Based on Freudenthal’s (1977) idea that mathematics – in order to be of human value – must be connected to reality, stay close to children and should be relevant to society, the use of realistic contexts became one of the determining characteristics of this approach to mathematics education. In RME, students should learn mathematics by developing and applying mathematical concepts and tools in daily-life problem situations that make sense to them.

On the one hand the adjective ‘realistic’ is definitely in agreement with how the teaching and learning of mathematics is seen within RME, but on the other hand this term is also confusing. In Dutch, the verb ‘zich realiseren’ means ‘to imagine’. In other words, the term ‘realistic’ refers more to the intention that students should be offered problem situations which they can imagine (see Van den Brink, 1973; Wijdeveld, 1980) than that it refers to the ‘realness’ or authenticity of problems. However, the latter does not mean that the connection to real life is not important. It only implies that the contexts are not necessarily restricted to real-world situations. The fantasy world of fairy tales and even the formal world of mathematics can be very suitable contexts for problems, as long as they are ‘real’ in the students’ minds. Apart from this often-arising misconception about the meaning of ‘realistic’ the use of this adjective to define a particular approach to mathematics education has an additional ‘shortcoming’. It does not reflect another essential feature of RME: the didactical use of models. In this article the focus will be on this aspect of RME.

In the first part of this position paper I will give general background information about the theory of RME and the role of models within this theory. Among other things, attention will be paid to the two ways of mathematizing that characterize RME, the different levels of understanding that can be distinguished and that typify the learning process, the way students can play an active role in developing models and how models can evolve during the teaching-learning process, and – as a result of this – can prompt and support level raising. In the second part of the article this general information will be made more concrete by concentrating on the content domain of percentage. A description is given of how the bar model can support the longitudinal process of learning percentage.

This description of the didactical use of the bar model is based on the development work carried out in the Mathematics in Context project, a project aimed at the development of a mathematics curriculum for the U.S. middle school (Romberg, 1997–1998). The project was funded by the National Science Foundation and executed by the Center for Research in Mathematical Sciences Education at the University ofWisconsin-Madison , and the Freudenthal Institute of Utrecht University. The designed curriculum reflects the mathematical content and teaching methods suggested by the ‘Curriculum and Evaluation Standards for School Mathematics’ (NCTM, 1989). This means that the philosophy of the curriculum and its development

is based on the belief that mathematics, like any other body of knowledge, is the product of human inventiveness and social activities. This philosophy has much in common with RME. It was Freudenthal’s (1987) belief that mathematical structures are not a fixed datum, but that they emerge from reality and expand continuously in individual and collective learning processes. In other words, in RME students are seen as active participants in the teaching-learning process that takes place within the social context of the classroom.

In addition to the foregoing, however, Freudenthal (1991) also emphasized that the process of re-invention should be a guided one. Students should be offered a learning environment in which they can construct mathematical knowledge and have possibilities of coming to higher levels of comprehension. This implies that scenarios should be developed that  have the potential to elicit this growth in understanding. The development of such a scenario for learning percentage was one of the goals of the Mathematics in Context project. Within this scenario the bar model was the main didactical tool to facilitate the students’ learning process.