History and Basic Philosophy
The development of what is now known as Realistic Mathematics Eduction (RME) started around 1970. The foundations were laid by Freudenthal and his colleagues at the former IOWO, the oldest processor 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 project’s first merit was that Dutch mathematics education was not affected by the New Math movement. The present form of RME has been mostly determined by Freudenthal’s (1977) view on mathematics. He felt mathematics must be connected to reality, stay close to children’ 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 the idea of mathematics as a human activity. Mathematics lesson should give students the “guided” opportunity to “re-invent” mathematics by doing it. This mean 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).
Misunderstanding of “Realistic”
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.
Key Principles of Realistic Mathematics Education
ü Guided Reinvention and Progressive Mathematizing
According to the reinvention principle, the students should be given the opportunity to experience a process similar to the process by which mathematics was invented. The history of mathematics can be used as a source of inspiration for course design. The reinvention principles can be inspired by informal solution procedures. Informal strategies of students can also be interpreted as anticipating more formal procedures. In this case, mathematizing similar solution procedures creates the opportunity for the reinvention process. In a general way one needs to find contextual problems that allow for a wide variety of solution procedures, preferably those which, considered together, already indicate a possible learning route through a process of progressive mathematization.
ü Didactical Phenomenology
According to the didactical phenomenology, situations where a given mathematical topic is applied are to be investigated for two reasons. Firstly, to reveal the kind of applications that have to be anticipated in instruction; secondly, to consider their suitability as points of impact for a process of progressive mathematization. If we see mathematics as historically evolved from solving practical problems, it is reasonable to expect to find the problems which gave rise to this evolving process in present day applications. Next, we can imagine that formal mathematics came into being in a process of generalizing and formalizing situation-specific problem-solving procedures and concepts about a variety of situations. Therefore, the goal of our phenomenological investigation is to find problem situations for which situation-specific approaches can be generalized, and to find situations that can evoke paradigmatic solution procedures that can be taken as the basis for vertical mathematization.
ü Self-developed Model
The third principle is found in the role which self-developed model play in bridging the gap between informal knowledge and formal mathematics. Whereas manipulatives are presented as pre-existing models in the information processing approach, in RME models are developed by the students themselves. This means that students develop models in solving problem. At first, a model is a model of situation that is familiar to the students. By a process of generalizing and formalizing, the model eventually becomes an entity on its own: It becomes possible that it is used as a model for mathematical reasoning (Gravemeijer, 1994; Streefland, 1985; Treffers, 1991)
There are five basic characteristics of RME. Here they are:
ü Use of Contextual Problem
The phenomena by which mathematics concept appear in reality should be the source of concept formation. The process of extracting the appropriate mathematical concept from a concrete situation is described by de Lange (1996) as a conceptual mathematization. This process forces pupils to: explore the situation or the context; find and identify the relevant mathematical elements; schematize and visualize in order to discover patterns; and develop a model resulting in a mathematical concept. By the process of reflecting and generalizing, the pupils will develop a more complete concept. It is then expected that the pupils will subsequently apply mathematical concepts to other aspects of their daily life, and by so doing, reinforce and strengthen the concept.
ü Use of Model or Bridging by vertical instruments
The term model refers to situational models and mathematical models that are developed by the pupil themselves. At first, a model is a model of situation that is familiar to the pupils. By a process of generalizing and formalizing, the model eventually becomes an entity on its own. It becomes possible that it is used as a model for mathematical reasoning. Here, self-developed models of the pupils serve to bridge the gap between informal and formal knowledge.
ü The Use of Pupil Own Creations and Contributions
Pupils should be asked to create concrete things. By making “free production”, pupils are forced to reflect on their learning process. Pupils show greater initiative when they are encouraged to construct and produce their own solutions. In addition, free productions can form an essential part of assessment. For example, pupils maybe asked to write an essay, to do an experiment, to collect data and draw conclusions, to design exercise that can be used in a test, or to design a test for other pupils in the classroom.
Interaction between pupils and between pupils and teachers is an essential part in instructional process. Explicit negotiation, intervention, discussion, cooperation and evaluation are essential elements in a constructive learning process in which the students’ informal methods are used as a vehicle to attain the formal ones. For instance, pupils are encouraged to discuss their strategies and to verify their own thinking rather than focusing on whether they have the right answer. Such activities can enable them to depend less on the teacher to tell them whether they are right or wrong. Hence, the pupils find opportunities to develop confidence in using mathematics.
The integration of mathematical strands or units is essential. It is often called the holistic approach, which incorporates applications, and implies that learning strands should not be dealt with as separate and distinct entities. Instead, an intertwining of learning strands is exploited in solving real life problems.
Gravemeijer, Kueno. 1994. Developing Realistic Mathematics Education. Freudenthal Institute, Utrecht.
Zulkardi. 2009. The “P” in PMRI: Progress and Problems. Proceeding of IICMA 2009.