MathEd Studies

Realistic Mathematics Education (RME) is an approach to mathematics education developed in The Netherlands by the math educators of the Freudenthal Institute. The development of what is now known as RME started around 1970. The foundations were laid by Hans Freudenthal and his colleagues. Hans Freudenthal stressed the idea of mathematics as a human activity and therefore he proposed that the teaching and learning of mathematics must be connected to reality, stay close to children’s experience and be relevant to society, in order to be of human value. Mathematics lessons according to Freudenthal should give students the ‘guided’ opportunity to ‘re-invent’ mathematics by doing it; the focal point should not be on mathematics as a closed system but on the activity, on the process of mathematization.

According to Van den Heuvel-Panhuizen, the reason 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.

Below is a list of principles each reflecting a part of the identity of RME identified by Heuvel-Panhuizen.

1. Activity principle

The idea of mathematization clearly refers to the concept of mathematics as an activity which, according to Freudenthal (1971, 1973), can best be learned by doing (see also Treffers, 1978, 1987). 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.

2. Reality principle

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’ (see Freudenthal, 1968).

3. Level principle

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 schematizations, to the acquisition of insight into the underlying principles and the discernment of even broader relationships.

4. Inter-twinement 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.

5. Interaction principle

Within RME, the learning of mathematics is considered as a social activity. Education should offer students opportunities to share their strategies and inventions with each other. 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.

6. Guidance principle

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 constructing process can emerge.

Reference:

Van den Heuvel-Panhuizen, M. (2000). Mathematics education in the Netherlands: A guided tour. Freudenthal Institute Cd-rom for ICME9. Utrecht: Utrecht University.

What is theory?

Mogan Niss in his paper The concept and role of theory in mathematics education offers this definition of theory:

A theory is a system of concepts and claims with certain properties, namely

The theory consists of an organised network of concepts (including ideas, notions, distinctions, terms etc.) and claims about some extensive domain, or a class of domains, of objects, situations and phenomena.
In the theory, the concepts are linked in a connected hierarchy (oftentimes of a logical or proto-logical nature), in which a certain set of concepts, taken to be basic, are used as building blocks in the formation of the other concepts in the hierarchy.
In the theory, the claims are either basic hypotheses, assumptions, or axioms, taken as fundamental (i.e. not subject to discussion within the boundaries of the theory itself), or statements obtained from the fundamental claims by means of formal or material (by “material” we mean experiential or experimental) derivation (including reasoning).

What are the theories put to use in mathematics education?

Niss: To begin with, four general observations pertaining to this question should be kept in mind.

Firstly, there is no such thing as a well-established unified “theory of mathematics education” which is supported by the majority of mathematics education researchers. On the contrary, different groups of researchers represent different schools of thought, some of which appear to be mutually incompatible if not outright contradictory. It is not likely that we shall get a unified theory of mathematics education in a foreseeable future, if ever.
The second observation is that many mathematics education researchers relate their work to explicitly invoked theories borrowed from other fields (or at least from authors who belong to other fields), and often do so in rather eclectic or vague ways. Only rarely are theories “homegrown” within mathematics education itself.
Thirdly, much discussion and debate in mathematics education research takes the shape of “fights” with and between theories. This may be potentially fruitful to the extent competing theories offer different perspectives on the same thing, whereas it is potentially futile, if not destructive, if the theories deal with different things and therefore are only competing in the superficial sense that “my topic object of study is more important than yours”.
Finally, the fourth observation is that quite a few mathematics education researchers do not explicitly invoke or employ any theory at all in their work. Many researchers who actually do invoke a theory in their publications do not seem to go beyond the mere invocation.

In other words, some theoretical framework may be referred to in the beginning or in the end of a paper without having any presence or bearing on what happens between the beginning and the end.

What should we strive at in theory building in mathematics education?

Niss: Imagine that there existed a full-fledged theory of mathematics education. What would it look like, and what minimum requirements would it have to fulfil? In order for it to be comprehensive enough to be worth its name, it would have to contain at least the following of sub-theories, each accounting for essential traits of mathematics education:

a sub-theory of mathematics as a discipline and a subject in all its dimensions, including is nature and role in society and culture;
a sub-theory of individuals’ affective notions, experiences, emotions, attitudes, and perspectives with regard to their actual and potential encounters with mathematics; a sub-theory of individuals’ cognitive notions, beliefs, experiences, and perceptions with regard to their actual and potential encounters with mathematics, and the outcomes thereof;
a sub-theory of the teaching of mathematics seen within all its institutional, societal, national, international, cultural and historical contexts;
a sub-theory of teachers of mathematics, individually and as communities, including their personal and educational backgrounds and professional identities and development.