Realistic Mathematics Education

work in progress

This text is based on the NORMA-lecture, by Marja van den Heuvel-Panhuizen, held in Kristiansand, Norway on 5-9 June 1998  (Quoted From Utrecht University Website)

RME in brief

Realistic Mathematics Education, or RME, is the Dutch answer to the world-wide felt need to reform the teaching and learning of mathematics. The roots of the Dutch reform movement go back to the early seventies when the first ideas for RME were conceptualized. It was a reaction to both the American “New Math” movement that was likely to flood our country in those days, and to the then prevailing Dutch approach to mathematics education, which often is labeled as “mechanistic mathematics education.”
Since the early days of RME much development work connected to developmental 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. This sounds like a superfluous statement, but it is not. Again and again, too optimistic thoughts are heard about educational innovations. The following statement indicates how we think about this: 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 completely conquered— 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. It is inherent to RME, with its founding idea of mathematics as a human activity, that it can never be considered a fixed and finished theory of mathematics education.

“Progress” issues to be dealt with

This self-renewing feature of RME explains why it is work in progress. But, there are at least two more aspects. One significant characteristic of RME, is the focus on the growth of the students’ knowledge and understanding of mathematics. RME continually works toward the progress of students. In this process, models which originate from context situations and which function as bridges to higher levels of understanding play a key role. Finally, considering the TIMSS results, it seems that RME really can elicit progress in achievements.

RME, History and founding principles

The development of what is now known as RME started almost thirty years ago. The foundations for it were laid by Freudenthal and his colleagues at the former IOWO, which is the oldest 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 is 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 — 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 said, that the difference between these two worlds is clear cut. Freudenthal 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.

Misunderstanding of “realistic”

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.

The realistic approach versus the mechanistic approach

The use of context problems is very significant in RME. This is in contrast with the traditional, mechanistic approach to mathematics education, which contains mostly bare, “naked” problems. If context problems are used in the mechanistic approach, 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 situation.
In RME this is different; Context problems function also as a source for the learning process. In other words, in RME, contexts 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 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 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 April 1998. It was he who in 1985*  detected this crucial mechanism in the growth of understanding. 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 as 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.

In summary, RME can be described by means of the following five characteristics (Treffers, 1987):

  • The use of contexts.
  • The use of models.
  • The use of students’ own productions and constructions.
  • The interactive character of the teaching process.
  • The intertwinement of various learning strands.

An overview of some trends in RME

RME is undeniable a child of its time. So all the issues which concern the community of mathematics education world-wide also affect the Dutch mathematics education community. As a consequence, you will certainly recognize a lot of similarities with your own agenda for further development. On the other hand, however, there might also be some dissimilarities.

To start with the overview:
* As mentioned is the issue of looking for ways of assessment that are in tune with our new approach to teaching.

Besides this topic there are more progressions to specify:

* Our greatest concern now is the implementation of RME teaching methods in classroom practice. For this reason professionalization of teachers and classroom research have a high priority. One of things we are working on is a course for teachers who want to become a mathematics co-ordinator at their school.
Besides this form of vertically organized teacher enhancement, efforts are also made to set up a network of teachers, including a website, for instance, for sharing ideas on teaching activities. Such a network can be regarded as a horizontally organized opportunity for teacher enhancement. In both forms, the earlier mentioned blueprints for longitudinal learning/teaching trajectories can play a key role.

* Constitutive input for further development is also coming from in-dept studies on how learning takes place in classrooms. In these studies the perspectives of both the individual learner and the learning community in a classroom are dealt with. In the first perspective the leading question is: how does a student come to understanding and what is the role of contexts and models? In the second one the questions to be answered are: what constitutes a learning community, and what is the role of the teacher and students in class?

* Another issue which is on our agenda for further development is the content of mathematics as a school subject. Within RME, there are still many questions about what is worthwhile to teach children and how particular content can best be taught, including the use of computers and ICT. Including these new media, however, does not mean that we already have clear thoughts about, for instance, the role of the calculator in primary school mathematics. This certainly will be one of the themes that need further elaboration. Of course this will be done in connection with the issue of numeracy which in its turn cannot be separated from mental arithmetic, estimation and doing algorithms.
Another content issue is geometry. Within this domain a lot of questions need to be answered about what to include in the primary school curriculum. In any case not the formal geometry that can be found in many mathematics curricula around the world.
A relatively new development within the number strand is the attention which is now devoted to the question regarding the role of practising in RME, like practising computational skills and number facts.

* Linked with this is the increased focus on vertical mathematization. Compared to several years ago, when the emphasis was strongly on solving problems within a real-life context, the two forms of mathematization are now becoming more and more balanced. Evidence for this can be found in the intensifying attention for problems within a mathematical context, mathematical investigations, and the so-called ‘productive practising’ (this means practising by means of student-generated problems). In a way this process towards a better balance between the two forms of mathematization is a topic which exceeds the elaborations in the field of the mathematical content. It is rather a progression that touches the heart of RME more in general.

* As a last point of progress we may consider the more differentiated attention for specific groups of students, like immigrants’ children — and more in general — special-needs students, but also adults with poor schooling who want to repair their shortage, and finally girls, who in The Netherlands have significantly lower achievement scores in mathematics than boys. Developmental research which is focused on particular populations of students is rather new in RME. Take for instance the gender issue. Probably we are the last country in the world in which on primary school level the differences in achievements between boys and girls were investigated. The same is true for developmental research aimed at students who have difficulties with mathematics. In this respect two important studies are now on the research agenda of the Freudenthal Institute: in one study a special-needs program, called ‘Jumping ahead’ (Menne, 1996/1997/1998), is being developed for early graders who are weak in mathematics; and in the other study the difficulties which immigrants’ children meet in mathematics in the beginning grades of secondary school are investigated (Van den Boer, 1997)

Progress in understanding —  a macro-didactic perspective

Until recently, three things were important for the macro-didactic tracing in Dutch mathematics education in primary school:

  • the mathematics textbooks series;
  • the “Proeve”; a document which describes the mathematical content to be taught in primary school;
  • the key goals to be reached by the end of primary school, described by the government.

The determining role of textbooks

In today’s world-wide reform of mathematics education, talking about textbooks — not to mention the use of them — often elicits a negative association. Actually, many reform movements are rather aimed at getting rid of textbooks. In The Netherlands, however, the contrary is the case. Here, the improvement of mathematics education is carried for a considerable part by the textbooks. In our country, textbooks have a determining role in mathematics education. Actually, they are the most important tool that guides the teachers’ teaching. This is true for both the content and the teaching methods, although for the latter the guidance is not sufficient enough to reach all teachers.
The determining role of textbooks, however, does not mean that teachers are a prisoner of their textbook. In The Netherlands, teachers are rather free in their teaching. They can make most of the educational decisions by themselves, or as a school team. Moreover, schools can decide by themselves which textbook series they use.
Currently, about eighty percent of the Dutch primary schools, use a mathematics textbook series which was inspired to a greater or lesser degree by RME. Also in this respect there was a lot of progress. Compared to ten, fifteen years ago this percentage has changed remarkably. At that time, only half of the schools worked with such a textbook series (De Jong, 1986).
The development of the textbook series was done by commercial publishers. In addition to using their own ideas, the textbook authors were free to use the ideas for teaching activities that resulted from the developmental research done at the Freudenthal Institute (and its predecessors) and at the SLO, the Dutch Institute for Curriculum Development.

The “Proeve” — a domain description of primary school mathematics

Important for the development of the textbooks is also the guidance which since the mid eighties is given by a series of publications, called the ‘Proeve ….’ * . Treffers is the main author of it. The documents contain descriptions of the various domains within mathematics as a school subject. The work on the ‘Proeve’ is still going on. Eventually, there will be descriptions for: the basic number skills; written algorithms; ratio and percentages; fractions and decimal numbers; measurement; and geometry. Although the ‘Proeve’ is written in a very accessible style with a lot of examples, it is not especially meant as a series for teachers. Instead, it is rather meant as a support for textbook authors, teacher trainers and school advisors. On the other hand, however, many of these experts on mathematics education were  also — and are still — important contributors to the realization of this series.
Looking back at our reform movement in mathematics education, it can be concluded that the reform proceeded in a very interactive and informal way. There was no interference from the government. Instead, developers and researchers, in collaboration with teacher trainers, school advisors and teachers, worked out teaching activities and learning strands. Later on, these were included in textbooks.

The key goals for mathematics education

Unlike many other countries, in The Netherlands there is — or maybe I have to say: there was — no centralized decision making regarding the curriculum, nor the textbooks and the testing on primary school level. For a long time, there has been only a general law text containing a list of subjects to be taught. What had to be taught within these subjects was almost completely the responsibility of the teachers and the school teams.
A few years ago, however, the policy of the government changed somewhat. In 1993, the Dutch ministry of education came with a list of attainment targets, called ‘Key Goals.’ These goals for each subject describe what has to be learned by the end of primary school. The students then are twelve years old. For mathematics the list consists of 23 goals, split up in six domains (see Table 1). The content of the list is in agreement with the ‘Proeve’ documents mentioned before.

Table 1, KEY GOALS FOR PRIMARY SCHOOL MATHEMATICS

General abilities 1 The students can count forward and backward with changing units.
2 The students can do addition tables and multiplication tables up to ten.
3 The students can do easy mental-arithmetic problems in a quick way with insight in the operations.
4 The students can estimate by determining the answer globally, also with fractions and decimals.
5 The students have insight in the structure of whole numbers and the place-value system of decimals.
6 The students can use the calculator with insight.
7 The students can convert into a mathematical problem, simple problems which are not presented in a mathematical way.
Written algorithms 8 The students can apply the standard algorithms, or variations of these, for the basic operations addition, subtraction, multiplication and division, in easy context situations.
Ratio and percentage 9 The students can compare ratios and percentages.
10 The students can do simple problems on ratio.
11 The students have understanding of the concept percentage and can carry out practical calculations with percentages presented in simple context situations.
12 The students understand the relation between ratios, fractions, and decimals.
Fractions 13 The students know that fractions and decimals can represent several different situations.
14 The students can locate fractions and decimals on a number line and can convert fractions into decimals; also with the help of a calculator.
15 The students can compare, add, subtract, devide, and multiply simple fractions in simple context situations by means of models.
Measurement 16 The students can read the time and calculate time intervals; also with the help of a calendar.
17 The students can do calculations with money in daily-life context situations.
18 The students have insight in the relation between the most important quantities and the corresponding units of measurement.
19 The students know the current units of measurement for length, area, volume, time, speed, weight, and temperature, and can apply these in simple context situations.
20 The students can read simple tables and diagrams and produce them based on own investigations of simple context situations.
Geometry 21 The students have some basic concepts with which they can organize and describe space in a geometrical way.
22 The students can reason geometrically using block buildings, ground plans, maps, pictures, and data about place, direction, distance, and scale.
23 The students can explain shadow images, can compound shapes, and can devise and identify nets of regular objects.

Compared to goal descriptions and programs from other countries it is notable that some widespread mathematical topics are not mentioned in this list, like, for instance, problem solving, probability, combinatorics, and logic **.
Another striking feature of the list is that it is so simple. This means that the teachers have a lot of freedom in interpreting the goals. At the same time, however, such a list does not give much support to teachers. As a result the list actually is a ‘dead’ document, mostly put away in a drawer when it arrives at school. Nevertheless, this first list of key goals was of importance for Dutch mathematics education. The publication of the list by the government confirmed and, in a way, validated the recent changes in our curriculum. The predominant changes are:

  • more attention is paid to mental arithmetic and estimation;
  • the formal operations with fractions are not any more in the core curriculum, the students only have to do operations with fractions in context situations;
  • geometry is officially included in the curriculum now;
  • and the same is true of the insightful use of calculators.

However, not all these changes have been completely implemented in our present classroom practice. This is especially true for geometry and the use of a calculator.

In the years after 1993, discussions emerged about these 23 key goals (see De Wit, 1997). Almost everybody agreed that they can never be sufficient to give support for improving classroom practice nor to control the outcome of education. The latter is conceived by the government as a powerful tool for guarding the quality of education. For both, the key goals were judged to fail. Simply stating goals is not enough in order to achieve these goals. For testing the outcome of education the key goals are also inappropriate. The complaints are that the goals are not formulated precisely enough to provide us with yardsticks for testing. These arguments were not only heard regarding mathematics, but in fact they are observed with respect to all the primary school subjects for which key goals were formulated.

For several years it was unclear which direction would be chosen to improve the key goals: for each grade a more detailed list of goals expressed in operationalized terms, or, a description which supports teaching rather than pure testing. In 1997, the government chose tentatively for the latter and asked the Freudenthal Institute to work it out for mathematics. The purpose of the project, which the Freudenthal Institute is carrying out together with the SLO, is to contribute to the enhancement of classroom practice — the one in the early grades to begin with. The reason for this choice was that at the same time the government took measures to reduce the class size in these grades.

* The complete title of this series is ‘Proeve van een nationaal programma voor het reken-wiskundeonderwijs op de basisschool [Design of a National Curriculum for mathematics education at primary school]’. The first part of it was published in 1989 (see Treffers, De Moor and Feijs, 1989).
** Problem solving is not located in one particular content goal but is expressed more or less in the general goals which go with this list of content goals. The other mathematical topics are not incorporated in our curriculum.

RME assessment

A very first requirement for assessment in order to have an assessment which is in tune with any reform of mathematics education can be found in the altered curriculum contents. In RME, however, there is more which asks for another way of assessment. To come to assessment which is in alignment with RME assessment has to do justice to all three pillars of RME:

  • its view on mathematics and the goals aimed for;
  • its view on how children learn mathematics;
  • its view on how mathematics has to be taught.

Developmental research on assessment based on the RME viewpoints carried out so far, has already produced some keys of how assessment can be improved, especially written assessment (De Lange, 1978; Van den Heuvel-Panhuizen, 1996). In contrast to the prevailing opinion that the improvement of assessment in mathematics can best be found by getting rid of the paper-and-pencil tests, our research brought to the fore that written assessment also has a future. Necessary, however, is a creative exploitation of the potential of written tests. More about this can be found in he listed publications.

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