Realistic Mathematics Education in The Netherlands 1980-1990
I. Realism in textbooks 1980-1990
A revolution in math education, which occurred in the Netherlands in the period 1980-1990 is a silent revolution, because not a single innovation experts to hear speaking or writing about it even almost a whisper in the media. What has taken place? The distribution of realistic textbooks for mathematics in primary schools increased compared with mechanistic textbooks. Realistic textbooks went from 5% in 1980 to no less than 75% in 1990.
How could this happen in The Netherlands in the past decade? The answer to this question must be sought in the developments of the seventies. This is thanks to the School Inspectorate which held back the introduction of the New Math textbooks. The Inspectorate still had the authority to do so in those days. And it did this at the recommendation of young Wiscobas group, and already before the founding of The IOWO, before 1971, and therefore even before the time when Freudenthal who is the founder of realistic mathematics education became involved with Wiskobas.
This was done because, First, it was felt, more should be clear about the desired direction of mathematics instruction. The second factor led to mathematics instruction of own signature, that at the end of the seventies was named realistic mathematics instruction. A third factor which explains this is the simple fact that the idea, the reasoning and the products Wiskobas proved to catch on.
In The Netherlands the development of curriculum and textbook did not primarily go to down, but sooner the other way around. One and other occurred initially mainly through the channels of the informal infrastructure instead of via the formal route in the educational provider system.
For good measure it should be added that in the eighties there was a great deal of progression or even progress in the further concretising of realistic mathematics instruction in learning strands about basic skill, mental arithmetic, column, arithmetic, fraction, ratio and percentages.
In addition, the revolution that occurred in mathematical education is also influenced by the emphasis on certain changes that emphasis shifted to the importance of elementary context problem, to the alignment of learning strands and the steering task of the teacher.
Didactical Background of Mathematics Program For Primary Education
This contribution outlines an instruction-theoretical framework of realistic mathematics education for primary school that was developed in The Netherlands in the period between 1970 and 1990. the development research that has led up to this realistic mathematics program is the fruit of ten years of IOWO and ten years of OW&OC effort, in co-operation with so many in the field of education. It has been especially the textbook authors who contributed to making this realistic mathematics program concrete. Today in 1991, approximately three-quarters of the primary school in The Netherlands use a realistic mathematics textbook series.
II. Instruction-theoretical framework of realistic mathematics education
We will discuss the five major learning and teaching principles that lie at the basis of such’ realistic’ courses.
- constructing and concretizing
The first learning principle is that first and foremost learning mathematics is a constructive activity. Something which contradicts the idea of learning as absorbing knowledge which is presented or transmitted. This construction characteristic is clearly visible in the outlined course: the pupils discover the division procedure for themselves.
2. levels and models
The learning of mathematical concept or skill is a process which is often stretched out over the long term and which moves at various levels of abstraction. To be able to achieve this raising in the level from informal context-bound arithmetic to formal arithmetic the pupil must at his disposal the tool to help bridge the gap between the concrete and abstract.
3. reflection and special assignments
The learning of mathematics and in particular the raising of the level of the learning process is promoted through reflection, therefore by considering own thought process that of others. The most important category however is the assignment to produce items of one’s own. This bring us to the third instruction principle: the pupils must constantly have the opportunity and be stimulated at important junctions in the course, to reflect on learning strands that have already been encountered and to anticipate on what lies ahead.
4. social context and interaction
Learning is not merely a solo activity but something that occurs in a society and is directed and stimulated by that socio-cultural context. For example is solving mathematics problem in group, because besides individual work the group also acts as a booster for learning.
5. structuring and interweaving
Learning mathematics does not consist of absorbing a collection of unrelated knowledge and skill elements, but is the construction of knowledge and skills to a structured entity.
Actually, all of the learning-instruction principles can be connected randomly, because every learning principle can basically be connected to each instruction principle.
III. Four directions in mathematics education
Besides the realistic direction in mathematics education there are others that can be distinguished, namely: the empiristic education, the mechanistic and the structuralistic. A brief characterization will follow for each on the basis of the long division problem, followed by a more general description via mathematising.
a. the empiristic approach to long division
We can be brief about the empiristic approach to long division because, in general, it is one that is not taught. Informal, context-bound aritmatic is the basis of instruction here. This educational design departs from the premise that all the while working with a great diversity of realistic problems the pupils will themselves be able to make the leap to the level of formal aritmatic.
b. the mechanistic approach to long division
What the methodology approach to long division consists of is something many of us know from own experience.
c. the structuralistic approach to long division
The problem with the stucturalistic set up of the learning strand for long division is that the algorithm is taught primarily at the formal arithmetic level. The characteristic for the structuralistic approach of education is that there insight is pursued. In this approach, real problems play no essential part in the learning of arithmetic in the initial phase.
d. Directions in the 5 by 5 learning-instruction structure.
In this issue, the mentioned principles have more relief. According to the realistic view, that the building up of elementary skill can take place via a process of reinvention on independent construction. The children are given the opportunity to bridge the gap between informal, context-bound work and the formal, standardized manner of operation, through the countructive contribution of the children themselves.
The four mentioned directions in mathematics education can be distinguished according to the presence or absence of the components of horizontal and vertical mathematisation. Of course, first, we will explain clearly what is meant by the horizontal and vertical mathematisation. Horizontal mathematising is the modeling of the problem situations thus that these can be approached with mathematical means, or in the bother words: it leads from the perceived world to the world of symbols. In other hand, vertical mathematising is directed at the perceived building and expansion of knowledge and skills within the subject system, the world of symbols.
vertical mathematising takes place with this structural material, by means of visual representations thereof, to operation with symbols.
IV. Example of realistic learning stands with an emphasis on vertical mathematising.
There are a number of examples of learning strands in primary school which fit into the outlined realistic structure. There are: counting, memorizing of addition and subtraction table to ten and twenty, addition and subtraction to one hundred, multiplication and division table, mental aritmatic, column aritmatic, ratio, and fraction.
In this case, we also highlight the location of the mechanistic, empiristic and structuralistic nature in each of the learning.