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Realistic Mathematics Education, work in progress
Monday, February 7, 2011 di 7:18 AM |

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.
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