Criticism of Raf Feys (from 1989 onwards) on constructivist & contextual approach to mathematics education from the Dutch Freudenthal Institute 

In my book' Calculating up to one hundred' (Wolters-Plantyn, 1998) and elsewhere, I made a comprehensive analysis of the disastrous aspects of' realistic arithmetic education', of contextual & constructivist mathematics. In this contribution, we only mention a number of conclusions.

The Freudenthal Institute made a caricature of het classic mathematical instruction and wrongly described it as purely mechanistic. However, it is well known that most people used to be able to calculate quickly. According to the classical didactics of the discipline, counting on inspiration (insight), but equally and even more so on transpiration (exercise, automating and memorizing, ready knowledge).

The insight into the processing etc. is not as difficult as the Freudenthals imagine and takes much less time (in the lower years of study) than calculating the smooth learning. For the notion of adding up and subtracting from it, one should not play endlessly in class van de Jan van den Brink's class bus. In addition to the way from knowing to being able, there is also the way from being able to know.

The misleading and artificial contradiction between realistic and mechanistic mathematics education does not do justice to the classical didactics of the subject, and the term' realistic' was given all possible meanings (application to reality, realization, etc.).

The strengths of conventional arithmetic thus ended up in the dark corner. This' redeeming' attitude is inherent to people who are exempted from the permanent revolution of education and want to remain exempt for the rest of their lives. Released people almost always come out with the paradigm of salvation instead of' renewal in continuity'.

The FI underestimates the great importance of the smooth and standardised master arithmetic, the fast and standardised numerical data, the smooth and standardised metering and the great importance of the knowledge available (table products, formulas for calculating the surface area and content, standard sizes and Greek system for metering...).

Smooth, skilful and automated calculation and ready knowledge is only possible with standardization and a lot of practice. The number of partial steps must be as small as possible because the working memory is limited.

The Freudenthals overemphasize the flexible calculation of the head and flexible numerical calculations according to their own method and/or context-related calculation methods. They wrongly call this' convenient' and mistakenly regard the other approaches as awkward and mechanistic. They also conceal the fact that such flexible counting on the back of standardized counting is so flexible. Only those who can calculate -40 raft, may realise that they can also calculate -39 raftily by first -40 and then +1. However, weaker students still have problems with such simple forms of flexible arithmetic.

In this way, the classic tables of multiplication are no longer rehearsed and elevated in grade 2. They are wrongly shifted to grade 3 and replaced by flexible calculation methods based on properties. Students then calculate for example 8 x 7 x 7 through 4 x 7 = 28.8 x 7 = 28 + 28 + 28 = 56. They make many mistakes and the calculation takes too much time.

The tables of x are taught classically in the 2nd year of study. Most students already realize grade 3 that 7 x 8 equals 7 x 8 x a group of 8. This insight is sufficient.

Flexible attributes are presented only in higher years of learning and in the context of larger tasks such as 13 x 7 where the application of the attributes brings a certain skill.

Criticism on constructivist principles:

- too much construction of individual pupils, too little mathematics as a cultural product, underestimation of the socio-cultural character and functional significance of mathematics.

- Too much respect for the student's own constructions and approaches: this makes learning short and fixed calculation methods, the guidance, the internalisation and automation of the arithmetic skills difficult. This also promotes the student's fixation on his own, informal constructions and primitive methods of calculation. - - - unilateral' bottom-up problem-solving', overemphasising of self-discovered and informal concepts and calculation methods - too little guidance and structuring by the teacher, too little' guided construction of knowledge'.

Few apprenticeships built up in stages.

Total superfluous introduction of colom-arithmetics that confuses the pupils with regard both to the ordinary capital accounts and to the figures that should normally also start at the beginning of group

3. When subtracting with deficits, for example, it becomes a fuss.

Traditional figures are neglected and Freudenthalers introduce a totally artificial alternative that has nothing to do with mathematical numerals - based on splitting numbers into hundreds, etc. The figures are transformed into a kind of long-drawn head arithmetic based on cute subtractions of bites. This is an approach with many partial results that is long-drawn-out and does not allow itself to be automated so that the figures can never become a skill.

Revaluation for classical metering and classical geometry - including knowledge of basic formulas for the calculation of surface area and content.

Too much and too long' pre-mathematics', too long' calculating in contexts' as an end in itself; too much contextualisation (context or situational calculation methods, etc.), too little decontextualisation. In this way, professional arithmetic and numeracy are slowed down by a connection to a specific context. An example. By linking subtraction to a linear context and to a calculation on the numerical line (a trajectory of 85 km, already covered 27 km, how many km I still have to cover), the basic insight into subtraction is obscured and the pupils are encouraged to interpret subtractions unilaterally as additional: 85 - 27 becomes: 27 + 3 + 3 + 10 + 10 + 10 + 10 + 10 + 5; and after that many more

No balanced and detailed vision of issues: too much criticism of classical issues, too few valid alternatives in realistic publications and methods. Too few applications (issues) also for meticulous arithmetic and too few difficult tasks. We also did not understand why the clear term' issues' should disappear. The difficulty in many context-related issues often lies more in the insufficient knowledge of the context (e. g. experience of parking with a car in terms of how many cars are parked at a parking space of 70 x 50 metres), the fact that the text is too long and too difficult and the fact that too many calculations are involved in one thing.

Wrong approach to visualisation and excessively long visual work. Fixing students on visual aids: students are allowed to use tools such as numerical, numerical and numerical tools for far too long.... This promotes, removes the visual support and calculates quickly and conveniently.

Gap between idealistic theory and practice. In a classroom with 20 students, it is not feasible to respond to individual ways of thinking and calculation.

Weak, but also better pupils are the victims.

The proponents of the realistic approach made exactly the same mistakes as those who were in favour of' modern mathematics' at the time. They only replaced one extreme by another. The' heavenly' (too abstract) New Math was replaced by the other extreme, by the' earthly', contextual and constructivist approach that pays too little attention to abstraction and generalization, and is stuck in the stage of pre-mathematics. The opponents were condemned. The criticism was silenced.