Mijn kritiek op contextuele en constructivistische aanpak van het Freudenthal Instituut vanaf 1987
Samenvatting kritiek van Raf Feys op realistische en constructivistische aanpak van FI
In boek Rekenen tot honderd ((Wolters-Plantyn, 199, en nu ook als e-book) en elders maakten we een uitvoerige
analyse van de nefaste aspecten van het zgn. realistisch reken-wiskundeonderwijs van het Nederlandse Freudenhtal Instituut, van hun contextueel en constructivistisch rekenen.
We vermelden hier
enkel een aantal conclusies. Vanaf 1987 namen we in het Nederlands taalgebied het voortouw in de strijd tegen het contextueel en constructivistisch rekenen van het F.I.
Het FI maakte vanaf 1980 een karikatuur van het rekenonderwijs anno 1970 en bestempelde
het ten onrechte als louter mechanistisch. Het is nochtans bekend dat de meeste mensen
vroeger vlot konden rekenen. De Nederlandse methodeFunctioneel Rekenen van Reynders
was bijvoorbeeld een degelijke methode, gebaseerd op een evenwichtige visie.
klassieke vakdidactiek berust degelijk rekenen op inspiratie (inzicht), maar evenzeer en
nog meer op transpiratie (inoefenen, automatiseren en memoriseren, parate kennis).
Het inzicht in bewerkingen e.d. is al bij al niet zo moeilijk als de Freudenthalers het voorstellen
en vergt (in de lagere leerjaren) veel minder tijd dan het vlot leren berekenen. Voor het begrip
optellen en aftrekken moet men niet eindeloos in klas autobusje spelen e.d.
Naast de weg van kennen naar kunnen, is er overigens ook de weg van kunnen naar kennen. Van Kunnen naar kennen was overigens de naam van de Vlaamse methode van Schneider rond
De misleidende en kunstmatige tegenstelling tussen realistisch en mechanistisch
rekenonderwijs doet geen recht aan de klassieke vakdidactiek en de term realistisch kreeg
alle mogelijke betekenissen (toepassen op realiteit, zich realiseren, enz.)
De sterke kanten van het klassieke rekenen belandden zo in de verdomhoek. Deze verlossende opstelling van onheilsprofeten die de verlossing uit de ellende prediken, is overigens inherent voor mensen die vrijgesteld worden voor de permanente
revolutie van het onderwijs en ook voor de rest van hun leven vrijgesteld willen blijven.
Vrijgestelden pakken bijna steeds uit met het verlossingsparadigma i.p.v. vernieuwing in
continuïteit. Zij zoeken werk voor de igen vernieuwingswinkel. En zo kreet ook het F.OI. steeds meer medewerkers.
Het FI onderschat het grote belang van het vlot en gestandaardiseerd hoofdrekenen, het vlot
en gestandaardiseerd cijferen, het vlot en gestandaardiseerd metend rekenen en het grote
belang van de parate kennis: tafelproducten, formules voor berekening van oppervlakte en
inhoud, standaardmaten en metriekstelsel voor metend rekenen
Vlot, vaardig en geautomatiseerd rekenen en parate kennis is maar mogelijk bij
standaardisering en veel oefenen. Het aantal deelstappen moet hierbij zo klein mogelijk zijn
omdat het werkgeheugen beperkt is.
De Freudenthalers overbeklemtonen het flexibel hoofdrekenen en flexibel cijferen volgens
eigenwijze en/of context- of opgave-gebonden berekeningswijzen.
Ze noemen dit ten onrechte handig en beschouwen de andere aanpakken ten onrechte als onhandig en mechanistisch.
Ze verzwijgen verder dat zulk flexibel rekenen op de rug zit van het gestandaardiseerd
rekenen. Enkel wie vlot -40 kan berekenen, beseft eventueel dat hij -39 ook vlot kan
berekenen via eerst -40 en vervolgens + 1.
Zwakkere leerlingen hebben echter toch nog
problemen met zulke eenvoudige vormen van flexibel rekenen.
Zo worden de klassieke tafels van vermenigvuldiging ook niet meer ingeoefend en
opgedreund en dit in groep 4. Ze worden ten onrechte verschoven naar groep 5 en er
vervangen door flexibele berekeningswijzen op basis van eigenschappen. Leerlingen
berekenen dan bijvoorbeeld 8 x 7 via 4 x 7 = 28, 8 x 7= 28 + 28 = 56. Ze maken veel fouten
en de berekening vergt te veel tijd. Ik probeerde o.a. Ter Heege hiervan te overtuigen, maar vruchteloos.
De tafels van x worden klassiek in het 2de leerjaar aangeleerd. De meeste leerlingen beseffen
ook al groep 3 dat 7 x 8 neerkomt op 7 x een groep van 8. Dit inzicht is voldoende.
Flexibel eigenschapsrekenen wordt pas in hogere leerjaren gepresenteerd en in de context
van grotere opgaven als 13 x 7 waar het toepassen van de eigenschappen een zekere
Kritiek op constructivistische uitgangspunten:
- te veel constructie van individuele leerling(en), te weinig wiskunde als cultuurproduct,
onderschatting van het socio-culturele karakter en functionele betekenis van de wiskunde.
veel respect voor de eigen constructies en aanpakken van de leerling: dit bemoeilijkt het leren
van korte en vaste berekeningswijzen, de begeleiding, de verinnerlijking en automatisatie van
de rekenvaardigheden. Dit bevordert ook de fixatie van de leerling op eigen, informele
constructies en primitieve rekenwijzen.
- eenzijdig bottom-up problem solving, overbeklemtoning van zelfontdekte en informele
begrippen en berekeningswijzen
- te weinig sturing en structurering door de leerkracht, te weinig guided construction of
-te weinig stapsgewijze opgebouwde leerlijnen.
Totaal overbodige invoering van het kolomsgewijs rekenen dat de leerlingen zowel in de war
brengt inzake het gewone hoofdrekenen als inzake het cijferen dat normaliter ook bij het
begin van groep 5 zou moeten starten. Bij het aftrekken met tekorten b.v. wordt het een
Het traditioneel cijferen wordt verwaarloosd en de Freudenthalers introduceren een totaal
gekunsteld alternatief dat niets meer te maken heeft met wiskundig cijferen gebaseerd op
splitsing van getal in hondertallen, enz.
Het cijferend delen verwordt tot een soort langdradig
hoofdrekenen op basis van schattend aftrekken van happen. Dit is een aanpak met veel
deelresultaten die langdradig is en die zich niet laat automatiseren zodat het cijferend delen
nooit een vaardigheid kan worden.
Onderwaardering voor het klassieke metend rekenen en voor de klassieke meetkunde met
inbegrip van de kennis van basisformules voor de berekening van oppervlakte en inhoud.
Te veel en te lang voor-wiskunde, te lang rekenen in contexten als doel op zich; te veel
contextualiseren (context- of situatiegebonden rekenwijzen e.d.), te weinig decontextualiseren.
Zo worden het vakmatig rekenen en het cijferen afgeremd door binding aan een specifieke
Een voorbeeld. Door de binding van de aftrekking aan een lineaire context en aan een
berekening op de getallenlijn (een traject van 85 km, al 27 km afgelegd, hoeveel km moet ik
nog afleggen) wordt het basisinzicht in aftrekken als wegnemen vertroebeld en stimuleert men
de leerlingen om aftrekken eenzijdig te interpreten als aanvullend optellen: 85 - 27 wordt dan:
27 + 3 + 10 +10+10+10 + 10 + 5; en achteraf moet men dan nog die vele tussenuitkomsten
Geen evenwichtige en uitgewerkte visie op vraagstukken: te veel kritiek op klassieke
vraagstukken,te weinig valabele alternatieven in realistische publicaties en methoden.
weinig toepassingen (vraagstukken) ook voor metend rekenen en te weinig moeilijke opgaven.
We begrepen ook niet waarom de duidelijke term vraagstukken moest verdwijnen. De
moeilijkheid bij veel context-vraagstukken ligt vaak eerder bij het onvoldoende kennen van de
context (b.v. ervaring van parkeren met een auto in opgave over hoeveel autos op parking
van 70 bij 50 meter), bij het feit dat de tekst te lang en te moeilijk is en bij het feit dat er te veel
berekeningen ineens bij betrokken zijn.
Foutieve benadering van de aanschouwelijkheid en te lang aanschouwelijk werken. Fixatie
van leerlingen op aanschouwelijke hulpmiddelen: de leerlingen mogen veel te lang gebruik
maken van hulpmiddelen als getallenlijn, rekenrek Dit bevordert, het loskomen van de
aanschouwelijke steun en het kort en handig uitrekenen De vele moeilijke (lange)
voorstellingswijzen van berekeningen op rekenrek en getallenlijn en de vele stappen
bemoeilijken een gestandaardiseerde en vlotte berekening.
Kloof tussen idealistische theorie en de praktijk. In een klas met 20 leerlingen is het inspelen
op individuele denkwijzen en berekeningswijzen niet haalbaar.
Zwakke, maar ook betere leerlingen zijn de dupe.
De voorstanders van de realistische aanpak begingen precies dezelfde fouten als de
voorstanders van de moderne wiskunde destijds. Ze vervingen enkel het ene extreem door
De hemelse (te abstracte & fomalistische) New Math werd vervangen door het andere extreem,
door de aardse, contextgebonden en constructivistische aanpak die al te weinig aandacht
heeft voor abstrahering en veralgemening en blijft steken in het stadium van de voorwiskunde. De tegenstanders werden verketterd. De kritiek werd doodgezwegen.
Raf Feys, pedagoog, ex-coördinator en docent Hoger Instituut voor Opvoedkunde -Brugge,
ex- afdelingshoofd lerarenopleiding -Torhout (Vlaanderen).
My cruisade against New Math (1970-1982) (& Constructivist Math (1988-.)
39 years ago, we succeeded in opening up the taboo surrounding formalistic New Mathematics in Belgium-Flemish
My cruisade against New Math (1970-1982) (& Constructivist Math (1988-.)
Introduction: Math-wars: New Math & Constructivistic -contextual Math
1+1=2 you would think, but curiously enough, the approach to mathematics and arithmetic education has been regularly debated over the past 50 years - also for primary education. Until about 1970, there was little discussion about arithmetic and mathematics education in primary school. There was a broad consensus, both among practitioners and among the professional didacticians. The mathematics curricula in the different countries were very similar. The vision of the practitioners has always remained more or less the same.
Since about 1970, mathematics wars have been fought. From 1970 onwards, we ourselves spent an enormous amount of time fighting two extreme visions that threaten classical arithmetic - and skills: from 1970 onwards the formalistic New Mathematics; and from 1988 the' constructivist & contextual mathematics' of the Dutch Freudenthal Institute (Utrecht) and the US Standards (1989). In this contribution, we limit ourselves to the fight against formalistic New Mathematics (NW) .( In the next article we show that constructivist mathematics unfortunately penetrated the learning curricula for the first grade secondary education.)
1. Breaking the taboo on criticism of the New Mathematics in Belgium-Flemish in 1982
Exactly 35 years ago, we succeeded in breaking the taboo on criticism of New Mathematics. In April 1982, we launched our campaign against the' New Mathematics' with the publication of a theme issue by Onderwijskrant with the challenging title: New Mathematics: een vlag op een modderschuit ( A flag on a mud boat, Onderwijskrant nr. 24). Partly because of the wide attention in the press, this publication provoked a huge number of positive reactions from teachers and ordinary citizens. A year later, a busy colloquium followed in the Congress Palace (Brussels) on' What Mathematics for 5- to 15-year-olds', where we took up the New Math supporters as prof. Roger Holvoet.
In May 1982, it became clear that mathematics was turned around. Since then, no more contributions have appeared about the many blessings of' new mathematics' (NM) .The taboo on criticism of the NM was' almost' broken through. In 1982, the inspector-general of technical education G. Smets wrote to us:"People at the top were bribed at the time to say nothing about New Mathematics" (see point 2). However, in 1982 we were not allowed to publicly mention his name. After the publication of' Moderne wiskunde: een vlag op een modderschuit in April 1982, however, we were subjected to much criticism from the corner of the propagandists of modern mathematics, from Papy sympathisants, from the Leuven professors Roger Holvoet and Alfred Warrinnier, from inspectors who had participated in modern mathematics methods, from the chief leader of the Catholic Education,
2. Breaking down taboo on New Math- religion: Top people were bribed to keep silent.
With our mathematics campaign of 1982, we wanted to break the taboo around the NM. As a result of the campaign, a number of people dared to express their thoughts about the NM for the first time. Since 1968, there has been a taboo on the New Math. Liège professors Pirard and Godfrind expressed similar criticism in La Libre Belgique, 11.03. 1980, as ours. And they also protested against the taboo on the NM. This was exactly what we ourselves have experienced since 1970 in Flanders. In their publication the Liège professors also showed that the NM was a formalistic theory that no longer referred to reality, was born from the brain of a few mathematicians, but was not interesting for primary and secondary school.
The reaction of prof. em. Karel Cuypers on our New Math- campaign was quite revelatory. Let us quote from his letter, which was later also included in' Person & Gemeenschap, September 1984. Cuypers: Since my initial sympathy for the New Math Renewal, which came to me as' miraculous', I have felt that the Papysts (the group around Brussels prof. Georges Papy supported by the Brussels education minister Vermeylen) as hypnotists have led the school world. Rarely has an educational innovation happened in such a climate of pervasive ideological engagement as the' new-math' phenomenon. All over the world, a force majeure was given to some prophets who could organize a spectacular show of persuasion with a hypnotic overthrow. Because of the enchantment that surrounds them, the secondary school teachers sat on the school benches to attend further training courses, which turned out to be remarkably theoretical and of little didactical help.. The situation had evolved so much that those who did not stand strong in the theory of sets did not even dare to take the floor, for fear of being placed ignorant or stupid against the wall.The many disgruntled teachers did not dare react openly either: congresses who did not agree were classed as conservative.
The inspector-general of technical education, G. Smets, wrote to us in a letter in response to mypublication' of 1982: Prof. Georges Papy had strong political relationships (including education minister Vermeylen) and ambitions. His lectures in Brussels and elsewhere were political meetings rather than scientific communications. His wife Frédérique also received large contributions from the then minister to experiment with new mathematics from nursery school. And then there were the many publishers who saw bread in a revolution of mathematics books. At the top a lot of people were literally bribed.
Also former inspector-mathematics E. H. Joniaux testified in a letter that the introduction of the NM thanks to the Ministry's nepotism. He wrote: Dear Mr Feys, at last, someone dared to rise up openly. New mathematics - and I have already said this since her first appearance - is the' philosophy' of mathematics, but not mathematics. And anyone who wants to teach this to children from 6 to 15 years old must have a lot of twists and turns in their brains. They wanted to fill the children with that - and this from kindergarten onwards.
Joniaux also caused me the critical contribution of the Liège professors Pirard and Godfrind, who had already been mentioned. They wrote: Many scientists, including noble prizewinners in physics, point out that their science is of no benefit at all with the collection theory, but with applicable mathematics. The scientists protest because they still have to teach their students many important aspects of the ABC of applicable mathematics:" We had already read ourselves in 1973 that the German noble prize winner Carl Von Weizsäcker, too, opposed the introduction of NM in education. Dutch prof. Hans Freudenthal succeeded in keeping the NM outside primary education in the Netherlands.
Pirard and Godfrind wrote:"Prof. Georges Papy, was not an inventor but rather an importer of the mathematics manuals of Revuez in France. Papy liked to describe mathematics as a poetic dream and said:' Mathematics is not science, but art and a dream. The mathematician is a child or a poet who makes his dream a reality' (Berkeley, VVW-Lcongress). According to Pirard and Godfrind: Many pupils experienced these mathematical dreams rather as a nightmare.
The supersonic rise of modern mathematics was thus only possible thanks to influence, sponsorship and reform pressure from Minister Vermeylen and a few senior officials, which gave Papy the monopoly on mathematics education and imposed the introduction. Policy makers also invested a huge amount of money in TV programmes, retraining, mathematics conferences and seminars of the Papy Group in luxury hotels in Knokke, etc. Prof. Papy presented the New Math as the mathematics of the third industrial revolution.
In debates on mathematics education, not only professors, but also we were very frankly silenced with such little choruses. We did not, so to speak, pay any attention to the future, to the mathematics of the third industrial revolution, the mathematics which, according to the new lighters in Japan, Russia... had already led to many economic successes. We stated in 1973 that in many countries the New Math was already on the retreat and that the MW would probably not even reach the 21st century. We did not find a hearing, and the New Math curriculum was also introduced in primary education in 1976 and presented as an enormous step forward, as a salvation from the misery of classical mathematics education.
3 Crusade against New Math :1970-1982
3.1 Our resistance during the period 1970-1982
In the years 1968-1969, we were very much captivated by the barnum advertising for new mathematics presented by the propagandists as the mathematics of the third industrial revolution. As a student, we followed several lectures at the KU-Leuven and some lessons from Alfred Vermandel. Our sympathy did not last long. From 1971 we distanced ourselves from formalistic & abstract NM
In the early seventies we did our best to convince those responsible for the educational umbrella organisations to not introduce new mathematics' into primary education. We did this also on the VLO Start Colloquium of 1 September 1973 in the Congress Palace Brussels. In October 1974 we published a contribution on the New Math in' Person and Community'. We wrote that the new draft curriculum from the first year of study onwards wanted to use a formalistic mathematics language, an unpalatable heap of new terms and notations; in short: superfluous thickdoing. We also mentioned that in countries such as the USA, Japan, Germany, the Netherlands, Germany, etc., there was already a lot of criticism of' new mathematics. In the US: Davis, Beberman, Rosenbloom, Page, Scott.... In Germany, Nobel Prize winner Carl von Weizsäcker took the lead, in the Netherlands prof. Mathematics Hans Freudenthal. In Flanders, we took the lead in this.
We also warned in 1974 that if we chose the wrong path of New Mathematics in primary education, it would be very difficult to leave it again in the short term. (It took 22 years before a new curriculum came into being in 1998 without Modern Mathematics.)
Unfortunately, the New Math was introduced into primary education in 1976, contrary to the views of the teachers. Our criticisms were haughtily swept away by the Papy Society, by academics, by curriculum designers, by mentors of mathematics.... We also noticed that not only teachers, but also inspectors, professors.... did not even dare to express their opinion; contradiction was not tolerated (see point 2).
Some of our criticisms of' Modern Mathematics
To formalistic, heavenly (floating) mathematics * to early abstraction * to much verbal bullying and verbal ballast * to the detriment of the application aspect of mathematics (calculating, memorizing, automatistion...) *to the detriment of the application aspect of mathematics (classical issues, metering arithmetic, etc.)
Little respect for the classical discipline of mathematics as a cultural product. Achievable for many students: Too many pupils have to switch to special needs education after the third year of study. Many parents are no longer able to accompany children. In point 5, we illustrate in detail how geometry was placed in the straitjacket of the NM and thus became totally formalistic.
As an alternative we chose in 1982 to update and dust off the many good elements and approaches from the mathematics tradition in our primary education, supplemented by a number of recent events such as three-dimensional geometric representations. We later implemented this in the 1998 curriculum.
3.2 Mathematics campaign 1982: NM: a flag on a mud boat breaks through taboo
In the years 1978-1982, a number of contributions appeared in which the supporters of' New Mathematics' broadly displayed the many blessings of this kind of mathematics. At the beginning of 1982, T. De Groote wrote triumphantly: Where calculation for most children used to be a whip blow, it can now become a fantastic experience for them in a fascinating world, and" And the great one further fantasized:"that the less gifted students now have a better place for themselves" (Person & Gemeenschap jg). 28, p. 35-36 In my contacts with the practice, however, I did not see a fascinating world show up, but false results in false realities and weaker pupils who gave up.
These contributions about the many blessings of the New Math for primary education were the incentive for me to launch a campaign against the Modern Mathematic' with a publication of Moderne wiskunde: een vlag op een moderschuit ( New Mathematics: a flag on a mud (muddle)boat) (Onderwijskrant no. 24) and the associated mathematics campaign, we were able to turn the maths time in 1982. Since then, no more contributions have appeared about the many blessings. However, a new curriculum was drawn up until 1998, in which the categories of modern mathematics were removed.
As a first step in the campaign, two thousand copies of the report' New Mathematics: a flag on a mud (muddle) boat' were distributed in April 1982. The campaign received a lot of reactions in the newspapers: De Morgen, Het Volk, Het Nieuwsblad, Libelle..... The articles about our campaign in four newspapers and two weeklies were very important for spreading the ideas and breaking the taboo. A number of people dared to express their opinions for the first time - also on paper. We received many enthusiastic reactions.
3.3 Later support from mathematics professors: Conversion Al Alfred Warrinnier (1987) et al.
In 1982 we still met with a lot of resistance because of a number of mathematics professors (Holvoet, Warrinnier, etc.), mathematics counsellors... Some of them converted afterwards.
The Leuven prof. Alfred Warrinnier sent his wife back in 1983 to the mathematical colloquium to lure me into the trap of asking whether the teacher wanted to define Feys exactly what mathematics meant in his opinion. A teacher may/cannot express a critical opinion on mathematics education. But in 1987 Warrinnier himself admitted that the introduction of modern mathematics was a bad thing - also in s. o. He wrote in De Standaard of 25 July 1987:The 11,12 and 13 year olds were not ready to deal with the very abstract undertone of the collections- relation-function building up, the algebraic structures, etc. The reform of mathematics education has de facto failed. Five years after our mathematics campaign, our university opponent of yesteryear was right. In 1982 we concentrated on primary education, but a few years later our criticism was also passed on to s. o. and a few mathematics professors took part (see point 4).
4 New Math: a child of structuralism of the 1930s - 40s.
4.1 Mathematics teachers later endorsed our criticism of 1982
In the publication M.W. een vlag op een modderschuit, we extensively illustrated that the New Math of the Bourbaki Group could not be separated from the structuralist and logical-formalistic trend in scientific thinking from the 1930s onwards and thus led to a formalistic approach. In section 4.2, we will elaborate on this in detail. But first we dwell for a moment on the (future) criticism of mathematics teachers who confirmed our earlier criticism.
In the weekly Intermediair' of 8 March 1994, the Leuven mathematics teachers Dirk Janssens and Dirk De Bock sit up the rise of the New Math. The movement for' modern mathematics' was typical of people who only believe in a theoretical approach: one would use a single starting point from which all parts of mathematics could be neatly constructed. This turned out to be an illusion afterwards. The NM was created from the most advanced positions of the discipline itself and only afterwards did it become part of the education system.
In the 1930s, a more or less revolutionary development took place. The so-called Bourbaki Group had ambitious plans to describe the full range of mathematics in a very systematic way, starting from axioms and the set-doctrine. They wanted to deliver a beautiful system, which there was no need to get a pin in between. It was not until later that this model of mathematics construction was chosen as a model for the structure of mathematics education.
It was rather shocking that this modern mathematics was never pedagogically substantiated. The failure of the whole experiment was ingrained in advance. This search for (formal) foundations is only useful for people who have already mastered a certain mathematical culture, but is therefore not yet suitable for teaching mathematics to those who do not know anything about it. That turned out to be an educational illusion. But this kind of pedagogical discussion was not held at the time, modern mathematics became absolutely compulsory for all secondary school pupils from 1968 onwards. In the 1970s, primary education also followed. ...... The urge for more and more abstraction (read: formalism) gradually made mathematics incomprehensible to the uninitiated. The fact that, for example, through a point outside a straight line, there is exactly one line that goes parallel to the given line, for example:' a line is a partition of a plane' and a term as length was introduced as a class of congruent lines:"This criticism from 1994 confirmed that our analysis of 1982 was also applicable to the New Math in the first degree secondary school.
Later in the magazine' Uitwiskeling' of 13 November 1994 we recorded analogous criticisms. One of the participants, Guido Roels (director of mathematics in the diocese of Ghent) answered the next day in Voor de dag , the question why it took so long before the mathematicians realised that New Mathematics was a mistake. According to Roels, this was because the mathematicians were fascinated by the fact that New Mathematics' so nicely put together' and did not see that this structure did not work in classroom. It was remarkable, however, that it took so long to realise this and that criticism abroad and our criticism were not heard since 1971.
4.2 Structuralist and logical-formalistic approach
In our publication of 1982 we also showed that Bourbaki-mathematics could not be separated from the structuralist and logical-formalistic trend in scientific thinking from the 1930s onwards. Structuralism as a scientific method attempted to discover the same patterns, patterns, patterns, structures, etc. in the most diverse phenomena. It developed' grammatical',' comprehensive' concepts and a formal logical language to name them. From the formalistic/grammatical approach, for example, one saw in the concepts' is parallel with' and' is multiple of' the same grammatical structure; both concepts followed this approach, for example, a case of' reflexive relations': a number is multiple of itself a parallel is parallel with itself - and a reflexive relationship was suggested with a' loop'.
The things known from reality (e. g. parallel, angle, multiples of numbers....) are deployed in artificially created relationships almost independently of their meaning; they are especially interesting as elements of a set, as cross-section, as a couple, reflexive relationship.... Pragmatically seen e. g. the notions' parallel' and' is multiple of have nothing in common.
They tried to approach and organize all concepts with the help of a formal logic and some sort of' grammatical' concepts. The structuralist approach used the deductive approach and the formal logic as scientific instruments. A reform of a structural and formalistic nature was therefore chosen. This leads to an erosion of the reality value of mathematics education.
The 'new mathematics' thus shifted to a new way of learning, in which the use of mathematics no longer takes place, but rather the learning of a structuralist grammar, which is central to the project. From our thesis on psychologist Jean Piaget, who was presented as the figurehead of modern mathematics at the time, we also referred in the' Modderschuit -1982 to the connection with structuralism within psychology. Piaget also used/misuse of the formal logic as a language to formulate his findings. In the philosophical work of prof. Wiener-Kreis on. Prof. Apostel sought for formal-logical systems (languages) to describe the laws and regulations in the most diverse scientific disciplines (linguistics, psychology, economics, etc.). The older Apostel postle took distance from this. Apostle became an ally in the fight against constructivist mathematics of the Dutch Freudenthal Institute around 1990.
We refer to a similar analysis of Eddy Daniëls in Intermediair, 8 March 1994. Daniëls: The inter-war was the phase in which they tried to forget the trenches of the first war. They wanted to focus all philosophical efforts on a completely deductive language that would eliminate all misunderstandings. "The logical-positivists of the Vienna County and the young Wittgenstein were also sick in this bed, according to him. According to Daniëls, the Bourbaki Group developed a formal mathematics theory that was fundamentally alienated from reality, which rather became oppressive instead of a liberating character. Because she designed a line of thought that literally suppressed the spontaneous urge to learn among children and young people.
5. Geometrics in a straightjacket of sets, relations = formalism & rubricitis
On the introduction of' New Math) in addition to the preservation of a number of classical subjects, we also receive a radical break with the traditional visual and functional approach: - a strictly logical-deductive structure; - the geometric concepts (flat, straight, rectangular, angle, triangle, rectangle, etc.) are put into the formal and abstract language of relations and collections; - abstract and hierarchical classification
Based on the option for a logical-deductive build-up responsible inspector R. Barbry, the reason why the design theory could only be started in the fourth year of study. He wrote: We only start in the fourth year of primary school with the formation of the plane pi, which is an infinite collection of points. Gradually, the main characteristics and richness of the pi plane are discovered by boundaries (subsets: rights, figures, etc.). We frequently refer to the language of sets and relationships. Only in the fourth year of study is the basis to start with the design theory, in order to be able to apply the collection and relational language (Barbry, 1978). New mathematics' overlooked the fact that children orientate themselves from birth and that the toddlers can and must learn to explore all kinds of figures in a visual way.
Terms in straitjacket of set theory, relations
Traditional notions were put into the straitjacket of the doctrine collections. Teachers had to explain that a (limited) segment is also an infinite set of points, because one can always make these points smaller. The parallelists were presented in a set with an empty cross-set (they have no points in common), and as a reflexive relationship with a loop arrow: after all, every line is parallel with itself.
An angle was defined and represented as the set of points of two half lines (belts of the angle) with the same starting point (angle). These points were presented with a set and the children had to learn that the points belonging to the classical corner sector do not belong to the angle (set). A triangle was often represented as' a closed broken line, consisting of three segments; the points within the perimeter of the triangle no longer belonged to the triangle, represented by a venndiagram.
Geometrics = classification
A considerable part of the formal education was taken up by the logical-hierarchical classification and deductive development of the network of flat and spatial figures. People always started from the more general (=empt) concepts. This means e. g. that the rectangle and the square (the more specific or filled up terms) were listed at the very end of the list. The curriculum of state education already stated as the objective for the second year of study: In the collection of polygons, classify with the criterion: parallelism-evenness of sides or corners; and can present in a venn diagram. From the new formalistic definitions (e. g. a square is a rectangle with four equal sides, a parallelogram with....) one could think of a virtually unlimited number of classification assignments.
A system of definitions and logical hierarchical classifications, choosing the order of the most general figures' large size, poor content) to the most special (rich content, small size). Where the more specific, rich and everyday figures were treated first (e. g., for example, the more specific, rich and everyday figures). square and rectangle with their visual characteristics, they now started from trapezium and parallelogram.
The children were taught to describe the square and recognize it as a special kind of rectangle, rhombus, parallelogram,.... The square was last mentioned and was described as a subset of a rectangle, a pane.... A rectangle thus became a trapezium with all angles straight, but at the same time a parallelogram with 4 (or at least one) right angles, etc. Such hierarchical (evident) descriptions were quite abstract and variable, much more complex than the previously based enumeration of the various intuitive concepts. We could no longer connect with the intuitive concepts that the children had already formed and that mainly relate to the richer and beautiful figures. It went so far that some curriculum designers recommended that the square logiots should no longer be called square blocks, but rather' tile', because according to modern mathematics, a square logi block was just as much a kind of rectangle, rhombus, parallelogram... A mathematics supervisor made the teachers even point out that toddlers were not talking about square, rectangle, triangle but respectively over tile, door and roof. And it was not until the fourth year of study that the geometric terms were allowed. However, the square was not allowed to be presented until last in the row and as a subset of the collection of quadrangles, trapezia, parallel icons, rectangles and windows.
6. What does New Math, as an untouchable religion, teach us about fads?
In this article we referred extensively to our mathematics campaign of 1982, the NM background and the New Math as a kind of religion that should not be criticised. Rages always display characteristics of religions. Those who do not participate are considered as renegades. New Math is one of the many rages in our education of the past 50 years.
We can learn a lot from it. The New Math- propagandists initially hanged a caricature of classical mathematics and the multi-faced methodical approaches. They wrongly gave the impression that it used to be just memory work. The new lighters grabbed the NM as the mathematics of the future, the mathematics of the third industrial revolution - just like many new lighters in recent years with the so-called Math of the 21st century
The supersonic rise of new mathematics was only possible thanks to the influence and pressure from the ministry (Minister Vermeylen and top officials) which led to the creation of prof. Papy & Co got the monopoly; and thanks to the many propaganda from all kinds of policymakers. Critics of the New Math, professors and even Directors-General and inspectors, were silenced from above. The Director-General for Technical Education, Smets, expressed his full support for our New Math campaign in 1982, but did not want his name to be mentioned. Today, censorship and self-censorship are greater than ever.
The New Math--new lighters did not only deal with ailments over our hopelessly outdated mathematics education, but also with myths about the excellent economic results of countries such as Japan, Russia, etc. which introduced modern mathematics.
Once the fad of new mathematics had passed, it wasn't easy to get back on track again. In primary education, many tried and tested approaches had been thrown into the dust and a break with the experience wisdom had emerged. We did manage to put the tried and tested values and approaches back at the heart of the 1998 curriculum as a curriculum author.
In the first degree secondary school unfortunately, they opted for the extreme of the heavenly, formalistic New Math the other extreme: the earthly, contextual and constructivist mathematics approach of the Dutch Freudenthal Institute (Utrecht) and the US Standards of 1989 (see next contribution). And so over the past 25 years, a new mathematics war has emerged in the Netherlands, the USA, Canada, and so on: constructivistic mathematics, which shows little appreciation for mathematics as a cultural discipline